## Section: New Results

### Recent results on Sparse Representations, Inverse Problems, and Dimension Reduction

Sparsity, low-rank, dimension-reduction, inverse problem, sparse recovery, scalability, compressive sensing

The team has had a substantial activity ranging from theoretical results to algorithmic design and software contributions in the fields of sparse representations, inverse problems, and dimension reduction, which is at the core of the ERC project PLEASE (Projections, Learning and Sparsity for Efficient Data Processing, see Section 9.2.1.1).

#### Theoretical results on Sparse Representations, Graph Signal Processing, and Dimension Reduction

Participants : Rémi Gribonval, Yann Traonmilin, Gilles Puy, Nicolas Tremblay, Pierre Vandergheynst.

Main collaboration: Mike Davies (University of Edinburgh), Pierre Borgnat (ENS Lyon), and members of the LTS2 lab of Pierre Vandergheynst at EPFL

**Stable recovery of low-dimensional cones in Hilbert spaces:**
Many inverse problems in signal processing deal with the robust estimation of unknown data from underdetermined linear observations. Low dimensional models, when combined with appropriate regularizers, have been shown to be efficient at performing this task. Sparse models with the ${\ell}_{1}$-norm or low rank models with the nuclear norm are examples of such successful combinations. Stable recovery guarantees in these settings have been established using a common tool adapted to each case: the notion of restricted isometry property (RIP). We established generic RIP-based guarantees for the stable recovery of cones (positively homogeneous model sets) with arbitrary regularizers. These guarantees were illustrated on selected examples. For block structured sparsity in the infinite dimensional setting, we used the guarantees for a family of regularizers which efficiency in terms of RIP constant can be controlled, leading to stronger and sharper guarantees than the state of the art. This has been published in a journal paper [21].

**Recipes for stable linear embeddings from Hilbert spaces to ${\mathbb{R}}^{m}$:**
We considered the problem of constructing a linear map from a Hilbert space (possibly infinite dimensional) to ${\mathbb{R}}^{m}$ that satisfies a restricted isometry property (RIP) on an arbitrary signal model set. We obtained a generic framework that handles a large class of low-dimensional subsets but also *unstructured* and *structured* linear maps. We provided a simple recipe to prove that a random linear map satisfies a general RIP on the model set with high probability. We also described a generic technique to construct linear maps that satisfy the RIP. Finally, we detailed how to use our results in several examples, which allow us to recover and extend many known compressive sampling results. This has been presented at the conference EUSIPCO 2015 [90], and a journal paper is under revision [91].

**Signal processing on graphs: from filtering to random sampling and robust PCA:**
Graph signal processing is an emerging field aiming at extending classical tools from signal processing (1D time series) and image processing (2D pixel grids, 3D voxel grids) to more loosely structured numerical data: collections of numerical values each associated to a vertex of a graph, where the graph encodes the underlying “topology” of proximities and distances. Since our pioneering contributions on this topic [4], the team regularly works on various aspects of graph signal processing, in collaboration with the LTS2 lab of Pierre Vandergheynst at EPFL. This year,
we studied the problem of sampling $k$-bandlimited signals on graphs. We proposed two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than $O\left(klog\right(k\left)\right)$ measurements are sufficient to ensure an accurate and stable recovery of all $k$-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we proposed a computationally efficient decoder to reconstruct $k$-bandlimited signals from their samples. We proved that it yields accurate reconstructions and that it is also stable to noise. Finally, we conducted several experiments to test these techniques. A journal paper has been published [17] accompanied by a toolbox for reproducible research (see Section 6.14). Other contributions from this year on the topic of graph signal processing include new subgraph-based filterbanks for graph signals [22], and new accelerated and robustified techniques for PCA on graphs [19], [20] (see also below our contributions in terms of new algorithms to obtain approximate Fast Graph Fourier Transforms [32], [53]).

