Joint Tensor Factorization and Outlying Slab Suppression with Applications

We consider factoring low-rank tensors in the presence of outlying slabs. This problem is important in practice, because data collected in many real-world applications, such as speech, fluorescence, and some social network data, fit this paradigm. Prior work tackles this problem by iteratively selecting a fixed number of slabs and fitting, a procedure which may not converge. We formulate this problem from a group-sparsity promoting point of view, and propose an alternating optimization framework to handle the corresponding $\ell_p$ ($0<p\leq 1$) minimization-based low-rank tensor factorization problem. The proposed algorithm features a similar per-iteration complexity as the plain trilinear alternating least squares (TALS) algorithm. Convergence of the proposed algorithm is also easy to analyze under the framework of alternating optimization and its variants. In addition, regularization and constraints can be easily incorporated to make use of \emph{a priori} information on the latent loading factors. Simulations and real data experiments on blind speech separation, fluorescence data analysis, and social network mining are used to showcase the effectiveness of the proposed algorithm

Tensor based singular spectrum analysis for automatic scoring of sleep EEG

A new supervised approach for decomposition of single channel signal mixtures is introduced in this paper. The performance of the traditional singular spectrum analysis (SSA) algorithm is significantly improved by applying tensor decomposition instead of traditional singular value decomposition (SVD). As another contribution to this subspace analysis method, the inherent frequency diversity of the data has been effectively exploited to highlight the subspace of interest. As an important application, sleep EEG has been analysed and the stages of sleep for the subjects in normal condition, with sleep restriction, and with sleep extension have been accurately estimated and compared with the results of sleep scoring by clinical experts

Blind identification of mixtures of quasi-stationary sources.

由於在盲語音分離的應用，線性準平穩源訊號混合的盲識別獲得了巨大的研究興趣。在這個問題上，我們利用準穩態源訊號的時變特性來識別未知的混合系統系數。傳統的方法有二：i)基於張量分解的平行因子分析(PARAFAC);ii)基於對多個矩陣的聯合對角化的聯合對角化算法(JD)。一般來說，PARAFAC和JD 都採用了源聯合的提取方法；即是說，對應所有訊號源的系統係數在升法上是用時進行識別的。在這篇論文中，我利用Khati-Rao(KR)子空間來設計一種新的盲識別算法。在我設計的算法中提出一種與傳統的方法不同的提法。在我設計的算法中，盲識別問題被分解成數個結構上相對簡單的子問題，分別對應不同的源。在超定混合模型，我們提出了一個專門的交替投影算法(AP)。由此產生的算法，不但能從經驗發現是非常有競爭力的，而且更有理論上的利落收斂保證。另外，作為一個有趣的延伸，該算法可循一個簡單的方式應用於欠混合模型。對於欠定混合模型，我們提出啟發式的秩最小化算法從而提高算法的速度。Blind identification of linear instantaneous mixtures of quasi-stationary sources (BI-QSS) has received great research interest over the past few decades, motivated by its application in blind speech separation. In this problem, we identify the unknown mixing system coefcients by exploiting the time-varying characteristics of quasi-stationary sources. Traditional BI-QSS methods fall into two main categories: i) Parallel Factor Analysis (PARAFAC), which is based on tensor decomposition; ii) Joint Diagonalization (JD), which is based on approximate joint diagonalization of multiple matrices. In both PARAFAC and JD, the joint-source formulation is used in general; i.e., the algorithms are designed to identify the whole mixing system simultaneously.In this thesis, I devise a novel blind identification framework using a Khatri-Rao (KR) subspace formulation. The proposed formulation is different from the traditional formulations in that it decomposes the blind identication problem into a number of per-source, structurally less complex subproblems. For the over determined mixing models, a specialized alternating projections algorithm is proposed for the KR subspace for¬mulation. The resulting algorithm is not only empirically found to be very competitive, but also has a theoretically neat convergence guarantee. Even better, the proposed algorithm can be applied to the underdetermined mixing models in a straightforward manner. Rank minimization heuristics are proposed to speed up the algorithm for the underdetermined mixing model. The advantages on employing the rank minimization heuristics are demonstrated by simulations.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Lee, Ka Kit.Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 72-76).Abstracts also in Chinese.Abstract --- p.iAcknowledgement --- p.iiChapter 1 --- Introduction --- p.1Chapter 2 --- Settings of Quasi-Stationary Signals based Blind Identification --- p.4Chapter 2.1 --- Signal Model --- p.4Chapter 2.2 --- Assumptions --- p.5Chapter 2.3 --- Local Covariance Model --- p.7Chapter 2.4 --- Noise Covariance Removal --- p.8Chapter 2.5 --- Prewhitening --- p.9Chapter 2.6 --- Summary --- p.10Chapter 3 --- Review on Some Existing BI-QSS Algorithms --- p.11Chapter 3.1 --- Joint Diagonalization --- p.11Chapter 3.1.1 --- Fast Frobenius Diagonalization [4] --- p.12Chapter 3.1.2 --- Pham’s JD [5, 6] --- p.14Chapter 3.2 --- Parallel Factor Analysis --- p.16Chapter 3.2.1 --- Tensor Decomposition [37] --- p.17Chapter 3.2.2 --- Alternating-Columns Diagonal-Centers [12] --- p.21Chapter 3.2.3 --- Trilinear Alternating Least-Squares [10, 11] --- p.23Chapter 3.3 --- Summary --- p.25Chapter 4 --- Proposed Algorithms --- p.26Chapter 4.1 --- KR Subspace Criterion --- p.27Chapter 4.2 --- Blind Identification using Alternating Projections --- p.29Chapter 4.2.1 --- All-Columns Identification --- p.31Chapter 4.3 --- Overdetermined Mixing Models (N > K): Prewhitened Alternating Projection Algorithm (PAPA) --- p.32Chapter 4.4 --- Underdetermined Mixing Models (N <K) --- p.34Chapter 4.4.1 --- Rank Minimization Heuristic --- p.34Chapter 4.4.2 --- Alternating Projections Algorithm with Huber Function Regularization --- p.37Chapter 4.5 --- Robust KR Subspace Extraction --- p.40Chapter 4.6 --- Summary --- p.44Chapter 5 --- Simulation Results --- p.47Chapter 5.1 --- General Settings --- p.47Chapter 5.2 --- Overdetermined Mixing Models --- p.49Chapter 5.2.1 --- Simulation 1 - Performance w.r.t. SNR --- p.49Chapter 5.2.2 --- Simulation 2 - Performance w.r.t. the Number of Available Frames M --- p.49Chapter 5.2.3 --- Simulation 3 - Performance w.r.t. the Number of Sources K --- p.50Chapter 5.3 --- Underdetermined Mixing Models --- p.52Chapter 5.3.1 --- Simulation 1 - Success Rate of KR Huber --- p.53Chapter 5.3.2 --- Simulation 2 - Performance w.r.t. SNR --- p.54Chapter 5.3.3 --- Simulation 3 - Performance w.r.t. M --- p.54Chapter 5.3.4 --- Simulation 4 - Performance w.r.t. N --- p.56Chapter 5.4 --- Summary --- p.56Chapter 6 --- Conclusion and Future Works --- p.58Chapter A --- Convolutive Mixing Model --- p.60Chapter B --- Proofs --- p.63Chapter B.1 --- Proof of Theorem 4.1 --- p.63Chapter B.2 --- Proof of Theorem 4.2 --- p.65Chapter B.3 --- Proof of Observation 4.1 --- p.65Chapter B.4 --- Proof of Proposition 4.1 --- p.66Chapter C --- Singular Value Thresholding --- p.67Chapter D --- Categories of Speech Sounds and Their Impact on SOSs-based BI-QSS Algorithms --- p.69Chapter D.1 --- Vowels --- p.69Chapter D.2 --- Consonants --- p.69Chapter D.1 --- Silent Pauses --- p.70Bibliography --- p.7

