Matrix sparsification is a well-known approach in the design of efficient algorithms, where one 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 Schatten $p$-norm implies (existentially and/or algorithmically) sparsification in 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 $\ell_p$-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 $p\neq q$.
翻译:在设计高效算法时,母体缩进是一种众所周知的方法,在设计高效算法时,人们可以使用一个基质$$美元,总价为1美元,包括光谱规范,作为特例。Achlioptas和McShelry [2007] 开始对光谱-诺尔姆蒸发进行一长串工作,目的是保证美元A'A'A ⁇ leq\ epsilon ⁇ A ⁇ A ⁇ $* ⁇ A ⁇ $,用于错误参数($\epsilon>0美元)。各种形式的基质近似激励考虑这一问题,根据一般价Scatten $p-norm的保证,其中包括光谱规范,作为特例$p ⁇ infty$。我们调查了固定但不同的 $p\ neq q q$之间的关系,也就是说,Scatten $ $-neqr 矢量答案是令人惊讶的。我们的主要发现是否定性的,对于正向矢的答案是:对于正反的答案是:对正质的答案是:对正质的答案是:对正质的答案是:我们的答案是几乎的对正质的。