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arxiv: 2604.04862 · v1 · submitted 2026-04-06 · 💻 cs.RO

Outlier-Robust Nonlinear Moving Horizon Estimation using Adaptive Loss Functions

Nestor Deniz , Guido Sanchez , Fernando Auat Cheein , Leonardo Giovanini This is my paper

Pith reviewed 2026-05-10 19:02 UTC · model grok-4.3

classification 💻 cs.RO
keywords moving horizon estimationadaptive loss functionsoutlier robustnessnonlinear estimationrobust optimizationstate estimationMHE
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The pith

An adaptive loss function lets moving horizon estimators automatically downweight outliers while reverting to standard least-squares on clean data.

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

The paper presents a framework for nonlinear moving horizon estimation that incorporates an adaptive robust loss function to reduce the impact of outliers in measurements. This loss is paired with a regularization term designed to prevent the estimator from collapsing into a trivial solution that ignores all data. A single tuning parameter controls the shape of the loss function, allowing adjustment of robustness. Simulations indicate that adaptation to outliers occurs in just a few iterations, and the method reverts to traditional L2 behavior when measurements contain no outliers. A sympathetic reader would care because this could enable more reliable state estimation in control and robotics applications without separate outlier-handling steps.

Core claim

The authors propose an adaptive robust loss function framework for MHE that integrates the loss with a regularization term to avoid naive solutions. The approach prioritizes the fitting of uncontaminated data and downweights contaminated ones. A tuning parameter is included to control the shape of the loss function and adjust the estimator's robustness to outliers. Simulation results show that adaptation occurs in just a few iterations, whereas the traditional L2 behavior predominates when the measurements are free of outliers.

What carries the argument

Adaptive robust loss function paired with a regularization term, governed by one tuning parameter that shapes the loss to balance outlier downweighting against standard fitting.

If this is right

  • Adaptation to the presence of outliers occurs within only a few iterations.
  • The estimator recovers traditional L2 behavior on measurements without outliers.
  • The regularization term prevents collapse to solutions that discard all measurements.
  • Contaminated measurements receive lower weight than uncontaminated ones during fitting.

Where Pith is reading between the lines

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

  • The single tuning parameter might be adjusted online if an auxiliary outlier-rate detector were added.
  • The framework could be tested on real sensor streams from mobile robots to see whether quick adaptation holds outside simulations.
  • Similar adaptive losses might be substituted into other receding-horizon or filtering schemes that currently rely on fixed robust costs.

Load-bearing premise

The added regularization term reliably blocks naive solutions that ignore measurements, and one tuning parameter can deliver the needed robustness without destabilizing the estimator.

What would settle it

Run the estimator on a dataset with a known high fraction of outliers and check whether estimates remain bounded and accurate or diverge, and separately verify on clean data whether the loss reverts exactly to squared-error behavior.

Figures

Figures reproduced from arXiv: 2604.04862 by Fernando Auat Cheein, Guido Sanchez, Leonardo Giovanini, Nestor Deniz.

Figure 1
Figure 1. Figure 1: Adaptive robust loss functions ρ(r, α, c) and φ(r, α, c) for different values of α [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a shows the reduction in normalized estimation error, averaged over 1000 trials in the first scenario, while [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the tractor’s x and y coordinates for one 0 20 40 60 80 100 t(s) -40 -20 0 20 40 60 x (m) Measurement True state Estimated state 0 20 40 60 80 100 t(s) -40 -30 -20 -10 0 10 20 30 y (m) Measurement True state Estimated state -5 0 5 10 15 20 25 x (m) -5 0 5 y (m) 9 10 11 12 13 -1.6 -1.4 -1.2 -1 -0.8 -0.6 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

In this work, we propose an adaptive robust loss function framework for MHE, integrating an adaptive robust loss function to reduce the impact of outliers with a regularization term that avoids naive solutions. The proposed approach prioritizes the fitting of uncontaminated data and downweights the contaminated ones. A tuning parameter is incorporated into the framework to control the shape of the loss function for adjusting the estimator's robustness to outliers. The simulation results demonstrate that adaptation occurs in just a few iterations, whereas the traditional behaviour $\mathrm{L_2}$ predominates when the measurements are free of outliers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper proposes an adaptive robust loss function framework for nonlinear moving horizon estimation (MHE). It combines an adaptive loss to down-weight outliers, a regularization term to avoid naive solutions, and a single tuning parameter to control the loss shape and robustness level. Simulations are presented to show that the estimator adapts in a few iterations when outliers are present and recovers standard L2 behavior on clean measurements.

