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Improved upper bounds for the rigidity of Kronecker products [article]

Bohdan Kivva
2021 arXiv   pre-print
At MFCS'77, Valiant introduced matrix rigidity as a tool to prove circuit lower bounds for linear functions and since then this notion received much attention and found applications in other areas of complexity  ...  In a recent paper (STOC'21), Alman gave a general non-rigidity result for such matrices: he showed that if an n× n matrix A (over any field) is a Kronecker product of d× d matrices M_1,..., M_k (so n=d  ...  Acknowledgments The author is grateful to his advisor László Babai for helpful discussions, his help in improving the organization of the paper, and for pointing out improvements to the results and  ... 
arXiv:2103.05631v2 fatcat:frualfjqv5cufo6yjr3t765jna

Lower Bounds for Matrix Factorization [article]

Mrinal Kumar, Ben Lee Volk
2019 arXiv   pre-print
which compute linear transformations, matrix rigidity and data structure lower bounds.  ...  In other words, any depth-d linear circuit computing the linear transformation M_n· x has size at least n^1+Ω(1/d).  ...  We also thank Rohit Gurjar, Nutan Limaye, Srikanth Srinivasan and Joel Tropp for helpful discussions.  ... 
arXiv:1904.01182v1 fatcat:7rswr2curbacxlx37ua53awwji

Arithmetic Circuits, Structured Matrices and (not so) Deep Learning [article]

Atri Rudra
2022 arXiv   pre-print
This survey presents a necessarily incomplete (and biased) overview of results at the intersection of arithmetic circuit complexity, structured matrices and deep learning.  ...  Most of this work has been experimental and in this survey, we formalize the research question and show how a recent work that combines arithmetic circuit complexity, structured matrices and deep learning  ...  Acknowledgments The material in Sections 2 and 3 are based on notes for AR's Open lectures for PhD students in computer science at University of Warsaw titled (Dense Structured) Matrix Vector Multiplication  ... 
arXiv:2206.12490v1 fatcat:ismimo43s5hpfdp7kiwzywei24

Fourier and Circulant Matrices Are Not Rigid

Zeev Dvir, Allen Liu, Michael Wagner
2019 Computational Complexity Conference  
The fourth and final part involves using the non-rigidity of the Fourier matrix to show that the group algebra matrix, given by Mxy = f (x − y) for x, y ∈ G, is not rigid for any function f and abelian  ...  In particular, we get that complex valued Fourier matrices, circulant matrices, and Toeplitz matrices are all not rigid and cannot be used to carry out Valiant's approach to proving circuit lower bounds  ...  On the other hand, this suggests that perhaps M G (f ) is a candidate for rigidity when all irreducible representations of G are large (for instance quasi-random groups [6] ).  ... 
doi:10.4230/lipics.ccc.2019.17 dblp:conf/coco/DvirL19 fatcat:4huwoz2l4jex7h7qplpwp4famm

On the smallest possible dimension and the largest possible margin of linear arrangements representing given concept classes

Jürgen Forster, Hans Ulrich Simon
2006 Theoretical Computer Science  
In addition, we present new results on the rigidity of matrices and briefly mention applications in complexity and learning theory.  ...  In particular, we study the embeddability of a given concept class into a class of Euclidean half spaces of low dimension, or of arbitrarily large dimension but realizing a large margin.  ...  The Hadamard matrix H n can be written as the nth Kronecker power of the matrix H 1 :  ... 
doi:10.1016/j.tcs.2005.10.015 fatcat:ppngiujzkver3mksbkt75sat4e

Fourier and Circulant Matrices are Not Rigid [article]

Zeev Dvir, Allen Liu
2021 arXiv   pre-print
In particular, we get that complex valued Fourier matrices, circulant matrices, and Toeplitz matrices are all not rigid and cannot be used to carry out Valiant's approach to proving circuit lower bounds  ...  In a surprising result, Alman and Williams (FOCS'19) showed that the (real valued) Hadamard matrix, which was conjectured to be rigid, is actually not very rigid.  ...  When G is nonabelian, it is no longer possible to diagonalize the matrix M G (f ) but there is a change of basis matrix A such that AM G (f )A * is block-diagonal where the diagonal blocks correspond to  ... 
arXiv:1902.07334v3 fatcat:dxmwlnwt25e6biq4yusexkovhq

