Zhenwen Dai

Mu Niu, Zhenwen Dai, Pokman Cheung et al.

This article presents a novel approach to construct Intrinsic Gaussian Processes for regression on unknown manifolds with probabilistic metrics (GPUM) in point clouds. In many real world applications, one often encounters high dimensional data (e.g. point cloud data) centred around some lower dimensional unknown manifolds. The geometry of manifold is in general different from the usual Euclidean geometry. Naively applying traditional smoothing methods such as Euclidean Gaussian Processes (GPs) to manifold valued data and so ignoring the geometry of the space can potentially lead to highly misleading predictions and inferences. A manifold embedded in a high dimensional Euclidean space can be well described by a probabilistic mapping function and the corresponding latent space. We investigate the geometrical structure of the unknown manifolds using the Bayesian Gaussian Processes latent variable models(BGPLVM) and Riemannian geometry. The distribution of the metric tensor is learned using BGPLVM. The boundary of the resulting manifold is defined based on the uncertainty quantification of the mapping. We use the the probabilistic metric tensor to simulate Brownian Motion paths on the unknown manifold. The heat kernel is estimated as the transition density of Brownian Motion and used as the covariance functions of GPUM. The applications of GPUM are illustrated in the simulation studies on the Swiss roll, high dimensional real datasets of WiFi signals and image data examples. Its performance is compared with the Graph Laplacian GP, Graph Matern GP and Euclidean GP.

Simon Damm, Dennis Forster, Dmytro Velychko et al.

The central objective function of a variational autoencoder (VAE) is its variational lower bound (the ELBO). Here we show that for standard (i.e., Gaussian) VAEs the ELBO converges to a value given by the sum of three entropies: the (negative) entropy of the prior distribution, the expected (negative) entropy of the observable distribution, and the average entropy of the variational distributions (the latter is already part of the ELBO). Our derived analytical results are exact and apply for small as well as for intricate deep networks for encoder and decoder. Furthermore, they apply for finitely and infinitely many data points and at any stationary point (including local maxima and saddle points). The result implies that the ELBO can for standard VAEs often be computed in closed-form at stationary points while the original ELBO requires numerical approximations of integrals. As a main contribution, we provide the proof that the ELBO for VAEs is at stationary points equal to entropy sums. Numerical experiments then show that the obtained analytical results are sufficiently precise also in those vicinities of stationary points that are reached in practice. Furthermore, we discuss how the novel entropy form of the ELBO can be used to analyze and understand learning behavior. More generally, we believe that our contributions can be useful for future theoretical and practical studies on VAE learning as they provide novel information on those points in parameters space that optimization of VAEs converges to.

Ke Ye, Mu Niu, Pokman Cheung et al.

Amidst the growing interest in nonparametric regression, we address a significant challenge in Gaussian processes(GP) applied to manifold-based predictors. Existing methods primarily focus on low dimensional constrained domains for heat kernel estimation, limiting their effectiveness in higher-dimensional manifolds. Our research proposes an intrinsic approach for constructing GP on general manifolds such as orthogonal groups, unitary groups, Stiefel manifolds and Grassmannian manifolds. Our methodology estimates the heat kernel by simulating Brownian motion sample paths using the exponential map, ensuring independence from the manifold's embedding. The introduction of our strip algorithm, tailored for manifolds with extra symmetries, and the ball algorithm, designed for arbitrary manifolds, constitutes our significant contribution. Both algorithms are rigorously substantiated through theoretical proofs and numerical testing, with the strip algorithm showcasing remarkable efficiency gains over traditional methods. This intrinsic approach delivers several key advantages, including applicability to high dimensional manifolds, eliminating the requirement for global parametrization or embedding. We demonstrate its practicality through regression case studies (torus knots and eight dimensional projective spaces) and by developing binary classifiers for real world datasets (gorilla skulls planar images and diffusion tensor images). These classifiers outperform traditional methods, particularly in limited data scenarios.

Mu Niu, Pokman Cheung, Lizhen Lin et al.

We propose a class of intrinsic Gaussian processes (in-GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregular shaped spaces arising as subsets or submanifolds of R, R2, R3 and beyond. For example, in-GPs can accommodate spatial domains arising as complex subsets of Euclidean space. in-GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the proposed approach is to utilise the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables in-GPs to be practically applied in great generality, while existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the in-GP approach is illustrated through simulation studies and data examples.

Mu Niu, Zhenwen Dai, Neil Lawrence et al.

Quantitative modeling of post-transcriptional regulation process is a challenging problem in systems biology. A mechanical model of the regulatory process needs to be able to describe the available spatio-temporal protein concentration and mRNA expression data and recover the continuous spatio-temporal fields. Rigorous methods are required to identify model parameters. A promising approach to deal with these difficulties is proposed using Gaussian process as a prior distribution over the latent function of protein concentration and mRNA expression. In this study, we consider a partial differential equation mechanical model with differential operators and latent function. Since the operators at stake are linear, the information from the physical model can be encoded into the kernel function. Hybrid Monte Carlo methods are employed to carry out Bayesian inference of the partial differential equation parameters and Gaussian process kernel parameters. The spatio-temporal field of protein concentration and mRNA expression are reconstructed without explicitly solving the partial differential equation.