A Relaxed Algorithm for Online Matrix Inversion

Storjohann, Arne; Yang, Shiyun

doi:10.1145/2733693.2733713

Cited by 5 publications

(5 citation statements)

References 6 publications

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“…We also show how to reproduce the result on "online linear systems" by Storjohann and Yang [SY15] via Theorem 1.1 and existing dynamic matrix inverse data structures. This again shows that many different results can be reproduced by reducing to dynamic matrix inverse, reducing the overall amount of proofs required in the area.…”

Section: Reproducing and Simplifying Further Resultsmentioning

confidence: 85%

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Unifying Matrix Data Structures: Simplifying and Speeding up Iterative Algorithms

Brand¹

2021

Symposium on Simplicity in Algorithms (SOSA)

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Many algorithms use data structures that maintain properties of matrices undergoing some changes. The applications are wide-ranging and include for example matchings, shortest paths, linear programming, semi-definite programming, convex hull and volume computation. Given the wide range of applications, the exact property these data structures must maintain varies from one application to another, forcing algorithm designers to invent them from scratch or modify existing ones. Thus it is not surprising that these data structures and their proofs are usually tailor-made for their specific application and that maintaining more complicated properties results in more complicated proofs.In this paper we present a unifying framework that captures a wide range of these data structures. The simplicity of this framework allows us to give short proofs for many existing data structures regardless of how complicated the to be maintained property is. We also show how the framework can be used to speed up existing iterative algorithms, such as the simplex algorithm.More formally, consider any rational function f (A 1 , ..., A d ) with input matrices A 1 , ..., A d . We show that the task of maintaining f (A 1 , ..., A d ) under updates to A 1 , ..., A d can be reduced to the much simpler problem of maintaining some matrix inverse M −1 under updates to M . The latter is a well studied problem called dynamic matrix inverse. By applying our reduction and using known algorithms for dynamic matrix inverse we can obtain fast data structures and iterative algorithms for much more general problems.

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Section: Reproducing and Simplifying Further Resultsmentioning

confidence: 85%

“…Storjohann and Yang [SY15] have given an O(n ω ) time algorithm for this problem. Here we show that the same result can be obtained by Corollary 4.2.…”

Section: Qr Decompositionmentioning

confidence: 99%

Unifying Matrix Data Structures: Simplifying and Speeding up Iterative Algorithms

Brand¹

2021

Symposium on Simplicity in Algorithms (SOSA)

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“…The decrease of the exponent of MM implies theoretical acceleration of the solution of a number of important problems in various areas of computations in Algebra and Computer Science, such as Boolean MM, computation of paths and distances in graphs, parsing context-free grammars, the solution of a nonsingular linear system of equations, computations of the inverse, determinant, characteristic and minimal polynomials, and various factorizations of a matrix. See [142], [34], [1, Sections 6.3-6.6], [24, pages 49-51], [18, Chapter 2], [3], [79], [48], [98], [160], [157], [158], [159], [86], [25], [89], [6], [54], [156], [132], [96], [4], [138], [140], [103], [104], [105], [125], and the bibliography therein and notice that some new important applications have been found very recently, e.g., in 4 papers at ISSAC 2016.…”

Section: Summary Of the Study Of The MM Exponents After 1978mentioning

confidence: 99%

“…Already Strassen in 1969 extended MM to matrix inversion (we sketch this in Examples 11.1 and 11.2 below), then Bunch and Hopcroft [34] extended MM to invertible LUP factorization, and further extensions followed in the eighties, for instance, in [82] to the computation of the matrix rank and of Gaussian elimination. Efficient algorithms whose complexity is sensitive to the rank or to the rank profile have only very recently been discovered [83,64,140]. One of the latest reductions to MM was that of the characteristic polynomial, which has extended the seminal work of [90] more than thirty years afterwards [126].…”

Section: Acceleration Of Computations Via Reductions To MMmentioning

confidence: 99%

Fast Matrix Multiplication and Symbolic Computation

Dumas,

Pan

2016

Preprint

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The complexity of matrix multiplication (hereafter MM) has been intensively studied since 1969, when Strassen surprisingly decreased the exponent 3 in the cubic cost of the straightforward classical MM to log 2 (7) ≈ 2.8074. Applications to some fundamental problems of Linear Algebra and Computer Science have been immediately recognized, but the researchers in Computer Algebra keep discovering more and more applications even today, with no sign of slowdown. We survey the unfinished history of decreasing the exponent towards its information lower bound 2, recall some important techniques discovered in this process and linked to other fields of computing, reveal sample surprising applications to fast computation of the inner products of two vectors and summation of integers, and discuss the curse of recursion, which separates the progress in fast MM into its most acclaimed and purely theoretical part and into valuable acceleration of MM of feasible sizes. Then, in the second part of our paper, we cover fast MM in realistic symbolic computations and discuss applications and implementation of fast exact matrix multiplication. We first review how most of exact linear algebra can be reduced to matrix multiplication over small finite fields. Then we highlight the differences in the design of approximate and exact implementations of fast MM, taking into account nowadays processor and memory hierarchies. In the concluding section we comment on current perspectives of the study of fast MM.

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“…We first propose quadratic, space and verification time, non-interactive practical certificates for the row or column rank profile and for the rank profile matrix that are rank-sensitive. Previously known certificates either had additional logarithmic factors to the quadratic complexities [14] or are not practical [16].…”

Section: Introductionmentioning

confidence: 99%

Certificates for Triangular Equivalence and Rank Profiles

Dumas

Lucas

Pernet

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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In this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a negligible overall overhead. We first provide quadratic time and space non-interactive certificates saving the logarithmic factors of previously known ones. Then we propose interactive certificates for the same problems whose Monte Carlo verification complexity requires a small constant number of matrixvector multiplications, a linear space, and a linear number of extra field operations. As an application we also give an interactive protocol, certifying the determinant of dense matrices, faster than the best previously known one.

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A Relaxed Algorithm for Online Matrix Inversion

Cited by 5 publications

References 6 publications

Unifying Matrix Data Structures: Simplifying and Speeding up Iterative Algorithms

Unifying Matrix Data Structures: Simplifying and Speeding up Iterative Algorithms

Fast Matrix Multiplication and Symbolic Computation

Certificates for Triangular Equivalence and Rank Profiles

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