Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Error Analysis of Triangular Optimal Transport Maps for Filtering

Mohammad Al-Jarrah, Bamdad Hosseini, Niyizhen Jin, Michele Martino, Amirhossein Taghvaei

TL;DR

This work presents a systematic analysis of estimation errors for a class of optimal transport based algorithms for filtering and data assimilation and applies these results in a filtering scenario to analyze the optimal transport filtering algorithm of Al-Jarrah et al. (2024, ICML).

Abstract

We present a systematic analysis of estimation errors for a class of optimal transport based algorithms for filtering and data assimilation. Along the way, we extend previous error analyses of Brenier maps to the case of conditional Brenier maps that arise in the context of simulation based inference. We then apply these results in a filtering scenario to analyze the optimal transport filtering algorithm of Al-Jarrah et al. (2024, ICML). An extension of that algorithm along with numerical benchmarks on various non-Gaussian and high-dimensional examples are provided to demonstrate its effectiveness and practical potential.

Error Analysis of Triangular Optimal Transport Maps for Filtering

TL;DR

This work presents a systematic analysis of estimation errors for a class of optimal transport based algorithms for filtering and data assimilation and applies these results in a filtering scenario to analyze the optimal transport filtering algorithm of Al-Jarrah et al. (2024, ICML).

Abstract

We present a systematic analysis of estimation errors for a class of optimal transport based algorithms for filtering and data assimilation. Along the way, we extend previous error analyses of Brenier maps to the case of conditional Brenier maps that arise in the context of simulation based inference. We then apply these results in a filtering scenario to analyze the optimal transport filtering algorithm of Al-Jarrah et al. (2024, ICML). An extension of that algorithm along with numerical benchmarks on various non-Gaussian and high-dimensional examples are provided to demonstrate its effectiveness and practical potential.
Paper Structure (38 sections, 29 theorems, 206 equations, 4 figures, 1 algorithm)

This paper contains 38 sections, 29 theorems, 206 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Summary of contributions
  3. Literature survey
  4. Notation
  5. Outline of the article
  6. Error analysis for conditional OT
  7. Review of conditional OT
  8. Problem setup and motivation
  9. Statement of main results and overview of proofs
  10. Slow rate
  11. Fast rate
  12. An example application
  13. Error analysis for OTF
  14. Problem setup
  15. Statement of main results and overview of proofs
  16. ...and 23 more sections

Key Result

Lemma 2.2

\newlabellem: triangular-maps-can-condition0 Let $T_{ { \begin{picture}(1,1) \polyline(1,0)(0,1)(0,0)(1,0)(.5,0) \end{picture} } }$ be a triangular map of the form def: triangular map and suppose $\eta$ is any measure such that $\eta_\mathcal{Y} = \nu_\mathcal{Y}$. If $T_{ { \begin{picture}

Figures (4)

  • Figure 1: The left figure shows the kernel density estimate function of the transported particles in comparison with the exact $\nu(\cdot\mid Y=1)$. The middle and right figures show the exact and approximate values of $\psi$ and $T$, respectively.
  • Figure 2: The left figure shows the $W_2$ distance as a function of dimension $n$ for a fixed number of particles $N=5000$. The middle figure shows the corresponding computational time as a function of dimension. The right figure shows the $W_2$ distance as a function of the number of particles $N$ for a fixed dimension $n=10$.
  • Figure 3: Numerical results for the Lorenz 63 example. The left three panels illustrate the true particle trajectory distribution and the corresponding distributions produced by each method. The right panel presents $W_2$ distances between each method and the true distribution over $100$ independent simulations.
  • Figure 4: Numerical results for the Lorenz 96 example. The left three panels depict the trajectory of one observed component $(U_t(1))$ and two unobserved components $(U_t(2), U_t(3))$ of the true state, along with the corresponding particle trajectories for each filtering method. The right panel displays the MSE of state estimation, averaged over 10 independent simulations.

Theorems & Definitions (62)

  • Definition 2.1: Triangular transport maps
  • Lemma 2.2: baptista2024conditional
  • Proposition 2.3: hosseini2025conditional Prop. 3.6
  • Proposition 2.5: hosseini2025conditional, carlier2016vector
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.10
  • Remark 2.11
  • Lemma 3.3
  • Theorem 3.4
  • ...and 52 more