Numerical linear algebra in the streaming model

We give near-optimal space bounds in the streaming model for linear algebra problems that include estimation of matrix products, linear regression, low-rank approximation, and approximation of matrix rank. In the streaming model, sketches of input matrices are maintained under updates of matrix entries; we prove results for turnstile updates, given in an arbitrary order. We give the first lower bounds known for the space needed by the sketches, for a given estimation error . We sharpen prior upper bounds, with respect to combinations of space, failure probability, and number of passes. The sketch we use for matrix A is simply ST A, where S is a sign matrix. Our results include the following upper and lower bounds on the bits of space needed for 1-pass algorithms. Here A is an n ? d matrix, B is an n ? d matrix, and c := d + d . These results are given for fixed failure probability; for failure probability > 0, the upper bounds require a factor of log(1/) more space. We assume the ...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Kenneth L. Clarkson, David P. Woodruff
• Keywords: Algorithms, Lower Bounds, Near-optimal Space Bounds, Prior Upper Bounds, STOC 2009