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Section: New Results

Recent results on sparse representations

Sparse approximation, high dimension, scalable algorithms, dictionary design, graph wavelets

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 FET-Open European project (FP7) SMALL (Sparse Models, Algorithms and Learning for Large-Scale Data, see section  8.2.1.1 ), the ANR project ECHANGE (ECHantillonnage Acoustique Nouvelle GEnération, see section  8.1.1.2 ), and the ERC project PLEASE (projections, Learning and Sparsity for Efficient Data Processing, see section  8.2.1.2 ).

A new framework for sparse representations: analysis sparse models

Participants : Rémi Gribonval, Sangnam Nam, Nancy Bertin, Srdjan Kitic.

Main collaboration: Mike Davies, Mehrdad Yaghoobi (Univ. Edinburgh), Michael Elad (The Technion).

In the past decade there has been a great interest in a synthesis-based model for signals, based on sparse and redundant representations. Such a model assumes that the signal of interest can be composed as a linear combination of few columns from a given matrix (the dictionary). An alternative analysis-based model can be envisioned, where an analysis operator multiplies the signal, leading to a cosparse outcome. Within the SMALL project, we initiated a research programme dedicated to this analysis model, in the context of a generic missing data problem (e.g., compressed sensing, inpainting, source separation, etc.). We obtained a uniqueness result for the solution of this problem, based on properties of the analysis operator and the measurement matrix. We also considered a number of pursuit algorithms for solving the missing data problem, including an L1-based and a new greedy method called GAP (Greedy Analysis Pursuit). Our simulations demonstrated the appeal of the analysis model, and the success of the pursuit techniques presented.

These results have been published in conferences and in a journal paper [42] . Other algorithms based on iterative cosparse projections [83] as well as extensions of GAP to deal with noise and structure in the cosparse representation have been developed, with applications to toy MRI reconstruction problems and acoustic source localization and reconstruction from few measurements [58] .

Theoretical results on sparse representations and dictionary learning

Participants : Rémi Gribonval, Sangnam Nam, Nancy Bertin.

Main collaboration: Karin Schnass (EPFL), Mike Davies (University of Edinburgh), Volkan Cevher (EPFL), Simon Foucart (Université Paris 5, Laboratoire Jacques-Louis Lions), Charles Soussen (Centre de recherche en automatique de Nancy (CRAN)), Jérôme Idier (Institut de Recherche en Communications et en Cybernétique de Nantes (IRCCyN)), Cédric Herzet (Equipe-projet FLUMINANCE (Inria - CEMAGREF, Rennes)), Morten Nielsen (Department of Mathematical Sciences [Aalborg]), Gilles Puy, Pierre Vandergheynst, Yves Wiaux (EPFL), Mehrdad Yaghoobi, Rodolphe Jenatton, Francis Bach (Equipe-projet SIERRA (Inria, Paris)), Boaz Ophir, Michael Elad (Technion), Mark D. Plumbley (Queen Mary, University of London).

Sparse recovery conditions for Orthogonal Least Squares : 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 / algorithms. We extended Tropp's analysis of Orthogonal Matching Pursuit (OMP) using the Exact Recovery Condition (ERC) to a first exact recovery analysis of Orthogonal Least Squares (OLS). We showed that when ERC is met, OLS is guaranteed to exactly recover the unknown support. Moreover, we provided a closer look at the analysis of both OMP and OLS when ERC is not fulfilled. We showed that there exist dictionaries for which some subsets are never recovered with OMP. This phenomenon, which also appears with ${\ell }_1$ minimization, does not occur for OLS. Finally, numerical experiments based on our theoretical analysis showed that none of the considered algorithms is uniformly better than the other. This work has been submitted for publication in a journal [86] . More recently, we obtained simpler coherence-based conditions [85] .

Performance guarantees for compressed sensing with spread spectrum techniques : We advocate a compressed sensing strategy that consists of multiplying the signal of interest by a wide bandwidth modulation before projection onto randomly selected vectors of an orthonormal basis. Firstly, in a digital setting with random modulation, considering a whole class of sensing bases including the Fourier basis, we prove that the technique is universal in the sense that the required number of measurements for accurate recovery is optimal and independent of the sparsity basis. This universality stems from a drastic decrease of coherence between the sparsity and the sensing bases, which for a Fourier sensing basis relates to a spread of the original signal spectrum by the modulation (hence the name "spread spectrum"). The approach is also efficient as sensing matrices with fast matrix multiplication algorithms can be used, in particular in the case of Fourier measurements. Secondly, these results are confirmed by a numerical analysis of the phase transition of the l1-minimization problem. Finally, we show that the spread spectrum technique remains effective in an analog setting with chirp modulation for application to realistic Fourier imaging. We illustrate these findings in the context of radio interferometry and magnetic resonance imaging. This work has been accepted for publication in a journal [45] .

