Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds
This work addresses matrix approximation efficiency for computational linear algebra applications, but it is incremental as it builds on existing Monte-Carlo methods with new bounds.
The paper tackles the problem of approximating the Gram matrix AA^T using a small number of weighted outer products of columns of A, providing exact computation conditions and probabilistic error bounds for a Monte-Carlo algorithm. It shows that the bounds depend on the stable rank or rank of A, not on matrix dimensions, and are validated through numerical experiments.
Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AA^T (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.