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A Polynomial Degree Bound on Equations of Non-rigid Matrices and Small Linear Circuits [article]

Mrinal Kumar, Ben Lee Volk
2020 arXiv   pre-print
Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [SV15] to construct low degree "universal" maps for non-rigid matrices and small linear circuits.  ...  We show that there is a defining equation of degree at most 𝗉𝗈𝗅𝗒(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field 𝔽, there is a  ...  This argument provides a (different) equation of polynomial degree for each irreducible component of the variety of non-rigid matrices. ♦ Degree Upper bound for Matrices with a Small Circuit In this  ... 
arXiv:2003.12938v2 fatcat:bmordmypzvbvlcnk332sd4z6ly

Lower Bounds for Matrix Factorization

Mrinal Kumar, Ben Lee Volk, Shubhangi Saraf
2020 Computational Complexity Conference  
circuits which compute linear transformations, matrix rigidity and data structure lower bounds.  ...  This improves upon the prior best lower bounds for this problem, which are barely super-linear, and were obtained by a long line of research based on the study of super-concentrators (albeit at the cost  ...  Quite informally, the intuition behind their lower bounds is that all small bounded depth linear circuits can be described as lying in the image of a low-degree polynomial map in a small number of variables  ... 
doi:10.4230/lipics.ccc.2020.5 dblp:conf/coco/0001V20 fatcat:5drgkcyleza4bjir5s2ryge57u

Lower Bounds for Matrix Factorization [article]

Mrinal Kumar, Ben Lee Volk
2019 arXiv   pre-print
circuits which compute linear transformations, matrix rigidity and data structure lower bounds.  ...  This improves upon the prior best lower bounds for this problem, which are barely super-linear, and were obtained by a long line of research based on the study of super-concentrators (albeit at the cost  ...  Acknowledgements We thank Swastik Kopparty for an insightful discussion on explicit construction of Sidon sets over finite fields.  ... 
arXiv:1904.01182v1 fatcat:7rswr2curbacxlx37ua53awwji

Probabilistic Rank and Matrix Rigidity [article]

Josh Alman, Ryan Williams
2017 arXiv   pre-print
We give surprising upper bounds on the rigidity of a family of matrices whose rigidity has been extensively studied, and was conjectured to be highly rigid.  ...  We also show non-trivial rigidity upper bounds for H_n with smaller target rank. Matrix Rigidity and Threshold Circuit Lower Bounds.  ...  s conjectures and results on non-rigidity at Banff (BIRS) in August 2016.  ... 
arXiv:1611.05558v2 fatcat:lqij2wpjzrduxoa3g5iswj6q2a

Recent Progress on Matrix Rigidity – A Survey [article]

C.Ramya
2020 arXiv   pre-print
In this survey, we present a selected set of results that highlight recent progress on matrix rigidity and its remarkable connections to other areas in theoretical computer science.  ...  Although we know rigid matrices exist, obtaining explicit constructions of rigid matrices have remained a long-standing open question.  ...  Acknowledgements I am grateful to Ramprasad Saptharishi for introducing to me the concept of matrix rigidity.  ... 
arXiv:2009.09460v1 fatcat:jq73yxwe75czhp7dbvdeymzldm

Complexity of Linear Circuits and Geometry

Fulvio Gesmundo, Jonathan D. Hauenstein, Christian Ikenmeyer, J. M. Landsberg
2015 Foundations of Computational Mathematics  
of matrices that are expected to have super-linear rigidity, and (4) prove results about the ideals and degrees of cones that are of interest in their own right.  ...  In particular, we (1) exhibit many non-obvious equations testing for (border) rigidity, (2) compute degrees of varieties associated with rigidity, (3) describe algebraic varieties associated with families  ...  Acknowledgments We thank the anonymous referees for very careful reading and numerous useful suggestions.  ... 
doi:10.1007/s10208-015-9258-8 fatcat:yfbantsj2bfzfnyzilreo3x2cm

On the Existence of Algebraically Natural Proofs [article]

Prerona Chatterjee, Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, Anamay Tengse
2021 arXiv   pre-print
For every constant c > 0, we show that there is a family P_N, c of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, that satisfies the following  ...  Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes.  ...  We also thank Ben Lee Volk for a careful reading of the paper and insightful comments which helped improve the presentation.  ... 
arXiv:2004.14147v2 fatcat:4ikspq4ebjeuxnscucjccfx4by

Complexity of linear circuits and geometry [article]

Fulvio Gesmundo, Jonathan Hauenstein, Christian Ikenmeyer, JM Landsberg
2015 arXiv   pre-print
We (i) exhibit many non-obvious equations testing for (border) rigidity, (ii) compute degrees of varieties associated to rigidity, (iii) describe algebraic varieties associated to families of matrices  ...  that are expected to have super-linear rigidity, and (iv) prove results about the ideals and degrees of cones that are of interest in their own right.  ...  Given a polynomial P on the space of n × n matrices that vanishes on all matrices of low rigidity (complexity), and a matrix A such that P (A) = 0, one obtains a lower bound on the rigidity (complexity  ... 
arXiv:1310.1362v2 fatcat:oxe5yy6xcnbzdngmda3petvlfu

