On denoising via penalized least-squares rules

Penalized least-squares (PELS) rules for signal denoising can be obtained via the use of various information criteria (AIC, BIC, etc.) or various minmax LS approaches. Let S denote the set of "significant" parameters in the denoising problem (which is to be determined), let n_s be the dimension of S, and let n_sp denote the penalty term of a PELS criterion. We show that, depending on the expression for p, the following cases can occur: type-1) If p does not depend on S, then denoising via the corresponding PELS rule is equivalent to simple thresholding; and type-2) If p depends on ns only, then the equivalence to thresholding no longer holds but the PELS rule can still be implemented quite efficiently. We also show that the use of BIC leads to an existing PELS rule of type-1 when the noise variance in the denoising problem is known, and to a novel PELS rule of type-2 when the noise variance is unknown.