# On Multilinear Forms: Bias, Correlation, and Tensor Rank

@inproceedings{Bhrushundi2018OnMF,
title={On Multilinear Forms: Bias, Correlation, and Tensor Rank},
author={Abhishek Bhrushundi and Prahladh Harsha and Pooya Hatami and Swastik Kopparty and Mrinal Kumar},
booktitle={Electron. Colloquium Comput. Complex.},
year={2018}
}
• Published in
Electron. Colloquium Comput…
1 April 2018
• Mathematics, Computer Science
In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors. 1. Correlation bounds : We show that a random $d$-linear form has exponentially low correlation with low-degree polynomials. More precisely, for $d \ll 2^{o(k)}$, we show that a random $d$-linear form $f(X_1,X_2, \dots, X_d) : \left(GF(2… 6 Citations The analytic rank of tensors and its applications The analytic rank of tensors and its applications, Discrete Analysis 2019:7, 10 pp. There are several arguments in additive combinatorics concerning two-variable functions taking values in a field Geometric rank of tensors and subrank of matrix multiplication • Mathematics, Computer Science Electron. Colloquium Comput. Complex. • 2020 It is proved that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs, and it is related to the analytic rank of Gowers and Wolf in an asymptotic fashion. Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors • Mathematics, Computer Science ArXiv • 2022 This paper proves for a generic bilinear map T : V × V → V where dim( V ) = n that θ ( √ n ) independent scalar multiplications can be reduced to T . C O ] 1 4 O ct 2 01 8 The analytic rank of tensors and its applications The analytic rank of a tensor, first defined by Gowers and Wolf in the context of higher-order Fourier analysis, is defined to be the logarithm of the bias of the tensor. We prove that it is a Subspaces of tensors with high analytic rank It is shown that for any subspace V⊆Fn×⋯×np of d-tensors, if dim(V)≥tnd−1, then there is subspace W⊆V of dimension at least t/(dr)−1 whose nonzero elements all have analytic rank Ωd,p(r). As an SUBSPACES OF TENSORS WITH HIGH ANALYTIC RANK It is shown that for any subspace V ⊆ Fn×···×n p of d-tensors, if dim(V ) ≥ tnd−1, then there is subspace W ⊆ V of dimension t/(dr) − 1 whose nonzero elements all have analytic rank Ωd,p(r). As an ## References SHOWING 1-10 OF 26 REFERENCES Tensor Rank: Some Lower and Upper Bounds • Computer Science, Mathematics 2011 IEEE 26th Annual Conference on Computational Complexity • 2011 This work explores tensor rank lower and upper bounds, focusing on explicit tensors, and 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. The Chasm at Depth Four, and Tensor Rank : Old results, new insights • Computer Science, Mathematics Electron. Colloquium Comput. Complex. • 2016 This paper abstracts the main ingredient to apply it to formulas and constant depth circuits, and shows more structured depth reductions for them, and extends this result for homogeneous formulas to show that the connection holds for any$d$such that$\omega(1) d \leq n^{o(1)}$. The analytic rank of tensors is subadditive, and its applications It is proved that the analytic rank of a tensor lower bounds the slice rank and partition rank of an tensor, quantities that arise in the study of Ramsey problems in product spaces and suggests a new approach to study some open problems in this domain. Random low-degree polynomials are hard to approximate • Mathematics, Computer Science computational complexity • 2011 It is proved that almost all degree d polynomials have only an exponentially small correlation with all polynmials of degree at most d − 1, for all degrees d up to Θ(n). Lower bounds for matrix product • Amir Shpilka • Computer Science, Mathematics Proceedings 2001 IEEE International Conference on Cluster Computing • 2001 We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two n /spl times/ n matrices over finite fields. In particular we obtain the A Lower Bound on the Complexity of Polynomial Multiplication over Finite Fields • M. Kaminski • Mathematics, Computer Science SIAM J. Comput. • 2005 It is shown that computing the coefficients of the product of two degree-n polynomials over a q-element field by means of a quadratic algorithm requires at least$(3+
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