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The Kalman-Bucy-Koopman filter converts nonlinear state estimation into a Riccati equation by parameterizing the Hamilton-Jacobi value function with principal Koopman eigenfunctions.

2026-06-30 04:46 UTC pith:5DR7NCOI

load-bearing objection The KBK filter gives a Koopman parameterization of the Mortensen HJ PDE that turns into a Riccati form under the span assumption, but that assumption looks like the load-bearing step and may only hold approximately. the 1 major comments →

arxiv 2606.30487 v1 pith:5DR7NCOI submitted 2026-06-29 math.OC math.DS

Discovering the Kalman-Bucy-Koopman Filter

Umesh Vaidya This is my paper
classification math.OC math.DS
keywords Koopman operatornonlinear filteringKalman-Bucy filterMortensen estimatorHamilton-Jacobi equationRiccati equationstate estimationspectral methods
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces the KBK filter to solve nonlinear state estimation by starting from the Mortensen maximum-likelihood estimator, whose solution satisfies a Hamilton-Jacobi PDE. The key step parameterizes that value function using principal eigenfunctions of the Koopman operator, which recasts the nonlinear PDE as a Riccati-type equation evolving in the transformed coordinates. The result is an operator-theoretic version of the classical Kalman-Bucy filter that still respects the original nonlinear dynamics. Supporting developments include path-integral methods to obtain the eigenfunctions, a characteristics-inspired basis for approximation, and derived error bounds on the value function and state estimates. Simulations indicate better accuracy than the extended Kalman filter and compatibility with data-driven settings.

Core claim

The proposed KBK filter provides a spectral, operator-theoretic realization of this nonlinear filtering problem by parameterizing the HJ value function in terms of principal Koopman eigenfunctions. This transformation converts the nonlinear estimation problem into a Riccati-type evolution in Koopman coordinates, yielding a linear-operator analogue of the classical Kalman-Bucy filter while preserving nonlinear structure in the original state variables.

What carries the argument

Parameterization of the Hamilton-Jacobi value function in terms of principal Koopman eigenfunctions, which converts the nonlinear PDE into Riccati evolution.

Load-bearing premise

That the principal Koopman eigenfunctions can be chosen or approximated so that the resulting Riccati evolution in the transformed coordinates exactly recovers the solution of the original Hamilton-Jacobi PDE for the Mortensen estimator.

What would settle it

A numerical test system where the state estimate produced by the KBK filter deviates from the true solution of the Mortensen Hamilton-Jacobi PDE by more than the derived error bounds.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The nonlinear estimation problem reduces to linear Riccati evolution while the original nonlinear structure is retained in the state variables.
  • Path-integral formulations compute the required principal Koopman eigenfunctions.
  • Dynamics-informed basis constructions yield approximations with explicit error bounds on value-function and state-estimation accuracy.
  • The framework operates without explicit model linearization and supports data-driven implementations.
  • Simulations show improved performance relative to the extended Kalman filter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If Koopman eigenfunctions are learned directly from trajectory data, the method could produce fully data-driven nonlinear filters without requiring an explicit dynamical model.
  • The same parameterization technique might extend to other optimal-control problems whose solutions are characterized by Hamilton-Jacobi equations.
  • The linear-operator structure in Koopman coordinates could allow reuse of existing Riccati solvers and stability analysis tools from linear filtering theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper introduces the Kalman-Bucy-Koopman (KBK) filter for nonlinear state estimation. It formulates the problem as the Mortensen maximum-likelihood estimator whose solution satisfies a Hamilton-Jacobi PDE, then parameterizes the value function in terms of principal Koopman eigenfunctions. This is claimed to convert the problem into a Riccati-type evolution in Koopman coordinates, producing a linear-operator analogue of the classical Kalman-Bucy filter while retaining nonlinear structure in the original variables. The manuscript develops a path-integral formulation for the eigenfunctions, a characteristics-inspired basis construction, derives theoretical error bounds for the value function and state estimates, and reports simulations showing improvement over the extended Kalman filter in data-driven settings.

