arXiv subjects

Patrick O. Perry

Publications and source records attributed to Patrick O. Perry.

12 recordsLinked to original sources

arXiv2020

Profile Likelihood Biclustering

Biclustering, the process of simultaneously clustering the rows and columns of a data matrix, is a popular and effective tool for finding structure in a high-dimensional dataset. Many biclustering procedures appear to work well in practice, but most do not have associated consistency guarantees. To address this shortcoming, we propose a new biclustering procedure based on profile likelihood. The procedure applies to a broad range of data modalities, including binary, count, and continuous observations. We prove that the procedure recovers the true row and column classes when the dimensions of the data matrix tend to infinity, even if the functional form of the data distribution is misspecified. The procedure requires computing a combinatorial search, which can be expensive in practice. Rather than performing this search directly, we propose a new heuristic optimization procedure based on the Kernighan-Lin heuristic, which has nice computational properties and performs well in simulations. We demonstrate our procedure with applications to congressional voting records, and microarray analysis.

stat.ME↗

arXiv2018

Fitting a deeply-nested hierarchical model to a large book review dataset using a moment-based estimator

We consider a particular instance of a common problem in recommender systems: using a database of book reviews to inform user-targeted recommendations. In our dataset, books are categorized into genres and sub-genres. To exploit this nested taxonomy, we use a hierarchical model that enables information pooling across across similar items at many levels within the genre hierarchy. The main challenge in deploying this model is computational: the data sizes are large, and fitting the model at scale using off-the-shelf maximum likelihood procedures is prohibitive. To get around this computational bottleneck, we extend a moment-based fitting procedure proposed for fitting single-level hierarchical models to the general case of arbitrarily deep hierarchies. This extension is an order of magnetite faster than standard maximum likelihood procedures. The fitting method can be deployed beyond recommender systems to general contexts with deeply-nested hierarchical generalized linear mixed models.

stat.ME↗

arXiv2017

Scaling Text with the Class Affinity Model

Probabilistic methods for classifying text form a rich tradition in machine learning and natural language processing. For many important problems, however, class prediction is uninteresting because the class is known, and instead the focus shifts to estimating latent quantities related to the text, such as affect or ideology. We focus on one such problem of interest, estimating the ideological positions of 55 Irish legislators in the 1991 Dáil confidence vote. To solve the Dáil scaling problem and others like it, we develop a text modeling framework that allows actors to take latent positions on a "gray" spectrum between "black" and "white" polar opposites. We are able to validate results from this model by measuring the influences exhibited by individual words, and we are able to quantify the uncertainty in the scaling estimates by using a sentence-level block bootstrap. Applying our method to the Dáil debate, we are able to scale the legislators between extreme pro-government and pro-opposition in a way that reveals nuances in their speeches not captured by their votes or party affiliations.

stat.ML↗

arXiv2017

Estimating the number of clusters using cross-validation

Many clustering methods, including k-means, require the user to specify the number of clusters as an input parameter. A variety of methods have been devised to choose the number of clusters automatically, but they often rely on strong modeling assumptions. This paper proposes a data-driven approach to estimate the number of clusters based on a novel form of cross-validation. The proposed method differs from ordinary cross-validation, because clustering is fundamentally an unsupervised learning problem. Simulation and real data analysis results show that the proposed method outperforms existing methods, especially in high-dimensional settings with heterogeneous or heavy-tailed noise. In a yeast cell cycle dataset, the proposed method finds a parsimonious clustering with interpretable gene groupings.

stat.ME↗

arXiv2015

Fast Moment-Based Estimation for Hierarchical Models

Hierarchical models allow for heterogeneous behaviours in a population while simultaneously borrowing estimation strength across all subpopulations. Unfortunately, existing likelihood-based methods for fitting hierarchical models have high computational demands, and these demands have limited their adoption in large-scale prediction and inference problems. This paper proposes a moment-based procedure for estimating the parameters of a hierarchical model which has its roots in a method originally introduced by Cochran in 1937. The method trades statistical efficiency for computational efficiency. It gives consistent parameter estimates, competitive prediction error performance, and substantial computational improvements. When applied to a large-scale recommender system application and compared to a standard maximum likelihood procedure, the method delivers competitive prediction performance while reducing the sequential computation time from hours to minutes.

stat.ME↗

arXiv2015

Degrees of freedom for combining regression with factor analysis

In the AGEMAP genomics study, researchers were interested in detecting genes related to age in a variety of tissue types. After not finding many age-related genes in some of the analyzed tissue types, the study was criticized for having low power. It is possible that the low power is due to the presence of important unmeasured variables, and indeed we find that a latent factor model appears to explain substantial variability not captured by measured covariates. We propose including the estimated latent factors in a multiple regression model. The key difficulty in doing so is assigning appropriate degrees of freedom to the estimated factors to obtain unbiased error variance estimators and enable valid hypothesis testing. When the number of responses is large relative to the sample size, treating the estimated factors like observed covariates leads to a downward bias in the variance estimates. Many ad-hoc solutions to this problem have been proposed in the literature without the backup of a careful theoretical analysis. Using recent results from random matrix theory, we derive a simple, easy to use expression for degrees of freedom. Our estimate gives a principled alternative to ad-hoc approaches in common use. Extensive simulation results show excellent agreement between the proposed estimator and its theoretical value. Applying our methodology to the AGEMAP genomics study, we found an order of magnitude increase in the number of significant genes. Although we focus on the AGEMAP study, the methods developed in this paper are widely applicable to other multivariate models, and thus are of independent interest.

