Xuanlong Nguyen

Ram Rajagopal, XuanLong Nguyen, Sinem Coleri Ergen et al.

Single fault sequential change point problems have become important in modeling for various phenomena in large distributed systems, such as sensor networks. But such systems in many situations present multiple interacting faults. For example, individual sensors in a network may fail and detection is performed by comparing measurements between sensors, resulting in statistical dependency among faults. We present a new formulation for multiple interacting faults in a distributed system. The formulation includes specifications of how individual subsystems composing the large system may fail, the information that can be shared among these subsystems and the interaction pattern between faults. We then specify a new sequential algorithm for detecting these faults. The main feature of the algorithm is that it uses composite stopping rules for a subsystem that depend on the decision of other subsystems. We provide asymptotic false alarm and detection delay analysis for this algorithm in the Bayesian setting and show that under certain conditions the algorithm is optimal. The analysis methodology relies on novel detailed comparison techniques between stopping times. We validate the approach with some simulations.

Jiacheng Zhu, Gregory Darnell, Agni Kumar et al.

Heart rate variability (HRV) is a practical and noninvasive measure of autonomic nervous system activity, which plays an essential role in cardiovascular health. However, using HRV to assess physiology status is challenging. Even in clinical settings, HRV is sensitive to acute stressors such as physical activity, mental stress, hydration, alcohol, and sleep. Wearable devices provide convenient HRV measurements, but the irregularity of measurements and uncaptured stressors can bias conventional analytical methods. To better interpret HRV measurements for downstream healthcare applications, we learn a personalized diurnal rhythm as an accurate physiological indicator for each individual. We develop Physiological Multitask-Learning (PhysioMTL) by harnessing Optimal Transport theory within a Multitask-learning (MTL) framework. The proposed method learns an individual-specific predictive model from heterogeneous observations, and enables estimation of an optimal transport map that yields a push forward operation onto the demographic features for each task. Our model outperforms competing MTL methodologies on unobserved predictive tasks for synthetic and two real-world datasets. Specifically, our method provides remarkable prediction results on unseen held-out subjects given only $20\%$ of the subjects in real-world observational studies. Furthermore, our model enables a counterfactual engine that generates the effect of acute stressors and chronic conditions on HRV rhythms.

Jiacheng Zhu, Jielin Qiu, Aritra Guha et al.

We propose to study and promote the robustness of a model as per its performance through the interpolation of training data distributions. Specifically, (1) we augment the data by finding the worst-case Wasserstein barycenter on the geodesic connecting subpopulation distributions of different categories. (2) We regularize the model for smoother performance on the continuous geodesic path connecting subpopulation distributions. (3) Additionally, we provide a theoretical guarantee of robustness improvement and investigate how the geodesic location and the sample size contribute, respectively. Experimental validations of the proposed strategy on \textit{four} datasets, including CIFAR-100 and ImageNet, establish the efficacy of our method, e.g., our method improves the baselines' certifiable robustness on CIFAR10 up to $7.7\%$, with $16.8\%$ on empirical robustness on CIFAR-100. Our work provides a new perspective of model robustness through the lens of Wasserstein geodesic-based interpolation with a practical off-the-shelf strategy that can be combined with existing robust training methods.

Dat Do, Nhat Ho, XuanLong Nguyen

As we collect additional samples from a data population for which a known density function estimate may have been previously obtained by a black box method, the increased complexity of the data set may result in the true density being deviated from the known estimate by a mixture distribution. To model this phenomenon, we consider the \emph{deviating mixture model} $(1-λ^{*})h_0 + λ^{*} (\sum_{i = 1}^{k} p_{i}^{*} f(x|θ_{i}^{*}))$, where $h_0$ is a known density function, while the deviated proportion $λ^{*}$ and latent mixing measure $G_{*} = \sum_{i = 1}^{k} p_{i}^{*} δ_{θ_i^{*}}$ associated with the mixture distribution are unknown. Via a novel notion of distinguishability between the known density $h_{0}$ and the deviated mixture distribution, we establish rates of convergence for the maximum likelihood estimates of $λ^{*}$ and $G^{*}$ under Wasserstein metric. Simulation studies are carried out to illustrate the theory.

