arXiv subjects

Carey E. Priebe

Publications and source records attributed to Carey E. Priebe.

At least 145 records · Page 8Linked to original sources

arXiv2016

Community Detection and Classification in Hierarchical Stochastic Blockmodels

We propose a robust, scalable, integrated methodology for community detection and community comparison in graphs. In our procedure, we first embed a graph into an appropriate Euclidean space to obtain a low-dimensional representation, and then cluster the vertices into communities. We next employ nonparametric graph inference techniques to identify structural similarity among these communities. These two steps are then applied recursively on the communities, allowing us to detect more fine-grained structure. We describe a hierarchical stochastic blockmodel---namely, a stochastic blockmodel with a natural hierarchical structure---and establish conditions under which our algorithm yields consistent estimates of model parameters and motifs, which we define to be stochastically similar groups of subgraphs. Finally, we demonstrate the effectiveness of our algorithm in both simulated and real data. Specifically, we address the problem of locating similar subcommunities in a partially reconstructed Drosophila connectome and in the social network Friendster.

stat.ML↗

arXiv2016

On the Power of Likelihood Ratio Tests in Dimension-Restricted Submodels

Likelihood ratio tests are widely used to test statistical hypotheses about parametric families of probability distributions. If interest is restricted to a subfamily of distributions, then it is natural to inquire if the restricted LRT is superior to the unrestricted LRT. Marden's general LRT conjecture posits that any restriction placed on the alternative hypothesis will increase power. The only published counterexample to this conjecture is rather technical and involves a restriction that maintains the dimension of the alternative. We formulate the dimension-restricted LRT conjecture, which posits that any restriction that replaces a parametric family with a subfamily of lower dimension will increase power. Under standard regularity conditions, we then demonstrate that the restricted LRT is asymptotically more powerful than the unrestricted LRT for local alternatives. Remarkably, however, even the dimension-restricted LRT conjecture fails in the case of finite samples. Our counterexamples involve subfamilies of multinomial distributions. In particular, our study of the Hardy-Weinberg subfamily of trinomial distributions provides a simple and elegant demonstration that restrictions may not increase power.

math.ST↗

arXiv2016

Limit theorems for eigenvectors of the normalized Laplacian for random graphs

We prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and furthermore the mean and the covariance matrix of each row are functions of the associated vertex's block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.

stat.ML↗

arXiv2016

A Model-based Semi-Supervised Clustering Methodology

We consider an extension of model-based clustering to the semi-supervised case, where some of the data are pre-labeled. We provide a derivation of the Bayesian Information Criterion (BIC) approximation to the Bayes factor in this setting. We then use the BIC to the select number of clusters and the variables useful for clustering. We demonstrate the efficacy of this adaptation of the model-based clustering paradigm through two simulation examples and a fly larvae behavioral dataset in which lines of neurons are clustered into behavioral groups.

stat.ME↗

arXiv2016

An SSD-based eigensolver for spectral analysis on billion-node graphs

Many eigensolvers such as ARPACK and Anasazi have been developed to compute eigenvalues of a large sparse matrix. These eigensolvers are limited by the capacity of RAM. They run in memory of a single machine for smaller eigenvalue problems and require the distributed memory for larger problems. In contrast, we develop an SSD-based eigensolver framework called FlashEigen, which extends Anasazi eigensolvers to SSDs, to compute eigenvalues of a graph with hundreds of millions or even billions of vertices in a single machine. FlashEigen performs sparse matrix multiplication in a semi-external memory fashion, i.e., we keep the sparse matrix on SSDs and the dense matrix in memory. We store the entire vector subspace on SSDs and reduce I/O to improve performance through caching the most recent dense matrix. Our result shows that FlashEigen is able to achieve 40%-60% performance of its in-memory implementation and has performance comparable to the Anasazi eigensolvers on a machine with 48 CPU cores. Furthermore, it is capable of scaling to a graph with 3.4 billion vertices and 129 billion edges. It takes about four hours to compute eight eigenvalues of the billion-node graph using 120 GB memory.

cs.DC↗

arXiv2016

Empirical Bayes Estimation for the Stochastic Blockmodel

Inference for the stochastic blockmodel is currently of burgeoning interest in the statistical community, as well as in various application domains as diverse as social networks, citation networks, brain connectivity networks (connectomics), etc. Recent theoretical developments have shown that spectral embedding of graphs yields tractable distributional results; in particular, a random dot product latent position graph formulation of the stochastic blockmodel informs a mixture of normal distributions for the adjacency spectral embedding. We employ this new theory to provide an empirical Bayes methodology for estimation of block memberships of vertices in a random graph drawn from the stochastic blockmodel, and demonstrate its practical utility. The posterior inference is conducted using a Metropolis-within-Gibbs algorithm. The theory and methods are illustrated through Monte Carlo simulation studies, both within the stochastic blockmodel and beyond, and experimental results on a Wikipedia data set are presented.