**Accelerated spectral clustering:**
We leveraged the proposed random sampling technique to propose a faster spectral clustering algorithm. Indeed, classical spectral clustering is based on the computation of the first $k$ eigenvectors of the similarity matrix' Laplacian, whose computation cost, even for sparse matrices, becomes prohibitive for large datasets. We showed that we can estimate the spectral clustering distance matrix without computing these eigenvectors: by graph filtering random signals. Also, we took advantage of the stochasticity of these random vectors to estimate the number of clusters $k$. We compared our method to classical spectral clustering on synthetic data, and showed that it reaches equal performance while being faster by a factor at least two for large datasets of real data. Two conference papers have been presented, at ICASSP 2016 [39] and ICML 2016 [40] and a toolbox for reproducible research has been released (see Section 6.4).

#### An Alternative Framework for Sparse Representations: Sparse “Analysis” Models

Participants : Rémi Gribonval, Nancy Bertin, Srdan Kitic, Clément Gaultier.

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.

Building on our pioneering work on the cosparse model [7] [73], [87] successful applications of this approach to sound source localization, audio declipping and brain imaging have been developed in 2015 and 2016. In addition, new applications to audio denoising were also introduced this year.

**Versatile cosparse regularization:** Digging the groove of previous years' results (comparison of the performance of several cosparse recovery algorithms in the context of sound source localization [77], demonstration of its efficiency in situations where usual methods fail ( [79], see paragraph 7.4.2), applicability to the hard declipping problem [78], application to EEG brain imaging [56]), a journal paper embedding the latest algorithms and results in sound source localization and brain source localization in a unified fashion was published this year [5]. This framework was also exploited to extend results on audio inpainting (see Section 7.3.2).

New results include experimental confirmation of robustness and versatility of the proposed scheme, and of its computational merits (convergence speed increasing with the amount of data). In a work presented in a workshop [44], we also proposed a multiscale strategy that aims at exploiting computational advantages of both sparse and cosparse regularization approaches, thanks to the simple yet effective all-zero initialization which the synthesis-based optimization can benefit from, while retaining the computational properties of the analysis-based approach for huge scale optimization problems arising in physics-driven settings.

**Parametric operator learning for cosparse calibration:**
In many inverse problems, a key challenge is to cope with unknown physical parameters of the problem such as the speed of sound or the boundary impedance. In the sound source localization problem, we previously showed that the unknown speed of sound can be learned jointly in the process of cosparse recovery, under mild conditions [58], [81]. This year, we extended the formulation to the case of unknown boundary impedance, and showing that a similar biconvex formulation and optimization could solve this new problem efficiently (conference paper published in ICASSP 2016 [29], see also Section 7.3.3).

#### Algorithmic and Theoretical results on Computational Representation Learning

Participants : Rémi Gribonval, Luc Le Magoarou, Nicolas Bellot, Adrien Leman, Cassio Fraga Dantas, Igal Rozenberg.

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 literature to learn a dictionary from a collection of training samples, classical dictionary learning is limited to small-scale problems. Inspired by usual fast transforms, we proposed a general dictionary structure that allows cheaper manipulation, and an algorithm to learn such dictionaries together with their fast implementation. The principle and its application to image denoising appeared at ICASSP 2015 [84] and an application to speedup linear inverse problems was published at EUSIPCO 2015 [83]. A Matlab library has been released (see Section 6.6) to reproduce the experiments from the comprehensive journal paper published this year [16], which additionally includes theoretical results on the improved sample complexity of learning such dictionaries. Pioneering identifiability results have been obtained in the Ph.D. thesis of Luc Le Magoarou on this topic [85].

We further explored the application of this technique to obtain fast approximations of Graph Fourier Transforms. A conference paper on this latter topic appeared in ICASSP 2016 [32], and a journal paper has been submitted [53] where we empirically show that $\mathcal{O}(nlogn)$ approximate implementations of Graph Fourier Transforms are possible for certain families of graphs. This opens the way to substantial accelerations for Fourier Transforms on large graphs.

A C++ software library has been developed (see Section 6.6) to release the resulting algorithms.