Ground Metric Learning on Graphs

Abstract

Optimal transport (OT) distances between probability distributions are parameterized by the ground metric they use between observations. Their relevance for real-life applications strongly hinges on whether that ground metric parameter is suitably chosen. The challenge of selecting it adaptively and algorithmically from prior knowledge, the so-called ground metric learning (GML) problem, has therefore appeared in various settings. In this paper, we consider the GML problem when the learned metric is constrained to be a geodesic distance on a graph that supports the measures of interest. This imposes a rich structure for candidate metrics, but also enables far more efficient learning procedures when compared to a direct optimization over the space of all metric matrices. We use this setting to tackle an inverse problem stemming from the observation of a density evolving with time; we seek a graph ground metric such that the OT interpolation between the starting and ending densities that result from that ground metric agrees with the observed evolution. This OT dynamic framework is relevant to model natural phenomena exhibiting displacements of mass, such as the evolution of the color palette induced by the modification of lighting and materials.

A Elements of proof for Proposition 1

The mapping $\phi _1 : \mathbf {w} \in \mathbb {R}^K \rightarrow {\varvec{M}} \in \mathbb {R}^{N^2}$ admits as adjoint Jacobian:

$$\begin{aligned} \left[ \partial \phi _1(\mathbf {w})\right] ^T({\varvec{X}})_{i,j} = -\frac{\varepsilon }{4S}\left( {\varvec{X}}_{i,j}+{\varvec{X}}_{j,i}-{\varvec{X}}_{i,i}-{\varvec{X}}_{j,j}\right) . \end{aligned}$$

(15)

The mapping $\phi _2 : {\varvec{M}} \in \mathbb {R}^{N^2} \rightarrow {\varvec{U}}={\varvec{M}}^{-1}\in \mathbb {R}^{N^2}$ admits as adjoint Jacobian:

$$\begin{aligned} \left[ \partial \phi _2({\varvec{M}})\right] ^T({\varvec{X}}) = -{\varvec{M}}^{-1}{\varvec{X}}{\varvec{M}}^{-1}. \end{aligned}$$

(16)

The mapping $\phi _3 : {\varvec{U}} \in \mathbb {R}^{N^2} \rightarrow {\varvec{V}}={\varvec{U}}^{S} \in \mathbb {R}^{N^2}$ admits as adjoint Jacobian:

$$\begin{aligned} \left[ \partial \phi _3({\varvec{U}})\right] ^T({\varvec{X}}) = \sum _{l=0}^{S-1}{\varvec{U}}^{l}{\varvec{X}}{\varvec{U}}^{S-l-1}. \end{aligned}$$

(17)

The mapping $\phi _4 : {\varvec{V}} \in \mathbb {R}^{N^2} \rightarrow \mathbf {y}={\varvec{V}}\mathbf {v} \in \mathbb {R}^{N}$ admits as adjoint Jacobian:

$$\begin{aligned} \left[ \partial \phi _4({\varvec{V}})\right] ^T(\mathbf {x}) = \mathbf {x}\mathbf {v}^T. \end{aligned}$$

(18)

Since $\Phi = \phi _4 \circ \phi _3 \circ \phi _2 \circ \phi _1$, we compose the adjoint Jacobians in the reverse order as follows:

$$\begin{aligned} \left[ \partial \Phi (\mathbf {w})\right] ^T({\varvec{g}})_{i,j} = \left[ \partial \phi _1\right] ^T\left[ \partial \phi _2\right] ^T\left[ \partial \phi _3\right] ^T\left[ \partial \phi _4\right] ^T(\mathbf {g})_{i,j}, \end{aligned}$$

(19)

to finally obtain:

$$\begin{aligned}&{[}\partial \Phi (\mathbf {w})]^T(\mathbf {g})_{i,j} = - \frac{\varepsilon }{4S}\sum _{\ell =0}^{S-1} \left( \mathbf {g}_i^\ell - \mathbf {g}_j^\ell \right) \left( \mathbf {v}_i^\ell - \mathbf {v}_j^\ell \right) , \end{aligned}$$

(20)

$$\begin{aligned}&\text {where}\quad \left\{ \begin{array}{l} \mathbf {g}^\ell {\mathop {=}\limits ^{\,\mathrm{def.}}}\,{\varvec{M}}^{\ell -S}\mathbf {g} \\ \mathbf {v}^\ell {\mathop {=}\limits ^{\,\mathrm{def.}}}\,{\varvec{M}}^{-\ell -1} \mathbf {v}. \end{array} \right. \end{aligned}$$

(21)