Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run

For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $Ω(1/(n^2 C_{\mathsf{PI}}))$, where $C_{\mathsf{PI}}$ is the Poincaré constant of the uniform distribution $π$ over $\mathcal{K}$. This implies that Hit-and-Run converges to a distribution within $χ^2$-divergence $\varepsilon$ of the uniform distribution $π$ in $O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))$ steps from any starting distribution $π_0$ with $M=χ^2(π_{0}\,\|\,π)$, thus refining the known bound of $O(n^2 R^2 \log(M/\varepsilon))$ by Lovász and Vempala (2004) in terms of the outer radius $R$; for nearly isotropic bodies, together with progress on the KLS conjecture, the complexity is $O(n^2\log n\log(M/\varepsilon))$, improving the dimension dependence from cubic to nearly quadratic while maintaining logarithmic dependence on the initial distance. It was an open problem to connect the convergence of Hit-and-Run to Poincaré/KLS constants as was done for the Ball walk by Kannan, Lovász and Simonovits (1997). Unlike Hit-and-Run, the Ball walk has an unavoidable linear dependence on (a stronger notion) of the initial warmness. We directly bound the spectral gap of the Hit-and-Run Markov chain by connecting it to functional isoperimetric constants, inspired by the recent analysis of In-and-Out. Rewriting the spectral gap in terms of dual certificates leads to the Babuška--Aziz constant studied in the analysis of PDEs; it is asymptotically bounded by the improved Poincaré constant, which we show can be bounded in terms of the usual Poincaré constant. The proof is based on duality and calculus, unlike known proofs of convergence for Hit-and-Run which are based on bounding the conductance. The same technique can be applied to Coordinate Hit-and-Run, resulting in a much improved mixing time of $O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))$.

Beyond the $d^{2.5}$-mixing bound for Dikin walks on polytopes

Inspired by interior-point methods (IPM) for structured convex optimization, Kannan and Narayanan introduced the Dikin walk for sampling uniformly from polytopes in 2009. As in IPMs, the Dikin walk is affine-invariant, and its convergence is governed by the barrier geometry used to define its local proposal. They showed that the Dikin walk with the logarithmic barrier for a polytope in $\mathbb{R}^{d}$ with $m$ linear inequalities mixes in $md$ iterations. In 2017, Chen, Dwivedi, Wainwright, and Yu improved this to $d^{2.5}$ using a Lewis-weight barrier, and conjectured that the correct mixing time should be $d^{2}$. We make progress toward this conjecture by improving the previous $d^{2.5}$-mixing bound. For exponential sampling over a polytope, we prove that the Dikin walk with a scaled Lee--Sidford metric mixes from a warm start in $d^{2.25}$ iterations. This also yields an improved cold-start complexity via a known annealing framework. The main technical ingredient is improved average self-concordance of the Lee--Sidford metric, which gives high acceptance probability for the Metropolis filter along a random Dikin proposal. While previous analyses were effectively limited to second-order control due to technical difficulties, we develop a principled higher-order analysis. The proof combines a selective higher-order expansion of recursive bottleneck terms, a moving orthonormal-frame calculus for higher derivatives of the Lewis weights, and Wiener-chaos decompositions via multiple stochastic integrals to control the resulting Gaussian polynomials.

A unified complexity bound for logconcave sampling

We give a simple, unified, and nearly tight bound for sampling arbitrary logconcave distributions from a warm start using the In-and-Out algorithm along with exponential lifting. The main new ingredient in the analysis is an improved bound on the Poincaré constant of a lifted distribution. As a consequence, the resulting convergence rate is nearly tight for both constrained settings (e.g., Gaussian restricted to a convex body) and well-conditioned settings (e.g., strongly logconcave and smooth densities).

Zeroth-order log-concave sampling

We study the zeroth-order query complexity of log-concave sampling, specifically uniform sampling from convex bodies using membership oracles. We propose a simple variant of the proximal sampler that achieves the query complexity with matched R\'enyi orders between the initial warmness and output guarantee. Specifically, for any $\varepsilon>0$ and $q\geq2$, the sampler, initialized at $\pi_{0}$, outputs a sample whose law is $\varepsilon$-close in $q$-R\'enyi divergence to $\pi$, the uniform distribution over a convex body in $\mathbb{R}^{d}$, using $\widetilde{O}(qM_{q}^{q/(q-1)}d^{2}\,\lVert\operatorname{cov}\pi\rVert\log\frac{1}{\varepsilon})$ membership queries, where $M_{q}=\lVert\text{d}\pi_{0}/\text{d}\pi\rVert_{L^{q}(\pi)}$. We further introduce a simple annealing scheme that produces a warm start in $q$-R\'enyi divergence (i.e., $M_{q}=O(1)$) using $\widetilde{O}(qd^{2}R^{3/2}\,\lVert\operatorname{cov}\pi\rVert^{1/4})$ queries, where $R^{2}=\mathbb{E}_{\pi}[|\cdot|^{2}]$. This interpolates between known complexities for warm-start generation in total variation and R\'enyi-infinity divergence. To relay a R\'enyi warmness across the annealing scheme, we establish hypercontractivity under simultaneous heat flow and translate it into an improved mixing guarantee for the proximal sampler under a logarithmic Sobolev inequality. These results extend naturally to general log-concave distributions accessible via evaluation oracles, incurring additional quadratic queries.

