Gatsby Computational Neuroscience Unit
Sainsbury Wellcome Centre
25 Howland Street
London W1T 4JG UK
+44 (0)7795 291 705
- Learning Interpretable Features to Compare Distributions
We presnt adaptive two-sample tests with maximum testing power and interpretable features, using two divergence measures: the maximum mean discrepancy (MMD), and differences of learned smooth features (the mean embedding (ME) test, NIPS 2016). In both cases, the key point is that variance matters: it is not enough to have a large empirical divergence; we also need to have high confidence in the value of our divergence. We use an optimized MMD discriminator to evaluate and troubleshoot a generative adversarial network (GAN), by detecting subtle differences in the distribution of GAN outputs and real hand-written digits which humans are unable to find (for instance, small imbalances in the proportions of certain digits, or minor distortions that are implausible in normal handwriting). We use the linear-time ME test to distinguish positive and negative emotions on a facial expression database, showing that a distinguishing feature reveals the facial areas most relevant to emotion.
Talk slides for the NIPS 2016 workshop on generative adversarial networks;more detailed slides from the Dagstuh workshop.
Papers: Adaptive MMD test paper ; linear-time ME test paper .
Software: Adaptive MMD test code; linear-time ME test code.
- Kernel Statistical Tests for Random Processes
We propose a kernel approach to statistical testing for random processes, which we illustrate using two settings: a two-sample test (comparison of marginal distributions of random processes), and an independence test. Our test statistics are in both cases straightforward V- statistics, derived from embeddings of probability measures to a reproducing kernel Hilbert space (RKHS). We use both shuffling and wild bootstrap approaches to construct provably consistent tests, and demonstrate that these are successful for cases where a naive permutation-based bootstrap fails. In experiments, the wild bootstrap gives strong performance on synthetic examples, on audio data, and in performance benchmarking for the Gibbs sampler.
Talk slides from the ISNPS 2016 conference.
Papers: A Wild Bootstrap for Degenerate Kernel Tests (NIPS 2014, uses wild bootstrap for null), and A Kernel Independence Test for Random Processes (ICML 2014, uses shuffling for null).
Software: Wild Bootstrap and Shuffling tests.
- A Kernel Test of Goodness of Fit
We describe a nonparametric statistical test for goodness-of-fit: given a set of samples, the test determines how likely it is that these were generated from a target density function. The measure of goodness-of-fit is a divergence constructed via Stein's method using functions from a Reproducing Kernel Hilbert Space. Our test statistic is based on an empirical estimate of this divergence, taking the form of a V-statistic in terms of the log gradients of the target density and the kernel. We derive a statistical test, both for i.i.d. and non-i.i.d. samples, where we estimate the null distribution quantiles using a wild bootstrap procedure. We apply our test to quantifying convergence of approximate Markov Chain Monte Carlo methods, statistical model criticism, and evaluating quality of fit vs model complexity in nonparametric density estimation.
Talk slides from ICML 2016 .
Paper and Code.
- Learning with probabilities as inputs, using kernels
Kernels have been widely applied when learning labels or regression values from very broad classes of inputs, for example images, strings, histograms, and graphs. In this talk, we deal with the case where inputs to a learning algorithm are probability distributions, or samples from distributions. The latter setting might include the task of labelling bags of image patches, or regressing from a collection of air samples to the pollution level.
We first show how to summarize probability distribution-valued inputs in terms of expected features, and to learn from kernels defined from these features. We demonstrate how to efficiently approximate the features to scale up our approach. We address two supervised learning settings: expectation propagation for general models, and classifying the direction of causal influence from observed samples of cause and effect. We also show, in the case of learning from samples from distributions, how many samples per distribution are needed for learning to occur. Time permitting, we might also cover the problem of learning with probability-valued outputs, for nonparametric Bayesian inference.
Talk slides from the NIPS 2015 workshop on Probabilistic Integration.
Papers: Learning for passing expectation propagation messages (UAI 2015), Two-Stage Sampled Learning Theory on Distributions (AISTATS 2015) and extended JMLR submission.
Software: Learning for passing expectation propagation messages and Two-Stage Sampled Learning Theory on Distributions
- Kernel Adaptive Metropolis-Hastings
Talk slides from the NIPS 2015 workshop on Scalable Monte Carlo
Papers: Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families (NIPS 2015), Kernel Adaptive Metropolis-Hastings (ICML 2014).
