Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds (1310.1502v3)

Published 5 Oct 2013 in math.NA, cs.LG, and stat.ML

Abstract: Given a real matrix A with n columns, the problem is to approximate the Gram product AAT by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AAT (in exact arithmetic) from c >= rank(A) columns depend on the right singular vector matrix of A. For a Monte-Carlo matrix multiplication algorithm by Drineas et al. that samples outer products, we present probabilistic bounds for the 2-norm relative error due to randomization. The bounds depend on the stable rank or the rank of A, but not on the matrix dimensions. Numerical experiments illustrate that the bounds are informative, even for stringent success probabilities and matrices of small dimension. We also derive bounds for the smallest singular value and the condition number of matrices obtained by sampling rows from orthonormal matrices.

Citations (41)

Summary

We haven't generated a summary for this paper yet.