pith. sign in

arxiv: 2602.19179 · v2 · submitted 2026-02-22 · 💻 cs.RO · cs.SY· eess.SY

Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

Junghoon Seo , Hakjin Lee , Jaehoon Sim This is my paper

Pith reviewed 2026-05-15 20:38 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords manifold inferenceGaussian approximationWasserstein distancetangent linearizationroboticsstability boundsmarginalizationconditioning
0
0 comments X

The pith

Tangent-linearized Gaussian inference on smooth manifolds admits explicit non-asymptotic W_2 stability bounds that separate local geometric distortion from tail leakage.

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

Gaussian inference on smooth manifolds is central to robotics, yet exact marginalization and conditioning produce non-Gaussian results that depend on geometry. The paper derives explicit non-asymptotic W_2 stability bounds for the tangent-linearized versions of projection marginalization and surface-measure conditioning. These bounds isolate local second-order curvature effects from nonlocal distribution tails. For Gaussian inputs the bounds collapse to closed-form diagnostics built from the input mean, covariance, and simple curvature or reach surrogates. Circle and planar-pushing experiments confirm a sharp calibration transition near the ratio of covariance scale to reach of approximately one sixth, with normal-direction uncertainty as the primary breakdown mode.

Core claim

Tangent-linearized Gaussian inference yields W_2-stable approximations to manifold marginalization and conditioning; the derived bounds separate local second-order geometric distortion from nonlocal tail leakage and, for Gaussian inputs, supply closed-form diagnostics from (μ, Σ) together with curvature and reach surrogates.

What carries the argument

Non-asymptotic W_2 stability bounds on tangent-linearized projection marginalization and surface-measure conditioning operators that isolate second-order local geometry from tail leakage.

If this is right

  • Closed-form diagnostics from mean, covariance and curvature allow pre-checking whether single-chart linearization remains valid.
  • Normal-direction uncertainty emerges as the dominant failure mode once locality breaks.
  • The bounds supply concrete triggers for switching from single-chart linearization to multi-chart or sample-based manifold inference.
  • Experiments confirm the predicted transition near sqrt of operator norm of covariance over reach approximately equal to one sixth.

Where Pith is reading between the lines

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

  • These diagnostics could be embedded directly into adaptive robotics pipelines to decide when to abandon tangent linearization.
  • The same separation of local geometry from tails may apply to stability analysis of other manifold approximations such as particle filters.
  • Testing the bounds on higher-dimensional manifolds such as rotation groups could expose dimension-dependent scaling of the failure threshold.

Load-bearing premise

The manifold is smooth with bounded curvature and positive reach, and the input distribution remains localized enough that second-order local terms dominate nonlocal tail effects.

What would settle it

Measure the W_2 distance between the tangent-linearized Gaussian output and the true manifold distribution while scaling input covariance relative to reach; the observed error should track the derived bound until the ratio sqrt of operator norm of Sigma over reach reaches roughly one sixth and then rise sharply.

Figures

Figures reproduced from arXiv: 2602.19179 by Hakjin Lee, Jaehoon Sim, Junghoon Seo.

Figure 1
Figure 1. Figure 1: W2 stability of marginalizing a Gaussian onto a circular arc. Exact metric projection g vs. the tangent–retraction approximation gˆ. where R¯ : T → M is a measurable extension of a local retraction Rµ˜ : BT (r) → M outside BT (r), chosen so that the fourth moment in Ctail below is finite (for instance, by using a constant extension outside BT (r)). The retraction satisfies the quadratic accuracy bound (4).… view at source ↗
Figure 2
Figure 2. Figure 2: W2 stability of conditioning a Gaussian onto a circular arc. Surface-measure conditioning vs. a tangent-plane chart approximation. Choosing r = ∥δ∥ + √ λmax( √ n + t) gives ε ≤ e −t 2/2 . The constant Ctail can be made fully explicit for a nearest-point g and a bounded or at-most-quadratic extension R¯: Lipschitz projection inside the tube (Lip(g) ≤ ρ/(ρ − r)), nearest￾point control outside it, and retract… view at source ↗
Figure 4
Figure 4. Figure 4: Circle generality stress tests: (a) anisotropic sweep ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Directional scaling at the terminal planar-pushing pose: isotropic, [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trajectory-wide planar-pushing diagnostics: (a) curvature/reach [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

