Abstract
The ℓ 1 relaxations of the sparse and cosparse representation problems which appear in the dictionary learning procedure are usually solved repeatedly (varying only the parameter vector), thus making them well-suited to a multi-parametric interpretation. The associated constrained optimization problems differ only through an affine term from one iteration to the next (i.e., the problem’s structure remains the same while only the current vector, which is to be (co)sparsely represented, changes). We exploit this fact by providing an explicit, piecewise affine with a polyhedral support, representation of the solution. Consequently, at runtime, the optimal solution (the (co)sparse representation) is obtained through a simple enumeration throughout the non-overlapping regions of the polyhedral partition and the application of an affine law. We show that, for a suitably large number of parameter instances, the explicit approach outperforms the classical implementation.
Highlights
We commonly represent m-dimensional data, or signals, through a collection of m linearly independent vectors that together form a base
We proposed a novel sparse representation method for both the spare and cosparse models by looking at the problem as a multi-parametric formulation and solving it through the Karush–Kuhn–Tucker conditions
We solve a large optimization problem once in the beginning, and for each signal, we enumerate through a set of possible regions where the solution might lie
Summary
We commonly represent m-dimensional data, or signals, through a collection of m linearly independent vectors that together form a base. In order to be able to retrieve the original signal, we need to have access to a proper base, containing the right s vectors that were used when the true signal was created. With such limited information, it is hard to pick such a base, especially for a large class of signals, which is why it is common to extend it with more vectors, making it a redundant base, called a frame or, as will be used through out this paper, a dictionary. The idea is to have more options from which to choose s vectors
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have