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,
full_jacobian=False,
):
assert not full_jacobian
try:
if torch.min(inputs) < left or torch.max(inputs) > right:
raise transforms.InputOutsideDomain()
except RuntimeError:
logger.error("Error in rational_quadratic_spline!")
logger.error(" Left: %s", left)
logger.error(" Right: %s", left)
logger.error(" Input shape: %s", inputs.size())
logger.error(" Input: %s", inputs)
raise
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 = various.searchsorted(cumheights, inputs)[..., None]
else:
bin_idx = various.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()
discriminant = torch.clamp(discriminant, min=0.0)
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
def quadratic_spline(
inputs,
unnormalized_widths,
unnormalized_heights,
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,
full_jacobian=False,
):
assert not full_jacobian
if not inverse and (torch.min(inputs) < left or torch.max(inputs) > right):
raise transforms.InputOutsideDomain()
elif inverse and (torch.min(inputs) < bottom or torch.max(inputs) > top):
raise transforms.InputOutsideDomain()
if inverse:
inputs = (inputs - bottom) / (top - bottom)
else:
inputs = (inputs - left) / (right - left)
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
unnorm_heights_exp = torch.exp(unnormalized_heights)
if unnorm_heights_exp.shape[-1] == num_bins - 1:
# Set boundary heights s.t. after normalization they are exactly 1.
first_widths = 0.5 * widths[..., 0]
last_widths = 0.5 * widths[..., -1]
numerator = (
0.5 * first_widths * unnorm_heights_exp[..., 0] +
0.5 * last_widths * unnorm_heights_exp[..., -1] + torch.sum(
((unnorm_heights_exp[..., :-1] + unnorm_heights_exp[..., 1:]) /
2) * widths[..., 1:-1],
dim=-1))
constant = numerator / (1 - 0.5 * first_widths - 0.5 * last_widths)
constant = constant[..., None]
unnorm_heights_exp = torch.cat(
[constant, unnorm_heights_exp, constant], dim=-1)
unnormalized_area = torch.sum(
((unnorm_heights_exp[..., :-1] + unnorm_heights_exp[..., 1:]) / 2) *
widths,
dim=-1)[..., None]
heights = unnorm_heights_exp / unnormalized_area
heights = min_bin_height + (1 - min_bin_height) * heights
bin_left_cdf = torch.cumsum(
((heights[..., :-1] + heights[..., 1:]) / 2) * widths, dim=-1)
bin_left_cdf[..., -1] = 1.0
bin_left_cdf = F.pad(bin_left_cdf, pad=(1, 0), mode="constant", value=0.0)
bin_locations = torch.cumsum(widths, dim=-1)
bin_locations[..., -1] = 1.0
bin_locations = F.pad(bin_locations,
pad=(1, 0),
mode="constant",
value=0.0)
if inverse:
bin_idx = various.searchsorted(bin_left_cdf, inputs)[..., None]
else:
bin_idx = various.searchsorted(bin_locations, inputs)[..., None]
input_bin_locations = bin_locations.gather(-1, bin_idx)[..., 0]
input_bin_widths = widths.gather(-1, bin_idx)[..., 0]
input_left_cdf = bin_left_cdf.gather(-1, bin_idx)[..., 0]
input_left_heights = heights.gather(-1, bin_idx)[..., 0]
input_right_heights = heights.gather(-1, bin_idx + 1)[..., 0]
a = 0.5 * (input_right_heights - input_left_heights) * input_bin_widths
b = input_left_heights * input_bin_widths
c = input_left_cdf
if inverse:
c_ = c - inputs
alpha = (-b + torch.sqrt(b.pow(2) - 4 * a * c_)) / (2 * a)
outputs = alpha * input_bin_widths + input_bin_locations
outputs = torch.clamp(outputs, 0, 1)
logabsdet = -torch.log(
(alpha *
(input_right_heights - input_left_heights) + input_left_heights))
else:
alpha = (inputs - input_bin_locations) / input_bin_widths
outputs = a * alpha.pow(2) + b * alpha + c
outputs = torch.clamp(outputs, 0, 1)
logabsdet = torch.log(
(alpha * (input_right_heights - input_left_heights) +
input_left_heights))
if inverse:
outputs = outputs * (right - left) + left
logabsdet = logabsdet - math.log(top - bottom) + math.log(right - left)
else:
outputs = outputs * (top - bottom) + bottom
logabsdet = logabsdet + math.log(top - bottom) - math.log(right - left)
return outputs, logabsdet
def linear_spline(inputs,
unnormalized_pdf,
inverse=False,
left=0.0,
right=1.0,
bottom=0.0,
top=1.0,
full_jacobian=False):
"""
Reference:
> Müller et al., Neural Importance Sampling, arXiv:1808.03856, 2018.
