Alfred O. Hero

Alfred O. Hero, Douglas Cochran

Sensor systems typically operate under resource constraints that prevent the simultaneous use of all resources all of the time. Sensor management becomes relevant when the sensing system has the capability of actively managing these resources; i.e., changing its operating configuration during deployment in reaction to previous measurements. Examples of systems in which sensor management is currently used or is likely to be used in the near future include autonomous robots, surveillance and reconnaissance networks, and waveform-agile radars. This paper provides an overview of the theory, algorithms, and applications of sensor management as it has developed over the past decades and as it stands today.

John A. Sidles, Joseph L. Garbini, Lee E. Harrell et al.

This article presents numerical recipes for simulating high-temperature and non-equilibrium quantum spin systems that are continuously measured and controlled. The notion of a spin system is broadly conceived, in order to encompass macroscopic test masses as the limiting case of large-j spins. The simulation technique has three stages: first the deliberate introduction of noise into the simulation, then the conversion of that noise into an equivalent continuous measurement and control process, and finally, projection of the trajectory onto a state-space manifold having reduced dimensionality and possessing a Kahler potential of multi-linear form. The resulting simulation formalism is used to construct a positive P-representation for the thermal density matrix. Single-spin detection by magnetic resonance force microscopy (MRFM) is simulated, and the data statistics are shown to be those of a random telegraph signal with additive white noise. Larger-scale spin-dust models are simulated, having no spatial symmetry and no spatial ordering; the high-fidelity projection of numerically computed quantum trajectories onto low-dimensionality Kahler state-space manifolds is demonstrated. The reconstruction of quantum trajectories from sparse random projections is demonstrated, the onset of Donoho-Stodden breakdown at the Candes-Tao sparsity limit is observed, a deterministic construction for sampling matrices is given, and methods for quantum state optimization by Dantzig selection are given.

Neophytos Charalambides, Hessam Mahdavifar, Mert Pilanci et al.

Linear regression is a fundamental and primitive problem in supervised machine learning, with applications ranging from epidemiology to finance. In this work, we propose methods for speeding up distributed linear regression. We do so by leveraging randomized techniques, while also ensuring security and straggler resiliency in asynchronous distributed computing systems. Specifically, we randomly rotate the basis of the system of equations and then subsample blocks, to simultaneously secure the information and reduce the dimension of the regression problem. In our setup, the basis rotation corresponds to an encoded encryption in an approximate gradient coding scheme, and the subsampling corresponds to the responses of the non-straggling servers in the centralized coded computing framework. This results in a distributive iterative stochastic approach for matrix compression and steepest descent.

Byoungwook Jang, Julia Nepper, Marc Chevrette et al.

Recent works in bandit problems adopted lasso convergence theory in the sequential decision-making setting. Even with fully observed contexts, there are technical challenges that hinder the application of existing lasso convergence theory: 1) proving the restricted eigenvalue condition under conditionally sub-Gaussian noise and 2) accounting for the dependence between the context variables and the chosen actions. This paper studies the effect of missing covariates on regret for stochastic linear bandit algorithms. Our work provides a high-probability upper bound on the regret incurred by the proposed algorithm in terms of covariate sampling probabilities, showing that the regret degrades due to missingness by at most $ζ_{min}^2$, where $ζ_{min}$ is the minimum probability of observing covariates in the context vector. We illustrate our algorithm for the practical application of experimental design for collecting gene expression data by a sequential selection of class discriminating DNA probes.

Conrad D. Hougen, Lance M. Kaplan, Magdalena Ivanovska et al.

In second-order uncertain Bayesian networks, the conditional probabilities are only known within distributions, i.e., probabilities over probabilities. The delta-method has been applied to extend exact first-order inference methods to propagate both means and variances through sum-product networks derived from Bayesian networks, thereby characterizing epistemic uncertainty, or the uncertainty in the model itself. Alternatively, second-order belief propagation has been demonstrated for polytrees but not for general directed acyclic graph structures. In this work, we extend Loopy Belief Propagation to the setting of second-order Bayesian networks, giving rise to Second-Order Loopy Belief Propagation (SOLBP). For second-order Bayesian networks, SOLBP generates inferences consistent with those generated by sum-product networks, while being more computationally efficient and scalable.

Conrad D. Hougen, Lance M. Kaplan, Federico Cerutti et al.

When the historical data are limited, the conditional probabilities associated with the nodes of Bayesian networks are uncertain and can be empirically estimated. Second order estimation methods provide a framework for both estimating the probabilities and quantifying the uncertainty in these estimates. We refer to these cases as uncer tain or second-order Bayesian networks. When such data are complete, i.e., all variable values are observed for each instantiation, the conditional probabilities are known to be Dirichlet-distributed. This paper improves the current state-of-the-art approaches for handling uncertain Bayesian networks by enabling them to learn distributions for their parameters, i.e., conditional probabilities, with incomplete data. We extensively evaluate various methods to learn the posterior of the parameters through the desired and empirically derived strength of confidence bounds for various queries.

Ekdeep Singh Lubana, Puja Trivedi, Conrad Hougen et al.

