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from functools import partial
import jax
import jax.numpy as np
from flax import linen as nn
from jax.nn.initializers import lecun_normal, normal
from .ssm_init import init_CV, init_VinvB, init_log_steps, trunc_standard_normal
# Discretization functions
def discretize_bilinear(Lambda, B_tilde, Delta):
""" Discretize a diagonalized, continuous-time linear SSM
using bilinear transform method.
Args:
Lambda (complex64): diagonal state matrix (P,)
B_tilde (complex64): input matrix (P, H)
Delta (float32): discretization step sizes (P,)
Returns:
discretized Lambda_bar (complex64), B_bar (complex64) (P,), (P,H)
"""
Identity = np.ones(Lambda.shape[0])
BL = 1 / (Identity - (Delta / 2.0) * Lambda)
Lambda_bar = BL * (Identity + (Delta / 2.0) * Lambda)
B_bar = (BL * Delta)[..., None] * B_tilde
return Lambda_bar, B_bar
def discretize_zoh(Lambda, B_tilde, Delta):
""" Discretize a diagonalized, continuous-time linear SSM
using zero-order hold method.
Args:
Lambda (complex64): diagonal state matrix (P,)
B_tilde (complex64): input matrix (P, H)
Delta (float32): discretization step sizes (P,)
Returns:
discretized Lambda_bar (complex64), B_bar (complex64) (P,), (P,H)
"""
Identity = np.ones(Lambda.shape[0])
Lambda_bar = np.exp(Lambda * Delta)
B_bar = (1/Lambda * (Lambda_bar-Identity))[..., None] * B_tilde
return Lambda_bar, B_bar
# Parallel scan operations
@jax.vmap
def binary_operator(q_i, q_j):
""" Binary operator for parallel scan of linear recurrence. Assumes a diagonal matrix A.
Args:
q_i: tuple containing A_i and Bu_i at position i (P,), (P,)
q_j: tuple containing A_j and Bu_j at position j (P,), (P,)
Returns:
new element ( A_out, Bu_out )
"""
A_i, b_i = q_i
A_j, b_j = q_j
return A_j * A_i, A_j * b_i + b_j
def apply_ssm(Lambda_bar, B_bar, C_tilde, input_sequence, conj_sym, bidirectional):
""" Compute the LxH output of discretized SSM given an LxH input.
Args:
Lambda_bar (complex64): discretized diagonal state matrix (P,)
B_bar (complex64): discretized input matrix (P, H)
C_tilde (complex64): output matrix (H, P)
input_sequence (float32): input sequence of features (L, H)
conj_sym (bool): whether conjugate symmetry is enforced
bidirectional (bool): whether bidirectional setup is used,
Note for this case C_tilde will have 2P cols
Returns:
ys (float32): the SSM outputs (S5 layer preactivations) (L, H)
"""
Lambda_elements = Lambda_bar * np.ones((input_sequence.shape[0],
Lambda_bar.shape[0]))
Bu_elements = jax.vmap(lambda u: B_bar @ u)(input_sequence)
_, xs = jax.lax.associative_scan(binary_operator, (Lambda_elements, Bu_elements))
if bidirectional:
_, xs2 = jax.lax.associative_scan(binary_operator,
(Lambda_elements, Bu_elements),
reverse=True)
xs = np.concatenate((xs, xs2), axis=-1)
if conj_sym:
return jax.vmap(lambda x: 2*(C_tilde @ x).real)(xs)
else:
return jax.vmap(lambda x: (C_tilde @ x).real)(xs)
class S5SSM(nn.Module):
Lambda_re_init: np.DeviceArray
Lambda_im_init: np.DeviceArray
V: np.DeviceArray
Vinv: np.DeviceArray
H: int
P: int
C_init: str
discretization: str
dt_min: float
dt_max: float
conj_sym: bool = True
clip_eigs: bool = False
bidirectional: bool = False
step_rescale: float = 1.0
""" The S5 SSM
Args:
Lambda_re_init (complex64): Real part of init diag state matrix (P,)
Lambda_im_init (complex64): Imag part of init diag state matrix (P,)
V (complex64): Eigenvectors used for init (P,P)
Vinv (complex64): Inverse eigenvectors used for init (P,P)
H (int32): Number of features of input seq
P (int32): state size
C_init (string): Specifies How C is initialized
Options: [trunc_standard_normal: sample from truncated standard normal
and then multiply by V, i.e. C_tilde=CV.
lecun_normal: sample from Lecun_normal and then multiply by V.
complex_normal: directly sample a complex valued output matrix
from standard normal, does not multiply by V]
conj_sym (bool): Whether conjugate symmetry is enforced
clip_eigs (bool): Whether to enforce left-half plane condition, i.e.
constrain real part of eigenvalues to be negative.
