A Survey on Metric Learning for Feature Vectors and Structured Data

The need for appropriate ways to measure the distance or similarity between data is ubiquitous in machine learning, pattern recognition and data mining, but handcrafting such good metrics for specific problems is generally difficult. This has led to the emergence of metric learning, which aims at automatically learning a metric from data and has attracted a lot of interest in machine learning and related fields for the past ten years. This survey paper proposes a systematic review of the metric learning literature, highlighting the pros and cons of each approach. We pay particular attention to Mahalanobis distance metric learning, a well-studied and successful framework, but additionally present a wide range of methods that have recently emerged as powerful alternatives, including nonlinear metric learning, similarity learning and local metric learning. Recent trends and extensions, such as semi-supervised metric learning, metric learning for histogram data and the derivation of generalization guarantees, are also covered. Finally, this survey addresses metric learning for structured data, in particular edit distance learning, and attempts to give an overview of the remaining challenges in metric learning for the years to come.Comment: Technical report, 59 pages. Changes in v2: fixed typos and improved presentation. Changes in v3: fixed typos. Changes in v4: fixed typos and new method

Clutter suppression in ultrasound: performance evaluation and review of low-rank and sparse matrix decomposition methods

Vessel diseases are often accompanied by abnormalities related to vascular shape and size. Therefore, a clear visualization of vasculature is of high clinical significance. Ultrasound color flow imaging (CFI) is one of the prominent techniques for flow visualization. However, clutter signals originating from slow-moving tissue are one of the main obstacles to obtain a clear view of the vascular network. Enhancement of the vasculature by suppressing the clutters is a significant and irreplaceable step for many applications of ultrasound CFI. Currently, this task is often performed by singular value decomposition (SVD) of the data matrix. This approach exhibits two well-known limitations. First, the performance of SVD is sensitive to the proper manual selection of the ranks corresponding to clutter and blood subspaces. Second, SVD is prone to failure in the presence of large random noise in the dataset. A potential solution to these issues is using decomposition into low-rank and sparse matrices (DLSM) framework. SVD is one of the algorithms for solving the minimization problem under the DLSM framework. Many other algorithms under DLSM avoid full SVD and use approximated SVD or SVD-free ideas which may have better performance with higher robustness and less computing time. In practice, these models separate blood from clutter based on the assumption that steady clutter represents a low-rank structure and that the moving blood component is sparse. In this paper, we present a comprehensive review of ultrasound clutter suppression techniques and exploit the feasibility of low-rank and sparse decomposition schemes in ultrasound clutter suppression. We conduct this review study by adapting 106 DLSM algorithms and validating them against simulation, phantom, and in vivo rat datasets. Two conventional quality metrics, signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR), are used for performance evaluation. In addition, computation times required by different algorithms for generating clutter suppressed images are reported. Our extensive analysis shows that the DLSM framework can be successfully applied to ultrasound clutter suppression

Inference, Computation, and Games

In this thesis, we use statistical inference and competitive games to design algorithms for computational mathematics. In the first part, comprising chapters two through six, we use ideas from Gaussian process statistics to obtain fast solvers for differential and integral equations. We begin by observing the equivalence of conditional (near-)independence of Gaussian processes and the (near-)sparsity of the Cholesky factors of its precision and covariance matrices. This implies the existence of a large class of dense matrices with almost sparse Cholesky factors, thereby greatly increasing the scope of application of sparse Cholesky factorization. Using an elimination ordering and sparsity pattern motivated by the screening effect in spatial statistics, we can compute approximate Cholesky factors of the covariance matrices of Gaussian processes admitting a screening effect in near-linear computational complexity. These include many popular smoothness priors such as the Matérn class of covariance functions. In the special case of Green's matrices of elliptic boundary value problems (with possibly unknown elliptic operators of arbitrarily high order, with possibly rough coefficients), we can use tools from numerical homogenization to prove the exponential accuracy of our method. This result improves the state-of-the-art for solving general elliptic integral equations and provides the first proof of an exponential screening effect. We also derive a fast solver for elliptic partial differential equations, with accuracy-vs-complexity guarantees that improve upon the state-of-the-art. Furthermore, the resulting solver is performant in practice, frequently beating established algebraic multigrid libraries such as AMGCL and Trilinos on a series of challenging problems in two and three dimensions. Finally, for any given covariance matrix, we obtain a closed-form expression for its optimal (in terms of Kullback-Leibler divergence) approximate inverse-Cholesky factorization subject to a sparsity constraint, recovering Vecchia approximation and factorized sparse approximate inverses. Our method is highly robust, embarrassingly parallel, and further improves our asymptotic results on the solution of elliptic integral equations. We also provide a way to apply our techniques to sums of independent Gaussian processes, resolving a major limitation of existing methods based on the screening effect. As a result, we obtain fast algorithms for large-scale Gaussian process regression problems with possibly noisy measurements. In the second part of this thesis, comprising chapters seven through nine, we study continuous optimization through the lens of competitive games. In particular, we consider competitive optimization, where multiple agents attempt to minimize conflicting objectives. In the single-agent case, the updates of gradient descent are minimizers of quadratically regularized linearizations of the loss function. We propose to generalize this idea by using the Nash equilibria of quadratically regularized linearizations of the competitive game as updates (linearize the game). We provide fundamental reasons why the natural notion of linearization for competitive optimization problems is given by the multilinear (as opposed to linear) approximation of the agents' loss functions. The resulting algorithm, which we call competitive gradient descent, thus provides a natural generalization of gradient descent to competitive optimization. By using ideas from information geometry, we extend CGD to competitive mirror descent (CMD) that can be applied to a vast range of constrained competitive optimization problems. CGD and CMD resolve the cycling problem of simultaneous gradient descent and show promising results on problems arising in constrained optimization, robust control theory, and generative adversarial networks. Finally, we point out the GAN-dilemma that refutes the common interpretation of GANs as approximate minimizers of a divergence obtained in the limit of a fully trained discriminator. Instead, we argue that GAN performance relies on the implicit competitive regularization (ICR) due to the simultaneous optimization of generator and discriminator and support this hypothesis with results on low-dimensional model problems and GANs on CIFAR10.</p

