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Po-Ling Loh

Po-Ling Loh appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.STStatistics TheoryMachine LearningInformation Theorymath.ITSocial and Information NetworksDiscrete Mathematicsmath.PRMethodologyApplicationsComputationmath.AG

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Published work

20 published item(s)

preprint2026arXiv

High-Dimensional Statistics: Reflections on Progress and Open Problems

Over the past two decades, the field of high-dimensional statistics has experienced substantial progress, driven largely by technological advances that have dramatically reduced the cost and effort for data collection and storage across a broad range of domains, including biology, medicine, astronomy, and the social and environmental sciences. Modern datasets are increasingly complex, often exhibiting rich dependency, heterogeneity, and other features that challenge traditional statistical methods. In response, high-dimensional statistics has evolved to address more sophisticated estimation and inference problems. This evolution has, in turn, fostered deep connections with and contributions to a wide range of research areas, including optimization, concentration of measure, random matrix theory, information theory, and theoretical computer science. Given the rapid pace of recent developments in high-dimensional statistics, our goal is to synthesize representative advances, highlight common themes and open problems, and point to important works that offer entry points into the field.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Node-private community estimation in stochastic block models: Tractable algorithms and lower bounds

We study the classical problem of community recovery in stochastic block models with a fixed number of communities, with a twist: We seek algorithms that are stable with respect to node-wise changes in the graph structure, formally defined as a differential privacy constraint. The algorithms we develop are based on spectral clustering, where we introduce privacy to the community recovery pipeline in the form of directly privatizing the adjacency matrix; private PCA; private convex optimization; private low-rank matrix estimation; and private approximate subspace estimation. Straightforward applications of existing private algorithms lead to a rapid increase in the privacy parameter $ε$ in order to ensure consistent estimation under node differential privacy, in contrast with the simpler setting of edge privacy. To alleviate these issues, we develop novel algorithms based on (1) sampling from an exponential mechanism with a Lipschitz extension and (2) a general framework for constructing smooth projections from the space of undirected graphs to the space of bounded-degree graphs, which can then be combined with various edge-private algorithms. Importantly, the methods we develop are all computable in polynomial-time as a function of the number of nodes in the graph. We also develop novel lower bounds on the growth rate of $ε$ required in order to achieve consistent community estimation under node privacy. On a technical note, our paper highlights the complications that arise when analyzing private algorithms under the non-standard scaling $ε\rightarrow \infty$ and proposes some solutions. We also provide a novel application of the HGR maximal correlation from information theory in the context of accuracy amplification in PAC learning, which may be of independent interest.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

On the identifiability of mixtures of ranking models

Mixtures of ranking models are standard tools for ranking problems. However, even the fundamental question of parameter identifiability is not fully understood: the identifiability of a mixture model with two Bradley-Terry-Luce (BTL) components has remained open. In this work, we show that popular mixtures of ranking models with two components (BTL, multinomial logistic models with slates of size 3, or Plackett-Luce) are generically identifiable, i.e., the ground-truth parameters can be identified except when they are from a pathological subset of measure zero. We provide a framework for verifying the number of solutions in a general family of polynomial systems using algebraic geometry, and apply it to these mixtures of ranking models to establish generic identifiability. The framework can be applied more broadly to other learning models and may be of independent interest.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Robust W-GAN-Based Estimation Under Wasserstein Contamination

Robust estimation is an important problem in statistics which aims at providing a reasonable estimator when the data-generating distribution lies within an appropriately defined ball around an uncontaminated distribution. Although minimax rates of estimation have been established in recent years, many existing robust estimators with provably optimal convergence rates are also computationally intractable. In this paper, we study several estimation problems under a Wasserstein contamination model and present computationally tractable estimators motivated by generative adversarial networks (GANs). Specifically, we analyze properties of Wasserstein GAN-based estimators for location estimation, covariance matrix estimation, and linear regression and show that our proposed estimators are minimax optimal in many scenarios. Finally, we present numerical results which demonstrate the effectiveness of our estimators.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Adaptive Estimation and Statistical Inference for High-Dimensional Graph-Based Linear Models

We consider adaptive estimation and statistical inference for high-dimensional graph-based linear models. In our model, the coordinates of regression coefficients correspond to an underlying undirected graph. Furthermore, the given graph governs the piecewise polynomial structure of the regression vector. In the adaptive estimation part, we apply graph-based regularization techniques and propose a family of locally adaptive estimators called the Graph-Piecewise-Polynomial-Lasso. We further study a one-step update of the Graph-Piecewise-Polynomial-Lasso for the problem of statistical inference. We develop the corresponding theory, which includes the fixed design and the sub-Gaussian random design. Finally, we illustrate the superior performance of our approaches by extensive simulation studies and conclude with an application to an Arabidopsis thaliana microarray dataset.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Boosting Algorithms for Estimating Optimal Individualized Treatment Rules

