arXiv subjects

Joshua Cape

Publications and source records attributed to Joshua Cape.

At least 19 recordsLinked to original sources

arXiv2025

Testing for latent structure via the Wilcoxon--Wigner random matrix of normalized rank statistics

This paper considers the problem of testing for latent structure in large symmetric data matrices. The goal here is to develop statistically principled methodology that is flexible in its applicability, computationally efficient, and insensitive to extreme data variation, thereby overcoming limitations facing existing approaches. To do so, we introduce and systematically study certain symmetric matrices, called Wilcoxon--Wigner random matrices, whose entries are normalized rank statistics derived from an underlying independent and identically distributed sample of absolutely continuous random variables. These matrices naturally arise as the matricization of one-sample problems in statistics and conceptually lie at the interface of nonparametrics, multivariate analysis, and data reduction. Among our results, we establish that the leading eigenvalue and corresponding eigenvector of Wilcoxon--Wigner random matrices admit asymptotically Gaussian fluctuations with explicit centering and scaling terms. These asymptotic results enable rigorous parameter-free and distribution-free spectral methodology for addressing two hypothesis testing problems, namely community detection and principal submatrix detection. Numerical examples illustrate the performance of the proposed approach. Throughout, our findings are juxtaposed with existing results based on the spectral properties of independent entry symmetric random matrices in signal-plus-noise data settings.

stat.ME↗

arXiv2025

Network Signflip Parallel Analysis for Selecting the Embedding Dimension

This paper investigates the problem of selecting the embedding dimension for large heterogeneous networks that have weakly distinguishable community structure. For a broad family of embeddings based on normalized adjacency matrices, we introduce a novel spectral method that compares the eigenvalues of the normalized adjacency matrix to those obtained after randomly signflipping its entries. The proposed method, called network signflip parallel analysis (NetFlipPA), is interpretable, simple to implement, data driven, and does not require users to carefully tune parameters. For large random graphs arising from degree-corrected stochastic blockmodels with weakly distinguishable community structure (and consequently, non-diverging eigenvalues), NetFlipPA provably recovers the spectral noise floor (i.e., the operator norm of the noise component of the normalized adjacency matrix). NetFlipPA thus provides a statistically rigorous randomization-based method for selecting the embedding dimension by keeping the eigenvalues whose magnitudes rise above the recovered spectral noise floor. Compared to traditional cutoff-based methods, the data-driven threshold used in NetFlipPA is provably effective under milder assumptions on the node degree heterogeneity and the number of node communities. Our main results rely on careful non-asymptotic perturbation analysis and leverage recent progress on local laws for nonhomogeneous Wigner-type random matrices.

math.ST↗

arXiv2025

Simultaneous estimation of connectivity and dimensionality in samples of networks

An overarching objective in contemporary statistical network analysis is extracting salient information from datasets consisting of multiple networks. To date, considerable attention has been devoted to node and network clustering, while comparatively less attention has been devoted to downstream connectivity estimation and parsimonious embedding dimension selection. Given a sample of potentially heterogeneous networks, this paper proposes a method to simultaneously estimate a latent matrix of connectivity probabilities and its embedding dimensionality or rank after first pre-estimating the number of communities and the node community memberships. The method is formulated as a convex optimization problem and solved using an alternating direction method of multipliers algorithm. We establish estimation error bounds under the Frobenius norm and nuclear norm for settings in which observable networks have blockmodel structure, even when node memberships are imperfectly recovered. When perfect membership recovery is possible and dimensionality is much smaller than the number of communities, the proposed method outperforms conventional averaging-based methods for estimating connectivity and dimensionality. Numerical studies empirically demonstrate the accuracy of our method across various scenarios. Additionally, analysis of a primate brain dataset demonstrates that posited connectivity is not necessarily full rank in practice, illustrating the need for flexible methodology.

stat.ME↗

arXiv2025

Extreme value theory for singular subspace estimation in the matrix denoising model

