Tropp's analysis of Orthogonal Matching Pursuit (OMP) using the Exact Recovery Condition (ERC) [1] is extended to a first exact recovery analysis of Orthogonal Least Squares (OLS). We show that when the ERC is met, OLS is guaranteed to exactly recover the unknown support in at most k iterations.Moreover, we provide a closer look at the analysis of both OMP and OLS when the ERC is not fulfilled.The existence of dictionaries for which some subsets are never recovered by OMP is proved. This phenomenon also appears with basis pursuit where support recovery depends on the sign patterns, but it does not occur for OLS. Finally, numerical experiments show that none of the considered algorithms is uniformly better than the other but for correlated dictionaries, guaranteed exact recovery may be obtained after fewer iterations for OLS than for OMP.

Due to the increasing availability of large-scale observation and simulation datasets, data-driven representations arise as efficient and relevant computation representations of dynamical systems for a wide range of applications, where modeldriven models based on ordinary differential equation remain the state-of-the-art approaches. In this work, we investigate neural networks (NN) as physically-sound data-driven representations of such systems. Reinterpreting Runge-Kutta methods as graphical models, we consider a residual NN architecture and introduce bilinear layers to embed non-linearities which are intrinsic features of dynamical systems. From numerical experiments for classic dynamical systems, we demonstrate the relevance of the proposed NN-based architecture both in terms of forecasting performance and model identification.

This paper is devoted to turbo synchronization, that is to say the use of soft information to estimate parameters like carrier phase, frequency offset or timing within a turbo receiver. It is shown how maximum-likelihood estimation of those synchronization parameters can be implemented by means of the iterative expectation-maximization (EM) algorithm [1]. Then we show that the EM algorithm iterations can be combined with those of a turbo receiver. This leads to a general theoretical framework for turbo synchronization. The soft decision-directed ad-hoc algorithm proposed in [2] for carrier phase recovery turns out to be a particular instance of this implementation. The proposed mathematical framework is illustrated by simulations reported for the particular case of carrier phase estimation combined with iterative demodulation and decoding [3]. 2933 0-7803-7802-4/03/$17.00

In this paper, we present a closed-form expression of a Bayesian Cramér-Rao lower bound for the estimation of a dynamical phase offset in a non-data-aided BPSK transmitting context. This kind of bound is derived considering two different scenarios: a first expression is obtained in an off-line context and then, a second expression in an on-line context logically follows. The SNR-asymptotic expressions of this bound drive us to introduce a new asymptotic bound, namely the Asymptotic Bayesian Cramér-Rao Bound. This bound is close to the classical Bayesian bound but is easier to evaluate.

To cite this version:Angélique Drémeau, Cédric Herzet, Laurent Daudet. Boltzmann machine and mean-field approximation for structured sparse decompositions. Accepté à IEEE Trans. On Signal Processing. 2012. AbstractTaking advantage of the structures inherent in many sparse decompositions constitutes a promising research axis. In this paper, we address this problem from a Bayesian point of view. We exploit a Boltzmann machine, allowing to take a large variety of structures into account, and focus on the resolution of a marginalized maximum a posteriori problem. To solve this problem, we resort to a mean-field approximation and the "variational Bayes Expectation-Maximization" algorithm. This approach results in a soft procedure making no hard decision on the support or the values of the sparse representation. We show that this characteristic leads to an improvement of the performance over state-of-the-art algorithms. Index TermsStructured sparse representation, Bernoulli-Gaussian model, Boltzmann machine, mean-field approximation.

To cite this version:Cédric Herzet, Charles Soussen, Jérôme Idier, Rémi Gribonval. Abstract-We address the exact recovery of a k-sparse vector in the noiseless setting when some partial information on the support is available. This partial information takes the form of either a subset of the true support or an approximate subset including wrong atoms as well. We derive a new sufficient and worst-case necessary (in some sense) condition for the success of some procedures based on ℓp-relaxation, Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS). Our result is based on the coherence µ of the dictionary and relaxes the wellknown condition µ < 1/(2k − 1) ensuring the recovery of any k-sparse vector in the non-informed setup. It reads µ < 1/(2k − g + b − 1) when the informed support is composed of g good atoms and b wrong atoms. We emphasize that our condition is complementary to some restrictedisometry based conditions by showing that none of them implies the other.Because this mutual coherence condition is common to all procedures, we carry out a finer analysis based on the Null Space Property (NSP) and the Exact Recovery Condition (ERC). Connections are established regarding the characterization of ℓp-relaxation procedures and OMP in the informed setup. First, we emphasize that the truncated NSP enjoys an ordering property when p is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the truncated NSP for the informed ℓ 1 problem, and the truncated NSP for p < 1.

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