Dictionary Learning for Sparse Representations With Applications to Blind Source Separation.

During the past decade, sparse representation has attracted much attention in the signal processing community. It aims to represent a signal as a linear combination of a small number of elementary signals called atoms. These atoms constitute a dictionary so that a signal can be expressed by the multiplication of the dictionary and a sparse coefficients vector. This leads to two main challenges that are studied in the literature, i.e. sparse coding (find the coding coefficients based on a given dictionary) and dictionary design (find an appropriate dictionary to fit the data). Dictionary design is the focus of this thesis. Traditionally, the signals can be decomposed by the predefined mathematical transform, such as discrete cosine transform (DCT), which forms the so-called analytical approach. In recent years, learning-based methods have been introduced to adapt the dictionary from a set of training data, leading to the technique of dictionary learning. Although this may involve a higher computational complexity, learned dictionaries have the potential to offer improved performance as compared with predefined dictionaries. Dictionary learning algorithm is often achieved by iteratively executing two operations: sparse approximation and dictionary update. We focus on the dictionary update step, where the dictionary is optimized with a given sparsity pattern. A novel framework is proposed to generalize benchmark mechanisms such as the method of optimal directions (MOD) and K-SVD where an arbitrary set of codewords and the corresponding sparse coefficients are simultaneously updated, hence the term simultaneous codeword optimization (SimCO). Moreover, its extended formulation ‘regularized SimCO’ mitigates the major bottleneck of dictionary update caused by the singular points. First and second order optimization procedures are designed to solve the primitive and regularized SimCO. In addition, a tree-structured multi-level representation of dictionary based on clustering is used to speed up the optimization process in the sparse coding stage. This novel dictionary learning algorithm is also applied for solving the underdetermined blind speech separation problem, leading to a multi-stage method, where the separation problem is reformulated as a sparse coding problem, with the dictionary being learned by an adaptive algorithm. Using mutual coherence and sparsity index, the performance of a variety of dictionaries for underdetermined speech separation is compared and analyzed, such as the dictionaries learned from speech mixtures and ground truth speech sources, as well as those predefined by mathematical transforms. Finally, we propose a new method for joint dictionary learning and source separation. Different from the multistage method, the proposed method can simultaneously estimate the mixing matrix, the dictionary and the sources in an alternating and blind manner. The advantages of all the proposed methods are demonstrated over the state-of-the-art methods using extensive numerical tests

Underdetermined Separation of Speech Mixture Based on Sparse Bayesian Learning

This paper describes a novel algorithm for underdetermined speech separation problem based on compressed sensing which is an emerging technique for efficient data reconstruction. The proposed algorithm consists of two steps. The unknown mixing matrix is firstly estimated from the speech mixtures in the transform domain by using K-means clustering algorithm. In the second step, the speech sources are recovered based on an autocalibration sparse Bayesian learning algorithm for speech signal. Numerical experiments including the comparison with other sparse representation approaches are provided to show the achieved performance improvement

Methods for learning adaptive dictionary in underdetermined speech separation

Underdetermined speech separation is a challenging problem that has been studied extensively in recent years. A promising method to this problem is based on the so-called sparse signal representation. Using this technique, we have recently developed a multi-stage algorithm, where the source signals are recovered using a pre-defined dictionary obtained by e.g. the discrete cosine transform (DCT). In this paper, instead of using the pre-defined dictionary, we present three methods for learning adaptive dictionaries for the reconstruction of source signals, and compare their performance with several state-of-the-art speech separation methods. © 2011 IEEE

Learning Overcomplete Dictionaries Based on Atom-by-Atom Updating

International audienceA dictionary learning algorithm learns a set of atoms from some training signals in such a way that each signal can be approximated as a linear combination of only a few atoms. Most dictionary learning algorithms use a two-stage iterative procedure. The first stage is to spars ely approximate the training signals over the current dictionary. The second stage is the update of the dictionary. In this paper we develop some atom-by-atom dictionary learning algorithms, which update the atoms sequentially. Specifically, we propose an efficient alternative to the well-known K-SVD algorithm, and show by various experiments that the proposed algorithm is much faster than K-SVD while its results are better. Moreover, we propose a novel algorithm that instead of alternating between the two dictionary learning stages, performs only the second stage. While in K-SVD each atom is updated along with the nonzero entries of its associated row vector in the coefficient matrix (which we name it its profile), in the new algorithm, each atom is updated along with the whole entries of its profile. As a result, contrary to K-SVD, the support of each profile can be changed while updating the dictionary. To further accelerate the convergence of this algorithm and to have a control on the cardinality of the representations, we then propose its two-stage counterpart by adding the sparse approximation stage. Experimental results on recovery of a known synthetic dictionary and dictionary learning for a class of auto-regressive signals demonstrate the promising performance of the proposed algorithms

