2021/06/16 by Zhili Feng, Feng, Zhili, Fred Roosta +3
Computer Science · Engineering · Mathematics · #Advanced Optimization Algorithms Research #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques #Tensor decomposition and applications #cs.LG #cs.NA #math.NA #stat.ML
paper · pdf · doi:10.48550/arxiv.2106.08544
arxiv created 2021/06/16 · openalex publication_date 2021/06/16 · arxiv updated 2021/06/17 · openalex created_date 2021/08/02 · openalex updated_date 2026/07/28
A variety of dimensionality reduction techniques have been applied for computations involving large matrices. The underlying matrix is randomly compressed into a smaller one, while approximately retaining many of its original properties. As a result, much of the expensive computation can be performed on the small matrix. The sketching of positive semidefinite (PSD) matrices is well understood, but there are many applications where the related matrices are not PSD, including Hessian matrices in non-convex optimization and covariance matrices in regression applications involving complex numbers. In this paper, we present novel dimensionality reduction methods for non-PSD matrices, as well as their ``square-roots", which involve matrices with complex entries. We show how these techniques can be used for multiple downstream tasks. In particular, we show how to use the proposed matrix sketching techniques for both convex and non-convex optimization, ℓp-regression for every 1 ≤ p ≤ ∞, and vector-matrix-vector queries.