Perspectives on CUR Decompositions

This note discusses an interesting matrix factorization called the CUR Decomposition. We illustrate various viewpoints of this method by comparing and contrasting them in different situations. Additionally, we offer a new characterization of CUR decompositions which synergizes these viewpoints and shows that they are indeed the same in the exact decomposition case.

[1]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[2]  David W. Lewis,et al.  Matrix theory , 1991 .

[3]  Laurent Demanet,et al.  Sublinear Randomized Algorithms for Skeleton Decompositions , 2011, SIAM J. Matrix Anal. Appl..

[4]  Akram Aldroubi,et al.  CUR Decompositions, Similarity Matrices, and Subspace Clustering , 2017, Front. Appl. Math. Stat..

[5]  Joel A. Tropp,et al.  Column subset selection, matrix factorization, and eigenvalue optimization , 2008, SODA.

[6]  Prabhat,et al.  Identifying important ions and positions in mass spectrometry imaging data using CUR matrix decompositions. , 2015, Analytical chemistry.

[7]  Xuelong Li,et al.  Deterministic Column-Based Matrix Decomposition , 2010, IEEE Transactions on Knowledge and Data Engineering.

[8]  R. Vidal,et al.  Sparse Subspace Clustering: Algorithm, Theory, and Applications. , 2013, IEEE transactions on pattern analysis and machine intelligence.

[9]  Petros Drineas,et al.  CUR matrix decompositions for improved data analysis , 2009, Proceedings of the National Academy of Sciences.

[10]  Per-Gunnar Martinsson,et al.  Efficient algorithms for cur and interpolative matrix decompositions , 2014, Adv. Comput. Math..

[11]  Yifan Hu,et al.  Drawing Large Graphs by Low‐Rank Stress Majorization , 2012, Comput. Graph. Forum.

[12]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[13]  Petros Drineas,et al.  FAST MONTE CARLO ALGORITHMS FOR MATRICES III: COMPUTING A COMPRESSED APPROXIMATE MATRIX DECOMPOSITION∗ , 2004 .

[14]  Alberto Mozo,et al.  Iterative column subset selection , 2017, Knowledge and Information Systems.

[15]  Ali Çivril,et al.  Column Subset Selection Problem is UG-hard , 2014, J. Comput. Syst. Sci..

[16]  R. Penrose.  On best approximate solutions of linear matrix equations , 1956, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  A. Cichocki,et al.  Generalizing the column–row matrix decomposition to multi-way arrays , 2010 .

[18]  Akram Aldroubi,et al.  Similarity matrix framework for data from union of subspaces , 2017, Applied and Computational Harmonic Analysis.

[19]  Luis Rademacher,et al.  Efficient Volume Sampling for Row/Column Subset Selection , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[20]  S. Muthukrishnan,et al.  Relative-Error CUR Matrix Decompositions , 2007, SIAM J. Matrix Anal. Appl..

[21]  Christos Boutsidis,et al.  An improved approximation algorithm for the column subset selection problem , 2008, SODA.

[22]  Danny C. Sorensen,et al.  A DEIM Induced CUR Factorization , 2014, SIAM J. Sci. Comput..

[23]  Subhash Khot.  On the power of unique 2-prover 1-round games , 2002, STOC '02.

[24]  Aditya Bhaskara,et al.  Greedy Column Subset Selection: New Bounds and Distributed Algorithms , 2016, ICML.

[25]  Rong Jin,et al.  An Explicit Sampling Dependent Spectral Error Bound for Column Subset Selection , 2015, ICML.

[26]  Miao Xu,et al.  CUR Algorithm for Partially Observed Matrices , 2014, ICML.

[27]  Yaroslav Shitov,et al.  Column subset selection is NP-complete , 2017, Linear Algebra and its Applications.

[28]  Petros Drineas,et al.  On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning , 2005, J. Mach. Learn. Res..

[29]  Qingshan Liu,et al.  Joint Active Learning with Feature Selection via CUR Matrix Decomposition , 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[30]  S. Goreinov,et al.  A Theory of Pseudoskeleton Approximations , 1997 .

[31]  S. Goreinov,et al.  Pseudo-skeleton approximations by matrices of maximal volume , 1997 .

[32]  A. Szalay,et al.  OBJECTIVE IDENTIFICATION OF INFORMATIVE WAVELENGTH REGIONS IN GALAXY SPECTRA , 2013, 1312.0637.

[33]  G. Strang.  Introduction to Linear Algebra , 1993 .

[34]  Michael W. Mahoney,et al.  CUR from a Sparse Optimization Viewpoint , 2010, NIPS.

[35]  G. W. Stewart,et al.  Four algorithms for the the efficient computation of truncated pivoted QR approximations to a sparse matrix , 1999, Numerische Mathematik.