No abstract

A regularized optimization problem over a large unstructured graph is studied, where the regularization term is tied to the graph geometry. Typical regularization examples include the total variation and the Laplacian regularizations over the graph. When applying the proximal gradient algorithm to solve this problem, there exist quite affordable methods to implement the proximity operator (backward step) in the special case where the graph is a simple path without loops. In this paper, an algorithm, referred to as "Snake", is proposed to solve such regularized problems over general graphs, by taking benefit of these fast methods. The algorithm consists in properly selecting random simple paths in the graph and performing the proximal gradient algorithm over these simple paths. This algorithm is an instance of a new general stochastic proximal gradient algorithm, whose convergence is proven. Applications to trend filtering and graph inpainting are provided among others. Numerical experiments are conducted over large graphs.

We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale realworld generative models such as DALL•E 2. Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution. In contrast to prior works, our results (1) hold for an L 2 -accurate score estimate (rather than L ∞ -accurate); (2) do not require restrictive functional inequality conditions that preclude substantial non-log-concavity; (3) scale polynomially in all relevant problem parameters; and (4) match state-of-the-art complexity guarantees for discretization of the Langevin diffusion, provided that the score error is sufficiently small. We view this as strong theoretical justification for the empirical success of SGMs. We also examine SGMs based on the critically damped Langevin diffusion (CLD). Contrary to conventional wisdom, we provide evidence that the use of the CLD does not reduce the complexity of SGMs.