Network pruning in Convolutional Neural Networks (CNNs) has been extensively investigated in recent years. To determine the impact of pruning a group of filters on a network's accuracy, state-of-the-art pruning methods consistently assume filters of a CNN are independent. This allows the importance of a group of filters to be estimated as the sum of importances of individual filters. However, overparameterization in modern networks results in highly correlated filters that invalidate this assumption, thereby resulting in incorrect importance estimates. To address this issue, we propose OrthoReg, a principled regularization strategy that enforces orthonormality on a network's filters to reduce inter-filter correlation, thereby allowing reliable, efficient determination of group importance estimates, improved trainability of pruned networks, and efficient, simultaneous pruning of large groups of filters. When used for iterative pruning on VGG-13, MobileNet-V1, and ResNet-34, OrthoReg consistently outperforms five baseline techniques, including the state-of-the-art, on CIFAR-100 and Tiny-ImageNet. For the recently proposed Early-Bird Ticket hypothesis, which claims networks become amenable to pruning early-on in training and can be pruned after a few epochs to minimize training expenditure, we find OrthoReg significantly outperforms prior work. Code available at https://github.com/EkdeepSLubana/OrthoReg.

Mohammadreza Tavasoli Naeini, Ali Bereyhi, Morteza Noshad et al.

This work invokes the notion of $f$-divergence to introduce a novel upper bound on the Bayes error rate of a general classification task. We show that the proposed bound can be computed by sampling from the output of a parameterized model. Using this practical interpretation, we introduce the Bayes optimal learning threshold (BOLT) loss whose minimization enforces a classification model to achieve the Bayes error rate. We validate the proposed loss for image and text classification tasks, considering MNIST, Fashion-MNIST, CIFAR-10, and IMDb datasets. Numerical experiments demonstrate that models trained with BOLT achieve performance on par with or exceeding that of cross-entropy, particularly on challenging datasets. This highlights the potential of BOLT in improving generalization.

Conrad D. Hougen, Karl T. Pazdernik, Alfred O. Hero

Interpretable topic modeling is essential for tracking how research interests evolve within co-author communities. In scientific corpora, where novelty is prized, identifying underrepresented niche topics is particularly important. However, contemporary models built from dense transformer embeddings tend to miss rare topics and therefore also fail to capture smooth temporal alignment. We propose a geometric method that integrates multimodal text and co-author network data, using Hellinger distances and Ward's linkage to construct a hierarchical topic dendrogram. This approach captures both local and global structure, supporting multiscale learning across semantic and temporal dimensions. Our method effectively identifies rare-topic structure and visualizes smooth topic drift over time. Experiments highlight the strength of interpretable bag-of-words models when paired with principled geometric alignment.

Mohammadreza Tavasoli Naeini, Ali Bereyhi, Morteza Noshad et al.

We introduce BOLT-GAN, a simple yet effective modification of the WGAN framework inspired by the Bayes Optimal Learning Threshold (BOLT). We show that with a Lipschitz continuous discriminator, BOLT-GAN implicitly minimizes a different metric distance than the Earth Mover (Wasserstein) distance and achieves better training stability. Empirical evaluations on four standard image generation benchmarks (CIFAR-10, CelebA-64, LSUN Bedroom-64, and LSUN Church-64) show that BOLT-GAN consistently outperforms WGAN, achieving 10-60% lower Frechet Inception Distance (FID). Our results suggest that BOLT is a broadly applicable principle for enhancing GAN training.

Neophytos Charalambides, Hessam Mahdavifar, Mert Pilanci et al.

In this work, we propose a method for speeding up linear regression distributively, while ensuring security. We leverage randomized sketching techniques, and improve straggler resilience in asynchronous systems. Specifically, we apply a random orthonormal matrix and then subsample in \textit{blocks}, to simultaneously secure the information and reduce the dimension of the regression problem. In our setup, the transformation corresponds to an encoded encryption in an \textit{approximate} gradient coding scheme, and the subsampling corresponds to the responses of the non-straggling workers; in a centralized coded computing network. We focus on the special case of the \textit{Subsampled Randomized Hadamard Transform}, which we generalize to block sampling; and discuss how it can be used to secure the data. We illustrate the performance through numerical experiments.

Davoud Ataee Tarzanagh, Laura Balzano, Alfred O. Hero

Inference of community structure in probabilistic graphical models may not be consistent with fairness constraints when nodes have demographic attributes. Certain demographics may be over-represented in some detected communities and under-represented in others. This paper defines a novel $\ell_1$-regularized pseudo-likelihood approach for fair graphical model selection. In particular, we assume there is some community or clustering structure in the true underlying graph, and we seek to learn a sparse undirected graph and its communities from the data such that demographic groups are fairly represented within the communities. In the case when the graph is known a priori, we provide a convex semidefinite programming approach for fair community detection. We establish the statistical consistency of the proposed method for both a Gaussian graphical model and an Ising model for, respectively, continuous and binary data, proving that our method can recover the graphs and their fair communities with high probability.

Yasin Yilmaz, Mehmet Aktukmak, Alfred O. Hero

The commonly used latent space embedding techniques, such as Principal Component Analysis, Factor Analysis, and manifold learning techniques, are typically used for learning effective representations of homogeneous data. However, they do not readily extend to heterogeneous data that are a combination of numerical and categorical variables, e.g., arising from linked GPS and text data. In this paper, we are interested in learning probabilistic generative models from high-dimensional heterogeneous data in an unsupervised fashion. The learned generative model provides latent unified representations that capture the factors common to the multiple dimensions of the data, and thus enable fusing multimodal data for various machine learning tasks. Following a Bayesian approach, we propose a general framework that combines disparate data types through the natural parameterization of the exponential family of distributions. To scale the model inference to millions of instances with thousands of features, we use the Laplace-Bernstein approximation for posterior computations involving nonlinear link functions. The proposed algorithm is presented in detail for the commonly encountered heterogeneous datasets with real-valued (Gaussian) and categorical (multinomial) features. Experiments on two high-dimensional and heterogeneous datasets (NYC Taxi and MovieLens-10M) demonstrate the scalability and competitive performance of the proposed algorithm on different machine learning tasks such as anomaly detection, data imputation, and recommender systems.

