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Learning from compressed observations

cs.IT cs.LG math.IT

The problem of statistical learning is to construct a predictor of a random variable $Y$ as a function of a related random variable $X$ on the basis of an i.i.d. training sample from the joint distribution of $(X,Y)$. Allowable predictors are drawn from some specified class, and the goal is to approach asymptotically the performance (expected loss) of the best predictor in the class. We consider the setting in which one has perfect observation of the $X$-part of the sample, while the $Y$-part has to be communicated at some finite bit rate. The encoding of the $Y$-values is allowed to depend on the $X$-values. Under suitable regularity conditions on the admissible predictors, the underlying family of probability distributions and the loss function, we give an information-theoretic characterization of achievable predictor performance in terms of conditional distortion-rate functions. The ideas are illustrated on the example of nonparametric regression in Gaussian noise.

Maxim Raginsky

10.1109/ITW.2007.4313111

0704.0671

Sensor Networks with Random Links: Topology Design for Distributed Consensus

cs.IT cs.LG math.IT

In a sensor network, in practice, the communication among sensors is subject to:(1) errors or failures at random times; (3) costs; and(2) constraints since sensors and networks operate under scarce resources, such as power, data rate, or communication. The signal-to-noise ratio (SNR) is usually a main factor in determining the probability of error (or of communication failure) in a link. These probabilities are then a proxy for the SNR under which the links operate. The paper studies the problem of designing the topology, i.e., assigning the probabilities of reliable communication among sensors (or of link failures) to maximize the rate of convergence of average consensus, when the link communication costs are taken into account, and there is an overall communication budget constraint. To consider this problem, we address a number of preliminary issues: (1) model the network as a random topology; (2) establish necessary and sufficient conditions for mean square sense (mss) and almost sure (a.s.) convergence of average consensus when network links fail; and, in particular, (3) show that a necessary and sufficient condition for both mss and a.s. convergence is for the algebraic connectivity of the mean graph describing the network topology to be strictly positive. With these results, we formulate topology design, subject to random link failures and to a communication cost constraint, as a constrained convex optimization problem to which we apply semidefinite programming techniques. We show by an extensive numerical study that the optimal design improves significantly the convergence speed of the consensus algorithm and can achieve the asymptotic performance of a non-random network at a fraction of the communication cost.

Soummya Kar and Jose M. F. Moura

10.1109/TSP.2008.920143

0704.0954

The on-line shortest path problem under partial monitoring

cs.LG cs.SC

The on-line shortest path problem is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1/\sqrt{n} and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with linear complexity in the number of rounds n and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen path at a total of m << n time instances. Another extension is shown where the decision maker competes against a time-varying path, a generalization of the problem of tracking the best expert. A version of the multi-armed bandit setting for shortest path is also discussed where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path. Applications to routing in packet switched networks along with simulation results are also presented.

Andras Gyorgy, Tamas Linder, Gabor Lugosi, Gyorgy Ottucsak

0704.1020

A neural network approach to ordinal regression

cs.LG cs.AI cs.NE

Ordinal regression is an important type of learning, which has properties of both classification and regression. Here we describe a simple and effective approach to adapt a traditional neural network to learn ordinal categories. Our approach is a generalization of the perceptron method for ordinal regression. On several benchmark datasets, our method (NNRank) outperforms a neural network classification method. Compared with the ordinal regression methods using Gaussian processes and support vector machines, NNRank achieves comparable performance. Moreover, NNRank has the advantages of traditional neural networks: learning in both online and batch modes, handling very large training datasets, and making rapid predictions. These features make NNRank a useful and complementary tool for large-scale data processing tasks such as information retrieval, web page ranking, collaborative filtering, and protein ranking in Bioinformatics.

Jianlin Cheng

0704.1028

Parametric Learning and Monte Carlo Optimization

cs.LG

This paper uncovers and explores the close relationship between Monte Carlo Optimization of a parametrized integral (MCO), Parametric machine-Learning (PL), and `blackbox' or `oracle'-based optimization (BO). We make four contributions. First, we prove that MCO is mathematically identical to a broad class of PL problems. This identity potentially provides a new application domain for all broadly applicable PL techniques: MCO. Second, we introduce immediate sampling, a new version of the Probability Collectives (PC) algorithm for blackbox optimization. Immediate sampling transforms the original BO problem into an MCO problem. Accordingly, by combining these first two contributions, we can apply all PL techniques to BO. In our third contribution we validate this way of improving BO by demonstrating that cross-validation and bagging improve immediate sampling. Finally, conventional MC and MCO procedures ignore the relationship between the sample point locations and the associated values of the integrand; only the values of the integrand at those locations are considered. We demonstrate that one can exploit the sample location information using PL techniques, for example by forming a fit of the sample locations to the associated values of the integrand. This provides an additional way to apply PL techniques to improve MCO.

David H. Wolpert and Dev G. Rajnarayan

0704.1274

Preconditioned Temporal Difference Learning

cs.LG cs.AI

This paper has been withdrawn by the author. This draft is withdrawn for its poor quality in english, unfortunately produced by the author when he was just starting his science route. Look at the ICML version instead: http://icml2008.cs.helsinki.fi/papers/111.pdf

Yao HengShuai

0704.1409

A Note on the Inapproximability of Correlation Clustering

cs.LG cs.DS

We consider inapproximability of the correlation clustering problem defined as follows: Given a graph $G = (V,E)$ where each edge is labeled either "+" (similar) or "-" (dissimilar), correlation clustering seeks to partition the vertices into clusters so that the number of pairs correctly (resp. incorrectly) classified with respect to the labels is maximized (resp. minimized). The two complementary problems are called MaxAgree and MinDisagree, respectively, and have been studied on complete graphs, where every edge is labeled, and general graphs, where some edge might not have been labeled. Natural edge-weighted versions of both problems have been studied as well. Let S-MaxAgree denote the weighted problem where all weights are taken from set S, we show that S-MaxAgree with weights bounded by $O(|V|^{1/2-\delta})$ essentially belongs to the same hardness class in the following sense: if there is a polynomial time algorithm that approximates S-MaxAgree within a factor of $\lambda = O(\log{|V|})$ with high probability, then for any choice of S', S'-MaxAgree can be approximated in polynomial time within a factor of $(\lambda + \epsilon)$, where $\epsilon > 0$ can be arbitrarily small, with high probability. A similar statement also holds for $S-MinDisagree. This result implies it is hard (assuming $NP \neq RP$) to approximate unweighted MaxAgree within a factor of $80/79-\epsilon$, improving upon a previous known factor of $116/115-\epsilon$ by Charikar et. al. \cite{Chari05}.

Jinsong Tan

0704.2092

Joint universal lossy coding and identification of stationary mixing sources

cs.IT cs.LG math.IT

The problem of joint universal source coding and modeling, treated in the context of lossless codes by Rissanen, was recently generalized to fixed-rate lossy coding of finitely parametrized continuous-alphabet i.i.d. sources. We extend these results to variable-rate lossy block coding of stationary ergodic sources and show that, for bounded metric distortion measures, any finitely parametrized family of stationary sources satisfying suitable mixing, smoothness and Vapnik-Chervonenkis learnability conditions admits universal schemes for joint lossy source coding and identification. We also give several explicit examples of parametric sources satisfying the regularity conditions.

Maxim Raginsky

0704.2644

Supervised Feature Selection via Dependence Estimation

cs.LG

We introduce a framework for filtering features that employs the Hilbert-Schmidt Independence Criterion (HSIC) as a measure of dependence between the features and the labels. The key idea is that good features should maximise such dependence. Feature selection for various supervised learning problems (including classification and regression) is unified under this framework, and the solutions can be approximated using a backward-elimination algorithm. We demonstrate the usefulness of our method on both artificial and real world datasets.

Le Song, Alex Smola, Arthur Gretton, Karsten Borgwardt, Justin Bedo

0704.2668

Equivalence of LP Relaxation and Max-Product for Weighted Matching in General Graphs

cs.IT cs.AI cs.LG cs.NI math.IT

Max-product belief propagation is a local, iterative algorithm to find the mode/MAP estimate of a probability distribution. While it has been successfully employed in a wide variety of applications, there are relatively few theoretical guarantees of convergence and correctness for general loopy graphs that may have many short cycles. Of these, even fewer provide exact ``necessary and sufficient'' characterizations. In this paper we investigate the problem of using max-product to find the maximum weight matching in an arbitrary graph with edge weights. This is done by first constructing a probability distribution whose mode corresponds to the optimal matching, and then running max-product. Weighted matching can also be posed as an integer program, for which there is an LP relaxation. This relaxation is not always tight. In this paper we show that \begin{enumerate} \item If the LP relaxation is tight, then max-product always converges, and that too to the correct answer. \item If the LP relaxation is loose, then max-product does not converge. \end{enumerate} This provides an exact, data-dependent characterization of max-product performance, and a precise connection to LP relaxation, which is a well-studied optimization technique. Also, since LP relaxation is known to be tight for bipartite graphs, our results generalize other recent results on using max-product to find weighted matchings in bipartite graphs.

Sujay Sanghavi

0705.0760

HMM Speaker Identification Using Linear and Non-linear Merging Techniques

cs.LG

Speaker identification is a powerful, non-invasive and in-expensive biometric technique. The recognition accuracy, however, deteriorates when noise levels affect a specific band of frequency. In this paper, we present a sub-band based speaker identification that intends to improve the live testing performance. Each frequency sub-band is processed and classified independently. We also compare the linear and non-linear merging techniques for the sub-bands recognizer. Support vector machines and Gaussian Mixture models are the non-linear merging techniques that are investigated. Results showed that the sub-band based method used with linear merging techniques enormously improved the performance of the speaker identification over the performance of wide-band recognizers when tested live. A live testing improvement of 9.78% was achieved

Unathi Mahola, Fulufhelo V. Nelwamondo, Tshilidzi Marwala

0705.1585

Statistical Mechanics of Nonlinear On-line Learning for Ensemble Teachers

cs.LG cond-mat.dis-nn

We analyze the generalization performance of a student in a model composed of nonlinear perceptrons: a true teacher, ensemble teachers, and the student. We calculate the generalization error of the student analytically or numerically using statistical mechanics in the framework of on-line learning. We treat two well-known learning rules: Hebbian learning and perceptron learning. As a result, it is proven that the nonlinear model shows qualitatively different behaviors from the linear model. Moreover, it is clarified that Hebbian learning and perceptron learning show qualitatively different behaviors from each other. In Hebbian learning, we can analytically obtain the solutions. In this case, the generalization error monotonically decreases. The steady value of the generalization error is independent of the learning rate. The larger the number of teachers is and the more variety the ensemble teachers have, the smaller the generalization error is. In perceptron learning, we have to numerically obtain the solutions. In this case, the dynamical behaviors of the generalization error are non-monotonic. The smaller the learning rate is, the larger the number of teachers is; and the more variety the ensemble teachers have, the smaller the minimum value of the generalization error is.

