Random matrices are fundamental objects in modern data science and machine learning. Although the statistical properties of many random matrix models have been studied extensively, their algorithmic properties remain comparatively underexplored. This short course will be divided into two parts. In the first part, we will discuss the behavior of iteration-wise outputs of various numerical linear algebra and optimization algorithms when their inputs are random matrices. In the second part, we will explore how these algorithms can be used inversely to study structural properties of random matrices—achieving statistical consistency while offering significantly improved computational efficiency. In particular, we only need to perform row and column multiplications, rather than computing eigenvalues and eigenvectors.

This talk is based on recent joint work primarily with Thomas Trogdon.