Comparison of Matrix Norm Sparsification

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 parallel to A' - A parallel to <= epsilon parallel to A parallel to for error parameter epsilon > 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 = infinity. We investigate the relation between fixed but different p not equal 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 l(p)-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 not equal q. In addition, our explicit constructions may be of independent interest.