Section: New Results
Recent results on sparse representations
The team has had a substantial activity ranging from theoretical results to algorithmic design and software contributions in the field of sparse representations, which is at the core of the Equipe Associée SPARS (see Section 8.1.1 ) initiated in 2006 between METISS and the LTS2 lab at EPFL as well as the FET-Open European project (FP7) SMALL (Sparse Models, Algorithms and Learning for Large-Scale Data, to begin in 2009, see Section 7.2.1 ) and the ANR project ECHANGE (ECHantillonnage Acoustique Nouvelle GEnération, see, Section 6.5.1 ).
Algorithmic breakthrough in sparse approximation : LoCOMP
Main collaborations: Pierre Vandergheynst (EPFL), Thomas Blumensath (Univ. Edinburgh), Emmanuel Ravelli, Laurent Daudet (LAM, Université Pierre et Marie Curie, Paris 6)
Our team had already made a substantial breakthrough in 2005 when first releasing the Matching Pursuit ToolKit (MPTK, see Section 5.3 ) which allowed for the first time the application of the Matching Pursuit algorithm to large scale data such as hours of CD-quality audio signals. In 2008, we designed a variant of Matching Pursuit called LoCOMP (ubiquitously for LOw Complexity Orthogonal Matching Pursuit or Local Orthogonal Matching Pursuit) speifically designed for shift-invariant dictionaries. LoCOMP has been shown to achieve an approximation quality very close to that of a full Orthonormal Matching Pursuit while retaining a much lower computational complexity of the order of that of Matching Pursuit. The complexity reduction is substantial, from one day of computation to 15 minutes for a typical audio signal  ,  , and the algorithm is being integrated into MPTK, in collaboration with Dr Thomas Blumensath. Moreover, joint experiments have been performed together with Dr Emmanuel Ravelli and Pr Laurent Daudet to assess the impact of this new algorithm on the audio codec developed at LAM which is based on MPTK.
Theoretical results on dictionary learning
Participant : Rémi Gribonval.
Main collaboration: Karin Schnass (EPFL)
While diverse heuristic techniques have been proposed in the litterature to learn a dictionary from a collection of training samples, there are little existing results which provide an adequate mathematical understanding of the behaviour of these techniques and their ability to recover an ideal dictionary from which the training samples may have been generated.
In 2008, we initiated a pioneering work on this topic, concentrating in particular on the fundamental theoretical question of the identifiability of the learned dictionary. Within the framework of the Ph.D. of Karin Schnass, we developed an analytic approach which was published at the conference ISCCSP 2008  and allowed us to describe "geometric" conditions which guarantee that a (non overcomplete) dictionary is "locally identifiable" by 1 minimization.
In a second step, we focused on estimating the number of sparse training samples which is typically sufficient to guarantee the identifiability (by 1 minimization), and obtained the following result, which is somewhat surprising considering that previous studies seemed to require a combinatorial number of training samples to guarantee the identifiability: the local identifiability condition is typically satisfied as soon as the number of training samples is roughly proportional to the ambient signal dimension. This second result was published at the conference EUSIPCO 2008  , and a journal paper has been submitted  .
Theoretical results on identification of sparse representations
Participant : Rémi Gribonval.
Main collaboration: Mike Davies (Univ. Edinburgh), Simon Foucart (Univ. Paris VI)
We pursued our investigation of conditions on an overcomplete dictionary which guarantee that certain ideal sparse decompositions can be recovered by some specific optimization principles. Our results from the previous years  ,  ,  concentrated on positive results for greedy algorithms and convex optimization (1 -minimization).
In contrast, in 2008, in collaboration with Pr Michael Davies, we concentrated on p -minimization, 0<p1 , and our results highlighted the pessimistic nature of sparse recovery analysis when recovery is predicted based on the restricted isometry constants (RIC) of the associated matrix (published in  ,  ). This year, we extended our analysis of the role of RIC to characterize the stability of p minimization with respect to the approximate recovery of vectors which are not exactly sparse  . Moreover, in collaboration with Dr Simon Foucart, we iidentified and solve an overlooked problem about the characterization of underdetermined systems of linear equations for which sparse solutions have minimal l1-norm. This characterization is known as the null space property. When the system has real coefficients, sparse solutions can be considered either as real or complex vectors, leading to two seemingly distinct null space properties. We proved that the two properties actually coincide by establishing a link with a problem about convex polygons in the real plane. Incidentally, we also show the equivalence between stable null space properties which account for the stable reconstruction by l1-minimization of vectors that are not exactly sparse  .
Shift-invariant dictionary learning algorithms and experiments with atrial signal extraction in ECG.
Main collaborations: Pierre Vandergheynst and Matthieu Lemay (EPFL)
In addition to our pioneering theoretical work on dictionary identifiability, we amplified the effort begun in 2007 on the design of dictionary learning algorithms for structured shift-invariant dictionaries. This work, performed in the framework of the Ph.D. of Boris Mailhé, was published at the conference EUSIPCO 2008  . The proposed approach was further developed to study the problem of ventricular cancellation and atrial modelling in the ECG of patients suffering from atrial fibrillation, in collaboration with Mathieu Lemay from EPFL  .