Abram Magner, Mayank Baranwal, Alfred O. Hero

Graph convolutional networks (GCNs) are a widely used method for graph representation learning. We investigate the power of GCNs, as a function of their number of layers, to distinguish between different random graph models on the basis of the embeddings of their sample graphs. In particular, the graph models that we consider arise from graphons, which are the most general possible parameterizations of infinite exchangeable graph models and which are the central objects of study in the theory of dense graph limits. We exhibit an infinite class of graphons that are well-separated in terms of cut distance and are indistinguishable by a GCN with nonlinear activation functions coming from a certain broad class if its depth is at least logarithmic in the size of the sample graph. These results theoretically match empirical observations of several prior works. Finally, we show a converse result that for pairs of graphons satisfying a degree profile separation property, a very simple GCN architecture suffices for distinguishability. To prove our results, we exploit a connection to random walks on graphs.

Abram Magner, Mayank Baranwal, Alfred O. Hero

Graph convolutional networks (GCNs) are a widely used method for graph representation learning. To elucidate the capabilities and limitations of GCNs, we investigate their power, as a function of their number of layers, to distinguish between different random graph models (corresponding to different class-conditional distributions in a classification problem) on the basis of the embeddings of their sample graphs. In particular, the graph models that we consider arise from graphons, which are the most general possible parameterizations of infinite exchangeable graph models and which are the central objects of study in the theory of dense graph limits. We give a precise characterization of the set of pairs of graphons that are indistinguishable by a GCN with nonlinear activation functions coming from a certain broad class if its depth is at least logarithmic in the size of the sample graph. This characterization is in terms of a degree profile closeness property. Outside this class, a very simple GCN architecture suffices for distinguishability. We then exhibit a concrete, infinite class of graphons arising from stochastic block models that are well-separated in terms of cut distance and are indistinguishable by a GCN. These results theoretically match empirical observations of several prior works. To prove our results, we exploit a connection to random walks on graphs. Finally, we give empirical results on synthetic and real graph classification datasets, indicating that indistinguishable graph distributions arise in practice.

Salimeh Yasaei Sekeh, Madan Ravi Ganesh, Shurjo Banerjee et al.

Online Streaming Feature Selection (OSFS) is a sequential learning problem where individual features across all samples are made available to algorithms in a streaming fashion. In this work, firstly, we assert that OSFS's main assumption of having data from all the samples available at runtime is unrealistic and introduce a new setting where features and samples are streamed concurrently called OSFS with Streaming Samples (OSFS-SS). Secondly, the primary OSFS method, SAOLA utilizes an unbounded mutual information measure and requires multiple comparison steps between the stored and incoming feature sets to evaluate a feature's importance. We introduce Geometric Online Adaption, an algorithm that requires relatively less feature comparison steps and uses a bounded conditional geometric dependency measure. Our algorithm outperforms several OSFS baselines including SAOLA on a variety of datasets. We also extend SAOLA to work in the OSFS-SS setting and show that GOA continues to achieve the best results. Thirdly, the current paradigm of the OSFS algorithm comparison is flawed. Algorithms are measured by comparing the number of features used and the accuracy obtained by the learner, two properties that are fundamentally at odds with one another. Without fixing a limit on either of these properties, the qualities of features obtained by different algorithms are incomparable. We try to rectify this inconsistency by fixing the maximum number of features available to the learner and comparing algorithms in terms of their accuracy. Additionally, we characterize the behaviour of SAOLA and GOA on feature sets derived from popular deep convolutional featurizers.

Joel W. LeBlanc, Brian J. Thelen, Alfred O. Hero

Many mathematical imaging problems are posed as non-convex optimization problems. When numerically tractable global optimization procedures are not available, one is often interested in testing ex post facto whether or not a locally convergent algorithm has found the globally optimal solution. When the problem is formulated in terms of maximizing the likelihood function under a statistical model for the measurements, one can construct a statistical test that a local maximum is in fact the global maximum. A one-sided test is proposed for the case that the statistical model is a member of the generalized location family of probability distributions, a condition often satisfied in imaging and other inverse problems. We propose a general method for improving the accuracy of the test by reparameterizing the likelihood function to embed its domain into a higher dimensional parameter space. We show that the proposed global maximum testing method results in improved accuracy and reduced computation for a physically-motivated joint-inverse problem arising in camera-blur estimation.

Salimeh Yasaei Sekeh, Alfred O. Hero

This paper proposes a geometric estimator of dependency between a pair of multivariate samples. The proposed estimator of dependency is based on a randomly permuted geometric graph (the minimal spanning tree) over the two multivariate samples. This estimator converges to a quantity that we call the geometric mutual information (GMI), which is equivalent to the Henze-Penrose divergence [1] between the joint distribution of the multivariate samples and the product of the marginals. The GMI has many of the same properties as standard MI but can be estimated from empirical data without density estimation; making it scalable to large datasets. The proposed empirical estimator of GMI is simple to implement, involving the construction of an MST spanning over both the original data and a randomly permuted version of this data. We establish asymptotic convergence of the estimator and convergence rates of the bias and variance for smooth multivariate density functions belonging to a Hölder class. We demonstrate the advantages of our proposed geometric dependency estimator in a series of experiments.

