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Zheng Tracy Ke

Publications and source records attributed to Zheng Tracy Ke.

44 records · Page 3Linked to original sources
arXiv2018

Network Global Testing by Counting Graphlets

Consider a large social network with possibly severe degree heterogeneity and mixed-memberships. We are interested in testing whether the network has only one community or there are more than one communities. The problem is known to be non-trivial, partially due to the presence of severe degree heterogeneity. We construct a class of test statistics using the numbers of short paths and short cycles, and the key to our approach is a general framework for canceling the effects of degree heterogeneity. The tests compare favorably with existing methods. We support our methods with careful analysis and numerical study with simulated data and a real data example.

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arXiv2017

A Sharp Lower Bound for Mixed-membership Estimation

Consider an undirected network with $n$ nodes and $K$ perceivable communities, where some nodes may have mixed memberships. We assume that for each node $1 \leq i \leq n$, there is a probability mass function $π_i$ defined over $\{1, 2, \ldots, K\}$ such that \[ π_i(k) = \mbox{the weight of node $i$ on community $k$}, \qquad 1 \leq k \leq K. \] The goal is to estimate $\{π_i, 1 \leq i \leq n\}$ (i.e., membership estimation). We model the network with the {\it degree-corrected mixed membership (DCMM)} model \cite{Mixed-SCORE}. Since for many natural networks, the degrees have an approximate power-law tail, we allow {\it severe degree heterogeneity} in our model. For any membership estimation $\{\hatπ_i, 1 \leq i \leq n\}$, since each $π_i$ is a probability mass function, it is natural to measure the errors by the average $\ell^1$-norm \[ \frac{1}{n} \sum_{i = 1}^n \| \hatπ_i - π_i\|_1. \] We also consider a variant of the $\ell^1$-loss, where each $\|\hatπ_i - π_i\|_1$ is re-weighted by the degree parameter $θ_i$ in DCMM (to be introduced). We present a sharp lower bound. We also show that such a lower bound is achievable under a broad situation. More discussion in this vein is continued in our forthcoming manuscript. The results are very different from those on community detection. For community detection, the focus is on the special case where all $π_i$ are degenerate; the goal is clustering, so Hamming distance is the natural choice of loss function, and the rate can be exponentially fast. The setting here is broader and more difficult: it is more natural to use the $\ell^1$-loss, and the rate is only polynomially fast.

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arXiv2017

Covariate Assisted Variable Ranking

Consider a linear model $y = X β+ z$, $z \sim N(0, σ^2 I_n)$. The Gram matrix $Θ= \frac{1}{n} X'X$ is non-sparse, but it is approximately the sum of two components, a low-rank matrix and a sparse matrix, where neither component is known to us. We are interested in the Rare/Weak signal setting where all but a small fraction of the entries of $β$ are nonzero, and the nonzero entries are relatively small individually. The goal is to rank the variables in a way so as to maximize the area under the ROC curve. We propose Factor-adjusted Covariate Assisted Ranking (FA-CAR) as a two-step approach to variable ranking. In the FA-step, we use PCA to reduce the linear model to a new one where the Gram matrix is approximately sparse. In the CAR-step, we rank variables by exploiting the local covariate structures. FA-CAR is easy to use and computationally fast, and it is effective in resolving signal cancellation, a challenge we face in regression models. FA-CAR is related to the recent idea of Covariate Assisted Screening and Estimation (CASE), but two methods are for different goals and are thus very different. We compare the ROC curve of FA-CAR with some other ranking ideas on numerical experiments, and show that FA-CAR has several advantages. Using a Rare/Weak signal model, we derive the convergence rate of the minimum sure-screening model size of FA-CAR. Our theoretical analysis contains several new ingredients, especially a new perturbation bound for PCA.

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arXiv2016

A Geometrical Approach to Topic Model Estimation

In the probabilistic topic models, the quantity of interest---a low-rank matrix consisting of topic vectors---is hidden in the text corpus matrix, masked by noise, and the Singular Value Decomposition (SVD) is a potentially useful tool for learning such a low-rank matrix. However, the connection between this low-rank matrix and the singular vectors of the text corpus matrix are usually complicated and hard to spell out, so how to use SVD for learning topic models faces challenges. In this paper, we overcome the challenge by revealing a surprising insight: there is a low-dimensional simplex structure which can be viewed as a bridge between the low-rank matrix of interest and the SVD of the text corpus matrix, and allows us to conveniently reconstruct the former using the latter. Such an insight motivates a new SVD approach to learning topic models, which we analyze with delicate random matrix theory and derive the rate of convergence. We support our methods and theory numerically, using both simulated data and real data.

