No abstract

Pseudoskeleton approximation and some other problems require the knowledge of sufficiently well-conditioned submatrix in a large-scale matrix. The quality of a submatrix can be measured by modulus of its determinant, also known as volume. In this paper we discuss a search algorithm for the maximum-volume submatrix which already proved to be useful in several matrix and tensor approximation algorithms. We investigate the behavior of this algorithm on random matrices and present some its applications, including maximization of a bivariate functional.

Theorem 3. The spectrum of the operator H is sem_ibounded from below.The results of the present paper considerably generalize results of [3], where the case of compactly supported potential p(x) was considered. w IntroductionIn [1] the following approach to fast multiplication by dense nonstructured matrices was developed. Ass,,me that the matrices An are generated by the formula An = [f(xi, yj)]inj=l, where xi, Vj are nodes of a (quasi-unlform) grid on a bounded domain of Euclidean space R d . In this case, for the class of the so-called asymptotically smooth functions f (for the definition and the corresponding theorems, see [1]), it is possible to indicate a method for partitioning the points xi and Vj into groups that lead to the decomposition of the matrix into blocks of small s-rank. Moreover, the complexity of the multiplication by the resultant approximant of the entire matrix A, is O(n lognlog a s-l).Although a practical realization of this idea (related, say, to algorithms of partial singular decomposition) [2] can be successful, this approach a pr/or/ assumes the calculation of all elements of the blocks, and hence O(n 2) operations. It is unclear whether a "good" low-rank approximation for a block can be constructed if only a small part of its elements is known. This is possible indeed, as was first shown in [3]. The development of the corresponding theory was performed in [4] (where, in particular, an experimental confirmation of the theory is presented for matrices arising in the nl~merical solution of the integral equation of an electric field). In more detail, if the s-rank of the matrix A 6 R '*xn is at most k, then A contains k cohlmns C 6 R '*xk and k rows R 6 R kx'* such that ][A -CaRl12 < ~(k, n)s,(1) where G 6 R kxk is determined by C and R, and the function T is bounded above by a polynomial of small degree.