Sunrit Chakraborty, Aritra Guha, Rayleigh Lei et al.

Analysis of heterogeneous patterns in complex spatio-temporal data finds usage across various domains in applied science and engineering, including training autonomous vehicles to navigate in complex traffic scenarios. Motivated by applications arising in the transportation domain, in this paper we develop a model for learning heterogeneous and dynamic patterns of velocity field data. We draw from basic nonparameric Bayesian modeling elements such as hierarchical Dirichlet process and infinite hidden Markov model, while the smoothness of each homogeneous velocity field element is captured with a Gaussian process prior. Of particular focus is a scalable approximate inference method for the proposed model; this is achieved by employing sequential MAP estimates from the infinite HMM model and an efficient sequential GP posterior computation technique, which is shown to work effectively on simulated data sets. Finally, we demonstrate the effectiveness of our techniques to the NGSIM dataset of complex multi-vehicle interactions.

Jiacheng Zhu, Aritra Guha, Dat Do et al.

We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator mapping a Hilbert space of functions to another. For numerous machine learning tasks, data can be naturally viewed as samples drawn from spaces of functions, such as curves and surfaces, in high dimensions. Optimal transport for functional data analysis provides a useful framework of treatment for such domains. { Since probability measures in infinite dimensional spaces generally lack absolute continuity (that is, with respect to non-degenerate Gaussian measures), the Monge map in the standard optimal transport theory for finite dimensional spaces may not exist. Our approach to the optimal transport problem in infinite dimensions is by a suitable regularization technique -- we restrict the class of transport maps to be a Hilbert-Schmidt space of operators.} To this end, we develop an efficient algorithm for finding the stochastic transport map between functional domains and provide theoretical guarantees on the existence, uniqueness, and consistency of our estimate for the Hilbert-Schmidt operator. We validate our method on synthetic datasets and examine the functional properties of the transport map. Experiments on real-world datasets of robot arm trajectories further demonstrate the effectiveness of our method on applications in domain adaptation.

Aritra Guha, Rayleigh Lei, Jiacheng Zhu et al.

Robust representation learning of temporal dynamic interactions is an important problem in robotic learning in general and automated unsupervised learning in particular. Temporal dynamic interactions can be described by (multiple) geometric trajectories in a suitable space over which unsupervised learning techniques may be applied to extract useful features from raw and high-dimensional data measurements. Taking a geometric approach to robust representation learning for temporal dynamic interactions, it is necessary to develop suitable metrics and a systematic methodology for comparison and for assessing the stability of an unsupervised learning method with respect to its tuning parameters. Such metrics must account for the (geometric) constraints in the physical world as well as the uncertainty associated with the learned patterns. In this paper we introduce a model-free metric based on the Procrustes distance for robust representation learning of interactions, and an optimal transport based distance metric for comparing between distributions of interaction primitives. These distance metrics can serve as an objective for assessing the stability of an interaction learning algorithm. They are also used for comparing the outcomes produced by different algorithms. Moreover, they may also be adopted as an objective function to obtain clusters and representative interaction primitives. These concepts and techniques will be introduced, along with mathematical properties, while their usefulness will be demonstrated in unsupervised learning of vehicle-to-vechicle interactions extracted from the Safety Pilot database, the world's largest database for connected vehicles.

Ryan Curtin, Ben Moseley, Hung Q. Ngo et al.

Conventional machine learning algorithms cannot be applied until a data matrix is available to process. When the data matrix needs to be obtained from a relational database via a feature extraction query, the computation cost can be prohibitive, as the data matrix may be (much) larger than the total input relation size. This paper introduces Rk-means, or relational k -means algorithm, for clustering relational data tuples without having to access the full data matrix. As such, we avoid having to run the expensive feature extraction query and storing its output. Our algorithm leverages the underlying structures in relational data. It involves construction of a small {\it grid coreset} of the data matrix for subsequent cluster construction. This gives a constant approximation for the k -means objective, while having asymptotic runtime improvements over standard approaches of first running the database query and then clustering. Empirical results show orders-of-magnitude speedup, and Rk-means can run faster on the database than even just computing the data matrix.