Hideto Utsumi, Seiji Miyoshi, Masato Okada

10.1143/JPSJ.76.114001

0705.2318

On the monotonization of the training set

cs.LG cs.AI

We consider the problem of minimal correction of the training set to make it consistent with monotonic constraints. This problem arises during analysis of data sets via techniques that require monotone data. We show that this problem is NP-hard in general and is equivalent to finding a maximal independent set in special orgraphs. Practically important cases of that problem considered in detail. These are the cases when a partial order given on the replies set is a total order or has a dimension 2. We show that the second case can be reduced to maximization of a quadratic convex function on a convex set. For this case we construct an approximate polynomial algorithm based on convex optimization.

Rustem Takhanov

0705.2765

Mixed membership stochastic blockmodels

stat.ME cs.LG math.ST physics.soc-ph stat.ML stat.TH

Observations consisting of measurements on relationships for pairs of objects arise in many settings, such as protein interaction and gene regulatory networks, collections of author-recipient email, and social networks. Analyzing such data with probabilisic models can be delicate because the simple exchangeability assumptions underlying many boilerplate models no longer hold. In this paper, we describe a latent variable model of such data called the mixed membership stochastic blockmodel. This model extends blockmodels for relational data to ones which capture mixed membership latent relational structure, thus providing an object-specific low-dimensional representation. We develop a general variational inference algorithm for fast approximate posterior inference. We explore applications to social and protein interaction networks.

Edoardo M Airoldi, David M Blei, Stephen E Fienberg, Eric P Xing

0705.4485

Loop corrections for message passing algorithms in continuous variable models

cs.AI cs.LG

In this paper we derive the equations for Loop Corrected Belief Propagation on a continuous variable Gaussian model. Using the exactness of the averages for belief propagation for Gaussian models, a different way of obtaining the covariances is found, based on Belief Propagation on cavity graphs. We discuss the relation of this loop correction algorithm to Expectation Propagation algorithms for the case in which the model is no longer Gaussian, but slightly perturbed by nonlinear terms.

Bastian Wemmenhove and Bert Kappen

0705.4566

A Novel Model of Working Set Selection for SMO Decomposition Methods

cs.LG cs.AI

In the process of training Support Vector Machines (SVMs) by decomposition methods, working set selection is an important technique, and some exciting schemes were employed into this field. To improve working set selection, we propose a new model for working set selection in sequential minimal optimization (SMO) decomposition methods. In this model, it selects B as working set without reselection. Some properties are given by simple proof, and experiments demonstrate that the proposed method is in general faster than existing methods.

Zhendong Zhao, Lei Yuan, Yuxuan Wang, Forrest Sheng Bao, Shunyi Zhang Yanfei Sun

10.1109/ICTAI.2007.99

0706.0585

Getting started in probabilistic graphical models

q-bio.QM cs.LG physics.soc-ph stat.ME stat.ML

Probabilistic graphical models (PGMs) have become a popular tool for computational analysis of biological data in a variety of domains. But, what exactly are they and how do they work? How can we use PGMs to discover patterns that are biologically relevant? And to what extent can PGMs help us formulate new hypotheses that are testable at the bench? This note sketches out some answers and illustrates the main ideas behind the statistical approach to biological pattern discovery.

Edoardo M Airoldi

10.1371/journal.pcbi.0030252

0706.2040

A tutorial on conformal prediction

cs.LG stat.ML

Conformal prediction uses past experience to determine precise levels of confidence in new predictions. Given an error probability $\epsilon$, together with a method that makes a prediction $\hat{y}$ of a label $y$, it produces a set of labels, typically containing $\hat{y}$, that also contains $y$ with probability $1-\epsilon$. Conformal prediction can be applied to any method for producing $\hat{y}$: a nearest-neighbor method, a support-vector machine, ridge regression, etc. Conformal prediction is designed for an on-line setting in which labels are predicted successively, each one being revealed before the next is predicted. The most novel and valuable feature of conformal prediction is that if the successive examples are sampled independently from the same distribution, then the successive predictions will be right $1-\epsilon$ of the time, even though they are based on an accumulating dataset rather than on independent datasets. In addition to the model under which successive examples are sampled independently, other on-line compression models can also use conformal prediction. The widely used Gaussian linear model is one of these. This tutorial presents a self-contained account of the theory of conformal prediction and works through several numerical examples. A more comprehensive treatment of the topic is provided in "Algorithmic Learning in a Random World", by Vladimir Vovk, Alex Gammerman, and Glenn Shafer (Springer, 2005).

Glenn Shafer and Vladimir Vovk

0706.3188

The Role of Time in the Creation of Knowledge

cs.LG cs.AI cs.IT math.IT

This paper I assume that in humans the creation of knowledge depends on a discrete time, or stage, sequential decision-making process subjected to a stochastic, information transmitting environment. For each time-stage, this environment randomly transmits Shannon type information-packets to the decision-maker, who examines each of them for relevancy and then determines his optimal choices. Using this set of relevant information-packets, the decision-maker adapts, over time, to the stochastic nature of his environment, and optimizes the subjective expected rate-of-growth of knowledge. The decision-maker's optimal actions, lead to a decision function that involves, over time, his view of the subjective entropy of the environmental process and other important parameters at each time-stage of the process. Using this model of human behavior, one could create psychometric experiments using computer simulation and real decision-makers, to play programmed games to measure the resulting human performance.

Roy E. Murphy

0707.0498

Clustering and Feature Selection using Sparse Principal Component Analysis

cs.AI cs.LG cs.MS

In this paper, we study the application of sparse principal component analysis (PCA) to clustering and feature selection problems. Sparse PCA seeks sparse factors, or linear combinations of the data variables, explaining a maximum amount of variance in the data while having only a limited number of nonzero coefficients. PCA is often used as a simple clustering technique and sparse factors allow us here to interpret the clusters in terms of a reduced set of variables. We begin with a brief introduction and motivation on sparse PCA and detail our implementation of the algorithm in d'Aspremont et al. (2005). We then apply these results to some classic clustering and feature selection problems arising in biology.

Ronny Luss, Alexandre d'Aspremont

0707.0701

Model Selection Through Sparse Maximum Likelihood Estimation

cs.AI cs.LG

We consider the problem of estimating the parameters of a Gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse. Our approach is to solve a maximum likelihood problem with an added l_1-norm penalty term. The problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes. We present two new algorithms for solving problems with at least a thousand nodes in the Gaussian case. Our first algorithm uses block coordinate descent, and can be interpreted as recursive l_1-norm penalized regression. Our second algorithm, based on Nesterov's first order method, yields a complexity estimate with a better dependence on problem size than existing interior point methods. Using a log determinant relaxation of the log partition function (Wainwright & Jordan (2006)), we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case. We test our algorithms on synthetic data, as well as on gene expression and senate voting records data.

Onureena Banerjee, Laurent El Ghaoui, Alexandre d'Aspremont

0707.0704

Optimal Solutions for Sparse Principal Component Analysis

cs.AI cs.LG

Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all target numbers of non zero coefficients, with total complexity O(n^3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n^3) per pattern. We discuss applications in subset selection and sparse recovery and show on artificial examples and biological data that our algorithm does provide globally optimal solutions in many cases.

Alexandre d'Aspremont, Francis Bach, Laurent El Ghaoui

0707.0705

A New Generalization of Chebyshev Inequality for Random Vectors

math.ST cs.LG math.PR stat.AP stat.TH

In this article, we derive a new generalization of Chebyshev inequality for random vectors. We demonstrate that the new generalization is much less conservative than the classical generalization.

Xinjia Chen

0707.0805

Clusters, Graphs, and Networks for Analysing Internet-Web-Supported Communication within a Virtual Community

cs.AI cs.LG

The proposal is to use clusters, graphs and networks as models in order to analyse the Web structure. Clusters, graphs and networks provide knowledge representation and organization. Clusters were generated by co-site analysis. The sample is a set of academic Web sites from the countries belonging to the European Union. These clusters are here revisited from the point of view of graph theory and social network analysis. This is a quantitative and structural analysis. In fact, the Internet is a computer network that connects people and organizations. Thus we may consider it to be a social network. The set of Web academic sites represents an empirical social network, and is viewed as a virtual community. The network structural properties are here analysed applying together cluster analysis, graph theory and social network analysis.

Xavier Polanco (INIST)

0707.1452

Universal Reinforcement Learning

cs.IT cs.LG math.IT

We consider an agent interacting with an unmodeled environment. At each time, the agent makes an observation, takes an action, and incurs a cost. Its actions can influence future observations and costs. The goal is to minimize the long-term average cost. We propose a novel algorithm, known as the active LZ algorithm, for optimal control based on ideas from the Lempel-Ziv scheme for universal data compression and prediction. We establish that, under the active LZ algorithm, if there exists an integer $K$ such that the future is conditionally independent of the past given a window of $K$ consecutive actions and observations, then the average cost converges to the optimum. Experimental results involving the game of Rock-Paper-Scissors illustrate merits of the algorithm.

Vivek F. Farias, Ciamac C. Moallemi, Tsachy Weissman, Benjamin Van Roy

0707.3087

Consistency of the group Lasso and multiple kernel learning

cs.LG

We consider the least-square regression problem with regularization by a block 1-norm, i.e., a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the 1-norm where all spaces have dimension one, where it is commonly referred to as the Lasso. In this paper, we study the asymptotic model consistency of the group Lasso. We derive necessary and sufficient conditions for the consistency of group Lasso under practical assumptions, such as model misspecification. When the linear predictors and Euclidean norms are replaced by functions and reproducing kernel Hilbert norms, the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection. Using tools from functional analysis, and in particular covariance operators, we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate, even when the necessary condition required for the non adaptive scheme is not satisfied.

Francis Bach (WILLOW Project - Inria/Ens)

0707.3390

Quantum Algorithms for Learning and Testing Juntas

quant-ph cs.LG

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses $O(k/\epsilon)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $\Omega(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.

Alp Atici, Rocco A. Servedio

10.1007/s11128-007-0061-6

0707.3479

Virtual screening with support vector machines and structure kernels

q-bio.QM cs.LG

Support vector machines and kernel methods have recently gained considerable attention in chemoinformatics. They offer generally good performance for problems of supervised classification or regression, and provide a flexible and computationally efficient framework to include relevant information and prior knowledge about the data and problems to be handled. In particular, with kernel methods molecules do not need to be represented and stored explicitly as vectors or fingerprints, but only to be compared to each other through a comparison function technically called a kernel. While classical kernels can be used to compare vector or fingerprint representations of molecules, completely new kernels were developed in the recent years to directly compare the 2D or 3D structures of molecules, without the need for an explicit vectorization step through the extraction of molecular descriptors. While still in their infancy, these approaches have already demonstrated their relevance on several toxicity prediction and structure-activity relationship problems.

