Neil D. Lawrence

Missing data in kernel PCA

Kernel Principal Component Analysis (KPCA) is a widely used technique for visualisation and feature extraction. Despite its success and flexibility, the lack of a probabilistic interpretation means that some problems, such as handling missing or corrupted data, are very hard to deal with. In this paper we exploit the probabilistic interpretation of linear PCA together with recent results on latent variable models in Gaussian Processes in order to introduce an objective function for KPCA. This in turn allows a principled approach to the missing data problem. Furthermore, this new approach can be extended to reconstruct corrupted test data using fixed kernel feature extractors. The experimental results show strong improvements over widely used heuristics.

Learning to learn with the informative vector machine

This paper describes an efficient method for learning the parameters of a Gaussian process (GP). The parameters are learned from multiple tasks which are assumed to have been drawn independently from the same GP prior. An efficient algorithm is obtained by extending the informative vector machine (IVM) algorithm to handle the multi-task learning case. The multi-task IVM (MTIVM) saves computation by greedily selecting the most informative examples from the separate tasks. The MT-IVM is also shown to be more efficient than random sub-sampling on an artificial data-set and more effective than the traditional IVM in a speaker dependent phoneme recognition task.

Local distance preservation in the GP-LVM through back constraints

The Gaussian process latent variable model (GP-LVM) is a generative approach to nonlinear low dimensional embedding, that provides a smooth probabilistic mapping from latent to data space. It is also a non-linear generalization of probabilistic PCA (PPCA) (Tipping & Bishop, 1999). While most approaches to non-linear dimensionality methods focus on preserving local distances in data space, the GP-LVM focusses on exactly the opposite. Being a smooth mapping from latent to data space, it focusses on keeping things apart in latent space that are far apart in data space. In this paper we first provide an overview of dimensionality reduction techniques, placing the emphasis on the kind of distance relation preserved. We then show how the GP-LVM can be generalized, through back constraints, to additionally preserve local distances. We give illustrative experiments on common data sets.

Non-linear matrix factorization with Gaussian processes

A popular approach to collaborative filtering is matrix factorization. In this paper we develop a non-linear probabilistic matrix factorization using Gaussian process latent variable models. We use stochastic gradient descent (SGD) to optimize the model. SGD allows us to apply Gaussian processes to data sets with millions of observations without approximate methods. We apply our approach to benchmark movie recommender data sets. The results show better than previous state-of-the-art performance.

Kernels for Vector-Valued Functions: A Review

Kernel methods are among the most popular techniques in machine learning. From a regularization perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. From a probabilistic perspective they are the key in the context of Gaussian processes, where the kernel function is known as the covariance function. Traditionally, kernel methods have been used in supervised learning problems with scalar outputs and indeed there has been a considerable amount of work devoted to designing and learning kernels. More recently there has been an increasing interest in methods that deal with multiple outputs, motivated partially by frameworks like multitask learning. In this monograph, we review different methods to design or learn valid kernel functions for multiple outputs, paying particular attention to the connection between probabilistic and functional methods.

Gaussian process latent variable models for human pose estimation

We describe a method for recovering 3D human body pose from silhouettes. Our model is based on learning a latent space using the Gaussian Process Latent Variable Model (GP-LVM) [1] encapsulating both pose and silhouette features Our method is generative, this allows us to model the ambiguities of a silhouette representation in a principled way. We learn a dynamical model over the latent space which allows us to disambiguate between ambiguous silhouettes by temporal consistency. The model has only two free parameters and has several advantages over both regression approaches and other generative methods. In addition to the application shown in this paper the suggested model is easily extended to multiple observation spaces without constraints on type.

Hierarchical Gaussian process latent variable models

The Gaussian process latent variable model (GP-LVM) is a powerful approach for probabilistic modelling of high dimensional data through dimensional reduction. In this paper we extend the GP-LVM through hierarchies. A hierarchical model (such as a tree) allows us to express conditional independencies in the data as well as the manifold structure. We first introduce Gaussian process hierarchies through a simple dynamical model, we then extend the approach to a more complex hierarchy which is applied to the visualisation of human motion data sets.

Topologically-constrained latent variable models

In dimensionality reduction approaches, the data are typically embedded in a Euclidean latent space. However for some data sets this is inappropriate. For example, in human motion data we expect latent spaces that are cylindrical or a toroidal, that are poorly captured with a Euclidean space. In this paper, we present a range of approaches for embedding data in a non-Euclidean latent space. Our focus is the Gaussian Process latent variable model. In the context of human motion modeling this allows us to (a) learn models with interpretable latent directions enabling, for example, style/content separation, and (b) generalise beyond the data set enabling us to learn transitions between motion styles even though such transitions are not present in the data.

Probabilistic inference of transcription factor concentrations and gene-specific regulatory activities

MOTIVATION Quantitative estimation of the regulatory relationship between transcription factors and genes is a fundamental stepping stone when trying to develop models of cellular processes. Recent experimental high-throughput techniques, such as Chromatin Immunoprecipitation (ChIP) provide important information about the architecture of the regulatory networks in the cell. However, it is very difficult to measure the concentration levels of transcription factor proteins and determine their regulatory effect on gene transcription. It is therefore an important computational challenge to infer these quantities using gene expression data and network architecture data. RESULTS We develop a probabilistic state space model that allows genome-wide inference of both transcription factor protein concentrations and their effect on the transcription rates of each target gene from microarray data. We use variational inference techniques to learn the model parameters and perform posterior inference of protein concentrations and regulatory strengths. The probabilistic nature of the model also means that we can associate credibility intervals to our estimates, as well as providing a tool to detect which binding events lead to significant regulation. We demonstrate our model on artificial data and on two yeast datasets in which the network structure has previously been obtained using ChIP data. Predictions from our model are consistent with the underlying biology and offer novel quantitative insights into the regulatory structure of the yeast cell. AVAILABILITY MATLAB code is available from http://umber.sbs.man.ac.uk/resources/puma

Joint Modelling of Confounding Factors and Prominent Genetic Regulators Provides Increased Accuracy in Genetical Genomics Studies

Expression quantitative trait loci (eQTL) studies are an integral tool to investigate the genetic component of gene expression variation. A major challenge in the analysis of such studies are hidden confounding factors, such as unobserved covariates or unknown subtle environmental perturbations. These factors can induce a pronounced artifactual correlation structure in the expression profiles, which may create spurious false associations or mask real genetic association signals. Here, we report PANAMA (Probabilistic ANAlysis of genoMic dAta), a novel probabilistic model to account for confounding factors within an eQTL analysis. In contrast to previous methods, PANAMA learns hidden factors jointly with the effect of prominent genetic regulators. As a result, this new model can more accurately distinguish true genetic association signals from confounding variation. We applied our model and compared it to existing methods on different datasets and biological systems. PANAMA consistently performs better than alternative methods, and finds in particular substantially more trans regulators. Importantly, our approach not only identifies a greater number of associations, but also yields hits that are biologically more plausible and can be better reproduced between independent studies. A software implementation of PANAMA is freely available online at http://ml.sheffield.ac.uk/qtl/.