Viet Huynh, Nhat Ho, Nhan Dam et al.

We propose a novel approach to the problem of multilevel clustering, which aims to simultaneously partition data in each group and discover grouping patterns among groups in a potentially large hierarchically structured corpus of data. Our method involves a joint optimization formulation over several spaces of discrete probability measures, which are endowed with Wasserstein distance metrics. We propose several variants of this problem, which admit fast optimization algorithms, by exploiting the connection to the problem of finding Wasserstein barycenters. Consistency properties are established for the estimates of both local and global clusters. Finally, experimental results with both synthetic and real data are presented to demonstrate the flexibility and scalability of the proposed approach.

Mikhail Yurochkin, Aritra Guha, Yuekai Sun et al.

We propose Dirichlet Simplex Nest, a class of probabilistic models suitable for a variety of data types, and develop fast and provably accurate inference algorithms by accounting for the model's convex geometry and low dimensional simplicial structure. By exploiting the connection to Voronoi tessellation and properties of Dirichlet distribution, the proposed inference algorithm is shown to achieve consistency and strong error bound guarantees on a range of model settings and data distributions. The effectiveness of our model and the learning algorithm is demonstrated by simulations and by analyses of text and financial data.

Mahmoud Abo Khamis, Ryan R. Curtin, Benjamin Moseley et al.

Motivated by fundamental applications in databases and relational machine learning, we formulate and study the problem of answering functional aggregate queries (FAQ) in which some of the input factors are defined by a collection of additive inequalities between variables. We refer to these queries as FAQ-AI for short. To answer FAQ-AI in the Boolean semiring, we define relaxed tree decompositions and relaxed submodular and fractional hypertree width parameters. We show that an extension of the InsideOut algorithm using Chazelle's geometric data structure for solving the semigroup range search problem can answer Boolean FAQ-AI in time given by these new width parameters. This new algorithm achieves lower complexity than known solutions for FAQ-AI. It also recovers some known results in database query answering. Our second contribution is a relaxation of the set of polymatroids that gives rise to the counting version of the submodular width, denoted by #subw. This new width is sandwiched between the submodular and the fractional hypertree widths. Any FAQ and FAQ-AI over one semiring can be answered in time proportional to #subw and respectively to the relaxed version of #subw. We present three applications of our FAQ-AI framework to relational machine learning: k-means clustering, training linear support vector machines, and training models using non-polynomial loss. These optimization problems can be solved over a database asymptotically faster than computing the join of the database relations.

Mikhail Yurochkin, Zhiwei Fan, Aritra Guha et al.

We develop new models and algorithms for learning the temporal dynamics of the topic polytopes and related geometric objects that arise in topic model based inference. Our model is nonparametric Bayesian and the corresponding inference algorithm is able to discover new topics as the time progresses. By exploiting the connection between the modeling of topic polytope evolution, Beta-Bernoulli process and the Hungarian matching algorithm, our method is shown to be several orders of magnitude faster than existing topic modeling approaches, as demonstrated by experiments working with several million documents in under two dozens of minutes.

Mikhail Yurochkin, Aritra Guha, XuanLong Nguyen

We propose new algorithms for topic modeling when the number of topics is unknown. Our approach relies on an analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex hull of vertices representing the latent topics. Our algorithms are shown in practice to have accuracy comparable to a Gibbs sampler in terms of topic estimation, which requires the number of topics be given. Moreover, they are one of the fastest among several state of the art parametric techniques. Statistical consistency of our estimator is established under some conditions.

