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Alman, Josh

doi:10.4086/toc.2021.v017a001

Cited by 6 publications

(8 citation statements)

References 30 publications

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“…Our barrier simplifies and generalizes previous barriers and tightly connects the barrier literature to central notions from the framework of Strassen [29,30,31,12]. Alman [1] reported very similar independent results shortly after our manuscript appeared on the arXiv, which we jointly presented at the Computational Complexity Conference 2019 in New Brunswick. For all the tensors mentioned, the barriers of Alman are identical to ours.…”

Section: Matrix Multiplication Barrierssupporting

confidence: 66%

“…We have not tried to optimize the barriers that we obtain, but focus instead on introducing the barrier itself. The barrier in [1] is very similar to ours, except for using asymptotic slice rank instead of asymptotic subrank.…”

Section: Comparison and Other Applicationsmentioning

confidence: 71%

“…Similarly, subrank is super-multiplicative under ⊗. Therefore, by Fekete's lemma, 1 the above limits exist and equal the respective infimum and supremum. We note that asymptotic rank and asymptotic subrank may equivalently be defined using border rank and border subrank, respectively, which are the approximative versions of rank and subrank (see, e. g., [7]).…”

Section: Tensors and Matrix Multiplicationmentioning

confidence: 92%

“…By this we mean that for any n the problem of multiplying n × n matrices can be reduced to the problem of multiplying n ω+o (1) pairs of numbers (and some additions, but they do not have an influence on the complexity). We will make these notions precise later.…”

Section: Our Barrier: Informal Explanationmentioning

confidence: 99%

“…Let p be a large prime. There is a subset A ⊆ Z/pZ (a Salem-Spencer set in Z/pZ) that is free of nontrivial three-term arithmetic progressions and that has size |A| = p 1−o (1) [25,5]. The support of the tensor p, p, p is the set of triples…”

Section: Tensors and Matrix Multiplicationmentioning

confidence: 99%

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Christandl¹,

Vrana²,

Zuiddam³

2021

Theory of Comput.

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Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent ω, is a central problem in algebraic complexity theory. The best upper bounds on ω, leading to the state-of-theart ω ≤ 2.37.., have been obtained via Strassen's laser method and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on ω.We introduce a new and more general barrier, providing stronger limitations than in previous work. Concretely, we introduce the notion of irreversibility of a tensor, and we prove (in some precise sense) that any approach that uses an irreversible tensor in an intermediate step (e. g., as a starting tensor in the laser method) cannot give ω = 2. In quantitative terms,

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Section: Matrix Multiplication Barrierssupporting

confidence: 66%

Section: Comparison and Other Applicationsmentioning

confidence: 71%

Section: Tensors and Matrix Multiplicationmentioning

confidence: 92%

Section: Our Barrier: Informal Explanationmentioning

confidence: 99%

Section: Tensors and Matrix Multiplicationmentioning

confidence: 99%

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Christandl¹,

Vrana²,

Zuiddam³

2021

Theory of Comput.

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Revisiting “Rapid multiplication of rectangular matrices”

Nissim,

Schwartz

2024

Research in Mathematics

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The tensor as an informational resource

Christandl

2024

PNAS Nexus

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A tensor is a multidimensional array of numbers that can be used to store data, encode a computational relation and represent quantum entanglement. In this sense a tensor can be viewed as valuable resource whose transformation can lead to an understanding of structure in data, computational complexity and quantum information. In order to facilitate the understanding of this resource, we propose a family of information-theoretically constructed preorders on tensors, which can be used to compare tensors with each other and to assess the existence of transformations between them. The construction places copies of a given tensor at the edges of a hypergraph and allows transformations at the vertices. A preorder is then induced by the transformations possible in a given growing sequence of hypergraphs. The new family of preorders generalises the asymptotic restriction preorder which Strassen defined in order to study the computational complexity of matrix multiplication. We derive general properties of the preorders and their associated asymptotic notions of tensor rank and view recent results on tensor rank non-additivity, tensor networks and algebraic complexity in this unifying frame. We hope that this work will provide a useful vantage point for exploring tensors in applied mathematics, physics and computer science, but also from a purely mathematical point of view.

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Cited by 6 publications

References 30 publications

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Revisiting “Rapid multiplication of rectangular matrices”

The tensor as an informational resource

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