Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Statistical Approaches for Signal Processing with Application to Automatic Singer Identification

In the music world, the oldest instrument is known as the singing voice that plays an important role in musical recordings. The singer\u27s identity serves as a primary aid for people to organize, browse, and retrieve music recordings. In this thesis, we focus on the problem of singer identification based on the acoustic features of singing voice. An automatic singer identification system is constructed and has achieved a very high identification accuracy. This system consists of three crucial parts: singing voice detection, background music removal and pattern recognition. These parts are introduced and explored in great details in this thesis. To be specific, in terms of the singing voice detection, we firstly study a traditional method, double GMM. Then an improved method, namely single GMM, is proposed. The experimental result shows that the detection accuracy of single GMM can be achieved as high as 96.42%. In terms of the background music removal, Non-negative Matrix Factorization (NMF) and Robust Principal Component Analysis (RPCA) are demonstrated. The evaluation result shows that RPCA outperforms NMF. In terms of pattern recognition, we explore the algorithms of Support Vector Machine (SVM) and Gaussian Mixture Model (GMM). Based on the experimental results, it turns out that the prediction accuracy of GMM classifier is about 16% higher than SVM

Ensemble learning of high dimension datasets

Ensemble learning, an approach in Machine Learning, makes decisions based on the collective decision of a committee of learners to solve complex tasks with minimal human intervention. Advances in computing technology have enabled researchers build datasets with the number of features in the order of thousands and enabled building more accurate predictive models. Unfortunately, high dimensional datasets are especially challenging for machine learning due to the phenomenon dubbed as the "curse of dimensionality". One approach to overcoming this challenge is ensemble learning using Random Subspace (RS) method, which has been shown to perform very well empirically however with few theoretical explanations to said effectiveness for classification tasks. In this thesis, we aim to provide theoretical insights into RS ensemble classifiers to give a more in-depth understanding of the theoretical foundations of other ensemble classifiers. We investigate the conditions for norm-preservations in RS projections. Insights into this provide us with the theoretical basis for RS in algorithms that are based on the geometry of the data (i.e. clustering, nearest-neighbour). We then investigate the guarantees for the dot products of two random vectors after RS projection. This guarantee is useful to capture the geometric structure of a classification problem. We will then investigate the accuracy of a majority vote ensemble using a generalized Polya-Urn model, and how the parameters of the model are derived from diversity measures. We will discuss the practical implications of the model, explore the noise tolerance of ensembles, and give a plausible explanation for the effectiveness of ensembles. We will provide empirical corroboration for our main results with both synthetic and real-world high-dimensional data. We will also discuss the implications of our theory on other applications (i.e. compressive sensing). Based on our results, we will propose a method of building ensembles for Deep Neural Network image classifications using RS projections without needing to retrain the neural network, which showed improved accuracy and very good robustness to adversarial examples. Ultimately, we hope that the insights gained in this thesis would make in-roads towards the answer to a key open question for ensemble classifiers, "When will an ensemble of weak learners outperform a single carefully tuned learner?

Sparse Modeling for Image and Vision Processing

In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio