arXiv subjects

Minh Tang

Publications and source records attributed to Minh Tang.

At least 37 records · Page 2Linked to original sources

arXiv2021

Valid Two-Sample Graph Testing via Optimal Transport Procrustes and Multiscale Graph Correlation with Applications in Connectomics

Testing whether two graphs come from the same distribution is of interest in many real world scenarios, including brain network analysis. Under the random dot product graph model, the nonparametric hypothesis testing frame-work consists of embedding the graphs using the adjacency spectral embedding (ASE), followed by aligning the embeddings using the median flip heuristic, and finally applying the nonparametric maximum mean discrepancy(MMD) test to obtain a p-value. Using synthetic data generated from Drosophila brain networks, we show that the median flip heuristic results in an invalid test, and demonstrate that optimal transport Procrustes (OTP) for alignment resolves the invalidity. We further demonstrate that substituting the MMD test with multiscale graph correlation(MGC) test leads to a more powerful test both in synthetic and in simulated data. Lastly, we apply this powerful test to the right and left hemispheres of the larval Drosophila mushroom body brain networks, and conclude that there is not sufficient evidence to reject the null hypothesis that the two hemispheres are equally distributed.

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arXiv2021

Supervised Dimensionality Reduction for Big Data

To solve key biomedical problems, experimentalists now routinely measure millions or billions of features (dimensions) per sample, with the hope that data science techniques will be able to build accurate data-driven inferences. Because sample sizes are typically orders of magnitude smaller than the dimensionality of these data, valid inferences require finding a low-dimensional representation that preserves the discriminating information (e.g., whether the individual suffers from a particular disease). There is a lack of interpretable supervised dimensionality reduction methods that scale to millions of dimensions with strong statistical theoretical guarantees.We introduce an approach, XOX, to extending principal components analysis by incorporating class-conditional moment estimates into the low-dimensional projection. The simplest ver-sion, "Linear Optimal Low-rank" projection (LOL), incorporates the class-conditional means. We prove, and substantiate with both synthetic and real data benchmarks, that LOL and its generalizations in the XOX framework lead to improved data representations for subsequent classification, while maintaining computational efficiency and scalability. Using multiple brain imaging datasets consisting of >150 million features, and several genomics datasets with>500,000 features, LOL outperforms other scalable linear dimensionality reduction techniques in terms of accuracy, while only requiring a few minutes on a standard desktop computer.

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arXiv2020

Two-sample Testing on Latent Distance Graphs With Unknown Link Functions

We propose a valid and consistent test for the hypothesis that two latent distance random graphs on the same vertex set have the same generating latent positions, up to some unidentifiable similarity transformations. Our test statistic is based on first estimating the edge probabilities matrices by truncating the singular value decompositions of the averaged adjacency matrices in each population and then computing a Spearman rank correlation coefficient between these estimates. Experimental results on simulated data indicate that the test procedure has power even when there is only one sample from each population, provided that the number of vertices is not too small. Application on a dataset of neural connectome graphs showed that we can distinguish between scans from different age groups while application on a dataset of epileptogenic recordings showed that we can discriminate between seizure and non-seizure events.

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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.

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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.

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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.

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arXiv2019

Limit theorems for out-of-sample extensions of the adjacency and Laplacian spectral embeddings

Graph embeddings, a class of dimensionality reduction techniques designed for relational data, have proven useful in exploring and modeling network structure. Most dimensionality reduction methods allow out-of-sample extensions, by which an embedding can be applied to observations not present in the training set. Applied to graphs, the out-of-sample extension problem concerns how to compute the embedding of a vertex that is added to the graph after an embedding has already been computed. In this paper, we consider the out-of-sample extension problem for two graph embedding procedures: the adjacency spectral embedding and the Laplacian spectral embedding. In both cases, we prove that when the underlying graph is generated according to a latent space model called the random dot product graph, which includes the popular stochastic block model as a special case, an out-of-sample extension based on a least-squares objective obeys a central limit theorem about the true latent position of the out-of-sample vertex. In addition, we prove a concentration inequality for the out-of-sample extension of the adjacency spectral embedding based on a maximum-likelihood objective. Our results also yield a convenient framework in which to analyze trade-offs between estimation accuracy and computational expense, which we explore briefly.

