Operator-norm bounds and a quadratic lower-growth example for the special Euclidean algebra se(3)

Sooraj K.C; Vivek Mishra

arxiv: 2605.31076 · v1 · pith:JDWBF2K3new · submitted 2026-05-29 · 🧮 math.FA

Operator-norm bounds and a quadratic lower-growth example for the special Euclidean algebra se(3)

Sooraj K.C , Vivek Mishra This is my paper

Pith reviewed 2026-06-28 20:44 UTC · model grok-4.3

classification 🧮 math.FA

keywords operator normexponential mapse(3)gradient Lipschitz constantLie algebraquadratic growthspecial Euclidean algebra

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The pith

The exponential map on se(3) satisfies an operator-norm bound of one plus the Frobenius norm of its argument.

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

The paper proves that for any element theta in the special Euclidean algebra se(3), the operator norm of its exponential is at most one plus the Frobenius norm of theta. This bound comes from the contraction property of the SO(3) left Jacobian. Using this, the authors obtain an explicit upper bound of 4.02 R squared on the gradient Lipschitz constant for functions on se(3). They also give an example objective where the Lipschitz constant is at least 0.0505 R squared for R at least 2. This shows se(3) has quadratic growth in its Lipschitz constants, between the bounded case for compact algebras and exponential growth for hyperbolic ones.

Core claim

Using the contraction property of the SO(3) left Jacobian, we show that the operator norm of exp(theta) is bounded by 1 plus the Frobenius norm of theta for all theta in se(3). This leads to an O(R^2) upper bound with constant 4.02 for the gradient Lipschitz constant, and we construct an objective J* with a matching quadratic lower bound of 0.0505 R^2.

What carries the argument

The operator-norm bound ||exp(theta)||_op <= 1 + ||theta||_F derived from the contraction property of the SO(3) left Jacobian.

If this is right

Gradient Lipschitz constants on se(3) grow at most quadratically with radius R.
There exist objectives on se(3) whose gradient Lipschitz constants grow at least quadratically with radius R.
The growth rate for se(3) is polynomial of degree two, distinct from both bounded and exponential cases in other Lie algebras.

Where Pith is reading between the lines

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

The bound may allow simpler convergence analysis for optimization methods that use the exponential map on rigid-body poses.
Analogous contraction arguments could be tested on other Lie algebras that combine compact and non-compact directions.
The numerical lower constant 0.0505 could be tightened or raised by choosing different test objectives.

Load-bearing premise

The contraction property of the SO(3) left Jacobian holds and suffices to establish the operator-norm bound for all theta in se(3).

What would settle it

Compute the operator norm of exp(theta) for some theta in se(3) with large Frobenius norm and check whether it exceeds 1 plus that norm, or compute the Lipschitz constant of the constructed objective J* and check whether it stays below 0.0505 R squared for R at least 2.

Figures

Figures reproduced from arXiv: 2605.31076 by Sooraj K.C, Vivek Mishra.

**Figure 1.** Figure 1: The three growth regimes for the gradient Lipschitz constant L(R; g) arising from exponential-map parameterizations on matrix Lie algebras. Compact algebras (e.g. so(n)) give bounded L; the special Euclidean algebra se(3) gives quadratic growth; algebras with hyperbolic elements (e.g. gl(n)) give exponential blow-up. 2.2. The special Euclidean algebra. The group SE(3) consists of orientation-preserving r… view at source ↗

read the original abstract

We prove operator-norm and gradient Lipschitz bounds for exponential-map parameterizations on the special Euclidean algebra se(3), providing an explicit example of intermediate polynomial growth behavior. Using the contraction property of the SO(3) left Jacobian, we show that ||exp(theta)||_op <= 1 + ||theta||_F for all theta in se(3). We then derive a self-contained O(R^2) upper bound for the gradient Lipschitz constant, with explicit constant 4.02, and construct an objective J* satisfying L_J*(R; se(3)) >= 0.0505 R^2 for R >= 2. These results place se(3) between compact Lie algebras, where the Lipschitz constant remains bounded, and Lie algebras with hyperbolic elements, where it grows exponentially. The upper and lower bounds are obtained for different objective classes; no minimax optimality claim is made.

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.

Desk Editor's Note private letter to a colleague

The paper supplies a new explicit J* with quadratic lower growth on se(3) and claims an operator-norm bound of 1 + ||theta||_F, but the latter rests on a step that needs explicit checking for the translational component.

read the letter

The punchline is that this work gives a concrete new example of an objective achieving quadratic lower growth in the gradient Lipschitz constant on se(3), with the specific coefficient 0.0505 for R at least 2, plus an explicit O(R^2) upper bound constant of 4.02. That example is the clearest addition.

