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Convergence analysis of kernel learning FBSDE filter

Yunzheng Lyu, Feng Bao

TL;DR

This work addresses nonlinear filtering by analyzing a kernel learning forward–backward SDE (FBSDE) filter that propagates prior state densities via a forward SDE and learns posterior densities through kernel density estimation (KDE). The authors establish a rigorous convergence framework across four components: KDE kernel learning, fixed-point backward iteration, Bayesian posterior updates, and the numerical discretization of the FBSDE system. They prove local convergence rates, including KDE $n^{-4/(4+d_x)}$ for Gaussian kernels and RMSE bounds $O(L^{-2/(4+d_x)})$ for kernel learning, and derive conditions for global convergence with a probability/likelihood ratio criterion. The results justify the empirical performance of the meshfree FBSDE filter in high dimensions, clarifying how convergence depends on bandwidth, iteration depth, sample size, and time discretization, with explicit uniform convergence guarantees under stated assumptions.

Abstract

Kernel learning forward backward SDE filter is an iterative and adaptive meshfree approach to solve the nonlinear filtering problem. It builds from forward backward SDE for Fokker-Planker equation, which defines evolving density for the state variable, and employs KDE to approximate density. This algorithm has shown more superior performance than mainstream particle filter method, in both convergence speed and efficiency of solving high dimension problems. However, this method has only been shown to converge empirically. In this paper, we present a rigorous analysis to demonstrate its local and global convergence, and provide theoretical support for its empirical results.

Convergence analysis of kernel learning FBSDE filter

TL;DR

This work addresses nonlinear filtering by analyzing a kernel learning forward–backward SDE (FBSDE) filter that propagates prior state densities via a forward SDE and learns posterior densities through kernel density estimation (KDE). The authors establish a rigorous convergence framework across four components: KDE kernel learning, fixed-point backward iteration, Bayesian posterior updates, and the numerical discretization of the FBSDE system. They prove local convergence rates, including KDE for Gaussian kernels and RMSE bounds for kernel learning, and derive conditions for global convergence with a probability/likelihood ratio criterion. The results justify the empirical performance of the meshfree FBSDE filter in high dimensions, clarifying how convergence depends on bandwidth, iteration depth, sample size, and time discretization, with explicit uniform convergence guarantees under stated assumptions.

Abstract

Kernel learning forward backward SDE filter is an iterative and adaptive meshfree approach to solve the nonlinear filtering problem. It builds from forward backward SDE for Fokker-Planker equation, which defines evolving density for the state variable, and employs KDE to approximate density. This algorithm has shown more superior performance than mainstream particle filter method, in both convergence speed and efficiency of solving high dimension problems. However, this method has only been shown to converge empirically. In this paper, we present a rigorous analysis to demonstrate its local and global convergence, and provide theoretical support for its empirical results.
Paper Structure (10 sections, 9 theorems, 66 equations)

This paper contains 10 sections, 9 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. FBSDE Filter
  3. FBSDE Filter Algorithm
  4. Convergence Analysis for KDE kernel learning FBSDE filter
  5. KDE kernel learning convergence analysis
  6. Fixed point iteration convergence analysis
  7. Bayesian update convergence analysis
  8. FBSDE numerical scheme convergence
  9. Unified Convergence Analysis
  10. Conclusions

Key Result

Theorem 4.1.2

Let p be integer, $p\geq1$. Let $f\in W_{p}^{(m)}(M)$ and $\sup_{x\in R^{d_{x}}}f(x)\leq \Lambda$. Kernel K satisfies properties (i)-(vi) and where $h=[\frac{Bd_{x}}{2mnM^2A}]^{\frac{1}{2m+d_{x}}}$ Then for $\forall x\in R^{d_{x}}$: with $\phi(2,p)=\frac{2m}{2m+d_{x}}$, which is irrelevant with p $D=\theta(M^2A)^{\frac{d_{x}}{2m+d_{x}}}(B)^{\frac{2m}{2m+d_{x}}}$$\theta=\frac{2m+d_{x}}{(4)^{\frac

Theorems & Definitions (9)

  • Theorem 4.1.2
  • Theorem 4.1.3
  • Lemma 4.1.4
  • Lemma 4.2.1
  • Lemma 4.3.1
  • Lemma 4.4.2
  • Lemma 4.5.1
  • Lemma 4.5.2
  • Theorem 4.5.3