Optimal shift invariant spaces and their Parseval frame generators

Abstract

Given a set of functions F = {f1, ..., fm} ⊂ L2 (Rd), we study the problem of finding the shift-invariant space V with n generators {φ1, ..., φn} that is "closest" to the functions of F in the sense thatV = under(arg min, V′ ∈ Vn) underover(∑, i = 1, m) wi {norm of matrix} fi - PV′ fi {norm of matrix}2, where wis are positive weights, and Vn is the set of all shift-invariant spaces that can be generated by n or less generators. The Eckart-Young theorem uses the singular value decomposition to provide a solution to a related problem in finite dimension. We transform the problem under study into an uncountable set of finite dimensional problems each of which can be solved using an extension of the Eckart-Young theorem. We prove that the finite dimensional solutions can be patched together and transformed to obtain the optimal shift-invariant space solution to the original problem, and we produce a Parseval frame for the optimal space. A typical application is the problem of finding a shift-invariant space model that describes a given class of signals or images (e.g., the class of chest X-rays), from the observation of a set of m signals or images f1, ..., fm, which may be theoretical samples, or experimental data.