Software: Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families, Kernel Adaptive Metropolis-Hastings.
- Nonparametric Bayesian inference using kernel distribution embeddings
A method is presented for approximate Bayesian inference, where explicit models for the prior and likelihood are unknown (or difficult to compute), but sampling from these distributions is possbile. The method expresses the prior as an element in a reproducing kernel Hilbert space, and the likelihood as a family of such elements, indexed by the conditioning variable. These distribution embeddings may be computed directly from training samples: in particular, the likelihood embedding is the solution of a mulitask learning problem with (potentially) infinite tasks, where each task is a different feature space dimension.
Kernel message passing can be applied to any domains where kernels are defined, handling challenging cases such as discrete variables with huge domains, or very complex, non-Gaussian continuous distributions. An empirical comparison with approximate Bayesian computation (ABC) shows better performance in high dimensions. Finally, the approach is applied to camera angle recovery from captured images, showing better performance than the extended Kalman filter.
Talk slides from the NIPS 2013 workshop on new directions in transfer and multi-task learning (the current slides are from an extended version given at the Lancaster Statistics Department).
IEEE Signal Processing Magazine 2013 Tutorial, JMLR 2013 Paper , ICML 2013 Paper , ICML 2012 Paper
- Optimal kernel choice for kernel hypothesis testing
We consider two nonparametric hypothesis testing problems: (1) Given samples from distributions p and q, a two-sample test determines whether to reject the null hypothesis p=q; and (2) Given a joint distribution p_xy over random variables x and y, an independence test determines whether to reject the null hypothesis of independence, p_xy = p_x p_y. In testing whether two distributions are identical, or whether two random variables are independent, we require a test statistic which is a measure of distance between probability distributions. One choice of test statistic is the maximum mean discrepancy (MMD), a distance between embeddings of the probability distributions in a reproducing kernel Hilbert space. The kernel used in obtaining these embeddings is critical in ensuring the test has high power, and correctly distinguishes unlike distributions with high probability.
In this talk, I will provide a tutorial overview of kernel distances on probabilities, and show how these may be used in two-sample and independence testing. I will then describe a strategy for optimal kernel choice, and compare it with earlier heuristics (including other multiple kernel learning approaches).
Talk slides from the CMU ML lunch series (also presented at Microsoft Research Cambridge), video (scroll down for the talk)
NIPS 2012 paper
- Hypothesis Testing and Bayesian Inference: New Applications of
In the early days of kernel machines research, the "kernel trick" was considered a useful way of constructing nonlinear learning algorithms from linear ones, by applying the linear algorithms to feature space mappings of the original data. More recently, it has become clear that a potentially more far reaching use of kernels is as a linear way of dealing with higher order statistics, by mapping probabilities to a suitable reproducing kernel Hilbert space (i.e., the feature space is an RKHS).
I will describe how probabilities can be mapped to kernel feature spaces, and how to compute distances between these mappings. A measure of strength of dependence between two random variables follows naturally from this distance. Applications that make use of kernel probability embeddings include:
Nonparametric two-sample testing and independence testing in complex (high dimensional) domains. In the latter case, we test whether text in English is translated from the French, as opposed to being random extracts on the same topic.
Inference on graphical models, in cases where the variable interactions are modeled nonparametrically (i.e., when parametric models are impractical or unknown).
Talk slides from a talk at the Oxford statistics department (also presented at Google Mountain View, Yahoo, Inria)
- Bayesian Inference with Kernels
An embedding of probability distributions into a reproducing kernel Hilbert space (RKHS) has been introduced: like the characteristic function, this provides a unique representation of a probability distribution in a high dimensional feature space. This representation forms the basis of an inference procedure on graphical models, where the likelihoods are represented as RKHS functions. The resulting algorithm is completely nonparametric: all aspects of the model are represented implicitly, and learned from a training sample. Both exact inference on trees and loopy belief propagation on pairwise Markov random fields are demonstrated.
