## Joint work with

Sarang Joshi

University of Utah

Martin Bauer

Florida State University

## Draw samples from non-uniform distribution on $$M$$

\mathrm{Prob}(M)=\{ \varrho\in\Omega^n(M)\mid \varrho>0, \int_M \varrho = 1\}
$\mathrm{Prob}(M)=\{ \varrho\in\Omega^n(M)\mid \varrho>0, \int_M \varrho = 1\}$

Smooth probability densities

Problem 1: given $$\mu\in\mathrm{Prob}(M)$$ generate $$N$$ samples from $$\mu$$

Most cases: use Monte-Carlo based methods

Special case here:

• $$M$$ low dimensional
• $$\mu$$ very non-uniform
• $$N$$ very large

transport map approach

might be useful

## Transport problem

Problem 2: given $$\mu\in\mathrm{Prob}(M)$$ find  $$\varphi\in\mathrm{Diff}(M)$$ such that

Method:

• $$N$$ samples $$x_1,\ldots,x_N$$ from uniform distribution $$\mu_0$$
• Compute $$y_i = \varphi(x_i)$$

Diffeomorphism $$\varphi$$ not unique!

\varphi_*\mu_0 = \mu
$\varphi_*\mu_0 = \mu$

## Optimal transport problem

Problem 3: given $$\mu\in\mathrm{Prob}(M)$$ find  $$\varphi\in\mathrm{Diff}(M)$$ minimizing

under constraint $$\varphi_*\mu_0 = \mu$$

Studied case: (Moselhy and Marzouk 2012, Reich 2013, ...)

• $$\mathrm{dist}$$ = $$L^2$$-Wasserstein distance
• $$\Rightarrow$$ optimal mass transport problem
• $$\Rightarrow$$ solve Monge-Ampere equation (heavily non-linear PDE)
E(\varphi) = \mathrm{dist}(\mathrm{id},\varphi)^2
$E(\varphi) = \mathrm{dist}(\mathrm{id},\varphi)^2$

Our notion:

• use optimal information transport

## Optimal information transport

Remarkable fact:

• solution to transport problem almost explicit in this setting

Right-invariant Riemannian $$H^1$$-metric on $$\mathrm{Diff}(M)$$

\displaystyle G_\mathrm{id}(u,v) = \int_M \langle -\Delta u,v\rangle\mu_0 + \ldots
$\displaystyle G_\mathrm{id}(u,v) = \int_M \langle -\Delta u,v\rangle\mu_0 + \ldots$

Use induced distance on $$\mathrm{Diff}(M)$$

## Riemannian submersion

\mathrm{Diff}(M)
$\mathrm{Diff}(M)$
\mathrm{Prob}(M)
$\mathrm{Prob}(M)$
\mathrm{Id}
$\mathrm{Id}$
\mu
$\mu$
\mu_0
$\mu_0$
\pi(\varphi)=\varphi^*\mu
$\pi(\varphi)=\varphi^*\mu$

$$H^1$$ metric

\mathrm{Hor}
$\mathrm{Hor}$

Fisher-Rao metric = explicit geodesics

## Horizontal lifting equations

Theorem: solution to optimal information transport is $$\varphi(1)$$ where $$\varphi(t)$$ fulfills

where $$\mu(t)$$ is Fisher-Rao geodesic between $$\mu_0$$ and $$\mu$$

\displaystyle \Delta f(t) = \frac{\dot\mu(t)}{\mu(t)}\circ\varphi(t)
$\displaystyle \Delta f(t) = \frac{\dot\mu(t)}{\mu(t)}\circ\varphi(t)$
\displaystyle v(t) = \nabla f(t)
$\displaystyle v(t) = \nabla f(t)$
\displaystyle \frac{d}{dt}\varphi(t)^{-1} = v(t)\circ\varphi(t)^{-1}
$\displaystyle \frac{d}{dt}\varphi(t)^{-1} = v(t)\circ\varphi(t)^{-1}$

Leads to numerical time-stepping scheme: Poisson problem at each time step

MATLAB code: github.com/kmodin/oit-random

## Simple 2D example

Warp computation time (256*256 gridsize, 100 time-steps): ~1s

Sample computation time (10^7 samples): < 1s

## Summary

Pros

• Can handle very non-uniform densities
• Draw samples ultra-fast once warp is generated

Cons

• Useless in high dimensions (dimensionality curse)

# THANKS!

Slides available at: slides.com/kmodin

MATLAB code available at: github.com/kmodin/oit-random

By Klas Modin

# Diffeomorphic random sampling using optimal information transport

Presentation given 2017-11-09 at the GSI'17 conference in Paris.

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