GP SSM

This page gathers references and materials related to the study of

• Gaussian Process (GP) State Space Models (SSM)
• Deep Learning
• Koopman Spectral Methods.

Gaussian Process Approaches

Basic Gaussian Process Info

• Rasmussen and Williams

Web Links

• Bibliography on GP Models in Dynamical Systems

Papers on GP-SSMs

• J.M. Wang, D.J. Fleet, A. Hertzmann, Gaussian Process Dynamical Models
• R. Turner, M.P. Deisenroth, C.E. Rasmussen, State-Space Inference and Learning with Gaussian Process;
• A. McHutchon, Nonlinear Modelling and Control Using Gaussian Processes (Ph.D. thesis, Cambridge University)
• J. Ko, D. Fox, GP-BayesFilters: Bayesian filtering using Gaussian Process Prediction and Observation Models
• F. Perez-Cruz, S.V. Vaerenbergh, J.J. Murrillo-Fuentes, M. Lazarro-Gredilla, and I. Santamaria, Gaussian Processes for Nonlinear Signal Processing;
• A. Svensson, A. Solin, S. Sarkka, T.B. Schon, Computationall Efficient Bayesian Learning of Gaussian Process State Space Models
• A.C. Damianou, M.K. Titsias, N.D. Lawrence, Variational Gaussian Process Dynamical Systems
• M.P. Deisenroth, D. Fox, C.E. Rasmussen, Gaussian Processes for Data-Efficient Learning in Robotics and Control;
• K. Jocikan, Dynamic GP Models: An Overview and Recent Developments;
• A. Solin, S. Sarkka, Hilbert Space Methods for Reduced-Rank Gaussian Process Regression; (ArXiv.1401.5508)
• C.L.C. Mattos, Z. Dai, A. Damianou, J. Forth, G.A. Barreto, N. Lawrence, Recorruent Gaussian Processes
• N.D. Lawrence, A.J. Moore, Hierarchical Gaussian Process Latent Variable Models
• M.K. Titsias, N.D. Lawrence, Bayesian Gaussian Process Latent Variable Model
• R. Calandra, J. Peters, C.E. Rasmussen, M.P. Deisenroth, Manifold Gaussian Processes for Regression
• F. Berkenkamp and A.P. Schoellig, Safe and Robust Learning Control with Gaussian Processes
• E.B. Fox, E.B. Sudderth, M.I. Jordan, A.S. Willsky, Sharing Features Among Dynamical Systems with Beta Processes
• J.M. Wang, D.j. Fleet, A. Hertzmann, Gaussian Process Dynamical Models for Human Motion
• E.D. Klenske, P. Hennig, Dual Control for Approximate Bayesian Reinforcement Learning
• Y. Pan and E.A. Theodorou, Data-Driven Differential Dynamic Programming Using Gaussian Processes
• F. Berkenkamp, R. Moriconi, A.P. Schoellig, A. Krause, Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes
• M.P. Deisenroth, J. Peters, C.E. Rasmussen, Approximate Dynamic Programming with Gaussian Processes
• R. Frigola, F. Lindsten, T.B. Schon, C.E. Rasmussen, Identification of Gaussian Process State-Space Models with Particle Stochastic Approximation EM

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• Z.Y. Wan and T.P. Sapsis, Reduced-Space Gaussian Process Regression for Data-Driven Forecast of Chaotic Dynamical Systems, arXiv:1611.01583
• T. Beckers, J. Umlauft, and S. Hirsche, Stable Model-Based Control with Gaussian Process Regression for Robot Manipulators,
• A. Marco, P. Hennig, S. Schaal, S. Trimpe, On the Design of LQR Kernels for Efficient Controller Learning, arXiv:1709.07089v1
• N. Gorbach, S. Bauer, J. Buhmann, Scalable Variational Inference for Dynamical Systems, NIPS 2017, Long Beach, CA, 2017.
• J. Umlauft, T. Beckers, M. Kimmel, S. Hirsche, Feedback Linearization Using Gaussian Processes
• F. Lindsten, M.I. Jordan, T.B. Schon, Particles Gibbs with Ancestor Sampling, J. Machine Learning Research, vo. 15, pp. 2145-2184.

Deep Learning

Papers on Deep Learning Theory

• This is the paper that we seek to understand in this reading group.
  • A. Achille and S. Soatto, Emergence of Invariance and Disentanglement in Deep Representations, arXiV:1706.01350v2, Oct. 2017.
• Here are some background papers (i.e., links to many of the references in the Achille and Soatto paper)
  • A. Achille and S. Soatto Information Dropout: Learning Optimal Representations Through Noisy Computation
  • A. Alemi, I. Fischer, K. Dillon, and K. Murphey, Deep Variational Information Bottleneck, arXiV:1612.00410
  • F. Anselmi, L. Rosasco, T. Poggio, On Invariance and Selectivity in Representation Learning
  • Y. Bengio Learning Deep Architectures for AI, 2009.
  • J. Bruna and S. Mallat, Classification with Scattering Operators, CVPR, 2011.
  • P. Chaudhari, A. Choromanska, S. Soatto, Y. LeCun, C. Baldassi, C. Borgs, J. Chayes, L. Sagun, R. Zecchina Entropy-SGD: Biasing Gradient Descent into Wide Valleys, Proceedings of the International Conference on Learning Representations (ICLR), 2017.
  • N. Tishby and N. Zaslavsky, Deep Learning and the Information Bottleneck Principle, arXiv:1503.02406, March, 2015.
  • D.-A. Clevert, T. Unterthiner, and S. Hochreiter, Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs), arXiv:1511.07289.
  • L. Dinh, R. Pascanu, S. Bengio, Y. Bengio Sharp Minima Can Generalize For Deep Nets, arXiv:1703.04933

• P. Chauderi, A.Oberman, S. Osher, S. Soatto, G. Ccarlier, Deep Relaxation: Partial Differential Equations for Optimizing Deep Neural Networks, arXiV:1704.04932v2

Review-like papers

• S. Mallat, Understanding Deep Convolutional Networks, Phil. Trans. Royal Society, A, vol. 374, May 15, 2017.
• M.D. Zeiler and R. Fergus, Visualizing and Understanding Deep Convolutional Networks

Papers on Scattering Networks

Scattering Networks are proposed by Stephan Mallat (of wavelet fame) to understand why deep nets work so well.

