Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms

In this work we study the behavior of the forward-backward (FB) algorithm when the proximity operator is replaced by a sub-iterative procedure to approximate a Gaussian denoiser, in a Plug-and-Play (PnP) fashion. In particular, we consider both analysis and synthesis Gaussian denoisers within a dictionary framework, obtained by unrolling dual-FB iterations or FB iterations, respectively. We analyze the associated minimization problems as well as the asymptotic behavior of the resulting FB-PnP iterations. In particular, we show that the synthesis Gaussian denoising problem can be viewed as a proximity operator. For each case, analysis and synthesis, we show that the FB-PnP algorithms solve the same problem whether we use only one or an infinite number of sub-iteration to solve the denoising problem at each iteration. To this aim, we show that each "one sub-iteration" strategy within the FB-PnP can be interpreted as a primal-dual algorithm when a warm-restart strategy is used. We further present similar results when using a Moreau-Yosida smoothing of the global problem, for an arbitrary number of sub-iterations. Finally, we provide numerical simulations to illustrate our theoretical results. In particular we first consider a toy compressive sensing example, as well as an image restoration problem in a deep dictionary framework.

Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals

When dealing with electro or magnetoencephalography records, many supervised prediction tasks are solved by working with covariance matrices to summarize the signals. Learning with these matrices requires using Riemanian geometry to account for their structure. In this paper, we propose a new method to deal with distributions of covariance matrices and demonstrate its computational efficiency on M/EEG multivariate time series. More specifically, we define a Sliced-Wasserstein distance between measures of symmetric positive definite matrices that comes with strong theoretical guarantees. Then, we take advantage of its properties and kernel methods to apply this distance to brain-age prediction from MEG data and compare it to state-of-the-art algorithms based on Riemannian geometry. Finally, we show that it is an efficient surrogate to the Wasserstein distance in domain adaptation for Brain Computer Interface applications.

Dictionary and prior learning with unrolled algorithms for unsupervised inverse problems

Inverse problems consist in recovering a signal given noisy observations. One classical resolution approach is to leverage sparsity and integrate prior knowledge of the signal to the reconstruction algorithm to get a plausible solution. Still, this prior might not be sufficiently adapted to the data. In this work, we study Dictionary and Prior learning from degraded measurements as a bi-level problem, and we take advantage of unrolled algorithms to solve approximate formulations of Synthesis and Analysis. We provide an empirical and theoretical analysis of automatic differentiation for Dictionary Learning to understand better the pros and cons of unrolling in this context. We find that unrolled algorithms speed up the recovery process for a small number of iterations by improving the gradient estimation. Then we compare Analysis and Synthesis by evaluating the performance of unrolled algorithms for inverse problems, without access to any ground truth data for several classes of dictionaries and priors. While Analysis can achieve good results,Synthesis is more robust and performs better. Finally, we illustrate our method on pattern and structure learning tasks from degraded measurements.

Estimation with Low-Rank Time-Frequency Synthesis Models

Many state-of-the-art signal decomposition techniques rely on a low-rank factorization of a time-frequency (t-f) transform. In particular, nonnegative matrix factorization (NMF) of the spectrogram has been considered in many audio applications. This is an analysis approach in the sense that the factorization is applied to the squared magnitude of the analysis coefficients returned by the t-f transform. In this paper we instead propose a synthesis approach, where low-rankness is imposed to the synthesis coefficients of the data signal over a given t-f dictionary (such as a Gabor frame). As such we offer a novel modeling paradigm that bridges t-f synthesis modeling and traditional analysis-based NMF approaches. The proposed generative model allows in turn to design more sophisticated multi-layer representations that can efficiently capture diverse forms of structure. Additionally, the generative modeling allows to exploit t-f low-rankness for compressive sensing. We present efficient iterative shrinkage algorithms to perform estimation in the proposed models and illustrate the capabilities of the new modeling paradigm over audio signal processing examples.

Social-sparsity brain decoders: faster spatial sparsity

Spatially-sparse predictors are good models for brain decoding: they give accurate predictions and their weight maps are interpretable as they focus on a small number of regions. However, the state of the art, based on total variation or graph-net, is computationally costly. Here we introduce sparsity in the local neighborhood of each voxel with social-sparsity, a structured shrinkage operator. We find that, on brain imaging classification problems, social-sparsity performs almost as well as total-variation models and better than graph-net, for a fraction of the computational cost. It also very clearly outlines predictive regions. We give details of the model and the algorithm.