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Title: | Deep Bayesian Supervised Learning given Hypercuboidally-shaped, Discontinuous Data, using Compound Tensor-Variate & Scalar-Variate Gaussian Processes |
Authors: | Wang, K Chakrabarty, D |
Keywords: | compound tensor-variate scalar-variate GPs;covariance kernel parametrisation;Lipschitz continuity;deep learning |
Issue Date: | 16-Apr-2018 |
Publisher: | Cornell University |
Citation: | Wang, K. and Chakrabarty, D. (2018) 'Deep Bayesian Supervised Learning given Hypercuboidally-shaped, Discontinuous Data, using Compound Tensor-Variate & Scalar-Variate Gaussian Processes', arXiv:1803.04582v2 [stat.ME], pp. 1 - 43. doi: 10.48550/arXiv.1803.04582. |
Abstract: | We undertake Bayesian learning of the high-dimensional functional relationship between a system parameter vector and an observable, that is in general tensor-valued. The ultimate aim is Bayesian inverse prediction of the system parameters, at which test data is recorded. We attempt such learning given hypercuboidally-shaped data that displays strong discontinuities, rendering learning challenging. We model the sought high-dimensional function, with a tensor-variate Gaussian Process (GP), and use three independent ways for learning covariance matrices of the resulting likelihood, which is Tensor-Normal. We demonstrate that the discontinuous data demands that implemented covariance kernels be non-stationary--achieved by modelling each kernel hyperparameter, as a function of the sample function of the invoked tensor-variate GP. Each such function can be shown to be temporally-evolving, and treated as a realisation from a distinct scalar-variate GP, with covariance described adaptively by collating information from a historical set of samples of chosen sample-size. We prove that deep-learning using 2-"layers", suffice, where the outer-layer comprises the tensor-variate GP, compounded with multiple scalar-variate GPs in the "inner-layer", and undertake inference with Metropolis-within-Gibbs. We apply our method to a cuboidally-shaped, discontinuous, real dataset, and subsequently perform forward prediction to generate data from our model, given our results--to perform model-checking. |
URI: | https://bura.brunel.ac.uk/handle/2438/20105 |
DOI: | https://doi.org/10.48550/arXiv.1803.04582 |
Other Identifiers: | https://arxiv.org/abs/1803.04582v2 ORCiD: Dalia Chakrabarty https://orcid.org/0000-0003-1246-4235 |
Appears in Collections: | Dept of Mathematics Research Papers |
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