Matrix rigidity of random toeplitz matrices

Oded Goldreich, Avishay Tal
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
This improves, for r = o(n/ log n log log n), over the Ω( n 2 r · log( n r )) bound that is known for many explicit matrices.  ...  We prove that random n-by-n Toeplitz (alternatively Hankel) matrices over F 2 have rigidity Ω( n 3 r 2 log n ) for rank r ≥ √ n, with high probability.  ...  Matrix Rigidity The "Matrix Rigidity Problem" (i.e., providing explicit matrices of high rigidity) is one of the most alluring problems in arithmetic circuits lower bounds.  ... 
doi:10.1145/2897518.2897633 dblp:conf/stoc/GoldreichT16 fatcat:xhkbzqr7mjf6be6embtsvnqd5e

Matrix rigidity of random Toeplitz matrices

Oded Goldreich, Avishay Tal
2016 Computational Complexity  
This improves, for r = o(n/ log n log log n), over the Ω( n 2 r · log( n r )) bound that is known for many explicit matrices.  ...  We prove that random n-by-n Toeplitz (alternatively Hankel) matrices over F 2 have rigidity Ω( n 3 r 2 log n ) for rank r ≥ √ n, with high probability.  ...  Matrix Rigidity The "Matrix Rigidity Problem" (i.e., providing explicit matrices of high rigidity) is one of the most alluring problems in arithmetic circuits lower bounds.  ... 
doi:10.1007/s00037-016-0144-9 fatcat:jwuajua4nnajlojbcchrucuzhe

Matrix Rigidity from the Viewpoint of Parameterized Complexity

Fedor V. Fomin, Daniel Lokshtanov, S. M. Meesum, Saket Saurabh, Meirav Zehavi
2018 SIAM Journal on Discrete Mathematics  
Rigidity is a classical concept in Computational Complexity Theory: constructions of rigid matrices are known to imply lower bounds of significant importance relating to arithmetic circuits.  ...  The steps of procedure Column-Reduction are all computable in polynomial time, and therefore Matrix-Reduction runs in polynomial time. We now prove the desired properties one by one.  ...  Valiant [22] presented the notion of the rigidity of a matrix as a means to prove lower bounds for linear algebraic circuits.  ... 
doi:10.1137/17m112258x fatcat:wbhtcizaxna3vegoyshtwxwzlu

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
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, Baur-Strassen  ...  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.  ...  Landsberg, Ketan Mulmuley, Toni Pitassi, Peter Scheiblechner, Chris Umans, Alasdair Urquhart, Ryan Williams, and Yiwei She for useful discussions. In particular, Williams suggested the  ... 
doi:10.1109/ccc.2014.35 dblp:conf/coco/Grochow14 fatcat:z5eytgp64jh2pclri4tlmky53e

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2015 Computational Complexity  
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, Baur-Strassen  ...  This enables us to expose a new viewpoint on GCT, whereby it is a natural unification and broad generalization of known results.  ...  Landsberg, Ketan Mulmuley, Toni Pitassi, Peter Scheiblechner, Chris Umans, Alasdair Urquhart, Ryan Williams, and Yiwei She for useful discussions. In particular, Williams suggested the  ... 
doi:10.1007/s00037-015-0103-x fatcat:ifoaveduubhmxlokr4pygqzjqq

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications [article]

Vishwas Bhargava, Sumanta Ghosh, Mrinal Kumar, Chandra Kanta Mohapatra
2022 arXiv   pre-print
and this algebraic structure naturally leads to the applications to data structure upper bounds for polynomial evaluation and to an upper bound on the rigidity of Vandermonde matrices.  ...  In a significant improvement to the state of art for this problem, Umans and Kedlaya & Umans gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields  ...  We also thank Ben Lund for helpful discussions and references on the finite fields Kakeya problem and for pointing us to the relevant literature on Furstenberg sets, both of which indirectly played a role  ... 
arXiv:2111.07572v3 fatcat:3uvva5p6lvaxnfpxltofxlwtj4

Learning Complexity vs. Communication Complexity

Nati Linial, Adi Shraibman
2008 2008 23rd Annual IEEE Conference on Computational Complexity  
This leads to the question of proving lower bounds on the rigidity of margin complexity.  ...  In the same way that matrix rigidity is related to rank, we introduce the notion of rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare.  ...  Now, AC 0 circuits are well approximated by low degree polynomials, as proved by Tarui [30] . Let C be an AC 0 circuit of size 2 polylog(m) acting on 2 m × 2 m 0 − 1 matrices φ 1 , . . . , φ s .  ... 
doi:10.1109/ccc.2008.28 dblp:conf/coco/LinialS08 fatcat:ressotde4rfq7cajswgvmdvvfm
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