Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes

• URL: http://arxiv.org/abs/2008.02386v1
• Date: Wed, 5 Aug 2020 22:28:53 GMT
• Title: Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes
• Authors: Panagiotis Tsilifis, Piyush Pandita, Sayan Ghosh, Valeria Andreoli, Thomas Vandeputte, Liping Wang
• Abstract summary: "Projection" mapping consists of an orthonormal matrix that is considered a priori unknown and needs to be inferred jointly with the GP parameters. We extend the proposed framework to multi-fidelity models using GPs including the scenarios of training multiple outputs together. The benefits of our proposed framework, are illustrated on the computationally challenging three-dimensional aerodynamic optimization of a last-stage blade for an industrial gas turbine.
• Score: 3.564709604457361
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a Bayesian approach to identify optimal transformations that map model input points to low dimensional latent variables. The "projection" mapping consists of an orthonormal matrix that is considered a priori unknown and needs to be inferred jointly with the GP parameters, conditioned on the available training data. The proposed Bayesian inference scheme relies on a two-step iterative algorithm that samples from the marginal posteriors of the GP parameters and the projection matrix respectively, both using Markov Chain Monte Carlo (MCMC) sampling. In order to take into account the orthogonality constraints imposed on the orthonormal projection matrix, a Geodesic Monte Carlo sampling algorithm is employed, that is suitable for exploiting probability measures on manifolds. We extend the proposed framework to multi-fidelity models using GPs including the scenarios of training multiple outputs together. We validate our framework on three synthetic problems with a known lower-dimensional subspace. The benefits of our proposed framework, are illustrated on the computationally challenging three-dimensional aerodynamic optimization of a last-stage blade for an industrial gas turbine, where we study the effect of an 85-dimensional airfoil shape parameterization on two output quantities of interest, specifically on the aerodynamic efficiency and the degree of reaction.

Related papers

• Low-rank Bayesian matrix completion via geodesic Hamiltonian Monte Carlo on Stiefel manifolds [0.18416014644193066]
  We present a new sampling-based approach for enabling efficient computation of low-rank Bayesian matrix completion. We show that our approach resolves the sampling difficulties encountered by standard Gibbs samplers for the common two-matrix factorization used in matrix completion. Numerical examples demonstrate superior sampling performance, including better mixing and faster convergence to a stationary distribution.
  arXiv  Detail & Related papers  (2024-10-27T03:12:53Z)
• A Statistical Machine Learning Approach for Adapting Reduced-Order Models using Projected Gaussian Process [4.658371840624581]
  Proper Orthogonal Decomposition (POD) computes optimal basis modes that span a low-dimensional subspace where the Reduced-Order Models (ROMs) reside. This paper proposes a Projected Gaussian Process (pGP) and formulates the problem of adapting POD basis as a supervised statistical learning problem. Numerical examples are presented to demonstrate the advantages of the proposed pGP for adapting POD basis against parameter changes.
  arXiv  Detail & Related papers  (2024-10-18T00:02:43Z)
• von Mises Quasi-Processes for Bayesian Circular Regression [57.88921637944379]
  We explore a family of expressive and interpretable distributions over circle-valued random functions. The resulting probability model has connections with continuous spin models in statistical physics. For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Markov Chain Monte Carlo sampling.
  arXiv  Detail & Related papers  (2024-06-19T01:57:21Z)
• Regularized Projection Matrix Approximation with Applications to Community Detection [1.3761665705201904]
  This paper introduces a regularized projection matrix approximation framework designed to recover cluster information from the affinity matrix. We investigate three distinct penalty functions, each specifically tailored to address bounded, positive, and sparse scenarios. Numerical experiments conducted on both synthetic and real-world datasets reveal that our regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.
  arXiv  Detail & Related papers  (2024-05-26T15:18:22Z)
• Object based Bayesian full-waveform inversion for shear elastography [0.0]
  We develop a computational framework to quantify uncertainty in shear elastography imaging of anomalies in tissues. We find the posterior probability of parameter fields representing the geometry of the anomalies and their shear moduli. We demonstrate the approach on synthetic two dimensional tests with smooth and irregular shapes.
  arXiv  Detail & Related papers  (2023-05-11T08:25:25Z)
• Gaussian process regression and conditional Karhunen-Lo\'{e}ve models for data assimilation in inverse problems [68.8204255655161]
  We present a model inversion algorithm, CKLEMAP, for data assimilation and parameter estimation in partial differential equation models. The CKLEMAP method provides better scalability compared to the standard MAP method.
  arXiv  Detail & Related papers  (2023-01-26T18:14:12Z)
• Manifold Gaussian Variational Bayes on the Precision Matrix [70.44024861252554]
  We propose an optimization algorithm for Variational Inference (VI) in complex models. We develop an efficient algorithm for Gaussian Variational Inference whose updates satisfy the positive definite constraint on the variational covariance matrix. Due to its black-box nature, MGVBP stands as a ready-to-use solution for VI in complex models.
  arXiv  Detail & Related papers  (2022-10-26T10:12:31Z)
• Sparse high-dimensional linear regression with a partitioned empirical Bayes ECM algorithm [62.997667081978825]
  We propose a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are used through the use of plug-in empirical Bayes estimates. The proposed approach is implemented in the R package probe.
  arXiv  Detail & Related papers  (2022-09-16T19:15:50Z)
• Counting Phases and Faces Using Bayesian Thermodynamic Integration [77.34726150561087]
  We introduce a new approach to reconstruction of the thermodynamic functions and phase boundaries in two-parametric statistical mechanics systems. We use the proposed approach to accurately reconstruct the partition functions and phase diagrams of the Ising model and the exactly solvable non-equilibrium TASEP.
  arXiv  Detail & Related papers  (2022-05-18T17:11:23Z)
• Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems [0.0]
  A novel multiscale Bayesian inference approach is introduced based on deep probabilistic generative models. The method allows high-dimensional parameter estimation while exhibiting stability, efficiency and accuracy.
  arXiv  Detail & Related papers  (2021-02-04T11:47:21Z)
• Estimation of Switched Markov Polynomial NARX models [75.91002178647165]
  We identify a class of models for hybrid dynamical systems characterized by nonlinear autoregressive (NARX) components. The proposed approach is demonstrated on a SMNARX problem composed by three nonlinear sub-models with specific regressors.
  arXiv  Detail & Related papers  (2020-09-29T15:00:47Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.