We present nonparametric algorithms for estimating optimal individualized treatment rules. The proposed algorithms are based on the XGBoost algorithm, which is known as one of the most powerful algorithms in the machine learning literature. Our main idea is to model the conditional mean of clinical outcome or the decision rule via additive regression trees, and use the boosting technique to estimate each single tree iteratively. Our approaches overcome the challenge of correct model specification, which is required in current parametric methods. The major contribution of our proposed algorithms is providing efficient and accurate estimation of the highly nonlinear and complex optimal individualized treatment rules that often arise in practice. Finally, we illustrate the superior performance of our algorithms by extensive simulation studies and conclude with an application to the real data from a diabetes Phase III trial.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Permutation Tests for Infection Graphs

We formulate and analyze a novel hypothesis testing problem for inferring the edge structure of an infection graph. In our model, a disease spreads over a network via contagion or random infection, where the random variables governing the rates of contracting the disease from neighbors or random infection are independent exponential random variables with unknown rate parameters. A subset of nodes is also censored uniformly at random. Given the statuses of nodes in the network, the goal is to determine the underlying graph. We present a procedure based on permutation testing, and we derive sufficient conditions for the validity of our test in terms of automorphism groups of the graphs corresponding to the null and alternative hypotheses. Further, the test is valid more generally for infection processes satisfying a basic symmetry condition. Our test is easy to compute and does not involve estimating unknown parameters governing the process. We also derive risk bounds for our permutation test in a variety of settings, and motivate our test statistic in terms of approximate equivalence to likelihood ratio testing and maximin tests. We conclude with an application to real data from an HIV infection network.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Analysis of centrality in sublinear preferential attachment trees via the CMJ branching process

We investigate centrality and root-inference properties in a class of growing random graphs known as sublinear preferential attachment trees. We show that a continuous time branching processes called the Crump-Mode-Jagers (CMJ) branching process is well-suited to analyze such random trees, and prove that almost surely, a unique terminal tree centroid emerges, having the property that it becomes more central than any other fixed vertex in the limit of the random growth process. Our result generalizes and extends previous work establishing persistent centrality in uniform and linear preferential attachment trees. We also show that centrality may be utilized to generate a finite-sized $1-ε$ confidence set for the root node, for any $ε> 0$ in a certain subclass of sublinear preferential attachment trees.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

High-dimensional robust precision matrix estimation: Cellwise corruption under $ε$-contamination

We analyze the statistical consistency of robust estimators for precision matrices in high dimensions. We focus on a contamination mechanism acting cellwise on the data matrix. The estimators we analyze are formed by plugging appropriately chosen robust covariance matrix estimators into the graphical Lasso and CLIME. Such estimators were recently proposed in the robust statistics literature, but only analyzed mathematically from the point of view of the breakdown point. This paper provides complementary high-dimensional error bounds for the precision matrix estimators that reveal the interplay between the dimensionality of the problem and the degree of contamination permitted in the observed distribution. We also show that although the graphical Lasso and CLIME estimators perform equally well from the point of view of statistical consistency, the breakdown property of the graphical Lasso is superior to that of CLIME. We discuss implications of our work for problems involving graphical model estimation when the uncontaminated data follow a multivariate normal distribution, and the goal is to estimate the support of the population-level precision matrix. Our error bounds do not make any assumptions about the the contaminating distribution and allow for a nonvanishing fraction of cellwise contamination.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Information-theoretic bounds for exact recovery in weighted stochastic block models using the Renyi divergence

We derive sharp thresholds for exact recovery of communities in a weighted stochastic block model, where observations are collected in the form of a weighted adjacency matrix, and the weight of each edge is generated independently from a distribution determined by the community membership of its endpoints. Our main result, characterizing the precise boundary between success and failure of maximum likelihood estimation when edge weights are drawn from discrete distributions, involves the Renyi divergence of order $\frac{1}{2}$ between the distributions of within-community and between-community edges. When the Renyi divergence is above a certain threshold, meaning the edge distributions are sufficiently separated, maximum likelihood succeeds with probability tending to 1; when the Renyi divergence is below the threshold, maximum likelihood fails with probability bounded away from 0. In the language of graphical channels, the Renyi divergence pinpoints the information-theoretic capacity of discrete graphical channels with binary inputs. Our results generalize previously established thresholds derived specifically for unweighted block models, and support an important natural intuition relating the intrinsic hardness of community estimation to the problem of edge classification. Along the way, we establish a general relationship between the Renyi divergence and the probability of success of the maximum likelihood estimator for arbitrary edge weight distributions. Finally, we discuss consequences of our bounds for the related problems of censored block models and submatrix localization, which may be seen as special cases of the framework developed in our paper.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

