A 5/2n/sup 2/-lower bound for the rank of n×n-matrix multiplication over arbitrary fields

Bläser, Markus

doi:10.1109/sffcs.1999.814576

Cited by 29 publications

(39 citation statements)

References 12 publications

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“…The best known lower bounds for bilinear circuits for problems such as polynomial multiplication or matrix multiplication are given in Refs. [BD80,Blä99,Shp03,Kam05].…”

Section: Sketchmentioning

confidence: 99%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

290

275

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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions.The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.

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“…The best known lower bounds for bilinear circuits for problems such as polynomial multiplication or matrix multiplication are given in Refs. [BD80,Blä99,Shp03,Kam05].…”

Section: Sketchmentioning

confidence: 99%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

290

275

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“…Conversely, most lower bounds on matrix multiplication seem to have a border rank lower bound at their heart. For example, Landsberg (2008, Section 6) showed that the tensor rank lower bound of Bläser (1999)-the then best known bound-implicitly uses the same key lemma that Strassen (1983) used to give a border rank lower bound. The currently best known lower bound on tensor rank (Landsberg 2014b;Massarenti & Raviolo 2012) also uses techniques from the best known lower bound on border rank (Landsberg & Ottaviani 2011).…”

Section: Assumptionmentioning

confidence: 99%

Unifying Known Lower Bounds via Geometric Complexity Theory

Grochow

2015

comput. complex.

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Abstract. We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representationtheoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on AC 0 [p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, BaurStrassen), the connected components technique (Ben-Or, Steele-Yao), depth 3 algebraic circuit lower bounds over finite fields (GrigorievKarpinski), lower bounds on permanent versus determinant (MignonRessayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (Bürgisser-Ikenmeyer, Landsberg-Ottaviani) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi, Agrawal-Vinay, Koiran, Tavenas), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification of known results and broad generalization of known techniques. It also shows that the framework of GCT is at least as powerful as previous methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT 394 Grochow cc 24 (2015) naturally provides new properties to consider in the search for new lower bounds.Keywords. Lower bounds, circuit complexity, algebraic complexity, geometric complexity theory, representation theory, algebraic geometry. Subject classification. 68Q17, 14L30, 13A50.Parts of this paper are written in a survey style in order to make it accessible to a wider audience. In particular, items marked Example or Fact are included only for expository purposes, and we make no claim to originality for those.This paper presupposes no knowledge of representation theory on the part of the reader. In fact, we use previous lower bounds together with our new viewpoint to motivate the use and definitions of representation theory and algebraic geometry in complexity theory.

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“…We interpret various prior results in the language of tensor rank. For the matrix multiplication (which corresponds to a tensor of size [n 2 ] × [n 2 ] × [n 2 ]), Shpilka [Shp03] showed that the tensor rank is at least 3n 2 − o(n 2 ) over F 2 , and Bläser [Blä99] earlier showed that over any field the tensor rank is at least 2.5n 2 − Θ(n). For polynomial multiplication (which corresponds to a tensor of size [2n Kaminski [Kam05] showed that the tensor rank over F q is known to be (3 + 1/Θ(q 3 ))n − o(n) and earlier work by Brown and Dobkin [BD80] showed that over F 2 the tensor rank is at least 3.52n.…”

Section: Prior Workmentioning

confidence: 99%

Tensor Rank: Some Lower and Upper Bounds

Alexeev

Forbes²,

Tsimerman

2011

2011 IEEE 26th Annual Conference on Computational Complexity

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Abstract. The results of Strassen [Str73] and Raz [Raz10] show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds.We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T : [n] d → F with rank at least 2n ⌊d/2⌋ + n − Θ(d lg n). This matches (over F2) or improves (all other fields) known lower bounds for d = 3 and improves (over any field) for odd d > 3.We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by counting, that there exists an order-3 permutation tensor with super-linear rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over large fields F, showing (among other things) that rank. We also show that if this upper bound is tight, then super-linear tensor rank lower bounds would follow. The second upper bound uses interpolation and only works for abelian G, showing that over any field F that rankIn either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank.We also explore monotone tensor rank. We give explicit 0/1 tensors T : [n] d → F that have tensor rank at most dn but have monotone tensor rank exactly n d−1 . This is a nearly optimal separation.

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A 5/2n/sup 2/-lower bound for the rank of n×n-matrix multiplication over arbitrary fields

Abstract: We prove a lower bound of ¾ Ò ¾ ¿Ò for the rank of Ò ¢Ò-matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.

Cited by 29 publications

References 12 publications

Arithmetic Circuits: A survey of recent results and open questions

Arithmetic Circuits: A survey of recent results and open questions

Unifying Known Lower Bounds via Geometric Complexity Theory

Tensor Rank: Some Lower and Upper Bounds

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