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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
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