PROJECT TITLE :

Sample Complexity of Dictionary Learning and Other Matrix Factorizations

ABSTRACT:

Several trendy tools in Machine Learning and Signal Processing, like sparse dictionary learning, principal element analysis, non-negative matrix factorization, $K$ -means that Clustering, and therefore on, depend upon the factorization of a matrix obtained by concatenating high-dimensional vectors from a coaching collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in apply by minimizing an empirical average over the considered collection. The focus of this paper is to produce sample complexity estimates to uniformly control how much the empirical average deviates from the expected value operate. Customary arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses many potential constraints on the factors (tensor product structure, shift-invariance, sparsity…), therefore providing a unified perspective on the sample complexity of many widely used matrix factorization schemes. The derived generalization bounds behave proportional to $(log (n)/n)^1/2$ with respect to the quantity of samples $n$ for the thought of matrix factorization techniques.