A Multicomponent proximal algorithm for Empirical Mode Decomposition

International audienceThe Empirical Mode Decomposition (EMD) is known to be a powerful tool adapted to the decomposition of a signal into a collection of intrinsic mode functions (IMF). A key procedure in the extraction of the IMFs is the sifting process whose main drawback is to depend on the choice of an interpolation method and to have no clear convergence guarantees. We propose a convex optimization procedure in order to replace the sifting process in the EMD. The considered method is based on proximal tools, which allow us to deal with a large class of constraints such as quasi-orthogonality or extrema-based constraints

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness

Adaptive Local Iterative Filtering for Signal Decomposition and Instantaneous Frequency analysis

Time-frequency analysis for non-linear and non-stationary signals is extraordinarily challenging. To capture features in these signals, it is necessary for the analysis methods to be local, adaptive and stable. In recent years, decomposition based analysis methods, such as the empirical mode decomposition (EMD) technique pioneered by Huang et al., were developed by different research groups. These methods decompose a signal into a finite number of components on which the time-frequency analysis can be applied more effectively. In this paper we consider the iterative filters (IFs) approach as an alternative to EMD. We provide sufficient conditions on the filters that ensure the convergence of IFs applied to any $L^2$ signal. Then we propose a new technique, the Adaptive Local Iterative Filtering (ALIF) method, which uses the IFs strategy together with an adaptive and data driven filter length selection to achieve the decomposition. Furthermore we design smooth filters with compact support from solutions of Fokker-Planck equations (FP filters) that can be used within both IFs and ALIF methods. These filters fulfill the derived sufficient conditions for the convergence of the IFs algorithm. Numerical examples are given to demonstrate the performance and stability of IFs and ALIF techniques with FP filters. In addition, in order to have a complete and truly local analysis toolbox for non-linear and non-stationary signals, we propose a new definition for the instantaneous frequency which depends exclusively on local properties of a signal

A Class of Randomized Primal-Dual Algorithms for Distributed Optimization

Based on a preconditioned version of the randomized block-coordinate forward-backward algorithm recently proposed in [Combettes,Pesquet,2014], several variants of block-coordinate primal-dual algorithms are designed in order to solve a wide array of monotone inclusion problems. These methods rely on a sweep of blocks of variables which are activated at each iteration according to a random rule, and they allow stochastic errors in the evaluation of the involved operators. Then, this framework is employed to derive block-coordinate primal-dual proximal algorithms for solving composite convex variational problems. The resulting algorithm implementations may be useful for reducing computational complexity and memory requirements. Furthermore, we show that the proposed approach can be used to develop novel asynchronous distributed primal-dual algorithms in a multi-agent context

Variational Downscaling, Fusion and Assimilation of Hydrometeorological States via Regularized Estimation

Improved estimation of hydrometeorological states from down-sampled observations and background model forecasts in a noisy environment, has been a subject of growing research in the past decades. Here, we introduce a unified framework that ties together the problems of downscaling, data fusion and data assimilation as ill-posed inverse problems. This framework seeks solutions beyond the classic least squares estimation paradigms by imposing proper regularization, which are constraints consistent with the degree of smoothness and probabilistic structure of the underlying state. We review relevant regularization methods in derivative space and extend classic formulations of the aforementioned problems with particular emphasis on hydrologic and atmospheric applications. Informed by the statistical characteristics of the state variable of interest, the central results of the paper suggest that proper regularization can lead to a more accurate and stable recovery of the true state and hence more skillful forecasts. In particular, using the Tikhonov and Huber regularization in the derivative space, the promise of the proposed framework is demonstrated in static downscaling and fusion of synthetic multi-sensor precipitation data, while a data assimilation numerical experiment is presented using the heat equation in a variational setting