Derivative Observations in Gaussian Process Models of Dynamic Systems

By Ercan Solak, Roderick Murray-Smith, William Edward Leithead, Douglas J. Leith, Carl E. Rasmussen. Neural Information Processing Systems, 2002

373 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/48a5c48e2aa8e1613501bb6001ae893a7c85f47f

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Citations in the corpus

Accelerating Bayesian Inference over Nonlinear Differential Equations with Gaussian Processes (2008, background): “where C ′′ φn denotes the auto- covariance for each state- derivative with C ′ φn and Cφn denoting the cross- covariances between the state and its derivative [13, 15].”
Transformations of Gaussian Process Priors (2004, methodology): “Solak et al. (2003) used the fact that the derivative of a Gaussian process is itself a Gaussian process to integrate function a nd derivative observations.”
Gaussian Process Dynamical Models (2005, background): “” Higher-order features can also be fused together with position information to reduce the Gaussian process prediction variance [15, 9].”
Many-Body Coarse-Grained Interactions Using Gaussian Approximation Potentials. (2016, unknown): “We can also train the Gaussian process from derivative observations [122, 123].”
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning) (2005, unknown): “11)), as shown in Sollich [1999] or Opper and Vivarelli [1999].”
Gaussian Processes for Regression and Optimisation (2007, methodology): “Interesting machine learning applications include Gaussian processes for reinforcement learning [60], the incorporation of derivative observations into Gaussian process models [73], Gaussian processes to speed up the evaluation of Bayesian integrals [58], and Gaussian process models of dynamical sys”
Approximate real-time optimal control based on sparse Gaussian process models (2014, background): “Since taking a derivative is a linear operation, the derivative of a GP is another, dependent, GP [17].”
Gaussian Process Training with Input Noise (2011, background): “If we expand up to the first order terms we get a linear model for the input noise, y = f(x) + Tx∂ f̄ + y (5) The probability of an observation y is therefore, P (y | f) = N (f, σ2y + ∂ T f̄ Σx ∂ f̄ ) (6) We keep the usual Gaussian Process prior, P (f | X) = N (0,K(X,X)), where K(X,X) is the N by N ”
Avoiding pathologies in very deep networks (2014, background): “input dimensions d1 and d2 of vector x are given by Solak et al. (2003):”
Gradient-based Hyperparameter Optimization through Reversible Learning (2015, background): “For example, Gaussian-process-based optimization methods could incorporate gradient information (Solak et al., 2003).”
Bayesian Solution Uncertainty Quantification for Differential Equations (2013, background): “GPs are a ﬂexible class of stochastic processes that can be deﬁned jointly with their derivatives in a consistent way through the speciﬁcation of an appropriate covariance operator (Solak et al., 2003).”
Predictive Entropy Search for Efficient Global Optimization of Black-box Functions (2014, background): “This initial cost is dominated by a matrix inversion necessary to pre-compute V for each hyperparameter sample”
Gaussian process model based predictive control (2004, methodology): “[14], [13], [3], [20], [9], [10] and some control algorithms based on such are described in [15], [4].”
Universal Multi-Task Kernels (2008, background): “Our next example called Hessian of Gaussian is motivated by the problem of learning gradients (Solak et al., 2002; Mukherjee and Zhou, 2006).”