Learning and inference with Wasserstein metrics

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Brain and Cognitive Sciences, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 131-143).This thesis develops new approaches for three problems in machine learning, using tools from the study of optimal transport (or Wasserstein) distances between probability distributions. Optimal transport distances capture an intuitive notion of similarity between distributions, by incorporating the underlying geometry of the domain of the distributions. Despite their intuitive appeal, optimal transport distances are often difficult to apply in practice, as computing them requires solving a costly optimization problem. In each setting studied here, we describe a numerical method that overcomes this computational bottleneck and enables scaling to real data. In the first part, we consider the problem of multi-output learning in the presence of a metric on the output domain. We develop a loss function that measures the Wasserstein distance between the prediction and ground truth, and describe an efficient learning algorithm based on entropic regularization of the optimal transport problem. We additionally propose a novel extension of the Wasserstein distance from probability measures to unnormalized measures, which is applicable in settings where the ground truth is not naturally expressed as a probability distribution. We show statistical learning bounds for both the Wasserstein loss and its unnormalized counterpart. The Wasserstein loss can encourage smoothness of the predictions with respect to a chosen metric on the output space. We demonstrate this property on a real-data image tagging problem, outperforming a baseline that doesn't use the metric. In the second part, we consider the probabilistic inference problem for diffusion processes. Such processes model a variety of stochastic phenomena and appear often in continuous-time state space models. Exact inference for diffusion processes is generally intractable. In this work, we describe a novel approximate inference method, which is based on a characterization of the diffusion as following a gradient flow in a space of probability densities endowed with a Wasserstein metric. Existing methods for computing this Wasserstein gradient flow rely on discretizing the underlying domain of the diffusion, prohibiting their application to problems in more than several dimensions. In the current work, we propose a novel algorithm for computing a Wasserstein gradient flow that operates directly in a space of continuous functions, free of any underlying mesh. We apply our approximate gradient flow to the problem of filtering a diffusion, showing superior performance where standard filters struggle. Finally, we study the ecological inference problem, which is that of reasoning from aggregate measurements of a population to inferences about the individual behaviors of its members. This problem arises often when dealing with data from economics and political sciences, such as when attempting to infer the demographic breakdown of votes for each political party, given only the aggregate demographic and vote counts separately. Ecological inference is generally ill-posed, and requires prior information to distinguish a unique solution. We propose a novel, general framework for ecological inference that allows for a variety of priors and enables efficient computation of the most probable solution. Unlike previous methods, which rely on Monte Carlo estimates of the posterior, our inference procedure uses an efficient fixed point iteration that is linearly convergent. Given suitable prior information, our method can achieve more accurate inferences than existing methods. We additionally explore a sampling algorithm for estimating credible regions.by Charles Frogner.Ph. D

An Unsupervised Cluster: Learning Water Customer Behavior Using Variation of Information on a Reconstructed Phase Space

The unsupervised clustering algorithm described in this dissertation addresses the need to divide a population of water utility customers into groups based on their similarities and differences, using only the measured flow data collected by water meters. After clustering, the groups represent customers with similar consumption behavior patterns and provide insight into ‘normal’ and ‘unusual’ customer behavior patterns. This research focuses upon individually metered water utility customers and includes both residential and commercial customer accounts serviced by utilities within North America. The contributions of this dissertation not only represent a novel academic work, but also solve a practical problem for the utility industry. This dissertation introduces a method of agglomerative clustering using information theoretic distance measures on Gaussian mixture models within a reconstructed phase space. The clustering method accommodates a utility’s limited human, financial, computational, and environmental resources. The proposed weighted variation of information distance measure for comparing Gaussian mixture models places emphasis upon those behaviors whose statistical distributions are more compact over those behaviors with large variation and contributes a novel addition to existing comparison options

Bayesian methods for non-gaussian data modeling and applications

Finite mixture models are among the most useful machine learning techniques and are receiving considerable attention in various applications. The use of finite mixture models in image and signal processing has proved to be of considerable interest in terms of both theoretical development and in their usefulness in several applications. In most of the applications, the Gaussian density is used in the mixture modeling of data. Although a Gaussian mixture may provide a reasonable approximation to many real-world distributions, it is certainly not always the best approximation especially in image and signal processing applications where we often deal with non-Gaussian data. In this thesis, we propose two novel approaches that may be used in modeling non-Gaussian data. These approaches use two highly flexible distributions, the generalized Gaussian distribution (GGD) and the general Beta distribution, in order to model the data. We are motivated by the fact that these distributions are able to fit many distributional shapes and then can be considered as a useful class of flexible models to address several problems and applications involving measurements and features having well-known marked deviation from the Gaussian shape. For the mixture estimation and selection problem, researchers have demonstrated that Bayesian approaches are fully optimal. The Bayesian learning allows the incorporation of prior knowledge in a formal coherent way that avoids overfitting problems. For this reason, we adopt different Bayesian approaches in order to learn our models parameters. First, we present a fully Bayesian approach to analyze finite generalized Gaussian mixture models which incorporate several standard mixtures, such as Laplace and Gaussian. This approach evaluates the posterior distribution and Bayes estimators using a Gibbs sampling algorithm, and selects the number of components in the mixture using the integrated likelihood. We also propose a fully Bayesian approach for finite Beta mixtures learning using a Reversible Jump Markov Chain Monte Carlo (RJMCMC) technique which simultaneously allows cluster assignments, parameters estimation, and the selection of the optimal number of clusters. We then validate the proposed methods by applying them to different image processing applications