stat.ME↗

arXiv2016

Semi-supervised K-means++

Traditionally, practitioners initialize the {\tt k-means} algorithm with centers chosen uniformly at random. Randomized initialization with uneven weights ({\tt k-means++}) has recently been used to improve the performance over this strategy in cost and run-time. We consider the k-means problem with semi-supervised information, where some of the data are pre-labeled, and we seek to label the rest according to the minimum cost solution. By extending the {\tt k-means++} algorithm and analysis to account for the labels, we derive an improved theoretical bound on expected cost and observe improved performance in simulated and real data examples. This analysis provides theoretical justification for a roughly linear semi-supervised clustering algorithm.

stat.ML↗

arXiv2016

Fidelity-Commensurability Tradeoff in Joint Embedding of Disparate Dissimilarities

In various data settings, it is necessary to compare observations from disparate data sources. We assume the data is in the dissimilarity representation and investigate a joint embedding method that results in a commensurate representation of disparate dissimilarities. We further assume that there are "matched" observations from different conditions which can be considered to be highly similar, for the sake of inference. The joint embedding results in the joint optimization of fidelity (preservation of within-condition dissimilarities) and commensurability (preservation of between-condition dissimilarities between matched observations). We show that the tradeoff between these two criteria can be made explicit using weighted raw stress as the objective function for multidimensional scaling. In our investigations, we use a weight parameter, $w$, to control the tradeoff, and choose match detection as the inference task. Our results show weights that are optimal (with respect to the inference task) are different than equal weights for commensurability and fidelity and the proposed weighted embedding scheme provides significant improvements in statistical power.

stat.ME↗

arXiv2015

A nonparametric two-sample hypothesis testing problem for random dot product graphs

We consider the problem of testing whether two finite-dimensional random dot product graphs have generating latent positions that are independently drawn from the same distribution, or distributions that are related via scaling or projection. We propose a test statistic that is a kernel-based function of the adjacency spectral embedding for each graph. We obtain a limiting distribution for our test statistic under the null and we show that our test procedure is consistent across a broad range of alternatives.

math.ST↗

arXiv2015

A model selection approach for clustering a multinomial sequence with non-negative factorization

We consider a problem of clustering a sequence of multinomial observations by way of a model selection criterion. We propose a form of a penalty term for the model selection procedure. Our approach subsumes both the conventional AIC and BIC criteria but also extends the conventional criteria in a way that it can be applicable also to a sequence of sparse multinomial observations, where even within a same cluster, the number of multinomial trials may be different for different observations. In addition, as a preliminary estimation step to maximum likelihood estimation, and more generally, to maximum $L_{q}$ estimation, we propose to use reduced rank projection in combination with non-negative factorization. We motivate our approach by showing that our model selection criterion and preliminary estimation step yield consistent estimates under simplifying assumptions. We also illustrate our approach through numerical experiments using real and simulated data.

stat.ML↗

arXiv2015

A Joint Graph Inference Case Study: the C.elegans Chemical and Electrical Connectomes

We investigate joint graph inference for the chemical and electrical connectomes of the \textit{Caenorhabditis elegans} roundworm. The \textit{C.elegans} connectomes consist of $253$ non-isolated neurons with known functional attributes, and there are two types of synaptic connectomes, resulting in a pair of graphs. We formulate our joint graph inference from the perspectives of seeded graph matching and joint vertex classification. Our results suggest that connectomic inference should proceed in the joint space of the two connectomes, which has significant neuroscientific implications.

stat.AP↗

arXiv2015

A semiparametric two-sample hypothesis testing problem for random dot product graphs

Two-sample hypothesis testing for random graphs arises naturally in neuroscience, social networks, and machine learning. In this paper, we consider a semiparametric problem of two-sample hypothesis testing for a class of latent position random graphs. We formulate a notion of consistency in this context and propose a valid test for the hypothesis that two finite-dimensional random dot product graphs on a common vertex set have the same generating latent positions or have generating latent positions that are scaled or diagonal transformations of one another. Our test statistic is a function of a spectral decomposition of the adjacency matrix for each graph and our test procedure is consistent across a broad range of alternatives. We apply our test procedure to real biological data: in a test-retest data set of neural connectome graphs, we are able to distinguish between scans from different subjects; and in the {\em C.elegans} connectome, we are able to distinguish between chemical and electrical networks. The latter example is a concrete demonstration that our test can have power even for small sample sizes. We conclude by discussing the relationship between our test procedure and generalized likelihood ratio tests.

stat.ME↗

arXiv2015

An Automated Images-to-Graphs Framework for High Resolution Connectomics

Reconstructing a map of neuronal connectivity is a critical challenge in contemporary neuroscience. Recent advances in high-throughput serial section electron microscopy (EM) have produced massive 3D image volumes of nanoscale brain tissue for the first time. The resolution of EM allows for individual neurons and their synaptic connections to be directly observed. Recovering neuronal networks by manually tracing each neuronal process at this scale is unmanageable, and therefore researchers are developing automated image processing modules. Thus far, state-of-the-art algorithms focus only on the solution to a particular task (e.g., neuron segmentation or synapse identification). In this manuscript we present the first fully automated images-to-graphs pipeline (i.e., a pipeline that begins with an imaged volume of neural tissue and produces a brain graph without any human interaction). To evaluate overall performance and select the best parameters and methods, we also develop a metric to assess the quality of the output graphs. We evaluate a set of algorithms and parameters, searching possible operating points to identify the best available brain graph for our assessment metric. Finally, we deploy a reference end-to-end version of the pipeline on a large, publicly available data set. This provides a baseline result and framework for community analysis and future algorithm development and testing. All code and data derivatives have been made publicly available toward eventually unlocking new biofidelic computational primitives and understanding of neuropathologies.

