Comparison of Matrix Norm Sparsification
Robert Krauthgamer, Shay Sapir

TL;DR
This paper explores how sparsification guarantees in different Schatten p-norms relate for matrices, revealing a surprising negative result for most p and q values, contrasting with vector cases.
Contribution
It provides the first analysis of the relationship between matrix sparsification in different Schatten p-norms, showing most p and q pairs do not imply each other.
Findings
Negative results for most p and q pairs in matrix Schatten norms
Contrasts with positive vector Schatten norm sparsification results
Explicit constructions of matrices demonstrating these properties
Abstract
A well-known approach in the design of efficient algorithms, called matrix sparsification, approximates a matrix with a sparse matrix . Achlioptas and McSherry [2007] initiated a long line of work on spectral-norm sparsification, which aims to guarantee that for error parameter . Various forms of matrix approximation motivate considering this problem with a guarantee according to the Schatten -norm for general , which includes the spectral norm as the special case . We investigate the relation between fixed but different , that is, whether sparsification in the Schatten -norm implies (existentially and/or algorithmically) sparsification in the Schatten with similar sparsity. An affirmative answer could be tremendously useful, as it will identify which value of to focus on. Our main…
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Taxonomy
TopicsSparse and Compressive Sensing Techniques · Stochastic Gradient Optimization Techniques · Mathematical Inequalities and Applications
