This work proposes a scalable probabilistic latent variable model based on Gaussian processes (Lawrence, 2004) in the context of multiple observation spaces. We focus on an application in astrophysics where data sets typically contain both observed spectral features and scientific properties of astrophysical objects such as galaxies or exoplanets. In our application, we study the spectra of very luminous galaxies known as quasars, along with their properties, such as the mass of their central supermassive black hole, accretion rate, and luminosity-resulting in multiple observation spaces. A single data point is then characterized by different classes of observations, each with different likelihoods. Our proposed model extends the baseline stochastic variational Gaussian process latent variable model (GPLVM) introduced by Lalchand et al. (2022) to this setting, proposing a seamless generative model where the quasar spectra and scientific labels can be generated simultaneously using a shared latent space as input to different sets of Gaussian process decoders, one for each observation space. Additionally, this framework enables training in a missing data setting where a large number of dimensions per data point may be unknown or unobserved. We demonstrate high-fidelity reconstructions of the spectra and scientific labels during test-time inference and briefly discuss the scientific interpretations of the results, along with the significance of such a generative model.

A BSTRACT The Dynamical Gaussian Process Latent V ariable Models provide an elegant nonparametric framework for learning the low dimensional representations of the high-dimensional time-series. Real world observational studies, however, are often ill-conditioned: the observations can be noisy, not assuming the luxury of relatively complete and equally spaced like those in time series. Such conditions make it difficult to learn reasonable representations in the high dimensional longitudinal data set by way of Gaussian Process Latent V ariable Model as well as other dimensionality reduction procedures. In this study, we approach the inference of Gaussian Process Dynamical Systems in Longitudinal scenario by augmenting the bound in the variational approximation to include systematic samples of the unseen observations. We demonstrate the usefulness of this approach on synthetic as well as the human motion capture data set. 1 I NTRODUCTION While it isn't trivial to find an unified definition of multivariate longitudinal data; the one definition being alluded to in Pullenayegum & Lim (2016) is the type of data being discussed in this work. Longitudinal designs track a repeated set of variables in experimental subjects over periods of time; however, unlike time series, which are often characterized by regular intervals of n-dimensional observations, longitudinal setups present inconsistent sampling frequencies and only a small subset of variables may be observed at any given time.

A mixture of Gaussians fit to a single curved or heavy-tailed cluster will report that the data contains many clusters. To produce more appropriate clusterings, we introduce a model which warps a latent mixture of Gaussians to produce nonparametric cluster shapes. The possibly low-dimensional latent mixture model allows us to summarize the properties of the high-dimensional clusters (or density manifolds) describing the data. The number of manifolds, as well as the shape and dimension of each manifold is automatically inferred. We derive a simple inference scheme for this model which analytically integrates out both the mixture parameters and the warping function. We show that our model is effective for density estimation, performs better than infinite Gaussian mixture models at recovering the true number of clusters, and produces interpretable summaries of high-dimensional datasets.

Density modeling is notoriously difficult for high dimensional data. One approach to the problem is to search for a lower dimensional manifold which captures the main characteristics of the data. Recently, the Gaussian Process Latent Variable Model (GPLVM) has successfully been used to find low dimensional manifolds in a variety of complex data. The GPLVM consists of a set of points in a low dimensional latent space, and a stochastic map to the observed space. We show how it can be interpreted as a density model in the observed space. However, the GPLVM is not trained as a density model and therefore yields bad density estimates. We propose a new training strategy and obtain improved generalisation performance and better density estimates in comparative evaluations on several benchmark data sets.