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Cycles of Nonzero Elements in Low Rank Matrices

Pavel Pudlák
2002 Combinatorica  
In this paper we show that for a n n real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles.  ...  We consider the problem of nding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science.  ...  In general, low rank does not imply the existence of nonzero elements (consider the zero matrix).  ... 
doi:10.1007/s004930200015 fatcat:u2kfnz5iy5az7juupgtsfk3xiq

Some structural properties of low-rank matrices related to computational complexity

Bruno Codenotti, Pavel Pudlák, Giovanni Resta
2000 Theoretical Computer Science  
We consider the problem of the presence of short cycles in the graphs of nonzero elements of matrices which have sublinear rank and nonzero entries on the main diagonal, and analyze the connection between  ...  In particular, we exhibit a family of matrices which shows that sublinear rank does not imply the existence of triangles.  ...  We are especially interested in odd alternating cycles as subgraphs of the graph of nonzero entries of low-rank matrices, because of a connection to matrix rigidity.  ... 
doi:10.1016/s0304-3975(99)00185-1 fatcat:bg2ixeeiyrccrcz7hozc7jb43u

On symmetric matrices with indeterminate leading diagonals

A. V. Seliverstov, V. A. Lyubetsky
2009 Problems of Information Transmission  
In particular, we show that the number of connected components of the graph of the matrix of such a quadratic form does not change when one edge of the graph is deleted.  ...  We consider properties of the matrix of a real quadratic form that takes a constant value on a sufficiently large set of vertices of a multidimensional cube centered at the origin given that the corresponding  ...  Here "simple" means that in the general case is of a lower rank than . Note that heuristic or approximate (though efficient) optimization algorithms for matrices of low ranks are known [4] . 2.  ... 
doi:10.1134/s0032946009030065 fatcat:gy5q6xgc4zbtdhekazdskfmvzm

A class of structured quasi-cyclic LDPC codes based on planar difference families

Shady M. Ibraheem, M. M. Abd Elrazzak, Salwa M. Serag Eldin, W. Saad, Atef E. Aboelazm
2013 2013 International Conference on Advanced Technologies for Communications (ATC 2013)  
The resulting codes have parity check matrices with column-weight greater than three, at least no 4-cycle and approximately full rank.  ...  They are at least no 4-cycle classes of codes. It can be shown that there is an improvement in their minimum distances along with the increasing in the column-weight of their parity-check matrices.  ...  It can be seen that the two proposed codes perform almost exactly as well in the low SNR region. The benefits of the full rank one appear in the error floor region.  ... 
doi:10.1109/atc.2013.6698188 fatcat:llxll43jzzcfzdl6hjq6ukvlxm

Resolvable 2-designs for regular low-density parity-check codes

S.J. Johnson, S.R. Weller
2003 IEEE Transactions on Communications  
This paper extends the class of low-density paritycheck (LDPC) codes that can be algebraically constructed.  ...  We present regular LDPC codes based on resolvable Steiner 2-designs which have Tanner graphs free of four-cycles.  ...  Neal for his online repository of LDPCrelated software.  ... 
doi:10.1109/tcomm.2003.816946 fatcat:kgwy4b3pxrdkvaa2zg6yq54obi

Separators and Structure Prediction in Sparse Orthogonal Factorization

J Gilbert
1997 Linear Algebra and its Applications  
In this paper we make this folk theorem precise: we prove tight upper and lower hounds on the nonzero counts of the two representations in terms of *  ...  In the factorization A = QR of a sparse matrix A, the orthogonal matrix Q can be represented either explicitly (as a matrix) or implicitly (as a sequence of Householder vectors).  ...  Every matrix with full column rank admits a row permutation that makes the diagonal elements nonzero [17] . Throughout the paper, we assume that the diagonal elements of A are nonzero.  ... 
doi:10.1016/s0024-3795(96)00473-9 fatcat:jnd2x27vrvagxcgkdbnyqrpruu

Separators and structure prediction in sparse orthogonal factorization

John R. Gilbert, Esmond G. Ng, Barry W. Peyton
1997 Linear Algebra and its Applications  
In this paper we make this folk theorem precise: we prove tight upper and lower hounds on the nonzero counts of the two representations in terms of *  ...  In the factorization A = QR of a sparse matrix A, the orthogonal matrix Q can be represented either explicitly (as a matrix) or implicitly (as a sequence of Householder vectors).  ...  Every matrix with full column rank admits a row permutation that makes the diagonal elements nonzero [17] . Throughout the paper, we assume that the diagonal elements of A are nonzero.  ... 
doi:10.1016/s0024-3795(97)80024-9 fatcat:rvbyiimorfbqrmsfvdvghx6aq4

Design of Nonbinary Quasi-Cyclic LDPC Cycle Codes

Rong-Hui Peng, Rong-Rong Chen
2007 2007 IEEE Information Theory Workshop  
Our construction utilizes the cycle elimination algorithm to remove short cycles in the normal graph and to select nonzero elements in the parity-check matrix to reduce the number of low-weight codewords  ...  In this paper, we study the design of nonbinary low-density parity-check (LDPC) cycle codes over Galois field GF(q).  ...  When the nonzero elements in a cycle are chosen such that the FRC is satisfied, the low weight codeword generated by this cycle can be eliminated.  ... 
doi:10.1109/itw.2007.4313042 fatcat:mx6wzqekejgodmahj26ysqysbu

