Structure vs. randomness for bilinear maps

Cohen, Alex; Moshkovitz, Guy

doi:10.1145/3406325.3451007

Cited by 9 publications

(8 citation statements)

References 15 publications

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“…We let SR(T ) be as in Definition 2.2, except the coefficients are allowed to come from F. Clearly SR(T ) ≤ SR(T ). One also has 2SR(T ) ≤ 3SR(T ) which is proved in [CM21].…”

Section: Closing the Loop For 3-tensors: Cohen And Moshkovitz's Argumentmentioning

confidence: 76%

“…Cohen and Moshkovitz [CM21] proved the following two theorems which, for 3-tensors over finite fields with at least 3 elements, show asymptotic equivalence between AR, GR, and SR when combined with Lovett's result. Our goal will be to give an intuitive sketch of their argument.…”

Section: Closing the Loop For 3-tensors: Cohen And Moshkovitz's Argumentmentioning

confidence: 95%

“…In Section 4 we give a rundown of the trivial relationships between the notions that come from their definitions. We then give sketches of results by Lovett [Lov19] (Section 5) and Cohen and Moshkovitz [CM21] (Section 6) which, put together, prove asymptotic equivalence between three key notions of rank over finite fields with three or more elements.…”

Section: Introductionmentioning

confidence: 99%

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Notions of Tensor Rank

Juvekar¹,

Nadjimzadah²

2022

Preprint

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Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses. In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.

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“…We let SR(T ) be as in Definition 2.2, except the coefficients are allowed to come from F. Clearly SR(T ) ≤ SR(T ). One also has 2SR(T ) ≤ 3SR(T ) which is proved in [CM21].…”

Section: Closing the Loop For 3-tensors: Cohen And Moshkovitz's Argumentmentioning

confidence: 76%

Section: Closing the Loop For 3-tensors: Cohen And Moshkovitz's Argumentmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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Notions of Tensor Rank

Juvekar¹,

Nadjimzadah²

2022

Preprint

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“…Many strong connections have been shown among these parameters. In particular, Derksen [Der20] showed that the G-stable rank is equal to the slice rank up to a constant factor, and Cohen-Moshkovitz [CM21a,CM21b] showed (over large fields) that the partition rank, analytic rank and geometric rank are equal up to a constant factor, the culmination of a long line of work on this topic [GT09, KL08, BL15, Jan18, Jan20, Mil19]. All of the aforementioned tensor parameters are lower bounded by the subrank.…”

Section: Related Workmentioning

confidence: 99%

Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors

Derksen¹,

Makam²,

Zuiddam³

2022

Preprint

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Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps like matrix multiplication in Strassen's work, or the determinant and permanent polynomials in Valiant's) can be reduced to each other (under the appropriate notion of reduction).In this paper we work in the setting of bilinear maps and with the usual notion of reduction that allows applying linear maps to the inputs and output of a bilinear map in order to compute another bilinear map. As our main result we determine precisely how many independent scalar multiplications can be reduced to a given bilinear map (this number is called the subrank, and extends the concept of matrix diagonalization to tensors), for essentially all (i.e. generic) bilinear maps. Namely, we prove for a generic bilinear map T : V × V → V where dim(V ) = n that θ( √ n) independent scalar multiplications can be reduced to T . Our result significantly improves on the previous upper bound from the work of Strassen (1991) and Bürgisser (1990) which was n 2/3+o(1) . Our result is very precise and tight up to an additive constant. Our full result is much more general and applies not only to bilinear maps and 3-tensors but also to k-tensors, for which we find that the generic subrank is θ(n 1/(k−1) ). Moreover, as an application we prove that the subrank is not additive under the direct sum.The subrank plays a central role in several areas of complexity theory (matrix multiplication algorithms, barrier results) and combinatorics (e.g., the cap set problem and sunflower problem). As a consequence of our result we obtain several large separations between the subrank and tensor methods that have received much interest recently, notably the slice rank (Tao, 2016), analytic rank (Gowers-Wolf, 2011 Lovett, 2018; Bhrushundi-Harsha-Hatami-Kopparty-Kumar, 2020), geometric rank (Kopparty-Moshkovitz-Zuiddam, 2020), and G-stable rank (Derksen, 2020).Our proofs of the lower bounds rely on a new technical result about an optimal decomposition of tensor space into structured subspaces, which we think may be of independent interest.

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“…Let Q ∈ Q be a map of degree e, assume w.l.o.g. that Q : V [e] → F, and suppose (to get a contradiction) that we have a non-trivial linear combination (5) ω∈{0,1} l ,0<|ω|≤e…”

Section: Proposition 71 (Almost Surely Vanishing Implies Vanishing) T...mentioning

confidence: 99%

Relative Rank and Regularization

Lampert,

Ziegler

2021

Preprint

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We introduce a new notion of rank -relative rank associated to a filtered collection of polynomials. When the filtration consists of one set the notion coincides with the Schmidt rank (also called strength). We also introduce the notion of relative bias. We prove a relation between these two notions over finite fields. This allows us to perform a regularization procedure in a number of steps that is polynomial in the initial number of polynomials. As an application we prove that any collection of homogeneous polynomials P = (P i ) c i=1 of degrees ≤ d in a polynomial ring over an algebraically closed field of characteristic > d is contained in an ideal I(Q), generated by a collection Q of homogeneous polynomials of degrees ≤ d which form a regular sequence, and Q is of size ≤ Ac A , where A = A(d) is independent of the number of variables.

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Structure vs. randomness for bilinear maps

Cited by 9 publications

References 15 publications

Notions of Tensor Rank

Notions of Tensor Rank

Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors

Relative Rank and Regularization

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