Multiresolution analysis on the symmetric group

There is no generally accepted way to define wavelets on permutations. We address this issue by introducing the notion of coset based multiresolution analysis (CMRA) on the symmetric group, find the corresponding wavelet functions, and describe a fast wavelet transform for sparse signals. We discuss potential applications in ranking, sparse approximation, and multi-object tracking.

[1]  D. Rockmore,et al.  Generalized FFT's- A survey of some recent results , 1996, Groups and Computation.

[2]  Leonidas J. Guibas,et al.  Fourier Theoretic Probabilistic Inference over Permutations , 2009, J. Mach. Learn. Res..

[3]  Michael Clausen,et al.  Fast Generalized Fourier Transforms , 1989, Theor. Comput. Sci..

[4]  Daniel N. Rockmore,et al.  Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II , 2015, Discrete Mathematics & Theoretical Computer Science.

[5]  Ronald R. Coifman,et al.  Multiscale Wavelets on Trees, Graphs and High Dimensional Data: Theory and Applications to Semi Supervised Learning , 2010, ICML.

[6]  Leonidas J. Guibas,et al.  Exploiting Probabilistic Independence for Permutations , 2009, AISTATS.

[7]  Pierre Vandergheynst,et al.  Wavelets on Graphs via Spectral Graph Theory , 2009, ArXiv.

[8]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[9]  R. Coifman,et al.  Diffusion Wavelets , 2004 .

[10]  Arthur D. Szlam,et al.  Diffusion wavelet packets , 2006 .

[11]  Tony Jebara,et al.  Multi-object tracking with representations of the symmetric group , 2007, AISTATS.

[12]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Leonidas J. Guibas,et al.  Fourier-Information Duality in the Identity Management Problem , 2011, ECML/PKDD.

[14]  Peter F. Stadler,et al.  Fast Fourier Transform for Fitness Landscapes , 2002 .

[15]  Devavrat Shah,et al.  Inferring rankings under constrained sensing , 2008, NIPS.