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

Vishwas Bhargava, Sumanta Ghosh, Mrinal Kumar, Chandra Kanta Mohapatra
2022 arXiv   pre-print
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.  ...  And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck , fast algorithms for the multivariate  ...  and for much encouragement.  ... 
arXiv:2111.07572v3 fatcat:3uvva5p6lvaxnfpxltofxlwtj4

On P vs. NP, Geometric Complexity Theory, Explicit Proofs and the Complexity Barrier [article]

Ketan D. Mulmuley
2009 arXiv   pre-print
NP and related problems. This article gives its complexity theoretic overview without assuming any background in algebraic geometry or representation theory.  ...  circuit of polylog(N ) depth and 2 polylog(N) size.  ...  Locally strongly explicit in this setting would mean computable by a constant depth circuit (or in a relaxed sense, say log log n depth circuit of 2 polylog(n) size).  ... 
arXiv:0908.1932v2 fatcat:ym652k52sjgxdf7klxx3eljney

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2015 Computational Complexity  
matrix rigidity (Valiant) and others.  ...  ), lower bounds on matrix multiplication (Bürgisser-Ikenmeyer) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi; Agrawal-Vinay; Koiran),  ...  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

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
, Agrawal-Vinay, Koiran, Tavenas), matrix rigidity (Valiant) and others.  ...  , Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (Bürgisser-Ikenmeyer, Landsberg-Ottaviani) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi  ...  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

SPA: Sparse Photorealistic Animation Using a Single RGB-D Camera

Kun Li, Jingyu Yang, Leijie Liu, Ronan Boulic, Yu-Kun Lai, Yebin Liu, Yubin Li, Eray Molla
2016 IEEE transactions on circuits and systems for video technology (Print)  
Photorealistic animation is a desirable technique for computer games and movie production.  ...  An adaptive model-guided texture synthesis method based on weighted low-rank matrix completion is proposed to be less sensitive to noise and outliers, which enables us to easily create photorealistic animation  ...  Xu for his helpful discussions and the distribution of data sets.  ... 
doi:10.1109/tcsvt.2016.2556419 fatcat:yw3node53ja6zcdqalrfgf3jl4

Proto-value Functions: A Laplacian Framework for Learning Representation and Control in Markov Decision Processes

Sridhar Mahadevan, Mauro Maggioni
2007 Journal of machine learning research  
continuous state spaces, including the Nyström extension for out-of-sample interpolation of eigenfunctions, and the use of Kronecker sum factorization to construct compact eigenfunctions in product spaces  ...  such as factored MDPs (vi) Finally, a series of illustrative discrete and continuous control tasks, which both illustrate the concepts and provide a benchmark for evaluating the proposed approach.  ...  This research also builds on recent work on manifold and spectral learning, including diffusion maps (Coifman et al., 2005a,b,c), ISOMAP (Tenenbaum et al., 2000) , LLE (Roweis and Saul, 2000) , and Laplacian  ... 
dblp:journals/jmlr/MahadevanM07 fatcat:ktren6grengbtlafouhzepw22u

Routed quantum circuits

Augustin Vanrietvelde, Hlér Kristjánsson, Jonathan Barrett
2021 Quantum  
This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces.  ...  We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits.  ...  In this picture, sequential composition is given by matrix products, parallel composition by tensor products of matrices, and taking the opposite relation corresponds to matrix transposition.  ... 
doi:10.22331/q-2021-07-13-503 fatcat:xr5asptqovhxhaukiqgyabh62y

Routed quantum circuits [article]

Augustin Vanrietvelde, Hlér Kristjánsson, Jonathan Barrett
2021 arXiv   pre-print
This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces.  ...  We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits.  ...  In this picture, sequential composition is given by matrix products, parallel composition by tensor products of matrices, and taking the opposite relation corresponds to matrix transposition.  ... 
arXiv:2011.08120v2 fatcat:zsntjrlfg5aqzlra5olqejzc7i

Robust Affordable 3D Haptic Sensation via Learning Deformation Patterns [article]

Huanbo Sun, Goerg Martius
2019 arXiv   pre-print
We compare data-driven approaches with model-based methods relying on geometric distance and information criteria such as Entropy and Mutual Information.  ...  Using machine learning techniques, we optimize the sensor number and their placement and are able to obtain high-precision force inference for a robotic limb using as few as 9 sensors.  ...  We assume that the surface is constructed such that it has a inside rigid support and a flexible outer shell.  ... 
arXiv:1902.09241v1 fatcat:mirr2nvexjfubgix6u5m76wxvy
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