New transformations of Cauchy matrices and Trummer's problem

Pan, Victor Y.; Tabanjeh, M. Abu; Chen, Z.Q.; Landowne, E. I.; Sadikou, Akimou

doi:10.1016/s0898-1221(98)00091-1

Cited by 14 publications

(8 citation statements)

References 19 publications

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“…The K c matrices within each group are coded into an MDS (S, K c ) code by a Cauchy encoding matrix to create S linear combinations of these K c matrices. Multiplication with a Cauchy encoding matrix corresponds to the well studied Trummer's problem [32] for which fast algorithms have been found in [33][34][35] that limit the encoding complexity to C eA = O( λκS log 2 S Kc ). The s th coded linear combination from each of the groups is sent to the s th server.…”

Section: Csa Codes: Main Resultsmentioning

confidence: 99%

Cross Subspace Alignment Codes for Coded Distributed Batch Computation

Jia

Jafar

2019

Preprint

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The goal of coded distributed computation is to efficiently distribute a computation task, such as matrix multiplication, N -linear computation, or multivariate polynomial evaluation, across S servers through a coding scheme, such that the response from any R servers (R is called the recovery threshold) is sufficient for the user to recover the desired computed value. Current state-of-art approaches are based on either exclusively matrix-partitioning (Entangled Polynomial (EP) Codes for matrix multiplication), or exclusively batch processing (Lagrange Coded Computing (LCC) for N -linear computations or multivariate polynomial evaluations). We present three related classes of codes, based on the idea of Cross-Subspace Alignment (CSA) which was introduced originally in the context of secure and private information retrieval. CSA codes are characterized by a Cauchy-Vandermonde matrix structure that facilitates interference alignment along Vandermonde terms, while the desired computations remain resolvable along the Cauchy terms. These codes are shown to unify, generalize and improve upon the state-ofart codes for distributed computing. First we introduce CSA codes for matrix multiplication, which yield LCC codes as a special case, and are shown to outperform LCC codes in general in download-limited settings. While matrix-partitioning approaches (EP codes) for distributed matrix multiplication have the advantage of flexible server computation latency, batch processing approaches (CSA, LCC) have significant advantages in communication costs as well as encoding and decoding complexity per matrix multiplication. In order to combine the benefits of these approaches, we introduce Generalized CSA (GCSA) codes for matrix multiplication that bridge the extremes of matrix-partitioning and batch processing approaches and demonstrate synergistic gains due to cross subspace alignment. Finally, we introduce N -CSA codes for N -linear distributed batch computations and multivariate batch polynomial evaluations. N -CSA codes include LCC codes as a special case, and are in general capable of outperforming LCC codes in download-constrained settings by upto a factor of N . Generalizations of N -CSA codes to include X-secure data and B-byzantine servers are also provided.

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Section: Csa Codes: Main Resultsmentioning

confidence: 99%

Cross Subspace Alignment Codes for Coded Distributed Batch Computation

Jia

Jafar

2019

Preprint

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“…The main goal in this paper is to enrich the power of the multipoint algorithm by introducing and proving some new expressions for Cauchy matrix via other Cauchy matrices [7], which we may vary by changing one of their basis vectors. Under a certain choice of such a vector, the solution of Trummer's problem will be simplified; thus, the power of the multipoint algorithm can be developed as we will see in the next section.…”

Section: New Transformation Of Cauchy Matricesmentioning

confidence: 99%

Fast and Numerically Stable Approximate Solution of Trummer’s Problem

Tabanjeh¹

2014

AJCM

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“…The concept of HSS matrices was first introduced in [4,5], which can be seen as an algebraic counterpart of the method in [28].…”

Section: B Parallel Hss Construction Algorithmmentioning

confidence: 99%

A Parallel matrix multiplication algorithm for some Cauchy-like matrices

Cheng

2014

2014 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC)

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In this paper we propose a fast matrix multiplication algorithm for Cauchy-like matrices with off-diagonal low-rank property, which uses hierarchically semiseparable (HSS) matrix techniques. We implemented this algorithm in shared memory environment by using OpenMP. The numerical results show it has good speedups and is much faster than the plain matrixmatrix multiplications. For matrices with big dimensions, our algorithm has over 15x speedups.

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New transformations of Cauchy matrices and Trummer's problem

Cited by 14 publications

References 19 publications

Cross Subspace Alignment Codes for Coded Distributed Batch Computation

Cross Subspace Alignment Codes for Coded Distributed Batch Computation

Fast and Numerically Stable Approximate Solution of Trummer’s Problem

A Parallel matrix multiplication algorithm for some Cauchy-like matrices

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