LOW-RANK MATRIX APPROXIMATIONS DO NOT NEED A SINGULAR VALUE GAP
LOW-RANK MATRIX APPROXIMATIONS DO NOT NEED A SINGULAR VALUE GAP
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DOI：
10.1137/18m1163658
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发表时间：
2019-01-01
影响因子：
1.5
通讯作者：
Ipsen, Ilse C. F.
中科院分区：
文献类型：
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作者：
Drineas, Petros;Ipsen, Ilse C. F.
可以从zz（t）a触发对真矩阵A的低级别近似值，其中z是带有正态列的矩阵，并且可以从A-zz（t）a的某些范围中估算近似值的精度。我们表明，在弗罗贝尼乌斯规范中计算a-zz（t）a，更通常，任何schatten p-norm都是一个良好的数学问题。而且，与主要的子空间计算相反，它不需要奇异的值差距。我们还表明，此问题对A和Z中的加性扰动有充分的条件（不敏感），并且在近似准确性的无标准中对尺寸变化或乘法扰动。对于特殊情况，当A确实具有单数值差距时，在低级别近似值和子空间角之间建立了连接。
Low-rank approximations to a real matrix A can be conputed from ZZ(T)A, where Z is a matrix with orthonormal columns, and the accuracy of the approximation can be estimated from some norm of A-ZZ(T)A. We show that computing A-ZZ(T)A in the two-norm, Frobenius norms, and more generally any Schatten p-norm is a well-posed mathematical problem; and, in contrast to dominant subspace computations, it does not require a singular value gap. We also show that this problem is well-conditioned (insensitive) to additive perturbations in A and Z, and to dimensionchanging or multiplicative perturbations in A-regardless of the accuracy of the approximation. For the special case when A does indeed have a singular values gap, connections are established between low-rank approximations and subspace angles.