q-bio.QM↗

arXiv2015

Spectral Clustering for Divide-and-Conquer Graph Matching

We present a parallelized bijective graph matching algorithm that leverages seeds and is designed to match very large graphs. Our algorithm combines spectral graph embedding with existing state-of-the-art seeded graph matching procedures. We justify our approach by proving that modestly correlated, large stochastic block model random graphs are correctly matched utilizing very few seeds through our divide-and-conquer procedure. We also demonstrate the effectiveness of our approach in matching very large graphs in simulated and real data examples, showing up to a factor of 8 improvement in runtime with minimal sacrifice in accuracy.

stat.ML↗

arXiv2015

On the Incommensurability Phenomenon

Suppose that two large, multi-dimensional data sets are each noisy measurements of the same underlying random process, and principle components analysis is performed separately on the data sets to reduce their dimensionality. In some circumstances it may happen that the two lower-dimensional data sets have an inordinately large Procrustean fitting-error between them. The purpose of this manuscript is to quantify this "incommensurability phenomenon." In particular, under specified conditions, the square Procrustean fitting-error of the two normalized lower-dimensional data sets is (asymptotically) a convex combination (via a correlation parameter) of the Hausdorff distance between the projection subspaces and the maximum possible value of the square Procrustean fitting-error for normalized data. We show how this gives rise to the incommensurability phenomenon, and we employ illustrative simulations as well as a real data experiment to explore how the incommensurability phenomenon may have an appreciable impact.

stat.ML↗

arXiv2015

FlashGraph: Processing Billion-Node Graphs on an Array of Commodity SSDs

Graph analysis performs many random reads and writes, thus, these workloads are typically performed in memory. Traditionally, analyzing large graphs requires a cluster of machines so the aggregate memory exceeds the graph size. We demonstrate that a multicore server can process graphs with billions of vertices and hundreds of billions of edges, utilizing commodity SSDs with minimal performance loss. We do so by implementing a graph-processing engine on top of a user-space SSD file system designed for high IOPS and extreme parallelism. Our semi-external memory graph engine called FlashGraph stores vertex state in memory and edge lists on SSDs. It hides latency by overlapping computation with I/O. To save I/O bandwidth, FlashGraph only accesses edge lists requested by applications from SSDs; to increase I/O throughput and reduce CPU overhead for I/O, it conservatively merges I/O requests. These designs maximize performance for applications with different I/O characteristics. FlashGraph exposes a general and flexible vertex-centric programming interface that can express a wide variety of graph algorithms and their optimizations. We demonstrate that FlashGraph in semi-external memory performs many algorithms with performance up to 80% of its in-memory implementation and significantly outperforms PowerGraph, a popular distributed in-memory graph engine.

cs.DC↗

arXiv2015

Graph Matching: Relax at Your Own Risk

Graph matching---aligning a pair of graphs to minimize their edge disagreements---has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and connectomics. Its attention can be partially attributed to its computational difficulty. Although many heuristics have previously been proposed in the literature to approximately solve graph matching, very few have any theoretical support for their performance. A common technique is to relax the discrete problem to a continuous problem, therefore enabling practitioners to bring gradient-descent-type algorithms to bear. We prove that an indefinite relaxation (when solved exactly) almost always discovers the optimal permutation, while a common convex relaxation almost always fails to discover the optimal permutation. These theoretical results suggest that initializing the indefinite algorithm with the convex optimum might yield improved practical performance. Indeed, experimental results illuminate and corroborate these theoretical findings, demonstrating that excellent results are achieved in both benchmark and real data problems by amalgamating the two approaches.

stat.ML↗

arXiv2014

Fast Approximate Quadratic Programming for Large (Brain) Graph Matching

Quadratic assignment problems (QAPs) arise in a wide variety of domains, ranging from operations research to graph theory to computer vision to neuroscience. In the age of big data, graph valued data is becoming more prominent, and with it, a desire to run algorithms on ever larger graphs. Because QAP is NP-hard, exact algorithms are intractable. Approximate algorithms necessarily employ an accuracy/efficiency trade-off. We developed a fast approximate quadratic assignment algorithm (FAQ). FAQ finds a local optima in (worst case) time cubic in the number of vertices, similar to other approximate QAP algorithms. We demonstrate empirically that our algorithm is faster and achieves a lower objective value on over 80% of the suite of QAP benchmarks, compared with the previous state-of-the-art. Applying the algorithms to our motivating example, matching C. elegans connectomes (brain-graphs), we find that FAQ achieves the optimal performance in record time, whereas none of the others even find the optimum.

math.OC↗