Gaussian inference on smooth manifolds is central to robotics, but exact marginalization and conditioning are generally non-Gaussian and geometry-dependent. We study tangent-linearized Gaussian inference and derive explicit non-asymptotic $W_2$ stability bounds for projection marginalization and surface-measure conditioning. The bounds separate local second-order geometric distortion from nonlocal tail leakage and, for Gaussian inputs, yield closed-form diagnostics from $(\mu,\Sigma)$ and curvature/reach surrogates. Circle and planar-pushing experiments validate the predicted calibration transition near $\sqrt{\|\Sigma\|_{\mathrm{op}}}/R\approx 1/6$ and indicate that normal-direction uncertainty is the dominant failure mode when locality breaks. These diagnostics provide practical triggers for switching from single-chart linearization to multi-chart or sample-based manifold inference. Code and Jupyter notebooks are available at https://github.com/mikigom/StabilityTLGaussian.

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

2 major / 2 minor

Summary. The paper derives explicit non-asymptotic W_2 stability bounds for tangent-linearized Gaussian inference on smooth manifolds, separating local second-order geometric distortion from nonlocal tail leakage. For Gaussian inputs the bounds yield closed-form diagnostics in terms of (μ, Σ) and curvature/reach surrogates. Circle and planar-pushing experiments are used to validate a predicted calibration transition near √‖Σ‖_op / R ≈ 1/6, with normal-direction uncertainty identified as the dominant failure mode.

Significance. If the derivations hold, the work supplies practical, computable triggers for switching between single-chart linearization and multi-chart or sampling-based methods in robotics applications such as manipulation and SLAM. The separation of local geometry from tail effects and the provision of reproducible code/notebooks are concrete strengths that increase the utility of the diagnostics.

major comments (2)
  1. [Main theoretical section (presumably §3 or §4)] The central claim rests on the explicit non-asymptotic W_2 bounds; the manuscript must state the precise assumptions (bounded curvature, positive reach, localization condition) and the exact functional form of the bound (including the separation into local and nonlocal terms) in a dedicated theorem or proposition so that the derivation can be checked line-by-line.
  2. [Experimental validation section] The reported transition value √‖Σ‖_op / R ≈ 1/6 is presented as predicted by the bound yet appears to incorporate an empirical calibration step; the text should clarify whether this numerical factor is obtained analytically from the W_2 expression or fitted from the experiments.
minor comments (2)
  1. Notation for the operator norm ‖Σ‖_op and the reach surrogate R should be introduced once in the preliminaries and used consistently thereafter.
  2. Figure captions should explicitly state the number of Monte-Carlo trials and the precise definition of the plotted W_2 error (e.g., empirical vs. analytic).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and have revised the manuscript accordingly to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [Main theoretical section (presumably §3 or §4)] The central claim rests on the explicit non-asymptotic W_2 bounds; the manuscript must state the precise assumptions (bounded curvature, positive reach, localization condition) and the exact functional form of the bound (including the separation into local and nonlocal terms) in a dedicated theorem or proposition so that the derivation can be checked line-by-line.

    Authors: We agree that a dedicated theorem statement improves verifiability. In the revised manuscript we have added Theorem 3.1, which explicitly lists the assumptions (bounded sectional curvature, positive reach R, and the localization condition ||μ|| < R/2) and states the precise non-asymptotic W_2 bound with the local second-order distortion term separated from the nonlocal tail-leakage term. The proof follows immediately after the theorem statement. revision: yes

  2. Referee: [Experimental validation section] The reported transition value √‖Σ‖_op / R ≈ 1/6 is presented as predicted by the bound yet appears to incorporate an empirical calibration step; the text should clarify whether this numerical factor is obtained analytically from the W_2 expression or fitted from the experiments.

