Projected support points: a new method for high-dimensional data reduction

In an era where big and high-dimensional data is readily available, data scientists are inevitably faced with the challenge of reducing this data for expensive downstream computation or analysis. To this end, we present here a new method for reducing high-dimensional big data into a representative point set, called projected support points (PSPs). A key ingredient in our method is the so-called sparsity-inducing (SpIn) kernel, which encourages the preservation of low-dimensional features when reducing high-dimensional data. We begin by introducing a unifying theoretical framework for data reduction, connecting PSPs with fundamental sampling principles from experimental design and Quasi-Monte Carlo. Through this framework, we then derive sparsity conditions under which the curse-of-dimensionality in data reduction can be lifted for our method. Next, we propose two algorithms for one-shot and sequential reduction via PSPs, both of which exploit big data subsampling and majorization-minimization for efficient optimization. Finally, we demonstrate the practical usefulness of PSPs in two real-world applications, the first for data reduction in kernel learning, and the second for reducing Markov Chain Monte Carlo (MCMC) chains

Optimally-Weighted Herding is Bayesian Quadrature

Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves performance superior to any other weighted herding method. We demonstrate empirically a rate of convergence faster than O(1/N). Our results also imply an upper bound on the empirical error of the Bayesian quadrature estimate.Comment: Accepted as an oral presentation at Uncertainty in Artificial Intelligence 2012. Updated to fix several typo

Predicting the Future Behavior of a Time-Varying Probability Distribution

We study the problem of predicting the future, though only in the probabilistic sense of estimating a future state of a time-varying probability distribution. This is not only an interesting academic problem, but solving this extrapolation problem also has many practical application, e.g. for training classifiers that have to operate under time-varying conditions. Our main contribution is a method for predicting the next step of the time-varying distribution from a given sequence of sample sets from earlier time steps. For this we rely on two recent machine learning techniques: embedding probability distributions into a reproducing kernel Hilbert space, and learning operators by vector-valued regression. We illustrate the working principles and the practical usefulness of our method by experiments on synthetic and real data. We also highlight an exemplary application: training a classifier in a domain adaptation setting without having access to examples from the test time distribution at training time

Reduced-Set Kernel Principal Components Analysis for Improving the Training and Execution Speed of Kernel Machines

This paper presents a practical, and theoretically well-founded, approach to improve the speed of kernel manifold learning algorithms relying on spectral decomposition. Utilizing recent insights in kernel smoothing and learning with integral operators, we propose Reduced Set KPCA (RSKPCA), which also suggests an easy-to-implement method to remove or replace samples with minimal effect on the empirical operator. A simple data point selection procedure is given to generate a substitute density for the data, with accuracy that is governed by a user-tunable parameter . The effect of the approximation on the quality of the KPCA solution, in terms of spectral and operator errors, can be shown directly in terms of the density estimate error and as a function of the parameter . We show in experiments that RSKPCA can improve both training and evaluation time of KPCA by up to an order of magnitude, and compares favorably to the widely-used Nystrom and density-weighted Nystrom methods

Kernel Mean Embedding of Distributions: A Review and Beyond

A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing kernel Hilbert space (RKHS) in which the whole arsenal of kernel methods can be extended to probability measures. It can be viewed as a generalization of the original "feature map" common to support vector machines (SVMs) and other kernel methods. While initially closely associated with the latter, it has meanwhile found application in fields ranging from kernel machines and probabilistic modeling to statistical inference, causal discovery, and deep learning. The goal of this survey is to give a comprehensive review of existing work and recent advances in this research area, and to discuss the most challenging issues and open problems that could lead to new research directions. The survey begins with a brief introduction to the RKHS and positive definite kernels which forms the backbone of this survey, followed by a thorough discussion of the Hilbert space embedding of marginal distributions, theoretical guarantees, and a review of its applications. The embedding of distributions enables us to apply RKHS methods to probability measures which prompts a wide range of applications such as kernel two-sample testing, independent testing, and learning on distributional data. Next, we discuss the Hilbert space embedding for conditional distributions, give theoretical insights, and review some applications. The conditional mean embedding enables us to perform sum, product, and Bayes' rules---which are ubiquitous in graphical model, probabilistic inference, and reinforcement learning---in a non-parametric way. We then discuss relationships between this framework and other related areas. Lastly, we give some suggestions on future research directions.Comment: 147 pages; this is a version of the manuscript after the review proces

