Overview of Constrained PARAFAC Models

In this paper, we present an overview of constrained PARAFAC models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition, or alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product based approach is also formulated in terms of the index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for $N^{th}$ order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models

Multimodal Data Fusion: An Overview of Methods, Challenges and Prospects

International audienceIn various disciplines, information about the same phenomenon can be acquired from different types of detectors, at different conditions, in multiple experiments or subjects, among others. We use the term "modality" for each such acquisition framework. Due to the rich characteristics of natural phenomena, it is rare that a single modality provides complete knowledge of the phenomenon of interest. The increasing availability of several modalities reporting on the same system introduces new degrees of freedom, which raise questions beyond those related to exploiting each modality separately. As we argue, many of these questions, or "challenges" , are common to multiple domains. This paper deals with two key questions: "why we need data fusion" and "how we perform it". The first question is motivated by numerous examples in science and technology, followed by a mathematical framework that showcases some of the benefits that data fusion provides. In order to address the second question, "diversity" is introduced as a key concept, and a number of data-driven solutions based on matrix and tensor decompositions are discussed, emphasizing how they account for diversity across the datasets. The aim of this paper is to provide the reader, regardless of his or her community of origin, with a taste of the vastness of the field, the prospects and opportunities that it holds

Block Coordinate Descent for Regularized Multi-convex Optimization

This thesis considers regularized block multi-convex optimization, where the feasible set and objective function are generally non-convex but convex in each block of variables. I review some of its interesting examples and propose a generalized block coordinate descent (BCD) method. The generalized BCD uses three different block-update schemes. Based on the property of one block subproblem, one can freely choose one of the three schemes to update the corresponding block of variables. Appropriate choices of block-update schemes can often speed up the algorithm and greatly save computing time. Under certain conditions, I show that any limit point satisfies the Nash equilibrium conditions. Furthermore, I establish its global convergence and estimate its asymptotic convergence rate by assuming a property based on the Kurdyka-{\L}ojasiewicz inequality. As a consequence, this thesis gives a global linear convergence result of cyclic block coordinate descent for strongly convex optimization. The proposed algorithms are adapted for factorizing nonnegative matrices and tensors, as well as completing them from their incomplete observations. The algorithms were tested on synthetic data, hyperspectral data, as well as image sets from the CBCL, ORL and Swimmer databases. Compared to the existing state-of-the-art algorithms, the proposed algorithms demonstrate superior performance in both speed and solution quality

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

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

Nonconvex Recovery of Low-complexity Models

Today we are living in the era of big data, there is a pressing need for efficient, scalable and robust optimization methods to analyze the data we create and collect. Although Convex methods offer tractable solutions with global optimality, heuristic nonconvex methods are often more attractive in practice due to their superior efficiency and scalability. Moreover, for better representations of the data, the mathematical model we are building today are much more complicated, which often results in highly nonlinear and nonconvex optimizations problems. Both of these challenges require us to go beyond convex optimization. While nonconvex optimization is extraordinarily successful in practice, unlike convex optimization, guaranteeing the correctness of nonconvex methods is notoriously difficult. In theory, even finding a local minimum of a general nonconvex function is NP-hard – nevermind the global minimum. This thesis aims to bridge the gap between practice and theory of nonconvex optimization, by developing global optimality guarantees for nonconvex problems arising in real-world engineering applications, and provable, efficient nonconvex optimization algorithms. First, this thesis reveals that for certain nonconvex problems we can construct a model specialized initialization that is close to the optimal solution, so that simple and efficient methods provably converge to the global solution with linear rate. These problem include sparse basis learning and convolutional phase retrieval. In addition, the work has led to the discovery of a broader class of nonconvex problems – the so-called ridable saddle functions. Those problems possess characteristic structures, in which (i) all local minima are global, (ii) the energy landscape does not have any ''flat'' saddle points. More interestingly, when data are large and random, this thesis reveals that many problems in the real world are indeed ridable saddle, those problems include complete dictionary learning and generalized phase retrieval. For each of the aforementioned problems, the benign geometric structure allows us to obtain global recovery guarantees by using efficient optimization methods with arbitrary initialization