A Novel Nonparametric Distance Estimator for Densities with Error Bounds

The use of a metric to assess distance between probability densities is an important practical problem. In this work, a particular metric induced by an a-divergence is studied. The Hellinger metric can be interpreted as a particular case within the framework of generalized Tsallis divergences and entropies. The nonparametric Parzen's density estimator emerges as a natural candidate to estimate the underlying probability density function, since it may account for data from different groups, or experiments with distinct instrumental precisions, i.e., non-independent and identically distributed (non-i.i.d.) data. However, the information theoretic derived metric of the nonparametric Parzen's density estimator displays infinite variance, limiting the direct use of resampling estimators. Based on measure theory, we present a change of measure to build a finite variance density allowing the use of resampling estimators. In order to counteract the poor scaling with dimension, we propose a new nonparametric two-stage robust resampling estimator of Hellinger's metric error bounds for heterocedastic data. The approach presents very promising results allowing the use of different covariances for different clusters with impact on the distance evaluation

Meta learning of bounds on the Bayes classifier error

Meta learning uses information from base learners (e.g. classifiers or estimators) as well as information about the learning problem to improve upon the performance of a single base learner. For example, the Bayes error rate of a given feature space, if known, can be used to aid in choosing a classifier, as well as in feature selection and model selection for the base classifiers and the meta classifier. Recent work in the field of f-divergence functional estimation has led to the development of simple and rapidly converging estimators that can be used to estimate various bounds on the Bayes error. We estimate multiple bounds on the Bayes error using an estimator that applies meta learning to slowly converging plug-in estimators to obtain the parametric convergence rate. We compare the estimated bounds empirically on simulated data and then estimate the tighter bounds on features extracted from an image patch analysis of sunspot continuum and magnetogram images.Comment: 6 pages, 3 figures, to appear in proceedings of 2015 IEEE Signal Processing and SP Education Worksho

Variational Analysis of Constrained M-Estimators

We propose a unified framework for establishing existence of nonparametric M-estimators, computing the corresponding estimates, and proving their strong consistency when the class of functions is exceptionally rich. In particular, the framework addresses situations where the class of functions is complex involving information and assumptions about shape, pointwise bounds, location of modes, height at modes, location of level-sets, values of moments, size of subgradients, continuity, distance to a "prior" function, multivariate total positivity, and any combination of the above. The class might be engineered to perform well in a specific setting even in the presence of little data. The framework views the class of functions as a subset of a particular metric space of upper semicontinuous functions under the Attouch-Wets distance. In addition to allowing a systematic treatment of numerous M-estimators, the framework yields consistency of plug-in estimators of modes of densities, maximizers of regression functions, level-sets of classifiers, and related quantities, and also enables computation by means of approximating parametric classes. We establish consistency through a one-sided law of large numbers, here extended to sieves, that relaxes assumptions of uniform laws, while ensuring global approximations even under model misspecification

Private Graphon Estimation for Sparse Graphs

We design algorithms for fitting a high-dimensional statistical model to a large, sparse network without revealing sensitive information of individual members. Given a sparse input graph $G$, our algorithms output a node-differentially-private nonparametric block model approximation. By node-differentially-private, we mean that our output hides the insertion or removal of a vertex and all its adjacent edges. If $G$ is an instance of the network obtained from a generative nonparametric model defined in terms of a graphon $W$, our model guarantees consistency, in the sense that as the number of vertices tends to infinity, the output of our algorithm converges to $W$ in an appropriate version of the $L_2$ norm. In particular, this means we can estimate the sizes of all multi-way cuts in $G$. Our results hold as long as $W$ is bounded, the average degree of $G$ grows at least like the log of the number of vertices, and the number of blocks goes to infinity at an appropriate rate. We give explicit error bounds in terms of the parameters of the model; in several settings, our bounds improve on or match known nonprivate results.Comment: 36 page

Demystifying Fixed k-Nearest Neighbor Information Estimators

Estimating mutual information from i.i.d. samples drawn from an unknown joint density function is a basic statistical problem of broad interest with multitudinous applications. The most popular estimator is one proposed by Kraskov and St\"ogbauer and Grassberger (KSG) in 2004, and is nonparametric and based on the distances of each sample to its $k^{\rm th}$ nearest neighboring sample, where $k$ is a fixed small integer. Despite its widespread use (part of scientific software packages), theoretical properties of this estimator have been largely unexplored. In this paper we demonstrate that the estimator is consistent and also identify an upper bound on the rate of convergence of the bias as a function of number of samples. We argue that the superior performance benefits of the KSG estimator stems from a curious "correlation boosting" effect and build on this intuition to modify the KSG estimator in novel ways to construct a superior estimator. As a byproduct of our investigations, we obtain nearly tight rates of convergence of the $\ell_2$ error of the well known fixed $k$ nearest neighbor estimator of differential entropy by Kozachenko and Leonenko.Comment: 55 pages, 8 figure