The paper does a solid job on the upper bound and the example construction. It places se(3) between the bounded-Lipschitz compact case and the exponential-growth hyperbolic case, and it keeps the derivations self-contained without fitted parameters. The explicit numbers make the claims easier to test.

The soft spot is the operator-norm claim. The abstract ties it directly to the contraction property of the SO(3) left Jacobian, but se(3) is a semidirect product, so exp(theta) produces a translation vector whose size depends on both rotational and translational velocities. The full matrix operator norm could pick up extra factors from that component. If the proof only controls the rotation block and does not separately bound the cross terms to stay inside 1 + ||theta||_F, the step is incomplete. The abstract does not show that verification, so the bound is not yet fully supported on the evidence given.

This is a narrow but useful paper for people working on gradient methods over SE(3) in robotics or vision who need concrete constants for analysis. A reader already familiar with Lie-group optimization would get the most out of the example and the numbers.

The work shows honest engagement with the growth-rate question and supplies verifiable claims. It deserves a serious referee to check the derivations, especially the extension of the Jacobian contraction to the full algebra.

Referee Report

1 major / 1 minor

Summary. The manuscript proves that ||exp(θ)||_op ≤ 1 + ||θ||_F for all θ ∈ se(3) by invoking the contraction property of the SO(3) left Jacobian. It then derives a self-contained O(R²) upper bound on the gradient Lipschitz constant with explicit constant 4.02 and constructs an explicit objective J* for which the Lipschitz constant satisfies L_{J*}(R; se(3)) ≥ 0.0505 R² for R ≥ 2. These place se(3) between the bounded-Lipschitz regime of compact Lie algebras and the exponential-growth regime of algebras containing hyperbolic elements.

Significance. If the operator-norm bound and the explicit constants hold, the work supplies concrete polynomial-growth examples that are useful for analyzing first-order methods on SE(3). The self-contained upper-bound derivation and the explicit numerical lower-bound coefficient are positive features; no optimality claim is asserted between the upper and lower constants.

major comments (1)

[Main theorem on operator-norm bound (abstract and §3)] The central operator-norm claim ||exp(θ)||_op ≤ 1 + ||θ||_F must be verified on the full semidirect-product structure se(3) = so(3) ⋉ ℝ³. The contraction property controls only the rotational block; the translational component t(ω,v) generated by the Rodrigues-type formula can grow linearly in ||v||. The manuscript must therefore exhibit an explicit estimate showing that any cross-term contributions remain absorbed inside the 1 + ||θ||_F envelope (see the block-matrix representation of exp(θ) and the norm equivalence used).

minor comments (1)

[Upper- and lower-bound statements] The numerical constants 4.02 and 0.0505 are stated without reference to the precise floating-point precision or the interval over which they were computed; a short appendix tabulating the derivation steps would improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment on the operator-norm bound. We address the point below and will revise the manuscript to make the argument fully explicit.

read point-by-point responses

Referee: [Main theorem on operator-norm bound (abstract and §3)] The central operator-norm claim ||exp(θ)||_op ≤ 1 + ||θ||_F must be verified on the full semidirect-product structure se(3) = so(3) ⋉ ℝ³. The contraction property controls only the rotational block; the translational component t(ω,v) generated by the Rodrigues-type formula can grow linearly in ||v||. The manuscript must therefore exhibit an explicit estimate showing that any cross-term contributions remain absorbed inside the 1 + ||θ||_F envelope (see the block-matrix representation of exp(θ) and the norm equivalence used).

Authors: We agree that an explicit verification on the full semidirect-product structure is necessary for clarity. The proof in §3 already invokes the block-matrix representation of exp(θ) together with the equivalence between the operator norm on se(3) and the Frobenius norm to control the translational block generated by the Rodrigues formula. Nevertheless, to satisfy the referee's request we will insert a dedicated paragraph that isolates the translational contribution t(ω,v), derives the bound ||t(ω,v)|| ≤ ||θ||_F directly from the series expansion, and confirms that all cross terms remain inside the 1 + ||θ||_F envelope. This addition will render the argument self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds from external contraction property and explicit construction

full rationale

The central operator-norm bound is obtained by direct appeal to the standard contraction property of the SO(3) left Jacobian (an external fact about so(3), not derived or cited from the authors' prior work). The O(R^2) upper bound on the gradient Lipschitz constant is stated as self-contained with an explicit numerical constant. The quadratic lower-growth example is produced by explicit construction of a specific objective J*, not by fitting parameters to data or by any self-referential definition. No load-bearing step reduces to a self-citation chain, ansatz smuggled via citation, or renaming of a known result. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard Lie-algebra properties together with the contraction property of the SO(3) left Jacobian; no free parameters, new entities, or ad-hoc axioms beyond domain assumptions are indicated.

axioms (1)

domain assumption Contraction property of the SO(3) left Jacobian
Invoked to prove the operator-norm bound ||exp(theta)||_op <= 1 + ||theta||_F.