Graphical Models for Multivariate Time-Series

Gaussian graphical models have received much attention in the last years, due to their flexibility and expression power. In particular, lots of interests have been devoted to graphical models for temporal data, or dynamical graphical models, to understand the relation of variables evolving in time. While powerful in modelling complex systems, such models suffer from computational issues both in terms of convergence rates and memory requirements, and may fail to detect temporal patterns in case the information on the system is partial. This thesis comprises two main contributions in the context of dynamical graphical models, tackling these two aspects: the need of reliable and fast optimisation methods and an increasing modelling power, which are able to retrieve the model in practical applications. The first contribution consists in a forward-backward splitting (FBS) procedure for Gaussian graphical modelling of multivariate time-series which relies on recent theoretical studies ensuring global convergence under mild assumptions. Indeed, such FBS-based implementation achieves, with fast convergence rates, optimal results with respect to ground truth and standard methods for dynamical network inference. The second main contribution focuses on the problem of latent factors, that influence the system while hidden or unobservable. This thesis proposes the novel latent variable time-varying graphical lasso method, which is able to take into account both temporal dynamics in the data and latent factors influencing the system. This is fundamental for the practical use of graphical models, where the information on the data is partial. Indeed, extensive validation of the method on both synthetic and real applications shows the effectiveness of considering latent factors to deal with incomplete information

Forecasting People Trajectories and Head Poses by Jointly Reasoning on Tracklets and Vislets

In this work, we explore the correlation between people trajectories and their head orientations. We argue that people trajectory and head pose forecasting can be modelled as a joint problem. Recent approaches on trajectory forecasting leverage short-term trajectories (aka tracklets) of pedestrians to predict their future paths. In addition, sociological cues, such as expected destination or pedestrian interaction, are often combined with tracklets. In this paper, we propose MiXing-LSTM (MX-LSTM) to capture the interplay between positions and head orientations (vislets) thanks to a joint unconstrained optimization of full covariance matrices during the LSTM backpropagation. We additionally exploit the head orientations as a proxy for the visual attention, when modeling social interactions. MX-LSTM predicts future pedestrians location and head pose, increasing the standard capabilities of the current approaches on long-term trajectory forecasting. Compared to the state-of-the-art, our approach shows better performances on an extensive set of public benchmarks. MX-LSTM is particularly effective when people move slowly, i.e. the most challenging scenario for all other models. The proposed approach also allows for accurate predictions on a longer time horizon.Comment: Accepted at IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2019. arXiv admin note: text overlap with arXiv:1805.0065

