Learning deep kernels for non-parametric two-sample tests

We propose a class of kernel-based two-sample tests, which aim to determine whether two sets of samples are drawn from the same distribution. Our tests are constructed from kernels parameterized by deep neural nets, trained to maximize test power. These tests adapt to variations in distribution smoothness and shape over space, and are especially suited to high dimensions and complex data. By contrast, the simpler kernels used in prior kernel testing work are spatially homogeneous, and adaptive only in lengthscale. We explain how this scheme includes popular classifier-based two-sample tests as a special case, but improves on them in general. We provide the first proof of consistency for the proposed adaptation method, which applies both to kernels on deep features and to simpler radial basis kernels or multiple kernel learning. In experiments, we establish the superior performance of our deep kernels in hypothesis testing on benchmark and real-world data

Advances in Non-parametric Hypothesis Testing with Kernels

Non-parametric statistical hypothesis testing procedures aim to distinguish the null hypothesis against the alternative with minimal assumptions on the model distributions. In recent years, the maximum mean discrepancy (MMD) has been developed as a measure to compare two distributions, which is applicable to two-sample problems and independence tests. With the aid of reproducing kernel Hilbert spaces (RKHS) that are rich-enough, MMD enjoys desirable statistical properties including characteristics, consistency, and maximal test power. Moreover, MMD receives empirical successes in complex tasks such as training and comparing generative models. Stein’s method also provides an elegant probabilistic tool to compare unnormalised distributions, which commonly appear in practical machine learning tasks. Combined with rich-enough RKHS, the kernel Stein discrepancy (KSD) has been developed as a proper discrepancy measure between distributions, which can be used to tackle one-sample problems (or goodness-of-fit tests). The existing development of KSD applies to a limited choice of domains, such as Euclidean space or finite discrete sets, and requires complete data observations, while the current MMD constructions are limited by the choice of simple kernels where the power of the tests suffer, e.g. high-dimensional image data. The main focus of this thesis is on the further advancement of kernel-based statistics for hypothesis testings. Firstly, Stein operators are developed that are compatible with broader data domains to perform the corresponding goodness-of-fit tests. Goodness-of-fit tests for general unnormalised densities on Riemannian manifolds, which are of the non-Euclidean topology, have been developed. In addition, novel non-parametric goodness-of-fit tests for data with censoring are studied. Then the tests for data observations with left truncation are studied, e.g. times of entering the hospital always happen before death time in the hospital, and we say the death time is truncated by the entering time. We test the notion of independence beyond truncation by proposing a kernelised measure for quasi-independence. Finally, we study the deep kernel architectures to improve the two-sample testing performances

Visual Dynamics: Probabilistic Future Frame Synthesis via Cross Convolutional Networks

We study the problem of synthesizing a number of likely future frames from a single input image. In contrast to traditional methods, which have tackled this problem in a deterministic or non-parametric way, we propose a novel approach that models future frames in a probabilistic manner. Our probabilistic model makes it possible for us to sample and synthesize many possible future frames from a single input image. Future frame synthesis is challenging, as it involves low- and high-level image and motion understanding. We propose a novel network structure, namely a Cross Convolutional Network to aid in synthesizing future frames; this network structure encodes image and motion information as feature maps and convolutional kernels, respectively. In experiments, our model performs well on synthetic data, such as 2D shapes and animated game sprites, as well as on real-wold videos. We also show that our model can be applied to tasks such as visual analogy-making, and present an analysis of the learned network representations.Comment: The first two authors contributed equally to this wor

Interpretable Distribution Features with Maximum Testing Power

Two semimetrics on probability distributions are proposed, given as the sum of differences of expectations of analytic functions evaluated at spatial or frequency locations (i.e, features). The features are chosen so as to maximize the distinguishability of the distributions, by optimizing a lower bound on test power for a statistical test using these features. The result is a parsimonious and interpretable indication of how and where two distributions differ locally. An empirical estimate of the test power criterion converges with increasing sample size, ensuring the quality of the returned features. In real-world benchmarks on high-dimensional text and image data, linear-time tests using the proposed semimetrics achieve comparable performance to the state-of-the-art quadratic-time maximum mean discrepancy test, while returning human-interpretable features that explain the test results