"""
assert not full_jacobian
if not inverse and (torch.min(inputs) < left or torch.max(inputs) > right):
raise transforms.InputOutsideDomain()
elif inverse and (torch.min(inputs) < bottom or torch.max(inputs) > top):
raise transforms.InputOutsideDomain()
if inverse:
inputs = (inputs - bottom) / (top - bottom)
else:
inputs = (inputs - left) / (right - left)
num_bins = unnormalized_pdf.size(-1)
pdf = F.softmax(unnormalized_pdf, dim=-1)
cdf = torch.cumsum(pdf, dim=-1)
cdf[..., -1] = 1.0
cdf = F.pad(cdf, pad=(1, 0), mode="constant", value=0.0)
if inverse:
inv_bin_idx = various.searchsorted(cdf, inputs)
bin_boundaries = torch.linspace(
0, 1, num_bins + 1).view([1] * inputs.dim() + [-1]).expand(
*inputs.shape, -1)
slopes = (cdf[..., 1:] - cdf[..., :-1]) / (bin_boundaries[..., 1:] -
bin_boundaries[..., :-1])
offsets = cdf[..., 1:] - slopes * bin_boundaries[..., 1:]
inv_bin_idx = inv_bin_idx.unsqueeze(-1)
input_slopes = slopes.gather(-1, inv_bin_idx)[..., 0]
input_offsets = offsets.gather(-1, inv_bin_idx)[..., 0]
outputs = (inputs - input_offsets) / input_slopes
outputs = torch.clamp(outputs, 0, 1)
logabsdet = -torch.log(input_slopes)
else:
bin_pos = inputs * num_bins
bin_idx = torch.floor(bin_pos).long()
bin_idx[bin_idx >= num_bins] = num_bins - 1
alpha = bin_pos - bin_idx.float()
input_pdfs = pdf.gather(-1, bin_idx[..., None])[..., 0]
outputs = cdf.gather(-1, bin_idx[..., None])[..., 0]
outputs += alpha * input_pdfs
outputs = torch.clamp(outputs, 0, 1)
bin_width = 1.0 / num_bins
logabsdet = torch.log(input_pdfs) - np.log(bin_width)
if inverse:
outputs = outputs * (right - left) + left
logabsdet = logabsdet - math.log(top - bottom) + math.log(right - left)
else:
outputs = outputs * (top - bottom) + bottom
logabsdet = logabsdet + math.log(top - bottom) - math.log(right - left)
return outputs, logabsdet
def cubic_spline(
inputs,
unnormalized_widths,
unnormalized_heights,
unnorm_derivatives_left,
unnorm_derivatives_right,
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,
eps=DEFAULT_EPS,
quadratic_threshold=DEFAULT_QUADRATIC_THRESHOLD,
full_jacobian=False,
):
"""
References:
> Blinn, J. F. (2007). How to solve a cubic equation, part 5: Back to numerics. IEEE Computer
Graphics and Applications, 27(3):78–89.
"""
assert not full_jacobian
if not inverse and (torch.min(inputs) < left or torch.max(inputs) > right):
raise transforms.InputOutsideDomain()
elif inverse and (torch.min(inputs) < bottom or torch.max(inputs) > top):
raise transforms.InputOutsideDomain()
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")
if inverse:
inputs = (inputs - bottom) / (top - bottom)
else:
inputs = (inputs - left) / (right - left)
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[..., -1] = 1
cumwidths = F.pad(cumwidths, pad=(1, 0), mode="constant", value=0.0)
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[..., -1] = 1
cumheights = F.pad(cumheights, pad=(1, 0), mode="constant", value=0.0)
slopes = heights / widths
min_something_1 = torch.min(torch.abs(slopes[..., :-1]),
torch.abs(slopes[..., 1:]))
min_something_2 = 0.5 * (widths[..., 1:] * slopes[..., :-1] +
widths[..., :-1] * slopes[..., 1:]) / (
widths[..., :-1] + widths[..., 1:])
min_something = torch.min(min_something_1, min_something_2)
derivatives_left = torch.sigmoid(unnorm_derivatives_left) * 3 * slopes[
..., 0][..., None]
derivatives_right = torch.sigmoid(unnorm_derivatives_right) * 3 * slopes[
..., -1][..., None]
derivatives = min_something * (torch.sign(slopes[..., :-1]) +
torch.sign(slopes[..., 1:]))
derivatives = torch.cat([derivatives_left, derivatives, derivatives_right],
dim=-1)
a = (derivatives[..., :-1] + derivatives[..., 1:] -
2 * slopes) / widths.pow(2)
b = (3 * slopes - 2 * derivatives[..., :-1] -
derivatives[..., 1:]) / widths
c = derivatives[..., :-1]
d = cumheights[..., :-1]
if inverse:
bin_idx = various.searchsorted(cumheights, inputs)[..., None]
else:
bin_idx = various.searchsorted(cumwidths, inputs)[..., None]
inputs_a = a.gather(-1, bin_idx)[..., 0]
inputs_b = b.gather(-1, bin_idx)[..., 0]
inputs_c = c.gather(-1, bin_idx)[..., 0]
inputs_d = d.gather(-1, bin_idx)[..., 0]
input_left_cumwidths = cumwidths.gather(-1, bin_idx)[..., 0]
input_right_cumwidths = cumwidths.gather(-1, bin_idx + 1)[..., 0]
if inverse:
