A Unified Framework for Sparse Non-Negative Least Squares using Multiplicative Updates and the Non-Negative Matrix Factorization Problem

We study the sparse non-negative least squares (S-NNLS) problem. S-NNLS occurs naturally in a wide variety of applications where an unknown, non-negative quantity must be recovered from linear measurements. We present a unified framework for S-NNLS based on a rectified power exponential scale mixture prior on the sparse codes. We show that the proposed framework encompasses a large class of S-NNLS algorithms and provide a computationally efficient inference procedure based on multiplicative update rules. Such update rules are convenient for solving large sets of S-NNLS problems simultaneously, which is required in contexts like sparse non-negative matrix factorization (S-NMF). We provide theoretical justification for the proposed approach by showing that the local minima of the objective function being optimized are sparse and the S-NNLS algorithms presented are guaranteed to converge to a set of stationary points of the objective function. We then extend our framework to S-NMF, showing that our framework leads to many well known S-NMF algorithms under specific choices of prior and providing a guarantee that a popular subclass of the proposed algorithms converges to a set of stationary points of the objective function. Finally, we study the performance of the proposed approaches on synthetic and real-world data.Comment: To appear in Signal Processin

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Exploring African tone scales

[TODO] Add abstract here

Blind audio-visual localization and separation via low-rank and sparsity

The ability to localize visual objects that are associated with an audio source and at the same time to separate the audio signal is a cornerstone in audio-visual signal-processing applications. However, available methods mainly focus on localizing only the visual objects, without audio separation abilities. Besides that, these methods often rely on either laborious preprocessing steps to segment video frames into semantic regions, or additional supervisions to guide their localization. In this paper, we aim to address the problem of visual source localization and audio separation in an unsupervised manner and avoid all preprocessing or post-processing steps. To this end, we devise a novel structured matrix decomposition method that decomposes the data matrix of each modality as a superposition of three terms: 1) a low-rank matrix capturing the background information; 2) a sparse matrix capturing the correlated components among the two modalities and, hence, uncovering the sound source in visual modality and the associated sound in audio modality; and 3) a third sparse matrix accounting for uncorrelated components, such as distracting objects in visual modality and irrelevant sound in audio modality. The generality of the proposed method is demonstrated by applying it onto three applications, namely: 1) visual localization of a sound source; 2) visually assisted audio separation; and 3) active speaker detection. Experimental results indicate the effectiveness of the proposed method on these application domains

Advances in Nonnegative Matrix Decomposition with Application to Cluster Analysis

Nonnegative Matrix Factorization (NMF) has found a wide variety of applications in machine learning and data mining. NMF seeks to approximate a nonnegative data matrix by a product of several low-rank factorizing matrices, some of which are constrained to be nonnegative. Such additive nature often results in parts-based representation of the data, which is a desired property especially for cluster analysis.  This thesis presents advances in NMF with application in cluster analysis. It reviews a class of higher-order NMF methods called Quadratic Nonnegative Matrix Factorization (QNMF). QNMF differs from most existing NMF methods in that some of its factorizing matrices occur twice in the approximation. The thesis also reviews a structural matrix decomposition method based on Data-Cluster-Data (DCD) random walk. DCD goes beyond matrix factorization and has a solid probabilistic interpretation by forming the approximation with cluster assigning probabilities only. Besides, the Kullback-Leibler divergence adopted by DCD is advantageous in handling sparse similarities for cluster analysis.  Multiplicative update algorithms have been commonly used for optimizing NMF objectives, since they naturally maintain the nonnegativity constraint of the factorizing matrix and require no user-specified parameters. In this work, an adaptive multiplicative update algorithm is proposed to increase the convergence speed of QNMF objectives.  Initialization conditions play a key role in cluster analysis. In this thesis, a comprehensive initialization strategy is proposed to improve the clustering performance by combining a set of base clustering methods. The proposed method can better accommodate clustering methods that need a careful initialization such as the DCD.  The proposed methods have been tested on various real-world datasets, such as text documents, face images, protein, etc. In particular, the proposed approach has been applied to the cluster analysis of emotional data

Generative-Discriminative Low Rank Decomposition for Medical Imaging Applications

