# Sparse representation by minimization of the norm like quantity - Handling the ill posed case.

The problem of finding the sparsest representation of a signal for a given dictionary is equivalent to minimization of the zero norm (or counting measure) which is defined as:

where are the coefficients of the atomic decomposition

 (29)

Thus, is equal to the number of nonzero coefficients in (29). However, sice minimization of is numerically intractable, the minimization of for has been considered [16]. Because the minimization of does not lead to a convex optimization problem, the most popular norm to minimize, when a sparse solution is required, is the 1-norm . Minimization of the 1-norm is considered the best convex approximant to the minimizer of [17,18]. In the context of signals splitting already stated, we are not particularly concerned about convexity so we have considered the minimization of , allowing for uncertainty in the available data [7]. This was implemented by a recursive process for incorporating constrains, which is equivalent to the procedure introduced in [19] and applied in [20].

Subsections