Faster logconcave sampling from a cold start in high dimension

We present a faster algorithm to generate a warm start for sampling an arbitrary logconcave density specified by an evaluation oracle, leading to the first sub-cubic sampling algorithms for inputs in (near-)isotropic position. A long line of prior work incurred a warm-start penalty of at least linear in the dimension, hitting a cubic barrier, even for the special case of uniform sampling from convex bodies. Our improvement relies on two key ingredients of independent interest. (1) We show how to sample given a warm start in weaker notions of distance, in particular $q$-R\'enyi divergence for $q=\widetilde{\mathcal{O}}(1)$, whereas previous analyses required stringent $\infty$-R\'enyi divergence (with the exception of Hit-and-Run, whose known mixing time is higher). This marks the first improvement in the required warmness since Lov\'asz and Simonovits (1991). (2) We refine and generalize the log-Sobolev inequality of Lee and Vempala (2018), originally established for isotropic logconcave distributions in terms of the diameter of the support, to logconcave distributions in terms of a geometric average of the support diameter and the largest eigenvalue of the covariance matrix.

Fast Tensor Completion via Approximate Richardson Iteration

We study tensor completion (TC) through the lens of low-rank tensor decomposition (TD). Many TD algorithms use fast alternating minimization methods, which solve highly structured linear regression problems at each step (e.g., for CP, Tucker, and tensor-train decompositions). However, such algebraic structure is lost in TC regression problems, making direct extensions unclear. To address this, we propose a lifting approach that approximately solves TC regression problems using structured TD regression algorithms as blackbox subroutines, enabling sublinear-time methods. We theoretically analyze the convergence rate of our approximate Richardson iteration based algorithm, and we demonstrate on real-world tensors that its running time can be 100x faster than direct methods for CP completion.

Sampling and Integration of Logconcave Functions by Algorithmic Diffusion

We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and matches the best-known complexities for the special case of uniform distributions on convex bodies. For the sampling problem, our output guarantees are significantly stronger than previously known, and lead to a streamlined analysis of statistical estimation based on dependent random samples.

Covariance estimation using Markov chain Monte Carlo

We investigate the complexity of covariance matrix estimation for Gibbs distributions based on dependent samples from a Markov chain. We show that when $\pi$ satisfies a Poincar\'e inequality and the chain possesses a spectral gap, we can achieve similar sample complexity using MCMC as compared to an estimator constructed using i.i.d. samples, with potentially much better query complexity. As an application of our methods, we show improvements for the query complexity in both constrained and unconstrained settings for concrete instances of MCMC. In particular, we provide guarantees regarding isotropic rounding procedures for sampling uniformly on convex bodies.

Rényi-infinity constrained sampling with $d^3$ membership queries

Uniform sampling over a convex body is a fundamental algorithmic problem, yet the convergence in KL or R\'enyi divergence of most samplers remains poorly understood. In this work, we propose a constrained proximal sampler, a principled and simple algorithm that possesses elegant convergence guarantees. Leveraging the uniform ergodicity of this sampler, we show that it converges in the R\'enyi-infinity divergence ($\mathcal R_\infty$) with no query complexity overhead when starting from a warm start. This is the strongest of commonly considered performance metrics, implying rates in $\{\mathcal R_q, \mathsf{KL}\}$ convergence as special cases. By applying this sampler within an annealing scheme, we propose an algorithm which can approximately sample $\varepsilon$-close to the uniform distribution on convex bodies in $\mathcal R_\infty$-divergence with $\widetilde{\mathcal{O}}(d^3\, \text{polylog} \frac{1}{\varepsilon})$ query complexity. This improves on all prior results in $\{\mathcal R_q, \mathsf{KL}\}$-divergences, without resorting to any algorithmic modifications or post-processing of the sample. It also matches the prior best known complexity in total variation distance.

In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies

We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in R\'enyi divergence (which implies TV, $\mathcal{W}_2$, KL, $\chi^2$). The proof departs from known approaches for polytime algorithms for the problem -- we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the stationary density.