Open Access Publications from the University of California

## On the Construction of Transport Maps for Scalable Inference and Optimal Communication with Applications in Global Health

• Author(s): Mesa, Diego Alberto
Using the theory of optimal transport, we also consider the dual problem of optimal communication. Many problems can naturally be cast as an optimal communication problem where a message W is signaled sequentially with feedback across a noisy channel. We model these problems as $W \in \cW \subset \reals^d$ and consider optimizing encoding strategies that map $W$ and $Y_1, \ldots, Y_{n-1}$ into $X_n$ that have a dynamical systems flavor. The decoder, with knowledge of the encoder's strategy, simply performs Bayesian updates to sequentially construct a posterior belief $\psi_n$ about the message after observations $Y_1,\ldots,Y_n$. In this thesis we use the theory of optimal transport to expand the Posterior Matching Scheme (PM) (Shayevitz and Feder) to address two unmet needs: (a) We develop a generalization to the PM scheme for arbitrary memoryless channels where $\cW \subset \reals^d$ for any $d \geq 1$. Specifically, we develop recursive encoding schemes that share the same mutual-information maximizing and iterative, time-invariant properties and reduce to the original scheme when $\cW=[0,1]$; (b) We define notions of reliability and achievability in a manner analogous to (Shayevitz
amp; Feder 2011) but in terms of almost-sure convergence of random variables. With this, we then develop necessary and sufficient conditions for the scheme to be reliable and/or attain optimal convergence rate (e.g. achieve capacity). We show that both of these conditions have the same necessary and sufficient condition: the ergodicity of a random process $(\tW_n)_{n \geq 1}$ within the encoder of a PM scheme. Using the theory of optimal transport, we construct schemes in (a), exploiting an invariability property implicit in these schemes to show the equivalent conditions in (b).