On model misspecification and KL separation for Gaussian graphical models

We establish bounds on the KL divergence between two multivariate Gaussian distributions in terms of the Hamming distance between the edge sets of the corresponding graphical models. We show that the KL divergence is bounded below by a constant when the graphs differ by at least one edge; this is essentially the tightest possible bound, since classes of graphs exist for which the edge discrepancy increases but the KL divergence remains bounded above by a constant. As a natural corollary to our KL lower bound, we also establish a sample size requirement for correct model selection via maximum likelihood estimation. Our results rigorize the notion that it is essential to estimate the edge structure of a Gaussian graphical model accurately in order to approximate the true distribution to close precision.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Persistence of centrality in random growing trees

We investigate properties of node centrality in random growing tree models. We focus on a measure of centrality that computes the maximum subtree size of the tree rooted at each node, with the most central node being the tree centroid. For random trees grown according to a preferential attachment model, a uniform attachment model, or a diffusion processes over a regular tree, we prove that a single node persists as the tree centroid after a finite number of steps, with probability 1. Furthermore, this persistence property generalizes to the top $K \ge 1$ nodes with respect to the same centrality measure. We also establish necessary and sufficient conditions for the size of an initial seed graph required to ensure persistence of a particular node with probability $1-ε$, as a function of $ε$: In the case of preferential and uniform attachment models, we derive bounds for the size of an initial hub constructed around the special node. In the case of a diffusion process over a regular tree, we derive bounds for the radius of an initial ball centered around the special node. Our necessary and sufficient conditions match up to constant factors for preferential attachment and diffusion tree models.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima

We provide novel theoretical results regarding local optima of regularized $M$-estimators, allowing for nonconvexity in both loss and penalty functions. Under restricted strong convexity on the loss and suitable regularity conditions on the penalty, we prove that \emph{any stationary point} of the composite objective function will lie within statistical precision of the underlying parameter vector. Our theory covers many nonconvex objective functions of interest, including the corrected Lasso for errors-in-variables linear models; regression for generalized linear models with nonconvex penalties such as SCAD, MCP, and capped-$\ell_1$; and high-dimensional graphical model estimation. We quantify statistical accuracy by providing bounds on the $\ell_1$-, $\ell_2$-, and prediction error between stationary points and the population-level optimum. We also propose a simple modification of composite gradient descent that may be used to obtain a near-global optimum within statistical precision $ε$ in $\log(1/ε)$ steps, which is the fastest possible rate of any first-order method. We provide simulation studies illustrating the sharpness of our theoretical results.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Statistical consistency and asymptotic normality for high-dimensional robust M-estimators

We study theoretical properties of regularized robust M-estimators, applicable when data are drawn from a sparse high-dimensional linear model and contaminated by heavy-tailed distributions and/or outliers in the additive errors and covariates. We first establish a form of local statistical consistency for the penalized regression estimators under fairly mild conditions on the error distribution: When the derivative of the loss function is bounded and satisfies a local restricted curvature condition, all stationary points within a constant radius of the true regression vector converge at the minimax rate enjoyed by the Lasso with sub-Gaussian errors. When an appropriate nonconvex regularizer is used in place of an l_1-penalty, we show that such stationary points are in fact unique and equal to the local oracle solution with the correct support---hence, results on asymptotic normality in the low-dimensional case carry over immediately to the high-dimensional setting. This has important implications for the efficiency of regularized nonconvex M-estimators when the errors are heavy-tailed. Our analysis of the local curvature of the loss function also has useful consequences for optimization when the robust regression function and/or regularizer is nonconvex and the objective function possesses stationary points outside the local region. We show that as long as a composite gradient descent algorithm is initialized within a constant radius of the true regression vector, successive iterates will converge at a linear rate to a stationary point within the local region. Furthermore, the global optimum of a convex regularized robust regression function may be used to obtain a suitable initialization. The result is a novel two-step procedure that uses a convex M-estimator to achieve consistency and a nonconvex M-estimator to increase efficiency.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Concavity of reweighted Kikuchi approximation

We analyze a reweighted version of the Kikuchi approximation for estimating the log partition function of a product distribution defined over a region graph. We establish sufficient conditions for the concavity of our reweighted objective function in terms of weight assignments in the Kikuchi expansion, and show that a reweighted version of the sum product algorithm applied to the Kikuchi region graph will produce global optima of the Kikuchi approximation whenever the algorithm converges. When the region graph has two layers, corresponding to a Bethe approximation, we show that our sufficient conditions for concavity are also necessary. Finally, we provide an explicit characterization of the polytope of concavity in terms of the cycle structure of the region graph. We conclude with simulations that demonstrate the advantages of the reweighted Kikuchi approach.

