R:SS Workshop on Geometry and Topology in Robotics

Gaussian Processes

on Riemannian Manifolds for Robotics

Alexander Terenin and Viacheslav Borovitskiy

Gaussian Processes

Gaussian Processes

Definition. A Gaussian process is random function $f : X \to \R$ such that for any $x_1,..,x_n$, the vector $f(x_1),..,f(x_n)$ is multivariate Gaussian.

Every GP is characterized by a mean $\mu(\.)$ and a kernel $k(\.,\.)$. We have $$ \htmlClass{fragment}{ f(\v{x}) \~ \f{N}(\v{\mu}_{\v{x}},\m{K}_{\v{x}\v{x}}) } $$ where $\v\mu_{\v{x}} = \mu(\v{x})$ and $\m{K}_{\v{x}\v{x}'} = k(\v{x},\v{x}')$.

Bayesian learning: $f \given \v{y}$.

Bayesian Optimization

Goal: minimize unknown function $\phi$ in as few evaluations as possible.

  1. Build GP posterior $f \given \v{y}$ using data $(x_1,\phi(x_1)),..,(x_n,\phi(x_n))$.
  2. Choose $$ \htmlClass{fragment}{ x_{n+1} = \argmax_{x\in\c{X}} \alpha_{f\given\v{y}}(x) } $$ to maximize the acquisition function $\alpha_{f\given\v{y}}$, for instance expected improvement $\alpha_{f\given\v{y}} = \E_{f\given\v{y}} \max(0, {\displaystyle\min_{i=1,..,n}} \phi(x_n) - f(x))$.

Automatic explore-exploit tradeoff.

Bayesian Optimization

Modeling Dynamical Systems with Uncertainty

$$ \htmlData{fragment-index=0,class=fragment}{ x_0 } \qquad \htmlData{fragment-index=1,class=fragment}{ x_1 = x_0 + f(x_0)\Delta t } \qquad \htmlData{fragment-index=2,class=fragment}{ x_2 = x_1 + f(x_1)\Delta t } \qquad \htmlData{fragment-index=3,class=fragment}{ .. } $$

Model-based Reinforcement Learning


Video from Deisenroth and Rasmussen (2011)

Geometry-aware Gaussian Processes

Matérn Gaussian Processes

$$ \htmlData{class=fragment fade-out,fragment-index=9}{ \footnotesize \mathclap{ k_\nu(x,x') = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)} \del{\sqrt{2\nu} \frac{\norm{x-x'}}{\kappa}}^\nu K_\nu \del{\sqrt{2\nu} \frac{\norm{x-x'}}{\kappa}} } } \htmlData{class=fragment d-print-none,fragment-index=9}{ \footnotesize \mathclap{ k_\infty(x,x') = \sigma^2 \exp\del{-\frac{\norm{x-x'}^2}{2\kappa^2}} } } $$ $\sigma^2$: variance $\kappa$: length scale $\nu$: smoothness
$\nu\to\infty$: recovers squared exponential kernel

$\nu = 1/2$

$\nu = 3/2$

$\nu = 5/2$

$\nu = \infty$

Riemannian Geometry

How should Matérn kernels generalize to this setting?

Geodesics

$$ k_\infty^{(d_g)}(x,x') = \sigma^2\exp\del{-\frac{d_g(x,x')^2}{2\kappa^2}} $$

Theorem. (Feragen et al.) Let $M$ be a complete Riemannian manifold without boundary. If $k_\infty^{(d_g)}$ is positive semi-definite for all $\kappa$, then $M$ is isometric to a Euclidean space.

Need a different candidate generalization


Feragen et al. (2015)

Stochastic Partial Differential Equations

$$ \htmlData{class=fragment,fragment-index=0}{ \underset{\t{Matérn}}{\undergroup{\del{\frac{2\nu}{\kappa^2} - \Delta}^{\frac{\nu}{2}+\frac{d}{4}} f = \c{W}}} } \qquad \htmlData{class=fragment,fragment-index=1}{ \underset{\t{squared exponential}}{\undergroup{\vphantom{\del{\frac{2\nu}{\kappa^2} - \Delta}^{\frac{\nu}{2}+\frac{d}{4}}} e^{-\frac{\kappa^2}{4}\Delta} f = \c{W}}} } $$ $\Delta$: Laplacian $\c{W}$: (rescaled) white noise

Generalizes well to the Riemannian setting

Whittle (1963)
Lindgren et al. (2011)

Riemannian Matérn Kernels: compact spaces

$$ k_\nu(x,x') = \frac{\sigma^2}{C_\nu} \sum_{n=0}^\infty \del{\frac{2\nu}{\kappa^2} - \lambda_n}^{\nu-\frac{d}{2}} f_n(x) f_n(x') $$

$\lambda_n, f_n$: Laplace–Beltrami eigenpairs

Analytic expressions for circle, sphere, ..

Riemannian Matérn Kernels

$k_{1/2}(\htmlStyle{color:rgb(255, 19, 0)!important}{\bullet},\.)$

Example: regression on the surface of a dragon

(a) Ground truth

(b) Posterior mean

(c) Standard deviation

(d) Posterior sample

Applications

Pendulum Dynamics: GP on a cylinder

(a) Ground truth

(b) Gaussian process

Bayesian Optimization in Robotics

Orientation: sphere
 

Manipulability:
SPD manifold

Joint postures: torus
 


Jaquier et al. (2020)

Matérn Gaussian Processes

 on Riemannian Manifolds

Viacheslav*
Borovitskiy

Alexander*
Terenin

Peter*
Mostowsky

Marc Peter
Deisenroth

*equal contribution

Matérn Gaussian Processes on Graphs

Viacheslav*
Borovitskiy

Iskander*
Azangulov

Alexander*
Terenin

Peter*
Mostowsky

Marc Peter
Deisenroth

Nicolas
Durrande

*equal contribution

Thank you!

Thank you!

V. Borovitskiy, I. Azangulov, A. Terenin, P. Mostowsky, M. P. Deisenroth. Matérn Gaussian Processes on Graphs. Artificial Intelligence and Statistics, 2021.

V. Borovitskiy, A. Terenin, P. Mostowsky, M. P. Deisenroth. Matérn Gaussian Processes on Riemannian Manifolds. Advances in Neural Information Processing Systems, 2020.