Multiple human tracking in RGB-depth data: A survey

© The Institution of Engineering and Technology. Multiple human tracking (MHT) is a fundamental task in many computer vision applications. Appearance-based approaches, primarily formulated on RGB data, are constrained and affected by problems arising from occlusions and/or illumination variations. In recent years, the arrival of cheap RGB-depth devices has led to many new approaches to MHT, and many of these integrate colour and depth cues to improve each and every stage of the process. In this survey, the authors present the common processing pipeline of these methods and review their methodology based (a) on how they implement this pipeline and (b) on what role depth plays within each stage of it. They identify and introduce existing, publicly available, benchmark datasets and software resources that fuse colour and depth data for MHT. Finally, they present a brief comparative evaluation of the performance of those works that have applied their methods to these datasets

Multiple view human activity recognition

Ankara : The Department of Computer Engineering and the Graduate School of Engineering and Science of Bilkent University, 2012.Thesis (Ph. D.) -- Bilkent University, 2012.Includes bibliographical references leaves 94-100.This thesis explores the human activity recognition problem when multiple views are available. We follow two main directions: we first present a system that performs volume matching using constructed 3D volumes from calibrated cameras, then we present a flexible system based on frame matching directly using multiple views. We examine the multiple view systems compared to single view systems, and measure the performance improvements in recognition using more views by various experiments. Initial part of the thesis introduces compact representations for volumetric data gained through reconstruction. The video frames recorded by many cameras with significant overlap are fused by reconstruction, and the reconstructed volumes are used as substitutes of action poses. We propose new pose descriptors over these three dimensional volumes. Our first descriptor is based on the histogram of oriented cylinders in various sizes and orientations. We then propose another descriptor which is view-independent, and which does not require pose alignment. We show the importance of discriminative pose representations within simpler activity classification schemes. Activity recognition framework based on volume matching presents promising results compared to the state-of-the-art. Volume reconstruction is one natural approach for multi camera data fusion, but there can be few cameras with overlapping views. In the second part of the thesis, we introduce an architecture that is adaptable to various number of cameras and features. The system collects and fuses activity judgments from cameras using a voting scheme. The architecture requires no camera calibration. Performance generally improves when there are more cameras and more features; training and test cameras do not need to overlap; camera drop in or drop out is handled easily with little penalty. Experiments support the performance penalties, and advantages for using multiple views versus single view.Pehlivan, SelenPh.D

Unsupervised Discovery and Representation of Subspace Trends in Massive Biomedical Datasets

The goal of this dissertation is to develop unsupervised algorithms for discovering previously unknown subspace trends in massive multivariate biomedical data sets without the benefit of prior information. A subspace trend is a sustained pattern of gradual/progressive changes within an unknown subset of feature dimensions. A fundamental challenge to subspace trend discovery is the presence of irrelevant data dimensions, noise, outliers, and confusion from multiple subspace trends driven by independent factors that are mixed in with each other. These factors can obscure the trends in traditional dimension reduction and projection based data visualizations. To overcome these limitations, we propose a novel graph-theoretic neighborhood similarity measure for sensing concordant progressive changes across data dimensions. Using this measure, we present an unsupervised algorithm for trend-relevant feature selection and visualization. Additionally, we propose to use an efficient online density-based representation to make the algorithm scalable for massive datasets. The representation not only assists in trend discovery, but also in cluster detection including rare populations. Our method has been successfully applied to diverse synthetic and real-world biomedical datasets, such as gene expression microarray and arbor morphology of neurons and microglia in brain tissue. Derived representations revealed biologically meaningful hidden subspace trend(s) that were obscured by irrelevant features and noise. Although our applications are mostly from the biomedical domain, the proposed algorithm is broadly applicable to exploratory analysis of high-dimensional data including visualization, hypothesis generation, knowledge discovery, and prediction in diverse other applications.Electrical and Computer Engineering, Department o

Robust density modelling using the student's t-distribution for human action recognition

The extraction of human features from videos is often inaccurate and prone to outliers. Such outliers can severely affect density modelling when the Gaussian distribution is used as the model since it is highly sensitive to outliers. The Gaussian distribution is also often used as base component of graphical models for recognising human actions in the videos (hidden Markov model and others) and the presence of outliers can significantly affect the recognition accuracy. In contrast, the Student's t-distribution is more robust to outliers and can be exploited to improve the recognition rate in the presence of abnormal data. In this paper, we present an HMM which uses mixtures of t-distributions as observation probabilities and show how experiments over two well-known datasets (Weizmann, MuHAVi) reported a remarkable improvement in classification accuracy. © 2011 IEEE