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preprint2014arXiv

Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses

We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Support recovery without incoherence: A case for nonconvex regularization

We demonstrate that the primal-dual witness proof method may be used to establish variable selection consistency and $\ell_\infty$-bounds for sparse regression problems, even when the loss function and/or regularizer are nonconvex. Using this method, we derive two theorems concerning support recovery and $\ell_\infty$-guarantees for the regression estimator in a general setting. Our results provide rigorous theoretical justification for the use of nonconvex regularization: For certain nonconvex regularizers with vanishing derivative away from the origin, support recovery consistency may be guaranteed without requiring the typical incoherence conditions present in $\ell_1$-based methods. We then derive several corollaries that illustrate the wide applicability of our method to analyzing composite objective functions involving losses such as least squares, nonconvex modified least squares for errors-in variables linear regression, the negative log likelihood for generalized linear models, and the graphical Lasso. We conclude with empirical studies to corroborate our theoretical predictions.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

High-dimensional learning of linear causal networks via inverse covariance estimation

We establish a new framework for statistical estimation of directed acyclic graphs (DAGs) when data are generated from a linear, possibly non-Gaussian structural equation model. Our framework consists of two parts: (1) inferring the moralized graph from the support of the inverse covariance matrix; and (2) selecting the best-scoring graph amongst DAGs that are consistent with the moralized graph. We show that when the error variances are known or estimated to close enough precision, the true DAG is the unique minimizer of the score computed using the reweighted squared l_2-loss. Our population-level results have implications for the identifiability of linear SEMs when the error covariances are specified up to a constant multiple. On the statistical side, we establish rigorous conditions for high-dimensional consistency of our two-part algorithm, defined in terms of a "gap" between the true DAG and the next best candidate. Finally, we demonstrate that dynamic programming may be used to select the optimal DAG in linear time when the treewidth of the moralized graph is bounded.

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preprint2012arXiv

High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity

Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependence, as well. We study these issues in the context of high-dimensional sparse linear regression, and propose novel estimators for the cases of noisy, missing and/or dependent data. Many standard approaches to noisy or missing data, such as those using the EM algorithm, lead to optimization problems that are inherently nonconvex, and it is difficult to establish theoretical guarantees on practical algorithms. While our approach also involves optimizing nonconvex programs, we are able to both analyze the statistical error associated with any global optimum, and more surprisingly, to prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers. On the statistical side, we provide nonasymptotic bounds that hold with high probability for the cases of noisy, missing and/or dependent data. On the computational side, we prove that under the same types of conditions required for statistical consistency, the projected gradient descent algorithm is guaranteed to converge at a geometric rate to a near-global minimizer. We illustrate these theoretical predictions with simulations, showing close agreement with the predicted scalings.

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preprint2012arXiv

The Synthesis and Analysis of Stochastic Switching Circuits

Stochastic switching circuits are relay circuits that consist of stochastic switches called pswitches. The study of stochastic switching circuits has widespread applications in many fields of computer science, neuroscience, and biochemistry. In this paper, we discuss several properties of stochastic switching circuits, including robustness, expressibility, and probability approximation. First, we study the robustness, namely, the effect caused by introducing an error of size εto each pswitch in a stochastic circuit. We analyze two constructions and prove that simple series-parallel circuits are robust to small error perturbations, while general series-parallel circuits are not. Specifically, the total error introduced by perturbations of size less than εis bounded by a constant multiple of εin a simple series-parallel circuit, independent of the size of the circuit. Next, we study the expressibility of stochastic switching circuits: Given an integer q and a pswitch set S=\{\frac{1}{q},\frac{2}{q},...,\frac{q-1}{q}\}, can we synthesize any rational probability with denominator q^n (for arbitrary n) with a simple series-parallel stochastic switching circuit? We generalize previous results and prove that when q is a multiple of 2 or 3, the answer is yes. We also show that when q is a prime number larger than 3, the answer is no. Probability approximation is studied for a general case of an arbitrary pswitch set S=\{s_1,s_2,...,s_{|S|}\}. In this case, we propose an algorithm based on local optimization to approximate any desired probability. The analysis reveals that the approximation error of a switching circuit decreases exponentially with an increasing circuit size.

0 citations0 reviewsNo code linkedOpen access