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arXiv2016

Phase Transitions for High Dimensional Clustering and Related Problems

Consider a two-class clustering problem where we observe $X_i = \ell_i μ+ Z_i$, $Z_i \stackrel{iid}{\sim} N(0, I_p)$, $1 \leq i \leq n$. The feature vector $μ\in R^p$ is unknown but is presumably sparse. The class labels $\ell_i\in\{-1, 1\}$ are also unknown and the main interest is to estimate them. We are interested in the statistical limits. In the two-dimensional phase space calibrating the rarity and strengths of useful features, we find the precise demarcation for the Region of Impossibility and Region of Possibility. In the former, useful features are too rare/weak for successful clustering. In the latter, useful features are strong enough to allow successful clustering. The results are extended to the case of colored noise using Le Cam's idea on comparison of experiments. We also extend the study on statistical limits for clustering to that for signal recovery and that for hypothesis testing. We compare the statistical limits for three problems and expose some interesting insight. We propose classical PCA and Important Features PCA (IF-PCA) for clustering. For a threshold $t > 0$, IF-PCA clusters by applying classical PCA to all columns of $X$ with an $L^2$-norm larger than $t$. We also propose two aggregation methods. For any parameter in the Region of Possibility, some of these methods yield successful clustering. We find an interesting phase transition for IF-PCA. Our results require delicate analysis, especially on post-selection Random Matrix Theory and on lower bound arguments.

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arXiv2015

QUADRO: A supervised dimension reduction method via Rayleigh quotient optimization

We propose a novel Rayleigh quotient based sparse quadratic dimension reduction method - named QUADRO (Quadratic Dimension Reduction via Rayleigh Optimization) - for analyzing high- dimensional data. Unlike in the linear setting where Rayleigh quotient optimization coincides with classification, these two problems are very different under nonlinear settings. In this paper, we clarify this difference and show that Rayleigh quotient optimization may be of independent scientific interests. One major challenge of Rayleigh quotient optimization is that the variance of quadratic statistics involves all fourth cross-moments of predictors, which are infeasible to compute for high-dimensional applications and may accumulate too many stochastic errors. This issue is resolved by considering a family of elliptical models. Moreover, for heavy-tail distributions, robust estimates of mean vectors and covariance matrices are employed to guarantee uniform convergence in estimating nonpolynomially many parameters, even though only the fourth moments are assumed. Methodologically, QUADRO is based on elliptical models which allow us to formulate the Rayleigh quotient maximization as a convex optimization problem. Computationally, we propose an efficient linearized augmented Lagrangian method to solve the constrained optimization problem. Theoretically, we provide explicit rates of convergence in terms of Rayleigh quotient under both Gaussian and general elliptical models. Thorough numerical results on both synthetic and real datasets are also provided to back up our theoretical results.

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arXiv2014

Covariate assisted screening and estimation

Consider a linear model $Y=Xβ+z$, where $X=X_{n,p}$ and $z\sim N(0,I_n)$. The vector $β$ is unknown but is sparse in the sense that most of its coordinates are $0$. The main interest is to separate its nonzero coordinates from the zero ones (i.e., variable selection). Motivated by examples in long-memory time series (Fan and Yao [Nonlinear Time Series: Nonparametric and Parametric Methods (2003) Springer]) and the change-point problem (Bhattacharya [In Change-Point Problems (South Hadley, MA, 1992) (1994) 28-56 IMS]), we are primarily interested in the case where the Gram matrix $G=X'X$ is nonsparse but sparsifiable by a finite order linear filter. We focus on the regime where signals are both rare and weak so that successful variable selection is very challenging but is still possible. We approach this problem by a new procedure called the covariate assisted screening and estimation (CASE). CASE first uses a linear filtering to reduce the original setting to a new regression model where the corresponding Gram (covariance) matrix is sparse. The new covariance matrix induces a sparse graph, which guides us to conduct multivariate screening without visiting all the submodels. By interacting with the signal sparsity, the graph enables us to decompose the original problem into many separated small-size subproblems (if only we know where they are!). Linear filtering also induces a so-called problem of information leakage, which can be overcome by the newly introduced patching technique. Together, these give rise to CASE, which is a two-stage screen and clean [Fan and Song Ann. Statist. 38 (2010) 3567-3604; Wasserman and Roeder Ann. Statist. 37 (2009) 2178-2201] procedure, where we first identify candidates of these submodels by patching and screening, and then re-examine each candidate to remove false positives.

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