Salimeh Yasaei Sekeh, Alfred O. Hero

Feature selection and reducing the dimensionality of data is an essential step in data analysis. In this work, we propose a new criterion for feature selection that is formulated as conditional information between features given the labeled variable. Instead of using the standard mutual information measure based on Kullback-Leibler divergence, we use our proposed criterion to filter out redundant features for the purpose of multiclass classification. This approach results in an efficient and fast non-parametric implementation of feature selection as it can be directly estimated using a geometric measure of dependency, the global Friedman-Rafsky (FR) multivariate run test statistic constructed by a global minimal spanning tree (MST). We demonstrate the advantages of our proposed feature selection approach through simulation. In addition the proposed feature selection method is applied to the MNIST data set.

Alexander Jung, Alfred O. Hero, Alexandru Mara et al.

We propose and analyze a method for semi-supervised learning from partially-labeled network-structured data. Our approach is based on a graph signal recovery interpretation under a clustering hypothesis that labels of data points belonging to the same well-connected subset (cluster) are similar valued. This lends naturally to learning the labels by total variation (TV) minimization, which we solve by applying a recently proposed primal-dual method for non-smooth convex optimization. The resulting algorithm allows for a highly scalable implementation using message passing over the underlying empirical graph, which renders the algorithm suitable for big data applications. By applying tools of compressed sensing, we derive a sufficient condition on the underlying network structure such that TV minimization recovers clusters in the empirical graph of the data. In particular, we show that the proposed primal-dual method amounts to maximizing network flows over the empirical graph of the dataset. Moreover, the learning accuracy of the proposed algorithm is linked to the set of network flows between data points having known labels. The effectiveness and scalability of our approach is verified by numerical experiments.

Salimeh Yasaei Sekeh, Brandon Oselio, Alfred O. Hero

In the context of supervised learning, meta learning uses features, metadata and other information to learn about the difficulty, behavior, or composition of the problem. Using this knowledge can be useful to contextualize classifier results or allow for targeted decisions about future data sampling. In this paper, we are specifically interested in learning the Bayes error rate (BER) based on a labeled data sample. Providing a tight bound on the BER that is also feasible to estimate has been a challenge. Previous work[1] has shown that a pairwise bound based on the sum of Henze-Penrose (HP) divergence over label pairs can be directly estimated using a sum of Friedman-Rafsky (FR) multivariate run test statistics. However, in situations in which the dataset and number of classes are large, this bound is computationally infeasible to calculate and may not be tight. Other multi-class bounds also suffer from computationally complex estimation procedures. In this paper, we present a generalized HP divergence measure that allows us to estimate the Bayes error rate with log-linear computation. We prove that the proposed bound is tighter than both the pairwise method and a bound proposed by Lin [2]. We also empirically show that these bounds are close to the BER. We illustrate the proposed method on the MNIST dataset, and show its utility for the evaluation of feature reduction strategies. We further demonstrate an approach for evaluation of deep learning architectures using the proposed bounds.

Salimeh Yasaei Sekeh, Morteza Noshad, Kevin R. Moon et al.

Bounding the best achievable error probability for binary classification problems is relevant to many applications including machine learning, signal processing, and information theory. Many bounds on the Bayes binary classification error rate depend on information divergences between the pair of class distributions. Recently, the Henze-Penrose (HP) divergence has been proposed for bounding classification error probability. We consider the problem of empirically estimating the HP-divergence from random samples. We derive a bound on the convergence rate for the Friedman-Rafsky (FR) estimator of the HP-divergence, which is related to a multivariate runs statistic for testing between two distributions. The FR estimator is derived from a multicolored Euclidean minimal spanning tree (MST) that spans the merged samples. We obtain a concentration inequality for the Friedman-Rafsky estimator of the Henze-Penrose divergence. We validate our results experimentally and illustrate their application to real datasets.

Hye Won Chung, Ji Oon Lee, Doyeon Kim et al.

Consider a query-based data acquisition problem that aims to recover the values of $k$ binary variables from parity (XOR) measurements of chosen subsets of the variables. Assume the response model where only a randomly selected subset of the measurements is received. We propose a method for designing a sequence of queries so that the variables can be identified with high probability using as few ($n$) measurements as possible. We define the query difficulty $\bar{d}$ as the average size of the query subsets and the sample complexity $n$ as the minimum number of measurements required to attain a given recovery accuracy. We obtain fundamental trade-offs between recovery accuracy, query difficulty, and sample complexity. In particular, the necessary and sufficient sample complexity required for recovering all $k$ variables with high probability is $n = c_0 \max\{k, (k \log k)/\bar{d}\}$ and the sample complexity for recovering a fixed proportion $(1-δ)k$ of the variables for $δ=o(1)$ is $n = c_1\max\{k, (k \log(1/δ))/\bar{d}\}$, where $c_0, c_1>0$.

Elizabeth Hou, Alfred O. Hero

In many real-world applications, data is not collected as one batch, but sequentially over time, and often it is not possible or desirable to wait until the data is completely gathered before analyzing it. Thus, we propose a framework to sequentially update a maximum margin classifier by taking advantage of the Maximum Entropy Discrimination principle. Our maximum margin classifier allows for a kernel representation to represent large numbers of features and can also be regularized with respect to a smooth sub-manifold, allowing it to incorporate unlabeled observations. We compare the performance of our classifier to its non-sequential equivalents in both simulated and real datasets.

