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Geodesic strong convexity does not imply forward invariance under gradient flow on SO(3): a certified counterexample

Dongming Wang, Wei Ren

arXiv:2608.28976v1eess.SY

TL;DR

Strong geodesic convexity need not make a geodesic ball forward invariant under gradient flow. The paper constructs and certifies a quadratic SO(3) counterexample, obtaining exact outward boundary motion and proving exit despite an interior minimizer. Its scope is a single-agent obstruction, while numerical excursion values are illustrative rather than certified.

  • Problem

    The paper asks whether strong geodesic convexity with an interior minimizer guarantees forward invariance of the operating ball under gradient flow.

  • Method

    The paper analyzes a principal-logarithmic-chart quadratic cost on SO(3), certifying its geodesic Hessian with rigorous ball arithmetic over exact rational inputs.

  • Results

    The certified example has Hess f ⪰ 0.172 I3 and exact outward boundary speed 21/500, with continuity proving that the flow exits the ball.

  • Takeaways & Limitations

    Strong convexity constrains the gradient toward the minimizer, not its projection onto the inward radial direction of a differently centered ball.

  • Takeaways & Limitations

    The obstruction is single-agent, and the numerical peak excursion near 0.3143 is not interval-certified.

Abstract

from arXiv · show

Let $mathcal{C}=\overline{\mathcal{B}}_ρ(R_c)$ be a geodesic ball of radius $ρ<π/2$ in SO(3) with the bi-invariant metric, and let $f$ be geodesically strongly convex on $\mathcal{C}$ with an interior minimizer. It is tempting to expect the gradient flow $\dot R=R(-\nabla f)^\wedge$ to keep $\mathcal{C}$ forward invariant: the flow is attracted to an interior point, and strong convexity appears to leave no room for outward motion. We show this expectation is false by an explicit, fully certified construction with $ρ=0.3$: a cost, quadratic in the principal logarithmic chart with off-diagonal coupling $0.7$, whose geodesic Hessian satisfies $\Hess f\succeqμI_3$ on all of $\mathcal{C}$ with a machine-certified modulus $μ\geq0.172$, rigorous ball arithmetic over exact rational inputs, yet whose descent velocity at a boundary point has the exact rational outward radial component $21/500$. A continuity corollary of the exact rate certifies that the flow exits the ball; numerical integration puts the peak excursion near $0.3143$ before convergence to the minimizer. The mechanism is elementary: strong convexity constrains the projection of the gradient onto the minimizer direction, not onto the inward radial direction. Code reproducing every certified constant and figure accompanies the note.

1 Introduction

The paper asks whether strong geodesic convexity guarantees forward invariance of a geodesic ball under gradient flow, and answers negatively with a certified SO(3) counterexample.

  • Motivation: Strong convexity does not ensure that a metric ball is forward invariant under gradient flow.A descent direction can point outward because the ball need not be a sublevel set of the cost.
  • Certified counterexample: The construction uses a quadratic principal-logarithmic-chart cost on B0.3(I3) with off-diagonal coupling 0.7.Its geodesic Hessian is certified on the entire ball.
  • Certified counterexample: The Hessian satisfies Hess f ⪰ 0.172 I3, while the boundary descent velocity has exact outward radial speed 21/500.The certification uses rigorous ball arithmetic over exact rational inputs.
  • Trajectory consequence: A continuity argument proves that the gradient flow exits the ball, with numerical integration placing the peak excursion near 0.3143.The numerical trajectory is explicitly presented as an illustration, while the exit itself is certified.
  • Repair: The paper also identifies inward-pointing boundary inequalities as the condition needed to restore strong invariance.Such inequalities hold automatically for suitable smooth radial costs.

2 Setup and main results

The paper constructs a chart-quadratic cost on SO(3) that is strongly convex with an interior minimizer yet violates the inward-pointing boundary condition and exits its ball.

  • Setup: The setup fixes the bi-invariant metric, center I3, radius ρ = 0.3, and principal chart containing the ball.A smooth cutoff extends the cost globally without changing statements on the ball.
  • Setup: The cost has a unique minimizer over the ball, with the minimizer strictly inside the chart ball.The construction uses a positive quadratic form in principal-log coordinates.
  • Boundary violation: At the boundary point Rb, the exact outward radial speed is 21/500 = 0.042 rad/s.The boundary violation is obtained by an exact block-structure argument.
  • Open family: For the family Hc = I3 + c(e1e1⊤ + e1e1⊤), outward motion occurs when a + cb > ρ while a2 + b2 < ρ2 keeps the minimizer inside.The certified instance is (a,b,c,ρ) = (0.23,0.16,0.7,0.3).
  • Main conclusion: The certified example satisfies strong convexity and bounded gradients, yet invariance still fails.The gradient bound is ∥∇f∥ ≤ 0.991 on C.

3 Second-order calculus in the principal chart

The second-order analysis differentiates the chart-quadratic cost along SO(3) geodesics, separating the pushed-forward chart Hessian from the correction induced by the inverse right Jacobian.

  • Right Jacobian: The inverse right Jacobian and three positive-coefficient scalar series control the chart derivatives throughout the principal domain.The series extensions make the formulas smooth at zero.
  • Geodesic differentiation: Along a geodesic, v(t) = Jr^-1(ϕ(t))ξ gives ϕ˙ = v and f˙ = q⊤Hv.Here q = ϕ − ϕ* and ξ is the constant body velocity.
  • Hessian formula: The exact second derivative contains a principal pushforward term and a correction from the derivative of Jr^-1.The formula reduces to Hess f = H at ϕ = 0.
  • Bounding the correction: The derivative bound exploits a sharpening that makes a key factor O(θ) rather than O(1).Without this sharpening, the certified Hessian budget would fail.
  • Endpoint certification: Because C1 has positive coefficients, it is increasing on [0, 0.3], so its supremum is attained at θ = 0.3.This reduces the rigorous certificate to endpoint evaluation without interval subdivision.

