arXiv subjects

Carey E. Priebe

Publications and source records attributed to Carey E. Priebe.

At least 91 records · Page 5Linked to original sources

arXiv2021

On spectral algorithms for community detection in stochastic blockmodel graphs with vertex covariates

In network inference applications, it is often desirable to detect community structure, namely to cluster vertices into groups, or blocks, according to some measure of similarity. Beyond mere adjacency matrices, many real networks also involve vertex covariates that carry key information about underlying block structure in graphs. To assess the effects of such covariates on block recovery, we present a comparative analysis of two model-based spectral algorithms for clustering vertices in stochastic blockmodel graphs with vertex covariates. The first algorithm uses only the adjacency matrix, and directly estimates the block assignments. The second algorithm incorporates both the adjacency matrix and the vertex covariates into the estimation of block assignments, and moreover quantifies the explicit impact of the vertex covariates on the resulting estimate of the block assignments. We employ Chernoff information to analytically compare the algorithms' performance and derive the information-theoretic Chernoff ratio for certain models of interest. Analytic results and simulations suggest that the second algorithm is often preferred: we can often better estimate the induced block assignments by first estimating the effect of vertex covariates. In addition, real data examples also indicate that the second algorithm has the advantages of revealing underlying block structure and taking observed vertex heterogeneity into account in real applications. Our findings emphasize the importance of distinguishing between observed and unobserved factors that can affect block structure in graphs.

cs.SI↗

arXiv2021

Leveraging semantically similar queries for ranking via combining representations

In modern ranking problems, different and disparate representations of the items to be ranked are often available. It is sensible, then, to try to combine these representations to improve ranking. Indeed, learning to rank via combining representations is both principled and practical for learning a ranking function for a particular query. In extremely data-scarce settings, however, the amount of labeled data available for a particular query can lead to a highly variable and ineffective ranking function. One way to mitigate the effect of the small amount of data is to leverage information from semantically similar queries. Indeed, as we demonstrate in simulation settings and real data examples, when semantically similar queries are available it is possible to gainfully use them when ranking with respect to a particular query. We describe and explore this phenomenon in the context of the bias-variance trade off and apply it to the data-scarce settings of a Bing navigational graph and the Drosophila larva connectome.

cs.LG↗

arXiv2021

Occam Factor for Gaussian Models With Unknown Variance Structure

We discuss model selection to determine whether the variance-covariance matrix of a multivariate Gaussian model with known mean should be considered to be a constant diagonal, a non-constant diagonal, or an arbitrary positive definite matrix. Of particular interest is the relationship between Bayesian evidence and the flexibility penalty due to Priebe and Rougier. For the case of an exponential family in canonical form equipped with a conjugate prior for the canonical parameter, flexibility may be exactly decomposed into the usual BIC likelihood penalty and a $O_p(1)$ term, the latter of which we explicitly compute. We also investigate the asymptotics of Bayes factors for linearly nested canonical exponential families equipped with conjugate priors; in particular, we find the exact rates at which Bayes factors correctly diverge in favor of the correct model: linearly and logarithmically in the number of observations when the full and nested models are true, respectively. Such theoretical considerations for the general case permit us to fully express the asymptotic behavior of flexibility and Bayes factors for the variance-covariance structure selection problem when we assume that the prior for the model precision is a member of the gamma/Wishart family of distributions or is uninformative. Simulations demonstrate evidence's immediate and superior performance in model selection compared to approximate criteria such as the BIC. We extend the framework to the multivariate Gaussian linear model with three data-driven examples.

stat.ME↗

arXiv2021

Graph matching between bipartite and unipartite networks: to collapse, or not to collapse, that is the question

Graph matching consists of aligning the vertices of two unlabeled graphs in order to maximize the shared structure across networks; when the graphs are unipartite, this is commonly formulated as minimizing their edge disagreements. In this paper, we address the common setting in which one of the graphs to match is a bipartite network and one is unipartite. Commonly, the bipartite networks are collapsed or projected into a unipartite graph, and graph matching proceeds as in the classical setting. This potentially leads to noisy edge estimates and loss of information. We formulate the graph matching problem between a bipartite and a unipartite graph using an undirected graphical model, and introduce methods to find the alignment with this model without collapsing. We theoretically demonstrate that our methodology is consistent, and provide non-asymptotic conditions that ensure exact recovery of the matching solution. In simulations and real data examples, we show how our methods can result in a more accurate matching than the naive approach of transforming the bipartite networks into unipartite, and we demonstrate the performance gains achieved by our method in simulated and real data networks, including a co-authorship-citation network pair, and brain structural and functional data.

