TY - GEN

T1 - Sparseness by iterative projections onto spheres

AU - Theis, Fabian J.

AU - Tanaka, Toshihisa

PY - 2006

Y1 - 2006

N2 - Many interesting signals share the property of being sparsely active. The search for such sparse components within a data set commonly involves a linear or nonlinear projection step in order to fulfill the sparseness constraints. In addition to the proximity measure used for the projection, the result of course is also intimately connected with the actual definition of the sparseness criterion. In this work, we introduce a novel sparseness measure and apply it to the problem of finding a sparse projection of a given signal. Here, sparseness is defined as the fixed ratio of p- over 2-norm, and existence and uniqueness of the projection holds. This framework extends previous work by Hoyer in the case of p = 1, where it is easy to give a deterministic, more or less closed-form solution. This is not possible for p ≠ 1, so we introduce an algorithm based on alternating projections onto spheres (POSH), which is similar to the projection onto convex sets (POCS). Although the assumption of convexity does not hold in our setting, we observe not only convergence of the algorithm, but also convergence to the correct minimal distance solution. Indications for a proof of this surprising property are given. Simulations confirm these results.

AB - Many interesting signals share the property of being sparsely active. The search for such sparse components within a data set commonly involves a linear or nonlinear projection step in order to fulfill the sparseness constraints. In addition to the proximity measure used for the projection, the result of course is also intimately connected with the actual definition of the sparseness criterion. In this work, we introduce a novel sparseness measure and apply it to the problem of finding a sparse projection of a given signal. Here, sparseness is defined as the fixed ratio of p- over 2-norm, and existence and uniqueness of the projection holds. This framework extends previous work by Hoyer in the case of p = 1, where it is easy to give a deterministic, more or less closed-form solution. This is not possible for p ≠ 1, so we introduce an algorithm based on alternating projections onto spheres (POSH), which is similar to the projection onto convex sets (POCS). Although the assumption of convexity does not hold in our setting, we observe not only convergence of the algorithm, but also convergence to the correct minimal distance solution. Indications for a proof of this surprising property are given. Simulations confirm these results.

UR - http://www.scopus.com/inward/record.url?scp=33947635664&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33947635664

SN - 142440469X

SN - 9781424404698

T3 - ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings

SP - V709-V712

BT - 2006 IEEE International Conference on Acoustics, Speech, and Signal Processing - Proceedings

T2 - 2006 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2006

Y2 - 14 May 2006 through 19 May 2006

ER -