Tensor Rank and Border Rank of Band Toeplitz Matrices

Бини, Дарио Андреа; Capovani, Milvio

doi:10.1137/0216021

Cited by 20 publications

(11 citation statements)

References 22 publications

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“…These perturbed maps can give rise to fast exact algorithms for matrix multiplication; see [7]. The border rank made appearances in the literature in the 1980s and early 1990s (see, e.g., [6,5,19,8,15,2,10,11,4,3,17,16,18,1,9,21]), but to our knowledge there has not been much progress on the question since then.…”

Section: Introductionmentioning

confidence: 99%

The border rank of the multiplication of $2\times 2$ matrices is seven

Landsberg

2005

J. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

The border rank of the multiplication of $2\times 2$ matrices is seven

Landsberg

2005

J. Amer. Math. Soc.

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“…Let Toep n (C) be the vector space of n × n Toeplitz matrices. The following result is well-known, proved in [2] using methods different from those we employ below. Our objective of including this is to provide another illustration of the generalized Cohn-Umans approach where a bilinear operation is embedded in an algebra, in this case, the algebra of circulant matrices Circ 2n (C) in Section 7.…”

Section: Toeplitz Matricesmentioning

confidence: 87%

Fast Structured Matrix Computations: Tensor Rank and Cohn–Umans Method

Lim

2016

Found Comput Math

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We discuss a generalization of the Cohn-Umans method, a potent technique developed for studying the bilinear complexity of matrix multiplication by embedding matrices into an appropriate group algebra. We investigate how the Cohn-Umans method may be used for bilinear operations other than matrix multiplication, with algebras other than group algebras, and we relate it to Strassen's tensor rank approach, the traditional framework for investigating bilinear complexity. To demonstrate the utility of the generalized method, we apply it to find the fastest algorithms for forming structured matrix-vector product, the basic operation underlying iterative algorithms for structured matrices. The structures we study include Toeplitz, Hankel, circulant, symmetric, skew-symmetric, f -circulant, block-Toeplitz-Toeplitz-block, triangular Toeplitz matrices, Toeplitzplus-Hankel, sparse/banded/triangular. Except for the case of skew-symmetric matrices, for which we have only upper bounds, the algorithms derived using the generalized Cohn-Umans method in all other instances are the fastest possible in the sense of having minimum bilinear complexity. We also apply this framework to a few other bilinear operations including matrix-matrix, commutator, simultaneous matrix products, and briefly discuss the relation between tensor nuclear norm and numerical stability.

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“…The algorithms for circulant [5] and Toeplitz [1] matrices are known but those for other structured matrices are new (as far as we know). In particular, the multilevel structured matrices in §VII include arbitrarily complicated nested structures, e.g., block BCCB matrices whose blocks are Toeplitz-plus-Hankel, a 3-level structure.…”

Section: Introductionmentioning

confidence: 99%

“…, a n−1 ) and b ∈ C is arbitrary. Using this embedding, we obtain Algorithm 2 for Toeplitz matrixvector product [1], [14] . Algorithm 2 Toeplitz matrix-vector product 1: Express the Toeplitz matrix A as (a 1 , .…”

Section: Introductionmentioning

confidence: 99%

Algorithms for structured matrix-vector product of optimal bilinear complexity

Lim

2016

2016 IEEE Information Theory Workshop (ITW)

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Abstract-We present explicit algorithms for computing structured matrix-vector products that are optimal in the sense of Strassen, i.e., using a provably minimum number of multiplications. These structures include Toeplitz/Hankel/circulant, symmetric, Toeplitz-plus-Hankel, sparse, and multilevel structures. The last category include BTTB, BHHB, BCCB but also any arbitrarily complicated nested structures built out of other structures.

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Tensor Rank and Border Rank of Band Toeplitz Matrices

Cited by 20 publications

References 22 publications

The border rank of the multiplication of $2\times 2$ matrices is seven

The border rank of the multiplication of $2\times 2$ matrices is seven

Fast Structured Matrix Computations: Tensor Rank and Cohn–Umans Method

Algorithms for structured matrix-vector product of optimal bilinear complexity

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