Abstract
A well-known approach in the design of efficient algorithms, called matrix sparsification, approximates a matrix A with a sparse matrix A′ . Achlioptas and McSherry (J ACM 54(2):9-es, 2007) initiated a long line of work on spectral-norm sparsification, which aims to guarantee that ‖ A′- A‖ ≤ ϵ‖ A‖ for error parameter ϵ> 0 . Various forms of matrix approximation motivate considering this problem with a guarantee according to the Schatten p-norm for general p, which includes the spectral norm as the special case p= ∞ . We investigate the relation between fixed but different p≠ q , that is, whether sparsification in the Schatten p-norm implies (existentially and/or algorithmically) sparsification in the Schatten q-norm with similar sparsity. An affirmative answer could be tremendously useful, as it will identify which value of p to focus on. Our main finding is a surprising contrast between this question and the analogous case of ℓp -norm sparsification for vectors: For vectors, the answer is affirmative for p< q and negative for p> q , but for matrices we answer negatively for almost all sufficiently distinct p≠ q . In addition, our explicit constructions may be of independent interest.
Original language | English |
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Pages (from-to) | 3957-3972 |
Number of pages | 16 |
Journal | Algorithmica |
Volume | 85 |
Issue number | 12 |
DOIs | |
State | Published - Dec 2023 |
All Science Journal Classification (ASJC) codes
- General Computer Science
- Applied Mathematics
- Computer Science Applications