We investigate statistical properties of a clustering algorithm that receives level set estimates from a kernel density estimator and then estimates the first split in the density level cluster tree if such a split is present or detects the absence of such a split. Key aspects of our analysis include finite sample guarantees, consistency, rates of convergence, and an adaptive data-driven strategy for chosing the kernel bandwidth. For the rates and the adaptivity we do not need continuity assumptions on the density such as H\"older continuity, but only require intuitive geometric assumptions of non-parametric nature.

The artificial neural network is a popular framework in machine learning. To empower individual neurons, we recently suggested that the current type of neurons could be upgraded to 2nd order counterparts, in which the linear operation between inputs to a neuron and the associated weights is replaced with a nonlinear quadratic operation. A single 2nd order neurons already has a strong nonlinear modeling ability, such as implementing basic fuzzy logic operations. In this paper, we develop a general backpropagation (BP) algorithm to train the network consisting of 2nd-order neurons. The numerical studies are performed to verify of the generalized BP algorithm.

The problem of subgroups is ubiquitous in scientific research (ex. disease heterogeneity, spatial distributions in ecology...), and piecewise regression is one way to deal with this phenomenon. Morse-Smale regression offers a way to partition the regression function based on level sets of a defined function and that function's basins of attraction. This topologically-based piecewise regression algorithm has shown promise in its initial applications, but the current implementation in the literature has been limited to elastic net and generalized linear regression. It is possible that nonparametric methods, such as random forest or conditional inference trees, may provide better prediction and insight through modeling interaction terms and other nonlinear relationships between predictors and a given outcome. This study explores the use of several machine learning algorithms within a Morse-Smale piecewise regression framework, including boosted regression with linear baselearners, homotopy-based LASSO, conditional inference trees, random forest, and a wide neural network framework called extreme learning machines. Simulations on Tweedie regression problems with varying Tweedie parameter and dispersion suggest that many machine learning approaches to Morse-Smale piecewise regression improve the original algorithm's performance, particularly for outcomes with lower dispersion and linear or a mix of linear and nonlinear predictor relationships. On a real actuarial problem, several of these new algorithms perform as good as or better than the original Morse-Smale regression algorithm, and most provide information on the nature of predictor relationships within each partition to provide insight into differences between dataset partitions.

Under a big-data setting, the storage and analysis of data can no longer be performed on a single machine, and in this case dividing data into many sub-samples becomes a critical 1 procedure for any numerical algorithm to be implemented. Distributed statistical estimation and distributed optimization have received increasing attention in recent years, and a flurry of research towards solving very large scale problems have emerged recently, such as Mcdonald et al. (2009); Zhang et al. (2013, 2015); Rosenblatt et al. (2016) and the references therein. In general, distributed algorithm can be classified into two families: data parallelism and task parallelism. Data parallelism aims at distributing the data across different parallel computing nodes or machines; and task parallelism distributes different tasks across parallel computing nodes. We are only concerned with data parallelism in this paper. In particular, we primarily consider the distributed estimation for partially linear models via using the standard divide and conquer strategy. Divide-and-conquer technology is a simple and communication-efficient way for handling big data, which is commonly used in the literature of statistical learning. To be precise, the whole data is randomly allocated among m machines, a local estimator is computed independently on each machine, and then the central node averages the local solutions into a global estimate. Partially linear models (PLM) (Hardle and Liang, 2007; Heckman, 1986), as the leading example of semiparametric models, are a class of important tools for modeling complex data, which retain model interpretation and flexibility simultaneously.

A Dirichlet $k$-partition of a domain $U \subseteq \mathbb{R}^d$ is a collection of $k$ pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has been posed on graphs with applications in data analysis. Both versions admit variational formulations: solutions are characterized by minimizers of the Dirichlet energy of mappings from $U$ into a singular space $\Sigma_k \subseteq \mathbb{R}^k$. In this paper, we extend results of N.\ Garc\'ia Trillos and D.\ Slep\v{c}ev to show that there exist solutions of the continuum problem arising as limits to solutions of a sequence of discrete problems. Specifically, a sequence of points $\{x_i\}_{i \in \mathbb{N}}$ from $U$ is sampled i.i.d.\ with respect to a given probability measure $\nu$ on $U$ and for all $n \in \mathbb{N}$, a geometric graph $G_n$ is constructed from the first $n$ points $x_1, x_2, \ldots, x_n$ and the pairwise distances between the points. With probability one with respect to the choice of points $\{x_i\}_{i \in \mathbb{N}}$, we show that as $n \to \infty$ the discrete Dirichlet energies for functions $G_n \to \Sigma_k$ $\Gamma$-converge to (a scalar multiple of) the continuum Dirichlet energy for functions $U \to \Sigma_k$ with respect to a metric coming from the theory of optimal transport. This, along with a compactness property for the aforementioned energies that we prove, implies the convergence of minimizers. When $\nu$ is the uniform distribution, our results also imply the statistical consistency statement that Dirichlet partitions of geometric graphs converge to partitions of the sampled space in the Hausdorff sense.