Mikhail Yurochkin, XuanLong Nguyen, Nikolaos Vasiloglou

We propose a Bayesian regression method that accounts for multi-way interactions of arbitrary orders among the predictor variables. Our model makes use of a factorization mechanism for representing the regression coefficients of interactions among the predictors, while the interaction selection is guided by a prior distribution on random hypergraphs, a construction which generalizes the Finite Feature Model. We present a posterior inference algorithm based on Gibbs sampling, and establish posterior consistency of our regression model. Our method is evaluated with extensive experiments on simulated data and demonstrated to be able to identify meaningful interactions in applications in genetics and retail demand forecasting.

Nhat Ho, XuanLong Nguyen, Mikhail Yurochkin et al.

We propose a novel approach to the problem of multilevel clustering, which aims to simultaneously partition data in each group and discover grouping patterns among groups in a potentially large hierarchically structured corpus of data. Our method involves a joint optimization formulation over several spaces of discrete probability measures, which are endowed with Wasserstein distance metrics. We propose a number of variants of this problem, which admit fast optimization algorithms, by exploiting the connection to the problem of finding Wasserstein barycenters. Consistency properties are established for the estimates of both local and global clusters. Finally, experiment results with both synthetic and real data are presented to demonstrate the flexibility and scalability of the proposed approach.

Mikhail Yurochkin, XuanLong Nguyen

We propose a geometric algorithm for topic learning and inference that is built on the convex geometry of topics arising from the Latent Dirichlet Allocation (LDA) model and its nonparametric extensions. To this end we study the optimization of a geometric loss function, which is a surrogate to the LDA's likelihood. Our method involves a fast optimization based weighted clustering procedure augmented with geometric corrections, which overcomes the computational and statistical inefficiencies encountered by other techniques based on Gibbs sampling and variational inference, while achieving the accuracy comparable to that of a Gibbs sampler. The topic estimates produced by our method are shown to be statistically consistent under some conditions. The algorithm is evaluated with extensive experiments on simulated and real data.

Nhat Ho, XuanLong Nguyen

Singularities of a statistical model are the elements of the model's parameter space which make the corresponding Fisher information matrix degenerate. These are the points for which estimation techniques such as the maximum likelihood estimator and standard Bayesian procedures do not admit the root-$n$ parametric rate of convergence. We propose a general framework for the identification of singularity structures of the parameter space of finite mixtures, and study the impacts of the singularity structures on minimax lower bounds and rates of convergence for the maximum likelihood estimator over a compact parameter space. Our study makes explicit the deep links between model singularities, parameter estimation convergence rates and minimax lower bounds, and the algebraic geometry of the parameter space for mixtures of continuous distributions. The theory is applied to establish concrete convergence rates of parameter estimation for finite mixture of skew-normal distributions. This rich and increasingly popular mixture model is shown to exhibit a remarkably complex range of asymptotic behaviors which have not been hitherto reported in the literature.

Hossein Keshavarz, Clayton Scott, XuanLong Nguyen

Gaussian random fields are a powerful tool for modeling environmental processes. For high dimensional samples, classical approaches for estimating the covariance parameters require highly challenging and massive computations, such as the evaluation of the Cholesky factorization or solving linear systems. Recently, Anitescu, Chen and Stein \cite{M.Anitescu} proposed a fast and scalable algorithm which does not need such burdensome computations. The main focus of this article is to study the asymptotic behavior of the algorithm of Anitescu et al. (ACS) for regular and irregular grids in the increasing domain setting. Consistency, minimax optimality and asymptotic normality of this algorithm are proved under mild differentiability conditions on the covariance function. Despite the fact that ACS's method entails a non-concave maximization, our results hold for any stationary point of the objective function. A numerical study is presented to evaluate the efficiency of this algorithm for large data sets.

