Fast matrix rank algorithms and applications

Cheung, Ho Yee; Kwok, Tsz Chiu; Lau, Lap Chi

doi:10.1145/2528404

Cited by 58 publications

(45 citation statements)

References 47 publications

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“…Our work in this paper is motivated by a recent breakthrough by Cheung, Kwok and Lau [2], who give an algorithm for computing the rank of A in time (r ω + |A|) 1+o (1) . Note that the +|A| term in the cost estimate is optimal since changing a single entry of A might modify the rank.…”

Section: Previous Workmentioning

confidence: 99%

“…However, the columns computed may not correspond to the column rank profile. For an extensive list of applications (with citations) of fast rank computation to combinatorial optimization and exact linear algebra problems we refer to [2].…”

Section: Previous Workmentioning

confidence: 99%

“…A linear independence oracle T for R, shown in Figure 2, is a perfect binary tree of height h := log 2 m, that is, with m leaf nodes. The nodes of the last level of the tree, from left to right, are associated with (i.e., store) the corresponding columns R [1], R [2], . .…”

Section: Definitionmentioning

confidence: 99%

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Linear independence oracles and applications to rectangular and low rank linear systems

Storjohann

Yang

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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Randomized algorithms are given for linear algebra problems on an input matrix A ∈ K n×m over a field K. We give an algorithm that simultaneously computes the row and column rank profiles of A in 2r 3 + (r 2 + n + m + |A|) 1+o(1) field operations from K, where r is the rank of A and |A| denotes the number of nonzero entries of A. Here, the +o(1) in cost estimates captures some missing log n and log m factors. The rank profiles algorithm is randomized of the Monte Carlo type: the correct answer will be returned with probability at least 1/2. Given a b ∈ K n×1 , we give an algorithm that either computes a particular solution vector x ∈ K m×1 to the system Ax = b, or produces an inconsistency certificate vector u ∈ K 1×n such that uA = 0 and ub = 0. The linear solver examines at most r + 1 rows and r columns of A and has running time 2r 3 + (r 2 + n + m + |R| + |C|)1+o (1) field operations from K, where |R| and |C| are the number of nonzero entries in the rows and columns, respectively, that are examined. The solver is randomized of the Las Vegas type: an incorrect result is never returned but the algorithm may report FAIL with probability at most 1/2. These cost estimates are achieved by making use of a novel randomized online data structure for the detection of linearly independent rows and columns.

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Section: Previous Workmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

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Linear independence oracles and applications to rectangular and low rank linear systems

Storjohann

Yang

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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“…In a surprising result, Cheung, Kwok and Lau [3] give a Monte Carlo algorithm for MaxIndependentRowSet with running time only (r ω + n + m + |A|) 1+o(1) field operations in K. Here, |A| denotes the number of nonzero entries of A and ω is the exponent for matrix multiplication. The following year, using an alternative technique [12], a Monte Carlo algorithm for RowRankProfile was presented that has running time (r 3 + n + m + |A|) 1+o(1) operations in K.…”

Section: Introductionmentioning

confidence: 99%

A Relaxed Algorithm for Online Matrix Inversion

Storjohann

Yang²

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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We consider a variation of the well known problem of computing the unique solution to a nonsingular system Ax = b of n linear equations over a field K. The variation assumes that A has generic rank profile and requires as output not only the single solution vector A −1 b ∈ K n×1 , but rather the solution to all leading principle subsystems. Most importantly, the rows of the augmented system A b are given one at a time from first to last, and as soon as the next row is given the solution to the next leading principal subsystem should be produced. We call this problem OnlineSystem. The obvious iterative algorithm for OnlineSystem has a cost in terms of field operations that is cubic in the dimension of A. In this paper we introduce a relaxed representation for the inverse and show how to obtain an algorithm for OnlineSystem that allows us to incorporate matrix multiplication. As an application we show how to introduce fast matrix multiplication into the inherently iterative algorithm for row rank profile computation presented previously by the authors.

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“…(Consider that changing a single entry of A can change the rank.) They also give an extension of their algorithm [1,Theorem 2.11] that can compute a set of r linearly independent rows and columns of A in the same running time. However, the rows computed may not correspond to the row rank profile, the lexicographically minimal list [i 1 , i 2 , .…”

Section: Introductionmentioning

confidence: 99%