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arXiv2019

A central limit theorem for an omnibus embedding of multiple random graphs and implications for multiscale network inference

Performing statistical analyses on collections of graphs is of import to many disciplines, but principled, scalable methods for multi-sample graph inference are few. Here we describe an "omnibus" embedding in which multiple graphs on the same vertex set are jointly embedded into a single space with a distinct representation for each graph. We prove a central limit theorem for this embedding and demonstrate how it streamlines graph comparison, obviating the need for pairwise subspace alignments. The omnibus embedding achieves near-optimal inference accuracy when graphs arise from a common distribution and yet retains discriminatory power as a test procedure for the comparison of different graphs. Moreover, this joint embedding and the accompanying central limit theorem are important for answering multiscale graph inference questions, such as the identification of specific subgraphs or vertices responsible for similarity or difference across networks. We illustrate this with a pair of analyses of connectome data derived from dMRI and fMRI scans of human subjects. In particular, we show that this embedding allows the identification of specific brain regions associated with population-level differences. Finally, we sketch how the omnibus embedding can be used to address pressing open problems, both theoretical and practical, in multisample graph inference.

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arXiv2019

Central Limit Theorems for Classical Multidimensional Scaling

Classical multidimensional scaling is a widely used method in dimensionality reduction and manifold learning. The method takes in a dissimilarity matrix and outputs a low-dimensional configuration matrix based on a spectral decomposition. In this paper, we present three noise models and analyze the resulting configuration matrices, or embeddings. In particular, we show that under each of the three noise models the resulting embedding gives rise to a central limit theorem. We also provide compelling simulations and real data illustrations of these central limit theorems. This perturbation analysis represents a significant advancement over previous results regarding classical multidimensional scaling behavior under randomness.

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arXiv2019

On a 'Two Truths' Phenomenon in Spectral Graph Clustering

Clustering is concerned with coherently grouping observations without any explicit concept of true groupings. Spectral graph clustering - clustering the vertices of a graph based on their spectral embedding - is commonly approached via K-means (or, more generally, Gaussian mixture model) clustering composed with either Laplacian or Adjacency spectral embedding (LSE or ASE). Recent theoretical results provide new understanding of the problem and solutions, and lead us to a 'Two Truths' LSE vs. ASE spectral graph clustering phenomenon convincingly illustrated here via a diffusion MRI connectome data set: the different embedding methods yield different clustering results, with LSE capturing left hemisphere/right hemisphere affinity structure and ASE capturing gray matter/white matter core-periphery structure.

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arXiv2018

The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference. Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.

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arXiv2018

Signal-plus-noise matrix models: eigenvector deviations and fluctuations

Estimating eigenvectors and low-dimensional subspaces is of central importance for numerous problems in statistics, computer science, and applied mathematics. This paper characterizes the behavior of perturbed eigenvectors for a range of signal-plus-noise matrix models encountered in both statistical and random matrix theoretic settings. We prove both first-order approximation results (i.e. sharp deviations) as well as second-order distributional limit theory (i.e. fluctuations). The concise methodology considered in this paper synthesizes tools rooted in two core concepts, namely (i) deterministic decompositions of matrix perturbations and (ii) probabilistic matrix concentration phenomena. We illustrate our theoretical results via simulation examples involving stochastic block model random graphs.

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arXiv2018

The Kato--Temple inequality and eigenvalue concentration with applications to graph inference

We present an adaptation of the Kato--Temple inequality for bounding perturbations of eigenvalues with applications to statistical inference for random graphs, specifically hypothesis testing and change-point detection. We obtain explicit high-probability bounds for the individual distances between certain signal eigenvalues of a graph's adjacency matrix and the corresponding eigenvalues of the model's edge probability matrix, even when the latter eigenvalues have multiplicity. Our results extend more broadly to the perturbation of singular values in the presence of quite general random matrix noise.