Dictionary learning : An important practical problem in sparse modeling is to choose the adequate dictionary to model a class of signals or images of interest. 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 [13] and allowed us to describe "geometric" conditions which guarantee that a (non overcomplete) dictionary is "locally identifiable" by ${\ell }^1$ minimization.

In a second step, we focused on estimating the number of sparse training samples which is typically sufficient to guarantee the identifiability (by ${\ell }^1$ 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. The outline of the second result was published in conferences [12] , [25] . These results have been published in the journal paper [15] .

Analysis Operator Learning for Overcomplete Cosparse Representations : Besides standard dictionary learning, we also considered learning in the context of the cosparse model. We consider the problem of learning a low-dimensional signal model from a collection of training samples. The mainstream approach would be to learn an overcomplete dictionary to provide good approximations of the training samples using sparse synthesis coefficients. This famous sparse model has a less well known counterpart, in analysis form, called the cosparse analysis model. In this new model, signals are characterized by their parsimony in a transformed domain using an overcomplete analysis operator. We consider two approaches to learn an analysis operator from a training corpus.

The first one uses a constrained optimization program based on L1 optimization. We derive a practical learning algorithm, based on projected subgradients, and demonstrate its ability to robustly recover a ground truth analysis operator, provided the training set is of sufficient size. A local optimality condition is derived, providing preliminary theoretical support for the well-posedness of the learning problem under appropriate conditions. Extensions to deal with noisy training samples are currently investigated, and a journal paper is under revision [87] .

In the second approach, analysis "atoms" are learned sequentially by identifying directions that are orthogonal to a subset of the training data. We demonstrate the effectiveness of the algorithm in three experiments, treating synthetic data and real images, showing a successful and meaningful recovery of the analysis operator.

Connections between sparse approximation and Bayesian estimation: Penalized least squares regression is often used for signal denoising and inverse problems, and is commonly interpreted in a Bayesian framework as a Maximum A Posteriori (MAP) estimator, the penalty function being the negative logarithm of the prior. For example, the widely used quadratic program (with an ${\ell }^1$ penalty) associated to the LASSO / Basis Pursuit Denoising is very often considered as MAP estimation under a Laplacian prior in the context of additive white Gaussian noise (AWGN) reduction.

A first result, which we published last year, highlights the fact that, while this is one possible Bayesian interpretation, there can be other equally acceptable Bayesian interpretations. Therefore, solving a penalized least squares regression problem with penalty $\phi \left(x\right)$ need not be interpreted as assuming a prior $C·exp(-\phi (x\left)\right)$ and using the MAP estimator. In particular, we showed that for any prior $P_X$, the minimum mean square error (MMSE) estimator is the solution of a penalized least square problem with some penalty $\phi \left(x\right)$, which can be interpreted as the MAP estimator with the prior $C·exp(-\phi (x\left)\right)$. Vice-versa, for certain penalties $\phi \left(x\right)$, the solution of the penalized least squares problem is indeed the MMSE estimator, with a certain prior $P_X$. In general $d{P}_X\left(x\right)\ne C·exp(-\phi \left(x\right))dx$.

A second result, obtained in collaboration with Prof. Mike Davies and Prof. Volkan Cevher (a paper is under revision) characterizes the "compressibility" of various probability distributions with applications to underdetermined linear regression (ULR) problems and sparse modeling. We identified simple characteristics of probability distributions whose independent and identically distributed (iid) realizations are (resp. are not) compressible, i.e., that can be approximated as sparse. We prove that many priors which MAP Bayesian interpretation is sparsity inducing (such as the Laplacian distribution or Generalized Gaussian distributions with exponent p<=1), are in a way inconsistent and do not generate compressible realizations. To show this, we identify non-trivial undersampling regions in ULR settings where the simple least squares solution outperform oracle sparse estimation in data error with high probability when the data is generated from a sparsity inducing prior, such as the Laplacian distribution [39] .