Transformations of Gaussian Process Priors

Gaussian process prior systems generally consist of noisy measurements of samples of the putatively Gaussian process of interest, where the samples serve to constrain the posterior estimate. Here we consider the case where the measurements are instead noisy weighted sums of samples. This framework incorporates measurements of derivative information and of filtered versions of the process, thereby allowing GPs to perform sensor fusion and tomography; allows certain group invariances (ie symmetries) to be weakly enforced; and under certain conditions suitable application allows the dataset to be dramatically reduced in size. The method is applied to a sparsely sampled image, where each sample is taken using a broad and non-monotonic point spread function. It is also applied to nonlinear dynamic system identification applications where a nonlinear function is followed by a known linear dynamic system, and where observed data can be a mixture of irregularly sampled higher derivatives of the signal of interest.

[1]  A. O'Hagan,et al.  Curve Fitting and Optimal Design for Prediction , 1978 .

[2]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[3]  Per Christian Hansen,et al.  Rank-Deficient and Discrete Ill-Posed Problems , 1996 .

[4]  Geoffrey E. Hinton,et al.  Evaluation of Gaussian processes and other methods for non-linear regression , 1997 .

[5]  Christopher K. I. Williams.  Prediction with Gaussian Processes: From Linear Regression to Linear Prediction and Beyond , 1999, Learning in Graphical Models.

[6]  Christopher K. I. Williams.  Computation with Infinite Neural Networks , 1998, Neural Computation.

[7]  W. Leithead,et al.  Analytic framework for blended multiple model systems using linear local models , 1999 .

[8]  Volker Tresp,et al.  A Bayesian Committee Machine , 2000, Neural Computation.

[9]  Christopher K. I. Williams,et al.  Using the Nyström Method to Speed Up Kernel Machines , 2000, NIPS.

[10]  B. Silverman,et al.  Functional Data Analysis , 1997 .

[11]  Daniel Sbarbaro,et al.  Nonlinear adaptive control using non-parametric Gaussian Process prior models , 2002 .

[12]  Carl E. Rasmussen,et al.  Derivative Observations in Gaussian Process Models of Dynamic Systems , 2002, NIPS.

[13]  Christopher M. Bishop,et al.  Bayesian Image Super-Resolution , 2002, NIPS.

[14]  Neil D. Lawrence,et al.  Fast Forward Selection to Speed Up Sparse Gaussian Process Regression , 2003, AISTATS.

[15]  Barak A. Pearlmutter,et al.  Filtered Gaussian Processes for Learning with Large Data-Sets , 2003, European Summer School on Multi-AgentControl.

[16]  Roderick Murray-Smith,et al.  Hierarchical Gaussian process mixtures for regression , 2005, Stat. Comput..

[17]  Roderick Murray-Smith,et al.  Learning with large data sets using filtered {G}aussian Process priors , 2005 .