We revisit the problem of denoising from noisy measurements where only the noise level is known, not the noise distribution. In multi-dimensions, independent noise $Z$ corrupts the signal $X$, resulting in the noisy measurement $Y = X + σZ$, where $σ\in (0, 1)$ is a known noise level. Our goal is to recover the underlying signal distribution $P_X$ from denoising $P_Y$. We propose and analyze universal denoisers that are agnostic to a wide range of signal and noise distributions. Our distributional denoisers offer order-of-magnitude improvements over the Bayes-optimal denoiser derived from Tweedie's formula, if the focus is on the entire distribution $P_X$ rather than on individual realizations of $X$. Our denoisers shrink $P_Y$ toward $P_X$ optimally, achieving $O(σ^4)$ and $O(σ^6)$ accuracy in matching generalized moments and density functions. Inspired by optimal transport theory, the proposed denoisers are optimal in approximating the Monge-Ampère equation with higher-order accuracy, and can be implemented efficiently via score matching. Let $q$ represent the density of $P_Y$; for optimal distributional denoising, we recommend replacing the Bayes-optimal denoiser, \[ \mathbf{T}^*(y) = y + σ^2 \nabla \log q(y), \] with denoisers exhibiting less aggressive distributional shrinkage, \[ \mathbf{T}_1(y) = y + \frac{σ^2}{2} \nabla \log q(y), \] \[ \mathbf{T}_2(y) = y + \frac{σ^2}{2} \nabla \log q(y) - \frac{σ^4}{8} \nabla \left( \frac{1}{2} \| \nabla \log q(y) \|^2 + \nabla \cdot \nabla \log q(y) \right) . \]

Abstract--Monotone gradient functions play a central role in solving the Monge formulation of the optimal transport (OT) problem, which arises in modern applications ranging from fluid dynamics to robot swarm control. When the transport cost is the squared Euclidean distance, Brenier's theorem guarantees that the unique optimal transport map satisfies a Monge-Amp ` ere equation and is the gradient of a convex function. In [1], [2], we proposed Monotone Gradient Networks (mGradNets), neural networks that directly parameterize the space of monotone gradient maps. In this work, we leverage mGradNets to directly learn the optimal transport mapping by minimizing a training loss function defined using the Monge-Amp ` ere equation. We empirically show that the structural bias of mGradNets facilitates the learning of optimal transport maps across both image morphing tasks and high-dimensional OT problems.

This paper introduces a novel neural network-based approach to solving the Monge-Amp\`ere equation with the transport boundary condition, specifically targeted towards optical design applications. We leverage multilayer perceptron networks to learn approximate solutions by minimizing a loss function that encompasses the equation's residual, boundary conditions, and convexity constraints. Our main results demonstrate the efficacy of this method, optimized using L-BFGS, through a series of test cases encompassing symmetric and asymmetric circle-to-circle, square-to-circle, and circle-to-flower reflector mapping problems. Comparative analysis with a conventional least-squares finite-difference solver reveals the competitive, and often superior, performance of our neural network approach on the test cases examined here. A comprehensive hyperparameter study further illuminates the impact of factors such as sampling density, network architecture, and optimization algorithm. While promising, further investigation is needed to verify the method's robustness for more complicated problems and to ensure consistent convergence. Nonetheless, the simplicity and adaptability of this neural network-based approach position it as a compelling alternative to specialized partial differential equation solvers.

In recent years, the knowledge surrounding diffusion models(DMs) has grown significantly, though several theoretical gaps remain. Particularly noteworthy is prior error, defined as the discrepancy between the termination distribution of the forward process and the initial distribution of the reverse process. To address these deficiencies, this paper explores the deeper relationship between optimal transport(OT) theory and DMs with discrete initial distribution. Specifically, we demonstrate that the two stages of DMs fundamentally involve computing time-dependent OT. However, unavoidable prior error result in deviation during the reverse process under quadratic transport cost. By proving that as the diffusion termination time increases, the probability flow exponentially converges to the gradient of the solution to the classical Monge-Amp\`ere equation, we establish a vital link between these fields. Therefore, static OT emerges as the most intrinsic single-step method for bridging this theoretical potential gap. Additionally, we apply these insights to accelerate sampling in both unconditional and conditional generation scenarios. Experimental results across multiple image datasets validate the effectiveness of our approach.