Mathematical Modeling and Analysis
Mathematical Modeling and Analysis
Decompositions of signals onto overcomplete sets of basis vectors may be made well defined by selecting an optimality criterion for the solutions. For example, minimum Euclidean norm solutions correspond to the those obtained by using the pseudo-inverse of the matrix with the columns consisting of the vectors in the basis set. Sparse signal representations are obtained by selecting solutions of maximal sparsity (i.e. the minimum number of non-zero coefficients in the solution). These representations have found a number of applications, including EEG (electroencephalography) and MEG (magnetoencephalography) estimation, time-frequency analysis, and spectrum estimation.
In certain applications it is useful to be able to estimate the error in the construction of the sparse representation. A number of recent uniqueness results provide conditions under which a signal has a unique sparse decomposition. These results, however, are of little assistance when the signal is known to include a noise component. Under these more realistic conditions, one would like to bound the reconstruction error in terms of the signal noise magnitude (that is, given a bound on the size of the noise in the signal, provide a bound on the maximum distance between any two appropriate sparse reprentations of that signal). The construction of such bounds is described in detail in a recent publication.
