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Variety Membership Testing, Algebraic Natural Proofs, and Geometric Complexity Theory [article]

Markus Bläser, Christian Ikenmeyer, Vladimir Lysikov, Anurag Pandey, Frank-Olaf Schreyer
2019 arXiv   pre-print
Our next result is the -hardness of membership testing in the minrank variety, hence we establish the -hardness of the orbit closure containment problem for 3-tensors.  ...  We study the variety membership testing problem in the case when the variety is given as an orbit closure and the ambient space is the set of all 3-tensors.  ...  Next, we phrase the slice rank problem in terms of orbit closures.  ... 
arXiv:1911.02534v1 fatcat:se5hnnuknvdfxeq4mgpkqaupx4

The ideal of the trifocal variety

Chris Aholt, Luke Oeding
2014 Mathematics of Computation  
Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety.  ...  An effective test for determining whether a given tensor is a trifocal tensor is also given.  ...  Acknowledgements We would like to thank Bernd Sturmfels for suggesting this problem to us, along with some suggestions for a few of the computations.  ... 
doi:10.1090/s0025-5718-2014-02842-1 fatcat:rgq7yjevi5a3nccniufatbu6ca

Tensor Rank, Invariants, Inequalities, and Applications

Elizabeth S. Allman, Peter D. Jarvis, John A. Rhodes, Jeremy G. Sumner
2013 SIAM Journal on Matrix Analysis and Applications  
subsets, one of which contains tensors of real rank n.  ...  Here we focus on the n × n × n tensors of rank n over C, which has as a dense subset the orbit of a single tensor under a natural group action.  ...  Much of the research in this work was conducted while ESA and JAR were in residence at the University of Tasmania as visiting scholars; they thank the university and the School of Mathematics and Physics  ... 
doi:10.1137/120899066 fatcat:lhxqarjomzfx3hcbnhlpq4idlu

Tensor Rank, Invariants, Inequalities, and Applications [article]

Elizabeth S. Allman, Peter D. Jarvis, John A. Rhodes, Jeremy G. Sumner
2012 arXiv   pre-print
subsets, one of which contains tensors of real rank n.  ...  Here we focus on the n× n× n tensors of rank n over C, which has as a dense subset the orbit of a single tensor under a natural group action.  ...  Finally, note that the closures of D(C) and D(R) also contain tensors of rank > n.  ... 
arXiv:1211.3461v1 fatcat:jspsnsmpfzbddnu5p4okqwlrsq

Weighted Slice Rank and a Minimax Correspondence to Strassen's Spectra [article]

Matthias Christandl, Vladimir Lysikov, Jeroen Zuiddam
2021 arXiv   pre-print
Structural and computational understanding of tensors is the driving force behind faster matrix multiplication algorithms, the unraveling of quantum entanglement, and the breakthrough on the cap set problem  ...  Our work advances and makes novel connections among two recent developments in the study of tensors, namely (1) the slice rank of tensors, a notion of rank for tensors that emerged from the resolution  ...  A tensor T ∈ V 1 ⊗ V 2 ⊗ V 3 is called semistable if the Zariski closure 7 of the orbit SL(V 1 ) × SL(V 2 ) × SL(V 3 ) • T does not contain 0.  ... 
arXiv:2012.14412v2 fatcat:klmtilalyvebhcy32ptdifry6m

Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory [article]

Peter Bürgisser and Ankit Garg and Rafael Oliveira and Michael Walter and Avi Wigderson
2017 arXiv   pre-print
Its main feature is that the local optimization domains are each a group of invertible matrices, together naturally acting on tensors, and the optimization problem is minimizing the norm of an input tensor  ...  This directly leads to progress on some of the problems in the areas above, and a unified view of others.  ...  MW acknowledges financial support by the NWO through Veni grant no. 680-47-459. PB acknowledges financial support from the DFG grant BU 1371 2-2.  ... 
arXiv:1711.08039v1 fatcat:mjkmfuo63jcy3hwgimur4dgpuu

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

Alwin Stegeman, Shmuel Friedland
2016 Linear and multilinear algebra  
For R=2 and real order-3 tensors it is shown that a best rank-2 approximation is also a local minimum of the best rank-(2,2,2) approximation problem.  ...  It is well known that a best rank-R approximation of order-3 tensors may not exist for R> 2.  ...  In that case one still has to compute a best approximation from the closure of the rank-2 set and decide whether it has rank 2 or 3.  ... 
doi:10.1080/03081087.2016.1234578 fatcat:el6iggrg25eqvmbnmzquvghygy

Maximum likelihood estimation of the Latent Class Model through model boundary decomposition [article]