True recommended for autoregressive task/unbounded sequence lengths
Discussed in https://arxiv.org/pdf/2206.11893.pdf.
bidirectional (bool): Whether model is bidirectional, if True, uses two C matrices
discretization: (string) Specifies discretization method
options: [zoh: zero-order hold method,
bilinear: bilinear transform]
dt_min: (float32): minimum value to draw timescale values from when
initializing log_step
dt_max: (float32): maximum value to draw timescale values from when
initializing log_step
step_rescale: (float32): allows for uniformly changing the timescale parameter, e.g. after training
on a different resolution for the speech commands benchmark
"""
def setup(self):
"""Initializes parameters once and performs discretization each time
the SSM is applied to a sequence
"""
if self.conj_sym:
# Need to account for case where we actually sample real B and C, and then multiply
# by the half sized Vinv and possibly V
local_P = 2*self.P
else:
local_P = self.P
# Initialize diagonal state to state matrix Lambda (eigenvalues)
self.Lambda_re = self.param("Lambda_re", lambda rng, shape: self.Lambda_re_init, (None,))
self.Lambda_im = self.param("Lambda_im", lambda rng, shape: self.Lambda_im_init, (None,))
if self.clip_eigs:
self.Lambda = np.clip(self.Lambda_re, None, -1e-4) + 1j * self.Lambda_im
else:
self.Lambda = self.Lambda_re + 1j * self.Lambda_im
# Initialize input to state (B) matrix
B_init = lecun_normal()
B_shape = (local_P, self.H)
self.B = self.param("B",
lambda rng, shape: init_VinvB(B_init,
rng,
shape,
self.Vinv),
B_shape)
B_tilde = self.B[..., 0] + 1j * self.B[..., 1]
# Initialize state to output (C) matrix
if self.C_init in ["trunc_standard_normal"]:
C_init = trunc_standard_normal
C_shape = (self.H, local_P, 2)
elif self.C_init in ["lecun_normal"]:
C_init = lecun_normal()
C_shape = (self.H, local_P, 2)
elif self.C_init in ["complex_normal"]:
C_init = normal(stddev=0.5 ** 0.5)
else:
raise NotImplementedError(
"C_init method {} not implemented".format(self.C_init))
if self.C_init in ["complex_normal"]:
if self.bidirectional:
C = self.param("C", C_init, (self.H, 2 * self.P, 2))
self.C_tilde = C[..., 0] + 1j * C[..., 1]
else:
C = self.param("C", C_init, (self.H, self.P, 2))
self.C_tilde = C[..., 0] + 1j * C[..., 1]
else:
if self.bidirectional:
self.C1 = self.param("C1",
lambda rng, shape: init_CV(C_init, rng, shape, self.V),
C_shape)
self.C2 = self.param("C2",
lambda rng, shape: init_CV(C_init, rng, shape, self.V),
C_shape)
C1 = self.C1[..., 0] + 1j * self.C1[..., 1]
C2 = self.C2[..., 0] + 1j * self.C2[..., 1]
self.C_tilde = np.concatenate((C1, C2), axis=-1)
else:
self.C = self.param("C",
lambda rng, shape: init_CV(C_init, rng, shape, self.V),
C_shape)
self.C_tilde = self.C[..., 0] + 1j * self.C[..., 1]
# Initialize feedthrough (D) matrix
self.D = self.param("D", normal(stddev=1.0), (self.H,))
# Initialize learnable discretization timescale value
self.log_step = self.param("log_step",
init_log_steps,
(self.P, self.dt_min, self.dt_max))
step = self.step_rescale * np.exp(self.log_step[:, 0])
# Discretize
if self.discretization in ["zoh"]:
self.Lambda_bar, self.B_bar = discretize_zoh(self.Lambda, B_tilde, step)
elif self.discretization in ["bilinear"]:
self.Lambda_bar, self.B_bar = discretize_bilinear(self.Lambda, B_tilde, step)
else:
raise NotImplementedError("Discretization method {} not implemented".format(self.discretization))
def __call__(self, input_sequence):
"""
Compute the LxH output of the S5 SSM given an LxH input sequence
using a parallel scan.
Args:
input_sequence (float32): input sequence (L, H)
Returns:
output sequence (float32): (L, H)
"""
ys = apply_ssm(self.Lambda_bar,
self.B_bar,
self.C_tilde,
input_sequence,
self.conj_sym,
self.bidirectional)
# Add feedthrough matrix output Du;
Du = jax.vmap(lambda u: self.D * u)(input_sequence)
return ys + Du
def init_S5SSM(H,
P,
Lambda_re_init,
Lambda_im_init,
V,
Vinv,
C_init,
discretization,
dt_min,
dt_max,
conj_sym,
clip_eigs,
bidirectional
):
"""Convenience function that will be used to initialize the SSM.
Same arguments as defined in S5SSM above."""
return partial(S5SSM,
H=H,
P=P,
Lambda_re_init=Lambda_re_init,
Lambda_im_init=Lambda_im_init,
V=V,
Vinv=Vinv,
C_init=C_init,
discretization=discretization,
dt_min=dt_min,
dt_max=dt_max,
conj_sym=conj_sym,
clip_eigs=clip_eigs,
bidirectional=bidirectional)
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