A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal

Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured "noises". As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations, based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field seismic data

Robust PCA as Bilinear Decomposition with Outlier-Sparsity Regularization

Principal component analysis (PCA) is widely used for dimensionality reduction, with well-documented merits in various applications involving high-dimensional data, including computer vision, preference measurement, and bioinformatics. In this context, the fresh look advocated here permeates benefits from variable selection and compressive sampling, to robustify PCA against outliers. A least-trimmed squares estimator of a low-rank bilinear factor analysis model is shown closely related to that obtained from an $\ell_0$-(pseudo)norm-regularized criterion encouraging sparsity in a matrix explicitly modeling the outliers. This connection suggests robust PCA schemes based on convex relaxation, which lead naturally to a family of robust estimators encompassing Huber's optimal M-class as a special case. Outliers are identified by tuning a regularization parameter, which amounts to controlling sparsity of the outlier matrix along the whole robustification path of (group) least-absolute shrinkage and selection operator (Lasso) solutions. Beyond its neat ties to robust statistics, the developed outlier-aware PCA framework is versatile to accommodate novel and scalable algorithms to: i) track the low-rank signal subspace robustly, as new data are acquired in real time; and ii) determine principal components robustly in (possibly) infinite-dimensional feature spaces. Synthetic and real data tests corroborate the effectiveness of the proposed robust PCA schemes, when used to identify aberrant responses in personality assessment surveys, as well as unveil communities in social networks, and intruders from video surveillance data.Comment: 30 pages, submitted to IEEE Transactions on Signal Processin

Optimization with Sparsity-Inducing Penalties

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view

Scalable Algorithms for Tractable Schatten Quasi-Norm Minimization

The Schatten-p quasi-norm $(0<p<1)$ is usually used to replace the standard nuclear norm in order to approximate the rank function more accurately. However, existing Schatten-p quasi-norm minimization algorithms involve singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each iteration, and thus may become very slow and impractical for large-scale problems. In this paper, we first define two tractable Schatten quasi-norms, i.e., the Frobenius/nuclear hybrid and bi-nuclear quasi-norms, and then prove that they are in essence the Schatten-2/3 and 1/2 quasi-norms, respectively, which lead to the design of very efficient algorithms that only need to update two much smaller factor matrices. We also design two efficient proximal alternating linearized minimization algorithms for solving representative matrix completion problems. Finally, we provide the global convergence and performance guarantees for our algorithms, which have better convergence properties than existing algorithms. Experimental results on synthetic and real-world data show that our algorithms are more accurate than the state-of-the-art methods, and are orders of magnitude faster.Comment: 16 pages, 5 figures, Appears in Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI), Phoenix, Arizona, USA, pp. 2016--2022, 201

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

A constrained-based optimization approach for seismic data recovery problems

Random and structured noise both affect seismic data, hiding the reflections of interest (primaries) that carry meaningful geophysical interpretation. When the structured noise is composed of multiple reflections, its adaptive cancellation is obtained through time-varying filtering, compensating inaccuracies in given approximate templates. The under-determined problem can then be formulated as a convex optimization one, providing estimates of both filters and primaries. Within this framework, the criterion to be minimized mainly consists of two parts: a data fidelity term and hard constraints modeling a priori information. This formulation may avoid, or at least facilitate, some parameter determination tasks, usually difficult to perform in inverse problems. Not only classical constraints, such as sparsity, are considered here, but also constraints expressed through hyperplanes, onto which the projection is easy to compute. The latter constraints lead to improved performance by further constraining the space of geophysically sound solutions.Comment: International Conference on Acoustics, Speech and Signal Processing (ICASSP 2014); Special session "Seismic Signal Processing

Non-convex regularization in remote sensing

In this paper, we study the effect of different regularizers and their implications in high dimensional image classification and sparse linear unmixing. Although kernelization or sparse methods are globally accepted solutions for processing data in high dimensions, we present here a study on the impact of the form of regularization used and its parametrization. We consider regularization via traditional squared (2) and sparsity-promoting (1) norms, as well as more unconventional nonconvex regularizers (p and Log Sum Penalty). We compare their properties and advantages on several classification and linear unmixing tasks and provide advices on the choice of the best regularizer for the problem at hand. Finally, we also provide a fully functional toolbox for the community.Comment: 11 pages, 11 figure

Sample Complexity of Dictionary Learning and other Matrix Factorizations

Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis (PCA), non-negative matrix factorization (NMF), $K$-means clustering, etc., rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity \ldots), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes. The derived generalization bounds behave proportional to $\sqrt{\log(n)/n}$ w.r.t.\ the number of samples $n$ for the considered matrix factorization techniques.Comment: to appea