12,047 research outputs found
Ensemble estimation of multivariate f-divergence
f-divergence estimation is an important problem in the fields of information
theory, machine learning, and statistics. While several divergence estimators
exist, relatively few of their convergence rates are known. We derive the MSE
convergence rate for a density plug-in estimator of f-divergence. Then by
applying the theory of optimally weighted ensemble estimation, we derive a
divergence estimator with a convergence rate of O(1/T) that is simple to
implement and performs well in high dimensions. We validate our theoretical
results with experiments.Comment: 14 pages, 6 figures, a condensed version of this paper was accepted
to ISIT 2014, Version 2: Moved the proofs of the theorems from the main body
to appendices at the en
Direct Ensemble Estimation of Density Functionals
Estimating density functionals of analog sources is an important problem in
statistical signal processing and information theory. Traditionally, estimating
these quantities requires either making parametric assumptions about the
underlying distributions or using non-parametric density estimation followed by
integration. In this paper we introduce a direct nonparametric approach which
bypasses the need for density estimation by using the error rates of k-NN
classifiers asdata-driven basis functions that can be combined to estimate a
range of density functionals. However, this method is subject to a non-trivial
bias that dramatically slows the rate of convergence in higher dimensions. To
overcome this limitation, we develop an ensemble method for estimating the
value of the basis function which, under some minor constraints on the
smoothness of the underlying distributions, achieves the parametric rate of
convergence regardless of data dimension.Comment: 5 page
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
Information Theoretic Structure Learning with Confidence
Information theoretic measures (e.g. the Kullback Liebler divergence and
Shannon mutual information) have been used for exploring possibly nonlinear
multivariate dependencies in high dimension. If these dependencies are assumed
to follow a Markov factor graph model, this exploration process is called
structure discovery. For discrete-valued samples, estimates of the information
divergence over the parametric class of multinomial models lead to structure
discovery methods whose mean squared error achieves parametric convergence
rates as the sample size grows. However, a naive application of this method to
continuous nonparametric multivariate models converges much more slowly. In
this paper we introduce a new method for nonparametric structure discovery that
uses weighted ensemble divergence estimators that achieve parametric
convergence rates and obey an asymptotic central limit theorem that facilitates
hypothesis testing and other types of statistical validation.Comment: 10 pages, 3 figure
Online estimation of discrete densities using classifier chains
We propose an approach to estimate a discrete joint density online, that is, the algorithm is only provided the current example, its current estimate, and a limited amount of memory. To design an online estimator for discrete densities, we use classifier chains to model dependencies among features. Each classifier in the chain estimates the probability of one particular feature. Because a single chain may not provide a reliable estimate, we also consider ensembles of classifier chains. Our experiments on synthetic data show that the approach is feasible and the estimated densities approach the true, known distribution with increasing amounts of data
Direct Estimation of Information Divergence Using Nearest Neighbor Ratios
We propose a direct estimation method for R\'{e}nyi and f-divergence measures
based on a new graph theoretical interpretation. Suppose that we are given two
sample sets and , respectively with and samples, where
is a constant value. Considering the -nearest neighbor (-NN)
graph of in the joint data set , we show that the average powered
ratio of the number of points to the number of points among all -NN
points is proportional to R\'{e}nyi divergence of and densities. A
similar method can also be used to estimate f-divergence measures. We derive
bias and variance rates, and show that for the class of -H\"{o}lder
smooth functions, the estimator achieves the MSE rate of
. Furthermore, by using a weighted ensemble
estimation technique, for density functions with continuous and bounded
derivatives of up to the order , and some extra conditions at the support
set boundary, we derive an ensemble estimator that achieves the parametric MSE
rate of . Our estimators are more computationally tractable than other
competing estimators, which makes them appealing in many practical
applications.Comment: 2017 IEEE International Symposium on Information Theory (ISIT
- …