Pierre Mah\'e (XRCE), Jean-Philippe Vert (CB)

0708.0171

Structure or Noise?

physics.data-an cond-mat.stat-mech cs.IT cs.LG math-ph math.IT math.MP math.ST nlin.CD stat.TH

We show how rate-distortion theory provides a mechanism for automated theory building by naturally distinguishing between regularity and randomness. We start from the simple principle that model variables should, as much as possible, render the future and past conditionally independent. From this, we construct an objective function for model making whose extrema embody the trade-off between a model's structural complexity and its predictive power. The solutions correspond to a hierarchy of models that, at each level of complexity, achieve optimal predictive power at minimal cost. In the limit of maximal prediction the resulting optimal model identifies a process's intrinsic organization by extracting the underlying causal states. In this limit, the model's complexity is given by the statistical complexity, which is known to be minimal for achieving maximum prediction. Examples show how theory building can profit from analyzing a process's causal compressibility, which is reflected in the optimal models' rate-distortion curve--the process's characteristic for optimally balancing structure and noise at different levels of representation.

Susanne Still, James P. Crutchfield

0708.0654

Cost-minimising strategies for data labelling : optimal stopping and active learning

cs.LG

Supervised learning deals with the inference of a distribution over an output or label space $\CY$ conditioned on points in an observation space $\CX$, given a training dataset $D$ of pairs in $\CX \times \CY$. However, in a lot of applications of interest, acquisition of large amounts of observations is easy, while the process of generating labels is time-consuming or costly. One way to deal with this problem is {\em active} learning, where points to be labelled are selected with the aim of creating a model with better performance than that of an model trained on an equal number of randomly sampled points. In this paper, we instead propose to deal with the labelling cost directly: The learning goal is defined as the minimisation of a cost which is a function of the expected model performance and the total cost of the labels used. This allows the development of general strategies and specific algorithms for (a) optimal stopping, where the expected cost dictates whether label acquisition should continue (b) empirical evaluation, where the cost is used as a performance metric for a given combination of inference, stopping and sampling methods. Though the main focus of the paper is optimal stopping, we also aim to provide the background for further developments and discussion in the related field of active learning.

Christos Dimitrakakis and Christian Savu-Krohn

0708.1242

Defensive forecasting for optimal prediction with expert advice

cs.LG

The method of defensive forecasting is applied to the problem of prediction with expert advice for binary outcomes. It turns out that defensive forecasting is not only competitive with the Aggregating Algorithm but also handles the case of "second-guessing" experts, whose advice depends on the learner's prediction; this paper assumes that the dependence on the learner's prediction is continuous.

Vladimir Vovk

0708.1503

Optimal Causal Inference: Estimating Stored Information and Approximating Causal Architecture

cs.IT cond-mat.stat-mech cs.LG math.IT math.ST stat.TH

We introduce an approach to inferring the causal architecture of stochastic dynamical systems that extends rate distortion theory to use causal shielding---a natural principle of learning. We study two distinct cases of causal inference: optimal causal filtering and optimal causal estimation. Filtering corresponds to the ideal case in which the probability distribution of measurement sequences is known, giving a principled method to approximate a system's causal structure at a desired level of representation. We show that, in the limit in which a model complexity constraint is relaxed, filtering finds the exact causal architecture of a stochastic dynamical system, known as the causal-state partition. From this, one can estimate the amount of historical information the process stores. More generally, causal filtering finds a graded model-complexity hierarchy of approximations to the causal architecture. Abrupt changes in the hierarchy, as a function of approximation, capture distinct scales of structural organization. For nonideal cases with finite data, we show how the correct number of underlying causal states can be found by optimal causal estimation. A previously derived model complexity control term allows us to correct for the effect of statistical fluctuations in probability estimates and thereby avoid over-fitting.

Susanne Still, James P. Crutchfield, Christopher J. Ellison

0708.1580

On Semimeasures Predicting Martin-Loef Random Sequences

cs.IT cs.LG math.IT math.PR

Solomonoff's central result on induction is that the posterior of a universal semimeasure M converges rapidly and with probability 1 to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a universal sequence predictor in case of unknown mu. Despite some nearby results and proofs in the literature, the stronger result of convergence for all (Martin-Loef) random sequences remained open. Such a convergence result would be particularly interesting and natural, since randomness can be defined in terms of M itself. We show that there are universal semimeasures M which do not converge for all random sequences, i.e. we give a partial negative answer to the open problem. We also provide a positive answer for some non-universal semimeasures. We define the incomputable measure D as a mixture over all computable measures and the enumerable semimeasure W as a mixture over all enumerable nearly-measures. We show that W converges to D and D to mu on all random sequences. The Hellinger distance measuring closeness of two distributions plays a central role.

Marcus Hutter and Andrej Muchnik

0708.2319

Continuous and randomized defensive forecasting: unified view

cs.LG

Defensive forecasting is a method of transforming laws of probability (stated in game-theoretic terms as strategies for Sceptic) into forecasting algorithms. There are two known varieties of defensive forecasting: "continuous", in which Sceptic's moves are assumed to depend on the forecasts in a (semi)continuous manner and which produces deterministic forecasts, and "randomized", in which the dependence of Sceptic's moves on the forecasts is arbitrary and Forecaster's moves are allowed to be randomized. This note shows that the randomized variety can be obtained from the continuous variety by smearing Sceptic's moves to make them continuous.

Vladimir Vovk

0708.2353

A Dichotomy Theorem for General Minimum Cost Homomorphism Problem

cs.LG cs.CC

In the constraint satisfaction problem ($CSP$), the aim is to find an assignment of values to a set of variables subject to specified constraints. In the minimum cost homomorphism problem ($MinHom$), one is additionally given weights $c_{va}$ for every variable $v$ and value $a$, and the aim is to find an assignment $f$ to the variables that minimizes $\sum_{v} c_{vf(v)}$. Let $MinHom(\Gamma)$ denote the $MinHom$ problem parameterized by the set of predicates allowed for constraints. $MinHom(\Gamma)$ is related to many well-studied combinatorial optimization problems, and concrete applications can be found in, for instance, defence logistics and machine learning. We show that $MinHom(\Gamma)$ can be studied by using algebraic methods similar to those used for CSPs. With the aid of algebraic techniques, we classify the computational complexity of $MinHom(\Gamma)$ for all choices of $\Gamma$. Our result settles a general dichotomy conjecture previously resolved only for certain classes of directed graphs, [Gutin, Hell, Rafiey, Yeo, European J. of Combinatorics, 2008].

Rustem Takhanov

0708.3226

Filtering Additive Measurement Noise with Maximum Entropy in the Mean

cs.LG

The purpose of this note is to show how the method of maximum entropy in the mean (MEM) may be used to improve parametric estimation when the measurements are corrupted by large level of noise. The method is developed in the context on a concrete example: that of estimation of the parameter in an exponential distribution. We compare the performance of our method with the bayesian and maximum likelihood approaches.

Henryk Gzyl and Enrique ter Horst

0709.0509

On Universal Prediction and Bayesian Confirmation

math.ST cs.IT cs.LG math.IT stat.ML stat.TH

The Bayesian framework is a well-studied and successful framework for inductive reasoning, which includes hypothesis testing and confirmation, parameter estimation, sequence prediction, classification, and regression. But standard statistical guidelines for choosing the model class and prior are not always available or fail, in particular in complex situations. Solomonoff completed the Bayesian framework by providing a rigorous, unique, formal, and universal choice for the model class and the prior. We discuss in breadth how and in which sense universal (non-i.i.d.) sequence prediction solves various (philosophical) problems of traditional Bayesian sequence prediction. We show that Solomonoff's model possesses many desirable properties: Strong total and weak instantaneous bounds, and in contrast to most classical continuous prior densities has no zero p(oste)rior problem, i.e. can confirm universal hypotheses, is reparametrization and regrouping invariant, and avoids the old-evidence and updating problem. It even performs well (actually better) in non-computable environments.

Marcus Hutter

0709.1516

Learning for Dynamic Bidding in Cognitive Radio Resources

cs.LG cs.GT

In this paper, we model the various wireless users in a cognitive radio network as a collection of selfish, autonomous agents that strategically interact in order to acquire the dynamically available spectrum opportunities. Our main focus is on developing solutions for wireless users to successfully compete with each other for the limited and time-varying spectrum opportunities, given the experienced dynamics in the wireless network. We categorize these dynamics into two types: one is the disturbance due to the environment (e.g. wireless channel conditions, source traffic characteristics, etc.) and the other is the impact caused by competing users. To analyze the interactions among users given the environment disturbance, we propose a general stochastic framework for modeling how the competition among users for spectrum opportunities evolves over time. At each stage of the dynamic resource allocation, a central spectrum moderator auctions the available resources and the users strategically bid for the required resources. The joint bid actions affect the resource allocation and hence, the rewards and future strategies of all users. Based on the observed resource allocation and corresponding rewards from previous allocations, we propose a best response learning algorithm that can be deployed by wireless users to improve their bidding policy at each stage. The simulation results show that by deploying the proposed best response learning algorithm, the wireless users can significantly improve their own performance in terms of both the packet loss rate and the incurred cost for the used resources.

Fangwen Fu, Mihaela van der Schaar

0709.2446

Mutual information for the selection of relevant variables in spectrometric nonlinear modelling

cs.LG cs.NE stat.AP

Data from spectrophotometers form vectors of a large number of exploitable variables. Building quantitative models using these variables most often requires using a smaller set of variables than the initial one. Indeed, a too large number of input variables to a model results in a too large number of parameters, leading to overfitting and poor generalization abilities. In this paper, we suggest the use of the mutual information measure to select variables from the initial set. The mutual information measures the information content in input variables with respect to the model output, without making any assumption on the model that will be used; it is thus suitable for nonlinear modelling. In addition, it leads to the selection of variables among the initial set, and not to linear or nonlinear combinations of them. Without decreasing the model performances compared to other variable projection methods, it allows therefore a greater interpretability of the results.

Fabrice Rossi (INRIA Rocquencourt / INRIA Sophia Antipolis), Amaury Lendasse (CIS), Damien Fran\c{c}ois (CESAME), Vincent Wertz (CESAME), Michel Verleysen (DICE - MLG)

10.1016/j.chemolab.2005.06.010

0709.3427

Fast Algorithm and Implementation of Dissimilarity Self-Organizing Maps

cs.NE cs.LG

In many real world applications, data cannot be accurately represented by vectors. In those situations, one possible solution is to rely on dissimilarity measures that enable sensible comparison between observations. Kohonen's Self-Organizing Map (SOM) has been adapted to data described only through their dissimilarity matrix. This algorithm provides both non linear projection and clustering of non vector data. Unfortunately, the algorithm suffers from a high cost that makes it quite difficult to use with voluminous data sets. In this paper, we propose a new algorithm that provides an important reduction of the theoretical cost of the dissimilarity SOM without changing its outcome (the results are exactly the same as the ones obtained with the original algorithm). Moreover, we introduce implementation methods that result in very short running times. Improvements deduced from the theoretical cost model are validated on simulated and real world data (a word list clustering problem). We also demonstrate that the proposed implementation methods reduce by a factor up to 3 the running time of the fast algorithm over a standard implementation.