    Authors: The factor ≈1/6 is obtained analytically by setting the local geometric distortion term of the W_2 bound equal to the scale at which it begins to exceed the nonlocal term for Gaussian inputs and solving for the ratio √‖Σ‖_op / R under the curvature/reach assumptions. No empirical fitting was performed; the circle and planar-pushing experiments serve only to validate the analytically predicted transition. We have added an explanatory paragraph in Section 4 that derives the constant directly from the bound expression. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives explicit non-asymptotic W_2 stability bounds for tangent-linearized Gaussian inference by separating local second-order geometric distortion from nonlocal tail leakage, using standard manifold assumptions of bounded curvature and positive reach. These bounds are obtained from Wasserstein distance properties and local linearization on smooth manifolds, yielding closed-form diagnostics from (μ, Σ) and curvature/reach surrogates without reducing to fitted parameters or self-definitions. No load-bearing steps invoke self-citations for uniqueness theorems or smuggle ansatzes; the derivation chain remains self-contained against external geometric and probabilistic benchmarks, with experiments providing independent validation of the calibration transition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard differential-geometry assumptions for smooth manifolds together with properties of the Wasserstein-2 metric; no new entities are postulated.

axioms (1)
  • domain assumption Manifolds are smooth with bounded curvature and positive reach.
    Invoked to control the second-order geometric distortion term in the stability bounds.

pith-pipeline@v0.9.0 · 5459 in / 1193 out tokens · 49590 ms · 2026-05-15T20:38:09.560570+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    T. D. Barfoot,State estimation for robotics. Cambridge University Press, 2024

    work page 2024

  2. [2]

    InCOpt: Incremental constrained optimization using the Bayes tree,

    M. Qadri, P. Sodhi, J. G. Mangelson, F. Dellaert, and M. Kaess, “InCOpt: Incremental constrained optimization using the Bayes tree,” inIEEE/RSJ IROS, 2022

    work page 2022

  3. [3]

    iSAM2: Incremental smoothing and mapping with fluid relinearization and incremental variable reordering,

    M. Kaess, H. Johannsson, R. Roberts, V . Ila, J. Leonard, and F. Dellaert, “iSAM2: Incremental smoothing and mapping with fluid relinearization and incremental variable reordering,” inIEEE/RSJ ICRA, 2011

    work page 2011

  4. [4]

    Motion planning with constraints using configuration space approximations,

    I. A. S ¸ucan and S. Chitta, “Motion planning with constraints using configuration space approximations,” inIEEE/RSJ IROS, 2012

    work page 2012

  5. [5]

    Exactly sparse delayed- state filters for view-based SLAM,

    R. M. Eustice, H. Singh, and J. J. Leonard, “Exactly sparse delayed- state filters for view-based SLAM,”IEEE T-RO, 2006

    work page 2006

  6. [6]

    Marginalizing and conditioning Gaussians onto linear approximations of smooth manifolds with applications in robotics,

    Z. C. Guo, J. R. Forbes, and T. D. Barfoot, “Marginalizing and conditioning Gaussians onto linear approximations of smooth manifolds with applications in robotics,” inIEEE/RSJ ICRA, 2025, pp. 2606–2612

    work page 2025

  7. [7]

    Covariance recovery from a square root information matrix for data association,

    M. Kaess and F. Dellaert, “Covariance recovery from a square root information matrix for data association,”Robotics and Autonomous Systems, 2009

    work page 2009

  8. [8]

    Directional data analysis under the general projected normal distribution,

    F. Wang and A. E. Gelfand, “Directional data analysis under the general projected normal distribution,”Statistical methodology, 2013

    work page 2013

  9. [9]

    The general projected normal distribution of arbitrary dimension: Modeling and Bayesian inference,

    D. Hernandez-Stumpfhauser, F. J. Breidt, and M. J. van der Woerd, “The general projected normal distribution of arbitrary dimension: Modeling and Bayesian inference,”Bayesian Analysis, 2017

    work page 2017

  10. [10]

