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transform.py

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+# adopted from https://github.com/bayesiains/nflows
+
+import numpy as np
+import torch
+from torch.nn import functional as F
+
+DEFAULT_MIN_BIN_WIDTH = 1e-3
+DEFAULT_MIN_BIN_HEIGHT = 1e-3
+DEFAULT_MIN_DERIVATIVE = 1e-3
+
+
+def piecewise_rational_quadratic_transform(
+    inputs,
+    unnormalized_widths,
+    unnormalized_heights,
+    unnormalized_derivatives,
+    inverse=False,
+    tails=None,
+    tail_bound=1.0,
+    min_bin_width=DEFAULT_MIN_BIN_WIDTH,
+    min_bin_height=DEFAULT_MIN_BIN_HEIGHT,
+    min_derivative=DEFAULT_MIN_DERIVATIVE,
+):
+    if tails is None:
+        spline_fn = rational_quadratic_spline
+        spline_kwargs = {}
+    else:
+        spline_fn = unconstrained_rational_quadratic_spline
+        spline_kwargs = {"tails": tails, "tail_bound": tail_bound}
+
+    outputs, logabsdet = spline_fn(
+        inputs=inputs,
+        unnormalized_widths=unnormalized_widths,
+        unnormalized_heights=unnormalized_heights,
+        unnormalized_derivatives=unnormalized_derivatives,
+        inverse=inverse,
+        min_bin_width=min_bin_width,
+        min_bin_height=min_bin_height,
+        min_derivative=min_derivative,
+        **spline_kwargs,
+    )
+    return outputs, logabsdet
+
+
+def searchsorted(bin_locations, inputs, eps=1e-6):
+    bin_locations[..., -1] += eps
+    return torch.sum(inputs[..., None] >= bin_locations, dim=-1) - 1
+
+
+def unconstrained_rational_quadratic_spline(
+    inputs,
+    unnormalized_widths,
+    unnormalized_heights,
+    unnormalized_derivatives,
+    inverse=False,
+    tails="linear",
+    tail_bound=1.0,
+    min_bin_width=DEFAULT_MIN_BIN_WIDTH,
+    min_bin_height=DEFAULT_MIN_BIN_HEIGHT,
+    min_derivative=DEFAULT_MIN_DERIVATIVE,
+):
+    inside_interval_mask = (inputs >= -tail_bound) & (inputs <= tail_bound)
+    outside_interval_mask = ~inside_interval_mask
+
+    outputs = torch.zeros_like(inputs)
+    logabsdet = torch.zeros_like(inputs)
+
+    if tails == "linear":
+        unnormalized_derivatives = F.pad(unnormalized_derivatives, pad=(1, 1))
+        constant = np.log(np.exp(1 - min_derivative) - 1)
+        unnormalized_derivatives[..., 0] = constant
+        unnormalized_derivatives[..., -1] = constant
+
+        outputs[outside_interval_mask] = inputs[outside_interval_mask]
+        logabsdet[outside_interval_mask] = 0
+    else:
+        raise RuntimeError("{} tails are not implemented.".format(tails))
+
+    outputs[inside_interval_mask], logabsdet[inside_interval_mask] = rational_quadratic_spline(
+        inputs=inputs[inside_interval_mask],
+        unnormalized_widths=unnormalized_widths[inside_interval_mask, :],
+        unnormalized_heights=unnormalized_heights[inside_interval_mask, :],
+        unnormalized_derivatives=unnormalized_derivatives[inside_interval_mask, :],
+        inverse=inverse,
+        left=-tail_bound,
+        right=tail_bound,
+        bottom=-tail_bound,
+        top=tail_bound,
+        min_bin_width=min_bin_width,
+        min_bin_height=min_bin_height,
+        min_derivative=min_derivative,
+    )
+
+    return outputs, logabsdet
+
+
+def rational_quadratic_spline(
+    inputs,
+    unnormalized_widths,
+    unnormalized_heights,
+    unnormalized_derivatives,
+    inverse=False,
+    left=0.0,
+    right=1.0,
+    bottom=0.0,
+    top=1.0,
+    min_bin_width=DEFAULT_MIN_BIN_WIDTH,
+    min_bin_height=DEFAULT_MIN_BIN_HEIGHT,
+    min_derivative=DEFAULT_MIN_DERIVATIVE,
+):
+    if torch.min(inputs) < left or torch.max(inputs) > right:
+        raise ValueError("Input to a transform is not within its domain")
+
+    num_bins = unnormalized_widths.shape[-1]
+