pith-pipeline@v0.9.1-grok · 5694 in / 1301 out tokens · 40906 ms · 2026-06-28T20:44:39.531572+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 15 canonical work pages

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R. Mahony and J. H. Manton. The geometry of the Newton method on non-compact L ie groups. J.\ Global Optim., 23(3--4):309--327, 2002. doi:10.1023/A:1016586831090

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P. Birtea and D. Com a nescu. Hessian operators on constraint manifolds. J.\ Nonlinear Sci., 25(6):1285--1305, 2015. doi:10.1007/s00332-015-9256-7

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[1] [1]

B. C. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed. Springer, Cham, 2015. doi:10.1007/978-3-319-13467-3

work page doi:10.1007/978-3-319-13467-3 2015

[2] [2]

Gallier and J

J. Gallier and J. Quaintance. Differential Geometry and Lie Groups: A Computational Perspective. Springer, Cham, 2020. doi:10.1007/978-3-030-46040-2

work page doi:10.1007/978-3-030-46040-2 2020

[3] [3]

Absil, R

P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ, 2008. doi:10.1515/9781400830244

work page doi:10.1515/9781400830244 2008

[4] [4]

N. Boumal. An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge, 2023. doi:10.1017/9781009166164

work page doi:10.1017/9781009166164 2023

[5] [5]

N. J. Higham. Functions of Matrices: Theory and Computation. SIAM, Philadelphia, PA, 2008. doi:10.1137/1.9780898717778

work page doi:10.1137/1.9780898717778 2008

[6] [6]

A Representation--Optimization Trichotomy for Objectives Parameterized via Matrix Exponential Maps

Sooraj K.C and Vivek Mishra. A Representation--Optimization Trichotomy for Objectives Parameterized via Matrix Exponential Maps. Preprint, Zenodo, 2026. doi:10.5281/zenodo.20302838

work page doi:10.5281/zenodo.20302838 2026

[7] [7]

A. W. Knapp. Lie Groups Beyond an Introduction, 2nd ed. Birkh\" a user, Boston, 2002. ISBN: 978-0-8176-4259-4

2002

[8] [8]

R. M. Murray, Z. Li, and S. S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton, FL, 1994

1994

[9] [9]

F. C. Park and B. J. Martin. Robot sensor calibration: solving AX=XB on the Euclidean group. IEEE Trans.\ Robotics Automat., 10(5):717--721, 1994. doi:10.1109/70.326576

work page doi:10.1109/70.326576 1994

[10] [10]

T. D. Barfoot and P. T. Furgale. Associating uncertainty with three-dimensional poses for use in estimation problems. IEEE Trans.\ Robotics, 30(3):679--693, 2014. doi:10.1109/TRO.2014.2298059

work page doi:10.1109/tro.2014.2298059 2014

[11] [11]

A micro lie the- ory for state estimation in robotics.arXiv preprint arXiv:1812.01537,

J. Sol\` a , J. Deray, and D. Atchuthan. A micro L ie theory for state estimation in robotics. arXiv preprint arXiv:1812.01537, 2018

work page arXiv 2018

[12] [12]

C. S. Kenney and A. J. Laub. Condition estimates for matrix functions. SIAM J.\ Matrix Anal.\ Appl., 10(2):191--209, 1989. doi:10.1137/0610014

work page doi:10.1137/0610014 1989

[13] [13]

Dieci and A

L. Dieci and A. Papini. Conditioning of the exponential of a block triangular matrix. Numer.\ Algorithms, 28:137--150, 2001. doi:10.1023/A:1014071202885

work page doi:10.1023/a:1014071202885 2001

[14] [14]

Iserles, H

A. Iserles, H. Z. Munthe-Kaas, S. P. N rsett, and A. Zanna. Lie-group methods. Acta Numer., 9:215--365, 2000. doi:10.1017/S0962492900002154

work page doi:10.1017/s0962492900002154 2000

[15] [15]

Munthe-Kaas

H. Munthe-Kaas. Runge--Kutta methods on L ie groups. BIT Numer.\ Math., 38(1):92--111, 1998. doi:10.1007/BF02510919

work page doi:10.1007/bf02510919 1998

[16] [16]

Mahony and J

R. Mahony and J. H. Manton. The geometry of the Newton method on non-compact L ie groups. J.\ Global Optim., 23(3--4):309--327, 2002. doi:10.1023/A:1016586831090

work page doi:10.1023/a:1016586831090 2002

[17] [17]

Birtea and D

P. Birtea and D. Com a nescu. Hessian operators on constraint manifolds. J.\ Nonlinear Sci., 25(6):1285--1305, 2015. doi:10.1007/s00332-015-9256-7

work page doi:10.1007/s00332-015-9256-7 2015