4 Certified evaluation

The certificate rigorously bounds the geodesic Hessian over the entire radius-0.3 ball using exact rational inputs, positive-coefficient series, and endpoint ball arithmetic, establishing strong convexity with modulus at least 0.172.

  • Certified bounds: 1.00375987 bounds κ(0.3), while β(0.3) lies in [0.08345860, 0.08345861] under outward-rounded propagation.The enclosure uses a positive-coefficient series and a tail bounded by 10^-11.
  • Endpoint method: sup[0,0.3] C1 = C1(0.3) ≤ 0.12909469 because the positive-coefficient construction makes C1 increasing.Thus the supremum requires only endpoint evaluation, not interval subdivision.
  • Certified bounds: 0.17267346 is the certified lower bound for the Hessian modulus before concluding Hess f ⪰ 0.172 I3 on C.The bound combines the exact λmin(H) value with certified correction terms.
  • Certification procedure: The computation uses Arb ball arithmetic at 256-bit precision with exact rational inputs and machine-verified inequalities.The reported inputs include 3/10 and 7/10, and the certificate is independently cross-checked numerically.
  • Certification procedure: Figure 1 compares sampled minimum Hessian eigenvalues across 4000 chart-ball points with Proposition 2’s certified lower bound.The caption reports a wide strong-convexity margin throughout the ball.

5 The exact boundary violation

At a boundary point, the certified descent velocity has an exact positive outward radial component despite pointing toward the interior minimizer. The violation extends over a boundary arc characterized by a negative quadratic form.

  • Geometric picture: The ϕ3 = 0 slice shows tilted level-set ellipses, the radius-0.3 operating disk, the interior minimizer, and the violating boundary arc.The plotted arrow is the chart velocity.
  • Exact boundary violation: 21/500 is the exact outward radial speed of the gradient flow at Rb.The result follows from a block-structure argument without transcendental evaluation.
  • Exact boundary violation: The velocity has a positive projection toward R∗ yet an outward radial component relative to the ball center.Thus attraction to the minimizer does not imply inward motion at the boundary.
  • Geometric picture: The violating arc is exactly the set where ϕ⊤H(ϕ − ϕ∗) is negative.Figure 3 displays an entire arc of violation around Rb in the e1–e2 plane.

6 The gradient flow exits the ball

The paper certifies that the gradient flow exits the geodesic ball immediately from a boundary point, while the later excursion and convergence behavior are numerical.

  • The exit itself is a theorem, not a numerical observation.Corollary 1 establishes immediate departure from the ball using the exact initial radial rate.
  • Figure 3 displays an open boundary arc where the radial gradient component violates the inward-pointing condition.
  • 21/500 is the exact positive outward radial component at the boundary point Rb.
  • The radial comparison cost started at the same boundary point never exits the ball.Its distance decreases monotonically toward the minimizer.
  • The numerical trajectory peaks near 0.3143 before converging toward the interior minimizer.The peak and subsequent convergence are explicitly identified as numerical observations.

7 Discussion: mechanism, scope, and the sufficient repair

The counterexample arises from misalignment between the minimizer direction and the ball’s inward radial direction, so strong convexity alone does not ensure boundary invariance. An inward-pointing boundary inequality supplies the additional sufficient condition.

  • Mechanism: Strong convexity controls the gradient component toward the minimizer, not the inward radial component of a ball centered elsewhere.At Rb, the two directions differ by 66.5°, and the coupling H12 = 0.7 produces the outward projection.
  • Scope: The obstruction applies to a single-agent flow and is graph-independent as a reason not to derive boundary behavior from convexity constants alone.Consensus terms may oppose outward descent on particular trajectories, so the paper does not claim every distributed protocol exits.
  • The sufficient repair: The inward-pointing condition e_c^T∇f(R) ≤ 0 on the boundary is sufficient for invariance under the single-valued gradient flow.The boundary radial derivative is the negative of this projection, enabling Nagumo’s theorem.
  • The sufficient repair: Radial costs with a nonnegative radial derivative and a minimizer inside the ball satisfy the inward-pointing condition automatically.
  • Reproducibility: The accompanying code reproduces the certified constants, exact violation, numerical checks, and figures from exact rational inputs and rigorous ball arithmetic.

A Explicit Tail Enclosures

The appendix certifies the truncated series and derived quantities over the full interval [0, 3/10] using positive-coefficient endpoint bounds and explicit remainder estimates.

  • All three quantities are evaluated at θ = 3/10 and bounded over [0, 3/10] by a positive-coefficient endpoint argument.The remaining analytic input is a remainder bound for each truncated series.
  • 7.4 × 10^-11 bounds the β′ tail after truncation through θ^5.The bound follows from shrinking consecutive tail terms and dominance by the k = 5 term.
  • κ is enclosed by evaluating its closed form with β’s truncation and the tail bound, adding radius below 10^-12.
  • 1.37 × 10^-12 and 3.66 × 10^-11 are the measured remainders for β and β′ at θ = 3/10.The high-precision numerical checks lie inside the displayed bounds.
  • κ(3/10) = 1.00375987 is reproduced to all shown digits by the verification script.
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