Hodge-Compositional Edge Gaussian Processes

TL;DR

Drawing upon the Hodge decomposition, classes of divergence-free and curl-free edge GPs are developed, suitable for various applications, and combined to create Hodge-compositional edge GPs, expressive enough to represent any edge function.

Abstract

We propose principled Gaussian processes (GPs) for modeling functions defined over the edge set of a simplicial 2-complex, a structure similar to a graph in which edges may form triangular faces. This approach is intended for learning flow-type data on networks where edge flows can be characterized by the discrete divergence and curl. Drawing upon the Hodge decomposition, we first develop classes of divergence-free and curl-free edge GPs, suitable for various applications. We then combine them to create \emph{Hodge-compositional edge GPs} that are expressive enough to represent any edge function. These GPs facilitate direct and independent learning for the different Hodge components of edge functions, enabling us to capture their relevance during hyperparameter optimization. To highlight their practical potential, we apply them for flow data inference in currency exchange, ocean currents and water supply networks, comparing them to alternative models.

Figures (5)

• Figure 1: (a) A $\mathrm{SC}_2$ where we shade (closed) triangles in green and denote reference orientations of edges/triangles by arrows. (b) An edge function ${\bm{f}}_1$ with its divergence (purple values on nodes) and curl (orange values in triangles). (c-e) Hodge decomposition: (c) gradient flow ${\bm{f}}_G={\bm{B}}_1^\top{\bm{f}}_0$, obtained as the gradient of some node function ${\bm{f}}_0$ (given in blue). It is curl-free: the net-circulation along each triangle is zero; (d) curl flow ${\bm{f}}_C={\bm{B}}_2{\bm{f}}_2$, induced by some circulating triangle signal ${\bm{f}}_2$ (given in red). It is div-free: the net-flow at each node is zero; and (e) harmonic flow ${\bm{f}}_H$, circulating around the 1-dimensional "hole" (open triangle $\{1,3,4\}$), where the net-flow on nodes and net-circulation in triangles are zero. All numbers are rounded to two decimal places.
• Figure 2: (Left) Matérn kernel functions $\Psi_{\Box}(\lambda)$ for $\Box=\{H,G,C\}$ in \ref{['eq.grad-curl-matern-kernel']} of gradient, curl and harmonic GPs in the eigen-spectrum $\lambda$ ranging in the min and man eigenvalues of ${\bm{L}}_1$. (Right) Matérn kernel function $\Psi(\lambda)$ of non-HC GP in \ref{['eq.simple_matern_diffusion']}.
• Figure 3: (a-d) Interpolating a smaller forex market (for better visibility) with train ratio $50\%$ where dashed (solid) edges are used for training (testing). (e) Learned Matérn kernels in the spectrum of the Hodge Laplacian, $\Psi(\lambda)$ for non-HC GP and $\Psi_\Box(\lambda)$ with $\Box=\{H,G,C\}$ for HC GPs.
• Figure 4: (a-b) Ground truth and interpolated ocean current in the vector field domain. (c) Standard deviation approximated by sampling 50 edge flows from the predictive posterior distribution and converted to the vector field domain. (d-e) The curl-free and div-free components directly obtained from the learned kernels. (f) Learned diffusion kernels $\Psi_G(\lambda)$ and $\Psi_C(\lambda)$ of the HC GP in the spectrum of the Hodge Laplacian.
• Figure 5: (a-e) Posterior mean and standard deviation (std) based on the Matérn node GPs, and the HC and non-HC Matérn edge GPs. Squared (Circled) nodes represent the node samples for training (testing). Dashed (solid) edges denote the edge samples for training (testing). (f) The learned edge GP kernels in the spectrum, $\Psi(\lambda)$ for non-HC GP and $\Psi_{H}(0), \Psi(\lambda)$ for HC GPs.

Theorems & Definitions (7)

• Theorem 1: hodge1989theory
• Proposition 2
• Proposition 3
• Definition 4
• Lemma 5
• Corollary 6
• Proposition 7