In this thesis, we propose a method that can be used to extract biomarkers from medical images toward early diagnosis of abnormalities. Surge of demand for biomarkers and availability of medical images in the recent years call for accurate, repeatable, and interpretable approaches for extracting meaningful imaging features. However, extracting such information from medical images is a challenging task because the number of pixels (voxels) in a typical image is in order of millions while even a large sample-size in medical image dataset does not usually exceed a few hundred. Nevertheless, depending on the nature of an abnormality, only a parsimonious subset of voxels is typically relevant to the disease; therefore various notions of sparsity are exploited in this thesis to improve the generalization performance of the prediction task. We propose a novel discriminative dimensionality reduction method that yields good classification performance on various datasets without compromising the clinical interpretability of the results. This is achieved by combining the modelling strength of generative learning framework and the classification performance of discriminative learning paradigm. Clinical interpretability can be viewed as an additional measure of evaluation and is also helpful in designing methods that account for the clinical prior such as association of certain areas in a brain to a particular cognitive task or connectivity of some brain regions via neural fibres. We formulate our method as a large-scale optimization problem to solve a constrained matrix factorization. Finding an optimal solution of the large-scale matrix factorization renders off-the-shelf solver computationally prohibitive; therefore, we designed an efficient algorithm based on the proximal method to address the computational bottle-neck of the optimization problem. Our formulation is readily extended for different scenarios such as cases where a large cohort of subjects has uncertain or no class labels (semi-supervised learning) or a case where each subject has a battery of imaging channels (multi-channel), \etc. We show that by using various notions of sparsity as feasible sets of the optimization problem, we can encode different forms of prior knowledge ranging from brain parcellation to brain connectivity

An investigation of the utility of monaural sound source separation via nonnegative matrix factorization applied to acoustic echo and reverberation mitigation for hands-free telephony

In this thesis we investigate the applicability and utility of Monaural Sound Source Separation (MSSS) via Nonnegative Matrix Factorization (NMF) for various problems related to audio for hands-free telephony. We first investigate MSSS via NMF as an alternative acoustic echo reduction approach to existing approaches such as Acoustic Echo Cancellation (AEC). To this end, we present the single-channel acoustic echo problem as an MSSS problem, in which the objective is to extract the users signal from a mixture also containing acoustic echo and noise. To perform separation, NMF is used to decompose the near-end microphone signal onto the union of two nonnegative bases in the magnitude Short Time Fourier Transform domain. One of these bases is for the spectral energy of the acoustic echo signal, and is formed from the in- coming far-end user’s speech, while the other basis is for the spectral energy of the near-end speaker, and is trained with speech data a priori. In comparison to AEC, the speaker extraction approach obviates Double-Talk Detection (DTD), and is demonstrated to attain its maximal echo mitigation performance immediately upon initiation and to maintain that performance during and after room changes for similar computational requirements. Speaker extraction is also shown to introduce distortion of the near-end speech signal during double-talk, which is quantified by means of a speech distortion measure and compared to that of AEC. Subsequently, we address Double-Talk Detection (DTD) for block-based AEC algorithms. We propose a novel block-based DTD algorithm that uses the available signals and the estimate of the echo signal that is produced by NMF-based speaker extraction to compute a suitably normalized correlation-based decision variable, which is compared to a fixed threshold to decide on doubletalk. Using a standard evaluation technique, the proposed algorithm is shown to have comparable detection performance to an existing conventional block-based DTD algorithm. It is also demonstrated to inherit the room change insensitivity of speaker extraction, with the proposed DTD algorithm generating minimal false doubletalk indications upon initiation and in response to room changes in comparison to the existing conventional DTD. We also show that this property allows its paired AEC to converge at a rate close to the optimum. Another focus of this thesis is the problem of inverting a single measurement of a non- minimum phase Room Impulse Response (RIR). We describe the process by which percep- tually detrimental all-pass phase distortion arises in reverberant speech filtered by the inverse of the minimum phase component of the RIR; in short, such distortion arises from inverting the magnitude response of the high-Q maximum phase zeros of the RIR. We then propose two novel partial inversion schemes that precisely mitigate this distortion. One of these schemes employs NMF-based MSSS to separate the all-pass phase distortion from the target speech in the magnitude STFT domain, while the other approach modifies the inverse minimum phase filter such that the magnitude response of the maximum phase zeros of the RIR is not fully compensated. Subjective listening tests reveal that the proposed schemes generally produce better quality output speech than a comparable inversion technique