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

Generalised Bayesian matrix factorisation models

Factor analysis and related models for probabilistic matrix factorisation are of central importance to the unsupervised analysis of data, with a colourful history more than a century long. Probabilistic models for matrix factorisation allow us to explore the underlying structure in data, and have relevance in a vast number of application areas including collaborative filtering, source separation, missing data imputation, gene expression analysis, information retrieval, computational finance and computer vision, amongst others. This thesis develops generalisations of matrix factorisation models that advance our understanding and enhance the applicability of this important class of models. The generalisation of models for matrix factorisation focuses on three concerns: widening the applicability of latent variable models to the diverse types of data that are currently available; considering alternative structural forms in the underlying representations that are inferred; and including higher order data structures into the matrix factorisation framework. These three issues reflect the reality of modern data analysis and we develop new models that allow for a principled exploration and use of data in these settings. We place emphasis on Bayesian approaches to learning and the advantages that come with the Bayesian methodology. Our port of departure is a generalisation of latent variable models to members of the exponential family of distributions. This generalisation allows for the analysis of data that may be real-valued, binary, counts, non-negative or a heterogeneous set of these data types. The model unifies various existing models and constructs for unsupervised settings, the complementary framework to the generalised linear models in regression. Moving to structural considerations, we develop Bayesian methods for learning sparse latent representations. We define ideas of weakly and strongly sparse vectors and investigate the classes of prior distributions that give rise to these forms of sparsity, namely the scale-mixture of Gaussians and the spike-and-slab distribution. Based on these sparsity favouring priors, we develop and compare methods for sparse matrix factorisation and present the first comparison of these sparse learning approaches. As a second structural consideration, we develop models with the ability to generate correlated binary vectors. Moment-matching is used to allow binary data with specified correlation to be generated, based on dichotomisation of the Gaussian distribution. We then develop a novel and simple method for binary PCA based on Gaussian dichotomisation. The third generalisation considers the extension of matrix factorisation models to multi-dimensional arrays of data that are increasingly prevalent. We develop the first Bayesian model for non-negative tensor factorisation and explore the relationship between this model and the previously described models for matrix factorisation.Supported by a Commonwealth Scholarship awarded by the Commonwealth Scholarship and Fellowship Programme (CSFP) [Award number ZACS-2207-363] Supported by award from the National Research Foundation, South Africa (NRF) [Award number SFH2007072200001

Learning Low-Dimensional Models for Heterogeneous Data

Modern data analysis increasingly involves extracting insights, trends and patterns from large and messy data collected from myriad heterogeneous sources. The scale and heterogeneity present exciting new opportunities for discovery, but also create a need for new statistical techniques and theory tailored to these settings. Traditional intuitions often no longer apply, e.g., when the number of variables measured is comparable to the number of samples obtained. A deeper theoretical understanding is needed to develop principled methods and guidelines for statistical data analysis. This dissertation studies the low-dimensional modeling of high-dimensional data in three heterogeneous settings. The first heterogeneity is in the quality of samples, and we consider the standard and ubiquitous low-dimensional modeling technique of Principal Component Analysis (PCA). We analyze how well PCA recovers underlying low-dimensional components from high-dimensional data when some samples are noisier than others (i.e., have heteroscedastic noise). Our analysis characterizes the penalty of heteroscedasticity for PCA, and we consider a weighted variant of PCA that explicitly accounts for heteroscedasticity by giving less weight to samples with more noise. We characterize the performance of weighted PCA for all choices of weights and derive optimal weights. The second heterogeneity is in the statistical properties of data, and we generalize the (increasingly) standard method of Canonical Polyadic (CP) tensor decomposition to allow for general statistical assumptions. Traditional CP tensor decomposition is most natural for data with all entries having Gaussian noise of homogeneous variance. Instead, the Generalized CP (GCP) tensor decomposition we propose allows for other statistical assumptions, and we demonstrate its flexibility on various datasets arising in social networks, neuroscience studies and weather patterns. Fitting GCP with alternative statistical assumptions provides new ways to explore trends in the data and yields improved predictions, e.g., of social network and mouse neural data. The third heterogeneity is in the class of samples, and we consider learning a mixture of low-dimensional subspaces. This model supposes that each sample comes from one of several (unknown) low-dimensional subspaces, that taken together form a union of subspaces (UoS). Samples from the same class come from the same subspace in the union. We consider an ensemble algorithm that clusters the samples, and analyze the approach to provide recovery guarantees. Finally, we propose a sequence of unions of subspaces (SUoS) model that systematically captures samples with heterogeneous complexity, and we describe some early ideas for learning and using SUoS models in patch-based image denoising.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/150043/1/dahong_1.pd

Motion capture data processing, retrieval and recognition.

