Matrix Probing and its Conditioning

When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to recover an approximation to A^-1. A basic question is whether this linear system is invertible and well-conditioned. In this paper, we show that if the Gram matrix of the B_j's is sufficiently well-conditioned and each B_j has a high numerical rank, then n {proportional} p log^2 n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging. We demonstrate numerically that matrix probing can also produce good preconditioners for inverting elliptic operators in variable media.

[1]  N. Higham,et al.  Computing A, log(A) and Related Matrix Functions by Contour Integrals , 2007 .

[2]  M. Shubin.  Pseudodifferential Operators and Spectral Theory , 1987 .

[3]  Joel A. Tropp,et al.  The restricted isometry property for time–frequency structured random matrices , 2011, ArXiv.

[4]  E. Chang.  Wavelet Foveation , 1999 .

[5]  H. Rauhut.  Compressive Sensing and Structured Random Matrices , 2009 .

[6]  Laurent Demanet,et al.  Matrix probing: A randomized preconditioner for the wave-equation Hessian , 2011, 1101.3615.

[7]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[8]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

[9]  Richard M. Karp,et al.  An introduction to randomized algorithms , 1991, Discret. Appl. Math..

[10]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[11]  Holger Rauhut,et al.  Circulant and Toeplitz matrices in compressed sensing , 2009, ArXiv.

[12]  Massimo Fornasier,et al.  Compressive Sensing and Structured Random Matrices , 2010 .

[13]  E. Giné,et al.  Decoupling: From Dependence to Independence , 1998 .

[14]  A. Buchholz.  Operator Khintchine inequality in non-commutative probability , 2001 .

[15]  Massimo Fornasier,et al.  Theoretical Foundations and Numerical Methods for Sparse Recovery , 2010, Radon Series on Computational and Applied Mathematics.

[16]  Nicholas J. Higham,et al.  Computing AAlpha, log(A), and Related Matrix Functions by Contour Integrals , 2008, SIAM J. Numer. Anal..

[17]  A. Edelman.  Eigenvalues and condition numbers of random matrices , 1988 .

[18]  V. Rokhlin,et al.  A randomized algorithm for the approximation of matrices , 2006 .

[19]  G. Pisier,et al.  Non commutative Khintchine and Paley inequalities , 1991 .

[20]  Richard M. Karp,et al.  Efficient Randomized Pattern-Matching Algorithms , 1987, IBM J. Res. Dev..

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

[22]  V. Rokhlin,et al.  A fast randomized algorithm for the approximation of matrices ✩ , 2007 .

[23]  Holger Rauhut,et al.  Sparsity in Time-Frequency Representations , 2007, ArXiv.

[24]  Holger Rauhut,et al.  Edinburgh Research Explorer Identification of Matrices Having a Sparse Representation , 2022 .

[25]  Nando de Freitas,et al.  An Introduction to MCMC for Machine Learning , 2004, Machine Learning.

[26]  David R. Karger,et al.  A new approach to the minimum cut problem , 1996, JACM.

[27]  T. Chan,et al.  A Survey of Preconditioners for Domain Decomposition. , 1985 .

[28]  Tony F. Chan,et al.  The Interface Probing Technique in Domain Decomposition , 1992, SIAM J. Matrix Anal. Appl..

[29]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[30]  Holger Rauhut,et al.  Compressive Sensing with structured random matrices , 2012 .

[31]  J. Tropp.  On the conditioning of random subdictionaries , 2008 .

[32]  E. Giné,et al.  On decoupling, series expansions, and tail behavior of chaos processes , 1993 .

[33]  Laurent Demanet,et al.  Discrete Symbol Calculus , 2008, SIAM Rev..