196 research outputs found
On accuracy of PDF divergence estimators and their applicability to representative data sampling
Generalisation error estimation is an important issue in machine learning. Cross-validation traditionally used for this purpose requires building multiple models and repeating the whole procedure many times in order to produce reliable error estimates. It is however possible to accurately estimate the error using only a single model, if the training and test data are chosen appropriately. This paper investigates the possibility of using various probability density function divergence measures for the purpose of representative data sampling. As it turned out, the first difficulty one needs to deal with is estimation of the divergence itself. In contrast to other publications on this subject, the experimental results provided in this study show that in many cases it is not possible unless samples consisting of thousands of instances are used. Exhaustive experiments on the divergence guided representative data sampling have been performed using 26 publicly available benchmark datasets and 70 PDF divergence estimators, and their results have been analysed and discussed
A note on Onicescu's informational energy and correlation coefficient in exponential families
The informational energy of Onicescu is a positive quantity that measures the
amount of uncertainty of a random variable like Shannon's entropy. In this
note, we report closed-form formula for Onicescu's informational energy and
correlation coefficient when the densities belong to an exponential family. We
also report as a byproduct a closed-form formula for the Cauchy-Schwarz
divergence between densities of an exponential family.Comment: 13 page
Convergence of Smoothed Empirical Measures with Applications to Entropy Estimation
This paper studies convergence of empirical measures smoothed by a Gaussian
kernel. Specifically, consider approximating , for
, by
, where is the empirical measure,
under different statistical distances. The convergence is examined in terms of
the Wasserstein distance, total variation (TV), Kullback-Leibler (KL)
divergence, and -divergence. We show that the approximation error under
the TV distance and 1-Wasserstein distance () converges at rate
in remarkable contrast to a typical
rate for unsmoothed (and ). For the
KL divergence, squared 2-Wasserstein distance (), and
-divergence, the convergence rate is , but only if
achieves finite input-output mutual information across the additive
white Gaussian noise channel. If the latter condition is not met, the rate
changes to for the KL divergence and , while
the -divergence becomes infinite - a curious dichotomy. As a main
application we consider estimating the differential entropy
in the high-dimensional regime. The distribution
is unknown but i.i.d samples from it are available. We first show that
any good estimator of must have sample complexity
that is exponential in . Using the empirical approximation results we then
show that the absolute-error risk of the plug-in estimator converges at the
parametric rate , thus establishing the minimax
rate-optimality of the plug-in. Numerical results that demonstrate a
significant empirical superiority of the plug-in approach to general-purpose
differential entropy estimators are provided.Comment: arXiv admin note: substantial text overlap with arXiv:1810.1158
Multi-modal filtering for non-linear estimation
Multi-modal densities appear frequently in time series and practical applications. However, they are not well represented by common state estimators, such as the Extended Kalman Filter and the Unscented Kalman Filter, which additionally suffer from the fact that uncertainty is often not captured sufficiently well. This can result in incoherent and divergent tracking performance. In this paper, we address these issues by devising a non-linear filtering algorithm where densities are represented by Gaussian mixture models, whose parameters are estimated in closed form. The resulting method exhibits a superior performance on nonlinear benchmarks. Ā© 2014 IEEE
-MLE: A fast algorithm for learning statistical mixture models
We describe -MLE, a fast and efficient local search algorithm for learning
finite statistical mixtures of exponential families such as Gaussian mixture
models. Mixture models are traditionally learned using the
expectation-maximization (EM) soft clustering technique that monotonically
increases the incomplete (expected complete) likelihood. Given prescribed
mixture weights, the hard clustering -MLE algorithm iteratively assigns data
to the most likely weighted component and update the component models using
Maximum Likelihood Estimators (MLEs). Using the duality between exponential
families and Bregman divergences, we prove that the local convergence of the
complete likelihood of -MLE follows directly from the convergence of a dual
additively weighted Bregman hard clustering. The inner loop of -MLE can be
implemented using any -means heuristic like the celebrated Lloyd's batched
or Hartigan's greedy swap updates. We then show how to update the mixture
weights by minimizing a cross-entropy criterion that implies to update weights
by taking the relative proportion of cluster points, and reiterate the mixture
parameter update and mixture weight update processes until convergence. Hard EM
is interpreted as a special case of -MLE when both the component update and
the weight update are performed successively in the inner loop. To initialize
-MLE, we propose -MLE++, a careful initialization of -MLE guaranteeing
probabilistically a global bound on the best possible complete likelihood.Comment: 31 pages, Extend preliminary paper presented at IEEE ICASSP 201
Multi-modal filtering for non-linear estimation
Multi-modal densities appear frequently in time series and practical applications. However, they are not well represented by common state estimators, such as the Extended Kalman Filter and the Unscented Kalman Filter, which additionally suffer from the fact that uncertainty is often not captured sufficiently well. This can result in incoherent and divergent tracking performance. In this paper, we address these issues by devising a non-linear filtering algorithm where densities are represented by Gaussian mixture models, whose parameters are estimated in closed form. The resulting method exhibits a superior performance on nonlinear benchmarks
Estimation and control of multi-object systems with high-fidenlity sensor models: A labelled random finite set approach
Principled and novel multi-object tracking algorithms are proposed, that have the ability to optimally process realistic sensor data, by accommodating complex observational phenomena such as merged measurements and extended targets. Additionally, a sensor control scheme based on a tractable, information theoretic objective is proposed, the goal of which is to optimise tracking performance in multi-object scenarios. The concept of labelled random finite sets is adopted in the development of these new techniques
- ā¦