随机近似Gram矩阵：精确计算与概率界
Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds
AI总结：
研究通过随机化方法近似Gram矩阵，提出基于稳定秩的概率误差界，适用于小维度矩阵和高成功概率场景。
摘要：
给定一个具有n列的实矩阵A，问题在于通过c<<n个加权外积近似Gram积AA^T。精确计算AA^T（在精确算术中）所需的条件取决于A的右奇异向量矩阵。对于Drineas等人提出的蒙特卡洛矩阵乘法算法，我们给出了由于随机化导致的2范数相对误差的概率界。这些界依赖于稳定秩或A的秩，而不是矩阵维度。数值实验表明，这些界在严格成功概率和小维度矩阵情况下仍具信息量。我们还推导了通过从正交矩阵中采样行所获得的矩阵的最小奇异值和条件数的界。
英文摘要：
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.
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