2016/04/29 by Li, Yi, Woodruff, David P.
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.1604.08679
For any real number p > 0, we nearly completely characterize the space complexity of estimating ‖A‖pp = ∑i=1n σip for n × n matrices A in which each row and each column has O(1) non-zero entries and whose entries are presented one at a time in a data stream model. Here the σi are the singular values of A, and when p ≥ 1, ‖A‖pp is the p-th power of the Schatten p-norm. We show that when p is not an even integer, to obtain a (1+ε)-approximation to ‖A‖pp with constant probability, any 1-pass algorithm requires n1-g(ε) bits of space, where g(ε) → 0 as ε→ 0 and ε> 0 is a constant independent of n. However, when p is an even integer, we give an upper bound of n1-2/p \textrmpoly(ε-1log n) bits of space, which holds even in the turnstile data stream model. The latter is optimal up to \textrmpoly(ε-1 log n) factors. Our results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices A: the previous best lower bound was Ω(log n) for p∈ (0,1), Ω(n1/p-1/2/log n) for p∈ [1,2) and Ω(n1-2/p) for p∈ (2,∞). We note for p ∈ (2, ∞), while our lower bound for even integers is the same, for other p in this range our lower bound is n1-g(ε), which is considerably stronger than the previous n1-2/p for small enough constant ε> 0. We obtain similar near-linear lower bounds for Ky-Fan norms, SVD entropy, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to our work.