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 [2007] initiated a long line of work on spectral-norm sparsification, which aims to guarantee that $\|A'-A\|\leq \epsilon \|A\|$ 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 $p\neq q$, 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 $p\neq q$. In addition, our explicit constructions may be of independent interest.

翻译：一种众所周知的算法设计高效方法——矩阵稀疏化——用稀疏矩阵$A'$逼近矩阵$A$。Achlioptas和McSherry [2007] 开创了一系列关于谱范数稀疏化的研究，旨在保证对于误差参数$\epsilon>0$有$\|A'-A\|\leq \epsilon \|A\|$。各种矩阵近似形式促使考虑针对一般$p$的Schatten $p$-范数保证下的该问题，其中谱范数是$p=\infty$的特例。我们研究了固定但不同的$p\neq q$之间的关系，即Schatten $p$-范数下的稀疏化是否（存在性和/或算法上）蕴含类似稀疏度的Schatten $q$-范数稀疏化。肯定的答案将极具价值，因为它将确定应重点关注哪个$p$值。我们的主要发现是，该问题与向量$\ell_p$-范数稀疏化的类似情况之间存在惊人对比：对于向量，当$p<q$时答案为肯定，当$p>q$时答案为否定；但对于矩阵，我们几乎对所有足够不同的$p\neq q$给出否定答案。此外，我们的显式构造可能具有独立研究价值。