Brieuc Conan-Guez (LITA), Fabrice Rossi (INRIA Rocquencourt / INRIA Sophia Antipolis), A\"icha El Golli (INRIA Rocquencourt / INRIA Sophia Antipolis)

10.1016/j.neunet.2006.05.002

0709.3461

Une adaptation des cartes auto-organisatrices pour des donn\'ees d\'ecrites par un tableau de dissimilarit\'es

cs.NE cs.LG

Many data analysis methods cannot be applied to data that are not represented by a fixed number of real values, whereas most of real world observations are not readily available in such a format. Vector based data analysis methods have therefore to be adapted in order to be used with non standard complex data. A flexible and general solution for this adaptation is to use a (dis)similarity measure. Indeed, thanks to expert knowledge on the studied data, it is generally possible to define a measure that can be used to make pairwise comparison between observations. General data analysis methods are then obtained by adapting existing methods to (dis)similarity matrices. In this article, we propose an adaptation of Kohonen's Self Organizing Map (SOM) to (dis)similarity data. The proposed algorithm is an adapted version of the vector based batch SOM. The method is validated on real world data: we provide an analysis of the usage patterns of the web site of the Institut National de Recherche en Informatique et Automatique, constructed thanks to web log mining method.

A\"icha El Golli (INRIA Rocquencourt / INRIA Sophia Antipolis), Fabrice Rossi (INRIA Rocquencourt / INRIA Sophia Antipolis), Brieuc Conan-Guez (LITA), Yves Lechevallier (INRIA Rocquencourt / INRIA Sophia Antipolis)

0709.3586

Self-organizing maps and symbolic data

cs.NE cs.LG

In data analysis new forms of complex data have to be considered like for example (symbolic data, functional data, web data, trees, SQL query and multimedia data, ...). In this context classical data analysis for knowledge discovery based on calculating the center of gravity can not be used because input are not $\mathbb{R}^p$ vectors. In this paper, we present an application on real world symbolic data using the self-organizing map. To this end, we propose an extension of the self-organizing map that can handle symbolic data.

A\"icha El Golli (INRIA Rocquencourt / INRIA Sophia Antipolis), Brieuc Conan-Guez (INRIA Rocquencourt / INRIA Sophia Antipolis), Fabrice Rossi (INRIA Rocquencourt / INRIA Sophia Antipolis)

0709.3587

Fast Selection of Spectral Variables with B-Spline Compression

cs.LG stat.AP

The large number of spectral variables in most data sets encountered in spectral chemometrics often renders the prediction of a dependent variable uneasy. The number of variables hopefully can be reduced, by using either projection techniques or selection methods; the latter allow for the interpretation of the selected variables. Since the optimal approach of testing all possible subsets of variables with the prediction model is intractable, an incremental selection approach using a nonparametric statistics is a good option, as it avoids the computationally intensive use of the model itself. It has two drawbacks however: the number of groups of variables to test is still huge, and colinearities can make the results unstable. To overcome these limitations, this paper presents a method to select groups of spectral variables. It consists in a forward-backward procedure applied to the coefficients of a B-Spline representation of the spectra. The criterion used in the forward-backward procedure is the mutual information, allowing to find nonlinear dependencies between variables, on the contrary of the generally used correlation. The spline representation is used to get interpretability of the results, as groups of consecutive spectral variables will be selected. The experiments conducted on NIR spectra from fescue grass and diesel fuels show that the method provides clearly identified groups of selected variables, making interpretation easy, while keeping a low computational load. The prediction performances obtained using the selected coefficients are higher than those obtained by the same method applied directly to the original variables and similar to those obtained using traditional models, although using significantly less spectral variables.

Fabrice Rossi (INRIA Rocquencourt / INRIA Sophia Antipolis), Damien Fran\c{c}ois (CESAME), Vincent Wertz (CESAME), Marc Meurens (BNUT), Michel Verleysen (DICE - MLG)

10.1016/j.chemolab.2006.06.007

0709.3639

Resampling methods for parameter-free and robust feature selection with mutual information

cs.LG stat.AP

Combining the mutual information criterion with a forward feature selection strategy offers a good trade-off between optimality of the selected feature subset and computation time. However, it requires to set the parameter(s) of the mutual information estimator and to determine when to halt the forward procedure. These two choices are difficult to make because, as the dimensionality of the subset increases, the estimation of the mutual information becomes less and less reliable. This paper proposes to use resampling methods, a K-fold cross-validation and the permutation test, to address both issues. The resampling methods bring information about the variance of the estimator, information which can then be used to automatically set the parameter and to calculate a threshold to stop the forward procedure. The procedure is illustrated on a synthetic dataset as well as on real-world examples.

Damien Fran\c{c}ois (CESAME), Fabrice Rossi (INRIA Rocquencourt / INRIA Sophia Antipolis), Vincent Wertz (CESAME), Michel Verleysen (DICE - MLG)

10.1016/j.neucom.2006.11.019

0709.3640

Evolving Classifiers: Methods for Incremental Learning

cs.LG cs.AI cs.NE

The ability of a classifier to take on new information and classes by evolving the classifier without it having to be fully retrained is known as incremental learning. Incremental learning has been successfully applied to many classification problems, where the data is changing and is not all available at once. In this paper there is a comparison between Learn++, which is one of the most recent incremental learning algorithms, and the new proposed method of Incremental Learning Using Genetic Algorithm (ILUGA). Learn++ has shown good incremental learning capabilities on benchmark datasets on which the new ILUGA method has been tested. ILUGA has also shown good incremental learning ability using only a few classifiers and does not suffer from catastrophic forgetting. The results obtained for ILUGA on the Optical Character Recognition (OCR) and Wine datasets are good, with an overall accuracy of 93% and 94% respectively showing a 4% improvement over Learn++.MT for the difficult multi-class OCR dataset.

Greg Hulley and Tshilidzi Marwala

0709.3965

Classification of Images Using Support Vector Machines

cs.LG cs.AI

Support Vector Machines (SVMs) are a relatively new supervised classification technique to the land cover mapping community. They have their roots in Statistical Learning Theory and have gained prominence because they are robust, accurate and are effective even when using a small training sample. By their nature SVMs are essentially binary classifiers, however, they can be adopted to handle the multiple classification tasks common in remote sensing studies. The two approaches commonly used are the One-Against-One (1A1) and One-Against-All (1AA) techniques. In this paper, these approaches are evaluated in as far as their impact and implication for land cover mapping. The main finding from this research is that whereas the 1AA technique is more predisposed to yielding unclassified and mixed pixels, the resulting classification accuracy is not significantly different from 1A1 approach. It is the authors conclusions that ultimately the choice of technique adopted boils down to personal preference and the uniqueness of the dataset at hand.

Gidudu Anthony, Hulley Greg and Marwala Tshilidzi

0709.3967

Prediction with expert advice for the Brier game

cs.LG

We show that the Brier game of prediction is mixable and find the optimal learning rate and substitution function for it. The resulting prediction algorithm is applied to predict results of football and tennis matches. The theoretical performance guarantee turns out to be rather tight on these data sets, especially in the case of the more extensive tennis data.

Vladimir Vovk and Fedor Zhdanov

0710.0485

Association Rules in the Relational Calculus

cs.DB cs.LG cs.LO

One of the most utilized data mining tasks is the search for association rules. Association rules represent significant relationships between items in transactions. We extend the concept of association rule to represent a much broader class of associations, which we refer to as \emph{entity-relationship rules.} Semantically, entity-relationship rules express associations between properties of related objects. Syntactically, these rules are based on a broad subclass of safe domain relational calculus queries. We propose a new definition of support and confidence for entity-relationship rules and for the frequency of entity-relationship queries. We prove that the definition of frequency satisfies standard probability axioms and the Apriori property.

Oliver Schulte, Flavia Moser, Martin Ester and Zhiyong Lu

0710.2083

The structure of verbal sequences analyzed with unsupervised learning techniques

cs.CL cs.AI cs.LG

Data mining allows the exploration of sequences of phenomena, whereas one usually tends to focus on isolated phenomena or on the relation between two phenomena. It offers invaluable tools for theoretical analyses and exploration of the structure of sentences, texts, dialogues, and speech. We report here the results of an attempt at using it for inspecting sequences of verbs from French accounts of road accidents. This analysis comes from an original approach of unsupervised training allowing the discovery of the structure of sequential data. The entries of the analyzer were only made of the verbs appearing in the sentences. It provided a classification of the links between two successive verbs into four distinct clusters, allowing thus text segmentation. We give here an interpretation of these clusters by applying a statistical analysis to independent semantic annotations.

Catherine Recanati (LIPN), Nicoleta Rogovschi (LIPN), Youn\`es Bennani (LIPN)

0710.2446

Consistency of trace norm minimization

cs.LG

Regularization by the sum of singular values, also referred to as the trace norm, is a popular technique for estimating low rank rectangular matrices. In this paper, we extend some of the consistency results of the Lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss. We also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled.

Francis Bach (WILLOW Project - Inria/Ens)

0710.2848

An efficient reduction of ranking to classification

cs.LG cs.IR

This paper describes an efficient reduction of the learning problem of ranking to binary classification. The reduction guarantees an average pairwise misranking regret of at most that of the binary classifier regret, improving a recent result of Balcan et al which only guarantees a factor of 2. Moreover, our reduction applies to a broader class of ranking loss functions, admits a simpler proof, and the expected running time complexity of our algorithm in terms of number of calls to a classifier or preference function is improved from $\Omega(n^2)$ to $O(n \log n)$. In addition, when the top $k$ ranked elements only are required ($k \ll n$), as in many applications in information extraction or search engines, the time complexity of our algorithm can be further reduced to $O(k \log k + n)$. Our reduction and algorithm are thus practical for realistic applications where the number of points to rank exceeds several thousands. Much of our results also extend beyond the bipartite case previously studied. Our rediction is a randomized one. To complement our result, we also derive lower bounds on any deterministic reduction from binary (preference) classification to ranking, implying that our use of a randomized reduction is essentially necessary for the guarantees we provide.