stat.ML↗

arXiv2021

On a complete and sufficient statistic for the correlated Bernoulli random graph model

Inference on vertex-aligned graphs is of wide theoretical and practical importance.There are, however, few flexible and tractable statistical models for correlated graphs, and even fewer comprehensive approaches to parametric inference on data arising from such graphs. In this paper, we consider the correlated Bernoulli random graph model (allowing different Bernoulli coefficients and edge correlations for different pairs of vertices), and we introduce a new variance-reducing technique -- called \emph{balancing} -- that can refine estimators for model parameters. Specifically, we construct a disagreement statistic and show that it is complete and sufficient; balancing can be interpreted as Rao-Blackwellization with this disagreement statistic. We show that for unbiased estimators of functions of model parameters, balancing generates uniformly minimum variance unbiased estimators (UMVUEs). However, even when unbiased estimators for model parameters do {\em not} exist -- which, as we prove, is the case with both the heterogeneity correlation and the total correlation parameters -- balancing is still useful, and lowers mean squared error. In particular, we demonstrate how balancing can improve the efficiency of the alignment strength estimator for the total correlation, a parameter that plays a critical role in graph matchability and graph matching runtime complexity.

math.ST↗

arXiv2021

Spectral inference for large Stochastic Blockmodels with nodal covariates

In many applications of network analysis, it is important to distinguish between observed and unobserved factors affecting network structure. To this end, we develop spectral estimators for both unobserved blocks and the effect of covariates in stochastic blockmodels. On the theoretical side, we establish asymptotic normality of our estimators for the subsequent purpose of performing inference. On the applied side, we show that computing our estimator is much faster than standard variational expectation--maximization algorithms and scales well for large networks. Monte Carlo experiments suggest that the estimator performs well under different data generating processes. Our application to Facebook data shows evidence of homophily in gender, role and campus-residence, while allowing us to discover unobserved communities. The results in this paper provide a foundation for spectral estimation of the effect of observed covariates as well as unobserved latent community structure on the probability of link formation in networks.

stat.ME↗

arXiv2021

Inducing a hierarchy for multi-class classification problems

In applications where categorical labels follow a natural hierarchy, classification methods that exploit the label structure often outperform those that do not. Un-fortunately, the majority of classification datasets do not come pre-equipped with a hierarchical structure and classical flat classifiers must be employed. In this paper, we investigate a class of methods that induce a hierarchy that can similarly improve classification performance over flat classifiers. The class of methods follows the structure of first clustering the conditional distributions and subsequently using a hierarchical classifier with the induced hierarchy. We demonstrate the effectiveness of the class of methods both for discovering a latent hierarchy and for improving accuracy in principled simulation settings and three real data applications.

stat.ML↗

arXiv2021

clusterNOR: A NUMA-Optimized Clustering Framework

Clustering algorithms are iterative and have complex data access patterns that result in many small random memory accesses. The performance of parallel implementations suffer from synchronous barriers for each iteration and skewed workloads. We rethink the parallelization of clustering for modern non-uniform memory architectures (NUMA) to maximizes independent, asynchronous computation. We eliminate many barriers, reduce remote memory accesses, and maximize cache reuse. We implement the 'Clustering NUMA Optimized Routines' (clusterNOR) extensible parallel framework that provides algorithmic building blocks. The system is generic, we demonstrate nine modern clustering algorithms that have simple implementations. clusterNOR includes (i) in-memory, (ii) semi-external memory, and (iii) distributed memory execution, enabling computation for varying memory and hardware budgets. For algorithms that rely on Euclidean distance, clusterNOR defines an updated Elkan's triangle inequality pruning algorithm that uses asymptotically less memory so that it works on billion-point data sets. clusterNOR extends and expands the scope of the 'knor' library for k-means clustering by generalizing underlying principles, providing a uniform programming interface and expanding the scope to hierarchical and linear algebraic classes of algorithms. The compound effect of our optimizations is an order of magnitude improvement in speed over other state-of-the-art solutions, such as Spark's MLlib and Apple's Turi.

