Relaxed Recovery Conditions for OMP/OLS by Exploiting both Coherence and Decay

We propose extended coherence-based conditions for exact sparse support recovery using orthogonal matching pursuit (OMP) and orthogonal least squares (OLS). Unlike standard uniform guarantees, we embed some information about the decay of the sparse vector coefficients in our conditions. As a result, the standard condition $\mu<1/(2k-1)$ (where $\mu$ denotes the mutual coherence and $k$ the sparsity level) can be weakened as soon as the non-zero coefficients obey some decay, both in the noiseless and the bounded-noise scenarios. Furthermore, the resulting condition is approaching $\mu<1/k$ for strongly decaying sparse signals. Finally, in the noiseless setting, we prove that the proposed conditions, in particular the bound $\mu<1/k$, are the tightest achievable guarantees based on mutual coherence

An Efficient Algorithm for Video Super-Resolution Based On a Sequential Model

In this work, we propose a novel procedure for video super-resolution, that is the recovery of a sequence of high-resolution images from its low-resolution counterpart. Our approach is based on a "sequential" model (i.e., each high-resolution frame is supposed to be a displaced version of the preceding one) and considers the use of sparsity-enforcing priors. Both the recovery of the high-resolution images and the motion fields relating them is tackled. This leads to a large-dimensional, non-convex and non-smooth problem. We propose an algorithmic framework to address the latter. Our approach relies on fast gradient evaluation methods and modern optimization techniques for non-differentiable/non-convex problems. Unlike some other previous works, we show that there exists a provably-convergent method with a complexity linear in the problem dimensions. We assess the proposed optimization method on {several video benchmarks and emphasize its good performance with respect to the state of the art.}Comment: 37 pages, SIAM Journal on Imaging Sciences, 201

Reduced-Order Modeling of Hidden Dynamics

International audienceThe objective of this paper is to investigate how noisy and incomplete observations can be integrated in the process of building a reduced-order model. This problematic arises in many scientific domains where there exists a need for accurate low-order descriptions of highly-complex phenomena, which can not be directly and/or deterministically observed. Within this context, the paper proposes a probabilistic framework for the construction of "POD-Galerkin" reduced-order models. Assuming a hidden Markov chain, the inference integrates the uncertainty of the hidden states relying on their posterior distribution. Simulations show the benefits obtained by exploiting the proposed framework

Safe rules for the identification of zeros in the solutions of the SLOPE problem

In this paper we propose a methodology to accelerate the resolution of the so-called ``Sorted L-One Penalized Estimation'' (SLOPE) problem. Our method leverages the concept of ``safe screening'', well-studied in the literature for \textit{group-separable} sparsity-inducing norms, and aims at identifying the zeros in the solution of SLOPE. More specifically, we introduce a family of $n!$ safe screening rules for this problem, where $n$ is the dimension of the primal variable, and propose a tractable procedure to verify if one of these tests is passed. Our procedure has a complexity $\mathcal{O}(n\log n + LT)$ where $T\leq n$ is a problem-dependent constant and $L$ is the number of zeros identified by the tests. We assess the performance of our proposed method on a numerical benchmark and emphasize that it leads to significant computational savings in many setups.Comment: 24 pages, 3 figure

Inverse Reduced-Order Modeling

International audienceWe propose a general probabilistic formulation of reduced-order modeling in the case the system state is hidden and characterized by some uncertainty. The objective is to integrate noisy and incomplete observations in the process of building a reduced-order model. We call this problematic inverse reduced-order modeling. This problematic arises in many scientific domains where there exists a need of accurate low-order descriptions of highly-complex phenomena, which can not be directly and/or deterministically observed. Among others, it concerns geophysical studies dealing with image data, which are important for the characterization of global warming or the prediction of natural disasters

Enhanced blind decoding of Tardos codes with new map-based functions

This paper presents a new decoder for probabilistic binary traitor tracing codes under the marking assumption. It is based on a binary hypothesis testing rule which integrates a collusion channel relaxation so as to obtain numerical and simple accusation functions. This decoder is blind as no estimation of the collusion channel prior to the accusation is required. Experimentations show that using the proposed decoder gives better performance than the well-known symmetric version of the Tardos decoder for common attack channels

Sparse representations in nested non-linear models

International audienceFollowing recent contributions in non-linear sparse represen-tations, this work focuses on a particular non-linear model, defined as the nested composition of functions. Recalling that most linear sparse representation algorithms can be straight-forwardly extended to non-linear models, we emphasize that their performance highly relies on an efficient computation of the gradient of the objective function. In the particular case of interest, we propose to resort to a well-known technique from the theory of optimal control to estimate the gradient. This computation is then implemented into the optimization procedure proposed byCan es et al., leading to a non-linear extension of it