# Modified coefficients for solving the cubic.
inputs_b_ = (inputs_b / inputs_a) / 3.0
inputs_c_ = (inputs_c / inputs_a) / 3.0
inputs_d_ = (inputs_d - inputs) / inputs_a
delta_1 = -inputs_b_.pow(2) + inputs_c_
delta_2 = -inputs_c_ * inputs_b_ + inputs_d_
delta_3 = inputs_b_ * inputs_d_ - inputs_c_.pow(2)
discriminant = 4.0 * delta_1 * delta_3 - delta_2.pow(2)
depressed_1 = -2.0 * inputs_b_ * delta_1 + delta_2
depressed_2 = delta_1
three_roots_mask = discriminant >= 0 # Discriminant == 0 might be a problem in practice.
one_root_mask = discriminant < 0
outputs = torch.zeros_like(inputs)
# Deal with one root cases.
p = various.cbrt((-depressed_1[one_root_mask] +
torch.sqrt(-discriminant[one_root_mask])) / 2.0)
q = various.cbrt((-depressed_1[one_root_mask] -
torch.sqrt(-discriminant[one_root_mask])) / 2.0)
outputs[one_root_mask] = (
p +
q) - inputs_b_[one_root_mask] + input_left_cumwidths[one_root_mask]
# Deal with three root cases.
theta = torch.atan2(torch.sqrt(discriminant[three_roots_mask]),
-depressed_1[three_roots_mask])
theta /= 3.0
cubic_root_1 = torch.cos(theta)
cubic_root_2 = torch.sin(theta)
root_1 = cubic_root_1
root_2 = -0.5 * cubic_root_1 - 0.5 * math.sqrt(3) * cubic_root_2
root_3 = -0.5 * cubic_root_1 + 0.5 * math.sqrt(3) * cubic_root_2
root_scale = 2 * torch.sqrt(-depressed_2[three_roots_mask])
root_shift = -inputs_b_[three_roots_mask] + input_left_cumwidths[
three_roots_mask]
root_1 = root_1 * root_scale + root_shift
root_2 = root_2 * root_scale + root_shift
root_3 = root_3 * root_scale + root_shift
root1_mask = ((input_left_cumwidths[three_roots_mask] - eps) <
root_1).float()
root1_mask *= (
root_1 < (input_right_cumwidths[three_roots_mask] + eps)).float()
root2_mask = ((input_left_cumwidths[three_roots_mask] - eps) <
root_2).float()
root2_mask *= (
root_2 < (input_right_cumwidths[three_roots_mask] + eps)).float()
root3_mask = ((input_left_cumwidths[three_roots_mask] - eps) <
root_3).float()
root3_mask *= (
root_3 < (input_right_cumwidths[three_roots_mask] + eps)).float()
roots = torch.stack([root_1, root_2, root_3], dim=-1)
masks = torch.stack([root1_mask, root2_mask, root3_mask], dim=-1)
mask_index = torch.argsort(masks, dim=-1,
descending=True)[..., 0][..., None]
outputs[three_roots_mask] = torch.gather(roots,
dim=-1,
index=mask_index).view(-1)
# Deal with a -> 0 (almost quadratic) cases.
quadratic_mask = inputs_a.abs() < quadratic_threshold
a = inputs_b[quadratic_mask]
b = inputs_c[quadratic_mask]
c = inputs_d[quadratic_mask] - inputs[quadratic_mask]
alpha = (-b + torch.sqrt(b.pow(2) - 4 * a * c)) / (2 * a)
outputs[quadratic_mask] = alpha + input_left_cumwidths[quadratic_mask]
shifted_outputs = outputs - input_left_cumwidths
logabsdet = -torch.log((3 * inputs_a * shifted_outputs.pow(2) +
2 * inputs_b * shifted_outputs + inputs_c))
else:
shifted_inputs = inputs - input_left_cumwidths
outputs = inputs_a * shifted_inputs.pow(
3) + inputs_b * shifted_inputs.pow(
2) + inputs_c * shifted_inputs + inputs_d
logabsdet = torch.log((3 * inputs_a * shifted_inputs.pow(2) +
2 * inputs_b * shifted_inputs + inputs_c))
if inverse:
outputs = outputs * (right - left) + left
logabsdet = logabsdet - math.log(top - bottom) + math.log(right - left)
else:
outputs = outputs * (top - bottom) + bottom
logabsdet = logabsdet + math.log(top - bottom) - math.log(right - left)
return outputs, logabsdet