Significance. If the central claims hold, the approach could provide a tunable, practical tool for outlier handling in robotic MHE applications where sensor data is frequently corrupted. The reported quick adaptation in simulations is a concrete empirical strength. However, the absence of theoretical convergence guarantees, error bounds, or comparisons to established robust MHE baselines (e.g., Huber or redescending losses) substantially limits the result's significance beyond the specific simulated cases.

major comments (3)
  1. Abstract: the central claim that the adaptive loss plus regularization reliably down-weights outliers while recovering L2 behavior and avoiding naive solutions lacks any derivation, stability bound, or analysis showing that the single tuning parameter prevents vanishing gradients or multiple local minima in the non-convex MHE program for general nonlinear dynamics.
  2. Simulation results: the reported outcomes are consistent with the stated behavior, yet no comparisons are provided against standard robust MHE formulations, making it impossible to quantify improvement or confirm that the adaptation is not an artifact of the chosen simulation setup.
  3. Framework: the regularization term is asserted to prevent naive solutions, but no explicit condition or bound is derived to guarantee this for arbitrary system nonlinearities, which is load-bearing for the claim that the method remains stable and unbiased.
minor comments (1)
  1. Abstract: the notation 'traditional behaviour L2' should be defined explicitly (e.g., as the quadratic loss) to avoid ambiguity for readers unfamiliar with the specific MHE formulation.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive comments and the recommendation for major revision. We address each major comment point by point below, indicating planned changes to the manuscript where appropriate. Our responses focus on clarifying the scope of the current work and strengthening the presentation without overstating its contributions.

read point-by-point responses

  1. Referee: Abstract: the central claim that the adaptive loss plus regularization reliably down-weights outliers while recovering L2 behavior and avoiding naive solutions lacks any derivation, stability bound, or analysis showing that the single tuning parameter prevents vanishing gradients or multiple local minima in the non-convex MHE program for general nonlinear dynamics.

    Authors: We agree that the abstract and introduction present the framework's behavior in terms that could be interpreted as implying general guarantees. The manuscript is an empirical study demonstrating the observed adaptation and L2 recovery in simulations; no derivations, stability bounds, or analysis of the non-convex program for arbitrary nonlinear dynamics are provided. In the revised version we will rewrite the abstract and relevant sections to state explicitly that the claims are supported only by the reported simulation results, and we will add a limitations paragraph noting the absence of theoretical analysis as an area for future work. revision: yes

  2. Referee: Simulation results: the reported outcomes are consistent with the stated behavior, yet no comparisons are provided against standard robust MHE formulations, making it impossible to quantify improvement or confirm that the adaptation is not an artifact of the chosen simulation setup.

    Authors: We acknowledge the lack of baseline comparisons. The current simulations illustrate the adaptation property and L2 recovery but do not benchmark against established robust estimators. In the revision we will add new simulation cases that directly compare the proposed adaptive loss against Huber loss and redescending loss formulations within the same MHE framework, using the same system models and outlier scenarios, to provide quantitative performance differences. revision: yes

  3. Referee: Framework: the regularization term is asserted to prevent naive solutions, but no explicit condition or bound is derived to guarantee this for arbitrary system nonlinearities, which is load-bearing for the claim that the method remains stable and unbiased.

    Authors: The regularization term is motivated by the structure of the MHE objective to discourage complete rejection of all measurements. We agree that no general condition or bound is derived that would guarantee this property for arbitrary nonlinear dynamics. In the revised manuscript we will expand the framework section with a clearer derivation of the regularization's effect on the cost for the considered classes of systems and will explicitly state that a general guarantee for all nonlinearities is not provided and remains an open question. revision: partial

standing simulated objections not resolved
  • Theoretical convergence guarantees, stability bounds, or analysis showing that the single tuning parameter prevents vanishing gradients or multiple local minima for general nonlinear dynamics.

Circularity Check

0 steps flagged

No significant circularity; new adaptive loss framework with explicit tuning parameter

full rationale

The paper introduces an adaptive robust loss function for nonlinear MHE as a new construction, controlled by a single tuning parameter that shapes the loss to down-weight outliers while recovering L2 behavior on clean data. Simulation results are reported as empirical outcomes showing rapid adaptation, not as predictions or derivations that reduce by the paper's own equations to values fitted from the same data. No load-bearing steps rely on self-citations, self-definitional loops, or renaming of known results. The framework and regularization term are presented as independent design choices whose performance is validated externally via simulation, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on one explicit free tuning parameter that shapes the loss, plus standard domain assumptions of MHE; the adaptive loss itself is a newly postulated functional form without independent falsifiable evidence outside the simulations.

free parameters (1)
  • tuning parameter
    Controls the shape of the loss function to adjust the estimator's robustness to outliers.
axioms (1)
  • domain assumption Nonlinear system dynamics and measurement models are known and can be used inside the moving-horizon optimization
    Standard premise of any MHE formulation invoked implicitly throughout the abstract.
invented entities (1)
  • adaptive robust loss function with regularization term no independent evidence
    purpose: To automatically downweight outliers while avoiding naive solutions that ignore all data
    New functional form introduced in this work; no external falsifiable handle provided.

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Reference graph

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