Sparsity from binary hypothesis testing and application to non-parametric estimation

This paper presents and discusses an alternative notion of sparsity. This notion derives from a theoretical result in binary hypothesis testing and slightly differs from the standard notion of sparsity introduced by Donoho and Johnstone. As an application of this alternative notion of sparsity and as an extension of the detection threshold recently proposed, level-dependent detection thresholds are introduced. The performance of level-dependent detection thresholds is illustrated in the context of non-parametric estimation by soft thresholding in the wavelet domain. Experimental results show that the resulting approach performs well in comparison with one of the best up-to-date parametric method. In connection with some results concerning the statistical properties of wavelet coefficients associated with strictly stationary random processes, prospects are suggested for estimating unknown signals in non-necessarily white or Gaussian noise.