On Approximating Matrix Norms in Data Streams

This paper presents a systematic study of the space complexity of estimating the Schatten p-norms of an n x n matrix in the turnstile streaming model. Both kinds of space complexities, bit complexity and sketching dimension, are considered. Furthermore, two sketching models, general linear sketching and bilinear sketching, are considered. When p is not an even integer, we show that any one-pass algorithm with constant success probability requires near-linear space in terms of bits. This lower bound holds even for sparse matrices, i.e., matrices with O(1) nonzero entries per row and per column. However, when p is an even integer, we give for sparse matrices an upper bound which, up to logarithmic factors, is the same as estimating the pth moment of an n-dimensional vector. These results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices. Similar near-linear lower bounds are obtained for Ky Fan norms, SVD entropy, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to this work. The results for general linear sketches give separations in the sketching complexity of Schatten p-norms with the corresponding vector p-norms, and rule out a table-lookup nearest-neighbor search for p = 1, making progress on a question of Andoni. The results for bilinear sketches are tight for the rank problem and nearly tight for p >= 2; the latter is the first general subquadratic upper bound for sketching the Schatten norms.