cs.DC↗

arXiv2020

A partition-based similarity for classification distributions

Herein we define a measure of similarity between classification distributions that is both principled from the perspective of statistical pattern recognition and useful from the perspective of machine learning practitioners. In particular, we propose a novel similarity on classification distributions, dubbed task similarity, that quantifies how an optimally-transformed optimal representation for a source distribution performs when applied to inference related to a target distribution. The definition of task similarity allows for natural definitions of adversarial and orthogonal distributions. We highlight limiting properties of representations induced by (universally) consistent decision rules and demonstrate in simulation that an empirical estimate of task similarity is a function of the decision rule deployed for inference. We demonstrate that for a given target distribution, both transfer efficiency and semantic similarity of candidate source distributions correlate with empirical task similarity.

stat.ML↗

arXiv2020

Simultaneous Dimensionality and Complexity Model Selection for Spectral Graph Clustering

Our problem of interest is to cluster vertices of a graph by identifying underlying community structure. Among various vertex clustering approaches, spectral clustering is one of the most popular methods because it is easy to implement while often outperforming more traditional clustering algorithms. However, there are two inherent model selection problems in spectral clustering, namely estimating both the embedding dimension and number of clusters. This paper attempts to address the issue by establishing a novel model selection framework specifically for vertex clustering on graphs under a stochastic block model. The first contribution is a probabilistic model which approximates the distribution of the extended spectral embedding of a graph. The model is constructed based on a theoretical result of asymptotic normality of the informative part of the embedding, and on a simulation result providing a conjecture for the limiting behavior of the redundant part of the embedding. The second contribution is a simultaneous model selection framework. In contrast with the traditional approaches, our model selection procedure estimates embedding dimension and number of clusters simultaneously. Based on our conjectured distributional model, a theorem on the consistency of the estimates of model parameters is presented, providing support for the validity of our method. Algorithms for our simultaneous model selection for vertex clustering are proposed, demonstrating superior performance in simulation experiments. We illustrate our method via application to a collection of brain graphs.

stat.ML↗

arXiv2020

Inference for multiple heterogeneous networks with a common invariant subspace

The development of models for multiple heterogeneous network data is of critical importance both in statistical network theory and across multiple application domains. Although single-graph inference is well-studied, multiple graph inference is largely unexplored, in part because of the challenges inherent in appropriately modeling graph differences and yet retaining sufficient model simplicity to render estimation feasible. This paper addresses exactly this gap, by introducing a new model, the common subspace independent-edge (COSIE) multiple random graph model, which describes a heterogeneous collection of networks with a shared latent structure on the vertices but potentially different connectivity patterns for each graph. The COSIE model encompasses many popular network representations, including the stochastic blockmodel. The model is both flexible enough to meaningfully account for important graph differences and tractable enough to allow for accurate inference in multiple networks. In particular, a joint spectral embedding of adjacency matrices - the multiple adjacency spectral embedding (MASE) - leads, in a COSIE model, to simultaneous consistent estimation of underlying parameters for each graph. Under mild additional assumptions, MASE estimates satisfy asymptotic normality and yield improvements for graph eigenvalue estimation and hypothesis testing. In both simulated and real data, the COSIE model and the MASE embedding can be deployed for a number of subsequent network inference tasks, including dimensionality reduction, classification, hypothesis testing and community detection. Specifically, when MASE is applied to a dataset of connectomes constructed through diffusion magnetic resonance imaging, the result is an accurate classification of brain scans by patient and a meaningful determination of heterogeneity across scans of different subjects.

stat.ME↗

arXiv2020

Maximum Likelihood Estimation and Graph Matching in Errorfully Observed Networks

Given a pair of graphs with the same number of vertices, the inexact graph matching problem consists in finding a correspondence between the vertices of these graphs that minimizes the total number of induced edge disagreements. We study this problem from a statistical framework in which one of the graphs is an errorfully observed copy of the other. We introduce a corrupting channel model, and show that in this model framework, the solution to the graph matching problem is a maximum likelihood estimator. Necessary and sufficient conditions for consistency of this MLE are presented, as well as a relaxed notion of consistency in which a negligible fraction of the vertices need not be matched correctly. The results are used to study matchability in several families of random graphs, including edge independent models, random regular graphs and small-world networks. We also use these results to introduce measures of matching feasibility, and experimentally validate the results on simulated and real-world networks.

