# GP SSM

From Robotics

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

- Gaussian Processes For Machine Learning, by Rasmussen and Williams
- Modeling and Control of Dynamic Systems Processes Using Gaussian Processes Models by J. Kocijan

### Web Links

### 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

- 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.

- A. Achille and S. Soatto, Emergence of Invariance and Disentanglement in Deep Representations,
- 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

- J. Schmidhuber, Deep Learning in Neural Networks: An Overview,
*arXiV:404.7828v4*, Oct. 2014. - 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

## 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. - E. Kaiser, N. Kutz, and S. Brunton, Sparse identification of nonlinear dynamics for model predictive control in the low-data limit
*arXiv preprint*arXiv:1711.05501 (2017).

#### 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.