Hilbert-space techniques for spectral representation in terms of overcomplete bases
| Authors | |
|---|---|
| Year of publication | 2002 |
| Type | Article in Proceedings |
| Conference | Proceedings of the Summer School DATASTAT'2001, Folia Fac. Sci. Nat. Univ. Masaryk. Brunensis, Mathematica 11 |
| MU Faculty or unit | |
| Citation | |
| Field | General mathematics |
| Keywords | functional approximation; kernel operators; frame and wavelet expansions; pseudoinverse operators |
| Description | Topics associated with the representation of objects from a separable Hilbert space in terms of an a priori given overcomplete system (dictionary) of its generators (atoms) are handled. First the procedure of finding such a representation is formulated and solved using the Hilbert-space technique of linear bounded operators and their generalized inverse. Afterwards the problem of finding its sparse representation is discussed, i.e. such representation where most information on the given object is concentrated in a fewest possible number of its nonzero (spectral) coefficients in that representation. This may be rephrased as a procedure for finding a subbasis which is in a certain sense optimal for the given object in the scope of the prescribed overcomplete system. In general the common approach based on Moore-Penrose pseudoinverse does not yield the desired sparse solutions. That is why alternate procedures are discussed, in particular from the point of view of their numerical stability and computational feasibility. |
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