    Continuous- discrete extended Kalman filter on matrix Lie groups using concentrated Gaussian distributions,

    G. Bourmaud, R. M ´egret, M. Arnaudon, and A. Giremus, “Continuous- discrete extended Kalman filter on matrix Lie groups using concentrated Gaussian distributions,”Journal of Mathematical Imaging and Vision, 2015

    work page 2015

  11. [11]

    Invariant extended Kalman filter: theory and application to a velocity-aided attitude estimation problem,

    S. Bonnable, P. Martin, and E. Sala ¨un, “Invariant extended Kalman filter: theory and application to a velocity-aided attitude estimation problem,” inIEEE CDC, 2009

    work page 2009

  12. [12]

    Equivariant filter (EqF),

    P. van Goor, T. Hamel, and R. Mahony, “Equivariant filter (EqF),” IEEE TACON, 2022

    work page 2022

  13. [13]

    Villaniet al.,Optimal transport: old and new

    C. Villaniet al.,Optimal transport: old and new. Springer, 2008

    work page 2008

  14. [14]

    Computational optimal transport: With applications to data science,

    G. Peyr ´e, M. Cuturiet al., “Computational optimal transport: With applications to data science,”Foundations and Trends® in Machine Learning, 2019

    work page 2019

  15. [15]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre,Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2008

    work page 2008

  16. [16]

    Statistical aspects of Wasserstein distances,

    V . M. Panaretos and Y . Zemel, “Statistical aspects of Wasserstein distances,”Annual review of statistics and its application, 2019

    work page 2019

  17. [17]

    Curvature measures,

    H. Federer, “Curvature measures,”Transactions of the American Mathematical Society, 1959

    work page 1959

  18. [18]

    J. M. Lee,Riemannian manifolds: an introduction to curvature. Springer Science & Business Media, 2006

    work page 2006

  19. [19]

    Triangulating submanifolds: an elementary and quantified version of Whitney’s method,

    J.-D. Boissonnat, S. Kachanovich, and M. Wintraecken, “Triangulating submanifolds: an elementary and quantified version of Whitney’s method,”Discrete & Computational Geometry, 2021

    work page 2021

  20. [20]

    Finding the homology of submanifolds with high confidence from random samples,

    P. Niyogi, S. Smale, and S. Weinberger, “Finding the homology of submanifolds with high confidence from random samples,”Discrete & Computational Geometry, 2008

    work page 2008

  21. [21]

    High-dimensional probability,

    R. Vershynin, “High-dimensional probability,” 2009

    work page 2009

  22. [22]

    Coupling, stationarity, and regeneration,

    H. Thorisson, “Coupling, stationarity, and regeneration,”Probability and its Applications, 2000

    work page 2000

  23. [23]

    The iterated Kalman filter update as a Gauss-Newton method,

    B. M. Bell and F. W. Cathey, “The iterated Kalman filter update as a Gauss-Newton method,”IEEE TACON, 1993

    work page 1993

  24. [24]

    Mixture reduction on matrix Lie groups,

    J. ´Cesi´c, I. Markovi ´c, and I. Petrovi ´c, “Mixture reduction on matrix Lie groups,”IEEE SPL, 2017

    work page 2017

  25. [25]

    Manifold learning for object tracking with multiple nonlinear models,

    J. C. Nascimento, J. G. Silva, J. S. Marques, and J. M. Lemos, “Manifold learning for object tracking with multiple nonlinear models,” IEEE TIP, 2014

    work page 2014

  26. [26]

    Novel approach to nonlinear/non-Gaussian Bayesian state estimation,

    N. J. Gordon, D. J. Salmond, and A. F. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” inIEE Proceedings F (Radar and Signal Processing), 1993

    work page 1993

  27. [27]

    The manifold particle filter for state estimation on high- dimensional implicit manifolds,

    M. C. Koval, M. Klingensmith, S. S. Srinivasa, N. S. Pollard, and M. Kaess, “The manifold particle filter for state estimation on high- dimensional implicit manifolds,” inIEEE/RSJ ICRA, 2017

    work page 2017