Topic area: Numerical Methods.
Title: Low rank matrix approximations in the age of machine learning.
Abstract
Approximation of a given matrix by a low-rank matrix is a famous problem in linear algebra and data compression that can be solved with the singular value decomposition, incomplete LU decomposition (Gaussian elimination) and cross approximation. However, data matrices in practical applications may have missing and noisy elements. This may happen, for example, for user i preference for movie j in a recommendation database, or for absorption of X-ray of energy i at pixel j in spectromicroscopy. We will look at some traditional algorithms for restoring low-rank matrices from data, advance them to new methods of machine learning (such as Adam with automatic differentiation and data splitting for reliable error estimation), and implement them in Python*. Time permitting, we will consider how low-rank matrices can be used for compressed approximation of tensors and multivariate functions.