Randomized matrix sparsification has proven to be a fruitful technique for producing faster algorithms in applications ranging from graph partitioning to semidefinite programming. In the decade or so of research into this technique, the focus has been—with few exceptions—on ensuring the quality of approximation in the spectral and Frobenius norms. For certain graph algorithms, however, the ∞→1 norm may be a more natural measure of performance. This paper addresses the problem of approximating a real matrix A by a sparse random matrix X with respect to several norms. It provides the first results on approximation error in the ∞→1 and ∞→2 norms, and it uses a result of Lata la to study approximation error in the spectral norm. These bounds hold for a reasonable family of random sparsification schemes, those which ensure that the entries of X are independent and average to the corresponding entries of A. Optimality of the ∞→1 and ∞→2 error estimates is established. Concentration results for the three norms hold when the entries of X are uniformly bounded. The spectral error bound is used to predict the performance of several sparsification and quantization schemes that have appeared in the literature; the results are competitive with the performance guarantees given by earlier scheme-specific analyses.
In @cite_10 , Rudelson and Vershynin take a different approach to the Monte Carlo methodology for low-rank approximation. They consider @math as a linear operator between finite-dimensional Banach spaces and apply techniques of probability in Banach spaces: decoupling, symmetrization, Slepian's lemma for Rademacher random variables, and a law of large numbers for operator-valued random variables. They show that, if @math can be approximated by any rank- @math matrix, then it is possible to obtain an accurate rank- @math approximation to @math by sampling @math rows of @math . Additionally, they quantify the behavior of the @math and @math norms of random submatrices.
@cite_10
1998058722
We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(rlog r) with a small error in the spectral norm, where r e VAV2F VAV22 is the numerical rank of A. The numerical rank is always bounded by, and is a stable relaxation of, the rank of A. This yields an asymptotically optimal guarantee in an algorithm for computing low-rank approximations of A. We also prove asymptotically optimal estimates on the spectral norm and the cut-norm of random submatrices of A. The result for the cut-norm yields a slight improvement on the best-known sample complexity for an approximation algorithm for MAX-2CSP problems. We use methods of Probability in Banach spaces, in particular the law of large numbers for operator-valued random variables.