Nir Ailon and Mehryar Mohri

0710.2889

Combining haplotypers

cs.LG cs.CE q-bio.QM

Statistically resolving the underlying haplotype pair for a genotype measurement is an important intermediate step in gene mapping studies, and has received much attention recently. Consequently, a variety of methods for this problem have been developed. Different methods employ different statistical models, and thus implicitly encode different assumptions about the nature of the underlying haplotype structure. Depending on the population sample in question, their relative performance can vary greatly, and it is unclear which method to choose for a particular sample. Instead of choosing a single method, we explore combining predictions returned by different methods in a principled way, and thereby circumvent the problem of method selection. We propose several techniques for combining haplotype reconstructions and analyze their computational properties. In an experimental study on real-world haplotype data we show that such techniques can provide more accurate and robust reconstructions, and are useful for outlier detection. Typically, the combined prediction is at least as accurate as or even more accurate than the best individual method, effectively circumventing the method selection problem.

Matti K\"a\"ari\"ainen, Niels Landwehr, Sampsa Lappalainen and Taneli Mielik\"ainen

0710.5116

A Tutorial on Spectral Clustering

cs.DS cs.LG

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

Ulrike von Luxburg

0711.0189

Building Rules on Top of Ontologies for the Semantic Web with Inductive Logic Programming

cs.AI cs.LG

Building rules on top of ontologies is the ultimate goal of the logical layer of the Semantic Web. To this aim an ad-hoc mark-up language for this layer is currently under discussion. It is intended to follow the tradition of hybrid knowledge representation and reasoning systems such as $\mathcal{AL}$-log that integrates the description logic $\mathcal{ALC}$ and the function-free Horn clausal language \textsc{Datalog}. In this paper we consider the problem of automating the acquisition of these rules for the Semantic Web. We propose a general framework for rule induction that adopts the methodological apparatus of Inductive Logic Programming and relies on the expressive and deductive power of $\mathcal{AL}$-log. The framework is valid whatever the scope of induction (description vs. prediction) is. Yet, for illustrative purposes, we also discuss an instantiation of the framework which aims at description and turns out to be useful in Ontology Refinement. Keywords: Inductive Logic Programming, Hybrid Knowledge Representation and Reasoning Systems, Ontologies, Semantic Web. Note: To appear in Theory and Practice of Logic Programming (TPLP)

Francesca A. Lisi

0711.1814

Empirical Evaluation of Four Tensor Decomposition Algorithms

cs.LG cs.CL cs.IR

Higher-order tensor decompositions are analogous to the familiar Singular Value Decomposition (SVD), but they transcend the limitations of matrices (second-order tensors). SVD is a powerful tool that has achieved impressive results in information retrieval, collaborative filtering, computational linguistics, computational vision, and other fields. However, SVD is limited to two-dimensional arrays of data (two modes), and many potential applications have three or more modes, which require higher-order tensor decompositions. This paper evaluates four algorithms for higher-order tensor decomposition: Higher-Order Singular Value Decomposition (HO-SVD), Higher-Order Orthogonal Iteration (HOOI), Slice Projection (SP), and Multislice Projection (MP). We measure the time (elapsed run time), space (RAM and disk space requirements), and fit (tensor reconstruction accuracy) of the four algorithms, under a variety of conditions. We find that standard implementations of HO-SVD and HOOI do not scale up to larger tensors, due to increasing RAM requirements. We recommend HOOI for tensors that are small enough for the available RAM and MP for larger tensors.

Peter D. Turney (National Research Council of Canada)

0711.2023

Inverse Sampling for Nonasymptotic Sequential Estimation of Bounded Variable Means

math.ST cs.LG math.PR stat.TH

In this paper, we consider the nonasymptotic sequential estimation of means of random variables bounded in between zero and one. We have rigorously demonstrated that, in order to guarantee prescribed relative precision and confidence level, it suffices to continue sampling until the sample sum is no less than a certain bound and then take the average of samples as an estimate for the mean of the bounded random variable. We have developed an explicit formula and a bisection search method for the determination of such bound of sample sum, without any knowledge of the bounded variable. Moreover, we have derived bounds for the distribution of sample size. In the special case of Bernoulli random variables, we have established analytical and numerical methods to further reduce the bound of sample sum and thus improve the efficiency of sampling. Furthermore, the fallacy of existing results are detected and analyzed.

Xinjia Chen

0711.2801

Image Classification Using SVMs: One-against-One Vs One-against-All

cs.LG cs.AI cs.CV

Support Vector Machines (SVMs) are a relatively new supervised classification technique to the land cover mapping community. They have their roots in Statistical Learning Theory and have gained prominence because they are robust, accurate and are effective even when using a small training sample. By their nature SVMs are essentially binary classifiers, however, they can be adopted to handle the multiple classification tasks common in remote sensing studies. The two approaches commonly used are the One-Against-One (1A1) and One-Against-All (1AA) techniques. In this paper, these approaches are evaluated in as far as their impact and implication for land cover mapping. The main finding from this research is that whereas the 1AA technique is more predisposed to yielding unclassified and mixed pixels, the resulting classification accuracy is not significantly different from 1A1 approach. It is the authors conclusion therefore that ultimately the choice of technique adopted boils down to personal preference and the uniqueness of the dataset at hand.

Gidudu Anthony, Hulley Gregg and Marwala Tshilidzi

0711.2914

Clustering with Transitive Distance and K-Means Duality

cs.LG

Recent spectral clustering methods are a propular and powerful technique for data clustering. These methods need to solve the eigenproblem whose computational complexity is $O(n^3)$, where $n$ is the number of data samples. In this paper, a non-eigenproblem based clustering method is proposed to deal with the clustering problem. Its performance is comparable to the spectral clustering algorithms but it is more efficient with computational complexity $O(n^2)$. We show that with a transitive distance and an observed property, called K-means duality, our algorithm can be used to handle data sets with complex cluster shapes, multi-scale clusters, and noise. Moreover, no parameters except the number of clusters need to be set in our algorithm.

Chunjing Xu, Jianzhuang Liu, Xiaoou Tang

0711.3594

Derivations of Normalized Mutual Information in Binary Classifications

cs.LG cs.IT math.IT

This correspondence studies the basic problem of classifications - how to evaluate different classifiers. Although the conventional performance indexes, such as accuracy, are commonly used in classifier selection or evaluation, information-based criteria, such as mutual information, are becoming popular in feature/model selections. In this work, we propose to assess classifiers in terms of normalized mutual information (NI), which is novel and well defined in a compact range for classifier evaluation. We derive close-form relations of normalized mutual information with respect to accuracy, precision, and recall in binary classifications. By exploring the relations among them, we reveal that NI is actually a set of nonlinear functions, with a concordant power-exponent form, to each performance index. The relations can also be expressed with respect to precision and recall, or to false alarm and hitting rate (recall).

Yong Wang, Bao-Gang Hu

0711.3675

Covariance and PCA for Categorical Variables

cs.LG

Covariances from categorical variables are defined using a regular simplex expression for categories. The method follows the variance definition by Gini, and it gives the covariance as a solution of simultaneous equations. The calculated results give reasonable values for test data. A method of principal component analysis (RS-PCA) is also proposed using regular simplex expressions, which allows easy interpretation of the principal components. The proposed methods apply to variable selection problem of categorical data USCensus1990 data. The proposed methods give appropriate criterion for the variable selection problem of categorical

Hirotaka Niitsuma and Takashi Okada

0711.4452

On the Relationship between the Posterior and Optimal Similarity

cs.LG

For a classification problem described by the joint density $P(\omega,x)$, models of $P(\omega\eq\omega'|x,x')$ (the ``Bayesian similarity measure'') have been shown to be an optimal similarity measure for nearest neighbor classification. This paper analyzes demonstrates several additional properties of that conditional distribution. The paper first shows that we can reconstruct, up to class labels, the class posterior distribution $P(\omega|x)$ given $P(\omega\eq\omega'|x,x')$, gives a procedure for recovering the class labels, and gives an asymptotically Bayes-optimal classification procedure. It also shows, given such an optimal similarity measure, how to construct a classifier that outperforms the nearest neighbor classifier and achieves Bayes-optimal classification rates. The paper then analyzes Bayesian similarity in a framework where a classifier faces a number of related classification tasks (multitask learning) and illustrates that reconstruction of the class posterior distribution is not possible in general. Finally, the paper identifies a distinct class of classification problems using $P(\omega\eq\omega'|x,x')$ and shows that using $P(\omega\eq\omega'|x,x')$ to solve those problems is the Bayes optimal solution.

Thomas M. Breuel

0712.0130

A Reactive Tabu Search Algorithm for Stimuli Generation in Psycholinguistics

cs.AI cs.CC cs.DM cs.LG

The generation of meaningless "words" matching certain statistical and/or linguistic criteria is frequently needed for experimental purposes in Psycholinguistics. Such stimuli receive the name of pseudowords or nonwords in the Cognitive Neuroscience literatue. The process for building nonwords sometimes has to be based on linguistic units such as syllables or morphemes, resulting in a numerical explosion of combinations when the size of the nonwords is increased. In this paper, a reactive tabu search scheme is proposed to generate nonwords of variables size. The approach builds pseudowords by using a modified Metaheuristic algorithm based on a local search procedure enhanced by a feedback-based scheme. Experimental results show that the new algorithm is a practical and effective tool for nonword generation.

Alejandro Chinea Manrique De Lara

0712.0451

Equations of States in Singular Statistical Estimation

cs.LG

Learning machines which have hierarchical structures or hidden variables are singular statistical models because they are nonidentifiable and their Fisher information matrices are singular. In singular statistical models, neither the Bayes a posteriori distribution converges to the normal distribution nor the maximum likelihood estimator satisfies asymptotic normality. This is the main reason why it has been difficult to predict their generalization performances from trained states. In this paper, we study four errors, (1) Bayes generalization error, (2) Bayes training error, (3) Gibbs generalization error, and (4) Gibbs training error, and prove that there are mathematical relations among these errors. The formulas proved in this paper are equations of states in statistical estimation because they hold for any true distribution, any parametric model, and any a priori distribution. Also we show that Bayes and Gibbs generalization errors are estimated by Bayes and Gibbs training errors, and propose widely applicable information criteria which can be applied to both regular and singular statistical models.

Sumio Watanabe

0712.0653

A Universal Kernel for Learning Regular Languages

cs.LG cs.DM

We give a universal kernel that renders all the regular languages linearly separable. We are not able to compute this kernel efficiently and conjecture that it is intractable, but we do have an efficient $\eps$-approximation.