Robust Learning from Multiple Information Sources

In the big data era, the ability to handle high-volume, high-velocity and high-variety information assets has become a basic requirement for data analysts. Traditional learning models, which focus on medium size, single source data, often fail to achieve reliable performance if data come from multiple heterogeneous sources (views). As a result, robust multi-view data processing methods that are insensitive to corruptions and anomalies in the data set are needed. This thesis develops robust learning methods for three problems that arise from real-world applications: robust training on a noisy training set, multi-view learning in the presence of between-view inconsistency and network topology inference using partially observed data. The central theme behind all these methods is the use of information-theoretic measures, including entropies and information divergences, as parsimonious representations of uncertainties in the data, as robust optimization surrogates that allows for efficient learning, and as flexible and reliable discrepancy measures for data fusion. More specifically, the thesis makes the following contributions: 1. We propose a maximum entropy-based discriminative learning model that incorporates the minimal entropy (ME) set anomaly detection technique. The resulting probabilistic model can perform both nonparametric classification and anomaly detection simultaneously. An efficient algorithm is then introduced to estimate the posterior distribution of the model parameters while selecting anomalies in the training data. 2. We consider a multi-view classification problem on a statistical manifold where class labels are provided by probabilistic density functions (p.d.f.) and may not be consistent among different views due to the existence of noise corruption. A stochastic consensus-based multi-view learning model is proposed to fuse predictive information for multiple views together. By exploring the non-Euclidean structure of the statistical manifold, a joint consensus view is constructed that is robust to single-view noise corruption and between-view inconsistency. 3. We present a method for estimating the parameters (partial correlations) of a Gaussian graphical model that learns a sparse sub-network topology from partially observed relational data. This model is applicable to the situation where the partial correlations between pairs of variables on a measured sub-network (internal data) are to be estimated when only summary information about the partial correlations between variables outside of the sub-network (external data) are available. The proposed model is able to incorporate the dependence structure between latent variables from external sources and perform latent feature selection efficiently. From a multi-view learning perspective, it can be seen as a two-view learning system given asymmetric information flow from both the internal view and the external view.PHDElectrical & Computer Eng PhDUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138599/1/tianpei_1.pd

Doctor of Philosophy

dissertationMachine learning is the science of building predictive models from data that automatically improve based on past experience. To learn these models, traditional learning algorithms require labeled data. They also require that the entire dataset fits in the memory of a single machine. Labeled data are available or can be acquired for small and moderately sized datasets but curating large datasets can be prohibitively expensive. Similarly, massive datasets are usually too huge to fit into the memory of a single machine. An alternative is to distribute the dataset over multiple machines. Distributed learning, however, poses new challenges as most existing machine learning techniques are inherently sequential. Additionally, these distributed approaches have to be designed keeping in mind various resource limitations of real-world settings, prime among them being intermachine communication. With the advent of big datasets machine learning algorithms are facing new challenges. Their design is no longer limited to minimizing some loss function but, additionally, needs to consider other resources that are critical when learning at scale. In this thesis, we explore different models and measures for learning with limited resources that have a budget. What budgetary constraints are posed by modern datasets? Can we reuse or combine existing machine learning paradigms to address these challenges at scale? How does the cost metrics change when we shift to distributed models for learning? These are some of the questions that have been investigated in this thesis. The answers to these questions hold the key to addressing some of the challenges faced when learning on massive datasets. In the first part of this thesis, we present three different budgeted scenarios that deal with scarcity of labeled data and limited computational resources. The goal is to leverage transfer information from related domains to learn under budgetary constraints. Our proposed techniques comprise semisupervised transfer, online transfer and active transfer. In the second part of this thesis, we study distributed learning with limited communication. We present initial sampling based results, as well as, propose communication protocols for learning distributed linear classifiers

Distributional Gradient Matching for Learning Uncertain Neural Dynamics Models

Differential equations in general and neural ODEs in particular are an essential technique in continuous-time system identification. While many deterministic learning algorithms have been designed based on numerical integration via the adjoint method, many downstream tasks such as active learning, exploration in reinforcement learning, robust control, or filtering require accurate estimates of predictive uncertainties. In this work, we propose a novel approach towards estimating epistemically uncertain neural ODEs, avoiding the numerical integration bottleneck. Instead of modeling uncertainty in the ODE parameters, we directly model uncertainties in the state space. Our algorithm - distributional gradient matching (DGM) - jointly trains a smoother and a dynamics model and matches their gradients via minimizing a Wasserstein loss. Our experiments show that, compared to traditional approximate inference methods based on numerical integration, our approach is faster to train, faster at predicting previously unseen trajectories, and in the context of neural ODEs, significantly more accurate.Comment: Published at NeurIPS 202

An enhanced augmented electric field integral equation and the calculation of casimir force using computational electromagnetics

In this dissertation, a surface integral equation formulation is developed for low-frequency problems by generalizing the existing augmented electric field integral equation from the perfect electric conductors to the dielectrics and general conductors. Detailed discussions of the basis functions and the pre-conditioner are provided for the dielectric problems, and a novel integration scheme for the evaluations of the matrix elements in the conductor problem is proposed. Then a broadband multilevel fast multipole algorithm (FMA) using a hybridization of the multipole and plane wave expansions is introduced. This high-accuracy algorithm is error controllable and stable at low frequencies. It reduces to the conventional diagonal FMA at higher frequencies. Therefore it can be regarded as a generalization of the dense FMA at low frequencies and the diagonal FMA at higher frequencies. Finally, the computational electromagnetic techniques are applied to the calculations of the Casimir force. The application of the integral equation method in Casimir force calculation is briefly reviewed and we proposed an efficient computing scheme using the randomized singular value decomposition and the hybrid FMA. As a result, the efficiency can be greatly enhanced for large problems