to

### Ensemble weighted kernel estimators for multivariate entropy estimation

The problem of estimation of entropy functionals of probability densities has received much attention in the information theory, machine learning and statistics communities. Kernel density plug-in estimators are simple, easy to implement and widely used for estimation of entropy. However, kernel plug-in estimators suffer from the curse of dimensionality, wherein the MSE rate of convergence is glacially slow - of order $O(T^{-{\gamma}/{d}})$, where $T$ is the number of samples, and $\gamma>0$ is a rate parameter. In this paper, it is shown that for sufficiently smooth densities, an ensemble of kernel plug-in estimators can be combined via a weighted convex combination, such that the resulting weighted estimator has a superior parametric MSE rate of convergence of order $O(T^{-1})$. Furthermore, it is shown that these optimal weights can be determined by solving a convex optimization problem which does not require training data or knowledge of the underlying density, and therefore can be performed offline. This novel result is remarkable in that, while each of the individual kernel plug-in estimators belonging to the ensemble suffer from the curse of dimensionality, by appropriate ensemble averaging we can achieve parametric convergence rates.

### Fast Conditional Density Estimation for Quantitative Structure-Activity Relationships

Many methods for quantitative structure-activity relationships (QSARs) deliver point estimates only, without quantifying the uncertainty inherent in the prediction. One way to quantify the uncertainy of a QSAR prediction is to predict the conditional density of the activity given the structure instead of a point estimate. If a conditional density estimate is available, it is easy to derive prediction intervals of activities. In this paper, we experimentally evaluate and compare three methods for conditional density estimation for their suitability in QSAR modeling. In contrast to traditional methods for conditional density estimation, they are based on generic machine learning schemes, more specifically, class probability estimators. Our experiments show that a kernel estimator based on class probability estimates from a random forest classifier is highly competitive with Gaussian process regression, while taking only a fraction of the time for training. Therefore, generic machine-learning based methods for conditional density estimation may be a good and fast option for quantifying uncertainty in QSAR modeling.

### Kernel Mean Estimation via Spectral Filtering

The problem of estimating the kernel mean in a reproducing kernel Hilbert space (RKHS) is central to kernel methods in that it is used by classical approaches (e.g., when centering a kernel PCA matrix), and it also forms the core inference step of modern kernel methods (e.g., kernel-based non-parametric tests) that rely on embedding probability distributions in RKHSs. Previous work [1] has shown that shrinkage can help in constructing "better" estimators of the kernel mean than the empirical estimator. The present paper studies the consistency and admissibility of the estimators in [1], and proposes a wider class of shrinkage estimators that improve upon the empirical estimator by considering appropriate basis functions. Using the kernel PCA basis, we show that some of these estimators can be constructed using spectral filtering algorithms which are shown to be consistent under some technical assumptions. Our theoretical analysis also reveals a fundamental connection to the kernel-based supervised learning framework.

### Ensemble weighted kernel estimators for multivariate entropy estimation

The problem of estimation of entropy functionals of probability densities has received much attention in the information theory, machine learning and statistics communities. Kernel density plug-in estimators are simple, easy to implement and widely used for estimation of entropy. However, kernel plug-in estimators suffer from the curse of dimensionality, wherein the MSE rate of convergence is glacially slow - of order $O(T {-{\gamma}/{d}})$, where $T$ is the number of samples, and $\gamma 0$ is a rate parameter. In this paper, it is shown that for sufficiently smooth densities, an ensemble of kernel plug-in estimators can be combined via a weighted convex combination, such that the resulting weighted estimator has a superior parametric MSE rate of convergence of order $O(T {-1})$. Furthermore, it is shown that these optimal weights can be determined by solving a convex optimization problem which does not require training data or knowledge of the underlying density, and therefore can be performed offline.

### A Spectral Series Approach to High-Dimensional Nonparametric Regression

A key question in modern statistics is how to make fast and reliable inferences for complex, high-dimensional data. While there has been much interest in sparse techniques, current methods do not generalize well to data with nonlinear structure. In this work, we present an orthogonal series estimator for predictors that are complex aggregate objects, such as natural images, galaxy spectra, trajectories, and movies. Our series approach ties together ideas from kernel machine learning, and Fourier methods. We expand the unknown regression on the data in terms of the eigenfunctions of a kernel-based operator, and we take advantage of orthogonality of the basis with respect to the underlying data distribution, P, to speed up computations and tuning of parameters. If the kernel is appropriately chosen, then the eigenfunctions adapt to the intrinsic geometry and dimension of the data. We provide theoretical guarantees for a radial kernel with varying bandwidth, and we relate smoothness of the regression function with respect to P to sparsity in the eigenbasis. Finally, using simulated and real-world data, we systematically compare the performance of the spectral series approach with classical kernel smoothing, k-nearest neighbors regression, kernel ridge regression, and state-of-the-art manifold and local regression methods.