Source separation when the input sources are discrete or haveconstant modulus

In this paper, we present a new method for the source separation problem when some prior information on the input sources is available. More specifically, we study the situation where the distributions of the input signals are discrete or are concentrated on a circle. The method is based on easy properties of Hankel forms and on the divisibility of Gaussian distributions. In both situations, we prove that the estimator converges in absence of noise or if we know the first moments of the noise up to its scale. Moreover, in the absence of noise, the estimate converges with a finite number of observations     

Error exponents for AR order testing

This paper is concerned with error exponents in testing problems raised by autoregressive (AR) modeling. The tests to be considered are variants of generalized likelihood ratio testing corresponding to traditional approaches to autoregressive moving-average (ARMA) modeling estimation. In several related problems, such as Markov order or hidden Markov model order estimation, optimal error exponents have been determined thanks to large deviations theory. AR order testing is specially challenging since the natural tests rely on quadratic forms of Gaussian processes. In sharp contrast with empirical measures of Markov chains, the large deviation principles (LDPs) satisfied by Gaussian quadratic forms do not always admit an information-theoretic representation. Despite this impediment, we prove the existence of nontrivial error exponents for Gaussian AR order testing. And furthermore, we exhibit situations where the exponents are optimal. These results are obtained by showing that the log-likelihood process indexed by AR models of a given order satisfy an LDP upper bound with a weakened information-theoretic representation.     

MEM pixel correlated solutions for generalized moment andinterpolation problems

In generalized moment problems (signed) measures are searched to fit given observations, or continuous functions are searched to fit given constraints. Known convex methods for solving such problems, and their stochastic interpretations via maximum entropy on the mean (MEM) and in a Bayesian sense are reviewed, with some improvements on previous results. Then the MEM and Bayesian approaches are extended to default models with a dependence structure, yielding new families of solutions. One family involves a transfer kernel, and allows using prior information such as modality, convexity, or Sobolev norms. Another family of solutions with possibly nonconvex criteria, is arrived at using default models with exchangeable random variables. The main technical tools are convex analysis and large deviations theory     

Optimal error exponents in hidden Markov models order estimation

We consider the estimation of the number of hidden states (the order) of a discrete-time finite-alphabet hidden Markov model (HMM). The estimators we investigate are related to code-based order estimators: penalized maximum-likelihood (ML) estimators and penalized versions of the mixture estimator introduced by Liu and Narayan (1994). We prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works. We prove a version of Stein's lemma for HMM order estimation and derive an upper bound on underestimation exponents. Then we prove that this upper bound can be achieved by the penalized ML estimator and by the penalized mixture estimator. The proof of the latter result gets around the elusive nature of the ML in HMM by resorting to large-deviation techniques for empirical processes. Finally, we prove that for any consistent HMM order estimator, for most HMM, the overestimation exponent is .     

Efficient Semiparametric Estimation of the Periods in a Superposition of Periodic Functions with Unknown Shape

Abstract. We consider the estimation of the periods of periodic functions when their shape is unknown and they are corrupted by Gaussian white noise. In the case of a single periodic function, we propose a consistent and asymptotically efficient semiparametric estimator of the period. We then study the case of a sum of two periodic functions of unknown shape with different periods and propose semiparametric estimators of their periods that are consistent and asymptotically Gaussian.     

On simultaneous signal estimation and parameter identificationusing a generalized likelihood approach

A common approach to blind deconvolution of Bernoulli-Gaussian processes consists of performing both signal restoration and hyperparameter identification through maximization of a single generalized likelihood criterion. It is shown on a simple example that the resulting hyperparameter estimates may not converge toward any meaningful value. Therefore, other more reliable approaches should be adopted whenever possible