On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication

Landsberg, J. M.; Ryder, Nicholas

doi:10.1080/10586458.2016.1162230

Cited by 4 publications

(5 citation statements)

References 16 publications

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“…In [22], an explicit algorithm shows that its border rank is ≤ 5 with this observation leading to an upper bound on the exponent of matrix multiplication ω of log 12 1000 ≈ 2.7799. Another reason for computing the border rank of such tensors arises from [67] where the border rank of the matrix multiplication tensor for matrices of size 2 × 2 and 2 × n is considered. The results in [67] build on computational results in [3,84].…”

Section: Matrix Multiplication With Zerosmentioning

confidence: 99%

“…Another reason for computing the border rank of such tensors arises from [67] where the border rank of the matrix multiplication tensor for matrices of size 2 × 2 and 2 × n is considered. The results in [67] build on computational results in [3,84].…”

Section: Matrix Multiplication With Zerosmentioning

confidence: 99%

“…Number 1 As mentioned above, the upper bound on border rank is provided in [22]. The lower bound is provided in [67,Prop. 3.2] which is based on an equation provided in [91].…”

Section: Numbermentioning

confidence: 99%

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Tensor decomposition and homotopy continuation

Bernardi

Daleo

Hauenstein

et al. 2017

Differential Geometry and its Applications

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A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties X1, . . . , X k ⊂ P N defined over C. After computing ranks over C, we also explore computing real ranks. A variety of examples are included to demonstrate the numerical algebraic geometric approaches.

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Section: Matrix Multiplication With Zerosmentioning

confidence: 99%

Section: Matrix Multiplication With Zerosmentioning

confidence: 99%

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Tensor decomposition and homotopy continuation

Bernardi

Daleo

Hauenstein

et al. 2017

Differential Geometry and its Applications

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“…Recently, there have been several papers analyzing the geometry and symmetries of algebraic algorithms for small matrices [Bur14,Bur15,LR16,LM16b,CILO16]. Here, we take this line of research one step further by using symmetries to discover new algorithms for multiplying matrices of small size.…”

Section: Introductionmentioning

confidence: 99%

Matrix multiplication algorithms from group orbits

Grochow,

Moore

2016

Preprint

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We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under some finite group action. We show how to use the representation theory of the corresponding group to derive simple constraints on the decomposition, which we solve by hand for n = 2, 3, 4, 5, recovering Strassen's algorithm (in a particularly symmetric form) and new algorithms for larger n. While these new algorithms do not improve the known upper bounds on tensor rank or the matrix multiplication exponent, they are beautiful in their own right, and we point out modifications of this idea that could plausibly lead to further improvements. Our constructions also suggest further patterns that could be mined for new algorithms, including a tantalizing connection with lattices. In particular, using lattices we give the most transparent proof to date of Strassen's algorithm; the same proof works for all n, to yield a decomposition with n 3 − n + 1 terms.

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“…A new lower bound for the border rank of matrix multiplication Theorem 6.1. When n ≥ 3, R(M red ⟨n⟩ ) ≥ 2n 2 − n. When n = 2, it was shown in [23] that R(M red ⟨2⟩ ) = 5. Theorem 6.1 combined with Corollary 5.3 implies:…”

mentioning

confidence: 99%

On the Geometry of Border Rank Decompositions for Matrix Multiplication and Other Tensors with Symmetry

Landsberg¹,

Michałek²

2017

SIAM J. Appl. Algebra Geometry

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Abstract. We present a new approach to study tensors with symmetry, via local algebraic geometry. Border rank decompositions for such tensors-in particular, matrix multiplication and the determinant polynomial-come in families. We prove that these families include representatives with normal forms. These normal forms will be useful to prove lower complexity bounds and possibly even to determine new decompositions. We derive a border rank version of the substitution method used in proving lower bounds for tensor rank. Applying these methods, we improve the lower bound on the border rank of matrix multiplication. We also point out difficulties that will be formidable obstacles to future progress on lower complexity bounds for tensors because of the "wild" structure of the Hilbert scheme of points.

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On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication

Abstract: We make an in-depth study of the known border rank (i.e. approximate) algorithms for the matrix multiplication tensor M ⟨n,2,2⟩ ∈ C 2n ⊗C 4 ⊗C 2n encoding the multiplication of an n × 2 matrix by a 2 × 2 matrix.

Cited by 4 publications

References 16 publications

Tensor decomposition and homotopy continuation

Tensor decomposition and homotopy continuation

Matrix multiplication algorithms from group orbits

On the Geometry of Border Rank Decompositions for Matrix Multiplication and Other Tensors with Symmetry

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