Filtering with State-Observation Examples via Kernel Monte Carlo Filter

This paper addresses the problem of filtering with a state-space model. Standard approaches for filtering assume that a probabilistic model for observations (i.e. the observation model) is given explicitly or at least parametrically. We consider a setting where this assumption is not satisfied; we assume that the knowledge of the observation model is only provided by examples of state-observation pairs. This setting is important and appears when state variables are defined as quantities that are very different from the observations. We propose Kernel Monte Carlo Filter, a novel filtering method that is focused on this setting. Our approach is based on the framework of kernel mean embeddings, which enables nonparametric posterior inference using the state-observation examples. The proposed method represents state distributions as weighted samples, propagates these samples by sampling, estimates the state posteriors by Kernel Bayes' Rule, and resamples by Kernel Herding. In particular, the sampling and resampling procedures are novel in being expressed using kernel mean embeddings, so we theoretically analyze their behaviors. We reveal the following properties, which are similar to those of corresponding procedures in particle methods: (1) the performance of sampling can degrade if the effective sample size of a weighted sample is small; (2) resampling improves the sampling performance by increasing the effective sample size. We first demonstrate these theoretical findings by synthetic experiments. Then we show the effectiveness of the proposed filter by artificial and real data experiments, which include vision-based mobile robot localization.Comment: 56 pages, 25 figures; Final version (accepted to Neural Computation

Learning to Herd Agents Amongst Obstacles: Training Robust Shepherding Behaviors using Deep Reinforcement Learning

Robotic shepherding problem considers the control and navigation of a group of coherent agents (e.g., a flock of bird or a fleet of drones) through the motion of an external robot, called shepherd. Machine learning based methods have successfully solved this problem in an empty environment with no obstacles. Rule-based methods, on the other hand, can handle more complex scenarios in which environments are cluttered with obstacles and allow multiple shepherds to work collaboratively. However, these rule-based methods are fragile due to the difficulty in defining a comprehensive set of rules that can handle all possible cases. To overcome these limitations, we propose the first known learning-based method that can herd agents amongst obstacles. By using deep reinforcement learning techniques combined with the probabilistic roadmaps, we train a shepherding model using noisy but controlled environmental and behavioral parameters. Our experimental results show that the proposed method is robust, namely, it is insensitive to the uncertainties originated from both environmental and behavioral models. Consequently, the proposed method has a higher success rate, shorter completion time and path length than the rule-based behavioral methods have. These advantages are particularly prominent in more challenging scenarios involving more difficult groups and strenuous passages

Bayesian Learning of Conditional Kernel Mean Embeddings for Automatic Likelihood-Free Inference

In likelihood-free settings where likelihood evaluations are intractable, approximate Bayesian computation (ABC) addresses the formidable inference task to discover plausible parameters of simulation programs that explain the observations. However, they demand large quantities of simulation calls. Critically, hyperparameters that determine measures of simulation discrepancy crucially balance inference accuracy and sample efficiency, yet are difficult to tune. In this paper, we present kernel embedding likelihood-free inference (KELFI), a holistic framework that automatically learns model hyperparameters to improve inference accuracy given limited simulation budget. By leveraging likelihood smoothness with conditional mean embeddings, we nonparametrically approximate likelihoods and posteriors as surrogate densities and sample from closed-form posterior mean embeddings, whose hyperparameters are learned under its approximate marginal likelihood. Our modular framework demonstrates improved accuracy and efficiency on challenging inference problems in ecology.Comment: To appear in the Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS) 2019, Naha, Okinawa, Japa

Model Selection for Simulator-based Statistical Models: A Kernel Approach

We propose a novel approach to model selection for simulator-based statistical models. The proposed approach defines a mixture of candidate models, and then iteratively updates the weight coefficients for those models as well as the parameters in each model simultaneously; this is done by recursively applying Bayes' rule, using the recently proposed kernel recursive ABC algorithm. The practical advantage of the method is that it can be used even when a modeler lacks appropriate prior knowledge about the parameters in each model. We demonstrate the effectiveness of the proposed approach with a number of experiments, including model selection for dynamical systems in ecology and epidemiology.Comment: 32 page

Improved Coresets for Kernel Density Estimates

We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation preserves the $L_\infty$ error between kernel density estimates within error $\epsilon$, with coreset size $2/\epsilon^2$, but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is unrestricted, we show this bound is tight for these kernels as well as a much broader set. This work provides a careful analysis of the iterative Frank-Wolfe algorithm adapted to this context, an algorithm called \emph{kernel herding}. This analysis unites a broad line of work that spans statistics, machine learning, and geometry. When the dimension $d$ is constant, we demonstrate much tighter bounds on the size of the coreset specifically for Gaussian kernels, showing that it is bounded by the size of the coreset for axis-aligned rectangles. Currently the best known constructive bound is $O(\frac{1}{\epsilon} \log^d \frac{1}{\epsilon})$, and non-constructively, this can be improved by $\sqrt{\log \frac{1}{\epsilon}}$. This improves the best constant dimension bounds polynomially for $d \geq 3$