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arXiv2018

On spectral embedding performance and elucidating network structure in stochastic block model graphs

Statistical inference on graphs often proceeds via spectral methods involving low-dimensional embeddings of matrix-valued graph representations, such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic information-theoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic block model graphs by way of Chernoff information. We investigate the relationship between spectral embedding performance and underlying network structure (e.g.~homogeneity, affinity, core-periphery, (un)balancedness) via a comprehensive treatment of the two-block stochastic block model and the class of $K$-block models exhibiting homogeneous balanced affinity structure. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, "Laplacian spectral embedding favors relatively sparse graphs, whereas adjacency spectral embedding favors not-too-sparse graphs." We also provide evidence in support of the claim that "adjacency spectral embedding favors core-periphery network structure."

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arXiv2018

The eigenvalues of stochastic blockmodel graphs

We derive the limiting distribution for the largest eigenvalues of the adjacency matrix for a stochastic blockmodel graph when the number of vertices tends to infinity. We show that, in the limit, these eigenvalues are jointly multivariate normal with bounded covariances. Our result extends the classic result of Füredi and Komlós on the fluctuation of the largest eigenvalue for Erdős-Rényi graphs.

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arXiv2017

Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectral

We establish asymptotic normality results for estimation of the block probability matrix $\mathbf{B}$ in stochastic blockmodel graphs using spectral embedding when the average degrees grows at the rate of $ω(\sqrt{n})$ in $n$, the number of vertices. As a corollary, we show that when $\mathbf{B}$ is of full-rank, estimates of $\mathbf{B}$ obtained from spectral embedding are asymptotically efficient. When $\mathbf{B}$ is singular the estimates obtained from spectral embedding can have smaller mean square error than those obtained from maximizing the log-likelihood under no rank assumption, and furthermore, can be almost as efficient as the true MLE that assume known $\mathrm{rk}(\mathbf{B})$. Our results indicate, in the context of stochastic blockmodel graphs, that spectral embedding is not just computationally tractable, but that the resulting estimates are also admissible, even when compared to the purportedly optimal but computationally intractable maximum likelihood estimation under no rank assumption.

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arXiv2017

Statistical inference on random dot product graphs: a survey

The random dot product graph (RDPG) is an independent-edge random graph that is analytically tractable and, simultaneously, either encompasses or can successfully approximate a wide range of random graphs, from relatively simple stochastic block models to complex latent position graphs. In this survey paper, we describe a comprehensive paradigm for statistical inference on random dot product graphs, a paradigm centered on spectral embeddings of adjacency and Laplacian matrices. We examine the analogues, in graph inference, of several canonical tenets of classical Euclidean inference: in particular, we summarize a body of existing results on the consistency and asymptotic normality of the adjacency and Laplacian spectral embeddings, and the role these spectral embeddings can play in the construction of single- and multi-sample hypothesis tests for graph data. We investigate several real-world applications, including community detection and classification in large social networks and the determination of functional and biologically relevant network properties from an exploratory data analysis of the Drosophila connectome. We outline requisite background and current open problems in spectral graph inference.

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arXiv2017

Robust Estimation from Multiple Graphs under Gross Error Contamination

Estimation of graph parameters based on a collection of graphs is essential for a wide range of graph inference tasks. In practice, weighted graphs are generally observed with edge contamination. We consider a weighted latent position graph model contaminated via an edge weight gross error model and propose an estimation methodology based on robust Lq estimation followed by low-rank adjacency spectral decomposition. We demonstrate that, under appropriate conditions, our estimator both maintains Lq robustness and wins the bias-variance tradeoff by exploiting low-rank graph structure. We illustrate the improvement offered by our estimator via both simulations and a human connectome data experiment.

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