# Highly Nonlinear Approximations for Sparse Signal Representation

# 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:

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**