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A flow transforms a simple base distribution into a complex target distribution. """ def __init__(self, base_distribution, transforms): super().__init__() self.base_dist = base_distribution self.transforms = nn.ModuleList(transforms) def forward(self, z0): """ Transform samples from base distribution. Returns: (zK, log_det_jacobian) """ z = z0 log_det_sum = 0.0 for transform in self.transforms: z, log_det = transform(z) log_det_sum += log_det return z, log_det_sum def log_prob(self, zK): """ Compute log probability of samples under the flow. Requires inverting all transformations. """ z = zK log_det_sum = 0.0 # Apply inverse transforms in reverse order for transform in reversed(self.transforms): z, log_det = transform.inverse(z) log_det_sum += log_det # z is now z0, compute base density log_prob_base = self.base_dist.log_prob(z).sum(-1) # Add log determinant (with correct sign for inverse) return log_prob_base + log_det_sum def sample(self, num_samples): """Sample from the flow distribution.""" z0 = self.base_dist.sample((num_samples,)) zK, _ = self.forward(z0) return zK73:T96a,import torch import torch.nn as nn import torch.nn.functional as F class PlanarFlow(nn.Module): """ Planar flow: f(z) = z + u * h(w^T z + b) Efficient O(d) Jacobian determinant computation. """ def __init__(self, dim): super().__init__() self.dim = dim self.w = nn.Parameter(torch.randn(dim)) self.u = nn.Parameter(torch.randn(dim)) self.b = nn.Parameter(torch.zeros(1)) def _get_u_hat(self): """Ensure invertibility: 1 + u^T ψ(z) > 0""" wTu = torch.dot(self.w, self.u) m_wTu = -1 + F.softplus(wTu) u_hat = self.u + (m_wTu - wTu) * self.w / (self.w.norm() ** 2 + 1e-8) return u_hat def forward(self, z): """Apply planar transformation.""" u_hat = self._get_u_hat() # Compute w^T z + b linear = F.linear(z, self.w.unsqueeze(0), self.b) # Apply transformation: z + u * tanh(w^T z + b) h = torch.tanh(linear) z_new = z + u_hat * h # Compute log |det Jacobian| # det = 1 + u^T * h'(w^T z + b) * w = 1 + u^T * ψ(z) h_prime = 1 - h ** 2 # derivative of tanh psi = h_prime * self.w log_det = torch.log(torch.abs(1 + psi @ u_hat) + 1e-8) return z_new, log_det class RadialFlow(nn.Module): """ Radial flow: f(z) = z + β * h(α, r) * (z - z0) Contracts or expands around a center point. """ def __init__(self, dim): super().__init__() self.dim = dim self.z0 = nn.Parameter(torch.zeros(dim)) self.log_alpha = nn.Parameter(torch.zeros(1)) self.beta = nn.Parameter(torch.zeros(1)) def forward(self, z): """Apply radial transformation.""" alpha = F.softplus(self.log_alpha) # Compute distance from center diff = z - self.z0 r = diff.norm(dim=-1, keepdim=True) + 1e-8 # h(α, r) = 1 / (α + r) h = 1 / (alpha + r) h_prime = -1 / (alpha + r) ** 2 # Transform z_new = z + self.beta * h * diff # Log determinant (derived from radial symmetry) d = self.dim log_det = (d - 1) * torch.log(torch.abs(1 + self.beta * h)) + \ torch.log(torch.abs(1 + self.beta * h + self.beta * h_prime * r)) return z_new, log_det.squeeze(-1)74:Tcc6,import torch import torch.nn as nn class MADE(nn.Module): """ Masked Autoencoder for Distribution Estimation. Produces autoregressive outputs where output_i depends only on input_{ 0** for all z. This is enforced through a reparameterization of **u**."},"type":"text"},{"data":{"content":"**Radial Flows:**\n\nRadial flows apply transformations that are radially symmetric around a reference point:\n\n$$f(z) = z + \\beta h(\\alpha, r)(z - z_0), \\quad r = |z - z_0|$$\n\nwhere **z₀** is the center, **α > 0** controls the smoothness, and **β** controls the magnitude. A common choice is:\n\n$$h(\\alpha, r) = \\frac{1}{\\alpha + r}$$\n\nRadial flows can either contract or expand regions of space around **z₀**, depending on the sign of **β**.\n\n**Computational Complexity:**\n\nBoth planar and radial flows have **O(d)** complexity for the Jacobian determinant, making them efficient for high-dimensional problems. However, their expressiveness is limited—each transformation can only \"bend\" the density in one direction (planar) or around one point (radial)."},"type":"text"},{"data":{"fileName":"planar_radial_flows.py","languages":[{"code":"$73","label":"Python","value":"python"}],"showLineNumbers":true},"type":"code"},{"data":{"icon":"Info","title":"Stacking Simple Flows","content":"While individual planar or radial flows have limited expressiveness, stacking many of them (typically 10-100 layers) can represent highly complex distributions. The Universal Approximation theorem for flows shows that with enough layers, any smooth density can be approximated arbitrarily well.","variant":"note"},"type":"callout"}],"titleType":"number","titleNumber":3},{"id":"autoregressive-flows","title":"Autoregressive Flows","content":[{"data":{"content":"**Autoregressive flows** represent a major advancement in flow design, achieving both high expressiveness and efficient computation. They exploit the autoregressive structure to ensure the Jacobian is triangular, enabling **O(d)** determinant computation.\n\n**Key Insight:**\n\nAn autoregressive transformation defines each output dimension as a function of only the preceding input dimensions:\n\n$$z_i = f_i(z_0^{(1)}, z_0^{(2)}, ..., z_0^{(i)})$$\n\nThis constraint makes the Jacobian matrix **triangular**, and the determinant is simply the product of diagonal elements:\n\n$$\\det \\frac{\\partial z}{\\partial z_0} = \\prod_{i=1}^{d} \\frac{\\partial z_i}{\\partial z_0^{(i)}}$$"},"type":"text"},{"data":{"content":"**Affine Autoregressive Flows (MADE-style):**\n\nThe most common autoregressive flow uses affine transformations:\n\n$$z_i = \\mu_i(z_0^{

Amortized Inference

Amortized Inference