Aiding Dictionary Learning Through Multi-Parametric Sparse Representation

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.

Fault Handling in Large Water Networks with Online Dictionary Learning

“…Our previous work [10,11] has shown encouraging results when adapting DL classification for FDI in water networks. In this paper we propose new methods that tackle large distribution networks and lift data dimensionality limitations by employing online DL strategies.…”

“…Dictionary learning (DL) [7] is an active field in the signal processing community with multiple applications such as denoising, compression, super-resolution, and classification. Recent studies have also shown good results when dealing with anomaly detection in general [21,22,23], and particularly when applied to the FDI problem in water networks [10,11].…”

“…Let us consider the residual matrix R defined as in 7 (11). To this matrix corresponds the fault signature matrix M obtained from the former by a binarization procedure [14]:…”

“…Considering the residual matrix R from (11) and the sensor selection I obtained in Section 3 we arrive to sub-matrix R I . To this we associate the fault labels L (the active fault index for each of the columns of R I ).…”

“…In fact, we use only a subset of columns from(11), the so-called training set, but for simplicity we abuse the notation 8. The "any" condition can be replaced with any selection criterion deemed necessary (e.g., "all entries", "the majority of the entries").…”