Lipschitz-Perturbation Brenier Latent Adapter
Implementation & benchmark of arXiv:2609.04052 — Caffarelli Estimates under Lipschitz Perturbations
Source paper: Caffarelli Estimates under Lipschitz Perturbations arXiv:2609.04052 ⓘ · analyzed Sep 4, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace an unconstrained latent-to-data map in a small generative model with a learned Brenier map T equal to the gradient of a convex potential, transporting a Gaussian latent distribution toward a density proportional to exp(-B) times the Gaussian density. Constrain B to be globally L-Lipschitz and use the resulting dimension-free Jacobian ceiling as an adaptive stability target. This should reduce pathological local expansion without requiring B itself to be convex or its Hessian to be bounded.
Formulas
Mathematical statement
Let gamma_d be the standard Gaussian probability measure on R^d, with density d gamma_d(x) = (2 pi)^(-d/2) exp(-|x|^2/2) d x. For a globally L-Lipschitz function B:R^d to R, define mu_B by d mu_B(x) = Z_B^(-1) exp(-B(x)) d gamma_d(x), where Z_B = integral of exp(-B) against gamma_d and |B(x)-B(y)| <= L |x-y|. The quadratic-cost Brenier map T = grad phi satisfies T#gamma_d = mu_B and has a globally Lipschitz representative with Lip(T) <= C(0,L), where log C(0,L) = 4 L^2 + log L + O(1) as L tends to infinity. Since phi belongs to C^(1,1), its Hessian exists almost everywhere and obeys 0 <= D^2 phi(x) <= C(0,L) Id in Loewner order. The transferable asset is that first-order control of B implies global differential control of the transport map, without assumptions that D^2 B is positive or small.
Implementation notes
(1) Integration point: use this as the final latent adapter in a VAE or as an initialization map for a diffusion model. Sample z from N(0,I_d), compute x=T_theta(z)=grad_z phi_theta(z), and pass x to the decoder or diffusion network. Parameterize phi_theta as an input-convex neural network: use nonnegative hidden-to-hidden weights and convex activations, then obtain T_theta by automatic differentiation. Parameterize B_psi with a spectrally normalized MLP so that its gradient norm is controllable. (2) Pseudocode: sample z; compute x=grad_z(phi_theta(z)); evaluate the generative loss; compute L_hat as the maximum minibatch value of norm(grad_x B_psi(x)); set log_C_hat=min(4*L_hat^2+log(max(1,L_hat)), log(C_max)); estimate the largest singular value s_hat of the Jacobian of T_theta using one or two power iterations with Jacobian-vector products; add lambda times relu(log(s_hat)-log_C_hat)^2 to the objective; update theta, psi, and decoder parameters. (3) The paper supplies the dimension-independent implication and the quadratic-in-L logarithmic scaling. Estimate L_hat, s_hat, and the multiplicative calibration factor empirically; do not treat the asymptotic expression as an exact finite-L constant. Clip log_C_hat to prevent overflow. (4) First experiment: train 2D and 16D VAEs on mixtures of Gaussians, comparing an unconstrained MLP decoder, an unconstrained convex-gradient adapter, and the proposed adapter. Measure held-out likelihood, reconstruction error, maximum and 99th-percentile Jacobian singular values, gradient explosions, and failed runs as dimension increases from 2 to 64. The expected signal is fewer unstable runs and smaller Jacobian tails at matched likelihood, with weaker degradation as dimension grows. A secondary experiment should initialize a diffusion sampler with T(z) and compare FID or sliced Wasserstein distance at equal sampling steps.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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