This paper studies fine-grained singular subspace estimation in the matrix denoising model where a deterministic low-rank signal matrix is additively perturbed by a stochastic matrix of Gaussian noise. We establish that the maximum Euclidean row norm (i.e., the two-to-infinity norm) of the aligned difference between the leading sample and population singular vectors approaches the Gumbel distribution in the large-matrix limit, under suitable signal-to-noise conditions and after appropriate centering and scaling. We apply our novel asymptotic distributional theory to test hypotheses of low-rank signal structure encoded in the leading singular vectors and their corresponding principal subspace. We provide de-biased estimators for the corresponding nuisance signal singular values and show that our proposed plug-in test statistic has desirable properties. Notably, compared to using the Frobenius norm subspace distance, our test statistic based on the two-to-infinity norm empirically has higher power to detect structured alternatives that differ from the null in only a few matrix entries or rows. Our main results are obtained by a novel synthesis of and technical analysis involving row-wise matrix perturbation analysis, extreme value theory, saddle point approximation methods, and random matrix theory. Our contributions complement the existing literature for matrix denoising focused on minimaxity, mean squared error analysis, unitarily invariant distances between subspaces, component-wise asymptotic distributional theory, and row-wise uniform error bounds. Numerical simulations illustrate our main results and demonstrate the robustness properties of our testing procedure to non-Gaussian noise distributions.

math.ST↗

arXiv2024

Robust spectral clustering with rank statistics

This paper analyzes the statistical performance of a robust spectral clustering method for latent structure recovery in noisy data matrices. We consider eigenvector-based clustering applied to a matrix of nonparametric rank statistics that is derived entrywise from the raw, original data matrix. This approach is robust in the sense that, unlike traditional spectral clustering procedures, it can provably recover population-level latent block structure even when the observed data matrix includes heavy-tailed entries and has a heterogeneous variance profile. Our main theoretical contributions are threefold and hold under flexible data generating conditions. First, we establish that robust spectral clustering with rank statistics can consistently recover latent block structure, viewed as communities of nodes in a graph, in the sense that unobserved community memberships for all but a vanishing fraction of nodes are correctly recovered with high probability when the data matrix is large. Second, we refine the former result and further establish that, under certain conditions, the community membership of any individual, specified node of interest can be asymptotically exactly recovered with probability tending to one in the large-data limit. Third, we establish asymptotic normality results associated with the truncated eigenstructure of matrices whose entries are rank statistics, made possible by synthesizing contemporary entrywise matrix perturbation analysis with the classical nonparametric theory of so-called simple linear rank statistics. Collectively, these results demonstrate the statistical utility of rank-based data transformations when paired with spectral techniques for dimensionality reduction. Additionally, for a dataset of human connectomes, our approach yields parsimonious dimensionality reduction and improved recovery of ground-truth neuroanatomical cluster structure.

stat.ML↗

arXiv2024

On varimax asymptotics in network models and spectral methods for dimensionality reduction

Varimax factor rotations, while popular among practitioners in psychology and statistics since being introduced by H. Kaiser, have historically been viewed with skepticism and suspicion by some theoreticians and mathematical statisticians. Now, work by K. Rohe and M. Zeng provides new, fundamental insight: varimax rotations provably perform statistical estimation in certain classes of latent variable models when paired with spectral-based matrix truncations for dimensionality reduction. We build on this newfound understanding of varimax rotations by developing further connections to network analysis and spectral methods rooted in entrywise matrix perturbation analysis. Concretely, this paper establishes the asymptotic multivariate normality of vectors in varimax-transformed Euclidean point clouds that represent low-dimensional node embeddings in certain latent space random graph models. We address related concepts including network sparsity, data denoising, and the role of matrix rank in latent variable parameterizations. Collectively, these findings, at the confluence of classical and contemporary multivariate analysis, reinforce methodology and inference procedures grounded in matrix factorization-based techniques. Numerical examples illustrate our findings and supplement our discussion.

math.ST↗

arXiv2024

On inference for modularity statistics in structured networks

This paper revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional results in the large-network limit for networks exhibiting nodal community structure. Our work facilitates testing for network differences and can be used in conjunction with existing theoretical guarantees for stochastic blockmodel random graphs. Our results are enabled by recent advances in the study of low-rank truncations of large network adjacency matrices. We provide confirmatory simulation studies and real data analysis pertaining to the network neuroscience study of psychosis, specifically schizophrenia. Collectively, this paper contributes to the limited existing literature to date on statistical inference for modularity-based network analysis. Supplemental materials for this article are available online.

stat.ME↗

arXiv2023

An Overview of Asymptotic Normality in Stochastic Blockmodels: Cluster Analysis and Inference

This paper provides a selective review of the statistical network analysis literature focused on clustering and inference problems for stochastic blockmodels and their variants. We survey asymptotic normality results for stochastic blockmodels as a means of thematically linking classical statistical concepts to contemporary research in network data analysis. Of note, multiple different forms of asymptotically Gaussian behavior arise in stochastic blockmodels and are useful for different purposes, pertaining to estimation and testing, the characterization of cluster structure in community detection, and understanding latent space geometry. This paper concludes with a discussion of open problems and ongoing research activities addressing asymptotic normality and its implications for statistical network modeling.

