A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2020; you can also visit the original URL.
The file type is `application/pdf`

.

## Filters

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

2019
*
arXiv
*
pre-print

Our next result is

arXiv:1911.02534v1
fatcat:se5hnnuknvdfxeq4mgpkqaupx4
*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*. ...##
###
The ideal of the trifocal variety

2014
*
Mathematics of Computation
*

Techniques from representation theory, symbolic computational algebra,

doi:10.1090/s0025-5718-2014-02842-1
fatcat:rgq7yjevi5a3nccniufatbu6ca
*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. ...##
###
Tensor Rank, Invariants, Inequalities, and Applications

2013
*
SIAM Journal on Matrix Analysis and Applications
*

subsets,

doi:10.1137/120899066
fatcat:lhxqarjomzfx3hcbnhlpq4idlu
*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 ...##
###
Tensor Rank, Invariants, Inequalities, and Applications
[article]

2012
*
arXiv
*
pre-print

subsets,

arXiv:1211.3461v1
fatcat:jspsnsmpfzbddnu5p4okqwlrsq
*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. ...##
###
Weighted Slice Rank and a Minimax Correspondence to Strassen's Spectra
[article]

2021
*
arXiv
*
pre-print

Structural

arXiv:2012.14412v2
fatcat:klmtilalyvebhcy32ptdifry6m
*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. ...##
###
Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory
[article]

2017
*
arXiv
*
pre-print

Its main feature is that

arXiv:1711.08039v1
fatcat:mjkmfuo63jcy3hwgimur4dgpuu
*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. ...##
###
On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

2016
*
Linear and multilinear algebra
*

For R=2

doi:10.1080/03081087.2016.1234578
fatcat:el6iggrg25eqvmbnmzquvghygy
*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. ...##
###
Maximum likelihood estimation of the Latent Class Model through model boundary decomposition
[article]

2018
*
arXiv
*
pre-print

Our theoretical study is complemented with a careful analysis

arXiv:1710.01696v2
fatcat:bapvsnt47zfehi5lqysq4jpp64
*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*. ...##
###
Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective
[article]

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 ...

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

2019
*
Journal of Algebraic Statistics
*

Our theoretical study is complemented with a careful analysis

doi:10.18409/jas.v10i1.75
fatcat:abhnkdilnfcjfokpyk3eaqurt4
*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*. ...##
###
Explicit lower bounds via geometric complexity theory

2013
*
Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13
*

We prove

doi:10.1145/2488608.2488627
dblp:conf/stoc/BurgisserI13
fatcat:2r2mz7zuwfcn5ajfyudmoketdi
*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 . ...##
###
Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras

2020
*
International Symposium on Mathematical Foundations of Computer Science
*

To obtain our result, we relate irreversibility to asymptotic

doi:10.4230/lipics.mfcs.2020.17
dblp:conf/mfcs/BlaserL20
fatcat:z4czwp3jtjcntlytr6qqydvhje
*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. ...##
###
Geometric rank of tensors and subrank of matrix multiplication
[article]

2020
*
arXiv
*
pre-print

We prove that

arXiv:2002.09472v2
fatcat:b37voq5wyngnnhrqz3rmw5humi
*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 )}. ...##
###
The Higgs Mechanism – Hasse Diagrams for Symplectic Singularities
[article]

2019
*
arXiv
*
pre-print

Most

arXiv:1908.04245v2
fatcat:2wuzf4pr45gvlfrlrjyrdcbdvi
*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. ...##
###
Recent progress on scaling algorithms and applications
[article]

2018
*
arXiv
*
pre-print

algorithms for

arXiv:1808.09669v1
fatcat:eohzepzccfhandivheqyvslej4
*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*...
« Previous

*Showing results 1 — 15 out of 1,930 results*