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Larry Wasserman

Publications and source records attributed to Larry Wasserman.

At least 145 records · Page 8Linked to original sources
arXiv2013

Density-sensitive semisupervised inference

Semisupervised methods are techniques for using labeled data $(X_1,Y_1),\ldots,(X_n,Y_n)$ together with unlabeled data $X_{n+1},\ldots,X_N$ to make predictions. These methods invoke some assumptions that link the marginal distribution $P_X$ of X to the regression function f(x). For example, it is common to assume that f is very smooth over high density regions of $P_X$. Many of the methods are ad-hoc and have been shown to work in specific examples but are lacking a theoretical foundation. We provide a minimax framework for analyzing semisupervised methods. In particular, we study methods based on metrics that are sensitive to the distribution $P_X$. Our model includes a parameter $α$ that controls the strength of the semisupervised assumption. We then use the data to adapt to $α$.

math.ST↗
arXiv2013

A Conformal Prediction Approach to Explore Functional Data

This paper applies conformal prediction techniques to compute simultaneous prediction bands and clustering trees for functional data. These tools can be used to detect outliers and clusters. Both our prediction bands and clustering trees provide prediction sets for the underlying stochastic process with a guaranteed finite sample behavior, under no distributional assumptions. The prediction sets are also informative in that they correspond to the high density region of the underlying process. While ordinary conformal prediction has high computational cost for functional data, we use the inductive conformal predictor, together with several novel choices of conformity scores, to simplify the computation. Our methods are illustrated on some real data examples.

stat.ML↗
arXiv2013

Distribution-Free Distribution Regression

`Distribution regression' refers to the situation where a response Y depends on a covariate P where P is a probability distribution. The model is Y=f(P) + mu where f is an unknown regression function and mu is a random error. Typically, we do not observe P directly, but rather, we observe a sample from P. In this paper we develop theory and methods for distribution-free versions of distribution regression. This means that we do not make distributional assumptions about the error term mu and covariate P. We prove that when the effective dimension is small enough (as measured by the doubling dimension), then the excess prediction risk converges to zero with a polynomial rate.

stat.ML↗
arXiv2013

Sparse Nonparametric Graphical Models

We present some nonparametric methods for graphical modeling. In the discrete case, where the data are binary or drawn from a finite alphabet, Markov random fields are already essentially nonparametric, since the cliques can take only a finite number of values. Continuous data are different. The Gaussian graphical model is the standard parametric model for continuous data, but it makes distributional assumptions that are often unrealistic. We discuss two approaches to building more flexible graphical models. One allows arbitrary graphs and a nonparametric extension of the Gaussian; the other uses kernel density estimation and restricts the graphs to trees and forests. Examples of both methods are presented. We also discuss possible future research directions for nonparametric graphical modeling.

stat.ML↗
arXiv2012

High Dimensional Semiparametric Gaussian Copula Graphical Models

In this paper, we propose a semiparametric approach, named nonparanormal skeptic, for efficiently and robustly estimating high dimensional undirected graphical models. To achieve modeling flexibility, we consider Gaussian Copula graphical models (or the nonparanormal) as proposed by Liu et al. (2009). To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This celebrating result suggests that the Gaussian copula graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare different estimators for their graph recovery performance under both ideal and noisy settings. The proposed methods are then applied on a large-scale genomic dataset to illustrate their empirical usefulness. The R language software package huge implementing the proposed methods is available on the Comprehensive R Archive Network: http://cran. r-project.org/.

stat.ML↗
arXiv2012

The Nonparanormal SKEPTIC

We propose a semiparametric approach, named nonparanormal skeptic, for estimating high dimensional undirected graphical models. In terms of modeling, we consider the nonparanormal family proposed by Liu et al (2009). In terms of estimation, we exploit nonparametric rank-based correlation coefficient estimators including the Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This result suggests that the nonparanormal graphical models are a safe replacement of the Gaussian graphical models, even when the data are Gaussian.

stat.ME↗
arXiv2012

Distribution Free Prediction Bands

We study distribution free, nonparametric prediction bands with a special focus on their finite sample behavior. First we investigate and develop different notions of finite sample coverage guarantees. Then we give a new prediction band estimator by combining the idea of "conformal prediction" (Vovk et al. 2009) with nonparametric conditional density estimation. The proposed estimator, called COPS (Conformal Optimized Prediction Set), always has finite sample guarantee in a stronger sense than the original conformal prediction estimator. Under regularity conditions the estimator converges to an oracle band at a minimax optimal rate. A fast approximation algorithm and a data driven method for selecting the bandwidth are developed. The method is illustrated first in simulated data. Then, an application shows that the proposed method gives desirable prediction intervals in an automatic way, as compared to the classical linear regression modeling.

stat.ME↗
arXiv2012

Differential Privacy for Functions and Functional Data

Differential privacy is a framework for privately releasing summaries of a database. Previous work has focused mainly on methods for which the output is a finite dimensional vector, or an element of some discrete set. We develop methods for releasing functions while preserving differential privacy. Specifically, we show that adding an appropriate Gaussian process to the function of interest yields differential privacy. When the functions lie in the same RKHS as the Gaussian process, then the correct noise level is established by measuring the "sensitivity" of the function in the RKHS norm. As examples we consider kernel density estimation, kernel support vector machines, and functions in reproducing kernel Hilbert spaces.