Morteza Noshad, Yu Zeng, Alfred O. Hero

The Mutual Information (MI) is an often used measure of dependency between two random variables utilized in information theory, statistics and machine learning. Recently several MI estimators have been proposed that can achieve parametric MSE convergence rate. However, most of the previously proposed estimators have the high computational complexity of at least $O(N^2)$. We propose a unified method for empirical non-parametric estimation of general MI function between random vectors in $\mathbb{R}^d$ based on $N$ i.i.d. samples. The reduced complexity MI estimator, called the ensemble dependency graph estimator (EDGE), combines randomized locality sensitive hashing (LSH), dependency graphs, and ensemble bias-reduction methods. We prove that EDGE achieves optimal computational complexity $O(N)$, and can achieve the optimal parametric MSE rate of $O(1/N)$ if the density is $d$ times differentiable. To the best of our knowledge EDGE is the first non-parametric MI estimator that can achieve parametric MSE rates with linear time complexity. We illustrate the utility of EDGE for the analysis of the information plane (IP) in deep learning. Using EDGE we shed light on a controversy on whether or not the compression property of information bottleneck (IB) in fact holds for ReLu and other rectification functions in deep neural networks (DNN).

Farshad Harirchi, Doohyun Kim, Omar A. Khalil et al.

Chemical kinetic mechanisms can be represented by sets of elementary reactions that are easily translated into mathematical terms using physicochemical relationships. The schematic representation of reactions captures the interactions between reacting species and products. Determining the minimal chemical interactions underlying the dynamic behavior of systems is a major task. In this paper, we introduce a novel approach for the identification of the influential reactions in chemical reaction networks for combustion applications, using a data-driven sparse-learning technique. The proposed approach identifies a set of influential reactions using species concentrations and reaction rates, with minimal computational cost without requiring additional data or simulations. The new approach is applied to analyze the combustion chemistry of H2 and C3H8 in a constant-volume homogeneous reactor. The influential reactions identified by the sparse-learning method are consistent with the current kinetics knowledge of chemical mechanisms. Additionally, we show that a reduced version of the parent mechanism can be generated as a combination of the influential reactions identified at different times and conditions and that for both H2 and C3H8 this reduced mechanism performs closely to the parent mechanism as a function of ignition delay over a wide range of conditions. Our results demonstrate the potential of the sparse-learning approach as an effective and efficient tool for mechanism analysis and mechanism reduction.

Farshad Harirchi, Omar A. Khalil, Sijia Liu et al.

In this paper, we propose an optimization-based sparse learning approach to identify the set of most influential reactions in a chemical reaction network. This reduced set of reactions is then employed to construct a reduced chemical reaction mechanism, which is relevant to chemical interaction network modeling. The problem of identifying influential reactions is first formulated as a mixed-integer quadratic program, and then a relaxation method is leveraged to reduce the computational complexity of our approach. Qualitative and quantitative validation of the sparse encoding approach demonstrates that the model captures important network structural properties with moderate computational load.

Tianpei Xie, Sijia Liu, Alfred O. Hero

Consider a social network where only a few nodes (agents) have meaningful interactions in the sense that the conditional dependency graph over node attribute variables (behaviors) is sparse. A company that can only observe the interactions between its own customers will generally not be able to accurately estimate its customers' dependency subgraph: it is blinded to any external interactions of its customers and this blindness creates false edges in its subgraph. In this paper we address the semiblind scenario where the company has access to a noisy summary of the complementary subgraph connecting external agents, e.g., provided by a consolidator. The proposed framework applies to other applications as well, including field estimation from a network of awake and sleeping sensors and privacy-constrained information sharing over social subnetworks. We propose a penalized likelihood approach in the context of a graph signal obeying a Gaussian graphical models (GGM). We use a convex-concave iterative optimization algorithm to maximize the penalized likelihood.

Gerrit J. J. van den Burg, Alfred O. Hero

We propose a new splitting criterion for a meta-learning approach to multiclass classifier design that adaptively merges the classes into a tree-structured hierarchy of increasingly difficult binary classification problems. The classification tree is constructed from empirical estimates of the Henze-Penrose bounds on the pairwise Bayes misclassification rates that rank the binary subproblems in terms of difficulty of classification. The proposed empirical estimates of the Bayes error rate are computed from the minimal spanning tree (MST) of the samples from each pair of classes. Moreover, a meta-learning technique is presented for quantifying the one-vs-rest Bayes error rate for each individual class from a single MST on the entire dataset. Extensive simulations on benchmark datasets show that the proposed hierarchical method can often be learned much faster than competing methods, while achieving competitive accuracy.

Morteza Noshad Iranzad, Alfred O. Hero

Meta learning of optimal classifier error rates allows an experimenter to empirically estimate the intrinsic ability of any estimator to discriminate between two populations, circumventing the difficult problem of estimating the optimal Bayes classifier. To this end we propose a weighted nearest neighbor (WNN) graph estimator for a tight bound on the Bayes classification error; the Henze-Penrose (HP) divergence. Similar to recently proposed HP estimators [berisha2016], the proposed estimator is non-parametric and does not require density estimation. However, unlike previous approaches the proposed estimator is rate-optimal, i.e., its mean squared estimation error (MSEE) decays to zero at the fastest possible rate of $O(1/M+1/N)$ where $M,N$ are the sample sizes of the respective populations. We illustrate the proposed WNN meta estimator for several simulated and real data sets.

Sijia Liu, Jie Chen, Pin-Yu Chen et al.

In this paper, we design and analyze a new zeroth-order online algorithm, namely, the zeroth-order online alternating direction method of multipliers (ZOO-ADMM), which enjoys dual advantages of being gradient-free operation and employing the ADMM to accommodate complex structured regularizers. Compared to the first-order gradient-based online algorithm, we show that ZOO-ADMM requires $\sqrt{m}$ times more iterations, leading to a convergence rate of $O(\sqrt{m}/\sqrt{T})$, where $m$ is the number of optimization variables, and $T$ is the number of iterations. To accelerate ZOO-ADMM, we propose two minibatch strategies: gradient sample averaging and observation averaging, resulting in an improved convergence rate of $O(\sqrt{1+q^{-1}m}/\sqrt{T})$, where $q$ is the minibatch size. In addition to convergence analysis, we also demonstrate ZOO-ADMM to applications in signal processing, statistics, and machine learning.

