Adaptive signals recovery by convex optimization

Abstract : We consider the problem of denoising a signal observed in Gaussian noise.In this problem, classical linear estimators are quasi-optimal provided that the set of possible signals is convex, compact, and known a priori. However, when the set is unspecified, designing an estimator which does not ``know'' the underlying structure of a signal yet has favorable theoretical guarantees of statistical performance remains a challenging problem. In this thesis, we study a new family of estimators for statistical recovery of signals satisfying certain time-invariance properties. Such signals are characterized by their harmonic structure, which is usually unknown in practice. We propose new estimators which are capable to exploit the unknown harmonic structure of a signal to reconstruct. We demonstrate that these estimators admit theoretical performance guarantees, in the form of oracle inequalities, in a variety of settings.We provide efficient algorithmic implementations of these estimators via first-order optimization algorithm with non-Euclidean geometry, and evaluate them on synthetic data, as well as some real-world signals and images.
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Dmitrii Ostrovskii. Adaptive signals recovery by convex optimization. Computation and Language [cs.CL]. Université Grenoble Alpes, 2018. English. ⟨NNT : 2018GREAM004⟩. ⟨tel-01767206v2⟩

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