Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gaussian Processes on Cellular Complexes

Mathieu Alain, So Takao, Brooks Paige, Marc Peter Deisenroth

TL;DR

Gaussian processes on cellular complexes are proposed, a generalisation of graphs that captures interactions between these higher-order cells and the derivation of two novel kernels, one that generalises the graph Mat\'ern kernel and one that additionally mixes information of different cell types.

Abstract

In recent years, there has been considerable interest in developing machine learning models on graphs to account for topological inductive biases. In particular, recent attention has been given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Matérn kernel and one that additionally mixes information of different cell types.

Gaussian Processes on Cellular Complexes

TL;DR

Gaussian processes on cellular complexes are proposed, a generalisation of graphs that captures interactions between these higher-order cells and the derivation of two novel kernels, one that generalises the graph Mat\'ern kernel and one that additionally mixes information of different cell types.

Abstract

In recent years, there has been considerable interest in developing machine learning models on graphs to account for topological inductive biases. In particular, recent attention has been given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Matérn kernel and one that additionally mixes information of different cell types.
Paper Structure (38 sections, 8 theorems, 93 equations, 15 figures, 5 tables)

This paper contains 38 sections, 8 theorems, 93 equations, 15 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Gaussian Processes
  3. Gaussian processes on graphs
  4. Modelling with Cellular Complexes
  5. Chains and Cochains
  6. Hodge Laplacian and Dirac Operator
  7. Numerical Representation
  8. Gaussian Processes on Cellular Complexes
  9. Gaussian Random Cochains
  10. GPs on Cellular Complexes
  11. Kernels on Cellular Complexes
  12. Matérn Kernel
  13. Reaction-diffusion Kernel
  14. Results
  15. Directed Edge Prediction
  16. ...and 23 more sections

Key Result

Theorem 8

A Gaussian random cochain $f : \Omega \rightarrow C^k(X)$ is fully characterised by a mean $\mu \in C^k(X)$ and a kernel $\kappa : C_k(X) \times C_k(X) \rightarrow \mathbb{R}$. Proof: Appendix app:grc-characterisation.

Figures (15)

  • Figure 1: Graph, simplicial complex, and cellular complex (specifically: polyhedral complex). A simplicial complex cannot represent arbitrary polygons like the pentagon in (\ref{['fig:cc']}).
  • Figure 2: A cellular complex is constructed by attaching boundaries of $k$-cells $e_\alpha^k$ to the $(k-1)$-skeleton $X^{k-1}$ via a continuous map $\phi^k_\alpha$.
  • Figure 3: The path going from vertex $v_0$ to $v_9$ (in red) can be expressed as a $1$-chain $c = e_1 + e_2 + e_5 + e_7 + e_9$, while $-c$ represents the reverse path going from $v_9$ to $v_0$. Boundaries of cells can be expressed as chains, whose direction is consistent with the cell orientation. For example, $\partial h_0 = e_1 + e_2 + e_5 - e_4 - e_3 - e_0$.
  • Figure 4: Probabilistic graphical structure of the reaction-diffusion GP. Interactions between vertices (green) and between edges (blue) are shown as well as the mixing between cochains of different orders (red). The cellular Matérn kernel does not have this mixing property.
  • Figure 5: Prediction of geostrophic current around the Southern tip of Africa using the CC-Matérn GP on edges. (Top left) Ground truth. (Top right) Predicted mean. (Bottom left) Absolute error. (Bottom right) Standard deviation. Orange dots are observed edges.
  • ...and 10 more figures

Theorems & Definitions (35)

  • Example 1
  • Example 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 8
  • Definition 9
  • Theorem 10
  • ...and 25 more