Semi-described and semi-supervised learning with Gaussian processes

Propagating input uncertainty through non-linear Gaussian process (GP) mappings is intractable. This hinders the task of training GPs using uncertain and partially observed inputs. In this paper we refer to this task as "semi-described learning". We then introduce a GP framework that solves both, the semi-described and the semi-supervised learning problems (where missing values occur in the outputs). Auto-regressive state space simulation is also recognised as a special case of semi-described learning. To achieve our goal we develop variational methods for handling semi-described inputs in GPs, and couple them with algorithms that allow for imputing the missing values while treating the uncertainty in a principled, Bayesian manner. Extensive experiments on simulated and real-world data study the problems of iterative forecasting and regression/classification with missing values. The results suggest that the principled propagation of uncertainty stemming from our framework can significantly improve performance in these tasks

Decomposing feature-level variation with Covariate Gaussian Process Latent Variable Models

The interpretation of complex high-dimensional data typically requires the use of dimensionality reduction techniques to extract explanatory low-dimensional representations. However, in many real-world problems these representations may not be sufficient to aid interpretation on their own, and it would be desirable to interpret the model in terms of the original features themselves. Our goal is to characterise how feature-level variation depends on latent low-dimensional representations, external covariates, and non-linear interactions between the two. In this paper, we propose to achieve this through a structured kernel decomposition in a hybrid Gaussian Process model which we call the Covariate Gaussian Process Latent Variable Model (c-GPLVM). We demonstrate the utility of our model on simulated examples and applications in disease progression modelling from high-dimensional gene expression data in the presence of additional phenotypes. In each setting we show how the c-GPLVM can extract low-dimensional structures from high-dimensional data sets whilst allowing a breakdown of feature-level variability that is not present in other commonly used dimensionality reduction approaches

DeepCoder: Semi-parametric Variational Autoencoders for Automatic Facial Action Coding

Human face exhibits an inherent hierarchy in its representations (i.e., holistic facial expressions can be encoded via a set of facial action units (AUs) and their intensity). Variational (deep) auto-encoders (VAE) have shown great results in unsupervised extraction of hierarchical latent representations from large amounts of image data, while being robust to noise and other undesired artifacts. Potentially, this makes VAEs a suitable approach for learning facial features for AU intensity estimation. Yet, most existing VAE-based methods apply classifiers learned separately from the encoded features. By contrast, the non-parametric (probabilistic) approaches, such as Gaussian Processes (GPs), typically outperform their parametric counterparts, but cannot deal easily with large amounts of data. To this end, we propose a novel VAE semi-parametric modeling framework, named DeepCoder, which combines the modeling power of parametric (convolutional) and nonparametric (ordinal GPs) VAEs, for joint learning of (1) latent representations at multiple levels in a task hierarchy1, and (2) classification of multiple ordinal outputs. We show on benchmark datasets for AU intensity estimation that the proposed DeepCoder outperforms the state-of-the-art approaches, and related VAEs and deep learning models.Comment: ICCV 2017 - accepte

Weakly-supervised Multi-output Regression via Correlated Gaussian Processes

Multi-output regression seeks to infer multiple latent functions using data from multiple groups/sources while accounting for potential between-group similarities. In this paper, we consider multi-output regression under a weakly-supervised setting where a subset of data points from multiple groups are unlabeled. We use dependent Gaussian processes for multiple outputs constructed by convolutions with shared latent processes. We introduce hyperpriors for the multinomial probabilities of the unobserved labels and optimize the hyperparameters which we show improves estimation. We derive two variational bounds: (i) a modified variational bound for fast and stable convergence in model inference, (ii) a scalable variational bound that is amenable to stochastic optimization. We use experiments on synthetic and real-world data to show that the proposed model outperforms state-of-the-art models with more accurate estimation of multiple latent functions and unobserved labels

Continual Multi-task Gaussian Processes

We address the problem of continual learning in multi-task Gaussian process (GP) models for handling sequential input-output observations. Our approach extends the existing prior-posterior recursion of online Bayesian inference, i.e.\ past posterior discoveries become future prior beliefs, to the infinite functional space setting of GP. For a reason of scalability, we introduce variational inference together with an sparse approximation based on inducing inputs. As a consequence, we obtain tractable continual lower-bounds where two novel Kullback-Leibler (KL) divergences intervene in a natural way. The key technical property of our method is the recursive reconstruction of conditional GP priors conditioned on the variational parameters learned so far. To achieve this goal, we introduce a novel factorization of past variational distributions, where the predictive GP equation propagates the posterior uncertainty forward. We then demonstrate that it is possible to derive GP models over many types of sequential observations, either discrete or continuous and amenable to stochastic optimization. The continual inference approach is also applicable to scenarios where potential multi-channel or heterogeneous observations might appear. Extensive experiments demonstrate that the method is fully scalable, shows a reliable performance and is robust to uncertainty error propagation over a plenty of synthetic and real-world datasets