+    if min_bin_width * num_bins > 1.0:
+        raise ValueError("Minimal bin width too large for the number of bins")
+    if min_bin_height * num_bins > 1.0:
+        raise ValueError("Minimal bin height too large for the number of bins")
+
+    widths = F.softmax(unnormalized_widths, dim=-1)
+    widths = min_bin_width + (1 - min_bin_width * num_bins) * widths
+    cumwidths = torch.cumsum(widths, dim=-1)
+    cumwidths = F.pad(cumwidths, pad=(1, 0), mode="constant", value=0.0)
+    cumwidths = (right - left) * cumwidths + left
+    cumwidths[..., 0] = left
+    cumwidths[..., -1] = right
+    widths = cumwidths[..., 1:] - cumwidths[..., :-1]
+
+    derivatives = min_derivative + F.softplus(unnormalized_derivatives)
+
+    heights = F.softmax(unnormalized_heights, dim=-1)
+    heights = min_bin_height + (1 - min_bin_height * num_bins) * heights
+    cumheights = torch.cumsum(heights, dim=-1)
+    cumheights = F.pad(cumheights, pad=(1, 0), mode="constant", value=0.0)
+    cumheights = (top - bottom) * cumheights + bottom
+    cumheights[..., 0] = bottom
+    cumheights[..., -1] = top
+    heights = cumheights[..., 1:] - cumheights[..., :-1]
+
+    if inverse:
+        bin_idx = searchsorted(cumheights, inputs)[..., None]
+    else:
+        bin_idx = searchsorted(cumwidths, inputs)[..., None]
+
+    input_cumwidths = cumwidths.gather(-1, bin_idx)[..., 0]
+    input_bin_widths = widths.gather(-1, bin_idx)[..., 0]
+
+    input_cumheights = cumheights.gather(-1, bin_idx)[..., 0]
+    delta = heights / widths
+    input_delta = delta.gather(-1, bin_idx)[..., 0]
+
+    input_derivatives = derivatives.gather(-1, bin_idx)[..., 0]
+    input_derivatives_plus_one = derivatives[..., 1:].gather(-1, bin_idx)[..., 0]
+
+    input_heights = heights.gather(-1, bin_idx)[..., 0]
+
+    if inverse:
+        a = (inputs - input_cumheights) * (
+            input_derivatives + input_derivatives_plus_one - 2 * input_delta
+        ) + input_heights * (input_delta - input_derivatives)
+        b = input_heights * input_derivatives - (inputs - input_cumheights) * (
+            input_derivatives + input_derivatives_plus_one - 2 * input_delta
+        )
+        c = -input_delta * (inputs - input_cumheights)
+
+        discriminant = b.pow(2) - 4 * a * c
+        assert (discriminant >= 0).all()
+
+        root = (2 * c) / (-b - torch.sqrt(discriminant))
+        outputs = root * input_bin_widths + input_cumwidths
+
+        theta_one_minus_theta = root * (1 - root)
+        denominator = input_delta + (
+            (input_derivatives + input_derivatives_plus_one - 2 * input_delta) * theta_one_minus_theta
+        )
+        derivative_numerator = input_delta.pow(2) * (
+            input_derivatives_plus_one * root.pow(2)
+            + 2 * input_delta * theta_one_minus_theta
+            + input_derivatives * (1 - root).pow(2)
+        )
+        logabsdet = torch.log(derivative_numerator) - 2 * torch.log(denominator)
+
+        return outputs, -logabsdet
+    else:
+        theta = (inputs - input_cumwidths) / input_bin_widths
+        theta_one_minus_theta = theta * (1 - theta)
+
+        numerator = input_heights * (input_delta * theta.pow(2) + input_derivatives * theta_one_minus_theta)
+        denominator = input_delta + (
+            (input_derivatives + input_derivatives_plus_one - 2 * input_delta) * theta_one_minus_theta
+        )
+        outputs = input_cumheights + numerator / denominator
+
+        derivative_numerator = input_delta.pow(2) * (
+            input_derivatives_plus_one * theta.pow(2)
+            + 2 * input_delta * theta_one_minus_theta
+            + input_derivatives * (1 - theta).pow(2)
+        )
+        logabsdet = torch.log(derivative_numerator) - 2 * torch.log(denominator)
+
+        return outputs, logabsdet