Hossein Keshavarz, Clayton Scott, XuanLong Nguyen

We study the problem of detecting a change in the mean of one-dimensional Gaussian process data. This problem is investigated in the setting of increasing domain (customarily employed in time series analysis) and in the setting of fixed domain (typically arising in spatial data analysis). We propose a detection method based on the generalized likelihood ratio test (GLRT), and show that our method achieves nearly asymptotically optimal rate in the minimax sense, in both settings. The salient feature of the proposed method is that it exploits in an efficient way the data dependence captured by the Gaussian process covariance structure. When the covariance is not known, we propose the plug-in GLRT method and derive conditions under which the method remains asymptotically near optimal. By contrast, the standard CUSUM method, which does not account for the covariance structure, is shown to be asymptotically optimal only in the increasing domain. Our algorithms and accompanying theory are applicable to a wide variety of covariance structures, including the Matern class, the powered exponential class, and others. The plug-in GLRT method is shown to perform well for maximum likelihood estimators with a dense covariance matrix.

Vijay Manikandan Janakiraman, XuanLong Nguyen, Dennis Assanis

In this article, a stochastic gradient based online learning algorithm for Extreme Learning Machines (ELM) is developed (SG-ELM). A stability criterion based on Lyapunov approach is used to prove both asymptotic stability of estimation error and stability in the estimated parameters suitable for identification of nonlinear dynamic systems. The developed algorithm not only guarantees stability, but also reduces the computational demand compared to the OS-ELM approach based on recursive least squares. In order to demonstrate the effectiveness of the algorithm on a real-world scenario, an advanced combustion engine identification problem is considered. The algorithm is applied to two case studies: An online regression learning for system identification of a Homogeneous Charge Compression Ignition (HCCI) Engine and an online classification learning (with class imbalance) for identifying the dynamic operating envelope of the HCCI Engine. The results indicate that the accuracy of the proposed SG-ELM is comparable to that of the state-of-the-art but adds stability and a reduction in computational effort.

Vijay Manikandan Janakiraman, XuanLong Nguyen, Dennis Assanis

Homogeneous charge compression ignition (HCCI) is a futuristic combustion technology that operates with a high fuel efficiency and reduced emissions. HCCI combustion is characterized by complex nonlinear dynamics which necessitates a model based control approach for automotive application. HCCI engine control is a nonlinear, multi-input multi-output problem with state and actuator constraints which makes controller design a challenging task. Typical HCCI controllers make use of a first principles based model which involves a long development time and cost associated with expert labor and calibration. In this paper, an alternative approach based on machine learning is presented using extreme learning machines (ELM) and nonlinear model predictive control (MPC). A recurrent ELM is used to learn the nonlinear dynamics of HCCI engine using experimental data and is shown to accurately predict the engine behavior several steps ahead in time, suitable for predictive control. Using the ELM engine models, an MPC based control algorithm with a simplified quadratic program update is derived for real time implementation. The working and effectiveness of the MPC approach has been analyzed on a nonlinear HCCI engine model for tracking multiple reference quantities along with constraints defined by HCCI states, actuators and operational limits.

Nhat Ho, XuanLong Nguyen

This paper studies identifiability and convergence behaviors for parameters of multiple types in finite mixtures, and the effects of model fitting with extra mixing components. First, we present a general theory for strong identifiability, which extends from the previous work of Nguyen [2013] and Chen [1995] to address a broad range of mixture models and to handle matrix-variate parameters. These models are shown to share the same Wasserstein distance based optimal rates of convergence for the space of mixing distributions --- $n^{-1/2}$ under $W_1$ for the exact-fitted and $n^{-1/4}$ under $W_2$ for the over-fitted setting, where $n$ is the sample size. This theory, however, is not applicable to several important model classes, including location-scale multivariate Gaussian mixtures, shape-scale Gamma mixtures and location-scale-shape skew-normal mixtures. The second part of this work is devoted to demonstrating that for these "weakly identifiable" classes, algebraic structures of the density family play a fundamental role in determining convergence rates of the model parameters, which display a very rich spectrum of behaviors. For instance, the optimal rate of parameter estimation in an over-fitted location-covariance Gaussian mixture is precisely determined by the order of a solvable system of polynomial equations --- these rates deteriorate rapidly as more extra components are added to the model. The established rates for a variety of settings are illustrated by a simulation study.

Vu Nguyen, Dinh Phung, XuanLong Nguyen et al.