stat.ML↗
arXiv2012

Frasian Inference

Don Fraser has given an interesting account of the agreements and disagreements between Bayesian posterior probabilities and confidence levels. In this comment I discuss some cases where the lack of such agreement is extreme. I then discuss a few cases where it is possible to have Bayes procedures with frequentist validity. Such frequentist-Bayesian---or Frasian---methods deserve more attention [arXiv:1112.5582].

stat.ME↗
arXiv2011

Minimax Rates for Homology Inference

Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In this paper, we consider the statistical problem of estimating the homology of a manifold from noisy samples under several different noise models. We derive upper and lower bounds on the minimax risk for this problem. Our upper bounds are based on estimators which are constructed from a union of balls of appropriate radius around carefully selected points. In each case we establish complementary lower bounds using Le Cam's lemma.

stat.ML↗
arXiv2011

Random Differential Privacy

We propose a relaxed privacy definition called {\em random differential privacy} (RDP). Differential privacy requires that adding any new observation to a database will have small effect on the output of the data-release procedure. Random differential privacy requires that adding a {\em randomly drawn new observation} to a database will have small effect on the output. We show an analog of the composition property of differentially private procedures which applies to our new definition. We show how to release an RDP histogram and we show that RDP histograms are much more accurate than histograms obtained using ordinary differential privacy. We finally show an analog of the global sensitivity framework for the release of functions under our privacy definition.

stat.ME↗
arXiv2011

Adaptive Semisupervised Inference

Semisupervised methods inevitably invoke some assumption that links the marginal distribution of the features to the regression function of the label. Most commonly, the cluster or manifold assumptions are used which imply that the regression function is smooth over high-density clusters or manifolds supporting the data. A generalization of these assumptions is that the regression function is smooth with respect to some density sensitive distance. This motivates the use of a density based metric for semisupervised learning. We analyze this setting and make the following contributions - (a) we propose a semi-supervised learner that uses a density-sensitive kernel and show that it provides better performance than any supervised learner if the density support set has a small condition number and (b) we show that it is possible to adapt to the degree of semi-supervisedness using data-dependent choice of a parameter that controls sensitivity of the distance metric to the density. This ensures that the semisupervised learner never performs worse than a supervised learner even if the assumptions fail to hold.

math.ST↗
arXiv2011

Efficient Nonparametric Conformal Prediction Regions

We investigate and extend the conformal prediction method due to Vovk,Gammerman and Shafer (2005) to construct nonparametric prediction regions. These regions have guaranteed distribution free, finite sample coverage, without any assumptions on the distribution or the bandwidth. Explicit convergence rates of the loss function are established for such regions under standard regularity conditions. Approximations for simplifying implementation and data driven bandwidth selection methods are also discussed. The theoretical properties of our method are demonstrated through simulations.

math.ST↗
arXiv2011

Minimax Manifold Estimation

We find the minimax rate of convergence in Hausdorff distance for estimating a manifold M of dimension d embedded in R^D given a noisy sample from the manifold. We assume that the manifold satisfies a smoothness condition and that the noise distribution has compact support. We show that the optimal rate of convergence is n^{-2/(2+d)}. Thus, the minimax rate depends only on the dimension of the manifold, not on the dimension of the space in which M is embedded.

stat.ML↗
arXiv2011

Structured, sparse regression with application to HIV drug resistance

We introduce a new version of forward stepwise regression. Our modification finds solutions to regression problems where the selected predictors appear in a structured pattern, with respect to a predefined distance measure over the candidate predictors. Our method is motivated by the problem of predicting HIV-1 drug resistance from protein sequences. We find that our method improves the interpretability of drug resistance while producing comparable predictive accuracy to standard methods. We also demonstrate our method in a simulation study and present some theoretical results and connections.

stat.ME↗
arXiv2011

An Unbiased Method of Modeling the Local Peculiar Velocity Field with Type Ia Supernovae

We apply statistically rigorous methods of nonparametric risk estimation to the problem of inferring the local peculiar velocity field from nearby supernovae (SNIa). We use two nonparametric methods - Weighted Least Squares (WLS) and Coefficient Unbiased (CU) - both of which employ spherical harmonics to model the field and use the estimated risk to determine at which multipole to truncate the series. We show that if the data are not drawn from a uniform distribution or if there is power beyond the maximum multipole in the regression, a bias is introduced on the coefficients using WLS. CU estimates the coefficients without this bias by including the sampling density making the coefficients more accurate but not necessarily modeling the velocity field more accurately. After applying nonparametric risk estimation to SNIa data, we find that there are not enough data at this time to measure power beyond the dipole. The WLS Local Group bulk flow is moving at 538 +- 86 km/s towards (l,b) = (258 +- 10 deg, 36 +- 11 deg) and the CU bulk flow is moving at 446 +- 101 km/s towards (l,b) = (273 +- 11 deg, 46 +- 8 deg). We find that the magnitude and direction of these measurements are in agreement with each other and previous results in the literature.

astro-ph.CO↗
arXiv2010

The Geometry of Nonparametric Filament Estimation

We consider the problem of estimating filamentary structure from planar point process data. We make some connections with computational geometry and we develop nonparametric methods for estimating the filaments. We show that, under weak conditions, the filaments have a simple geometric representation as the medial axis of the data distribution's support. Our methods convert an estimator of the support's boundary into an estimator of the filaments. We also find the rates of convergence of our estimators.

math.ST↗