# Signals discrimination by subspace selection

We discuss now the possibility of extracting the signal , from a mixture when the subspaces and are not well separated and the oblique projector onto along fails to yield the right signals separation. For this we introduce the following hypothesis on the class of signals to be considered:
We assume that the signal of interest is -sparse in a spanning set for
This implies that given there exists a subset of elements characterized by the set of indices , of cardinality , spanning the subspace and such that Thus, the hypothesis generates an, in general, intractable problem because the number of possible subspaces spanned by vectors out of is a combinatorial number .

The techniques developed within the project aim at reducing complexity by making the search for the right subspace signal dependent.

Given a signal , and assuming that the subspaces and , are known, the goal is to find spanning and such that . The cardinality of the subset of indices should be such that construction of is well posed. This assumption characterizes the class of signals that can be handled by the proposed approaches.

Under the stated hypothesis, if the subspace were known, one would have

 (23)

However, if the computation of is an ill posed problem, which is the situation we are considering, is not available. In order to look for the subset of indices yielding one may proceed as follows: Applying on every term of (23) and using the properties and , where , (23) becomes

 (24)

Since is given and , the left hand side of (23) is available and we can search for the set , in a stepwise manner by adaptive pursuit approaches.

Subsections