Conditional Optimal Transport on Function Spaces

TL;DR

A theory of constrained optimal transport problems that describe block-triangular Monge maps that characterize conditional measures along with their Kantorovich relaxations generalizes the theory of optimal triangular transport to infinite-dimensional Hilbert spaces with general cost functions.

Abstract

We present a systematic study of conditional triangular transport maps in function spaces from the perspective of optimal transportation and with a view towards amortized Bayesian inference. More specifically, we develop a theory of constrained optimal transport problems that describe block-triangular Monge maps that characterize conditional measures along with their Kantorovich relaxations. This generalizes the theory of optimal triangular transport to separable infinite-dimensional function spaces with general cost functions. We further tailor our results to the case of Bayesian inference problems and obtain regularity estimates on the conditioning maps from the prior to the posterior. Finally, we present numerical experiments that demonstrate the computational applicability of our theoretical results for amortized and likelihood-free inference of functional parameters.

Figures (5)

• Figure 1: A depiction of conditional sampling using triangular maps obtained from \ref{['alg:plugin']}. The left panel shows a scatter plot of samples from $\nu$ that were used to solve the conditional OT problem while the blue, red, and black lines denote slices along which conditional samples are generated. The other panels show the conditional histograms with dotted lines showing the true histograms and solid lines showing the numerical approximations.
• Figure 1: Conditional OT experiments on two dimensional benchmarks using \ref{['alg:plugin']}: (Left column) the empirical samples from $\nu$. The blue and black lines indicate slices along which we condition the measures. (Middle column) comparison between the ground truth histogram indicated with dotted lines and the conditional OT histogram indicated with solid lines. (Right column) The conditioning maps $\widehat{T}_\mathcal{U}(y, \cdot)$ for each slice computed using two nearest neighbor interpolation.
• Figure 2: Summary of our experiments for Darcy flow inverse problem comparing pCN and WaMGAN. From left, the first column shows the ground truth field $u^\dagger$, second column shows the corresponding $p(u^\dagger)$ and measurement locations $x_j$ denoted as black dots. Third and fourth columns show posterior means for WaMGAN and pCN respectively while the fifth and sixth columns show pointwise variances of the posteriors in the same order. In all cases we used the same color range for pCN and WaMGAN to ensure proper comparison between the two methods.
• Figure 3: Scatterplot comparison of pointwise posterior variances calculated by WaMGAN and pCN for the four examples in \ref{['fig:darcy-summary-stats']}. The diagonal line represents perfect agreement between the two methods.
• Figure 4: A comparison of posterior samples from pCN (top row) vs WaMGAN (bottom row) for one of our experiments.

Theorems & Definitions (37)

• Definition 1.2
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4: villani-OT
• Proposition 2.5: villani-OT
• Definition 2.6