Kernel message passing can be applied to general domains where kernels are defined, handling challenging cases such as discrete variables with huge domains, or very complex, non-Gaussian continuous distributions. We apply kernel message passing and competing approaches to cross-language document retrieval, depth prediction from still images, protein configuration prediction, and paper topic inference from citation networks: these are all large-scale problems, with continuous-valued or structured random variables having complex underlying probability distributions. In all cases, kernel message passing performs outstandingly, being orders of magnitude faster than state-of-the-art nonparametric alternatives, and returning more accurate results.
Talk slides (with proofs in supplementary slides at the end) and video from the CMU machine learning lunch series (scroll down to the talk)
AISTATS 2010 paper for inference on trees; AISTATS 2011 paper for inference on loopy graphs
NIPS 2010 Workshop on Challenges of Data Visualization (invited talk); also presented at the University of Cambridge, MSR Cambridge, University of Bristol, and CMU.
- Consistent Nonparametric Tests of Independence: L1, Log-Likelihood and Kernel
Three simple and explicit procedures for testing the independence of two multi-dimensional random variables are described. Two of the associated test statistics (L1, log-likelihood) are defined when the empirical distribution of the variables is restricted to finite partitions. A third test statistic is defined as a kernel-based independence measure. Two kinds of tests are provided. Distribution-free strong consistent tests are derived on the basis of large deviation bounds on the test statistics: these tests make almost surely no Type I or Type II error after a random sample size. Asymptotically alpha-level tests are obtained from the limiting distribution of the test statistics. For the latter tests, the Type I error converges to a fixed non-zero value alpha, and the Type II error drops to zero, for increasing sample size. All tests reject the null hypothesis of independence if the test statistics become large. The performance of the tests is evaluated experimentally on benchmark data.
JMLR paper and earlier ALT paper for the L1 and log-likelihood tests, NIPS paper for the kernel test
Prague Stochastics 2010
- Covariate Shift by Kernel Mean Matching
Assume we are given sets of observations of training and test data, where (unlike in the classical setting) the training and test distributions are allowed to differ. Thus for learning purposes, we face the problem of re-weighting the training data such that its distribution more closely matches that of the test data. We consider specifically the case where the difference in training and test distributions occurs only in the marginal distribution of the covariates: the conditional distribution of the outputs given the covariates is unchanged. We achieve covariate shift correction by matching covariate distributions between training and test sets in a high dimensional feature space (specifically, a reproducing kernel Hilbert space). This approach does not require distribution estimation, making it suited to high dimensions and structured data, where distribution estimates may not be practical.
We first describe the general setting of covariate shift correction, and the importance weighting approach. While direct density estimation provides an estimate of the importance weights, this has two potential disadvantages: it may not offer the best bias/variance tradeoff, and density estimation might be difficult on complex, high dimensional domains (such as text). We then describe how distributions may be mapped to reproducing kernel Hilbert spaces (RKHS), and review distances between such mappings. We demonstrate a transfer learning algorithm that reweights the training points such that their RKHS mapping matches that of the (unlabeled) test points. The sample weights are obtained by a simple quadratic programming procedure. Our correction method yields its greatest and most consistent advantages when the learning algorithm returns a classifier/regressor that is "simpler" than the data might suggest. On the other hand, even an ideal sample reweighting may not be of practical benefit given a sufficiently powerful learning algorithm (if available).
Talk slides and video
Book chapter and NIPS paper
NIPS 2009 Workshop on Transfer Learning for Structured Data (invited talk)
- Introduction to Independent Component Analysis
Independent component analysis (ICA) is a technique for extracting underlying sources of information from linear mixtures of these sources, based only on the assumption that the sources are independent of each other. To illustrate the idea, we might be in a room containing several people (the sources) talking simultaneously, with microphones picking up multiple conversations at once (the mixtures), and we might wish to automatically recover the original separate conversations from these mixtures. More broadly, ICA is used in a very wide variety of applications, including signal extraction from EEG, image processing, bioinformatics, and economics. I will present an introduction to ICA, which includes a description of principal component analysis (PCA), and how ICA differs from PCA, the maximum likelihood approach, the case where fixed nonlinearities are used as heuristics for source extraction, some more modern information theoretic approaches, and a kernel-based method. I will also cover two optimization strategies, and provide a comparison of the various approaches on benchmark data, to reveal the strengths and failure modes of different ICA algorithms (with a focus on modern, recently published methods).
Software for Fast Kernel ICA demo
MLD student research symposium (invited talk)