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 \(\Vert A'-A\Vert \le \epsilon \Vert A\Vert \) 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 \(p=\infty \). We investigate the relation between fixed but different \(p\ne q\), that is, whether sparsification in the Schatten p-norm implies (existentially and/or algorithmically) sparsification in the Schatten \(q\text {-norm}\) 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 \(\ell _p\)-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 \(p\ne q\). In addition, our explicit constructions may be of independent interest.