Pin-Yu Chen, Alfred O. Hero

Multilayer graphs are commonly used for representing different relations between entities and handling heterogeneous data processing tasks. Non-standard multilayer graph clustering methods are needed for assigning clusters to a common multilayer node set and for combining information from each layer. This paper presents a multilayer spectral graph clustering (SGC) framework that performs convex layer aggregation. Under a multilayer signal plus noise model, we provide a phase transition analysis of clustering reliability. Moreover, we use the phase transition criterion to propose a multilayer iterative model order selection algorithm (MIMOSA) for multilayer SGC, which features automated cluster assignment and layer weight adaptation, and provides statistical clustering reliability guarantees. Numerical simulations on synthetic multilayer graphs verify the phase transition analysis, and experiments on real-world multilayer graphs show that MIMOSA is competitive or better than other clustering methods.

Sijia Liu, Pin-Yu Chen, Alfred O. Hero

We consider the problem of accelerating distributed optimization in multi-agent networks by sequentially adding edges. Specifically, we extend the distributed dual averaging (DDA) subgradient algorithm to evolving networks of growing connectivity and analyze the corresponding improvement in convergence rate. It is known that the convergence rate of DDA is influenced by the algebraic connectivity of the underlying network, where better connectivity leads to faster convergence. However, the impact of network topology design on the convergence rate of DDA has not been fully understood. In this paper, we begin by designing network topologies via edge selection and scheduling. For edge selection, we determine the best set of candidate edges that achieves the optimal tradeoff between the growth of network connectivity and the usage of network resources. The dynamics of network evolution is then incurred by edge scheduling. Further, we provide a tractable approach to analyze the improvement in the convergence rate of DDA induced by the growth of network connectivity. Our analysis reveals the connection between network topology design and the convergence rate of DDA, and provides quantitative evaluation of DDA acceleration for distributed optimization that is absent in the existing analysis. Lastly, numerical experiments show that DDA can be significantly accelerated using a sequence of well-designed networks, and our theoretical predictions are well matched to its empirical convergence behavior.

Alan Wisler, Visar Berisha, Andreas Spanias et al.

A number of fundamental quantities in statistical signal processing and information theory can be expressed as integral functions of two probability density functions. Such quantities are called density functionals as they map density functions onto the real line. For example, information divergence functions measure the dissimilarity between two probability density functions and are useful in a number of applications. Typically, estimating these quantities requires complete knowledge of the underlying distribution followed by multi-dimensional integration. Existing methods make parametric assumptions about the data distribution or use non-parametric density estimation followed by high-dimensional integration. In this paper, we propose a new alternative. We introduce the concept of "data-driven basis functions" - functions of distributions whose value we can estimate given only samples from the underlying distributions without requiring distribution fitting or direct integration. We derive a new data-driven complete basis that is similar to the deterministic Bernstein polynomial basis and develop two methods for performing basis expansions of functionals of two distributions. We also show that the new basis set allows us to approximate functions of distributions as closely as desired. Finally, we evaluate the methodology by developing data driven estimators for the Kullback-Leibler divergences and the Hellinger distance and by constructing empirical estimates of tight bounds on the Bayes error rate.

Morteza Noshad, Kevin R. Moon, Salimeh Yasaei Sekeh et al.

We propose a direct estimation method for Rényi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets $X$ and $Y$, respectively with $N$ and $M$ samples, where $η:=M/N$ is a constant value. Considering the $k$-nearest neighbor ($k$-NN) graph of $Y$ in the joint data set $(X,Y)$, we show that the average powered ratio of the number of $X$ points to the number of $Y$ points among all $k$-NN points is proportional to Rényi divergence of $X$ and $Y$ densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of $γ$-Hölder smooth functions, the estimator achieves the MSE rate of $O(N^{-2γ/(γ+d)})$. Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order $d$, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of $O(1/N)$. Our estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.

Elizabeth Hou, Kumar Sricharan, Alfred O. Hero

Data-driven anomaly detection methods suffer from the drawback of detecting all instances that are statistically rare, irrespective of whether the detected instances have real-world significance or not. In this paper, we are interested in the problem of specifically detecting anomalous instances that are known to have high real-world utility, while ignoring the low-utility statistically anomalous instances. To this end, we propose a novel method called Latent Laplacian Maximum Entropy Discrimination (LatLapMED) as a potential solution. This method uses the EM algorithm to simultaneously incorporate the Geometric Entropy Minimization principle for identifying statistical anomalies, and the Maximum Entropy Discrimination principle to incorporate utility labels, in order to detect high-utility anomalies. We apply our method in both simulated and real datasets to demonstrate that it has superior performance over existing alternatives that independently pre-process with unsupervised anomaly detection algorithms before classifying.

Kristjan Greenewald, Stephen Kelley, Brandon Oselio et al.

Recent work in distance metric learning has focused on learning transformations of data that best align with specified pairwise similarity and dissimilarity constraints, often supplied by a human observer. The learned transformations lead to improved retrieval, classification, and clustering algorithms due to the better adapted distance or similarity measures. Here, we address the problem of learning these transformations when the underlying constraint generation process is nonstationary. This nonstationarity can be due to changes in either the ground-truth clustering used to generate constraints or changes in the feature subspaces in which the class structure is apparent. We propose Online Convex Ensemble StrongLy Adaptive Dynamic Learning (OCELAD), a general adaptive, online approach for learning and tracking optimal metrics as they change over time that is highly robust to a variety of nonstationary behaviors in the changing metric. We apply the OCELAD framework to an ensemble of online learners. Specifically, we create a retro-initialized composite objective mirror descent (COMID) ensemble (RICE) consisting of a set of parallel COMID learners with different learning rates, and demonstrate parameter-free RICE-OCELAD metric learning on both synthetic data and a highly nonstationary Twitter dataset. We show significant performance improvements and increased robustness to nonstationary effects relative to previously proposed batch and online distance metric learning algorithms.

