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Wing Hung Wong

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3 papers

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JMLR Journal 2023 Journal Article

Convergence Rates of a Class of Multivariate Density Estimation Methods Based on Adaptive Partitioning

Linxi Liu
Dangna Li
Wing Hung Wong

Density estimation is a building block for many other statistical methods, such as classification, nonparametric testing, and data compression. In this paper, we focus on a non-parametric approach to multivariate density estimation, and study its asymptotic properties under both frequentist and Bayesian settings. The estimated density function is obtained by considering a sequence of approximating spaces to the space of densities. These spaces consist of piecewise constant density functions supported by binary partitions with increasing complexity. To obtain an estimate, the partition is learned by maximizing either the likelihood of the corresponding histogram on that partition, or the marginal posterior probability of the partition under a suitable prior. We analyze the convergence rate of the maximum likelihood estimator and the posterior concentration rate of the Bayesian estimator, and conclude that for a relatively rich class of density functions the rate does not directly depend on the dimension. We also show that the Bayesian method can adapt to the unknown smoothness of the density function. The method is applied to several specific function classes and explicit rates are obtained. These include spatially sparse functions, functions of bounded variation, and Hölder continuous functions. We also introduce an ensemble approach, obtained by aggregating multiple density estimates fit under carefully designed perturbations, and show that for density functions lying in a Hölder space ($\mathcal{H}^{1, \beta}, 0 [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

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NeurIPS Conference 2017 Conference Paper

Convergence rates of a partition based Bayesian multivariate density estimation method

Linxi Liu
Dangna Li
Wing Hung Wong

We study a class of non-parametric density estimators under Bayesian settings. The estimators are obtained by adaptively partitioning the sample space. Under a suitable prior, we analyze the concentration rate of the posterior distribution, and demonstrate that the rate does not directly depend on the dimension of the problem in several special cases. Another advantage of this class of Bayesian density estimators is that it can adapt to the unknown smoothness of the true density function, thus achieving the optimal convergence rate without artificial conditions on the density. We also validate the theoretical results on a variety of simulated data sets.

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NeurIPS Conference 2016 Conference Paper

Density Estimation via Discrepancy Based Adaptive Sequential Partition

Dangna Li
Kun Yang
Wing Hung Wong

Given $iid$ observations from an unknown continuous distribution defined on some domain $\Omega$, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function. Our density estimate is a piecewise constant function defined on a binary partition of $\Omega$. The key ingredient of the algorithm is to use discrepancy, a concept originates from Quasi Monte Carlo analysis, to control the partition process. The resulting algorithm is simple, efficient, and has provable convergence rate. We demonstrate empirically its efficiency as a density estimation method. We also show how it can be utilized to find good initializations for k-means.

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