This paper considers the actor-critic contextual bandit for the mobile health (mHealth) intervention. The state-of-the-art decision-making methods in mHealth generally assume that the noise in the dynamic system follows the Gaussian distribution. Those methods use the least-square-based algorithm to estimate the expected reward, which is prone to the existence of outliers. To deal with the issue of outliers, we propose a novel robust actor-critic contextual bandit method for the mHealth intervention. In the critic updating, the capped-$\ell_{2}$ norm is used to measure the approximation error, which prevents outliers from dominating our objective. A set of weights could be achieved from the critic updating. Considering them gives a weighted objective for the actor updating. It provides the badly noised sample in the critic updating with zero weights for the actor updating. As a result, the robustness of both actor-critic updating is enhanced. There is a key parameter in the capped-$\ell_{2}$ norm. We provide a reliable method to properly set it by making use of one of the most fundamental definitions of outliers in statistics. Extensive experiment results demonstrate that our method can achieve almost identical results compared with the state-of-the-art methods on the dataset without outliers and dramatically outperform them on the datasets noised by outliers.

Choosing the best-performing optimizer(s) out of a portfolio of optimization algorithms is usually a difficult and complex task. It gets even worse, if the underlying functions are unknown, i.e., so-called Black-Box problems, and function evaluations are considered to be expensive. In the case of continuous single-objective optimization problems, Exploratory Landscape Analysis (ELA) - a sophisticated and effective approach for characterizing the landscapes of such problems by means of numerical values before actually performing the optimization task itself - is advantageous. Unfortunately, until now it has been quite complicated to compute multiple ELA features simultaneously, as the corresponding code has been - if at all - spread across multiple platforms or at least across several packages within these platforms. This article presents a broad summary of existing ELA approaches and introduces flacco, an R-package for feature-based landscape analysis of continuous and constrained optimization problems. Although its functions neither solve the optimization problem itself nor the related "Algorithm Selection Problem (ASP)", it offers easy access to an essential ingredient of the ASP by providing a wide collection of ELA features on a single platform - even within a single package. In addition, flacco provides multiple visualization techniques, which enhance the understanding of some of these numerical features, and thereby make certain landscape properties more comprehensible. On top of that, we will introduce the package's build-in, as well as web-hosted and hence platform-independent, graphical user interface (GUI), which facilitates the usage of the package - especially for people who are not familiar with R - making it a very convenient toolbox when working towards algorithm selection of continuous single-objective optimization problems.

Bayesian models that mix multiple Dirichlet prior parameters, called Multi-Dirichlet priors (MD) in this paper, are gaining popularity. Inferring mixing weights and parameters of mixed prior distributions seems tricky, as sums over Dirichlet parameters complicate the joint distribution of model parameters. This paper shows a novel auxiliary variable scheme which helps to simplify the inference for models involving hierarchical MDs and MDPs. Using this scheme, it is easy to derive fully collapsed inference schemes which allow for an efficient inference.

Sampling from the posterior distribution using Markov chain Monte Carlo (MCMC) methods can require an exhaustive number of iterations to fully explore the correct posterior. This is often the case when the posterior of interest is multi-modal, as the MCMC sampler can become trapped in a local mode for a large number of iterations. In this paper, we introduce the pseudo-extended MCMC method as an approach for improving the mixing of the MCMC sampler in complex posterior distributions. The pseudo-extended method augments the state-space of the posterior using pseudo-samples as auxiliary variables, where on the extended space, the MCMC sampler is able to easily move between the well-separated modes of the posterior. We apply the pseudo-extended method within an Hamiltonian Monte Carlo sampler and show that by using the No U-turn algorithm (Hoffman and Gelman, 2014), our proposed sampler is completely tuning free. We compare the pseudo-extended method against well-known tempered MCMC algorithms and show the advantages of the new sampler on a number of challenging examples from the statistics literature.

Malaria is a serious infectious disease that is responsible for over half million deaths yearly worldwide. The major cause of these mortalities is late or inaccurate diagnosis. Manual microscopy is currently considered as the dominant diagnostic method for malaria. However, it is time consuming and prone to human errors. The aim of this paper is to automate the diagnosis process and minimize the human intervention. We have developed the hardware and software for a cost-efficient malaria diagnostic system. This paper describes the manufactured hardware and also proposes novel software to handle parasite detection and life-stage identification. A motorized microscope is developed to take images from Giemsa-stained blood smears. A patch-based unsupervised statistical clustering algorithm is proposed which offers a novel method for classification of different regions within blood images. The proposed method provides better robustness against different imaging settings. The core of the proposed algorithm is a model called Mixture of Independent Component Analysis. A manifold based optimization method is proposed that facilitates the application of the model for high dimensional data usually acquired in medical microscopy. The method was tested on 600 blood slides with various imaging conditions. The speed of the method is higher than current supervised systems while its accuracy is comparable to or better than them.