Elizabeth S. Allman, Hector Baños Cervantes, Robin Evans, Serkan Hoşten, Kaie Kubjas, Daniel Lemke, John A. Rhodes, Piotr Zwiernik
2018 arXiv   pre-print
Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function  ...  We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator.  ...  For example, if a (1) 11 = 0 then a (1) 12 = 1, and the slice (p 1j 2 ···jn ) of the tensor P is a binary tensor of rank one.  ... 
arXiv:1710.01696v2 fatcat:bapvsnt47zfehi5lqysq4jpp64

Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective [article]

Pascal Koiran
2019 arXiv   pre-print
The main open problem that arises from this work is to obtain a complete description of the closures. This question is akin to that of characterizing border rank of tensors in algebraic complexity.  ...  , and we begin a study of their closures.  ...  Acknowledgements Nicolas Ressayre made some useful comments on an early version of this paper. I would also like to thank Kevin O'Meara for the encouragements, and Roger Horn  ... 
arXiv:1905.05094v3 fatcat:37qm6zwxfffxdgngsrc3azcr6a

Maximum likelihood estimation of the Latent Class Model through model boundary decomposition

Elizabeth Allman, Hector Banos Cervantes, Serkan Hosten, Kaie Kubjas, Daniel Lemke, John Rhodes, Piotr Zwiernik
2019 Journal of Algebraic Statistics  
Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function  ...  We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator.  ...  For example, if a (1) 11 = 0 then a (1) 12 = 1, and the slice (p 1j 2 ···jn ) of the tensor P is a binary tensor of rank one.  ... 
doi:10.18409/jas.v10i1.75 fatcat:abhnkdilnfcjfokpyk3eaqurt4

Explicit lower bounds via geometric complexity theory

Peter Bürgisser, Christian Ikenmeyer
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
We prove the lower bound R(Mm) ≥ 3 2 m 2 − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense the geometric complexity  ...  This description results from analyzing the process of polarization and Schur-Weyl duality.  ...  This work contains results from the PhD thesis of the second author [13] . ORBIT CLOSURE PROBLEMS Border Rank Consider W := 3 C m 2 .  ... 
doi:10.1145/2488608.2488627 dblp:conf/stoc/BurgisserI13 fatcat:2r2mz7zuwfcn5ajfyudmoketdi

Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras

Markus Bläser, Vladimir Lysikov, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science  
To obtain our result, we relate irreversibility to asymptotic slice rank and instability of tensors and prove that the instability of block tensors can often be decided by looking only on the sizes of  ...  Determining the exponent of matrix multiplication ω is one of the central open problems in algebraic complexity theory.  ...  A tensor t ∈ V 1 ⊗ V 2 ⊗ V 3 is called unstable if 0 is contained in the (Zariski) closure of the SL(V 1 ) × SL(V 2 ) × SL(V 3 ) orbit of t, and semistable otherwise.  ... 
doi:10.4230/lipics.mfcs.2020.17 dblp:conf/mfcs/BlaserL20 fatcat:z4czwp3jtjcntlytr6qqydvhje

Geometric rank of tensors and subrank of matrix multiplication [article]

Swastik Kopparty, Guy Moshkovitz, Jeroen Zuiddam
2020 arXiv   pre-print
We prove that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs.  ...  We prove that the geometric rank is smaller than the slice rank of Tao, and relate geometric rank to the analytic rank of Gowers and Wolf in an asymptotic fashion.  ...  The set {T | GR(T ) ≤ GR(T )} is Zariski closed by Lemma 5.3. It contains the orbit G · T and hence also its Zariski closure G · T , that is, {T | T T } = G · T ⊆ {T | GR(T ) ≤ GR(T )}.  ... 
arXiv:2002.09472v2 fatcat:b37voq5wyngnnhrqz3rmw5humi

The Higgs Mechanism – Hasse Diagrams for Symplectic Singularities [article]

Antoine Bourget, Santiago Cabrera, Julius F. Grimminger, Amihay Hanany, Marcus Sperling, Anton Zajac, Zhenghao Zhong
2019 arXiv   pre-print
Most of the Hasse diagrams we obtain extend beyond the cases of nilpotent orbit closures known in the mathematics literature.  ...  Higgs branch by a Hasse diagram with symplectic leaves and transverse slices, thus refining the analysis and extending it to non-Lagrangian theories.  ...  Acknowledgements We are indebted to Gabi Zafrir for many insightful discussions and clarifications.  ... 
arXiv:1908.04245v2 fatcat:2wuzf4pr45gvlfrlrjyrdcbdvi

Recent progress on scaling algorithms and applications [article]

Ankit Garg, Rafael Oliveira
2018 arXiv   pre-print
algorithms for the scaling problems.  ...  Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering.  ...  Acknowledgements We thank Vikraman Arvind for inviting us to write this survey for the EATCS complexity column (June 2018 issue) and Avi Wigderson for providing helpful comments on an earlier version of  ... 
arXiv:1808.09669v1 fatcat:eohzepzccfhandivheqyvslej4
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