Dataflow acceleration of Krylov subspace sparse banded problems

Pavel Burovskiy, Stephen Girdlestone, Craig Davies, Spencer Sherwin, Wayne Luk
2014 2014 24th International Conference on Field Programmable Logic and Applications (FPL)  
Most of the efforts in the FPGA community related to sparse linear algebra focus on increasing the degree of internal parallelism in matrix-vector multiply kernels.  ...  Our approach enables trade-off between the number k of overlapped matrix power actions and the level of parallelism in a PE.  ...  ACKNOWLEDGMENT This work is supported in part by the European Union Seventh Framework Programme under grant agreement number 257906, 287804 and 318521, by the UK EPSRC, by the Maxeler University Programme  ... 
doi:10.1109/fpl.2014.6927453 dblp:conf/fpl/BurovskiyGDSL14 fatcat:kqiawknxq5htzdxe6zfnmua3ue

Alternative Life‐History Pathways and the Elasticity of Stochastic Matrix Models

David Claessen
2005 American Naturalist  
If the life cycle contains nonoverlapping, alternative life-history pathways, the ranking in terms of elasticity of the most critical vital rates may be reversed in stochastic and the corresponding average  ...  This note shows that the results of loop analysis, which have been proved for constant matrices only, apply to stochastic matrices as well if elasticity is defined as the effect of a proportional perturbation  ...  The critical remarks of one reviewer in particular helped to greatly improve the manuscript. This work was supported by the Biotechnology and Biological Sciences Research Council.  ... 
doi:10.1086/427091 pmid:15729645 fatcat:5rpd2bs6zvgctb2yap7zl5zwhi

Diagonal entry restrictions in minimum rank matrices

Wayne Barrett, Nicole Malloy, Curtis Nelson, William Sexton, John Sinkovic
2013 The Electronic Journal of Linear Algebra  
This completes our classification of nil, neutral, and nonzero vertices of "extreme" graphs, those with low or high minimum rank.  ...  In this paper, we continue our investigation and present results that specify when diagonal entries of minimum rank matrices must be zero, nonzero, or neither.  ...  In Sections 7 and 8, we completed the classification of nil, nonzero and neutral vertices for graphs of extreme minimum ranks, that is, graphs whose minimum rank is 1, 2, n−1, or n−2.  ... 
doi:10.13001/1081-3810.1655 fatcat:s6tbl24uorc7dc5l35ketmclbi

Quasi-cyclic unit memory convolutional codes

J. Justesen, E. Paaske, M. Ballan
1990 IEEE Transactions on Information Theory  
Unit memory convolutional codes with generator matrices, which are composed of circulant suhmatrices, are introduced.  ...  Equivalences among such codes and consider some of the basic structural properties are discussed. In particular, catastrophic encoders and minimal encoders are characterized and dual codes treated.  ...  If the new row is nonzero, the rank of the binary image of the first row is the sum of the ranks of the leading circulant and the rank of the image of the new row.  ... 
doi:10.1109/18.54876 fatcat:veo7qiyf7nddfmrxdulqnzn3hi

Typical ranks in symmetric matrix completion [article]

Daniel Irving Bernstein, Grigoriy Blekherman, Kisun Lee
2020 arXiv   pre-print
We study the problem of low-rank matrix completion for symmetric matrices.  ...  We give a combinatorial description of the patterns of specified entires of n× n symmetric matrices that have n as a typical rank.  ...  Acknowledgements We thank the Institute for Computational and Experimental Research in Mathematics (ICERM) in Provicence, Rhode Island, for supporting all three authors during the Fall 2018 semester.  ... 
arXiv:1909.06593v2 fatcat:cvktkuxgobh5fmzne4fxawp6ja

Families of LDPC Codes Derived from Nonprimitive BCH Codes and Cyclotomic Cosets [article]

Salah A. Aly
2008 arXiv   pre-print
Low-density parity check (LDPC) codes are an important class of codes with many applications.  ...  The constructed codes have high rates and are free of cycles of length four; consequently, they can be decoded using standard iterative decoding algorithms.  ...  The matrices A(h ij ) ′ s are µ × µ circulant permutation matrices based on some primitive elements h ij as shown in Definition 1. B.  ... 
arXiv:0802.4079v1 fatcat:qejy67vinjap5mro4vhx3uquaq

On graphs and algebraic graphs that do not contain cycles of length 4

Noga Alon, H. Tracy Hall, Christian Knauer, Rom Pinchasi, Raphael Yuster
2010 Journal of Graph Theory  
We also derive a lower bound on the rank of the adjacency matrix of a general abstract graph using the number of 4-cycles and a parameter which measures how close the graph is to being regular.  ...  From this we derive a rank bound for the adjacency matrix A of any simple graph with n vertices and E edges which does not contain a copy of K 2,r : rank(A) ≥ E−2n(r+1) r 2 √ n .  ...  We will try to bound the number of nonzero entries in those matrices in terms of n and d. Such questions can be considered as part of the study of sign patterns of matrices.  ... 
doi:10.1002/jgt.20542 fatcat:fkjquevm3bag7n4f5b6wzf4dum
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