Leonid (Aryeh) Kontorovich

0712.0840

Automatic Pattern Classification by Unsupervised Learning Using Dimensionality Reduction of Data with Mirroring Neural Networks

cs.LG cs.AI cs.NE

This paper proposes an unsupervised learning technique by using Multi-layer Mirroring Neural Network and Forgy's clustering algorithm. Multi-layer Mirroring Neural Network is a neural network that can be trained with generalized data inputs (different categories of image patterns) to perform non-linear dimensionality reduction and the resultant low-dimensional code is used for unsupervised pattern classification using Forgy's algorithm. By adapting the non-linear activation function (modified sigmoidal function) and initializing the weights and bias terms to small random values, mirroring of the input pattern is initiated. In training, the weights and bias terms are changed in such a way that the input presented is reproduced at the output by back propagating the error. The mirroring neural network is capable of reducing the input vector to a great degree (approximately 1/30th the original size) and also able to reconstruct the input pattern at the output layer from this reduced code units. The feature set (output of central hidden layer) extracted from this network is fed to Forgy's algorithm, which classify input data patterns into distinguishable classes. In the implementation of Forgy's algorithm, initial seed points are selected in such a way that they are distant enough to be perfectly grouped into different categories. Thus a new method of unsupervised learning is formulated and demonstrated in this paper. This method gave impressive results when applied to classification of different image patterns.

Dasika Ratna Deepthi, G.R.Aditya Krishna and K. Eswaran

0712.0938

Reconstruction of Markov Random Fields from Samples: Some Easy Observations and Algorithms

cs.CC cs.LG

Markov random fields are used to model high dimensional distributions in a number of applied areas. Much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the Markov random fields. We analyze a simple algorithm for reconstructing the underlying graph defining a Markov random field on $n$ nodes and maximum degree $d$ given observations. We show that under mild non-degeneracy conditions it reconstructs the generating graph with high probability using $\Theta(d \epsilon^{-2}\delta^{-4} \log n)$ samples where $\epsilon,\delta$ depend on the local interactions. For most local interaction $\eps,\delta$ are of order $\exp(-O(d))$. Our results are optimal as a function of $n$ up to a multiplicative constant depending on $d$ and the strength of the local interactions. Our results seem to be the first results for general models that guarantee that {\em the} generating model is reconstructed. Furthermore, we provide explicit $O(n^{d+2} \epsilon^{-2}\delta^{-4} \log n)$ running time bound. In cases where the measure on the graph has correlation decay, the running time is $O(n^2 \log n)$ for all fixed $d$. We also discuss the effect of observing noisy samples and show that as long as the noise level is low, our algorithm is effective. On the other hand, we construct an example where large noise implies non-identifiability even for generic noise and interactions. Finally, we briefly show that in some simple cases, models with hidden nodes can also be recovered.

Guy Bresler, Elchanan Mossel, Allan Sly

0712.1402

A New Theoretic Foundation for Cross-Layer Optimization

cs.NI cs.LG

Cross-layer optimization solutions have been proposed in recent years to improve the performance of network users operating in a time-varying, error-prone wireless environment. However, these solutions often rely on ad-hoc optimization approaches, which ignore the different environmental dynamics experienced at various layers by a user and violate the layered network architecture of the protocol stack by requiring layers to provide access to their internal protocol parameters to other layers. This paper presents a new theoretic foundation for cross-layer optimization, which allows each layer to make autonomous decisions individually, while maximizing the utility of the wireless user by optimally determining what information needs to be exchanged among layers. Hence, this cross-layer framework does not change the current layered architecture. Specifically, because the wireless user interacts with the environment at various layers of the protocol stack, the cross-layer optimization problem is formulated as a layered Markov decision process (MDP) in which each layer adapts its own protocol parameters and exchanges information (messages) with other layers in order to cooperatively maximize the performance of the wireless user. The message exchange mechanism for determining the optimal cross-layer transmission strategies has been designed for both off-line optimization and on-line dynamic adaptation. We also show that many existing cross-layer optimization algorithms can be formulated as simplified, sub-optimal, versions of our layered MDP framework.

Fangwen Fu and Mihaela van der Schaar

0712.2497

Density estimation in linear time

cs.LG

We consider the problem of choosing a density estimate from a set of distributions F, minimizing the L1-distance to an unknown distribution (Devroye, Lugosi 2001). Devroye and Lugosi analyze two algorithms for the problem: Scheffe tournament winner and minimum distance estimate. The Scheffe tournament estimate requires fewer computations than the minimum distance estimate, but has strictly weaker guarantees than the latter. We focus on the computational aspect of density estimation. We present two algorithms, both with the same guarantee as the minimum distance estimate. The first one, a modification of the minimum distance estimate, uses the same number (quadratic in |F|) of computations as the Scheffe tournament. The second one, called ``efficient minimum loss-weight estimate,'' uses only a linear number of computations, assuming that F is preprocessed. We also give examples showing that the guarantees of the algorithms cannot be improved and explore randomized algorithms for density estimation.

Satyaki Mahalanabis, Daniel Stefankovic

0712.2869

Graph kernels between point clouds

cs.LG

Point clouds are sets of points in two or three dimensions. Most kernel methods for learning on sets of points have not yet dealt with the specific geometrical invariances and practical constraints associated with point clouds in computer vision and graphics. In this paper, we present extensions of graph kernels for point clouds, which allow to use kernel methods for such ob jects as shapes, line drawings, or any three-dimensional point clouds. In order to design rich and numerically efficient kernels with as few free parameters as possible, we use kernels between covariance matrices and their factorizations on graphical models. We derive polynomial time dynamic programming recursions and present applications to recognition of handwritten digits and Chinese characters from few training examples.

Francis Bach (WILLOW Project - Inria/Ens)

0712.3402

Improving the Performance of PieceWise Linear Separation Incremental Algorithms for Practical Hardware Implementations

cs.NE cs.AI cs.LG

In this paper we shall review the common problems associated with Piecewise Linear Separation incremental algorithms. This kind of neural models yield poor performances when dealing with some classification problems, due to the evolving schemes used to construct the resulting networks. So as to avoid this undesirable behavior we shall propose a modification criterion. It is based upon the definition of a function which will provide information about the quality of the network growth process during the learning phase. This function is evaluated periodically as the network structure evolves, and will permit, as we shall show through exhaustive benchmarks, to considerably improve the performance(measured in terms of network complexity and generalization capabilities) offered by the networks generated by these incremental models.

Alejandro Chinea Manrique De Lara, Juan Manuel Moreno, Arostegui Jordi Madrenas, Joan Cabestany

0712.3654

Improved Collaborative Filtering Algorithm via Information Transformation

cs.LG cs.CY

In this paper, we propose a spreading activation approach for collaborative filtering (SA-CF). By using the opinion spreading process, the similarity between any users can be obtained. The algorithm has remarkably higher accuracy than the standard collaborative filtering (CF) using Pearson correlation. Furthermore, we introduce a free parameter $\beta$ to regulate the contributions of objects to user-user correlations. The numerical results indicate that decreasing the influence of popular objects can further improve the algorithmic accuracy and personality. We argue that a better algorithm should simultaneously require less computation and generate higher accuracy. Accordingly, we further propose an algorithm involving only the top-$N$ similar neighbors for each target user, which has both less computational complexity and higher algorithmic accuracy.

Jian-Guo Liu, Bing-Hong Wang, Qiang Guo

10.1142/S0129183109013613

0712.3807

Online EM Algorithm for Latent Data Models

stat.CO cs.LG

In this contribution, we propose a generic online (also sometimes called adaptive or recursive) version of the Expectation-Maximisation (EM) algorithm applicable to latent variable models of independent observations. Compared to the algorithm of Titterington (1984), this approach is more directly connected to the usual EM algorithm and does not rely on integration with respect to the complete data distribution. The resulting algorithm is usually simpler and is shown to achieve convergence to the stationary points of the Kullback-Leibler divergence between the marginal distribution of the observation and the model distribution at the optimal rate, i.e., that of the maximum likelihood estimator. In addition, the proposed approach is also suitable for conditional (or regression) models, as illustrated in the case of the mixture of linear regressions model.

Olivier Capp\'e (LTCI), Eric Moulines (LTCI)

10.1111/j.1467-9868.2009.00698.x

0712.4273

Staring at Economic Aggregators through Information Lenses

cs.IT cs.LG math.IT math.OC

It is hard to exaggerate the role of economic aggregators -- functions that summarize numerous and / or heterogeneous data -- in economic models since the early XX$^{th}$ century. In many cases, as witnessed by the pioneering works of Cobb and Douglas, these functions were information quantities tailored to economic theories, i.e. they were built to fit economic phenomena. In this paper, we look at these functions from the complementary side: information. We use a recent toolbox built on top of a vast class of distortions coined by Bregman, whose application field rivals metrics' in various subfields of mathematics. This toolbox makes it possible to find the quality of an aggregator (for consumptions, prices, labor, capital, wages, etc.), from the standpoint of the information it carries. We prove a rather striking result. From the informational standpoint, well-known economic aggregators do belong to the \textit{optimal} set. As common economic assumptions enter the analysis, this large set shrinks, and it essentially ends up \textit{exactly fitting} either CES, or Cobb-Douglas, or both. To summarize, in the relevant economic contexts, one could not have crafted better some aggregator from the information standpoint. We also discuss global economic behaviors of optimal information aggregators in general, and present a brief panorama of the links between economic and information aggregators. Keywords: Economic Aggregators, CES, Cobb-Douglas, Bregman divergences

Richard Nock, Nicolas Sanz, Fred Celimene, Frank Nielsen

0801.0390

Online variants of the cross-entropy method

cs.LG

The cross-entropy method is a simple but efficient method for global optimization. In this paper we provide two online variants of the basic CEM, together with a proof of convergence.

Istvan Szita and Andras Lorincz

0801.1988

Factored Value Iteration Converges

cs.AI cs.LG

In this paper we propose a novel algorithm, factored value iteration (FVI), for the approximate solution of factored Markov decision processes (fMDPs). The traditional approximate value iteration algorithm is modified in two ways. For one, the least-squares projection operator is modified so that it does not increase max-norm, and thus preserves convergence. The other modification is that we uniformly sample polynomially many samples from the (exponentially large) state space. This way, the complexity of our algorithm becomes polynomial in the size of the fMDP description length. We prove that the algorithm is convergent. We also derive an upper bound on the difference between our approximate solution and the optimal one, and also on the error introduced by sampling. We analyze various projection operators with respect to their computation complexity and their convergence when combined with approximate value iteration.

Istvan Szita and Andras Lorincz

0801.2069

The optimal assignment kernel is not positive definite

cs.LG

We prove that the optimal assignment kernel, proposed recently as an attempt to embed labeled graphs and more generally tuples of basic data to a Hilbert space, is in fact not always positive definite.