Tensor and Matrix Inversions with Applications

Higher order tensor inversion is possible for even order. We have shown that a tensor group endowed with the Einstein (contracted) product is isomorphic to the general linear group of degree $n$. With the isomorphic group structures, we derived new tensor decompositions which we have shown to be related to the well-known canonical polyadic decomposition and multilinear SVD. Moreover, within this group structure framework, multilinear systems are derived, specifically, for solving high dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense, that is, when the tensor has an odd number of modes or when the tensor has distinct dimensions in each modes. With the notion of tensor inversion, multilinear systems are solvable. Numerically we solve multilinear systems using iterative techniques, namely biconjugate gradient and Jacobi methods in tensor format

Tensors: a Brief Introduction

International audienceTensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor decomposition plays a central role in identification of underdetermined mixtures. Despite some similarities, CP and Singular value Decomposition (SVD) are quite different. More generally, tensors and matrices enjoy different properties, as pointed out in this brief survey

Decentralized Ambient System Identification of Structures

Many of the existing ambient modal identification methods based on vibration data process information centrally to calculate the modal properties. Such methods demand relatively large memory and processing capabilities to interrogate the data. With the recent advances in wireless sensor technology, it is now possible to process information on the sensor itself. The decentralized information so obtained from individual sensors can be combined to estimate the global modal information of the structure. The main objective of this thesis is to present a new class of decentralized algorithms that can address the limitations stated above. The completed work in this regard involves casting the identification problem within the framework of underdetermined blind source separation (BSS). Time-frequency transformations of measurements are carried out, resulting in a sparse representation of the signals. Stationary wavelet packet transform (SWPT) is used as the primary means to obtain a sparse representation in the time-frequency domain. Several partial setups are used to obtain the partial modal information, which are then combined to obtain the global structural mode information. Most BSS methods in the context of modal identification assume that the excitation is white and do not contain narrow band excitation frequencies. However, this assumption is not satisfied in many situations (e.g., pedestrian bridges) when the excitation is a superposition of narrow-band harmonic(s) and broad-band disturbance. Under such conditions, traditional BSS methods yield sources (modes) without any indication as to whether the identified source(s) is a system or an excitation harmonic. In this research, a novel under-determined BSS algorithm is developed involving statistical characterization of the sources which are used to delineate the sources corresponding to external disturbances versus intrinsic modes of the system. Moreover, the issue of computational burden involving an over-complete dictionary of sparse bases is alleviated through a new underdetermined BSS method based on a tensor algebra tool called PARAllel FACtor (PARAFAC) decomposition. At the core of this method, the wavelet packet decomposition coefficients are used to form a covariance tensor, followed by PARAFAC tensor decomposition to separate the modal responses. Finally, the proposed methods are validated using measurements obtained from both wired and wireless sensors on laboratory scale and full scale buildings and bridges