Character animation plays an essential role in the area of featured film and computer games. Manually creating character animation by animators is both tedious and inefficient, where motion capture techniques (MoCap) have been developed and become the most popular method for creating realistic character animation products. Commercial MoCap systems are expensive and the capturing process itself usually requires an indoor studio environment. Procedural animation creation is often lacking extensive user control during the generation progress. Therefore, efficiently and effectively reusing MoCap data can brings significant benefits, which has motivated wider research in terms of machine learning based MoCap data processing. A typical work flow of MoCap data reusing can be divided into 3 stages: data capture, data management and data reusing. There are still many challenges at each stage. For instance, the data capture and management often suffer from data quality problems. The efficient and effective retrieval method is also demanding due to the large amount of data being used. In addition, classification and understanding of actions are the fundamental basis of data reusing. This thesis proposes to use machine learning on MoCap data for reusing purposes, where a frame work of motion capture data processing is designed. The modular design of this framework enables motion data refinement, retrieval and recognition. The first part of this thesis introduces various methods used in existing motion capture processing approaches in literature and a brief introduction of relevant machine learning methods used in this framework. In general, the frameworks related to refinement, retrieval, recognition are discussed. A motion refinement algorithm based on dictionary learning will then be presented, where kinematical structural and temporal information are exploited. The designed optimization method and data preprocessing technique can ensure a smooth property for the recovered result. After that, a motion refinement algorithm based on matrix completion is presented, where the low-rank property and spatio-temporal information is exploited. Such model does not require preparing data for training. The designed optimization method outperforms existing approaches in regard to both effectiveness and efficiency. A motion retrieval method based on multi-view feature selection is also proposed, where the intrinsic relations between visual words in each motion feature subspace are discovered as a means of improving the retrieval performance. A provisional trace-ratio objective function and an iterative optimization method are also included. A non-negative matrix factorization based motion data clustering method is proposed for recognition purposes, which aims to deal with large scale unsupervised/semi-supervised problems. In addition, deep learning models are used for motion data recognition, e.g. 2D gait recognition and 3D MoCap recognition. To sum up, the research on motion data refinement, retrieval and recognition are presented in this thesis with an aim to tackle the major challenges in motion reusing. The proposed motion refinement methods aim to provide high quality clean motion data for downstream applications. The designed multi-view feature selection algorithm aims to improve the motion retrieval performance. The proposed motion recognition methods are equally essential for motion understanding. A collection of publications by the author of this thesis are noted in publications section

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Shallow Representations, Profound Discoveries : A methodological study of game culture in social media

This thesis explores the potential of representation learning techniques in game studies, highlighting their effectiveness and addressing challenges in data analysis. The primary focus of this thesis is shallow representation learning, which utilizes simpler model architectures but is able to yield effective modeling results. This thesis investigates the following research objectives: disentangling the dependencies of data, modeling temporal dynamics, learning multiple representations, and learning from heterogeneous data. The contributions of this thesis are made from two perspectives: empirical analysis and methodology development, to address these objectives. Chapters 1 and 2 provide a thorough introduction, motivation, and necessary background information for the thesis, framing the research and setting the stage for subsequent publications. Chapters 3 to 5 summarize the contribution of the 6 publications, each of which contributes to demonstrating the effectiveness of representation learning techniques in addressing various analytical challenges. In Chapter 1 and 2, the research objects and questions are also motivated and described. In particular, Introduction to the primary application field game studies is provided and the connections of data analysis and game culture is highlighted. Basic notion of representation learning, and canonical techniques such as probabilistic principal component analysis, topic modeling, and embedding models are described. Analytical challenges and data types are also described to motivate the research of this thesis. Chapter 3 presents two empirical analyses conducted in Publication I and II that present empirical data analysis on player typologies and temporal dynamics of player perceptions. The first empirical analysis takes the advantage of a factor model to offer a flexible player typology analysis. Results and analytical framework are particularly useful for personalized gamification. The Second empirical analysis uses topic modeling to analyze the temporal dynamic of player perceptions of the game No Man’s Sky in relation to game changes. The results reflect a variety of player perceptions including general gaming activities, game mechanic. Moreover, a set of underlying topics that are directly related to game updates and changes are extracted and the temporal dynamics of them have reflected that players responds differently to different updates and changes. Chapter 4 presents two method developments that are related to factor models. The first method, DNBGFA, developed in Publication III, is a matrix factorization model for modeling the temporal dynamics of non-negative matrices from multiple sources. The second mothod, CFTM, developed in Publication IV introduces a factor model to a topic model to handle sophisticated document-level covariates. The develeopd methods in Chapter 4 are also demonstrated for analyzing text data. Chapter 5 summarizes Publication V and Publication VI that develop embedding models. Publication V introduces Bayesian non-parametric to a graph embedding model to learn multiple representations for nodes. Publication VI utilizes a Gaussian copula model to deal with heterogeneous data in representation learning. The develeopd methods in Chapter 5 are also demonstrated for data analysis tasks in the context of online communities. Lastly, Chapter 6 renders discussions and conclusions. Contributions of this thesis are highlighted, limitations, ongoing challenges, and potential future research directions are discussed