Jean-Philippe Vert (CB)

0801.4061

Information Width

cs.DM cs.IT cs.LG math.IT

Kolmogorov argued that the concept of information exists also in problems with no underlying stochastic model (as Shannon's information representation) for instance, the information contained in an algorithm or in the genome. He introduced a combinatorial notion of entropy and information $I(x:\sy)$ conveyed by a binary string $x$ about the unknown value of a variable $\sy$. The current paper poses the following questions: what is the relationship between the information conveyed by $x$ about $\sy$ to the description complexity of $x$ ? is there a notion of cost of information ? are there limits on how efficient $x$ conveys information ? To answer these questions Kolmogorov's definition is extended and a new concept termed {\em information width} which is similar to $n$-widths in approximation theory is introduced. Information of any input source, e.g., sample-based, general side-information or a hybrid of both can be evaluated by a single common formula. An application to the space of binary functions is considered.

Joel Ratsaby

0801.4790

On the Complexity of Binary Samples

cs.DM cs.AI cs.LG

Consider a class $\mH$ of binary functions $h: X\to\{-1, +1\}$ on a finite interval $X=[0, B]\subset \Real$. Define the {\em sample width} of $h$ on a finite subset (a sample) $S\subset X$ as $\w_S(h) \equiv \min_{x\in S} |\w_h(x)|$, where $\w_h(x) = h(x) \max\{a\geq 0: h(z)=h(x), x-a\leq z\leq x+a\}$. Let $\mathbb{S}_\ell$ be the space of all samples in $X$ of cardinality $\ell$ and consider sets of wide samples, i.e., {\em hypersets} which are defined as $A_{\beta, h} = \{S\in \mathbb{S}_\ell: \w_{S}(h) \geq \beta\}$. Through an application of the Sauer-Shelah result on the density of sets an upper estimate is obtained on the growth function (or trace) of the class $\{A_{\beta, h}: h\in\mH\}$, $\beta>0$, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples $S\in\mathbb{S}_\ell$ of cardinality $m$. The estimate is $2\sum_{i=0}^{2\lfloor B/(2\beta)\rfloor}{m-\ell\choose i}$.

Joel Ratsaby

0801.4794

New Estimation Procedures for PLS Path Modelling

cs.LG

Given R groups of numerical variables X1, ... XR, we assume that each group is the result of one underlying latent variable, and that all latent variables are bound together through a linear equation system. Moreover, we assume that some explanatory latent variables may interact pairwise in one or more equations. We basically consider PLS Path Modelling's algorithm to estimate both latent variables and the model's coefficients. New "external" estimation schemes are proposed that draw latent variables towards strong group structures in a more flexible way. New "internal" estimation schemes are proposed to enable PLSPM to make good use of variable group complementarity and to deal with interactions. Application examples are given.

Xavier Bry (I3M)

0802.1002

Learning Balanced Mixtures of Discrete Distributions with Small Sample

cs.LG stat.ML

We study the problem of partitioning a small sample of $n$ individuals from a mixture of $k$ product distributions over a Boolean cube $\{0, 1\}^K$ according to their distributions. Each distribution is described by a vector of allele frequencies in $\R^K$. Given two distributions, we use $\gamma$ to denote the average $\ell_2^2$ distance in frequencies across $K$ dimensions, which measures the statistical divergence between them. We study the case assuming that bits are independently distributed across $K$ dimensions. This work demonstrates that, for a balanced input instance for $k = 2$, a certain graph-based optimization function returns the correct partition with high probability, where a weighted graph $G$ is formed over $n$ individuals, whose pairwise hamming distances between their corresponding bit vectors define the edge weights, so long as $K = \Omega(\ln n/\gamma)$ and $Kn = \tilde\Omega(\ln n/\gamma^2)$. The function computes a maximum-weight balanced cut of $G$, where the weight of a cut is the sum of the weights across all edges in the cut. This result demonstrates a nice property in the high-dimensional feature space: one can trade off the number of features that are required with the size of the sample to accomplish certain tasks like clustering.

Shuheng Zhou

0802.1244

Bayesian Nonlinear Principal Component Analysis Using Random Fields

cs.CV cs.LG

We propose a novel model for nonlinear dimension reduction motivated by the probabilistic formulation of principal component analysis. Nonlinearity is achieved by specifying different transformation matrices at different locations of the latent space and smoothing the transformation using a Markov random field type prior. The computation is made feasible by the recent advances in sampling from von Mises-Fisher distributions.

Heng Lian

0802.1258

A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization

cs.LG

We present a general approach for collaborative filtering (CF) using spectral regularization to learn linear operators from "users" to the "objects" they rate. Recent low-rank type matrix completion approaches to CF are shown to be special cases. However, unlike existing regularization based CF methods, our approach can be used to also incorporate information such as attributes of the users or the objects -- a limitation of existing regularization based CF methods. We then provide novel representer theorems that we use to develop new estimation methods. We provide learning algorithms based on low-rank decompositions, and test them on a standard CF dataset. The experiments indicate the advantages of generalizing the existing regularization based CF methods to incorporate related information about users and objects. Finally, we show that certain multi-task learning methods can be also seen as special cases of our proposed approach.

Jacob Abernethy, Francis Bach (INRIA Rocquencourt), Theodoros Evgeniou, Jean-Philippe Vert (CB)

0802.1430

Combining Expert Advice Efficiently

cs.LG cs.DS cs.IT math.IT

We show how models for prediction with expert advice can be defined concisely and clearly using hidden Markov models (HMMs); standard HMM algorithms can then be used to efficiently calculate, among other things, how the expert predictions should be weighted according to the model. We cast many existing models as HMMs and recover the best known running times in each case. We also describe two new models: the switch distribution, which was recently developed to improve Bayesian/Minimum Description Length model selection, and a new generalisation of the fixed share algorithm based on run-length coding. We give loss bounds for all models and shed new light on their relationships.

Wouter Koolen and Steven de Rooij

0802.2015

A Radar-Shaped Statistic for Testing and Visualizing Uniformity Properties in Computer Experiments

cs.LG math.ST stat.TH

In the study of computer codes, filling space as uniformly as possible is important to describe the complexity of the investigated phenomenon. However, this property is not conserved by reducing the dimension. Some numeric experiment designs are conceived in this sense as Latin hypercubes or orthogonal arrays, but they consider only the projections onto the axes or the coordinate planes. In this article we introduce a statistic which allows studying the good distribution of points according to all 1-dimensional projections. By angularly scanning the domain, we obtain a radar type representation, allowing the uniformity defects of a design to be identified with respect to its projections onto straight lines. The advantages of this new tool are demonstrated on usual examples of space-filling designs (SFD) and a global statistic independent of the angle of rotation is studied.

Jessica Franco, Laurent Carraro, Olivier Roustant, Astrid Jourdan (LMA-PAU)

0802.2158

Compressed Counting

cs.IT cs.CC cs.DM cs.DS cs.LG math.IT

Counting is among the most fundamental operations in computing. For example, counting the pth frequency moment has been a very active area of research, in theoretical computer science, databases, and data mining. When p=1, the task (i.e., counting the sum) can be accomplished using a simple counter. Compressed Counting (CC) is proposed for efficiently computing the pth frequency moment of a data stream signal A_t, where 0<p<=2. CC is applicable if the streaming data follow the Turnstile model, with the restriction that at the time t for the evaluation, A_t[i]>= 0, which includes the strict Turnstile model as a special case. For natural data streams encountered in practice, this restriction is minor. The underly technique for CC is what we call skewed stable random projections, which captures the intuition that, when p=1 a simple counter suffices, and when p = 1+/\Delta with small \Delta, the sample complexity of a counter system should be low (continuously as a function of \Delta). We show at small \Delta the sample complexity (number of projections) k = O(1/\epsilon) instead of O(1/\epsilon^2). Compressed Counting can serve a basic building block for other tasks in statistics and computing, for example, estimation entropies of data streams, parameter estimations using the method of moments and maximum likelihood. Finally, another contribution is an algorithm for approximating the logarithmic norm, \sum_{i=1}^D\log A_t[i], and logarithmic distance. The logarithmic distance is useful in machine learning practice with heavy-tailed data.

Ping Li

0802.2305

Sign Language Tutoring Tool

cs.LG cs.HC

In this project, we have developed a sign language tutor that lets users learn isolated signs by watching recorded videos and by trying the same signs. The system records the user's video and analyses it. If the sign is recognized, both verbal and animated feedback is given to the user. The system is able to recognize complex signs that involve both hand gestures and head movements and expressions. Our performance tests yield a 99% recognition rate on signs involving only manual gestures and 85% recognition rate on signs that involve both manual and non manual components, such as head movement and facial expressions.

Oya Aran, Ismail Ari, Alexandre Benoit (GIPSA-lab), Ana Huerta Carrillo, Fran\c{c}ois-Xavier Fanard (TELE), Pavel Campr, Lale Akarun, Alice Caplier (GIPSA-lab), Michele Rombaut (GIPSA-lab), Bulent Sankur

0802.2428

Pure Exploration for Multi-Armed Bandit Problems

math.ST cs.LG stat.TH

We consider the framework of stochastic multi-armed bandit problems and study the possibilities and limitations of forecasters that perform an on-line exploration of the arms. These forecasters are assessed in terms of their simple regret, a regret notion that captures the fact that exploration is only constrained by the number of available rounds (not necessarily known in advance), in contrast to the case when the cumulative regret is considered and when exploitation needs to be performed at the same time. We believe that this performance criterion is suited to situations when the cost of pulling an arm is expressed in terms of resources rather than rewards. We discuss the links between the simple and the cumulative regret. One of the main results in the case of a finite number of arms is a general lower bound on the simple regret of a forecaster in terms of its cumulative regret: the smaller the latter, the larger the former. Keeping this result in mind, we then exhibit upper bounds on the simple regret of some forecasters. The paper ends with a study devoted to continuous-armed bandit problems; we show that the simple regret can be minimized with respect to a family of probability distributions if and only if the cumulative regret can be minimized for it. Based on this equivalence, we are able to prove that the separable metric spaces are exactly the metric spaces on which these regrets can be minimized with respect to the family of all probability distributions with continuous mean-payoff functions.

S\'ebastien Bubeck (INRIA Futurs), R\'emi Munos (INRIA Futurs), Gilles Stoltz (DMA, GREGH)

0802.2655

Knowledge Technologies

cs.CY cs.AI cs.LG cs.SE

Several technologies are emerging that provide new ways to capture, store, present and use knowledge. This book is the first to provide a comprehensive introduction to five of the most important of these technologies: Knowledge Engineering, Knowledge Based Engineering, Knowledge Webs, Ontologies and Semantic Webs. For each of these, answers are given to a number of key questions (What is it? How does it operate? How is a system developed? What can it be used for? What tools are available? What are the main issues?). The book is aimed at students, researchers and practitioners interested in Knowledge Management, Artificial Intelligence, Design Engineering and Web Technologies. During the 1990s, Nick worked at the University of Nottingham on the application of AI techniques to knowledge management and on various knowledge acquisition projects to develop expert systems for military applications. In 1999, he joined Epistemics where he worked on numerous knowledge projects and helped establish knowledge management programmes at large organisations in the engineering, technology and legal sectors. He is author of the book "Knowledge Acquisition in Practice", which describes a step-by-step procedure for acquiring and implementing expertise. He maintains strong links with leading research organisations working on knowledge technologies, such as knowledge-based engineering, ontologies and semantic technologies.

