更快的求逆及其他黑箱矩阵计算：利用高效块投影
Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections
AI总结：
本文证明高效块投影在足够大域上存在，从而确立Eberly等人稀疏线性系统求解算法的正确性，并据此改进稀疏矩阵求逆、零空间基及行列式等黑箱计算的复杂度界，算法均为Las Vegas概率型。
摘要：
块投影在[Eberly等人2006]中已被用于获得求解稀疏线性方程组的有效算法。假设输入矩阵可以与常数大小条目的向量在softO(n)次机器操作内相乘，则得到softO(n^(2.5))次机器操作的界。不幸的是，该算法的正确性依赖于高效块投影的存在性，而这只是一个猜想。在本文中，我们通过证明在足够大的域上高效块投影的存在性，确立了[Eberly等人2006]中算法的正确性。我们通过推导几个矩阵问题成本的改进界来展示这些投影的实用性，特别考虑可以通过softO(n)次域操作与向量相乘的“稀疏”矩阵。我们展示了如何在域F上以期望的softO(n^(2.27))次操作计算稀疏矩阵的逆。稀疏矩阵零空间的一组基及其秩的认证，以相同的成本获得。将这一结果应用于Kaltofen和Villard的用于整数矩阵行列式和Smith标准型的Baby-Steps/Giant-Steps算法，得到需要softO(n^(2.66))次机器操作的算法。所推导的算法均为Las Vegas类型的概率算法。
英文摘要：
Block projections have been used, in [Eberly et al. 2006], to obtain an efficient algorithm to find solutions for sparse systems of linear equations. A bound of softO(n^(2.5)) machine operations is obtained assuming that the input matrix can be multiplied by a vector with constant-sized entries in softO(n) machine operations. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections, and this has been conjectured. In this paper we establish the correctness of the algorithm from [Eberly et al. 2006] by proving the existence of efficient block projections over sufficiently large fields. We demonstrate the usefulness of these projections by deriving improved bounds for the cost of several matrix problems, considering, in particular, ``sparse'' matrices that can be be multiplied by a vector using softO(n) field operations. We show how to compute the inverse of a sparse matrix over a field F using an expected number of softO(n^(2.27)) operations in F. A basis for the null space of a sparse matrix, and a certification of its rank, are obtained at the same cost. An application to Kaltofen and Villard's Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an integer matrix yields algorithms requiring softO(n^(2.66)) machine operations. The derived algorithms are all probabilistic of the Las Vegas type.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
