An Exact Tree Projection Algorithm for Wavelets

We propose a dynamic programming algorithm for projection onto wavelet tree structures. In contrast to other recently proposed algorithms which only give approximate tree projections for a given sparsity, our algorithm is guaranteed to calculate the projection exactly. We also prove that our algorithm has O(Nk) complexity, where N is the signal dimension and k is the sparsity of the tree approximation.

[1]  Douglas L. Jones,et al.  A signal-dependent time-frequency representation: fast algorithm for optimal kernel design , 1994, IEEE Trans. Signal Process..

[2]  Ted K. Ralphs,et al.  Integer and Combinatorial Optimization , 2013 .

[3]  Volkan Cevher,et al.  Tractability of interpretability via selection of group-sparse models , 2013, 2013 IEEE International Symposium on Information Theory.

[4]  Volkan Cevher,et al.  Model-Based Compressive Sensing , 2008, IEEE Transactions on Information Theory.

[5]  Mike E. Davies,et al.  Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces , 2009, IEEE Transactions on Information Theory.

[6]  D. Donoho.  CART AND BEST-ORTHO-BASIS: A CONNECTION' , 1997 .

[7]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[8]  Jerome M. Shapiro,et al.  Embedded image coding using zerotrees of wavelet coefficients , 1993, IEEE Trans. Signal Process..

[9]  D. Donoho,et al.  Smooth Multiwavelet Duals of Alpert Bases by Moment-Interpolating Refinement , 2000 .

[10]  Ivan Bratko,et al.  Trading Accuracy for Simplicity in Decision Trees , 1994, Machine Learning.

[11]  I. Daubechies,et al.  Tree Approximation and Optimal Encoding , 2001 .

[12]  Ronald R. Coifman,et al.  Entropy-based algorithms for best basis selection , 1992, IEEE Trans. Inf. Theory.

[13]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition , 2008 .

[14]  Wei-Yin Loh,et al.  Classification and regression trees , 2011, WIREs Data Mining Knowl. Discov..

[15]  R. Bellman.  Dynamic programming. , 1957, Science.

[16]  Richard G. Baraniuk,et al.  Optimal tree approximation with wavelets , 1999, Optics & Photonics.