Significance. If the parameterization step produces a closed Riccati evolution (or one with rigorously controlled residuals) that recovers the Mortensen estimator, the work would supply a spectral, operator-theoretic route to nonlinear filtering that avoids explicit linearization and supports data-driven operation. The explicit derivation of error bounds and the path-integral plus characteristics-based construction for the eigenfunctions are concrete strengths that render the claims falsifiable and potentially reproducible.

major comments (1)
  1. [Abstract] Abstract, paragraph describing the parameterization step: the claim that substitution of principal Koopman eigenfunctions 'converts the nonlinear estimation problem into a Riccati-type evolution' requires that the non-quadratic term arising from the observation likelihood in the HJ PDE remains closed (or produces only controlled residuals) within the chosen eigenbasis. Koopman eigenfunctions diagonalize the uncontrolled dynamics generator, but nothing in the abstract guarantees compatibility of the maximum-likelihood cost term without approximation; the existence of derived error bounds indicates the reduction is not exact, which undercuts the assertion of a 'linear-operator analogue'.
minor comments (1)
  1. [Abstract] The abstract states that 'simulation results demonstrate improved performance over the extended Kalman filter' but provides no quantitative metrics, test cases, or figure references, making it impossible to assess the magnitude or statistical significance of the reported gains.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough reading and constructive feedback on the manuscript. We address the single major comment below and agree that clarification is warranted.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph describing the parameterization step: the claim that substitution of principal Koopman eigenfunctions 'converts the nonlinear estimation problem into a Riccati-type evolution' requires that the non-quadratic term arising from the observation likelihood in the HJ PDE remains closed (or produces only controlled residuals) within the chosen eigenbasis. Koopman eigenfunctions diagonalize the uncontrolled dynamics generator, but nothing in the abstract guarantees compatibility of the maximum-likelihood cost term without approximation; the existence of derived error bounds indicates the reduction is not exact, which undercuts the assertion of a 'linear-operator analogue'.

    Authors: We agree that the abstract phrasing risks implying an exact closed-form reduction. The principal Koopman eigenfunctions diagonalize the drift generator, so that the value-function evolution takes a Riccati structure in the spectral coordinates for the dynamics contribution; the non-quadratic observation term is projected onto the finite basis and therefore produces a controlled residual whose size is bounded by the error analysis already derived in the paper. The KBK construction is therefore a linear-operator analogue with rigorously quantified approximation error rather than an exact equivalence. We will revise the abstract to replace 'converts' with 'approximately converts via a spectral parameterization whose error is controlled by the derived bounds' (or equivalent wording) to remove any ambiguity. revision: yes

Circularity Check

0 steps flagged

No circularity: parameterization presented as ansatz with explicit error bounds

full rationale

The paper's core step is an explicit methodological choice to parameterize the Mortensen HJ value function in the span of principal Koopman eigenfunctions, converting the PDE into a Riccati equation in the new coordinates. This is introduced as a spectral realization with a path-integral construction for the eigenfunctions and a characteristics-inspired approximation, accompanied by derived theoretical error bounds on both value function and state estimates. No equation is shown to reduce to its own input by definition, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness result is imported solely via self-citation. The construction is therefore self-contained as a proposed framework rather than a tautological restatement of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5704 in / 1106 out tokens · 43720 ms · 2026-06-30T04:46:58.476697+00:00 · methodology

0 comments
read the original abstract

This paper introduces the Kalman-Bucy-Koopman (KBK) filter, a novel framework for nonlinear state estimation grounded in Koopman operator spectral theory. The nonlinear estimation problem is formulated as a maximum-likelihood (Mortensen) estimator whose solution is characterized by a Hamilton-Jacobi (HJ) partial differential equation. The proposed KBK filter provides a spectral, operator-theoretic realization of this nonlinear filtering problem by parameterizing the HJ value function in terms of principal Koopman eigenfunctions. This transformation converts the nonlinear estimation problem into a Riccati-type evolution in Koopman coordinates, yielding a linear-operator analogue of the classical Kalman-Bucy filter while preserving nonlinear structure in the original state variables. We develop a path-integral formulation for computing principal Koopman eigenfunctions and introduce a dynamics-informed, characteristics-inspired basis construction for their approximation. Theoretical error bounds are derived for value-function and state-estimation approximations. Simulation results demonstrate improved performance over the extended Kalman filter and illustrate the ability of the KBK framework to operate in data-driven settings without explicit model linearization.

Figures

Figures reproduced from arXiv: 2606.30487 by Umesh Vaidya.

Figure 1
Figure 1. Figure 1: Comparison of the error estimates between the KBK, [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of the state density function, [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the KBK with the EKF filter: State error [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the KBK with the EKF filter [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: KBK filter for the estimation of friction coefficient [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

discussion (0)

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