Related papers: Gradient flow on extensive-rank positive semi-defi…
This paper establishes risk convergence and asymptotic weight matrix alignment --- a form of implicit regularization --- of gradient flow and gradient descent when applied to deep linear networks on linearly separable data. In more detail,…
We study a class of deterministic flows in ${\mathbb R}^{d\times k}$, parametrized by a random matrix ${\boldsymbol X}\in {\mathbb R}^{n\times d}$ with i.i.d. centered subgaussian entries. We characterize the asymptotic behavior of these…
A finite element method for solving nonlinear differential equations on a grid, with potential applicability to computational fluid dynamics (CFD), is developed and tested. The current method facilitates the computation of solutions of a…
We consider a nonlinear fourth-order diffusion equation that arises in denoising of image densities. We propose an implicit time-stepping scheme that employs a primal-dual method for computing the subgradient of the total variation…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
We describe novel subgradient methods for a broad class of matrix optimization problems involving nuclear norm regularization. Unlike existing approaches, our method executes very cheap iterations by combining low-rank stochastic…
Plug-and-Play methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser. Although Plug-and-Play methods can lead to tremendous visual performance for various image…
Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often…
Generative models based on flow matching have emerged as a powerful paradigm for inverse problems, offering straighter trajectories and faster sampling compared to diffusion models. However, existing approaches often necessitate…
We develop a novel framework to study smooth and strongly convex optimization algorithms, both deterministic and stochastic. Focusing on quadratic functions we are able to examine optimization algorithms as a recursive application of linear…
We study the convergence of gradient flows related to learning deep linear neural networks (where the activation function is the identity map) from data. In this case, the composition of the network layers amounts to simply multiplying the…
In this paper we estimate the worst rate of exponential decay of degenerate gradient flows $\dot x = -S x$, issued from adaptive control theory. Under persistent excitation assumptions on the positive semi-definite matrix $S$, we provide…
We study the effect of fluid flow on three-dimensional (3D) dendrite growth using a phase-field model on an adaptive finite element grid. In order to simulate 3D fluid flow, we use an averaging method for the flow problem coupled to the…
Discrete flow-based models are a recently proposed class of generative models that learn invertible transformations for discrete random variables. Since they do not require data dequantization and maximize an exact likelihood objective,…
We introduce new multilevel methods for solving large-scale unconstrained optimization problems. Specifically, the philosophy of multilevel methods is applied to Newton-type methods that regularize the Newton sub-problem using second order…
A finite element method for the evolution of a two-phase membrane in a sharp interface formulation is introduced. The evolution equations are given as an $L^2$--gradient flow of an energy involving an elastic bending energy and a line…
We address the problem of simultaneously recovering a sequence of point source signals from observations limited to the low-frequency end of the spectrum of their summed convolution, where the point spread functions (PSFs) are unknown. By…
While the optimization landscape of policy gradient methods has been recently investigated for partially observed linear systems in terms of both static output feedback and dynamical controllers, they only provide convergence guarantees to…
This article is a follow up of our submitted paper [11] in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to…
Finding latent structures in data is drawing increasing attention in diverse fields such as image and signal processing, fluid dynamics, and machine learning. In this work we examine the problem of finding the main modes of gradient flows.…