Matrix Norms in Data Streams: Faster, Multi-Pass and Row-Order

Given the prevalence of large scale linear algebra problems in machine learning, recently there has been considerable effort in characterizing which functions can be approximated efficiently of a matrix in the data stream model. We study a number of aspects of estimating matrix norms - an important class of matrix functions - in a stream that have not previously been considered: (1) multi-pass algorithms, (2) algorithms that see the underlying matrix one row at a time, and (3) time-efficient algorithms. Our multi-pass and row-order algorithms use less memory than what is provably required in the single-pass and entrywise-update models, and thus give separations between these models (in terms of memory). Moreover, all of our algorithms are considerably faster than previous ones. We also prove a number of lower bounds, and obtain for instance, a near-complete characterization of the memory required of row-order algorithms for estimating Schatten p-norms of sparse matrices. We complement our results with numerical experiments.