• S. Mallat Group Invariant Scattering, 2012.
• J. Anden, S. Mallat, Deep Scattering Spectrum, 2015.
• J. Bruna, S. Mallat, Invariant Scattering Convolution Networks

Classics

• K. Hornik, M. Stinchcombe, H. White, Universal Approximation of an Unknown Mapping and Its Derivatives Using Multi-Layer Feedforward Networks, Neural networks, vol. 3, 1990.

Web Links

• Deeplearning.net; On-line Deep Learning TextBook

Koopman Spectral Method

Papers on Koopman Spectral methods

• S. Brunton, J. Proctor, N. Kutz, Discovering Governing Equations from Data: Sparse Identification of Nonlinear Dyanmical Systems, arXiv:1509.03580v1 (also, PNAS Version of the paper).
• S. L. Brunton, B.W.Brunton, J.L. Proctor, J.N. Kutz, Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control, PLOS One, vol. 11, no. 2, 2016.
• M. Budisic, R. Mohr, I. Mezic, Applied Koopmanism, Chaos, vol. 22, 2012.
• M.O. Williams, C.W. Rowley, I. G. Kevrekidis, A Kernel-Based Method for Data-Driven Koopman Spectral Analysis, arXiv:1411.2260v4
• J.L. Proctor, S.L. Brunton, J.N. Kutz, Dynamic Mode Decomposition with Control, SIAM J. Applied Dynamical Systems, vol. 15, no. 1, pp. 142-161, 2016.
• I. Mezic, On the Applications of the Theory of the Koopman Operator in Dynamical Systems and Control Theory, Proc. IEEE Conf. Decision Control, 2015
• M.O. Williams, C. Rowley, I.G. Kevrekidis, A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition, J. Nonlinear Science, vol. 25, 2015.
• D. Giannakis, Data-Driven Spectral Decomposition and Forecasting of ergodic dynamical systems, arXiv:1507.02338v2
• I. Mezic and A. Surana, Koopman Mode Decomposition for Periodic/Quasi-Periodic Time Dependence, IFAC Papers Online, 48-18, pp. 690697, 2016.
• H. Schaeffer, Learning Partial Differential Equations via Data Discovery and Sparse Optimization, J. Royal Society, Proceedings A, 2017
• J.N. Kutz, X. Fu, S.L. Brunton, Multi-resolution Dynamic Mode Decomposition, arXiv:1506.00564

Papers which are more oriented toward control

• D. Goswami and D.A. Paley, Global Bilinearization and Controllability of Control-Affine Nonlinear Systems: A Koopman Spectral Approach,
• A. Surana, Koopman Based Operator Synthesis for Control-Affine Noninear Systems, Proc. IEEE Conf. Decision and Control, 2016
• A. Surana and A. Banaszuk, Linear Observer Synthesis for Nonlinear Systems Using Koopman Operator Framework, IFAC Papers Online, 49-18, pp. 716-723, 2016.

Papers which are more oriented toward fluids

• I. Mezic, Analysis of Fluid Flows via Spectral Properties of the Koopman Operator, Annual Review of Fluids, vol. 45, 357-378, 2013.
• C.W. Rowley and S.T.M. Dawson, Model Reduction for Flow Analysis and Control, Annual Review Fluids, 49:387-417, 2017.
• M.S Hemati, C.W. Rowley, E.A. Deem, L.N. Cattafesta, De-biasing the Dynamic Mode Decomposition for applied Koopman spectral analysis of noisy data sets, arXiv:1502.03854v2
• J.H. Tu, Dynamic Mode Decomposition, Theory and Applications, Ph.D. Thesis, Princeton, 2013.
• S Bagheri, Analysis and Control of Transitional Shear Flows Using Global Modes, Ph.D. Thesis, Royal Inst. Technology, Sweden, 2010.

Some Early Papers

• Koopman's original paper: Dynamical Systems of Continuous Spectra, PNAS , vol. 18, 1932.
• J. Ding, The Point Spectrum of Frobenius-Perron and Koopman Operators, Proc. AMS, vol. 126, no. 5, 1998.
• I. Mezic and A. Banaszuk, Comparison of Systems with Complex Behavior, Physica D, vol. 197, pp. 101-133, 2004.
• Y. Lan and I. Mezic, Linearization in the Large of Nonlinear Systems and Koopman Operator Spectrum, Physica D, 2013

Other Papers

• E.Berger, M. Sastuba, D. Vogt, B. Jung, H.B. Amor, Estimation of Perturbations in Robotic Behavior using Dynamic Mode Decomposition, Advanced Robotics, vol. 25, no. 5, 2015.
• S. Wang, Z. Qiao, Nuclear Norm Regularized Dynamic Mode Decomposition, IET Signal Processing, 2016.