stat.ML↗

arXiv2020

On estimation and inference in latent structure random graphs

We define a latent structure model (LSM) random graph as a random dot product graph (RDPG) in which the latent position distribution incorporates both probabilistic and geometric constraints, delineated by a family of underlying distributions on some fixed Euclidean space, and a structural support submanifold from which the latent positions for the graph are drawn. For a one-dimensional latent structure model with known structural support, we show how spectral estimates of the latent positions of an RDPG can be used for efficient estimation of the paramaters of the LSM. We describe how to estimate or learn the structural support in cases where it is unknown, with an illustrative focus on graphs with latent positions along the Hardy-Weinberg curve. Finally, we use the latent structure model formulation to test bilateral homology in the Drosophila connectome.

stat.ME↗

arXiv2020

Vertex Nomination, Consistent Estimation, and Adversarial Modification

Given a pair of graphs $G_1$ and $G_2$ and a vertex set of interest in $G_1$, the vertex nomination (VN) problem seeks to find the corresponding vertices of interest in $G_2$ (if they exist) and produce a rank list of the vertices in $G_2$, with the corresponding vertices of interest in $G_2$ concentrating, ideally, at the top of the rank list. In this paper, we define and derive the analogue of Bayes optimality for VN with multiple vertices of interest, and we define the notion of maximal consistency classes in vertex nomination. This theory forms the foundation for a novel VN adversarial contamination model, and we demonstrate with real and simulated data that there are VN schemes that perform effectively in the uncontaminated setting, and adversarial network contamination adversely impacts the performance of our VN scheme. We further define a network regularization method for mitigating the impact of the adversarial contamination, and we demonstrate the effectiveness of regularization in both real and synthetic data.

stat.ML↗

arXiv2020

On Two Distinct Sources of Nonidentifiability in Latent Position Random Graph Models

Two separate and distinct sources of nonidentifiability arise naturally in the context of latent position random graph models, though neither are unique to this setting. In this paper we define and examine these two nonidentifiabilities, dubbed subspace nonidentifiability and model-based nonidentifiability, in the context of random graph inference. We give examples where each type of nonidentifiability comes into play, and we show how in certain settings one need worry about one or the other type of nonidentifiability. Then, we characterize the limit for model-based nonidentifiability both with and without subspace nonidentifiability. We further obtain additional limiting results for covariances and $U$-statistics of stochastic block models and generalized random dot product graphs.

math.ST↗

arXiv2020

Numerical tolerance for spectral decompositions of random matrices

We precisely quantify the impact of statistical error in the quality of a numerical approximation to a random matrix eigendecomposition, and under mild conditions, we use this to introduce an optimal numerical tolerance for residual error in spectral decompositions of random matrices. We demonstrate that terminating an eigendecomposition algorithm when the numerical error and statistical error are of the same order results in computational savings with no loss of accuracy. We also repair a flaw in a ubiquitous termination condition, one in wide employ in several computational linear algebra implementations. We illustrate the practical consequences of our stopping criterion with an analysis of simulated and real networks. Our theoretical results and real-data examples establish that the tradeoff between statistical and numerical error is of significant import for data science.

stat.CO↗

arXiv2020

Alignment Strength and Correlation for Graphs

When two graphs have a correlated Bernoulli distribution, we prove that the alignment strength of their natural bijection strongly converges to a novel measure of graph correlation $ρ_T$ that neatly combines intergraph with intragraph distribution parameters. Within broad families of the random graph parameter settings, we illustrate that exact graph matching runtime and also matchability are both functions of $ρ_T$, with thresholding behavior starkly illustrated in matchability.

math.CO↗

arXiv2020

Sparse Representation Classification Beyond L1 Minimization and the Subspace Assumption

The sparse representation classifier (SRC) has been utilized in various classification problems, which makes use of L1 minimization and works well for image recognition satisfying a subspace assumption. In this paper we propose a new implementation of SRC via screening, establish its equivalence to the original SRC under regularity conditions, and prove its classification consistency under a latent subspace model and contamination. The results are demonstrated via simulations and real data experiments, where the new algorithm achieves comparable numerical performance and significantly faster.

stat.ML↗