Alexander Jung, Alfred O. Hero, Alexandru Mara et al.

This work proposes a novel method for semi-supervised learning from partially labeled massive network-structured datasets, i.e., big data over networks. We model the underlying hypothesis, which relates data points to labels, as a graph signal, defined over some graph (network) structure intrinsic to the dataset. Following the key principle of supervised learning, i.e., similar inputs yield similar outputs, we require the graph signals induced by labels to have small total variation. Accordingly, we formulate the problem of learning the labels of data points as a non-smooth convex optimization problem which amounts to balancing between the empirical loss, i.e., the discrepancy with some partially available label information, and the smoothness quantified by the total variation of the learned graph signal. We solve this optimization problem by appealing to a recently proposed preconditioned variant of the popular primal-dual method by Pock and Chambolle, which results in a sparse label propagation algorithm. This learning algorithm allows for a highly scalable implementation as message passing over the underlying data graph. By applying concepts of compressed sensing to the learning problem, we are also able to provide a transparent sufficient condition on the underlying network structure such that accurate learning of the labels is possible. We also present an implementation of the message passing formulation allows for a highly scalable implementation in big data frameworks.

Alexander Jung, Alfred O. Hero, Alexandru Mara et al.

We propose a scalable method for semi-supervised (transductive) learning from massive network-structured datasets. Our approach to semi-supervised learning is based on representing the underlying hypothesis as a graph signal with small total variation. Requiring a small total variation of the graph signal representing the underlying hypothesis corresponds to the central smoothness assumption that forms the basis for semi-supervised learning, i.e., input points forming clusters have similar output values or labels. We formulate the learning problem as a nonsmooth convex optimization problem which we solve by appealing to Nesterovs optimal first-order method for nonsmooth optimization. We also provide a message passing formulation of the learning method which allows for a highly scalable implementation in big data frameworks.

Tianpei Xie, Nasser. M. Narabadi, Alfred O. Hero

In this paper, we propose a general framework to learn a robust large-margin binary classifier when corrupt measurements, called anomalies, caused by sensor failure might be present in the training set. The goal is to minimize the generalization error of the classifier on non-corrupted measurements while controlling the false alarm rate associated with anomalous samples. By incorporating a non-parametric regularizer based on an empirical entropy estimator, we propose a Geometric-Entropy-Minimization regularized Maximum Entropy Discrimination (GEM-MED) method to learn to classify and detect anomalies in a joint manner. We demonstrate using simulated data and a real multimodal data set. Our GEM-MED method can yield improved performance over previous robust classification methods in terms of both classification accuracy and anomaly detection rate.

Pin-Yu Chen, Alfred O. Hero

Multilayer graphs are commonly used for representing different relations between entities and handling heterogeneous data processing tasks. New challenges arise in multilayer graph clustering for assigning clusters to a common multilayer node set and for combining information from each layer. This paper presents a theoretical framework for multilayer spectral graph clustering of the nodes via convex layer aggregation. Under a novel multilayer signal plus noise model, we provide a phase transition analysis that establishes the existence of a critical value on the noise level that permits reliable cluster separation. The analysis also specifies analytical upper and lower bounds on the critical value, where the bounds become exact when the clusters have identical sizes. Numerical experiments on synthetic multilayer graphs are conducted to validate the phase transition analysis and study the effect of layer weights and noise levels on clustering reliability.

Pin-Yu Chen, Thibaut Gensollen, Alfred O. Hero

One of the longstanding problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. In this paper, we propose AMOS, an automated model order selection algorithm for SGC. Based on a recent analysis of clustering reliability for SGC under the random interconnection model, AMOS works by incrementally increasing the number of clusters, estimating the quality of identified clusters, and providing a series of clustering reliability tests. Consequently, AMOS outputs clusters of minimal model order with statistical clustering reliability guarantees. Comparing to three other automated graph clustering methods on real-world datasets, AMOS shows superior performance in terms of multiple external and internal clustering metrics.

Kevin R. Moon, Morteza Noshad, Salimeh Yasaei Sekeh et al.

Information theoretic measures (e.g. the Kullback Liebler divergence and Shannon mutual information) have been used for exploring possibly nonlinear multivariate dependencies in high dimension. If these dependencies are assumed to follow a Markov factor graph model, this exploration process is called structure discovery. For discrete-valued samples, estimates of the information divergence over the parametric class of multinomial models lead to structure discovery methods whose mean squared error achieves parametric convergence rates as the sample size grows. However, a naive application of this method to continuous nonparametric multivariate models converges much more slowly. In this paper we introduce a new method for nonparametric structure discovery that uses weighted ensemble divergence estimators that achieve parametric convergence rates and obey an asymptotic central limit theorem that facilitates hypothesis testing and other types of statistical validation.

Sundeep Prabhakar Chepuri, Sijia Liu, Geert Leus et al.

