Sampling, Privacy, and Spectral Geometry: Insights from Low-Rank Approximation

Abstract

This talk explores how problems in private optimization—specifically, low-rank matrix approximation—give rise to novel tools and results in sampling and statistical physics. I will present two recent advances:

Sampling from Harish-Chandra–Itzykson–Zuber (HCIZ) Distributions via Private Optimization:
We introduce an efficient algorithm for computing private low-rank approximations and show how its structure enables efficient sampling from HCIZ measures, which are central to mathematical physics and random matrix theory.

Spectral Sampling and Utility of the Gaussian Mechanism:
We provide a new analysis of the Gaussian Mechanism for differential privacy through the lens of Dyson Brownian motion, yielding refined spectral sampling guarantees and new bounds on eigenvalue gaps in random matrices.

These results illustrate how sampling tasks arising from privacy constraints can lead to powerful connections between random matrix theory, optimization, sampling, and statistical physics.