Low-Coherence Frames From Group Fourier Matrices

Many problems in areas such as compressive sensing and coding theory seek to design a set of equal-norm vectors with large angular separation. This idea is essentially equivalent to constructing a frame with low coherence. The elements of such frames can in turn be used to build high-performance spherical codes, quantum measurement operators, and compressive sensing measurement matrices, to name a few applications. In this paper, we allude to the group-frame construction first described by Slepian and further explored in the works of Vale and Waldron. We present a method for selecting representations of a finite group to construct a group frame that achieves low coherence. Our technique produces a tight frame with a small number of distinct inner product values between the frame elements, in a sense approximating a Grassmannian frame. We identify special cases in which our construction yields some previously known frames with optimal coherence meeting the Welch lower bound, and other cases in which the entries of our frame vectors come from small alphabets. In particular, we apply our technique to the problem choosing a subset of rows of an Hadamard matrix so that the resulting columns form a low-coherence frame. Finally, we give an explicit calculation of the average coherence of our frames, and find regimes in which they satisfy the strong coherence property described by Mixon, Bajwa, and Calderbank.

[1]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Shayne Waldron,et al.  Tight frames generated by finite nonabelian groups , 2008, Numerical Algorithms.

[3]  Joseph M. Renes,et al.  Symmetric informationally complete quantum measurements , 2003, quant-ph/0310075.

[4]  Cunsheng Ding,et al.  Complex Codebooks From Combinatorial Designs , 2006, IEEE Transactions on Information Theory.

[5]  P. Casazza,et al.  Frames of subspaces , 2003, math/0311384.

[6]  John J. Benedetto,et al.  Geometric Properties of Grassmannian Frames for R 2 and R 3 , 2004 .

[7]  Yonina C. Eldar,et al.  Optimal tight frames and quantum measurement , 2002, IEEE Trans. Inf. Theory.

[8]  A. J. Scott.  Tight informationally complete quantum measurements , 2006, quant-ph/0604049.

[9]  Peter G. Casazza,et al.  Constructing tight fusion frames , 2011 .

[10]  Bernhard G. Bodmann,et al.  The road to equal-norm Parseval frames , 2010 .

[11]  I. Blake,et al.  Group Codes for the Gaussian Channel , 1975 .

[12]  Babak Hassibi,et al.  Frames from generalized group fourier transforms and SL2(Fq) , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[13]  John J. Benedetto,et al.  Finite Normalized Tight Frames , 2003, Adv. Comput. Math..

[14]  Robert W. Heath,et al.  Linear dispersion codes for MIMO systems based on frame theory , 2002, IEEE Trans. Signal Process..

[15]  Thomas Strohmer,et al.  GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION , 2003, math/0301135.

[16]  John J. Benedetto,et al.  Geometric Properties of Grassmannian Frames for and , 2006, EURASIP J. Adv. Signal Process..

[17]  Cunsheng Ding,et al.  A Generic Construction of Complex Codebooks Meeting the Welch Bound , 2007, IEEE Transactions on Information Theory.

[18]  Shun’ichi Tanaka,et al.  Construction and classification of irreducible representations of special linear group of the second order over a finite field , 1967 .

[19]  S. Waldron,et al.  Tight Frames and Their Symmetries , 2004 .

[20]  Matthew J. Hirn.  The number of harmonic frames of prime order , 2010 .

[21]  Randall R. Holmes.  Linear Representations of Finite Groups , 2008 .

[22]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[23]  Charles J. Colbourn,et al.  Combinatorial Designs , 1999, Handbook of Discrete and Combinatorial Mathematics.

[24]  J. Bourgain,et al.  Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order , 2003 .

[25]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[26]  Peter G. Casazza,et al.  Equal-Norm Tight Frames with Erasures , 2003, Adv. Comput. Math..

[27]  J. Seidel,et al.  SPHERICAL CODES AND DESIGNS , 1991 .

[28]  P G Cazassa,et al.  FRAMES OF SUBSPACES. WAVELETS, FRAMES AND OPERATOR THEORY , 2004 .

[29]  Shayne Waldron,et al.  A classification of the harmonic frames up to unitary equivalence , 2011 .

[30]  Shamgar Gurevich,et al.  The finite harmonic oscillator and its associated sequences , 2008, Proceedings of the National Academy of Sciences.

[31]  M. Huxley.  Area, Lattice Points, and Exponential Sums , 1996 .

[32]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[33]  Guang Gong,et al.  A new binary sequence family with low correlation and large size , 2006, IEEE Transactions on Information Theory.

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

[35]  Dustin G. Mixon,et al.  Steiner equiangular tight frames , 2010, 1009.5730.

[36]  A. Robert Calderbank,et al.  Why Gabor frames? Two fundamental measures of coherence and their role in model selection , 2010, Journal of Communications and Networks.

[37]  Babak Hassibi,et al.  Group Frames With Few Distinct Inner Products and Low Coherence , 2015, IEEE Transactions on Signal Processing.

[38]  P. Casazza.  THE ART OF FRAME THEORY , 1999, math/9910168.

[39]  A. Terras.  Fourier Analysis on Finite Groups and Applications: Index , 1999 .

[40]  Lloyd R. Welch,et al.  Lower bounds on the maximum cross correlation of signals (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[41]  A. Robert Calderbank,et al.  Frame coherence and sparse signal processing , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[42]  N. M. Korobov.  Exponential Sums and their Applications , 1992 .

[43]  Georgios B. Giannakis,et al.  Achieving the Welch bound with difference sets , 2005, IEEE Transactions on Information Theory.

[44]  Babak Hassibi,et al.  Frames from groups: Generalized bounds and dihedral groups , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[45]  Babak Hassibi,et al.  Frames, group codes, and subgroups of (Z/pZ)× , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[46]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[47]  P. Casazza,et al.  Fusion frames and distributed processing , 2006, math/0605374.

[48]  Dustin G. Mixon,et al.  Tables of the existence of equiangular tight frames , 2015, ArXiv.

[49]  Shayne Waldron,et al.  The symmetry group of a finite frame , 2010 .

[50]  John J. Benedetto,et al.  Geometric Properties of Grassmannian Frames for Open image in new window and Open image in new window , 2006 .

[51]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[52]  Peter G. Casazza,et al.  Finite Frames: Theory and Applications , 2012 .

[53]  S. Waldron.  An Introduction to Finite Tight Frames , 2018 .

[54]  J. Tropp,et al.  SIGNAL RECOVERY FROM PARTIAL INFORMATION VIA ORTHOGONAL MATCHING PURSUIT , 2005 .

[55]  Babak Hassibi,et al.  On frames from abelian group codes , 2013, 2013 IEEE International Symposium on Information Theory.

[56]  Shayne Waldron,et al.  On computing all harmonic frames of n vectors in $\C^d$ , 2006 .

[57]  Elwyn R. Berlekamp.  The Weight Enumerators for Certain Subcodes of the Second Order Binary Reed-Muller Codes , 1970, Inf. Control..

[58]  Robert W. Heath,et al.  Space-time signaling and frame theory , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).