Zhiyang Xun

Shivam Gupta, Aditya Parulekar, Eric Price et al.

Diffusion models have become the most popular approach to deep generative modeling of images, largely due to their empirical performance and reliability. From a theoretical standpoint, a number of recent works have studied the iteration complexity of sampling, assuming access to an accurate diffusion model. In this work, we focus on understanding the sample complexity of training such a model; how many samples are needed to learn an accurate diffusion model using a sufficiently expressive neural network? Prior work showed bounds polynomial in the dimension, desired Total Variation error, and Wasserstein error. We show an exponential improvement in the dependence on Wasserstein error and depth, along with improved dependencies on other relevant parameters.

Eric Price, Zhiyang Xun

In streaming PCA, we see a stream of vectors $x_1, \dotsc, x_n \in \mathbb{R}^d$ and want to estimate the top eigenvector of their covariance matrix. This is easier if the spectral ratio $R = λ_1 / λ_2$ is large. We ask: how large does $R$ need to be to solve streaming PCA in $\widetilde{O}(d)$ space? Existing algorithms require $R = \widetildeΩ(d)$. We show: (1) For all mergeable summaries, $R = \widetildeΩ(\sqrt{d})$ is necessary. (2) In the insertion-only model, a variant of Oja's algorithm gets $o(1)$ error for $R = O(\log n \log d)$. (3) No algorithm with $o(d^2)$ space gets $o(1)$ error for $R = O(1)$. Our analysis is the first application of Oja's algorithm to adversarial streams. It is also the first algorithm for adversarial streaming PCA that is designed for a spectral, rather than Frobenius, bound on the tail; and the bound it needs is exponentially better than is possible by adapting a Frobenius guarantee.

Zhiyang Xun, Shivam Gupta, Eric Price

Given a noisy linear measurement $y = Ax + ξ$ of a distribution $p(x)$, and a good approximation to the prior $p(x)$, when can we sample from the posterior $p(x \mid y)$? Posterior sampling provides an accurate and fair framework for tasks such as inpainting, deblurring, and MRI reconstruction, and several heuristics attempt to approximate it. Unfortunately, approximate posterior sampling is computationally intractable in general. To sidestep this hardness, we focus on (local or global) log-concave distributions $p(x)$. In this regime, Langevin dynamics yields posterior samples when the exact scores of $p(x)$ are available, but it is brittle to score--estimation error, requiring an MGF bound (sub-exponential error). By contrast, in the unconditional setting, diffusion models succeed with only an $L^2$ bound on the score error. We prove that combining diffusion models with an annealed variant of Langevin dynamics achieves conditional sampling in polynomial time using merely an $L^4$ bound on the score error.

Zhiyang Xun, Eric Price

Diffusion models generate samples by iteratively querying learned score estimates. A rapidly growing literature focuses on accelerating sampling by minimizing the number of score evaluations, yet the information-theoretic limits of such acceleration remain unclear. In this work, we establish the first score query lower bounds for diffusion sampling. We prove that for $d$-dimensional distributions, given access to score estimates with polynomial accuracy $\varepsilon=d^{-O(1)}$ (in any $L^p$ sense), any sampling algorithm requires $\widetildeΩ(\sqrt{d})$ adaptive score queries. In particular, our proof shows that any sampler must search over $\widetildeΩ(\sqrt{d})$ distinct noise levels, providing a formal explanation for why multiscale noise schedules are necessary in practice.

Shivam Gupta, Ajil Jalal, Aditya Parulekar et al.

Diffusion models are a remarkably effective way of learning and sampling from a distribution $p(x)$. In posterior sampling, one is also given a measurement model $p(y \mid x)$ and a measurement $y$, and would like to sample from $p(x \mid y)$. Posterior sampling is useful for tasks such as inpainting, super-resolution, and MRI reconstruction, so a number of recent works have given algorithms to heuristically approximate it; but none are known to converge to the correct distribution in polynomial time. In this paper we show that posterior sampling is computationally intractable: under the most basic assumption in cryptography -- that one-way functions exist -- there are instances for which every algorithm takes superpolynomial time, even though unconditional sampling is provably fast. We also show that the exponential-time rejection sampling algorithm is essentially optimal under the stronger plausible assumption that there are one-way functions that take exponential time to invert.