stat.ME↗

arXiv2012

Point process modeling for directed interaction networks

Network data often take the form of repeated interactions between senders and receivers tabulated over time. A primary question to ask of such data is which traits and behaviors are predictive of interaction. To answer this question, a model is introduced for treating directed interactions as a multivariate point process: a Cox multiplicative intensity model using covariates that depend on the history of the process. Consistency and asymptotic normality are proved for the resulting partial-likelihood-based estimators under suitable regularity conditions, and an efficient fitting procedure is described. Multicast interactions--those involving a single sender but multiple receivers--are treated explicitly. The resulting inferential framework is then employed to model message sending behavior in a corporate e-mail network. The analysis gives a precise quantification of which static shared traits and dynamic network effects are predictive of message recipient selection.

stat.ME↗

arXiv2012

Null models for network data

The analysis of datasets taking the form of simple, undirected graphs continues to gain in importance across a variety of disciplines. Two choices of null model, the logistic-linear model and the implicit log-linear model, have come into common use for analyzing such network data, in part because each accounts for the heterogeneity of network node degrees typically observed in practice. Here we show how these both may be viewed as instances of a broader class of null models, with the property that all members of this class give rise to essentially the same likelihood-based estimates of link probabilities in sparse graph regimes. This facilitates likelihood-based computation and inference, and enables practitioners to choose the most appropriate null model from this family based on application context. Comparative model fits for a variety of network datasets demonstrate the practical implications of our results.

math.ST↗

arXiv2011

Regularized Laplacian Estimation and Fast Eigenvector Approximation

Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistical interpretation of their approximation procedure. Our interpretation will be analogous to the manner in which $\ell_2$-regularized or $\ell_1$-regularized $\ell_2$-regression (often called Ridge regression and Lasso regression, respectively) can be interpreted in terms of a Gaussian prior or a Laplace prior, respectively, on the coefficient vector of the regression problem. Our framework will imply that the solutions to the Mahoney-Orecchia regularized SDP can be interpreted as regularized estimates of the pseudoinverse of the graph Laplacian. Conversely, it will imply that the solution to this regularized estimation problem can be computed very quickly by running, e.g., the fast diffusion-based PageRank procedure for computing an approximation to the first nontrivial eigenvector of the graph Laplacian. Empirical results are also provided to illustrate the manner in which approximate eigenvector computation \emph{implicitly} performs statistical regularization, relative to running the corresponding exact algorithm.

cs.DS↗

arXiv2010

Minimax rank estimation for subspace tracking

Rank estimation is a classical model order selection problem that arises in a variety of important statistical signal and array processing systems, yet is addressed relatively infrequently in the extant literature. Here we present sample covariance asymptotics stemming from random matrix theory, and bring them to bear on the problem of optimal rank estimation in the context of the standard array observation model with additive white Gaussian noise. The most significant of these results demonstrates the existence of a phase transition threshold, below which eigenvalues and associated eigenvectors of the sample covariance fail to provide any information on population eigenvalues. We then develop a decision-theoretic rank estimation framework that leads to a simple ordered selection rule based on thresholding; in contrast to competing approaches, however, it admits asymptotic minimax optimality and is free of tuning parameters. We analyze the asymptotic performance of our rank selection procedure and conclude with a brief simulation study demonstrating its practical efficacy in the context of subspace tracking.

stat.ME↗

arXiv2009

Cross-Validation for Unsupervised Learning

Cross-validation (CV) is a popular method for model-selection. Unfortunately, it is not immediately obvious how to apply CV to unsupervised or exploratory contexts. This thesis discusses some extensions of cross-validation to unsupervised learning, specifically focusing on the problem of choosing how many principal components to keep. We introduce the latent factor model, define an objective criterion, and show how CV can be used to estimate the intrinsic dimensionality of a data set. Through both simulation and theory, we demonstrate that cross-validation is a valuable tool for unsupervised learning.

stat.ME↗

arXiv2009

Bi-cross-validation of the SVD and the nonnegative matrix factorization

This article presents a form of bi-cross-validation (BCV) for choosing the rank in outer product models, especially the singular value decomposition (SVD) and the nonnegative matrix factorization (NMF). Instead of leaving out a set of rows of the data matrix, we leave out a set of rows and a set of columns, and then predict the left out entries by low rank operations on the retained data. We prove a self-consistency result expressing the prediction error as a residual from a low rank approximation. Random matrix theory and some empirical results suggest that smaller hold-out sets lead to more over-fitting, while larger ones are more prone to under-fitting. In simulated examples we find that a method leaving out half the rows and half the columns performs well.

stat.AP↗