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In this paper, we bridge Variational Autoencoders (VAEs) [17] and kernel density estimations (KDEs) [25 ],[23] by approximating the posterior by KDEs and deriving an upper bound of the Kullback-Leibler (KL) divergence in the evidence lower bound (ELBO). The flexibility of KDEs makes the optimization of posteriors in VAEs possible, which not only addresses the limitations of Gaussian latent space in vanilla VAE but also provides a new perspective of estimating the KL-divergence in ELBO. Under appropriate conditions [ 9],[3 ], we show that the Epanechnikov kernel is the optimal choice in minimizing the derived upper bound of KL-divergence asymptotically. Compared with Gaussian kernel, Epanechnikov kernel has compact support which should make the generated sample less noisy and blurry. The implementation of Epanechnikov kernel in ELBO is straightforward as it lies in the "location-scale" family of distributions where the reparametrization tricks can be directly employed. A series of experiments on benchmark datasets such as MNIST, Fashion-MNIST, CIFAR-10 and CelebA further demonstrate the superiority of Epanechnikov Variational Autoenocoder (EVAE) over vanilla VAE in the quality of reconstructed images, as measured by the FID score and Sharpness[27].

We propose a novel ensemble method called Riemann-Lebesgue Forest (RLF) for regression. The core idea in RLF is to mimic the way how a measurable function can be approximated by partitioning its range into a few intervals. With this idea in mind, we develop a new tree learner named Riemann-Lebesgue Tree (RLT) which has a chance to perform Lebesgue type cutting,i.e splitting the node from response $Y$ at certain non-terminal nodes. We show that the optimal Lebesgue type cutting results in larger variance reduction in response $Y$ than ordinary CART \cite{Breiman1984ClassificationAR} cutting (an analogue of Riemann partition). Such property is beneficial to the ensemble part of RLF. We also generalize the asymptotic normality of RLF under different parameter settings. Two one-dimensional examples are provided to illustrate the flexibility of RLF. The competitive performance of RLF against original random forest \cite{Breiman2001RandomF} is demonstrated by experiments in simulation data and real world datasets.

Seasonal influenza causes on average 425,000 hospitalizations and 32,000 deaths per year in the United States. Forecasts of influenza-like illness (ILI) -- a surrogate for the proportion of patients infected with influenza -- support public health decision making. The goal of an ensemble forecast of ILI is to increase accuracy and calibration compared to individual forecasts and to provide a single, cohesive prediction of future influenza. However, an ensemble may be composed of models that produce similar forecasts, causing issues with ensemble forecast performance and non-identifiability. To improve upon the above issues we propose a novel Cluster-Aggregate-Pool or `CAP' ensemble algorithm that first clusters together individual forecasts, aggregates individual models that belong to the same cluster into a single forecast (called a cluster forecast), and then pools together cluster forecasts via a linear pool. When compared to a non-CAP approach, we find that a CAP ensemble improves calibration by approximately 10% while maintaining similar accuracy to non-CAP alternatives. In addition, our CAP algorithm (i) generalizes past ensemble work associated with influenza forecasting and introduces a framework for future ensemble work, (ii) automatically accounts for missing forecasts from individual models, (iii) allows public health officials to participate in the ensemble by assigning individual models to clusters, and (iv) provide an additional signal about when peak influenza may be near.

With the insight of variance-bias decomposition, we design a new hybrid bagging-boosting algorithm named SBPMT for classification problems. For the boosting part of SBPMT, we propose a new tree model called Probit Model Tree (PMT) as base classifiers in AdaBoost procedure. For the bagging part, instead of subsampling from the dataset at each step of boosting, we perform boosted PMTs on each subagged dataset and combine them into a powerful "committee", which can be viewed an incomplete U-statistic. Our theoretical analysis shows that (1) SBPMT is consistent under certain assumptions, (2) Increase the subagging times can reduce the generalization error of SBPMT to some extent and (3) Large number of ProbitBoost iterations in PMT can benefit the performance of SBPMT with fewer steps in the AdaBoost part. Those three properties are verified by a famous simulation designed by Mease and Wyner (2008). The last two points also provide a useful guidance in model tuning. A comparison of performance with other state-of-the-art classification methods illustrates that the proposed SBPMT algorithm has competitive prediction power in general and performs significantly better in some cases.

The threshold model has been widely adopted as a classic model for studying contagion processes on social networks. We consider asymmetric individual interactions in social networks and introduce a persuasion mechanism into the threshold model. Specifically, we study a combination of adoption and persuasion in cascading processes on complex networks. It is found that with the introduction of the persuasion mechanism, the system may become more vulnerable to global cascades, and the effects of persuasion tend to be more significant in heterogeneous networks than those in homogeneous networks: a comparison between heterogeneous and homogeneous networks shows that under weak persuasion, heterogeneous networks tend to be more robust against random shocks than homogeneous networks; whereas under strong persuasion, homogeneous networks are more stable. Finally, we study the effects of adoption and persuasion threshold heterogeneity on systemic stability. Though both heterogeneities give rise to global cascades, the adoption heterogeneity has an overwhelmingly stronger impact than the persuasion heterogeneity when the network connectivity is sufficiently dense., Comment: 12 pages, 7 figures