In this talk, I will discuss the following connections between private optimization and statistical physics in the context of the low-rank matrix approximation problem:

1) An efficient algorithm to privately compute a low-rank approximation and how it leads to an efficient way to sample from Harish-Chandra-Itzykson-Zuber densities studied in physics and mathematics, and

2) An improved analysis of the "utility" of the  "Gaussian Mechanism" for private low-rank approximation using Dyson Brownian motion.