We present a Bayesian nonparametric framework for multilevel clustering which utilizes group-level context information to simultaneously discover low-dimensional structures of the group contents and partitions groups into clusters. Using the Dirichlet process as the building block, our model constructs a product base-measure with a nested structure to accommodate content and context observations at multiple levels. The proposed model possesses properties that link the nested Dirichlet processes (nDP) and the Dirichlet process mixture models (DPM) in an interesting way: integrating out all contents results in the DPM over contexts, whereas integrating out group-specific contexts results in the nDP mixture over content variables. We provide a Polya-urn view of the model and an efficient collapsed Gibbs inference procedure. Extensive experiments on real-world datasets demonstrate the advantage of utilizing context information via our model in both text and image domains.

Arash A. Amini, XuanLong Nguyen

We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors of exact and approximate message-passing algorithms that arise in a sequential change point detection problem formulated via a latent variable directed graphical model. The sequential inference algorithm and its supporting theory are illustrated by simulated examples.

Vijay Manikandan Janakiraman, XuanLong Nguyen, Jeff Sterniak et al.

Advanced combustion technologies such as homogeneous charge compression ignition (HCCI) engines have a narrow stable operating region defined by complex control strategies such as exhaust gas recirculation (EGR) and variable valve timing among others. For such systems, it is important to identify the operating envelope or the boundary of stable operation for diagnostics and control purposes. Obtaining a good model of the operating envelope using physics becomes intractable owing to engine transient effects. In this paper, a machine learning based approach is employed to identify the stable operating boundary of HCCI combustion directly from experimental data. Owing to imbalance in class proportions in the data, two approaches are considered. A re-sampling (under-sampling, over-sampling) based approach is used to develop models using existing algorithms while a cost-sensitive approach is used to modify the learning algorithm without modifying the data set. Support vector machines and recently developed extreme learning machines are used for model development and results compared against linear classification methods show that cost-sensitive versions of ELM and SVM algorithms are well suited to model the HCCI operating envelope. The prediction results indicate that the models have the potential to be used for predicting HCCI instability based on sensor measurement history.

XuanLong Nguyen

This paper studies posterior concentration behavior of the base probability measure of a Dirichlet measure, given observations associated with the sampled Dirichlet processes, as the number of observations tends to infinity. The base measure itself is endowed with another Dirichlet prior, a construction known as the hierarchical Dirichlet processes (Teh et al. [J. Amer. Statist. Assoc. 101 (2006) 1566-1581]). Convergence rates are established in transportation distances (i.e., Wasserstein metrics) under various conditions on the geometry of the support of the true base measure. As a consequence of the theory, we demonstrate the benefit of "borrowing strength" in the inference of multiple groups of data - a powerful insight often invoked to motivate hierarchical modeling. In certain settings, the gain in efficiency due to the latent hierarchy can be dramatic, improving from a standard nonparametric rate to a parametric rate of convergence. Tools developed include transportation distances for nonparametric Bayesian hierarchies of random measures, the existence of tests for Dirichlet measures, and geometric properties of the support of Dirichlet measures.

Arash Ali Amini, XuanLong Nguyen

We propose a probabilistic formulation that enables sequential detection of multiple change points in a network setting. We present a class of sequential detection rules for certain functionals of change points (minimum among a subset), and prove their asymptotic optimality properties in terms of expected detection delay time. Drawing from graphical model formalism, the sequential detection rules can be implemented by a computationally efficient message-passing protocol which may scale up linearly in network size and in waiting time. The effectiveness of our inference algorithm is demonstrated by simulations.

XuanLong Nguyen

We study the posterior contraction behavior of the latent population structure that arises in admixture models as the amount of data increases. We adopt the geometric view of admixture models - alternatively known as topic models - as a data generating mechanism for points randomly sampled from the interior of a (convex) population polytope, whose extreme points correspond to the population structure variables of interest. Rates of posterior contraction are established with respect to Hausdorff metric and a minimum matching Euclidean metric defined on polytopes. Tools developed include posterior asymptotics of hierarchical models and arguments from convex geometry.