Research
Papers and Preprints
Singular-value gap of nonreversible Markov processes
Preprint, 2026.
We consider a generalization of the spectral gap of reversible Markov generators to nonreversible processes, following the recent work of Sourav Chatterjee (2025) on nonreversible finite-state Markov chains. Extending Chatterjee’s observations, we find that this spectral quantity that we call the singular-value gap characterizes the convergence of empirical averages, providing upper and lower bounds for finite time variance uniformly over \(L^2\) -functions. A key observation is that when the singular-value gap is positive, the generator is invertible on the \(L^2\) -orthogonal complement of constant functions. In particular, the Poisson equation \(−Lf = g\) can be solved, which enables our proof and connects our results to asymptotic variance and associated central limit theorems.
We also compare the singular-value gap with the spectral gap of the reversibilized process, the mixing time in total-variation distance, and the Cheeger constant. Several examples are provided throughout the text. Among other potential applications of the singular-value gap, these examples illustrate that a positive singular-value gap can help with variance reduction for observable classes in MCMC sampling, uncover slow-mixing mechanisms, and certify convergence of empirical averages for diffusion operators with complicated spectrum.
A convex geometric study of quantum codes
Undergraduate summer research, 2022.
This paper explores a generalization of quantum error detection and correction in representations of Lie algebras, specifically in a class of irreducible representations of \(\mathfrak{sl}_n \mathbb{C}\). The central method introduced is a two-step construction that transforms the problem of finding quantum codes into a classical problem of convex geometry. The purpose of this paper is also two-fold. First, we wish to build up background for this rarely studied topic and explain essential aspects of classical and quantum codes that motivate our study. Second, we discuss the interplay of irreducible representations of \(\mathfrak{sl}_n \mathbb{C}\) with the so-called Lie type graph metric and present code constructions in representations of \(\mathfrak{sl}_n \mathbb{C}\) for \(n \leq 4\).
Projects
Stochastic localization for generative modeling
Stanford CS229 project, 2026
The theoretical work by Andrea Montanari (2023) shows that the seemingly canonical method of sampling by reversing a diffusion sits inside a much wider class of sampling methods inspired by the technique of stochastic localization in Markov chain mixing time analysis. This wide class of sampling schemes is based on estimating the posterior mean of a desired sample conditioned on an observation process. When the data distribution contains low-dimensional global structure, such as class identity, coarse layout, or global color statistics, an isotropic noising process may require the denoiser to infer that structure indirectly from noisy local observations. Thus, fixing the neural network architecture used for score-matching, models that use isotropic diffusion often fail to learn certain latent structures of the data. On the other hand, models that use carefully engineered observation processes can simplify the denoising task and improve sampling in controlled examples, as demonstrated in sections 5,6,7 of Montanari (2023).
Since Montanari demonstrated the superiority of tailored observation processes on synthetic data and Gaussian mixtures, we find it interesting and useful to test whether we can use similar ideas to improve sampling results in real-world datasets. On CIFAR-10, we study whether discrete or continuous observations can expose class-like or low-frequency image structures to a small U-Net denoiser. The discrete observations include random labels, k-means pseudo-labels, and a true-label oracle. The continuous observations are low-resolution versions of the image, injected either through the time embedding or as spatial channels. On a landscape subset of the MIT image set Places, we study if explicit global color observations improve coherence in scene generation, so that the generated image is distinctly a beach or a mountain.