Sparse and low-rank methods in structural system identification and monitoring

This paper presents sparse and low-rank methods for explicit modeling and harnessing the data structure to address the inverse problems in structural dynamics, identification, and data-driven health monitoring. In particular, it is shown that the structural dynamic features and damage information, intrinsic within the structural vibration response measurement data, possesses sparse and low-rank structure, which can be effectively modeled and processed by emerging mathematical tools such as sparse representation (SR), and low-rank matrix decomposition. It is also discussed that explicitly modeling and harnessing the sparse and low-rank data structure could benefit future work in developing data-driven approaches towards rapid, unsupervised, and effective system identification, damage detection, as well as massive SHM data sensing and management

Sparsity and adaptivity for the blind separation of partially correlated sources

Blind source separation (BSS) is a very popular technique to analyze multichannel data. In this context, the data are modeled as the linear combination of sources to be retrieved. For that purpose, standard BSS methods all rely on some discrimination principle, whether it is statistical independence or morphological diversity, to distinguish between the sources. However, dealing with real-world data reveals that such assumptions are rarely valid in practice: the signals of interest are more likely partially correlated, which generally hampers the performances of standard BSS methods. In this article, we introduce a novel sparsity-enforcing BSS method coined Adaptive Morphological Component Analysis (AMCA), which is designed to retrieve sparse and partially correlated sources. More precisely, it makes profit of an adaptive re-weighting scheme to favor/penalize samples based on their level of correlation. Extensive numerical experiments have been carried out which show that the proposed method is robust to the partial correlation of sources while standard BSS techniques fail. The AMCA algorithm is evaluated in the field of astrophysics for the separation of physical components from microwave data.Comment: submitted to IEEE Transactions on signal processin

Anomaly detection: sparse representation for high dimensional data

In this thesis, I investigated in three different anomaly aware sparse representation approaches. The first approach focuses on algorithmic development for the low-rank matrix completion problem. It has been shown that in the l0-search for low- rank matrix completion, the singular points in the objective function are the major reasons for failures. While different methods have been proposed to handle singular points, rigorous analysis has shown that there is a need for further improvement. To address the singularity issue, we propose a new objective function that is continuous everywhere. The new objective function is a good approximation of the original objective function in the sense that in the limit, the lower level sets of the new objective function are the closure of those of the original objective function. We formulate the matrix completion problem as the minimization of the new objective function and design a quasi-Newton method to solve it. Simulations demonstrate that the new method achieves excellent numerical performance. The second part discusses dictionary learning algorithms to solve the blind source separation (BSS) problem. For the proof of concepts, the focus is on the scenario where the number of mixtures is not less than that of sources. Based on the assumption that the sources are sparsely represented by some dictionaries, we present a joint source separation and dictionary learning algorithm (SparseBSS) to separate the noise corrupted mixed sources with very little extra information. We also discuss the singularity issue in the dictionary learning process which is one major reason for algorithm failure. Finally, two approaches are presented to address the singularity issue. The last approach focuses on algorithmic approaches to solve the robust face recognition problem where the test face image can be corrupted by arbitrary sparse noise. The standard approach is to formulate the problem as a sparse recovery problem and solve it using l1-minimization. As an alternative, the approximate message passing (AMP) algorithm had been tested but resulted in pessimistic results. The contribution of this part is to successfully solve the robust face recognition problem using the AMP framework. The recently developed adaptive damping technique has been adopted to address the issue that AMP normally only works well with Gaussian matrices. Statistical models are designed to capture the nature of the signal more authentically. Expectation maximization (EM) method has been used to learn the unknown hyper-parameters of the statistical model in an online fashion. Simulations demonstrate that our method achieves better recognition performance than the already impressive benchmark l1-minimization, is robust to the initial values of hyper-parameters, and exhibits low computational cost.Open Acces