math.ST↗

arXiv2023

A statistical framework for GWAS of high dimensional phenotypes using summary statistics, with application to metabolite GWAS

The recent explosion of genetic and high dimensional biobank and 'omic' data has provided researchers with the opportunity to investigate the shared genetic origin (pleiotropy) of hundreds to thousands of related phenotypes. However, existing methods for multi-phenotype genome-wide association studies (GWAS) do not model pleiotropy, are only applicable to a small number of phenotypes, or provide no way to perform inference. To add further complication, raw genetic and phenotype data are rarely observed, meaning analyses must be performed on GWAS summary statistics whose statistical properties in high dimensions are poorly understood. We therefore developed a novel model, theoretical framework, and set of methods to perform Bayesian inference in GWAS of high dimensional phenotypes using summary statistics that explicitly model pleiotropy, beget fast computation, and facilitate the use of biologically informed priors. We demonstrate the utility of our procedure by applying it to metabolite GWAS, where we develop new nonparametric priors for genetic effects on metabolite levels that use known metabolic pathway information and foster interpretable inference at the pathway level.

stat.ME↗

arXiv2022

Spectral embedding and the latent geometry of multipartite networks

Spectral embedding finds vector representations of the nodes of a network, based on the eigenvectors of a properly constructed matrix, and has found applications throughout science and technology. Many networks are multipartite, meaning that they contain nodes of fundamentally different types, e.g. drugs, diseases and proteins, and edges are only observed between nodes of different types. When the network is multipartite, this paper demonstrates that the node representations obtained via spectral embedding lie near type-specific low-dimensional subspaces of a higher-dimensional ambient space. For this reason we propose a follow-on step after spectral embedding, to recover node representations in their intrinsic rather than ambient dimension, proving uniform consistency under a low-rank, inhomogeneous random graph model. We demonstrate the performance of our procedure on a large 6-partite biomedical network relevant for drug discovery.

stat.ME↗

arXiv2020

Multiple Network Embedding for Anomaly Detection in Time Series of Graphs

This paper considers the graph signal processing problem of anomaly detection in time series of graphs. We examine two related, complementary inference tasks: the detection of anomalous graphs within a time series, and the detection of temporally anomalous vertices. We approach these tasks via the adaptation of statistically principled methods for joint graph inference, specifically \emph{multiple adjacency spectral embedding} (MASE). We demonstrate that our method is effective for our inference tasks. Moreover, we assess the performance of our method in terms of the underlying nature of detectable anomalies. We further provide the theoretical justification for our method and insight into its use. Applied to the Enron communication graph, a large-scale commercial search engine time series of graphs, and a larval Drosophila connectome data, our approaches demonstrate their applicability and identify the anomalous vertices beyond just large degree change.

stat.ME↗

arXiv2020

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↗

arXiv2019

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↗

arXiv2019

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↗

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.

stat.ML↗

arXiv2019

Bayesian Estimation of Sparse Spiked Covariance Matrices in High Dimensions

We propose a Bayesian methodology for estimating spiked covariance matrices with jointly sparse structure in high dimensions. The spiked covariance matrix is reparametrized in terms of the latent factor model, where the loading matrix is equipped with a novel matrix spike-and-slab LASSO prior, which is a continuous shrinkage prior for modeling jointly sparse matrices. We establish the rate-optimal posterior contraction for the covariance matrix with respect to the operator norm as well as that for the principal subspace with respect to the projection operator norm loss. We also study the posterior contraction rate of the principal subspace with respect to the two-to-infinity norm loss, a novel loss function measuring the distance between subspaces that is able to capture element-wise eigenvector perturbations. We show that the posterior contraction rate with respect to the two-to-infinity norm loss is tighter than that with respect to the routinely used projection operator norm loss under certain low-rank and bounded coherence conditions. In addition, a point estimator for the principal subspace is proposed with the rate-optimal risk bound with respect to the projection operator norm loss. These results are based on a collection of concentration and large deviation inequalities for the matrix spike-and-slab LASSO prior. The numerical performance of the proposed methodology is assessed through synthetic examples and the analysis of a real-world face data example.

stat.ME↗

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.

math.ST↗

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.

math.ST↗