In this paper, we are interested in learning the underlying graph structure behind training data. Solving this basic problem is essential to carry out any graph signal processing or machine learning task. To realize this, we assume that the data is smooth with respect to the graph topology, and we parameterize the graph topology using an edge sampling function. That is, the graph Laplacian is expressed in terms of a sparse edge selection vector, which provides an explicit handle to control the sparsity level of the graph. We solve the sparse graph learning problem given some training data in both the noiseless and noisy settings. Given the true smooth data, the posed sparse graph learning problem can be solved optimally and is based on simple rank ordering. Given the noisy data, we show that the joint sparse graph learning and denoising problem can be simplified to designing only the sparse edge selection vector, which can be solved using convex optimization.

Pin-Yu Chen, Alfred O. Hero

One of the longstanding open problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. We propose automated model order selection (AMOS), a solution to the SGC model selection problem under a random interconnection model (RIM) using a novel selection criterion that is based on an asymptotic phase transition analysis. AMOS can more generally be applied to discovering hidden block diagonal structure in symmetric non-negative matrices. Numerical experiments on simulated graphs validate the phase transition analysis, and real-world network data is used to validate the performance of the proposed model selection procedure.

Pin-Yu Chen, Sutanay Choudhury, Alfred O. Hero

Many modern datasets can be represented as graphs and hence spectral decompositions such as graph principal component analysis (PCA) can be useful. Distinct from previous graph decomposition approaches based on subspace projection of a single topological feature, e.g., the Fiedler vector of centered graph adjacency matrix (graph Laplacian), we propose spectral decomposition approaches to graph PCA and graph dictionary learning that integrate multiple features, including graph walk statistics, centrality measures and graph distances to reference nodes. In this paper we propose a new PCA method for single graph analysis, called multi-centrality graph PCA (MC-GPCA), and a new dictionary learning method for ensembles of graphs, called multi-centrality graph dictionary learning (MC-GDL), both based on spectral decomposition of multi-centrality matrices. As an application to cyber intrusion detection, MC-GPCA can be an effective indicator of anomalous connectivity pattern and MC-GDL can provide discriminative basis for attack classification.

Pin-Yu Chen, Baichuan Zhang, Mohammad Al Hasan et al.

The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used for spectral clustering and community detection. However, in real-life applications the number of clusters or communities (say, $K$) is generally unknown a-priori. Consequently, the majority of the existing methods either choose $K$ heuristically or they repeat the clustering method with different choices of $K$ and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the $K$-th eigenpairs of the Laplacian matrix given a collection of all the $K-1$ smallest eigenpairs. Our proposed method adapts the Laplacian matrix such that the batch eigenvalue decomposition problem transforms into an efficient sequential leading eigenpair computation problem. As a practical application, we consider user-guided spectral clustering. Specifically, we demonstrate that users can utilize the proposed incremental method for effective eigenpair computation and determining the desired number of clusters based on multiple clustering metrics.

Stephen V. Gliske, Kevin R. Moon, William C. Stacey et al.

High frequency oscillations (HFOs) are a promising biomarker of epileptic brain tissue and activity. HFOs additionally serve as a prototypical example of challenges in the analysis of discrete events in high-temporal resolution, intracranial EEG data. Two primary challenges are 1) dimensionality reduction, and 2) assessing feasibility of classification. Dimensionality reduction assumes that the data lie on a manifold with dimension less than that of the feature space. However, previous HFO analyses have assumed a linear manifold, global across time, space (i.e. recording electrode/channel), and individual patients. Instead, we assess both a) whether linear methods are appropriate and b) the consistency of the manifold across time, space, and patients. We also estimate bounds on the Bayes classification error to quantify the distinction between two classes of HFOs (those occurring during seizures and those occurring due to other processes). This analysis provides the foundation for future clinical use of HFO features and buides the analysis for other discrete events, such as individual action potentials or multi-unit activity.

Ko-Jen Hsiao, Kevin S. Xu, Jeff Calder et al.

We consider the problem of identifying patterns in a data set that exhibit anomalous behavior, often referred to as anomaly detection. Similarity-based anomaly detection algorithms detect abnormally large amounts of similarity or dissimilarity, e.g.~as measured by nearest neighbor Euclidean distances between a test sample and the training samples. In many application domains there may not exist a single dissimilarity measure that captures all possible anomalous patterns. In such cases, multiple dissimilarity measures can be defined, including non-metric measures, and one can test for anomalies by scalarizing using a non-negative linear combination of them. If the relative importance of the different dissimilarity measures are not known in advance, as in many anomaly detection applications, the anomaly detection algorithm may need to be executed multiple times with different choices of weights in the linear combination. In this paper, we propose a method for similarity-based anomaly detection using a novel multi-criteria dissimilarity measure, the Pareto depth. The proposed Pareto depth analysis (PDA) anomaly detection algorithm uses the concept of Pareto optimality to detect anomalies under multiple criteria without having to run an algorithm multiple times with different choices of weights. The proposed PDA approach is provably better than using linear combinations of the criteria and shows superior performance on experiments with synthetic and real data sets.

Tianpei Xie, Nasser M. Nasrabadi, Alfred O. Hero

In this paper, we propose a general framework to learn a robust large-margin binary classifier when corrupt measurements, called anomalies, caused by sensor failure might be present in the training set. The goal is to minimize the generalization error of the classifier on non-corrupted measurements while controlling the false alarm rate associated with anomalous samples. By incorporating a non-parametric regularizer based on an empirical entropy estimator, we propose a Geometric-Entropy-Minimization regularized Maximum Entropy Discrimination (GEM-MED) method to learn to classify and detect anomalies in a joint manner. We demonstrate using simulated data and a real multimodal data set. Our GEM-MED method can yield improved performance over previous robust classification methods in terms of both classification accuracy and anomaly detection rate.