Nick Milton

0802.3789

What Can We Learn Privately?

cs.LG cs.CC cs.CR cs.DB

Learning problems form an important category of computational tasks that generalizes many of the computations researchers apply to large real-life data sets. We ask: what concept classes can be learned privately, namely, by an algorithm whose output does not depend too heavily on any one input or specific training example? More precisely, we investigate learning algorithms that satisfy differential privacy, a notion that provides strong confidentiality guarantees in contexts where aggregate information is released about a database containing sensitive information about individuals. We demonstrate that, ignoring computational constraints, it is possible to privately agnostically learn any concept class using a sample size approximately logarithmic in the cardinality of the concept class. Therefore, almost anything learnable is learnable privately: specifically, if a concept class is learnable by a (non-private) algorithm with polynomial sample complexity and output size, then it can be learned privately using a polynomial number of samples. We also present a computationally efficient private PAC learner for the class of parity functions. Local (or randomized response) algorithms are a practical class of private algorithms that have received extensive investigation. We provide a precise characterization of local private learning algorithms. We show that a concept class is learnable by a local algorithm if and only if it is learnable in the statistical query (SQ) model. Finally, we present a separation between the power of interactive and noninteractive local learning algorithms.

Shiva Prasad Kasiviswanathan, Homin K. Lee, Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith

0803.0924

Privacy Preserving ID3 over Horizontally, Vertically and Grid Partitioned Data

cs.DB cs.LG

We consider privacy preserving decision tree induction via ID3 in the case where the training data is horizontally or vertically distributed. Furthermore, we consider the same problem in the case where the data is both horizontally and vertically distributed, a situation we refer to as grid partitioned data. We give an algorithm for privacy preserving ID3 over horizontally partitioned data involving more than two parties. For grid partitioned data, we discuss two different evaluation methods for preserving privacy ID3, namely, first merging horizontally and developing vertically or first merging vertically and next developing horizontally. Next to introducing privacy preserving data mining over grid-partitioned data, the main contribution of this paper is that we show, by means of a complexity analysis that the former evaluation method is the more efficient.

Bart Kuijpers, Vanessa Lemmens, Bart Moelans and Karl Tuyls

0803.1555

Figuring out Actors in Text Streams: Using Collocations to establish Incremental Mind-maps

cs.CL cs.LG

The recognition, involvement, and description of main actors influences the story line of the whole text. This is of higher importance as the text per se represents a flow of words and expressions that once it is read it is lost. In this respect, the understanding of a text and moreover on how the actor exactly behaves is not only a major concern: as human beings try to store a given input on short-term memory while associating diverse aspects and actors with incidents, the following approach represents a virtual architecture, where collocations are concerned and taken as the associative completion of the actors' acting. Once that collocations are discovered, they become managed in separated memory blocks broken down by the actors. As for human beings, the memory blocks refer to associative mind-maps. We then present several priority functions to represent the actual temporal situation inside a mind-map to enable the user to reconstruct the recent events from the discovered temporal results.

T. Rothenberger, S. Oez, E. Tahirovic, C. Schommer

0803.2856

Robustness and Regularization of Support Vector Machines

cs.LG cs.AI

We consider regularized support vector machines (SVMs) and show that they are precisely equivalent to a new robust optimization formulation. We show that this equivalence of robust optimization and regularization has implications for both algorithms, and analysis. In terms of algorithms, the equivalence suggests more general SVM-like algorithms for classification that explicitly build in protection to noise, and at the same time control overfitting. On the analysis front, the equivalence of robustness and regularization, provides a robust optimization interpretation for the success of regularized SVMs. We use the this new robustness interpretation of SVMs to give a new proof of consistency of (kernelized) SVMs, thus establishing robustness as the reason regularized SVMs generalize well.

Huan Xu, Constantine Caramanis and Shie Mannor

0803.3490

Recorded Step Directional Mutation for Faster Convergence

cs.NE cs.LG

Two meta-evolutionary optimization strategies described in this paper accelerate the convergence of evolutionary programming algorithms while still retaining much of their ability to deal with multi-modal problems. The strategies, called directional mutation and recorded step in this paper, can operate independently but together they greatly enhance the ability of evolutionary programming algorithms to deal with fitness landscapes characterized by long narrow valleys. The directional mutation aspect of this combined method uses correlated meta-mutation but does not introduce a full covariance matrix. These new methods are thus much more economical in terms of storage for problems with high dimensionality. Additionally, directional mutation is rotationally invariant which is a substantial advantage over self-adaptive methods which use a single variance per coordinate for problems where the natural orientation of the problem is not oriented along the axes.

Ted Dunning

0803.3838

Support Vector Machine Classification with Indefinite Kernels

cs.LG cs.AI

We propose a method for support vector machine classification using indefinite kernels. Instead of directly minimizing or stabilizing a nonconvex loss function, our algorithm simultaneously computes support vectors and a proxy kernel matrix used in forming the loss. This can be interpreted as a penalized kernel learning problem where indefinite kernel matrices are treated as a noisy observations of a true Mercer kernel. Our formulation keeps the problem convex and relatively large problems can be solved efficiently using the projected gradient or analytic center cutting plane methods. We compare the performance of our technique with other methods on several classic data sets.

Ronny Luss, Alexandre d'Aspremont

0804.0188

A Unified Semi-Supervised Dimensionality Reduction Framework for Manifold Learning

cs.LG cs.AI

We present a general framework of semi-supervised dimensionality reduction for manifold learning which naturally generalizes existing supervised and unsupervised learning frameworks which apply the spectral decomposition. Algorithms derived under our framework are able to employ both labeled and unlabeled examples and are able to handle complex problems where data form separate clusters of manifolds. Our framework offers simple views, explains relationships among existing frameworks and provides further extensions which can improve existing algorithms. Furthermore, a new semi-supervised kernelization framework called ``KPCA trick'' is proposed to handle non-linear problems.

Ratthachat Chatpatanasiri and Boonserm Kijsirikul

0804.0924

Bolasso: model consistent Lasso estimation through the bootstrap

cs.LG math.ST stat.ML stat.TH

We consider the least-square linear regression problem with regularization by the l1-norm, a problem usually referred to as the Lasso. In this paper, we present a detailed asymptotic analysis of model consistency of the Lasso. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection (i.e., variable selection). For a specific rate decay, we show that the Lasso selects all the variables that should enter the model with probability tending to one exponentially fast, while it selects all other variables with strictly positive probability. We show that this property implies that if we run the Lasso for several bootstrapped replications of a given sample, then intersecting the supports of the Lasso bootstrap estimates leads to consistent model selection. This novel variable selection algorithm, referred to as the Bolasso, is compared favorably to other linear regression methods on synthetic data and datasets from the UCI machine learning repository.

Francis Bach (INRIA Rocquencourt)

0804.1302

On Kernelization of Supervised Mahalanobis Distance Learners

cs.LG cs.AI

This paper focuses on the problem of kernelizing an existing supervised Mahalanobis distance learner. The following features are included in the paper. Firstly, three popular learners, namely, "neighborhood component analysis", "large margin nearest neighbors" and "discriminant neighborhood embedding", which do not have kernel versions are kernelized in order to improve their classification performances. Secondly, an alternative kernelization framework called "KPCA trick" is presented. Implementing a learner in the new framework gains several advantages over the standard framework, e.g. no mathematical formulas and no reprogramming are required for a kernel implementation, the framework avoids troublesome problems such as singularity, etc. Thirdly, while the truths of representer theorems are just assumptions in previous papers related to ours, here, representer theorems are formally proven. The proofs validate both the kernel trick and the KPCA trick in the context of Mahalanobis distance learning. Fourthly, unlike previous works which always apply brute force methods to select a kernel, we investigate two approaches which can be efficiently adopted to construct an appropriate kernel for a given dataset. Finally, numerical results on various real-world datasets are presented.

Ratthachat Chatpatanasiri, Teesid Korsrilabutr, Pasakorn Tangchanachaianan and Boonserm Kijsirikul

0804.1441

Isotropic PCA and Affine-Invariant Clustering

cs.LG cs.CG

We present a new algorithm for clustering points in R^n. The key property of the algorithm is that it is affine-invariant, i.e., it produces the same partition for any affine transformation of the input. It has strong guarantees when the input is drawn from a mixture model. For a mixture of two arbitrary Gaussians, the algorithm correctly classifies the sample assuming only that the two components are separable by a hyperplane, i.e., there exists a halfspace that contains most of one Gaussian and almost none of the other in probability mass. This is nearly the best possible, improving known results substantially. For k > 2 components, the algorithm requires only that there be some (k-1)-dimensional subspace in which the emoverlap in every direction is small. Here we define overlap to be the ratio of the following two quantities: 1) the average squared distance between a point and the mean of its component, and 2) the average squared distance between a point and the mean of the mixture. The main result may also be stated in the language of linear discriminant analysis: if the standard Fisher discriminant is small enough, labels are not needed to estimate the optimal subspace for projection. Our main tools are isotropic transformation, spectral projection and a simple reweighting technique. We call this combination isotropic PCA.

S. Charles Brubaker and Santosh S. Vempala

0804.3575

Multiple Random Oracles Are Better Than One

cs.LG

We study the problem of learning k-juntas given access to examples drawn from a number of different product distributions. Thus we wish to learn a function f : {-1,1}^n -> {-1,1} that depends on k (unknown) coordinates. While the best known algorithms for the general problem of learning a k-junta require running time of n^k * poly(n,2^k), we show that given access to k different product distributions with biases separated by \gamma>0, the functions may be learned in time poly(n,2^k,\gamma^{-k}). More generally, given access to t <= k different product distributions, the functions may be learned in time n^{k/t} * poly(n,2^k,\gamma^{-k}). Our techniques involve novel results in Fourier analysis relating Fourier expansions with respect to different biases and a generalization of Russo's formula.

Jan Arpe and Elchanan Mossel

0804.3817

Dependence Structure Estimation via Copula

cs.LG cs.IR stat.ME

Dependence strucuture estimation is one of the important problems in machine learning domain and has many applications in different scientific areas. In this paper, a theoretical framework for such estimation based on copula and copula entropy -- the probabilistic theory of representation and measurement of statistical dependence, is proposed. Graphical models are considered as a special case of the copula framework. A method of the framework for estimating maximum spanning copula is proposed. Due to copula, the method is irrelevant to the properties of individual variables, insensitive to outlier and able to deal with non-Gaussianity. Experiments on both simulated data and real dataset demonstrated the effectiveness of the proposed method.

Jian Ma and Zengqi Sun

0804.4451