[Paper Review] The inverse moment problem for convex polytopes: implementation aspects

Nick Gravin, Danny Nguyen+2 more|arXiv (Cornell University)|Sep 10, 2014

Computational Geometry and Mesh Generation3 references3 citations

TL;DR

This paper presents an efficient computational implementation of the inverse moment problem for convex polytopes, reconstructing N-vertex polytopes in ℝᵈ from O(Nd) moments using a modified Prony method with complex projections. It demonstrates that complex moment analysis significantly improves numerical stability and vertex recovery precision, especially under finite-precision arithmetic, though sensitivity to numerical errors remains a key challenge for practical deployment.

ABSTRACT

We give a detailed technical report on the implementation of the algorithm presented in Gravin et al. (Discrete & Computational Geometry'12) for reconstructing an $N$-vertex convex polytope $P$ in $\mathbb{R}^d$ from the knowledge of $O(Nd)$ of its moments.

Motivation & Objective

• To implement and optimize the inverse moment algorithm from [GLPR12] for reconstructing convex polytopes from moment data.
• To evaluate the performance and numerical stability of the algorithm under exact rational arithmetic and finite-precision floating-point arithmetic.
• To improve vertex projection recovery by using complex-valued directions instead of real-only projections.
• To assess the sensitivity of the method to numerical errors in moment measurements.
• To establish practical guidelines for precision requirements in real-world applications.

Proposed method

• The algorithm uses axial moments μⱼ(𝐳) along a direction 𝐳 ∈ ℝᵈ to construct a Hankel matrix from moment data.
• It applies a variant of the Prony method to solve for the projections ⟨𝐯ᵢ, 𝐳⟩ of vertices onto 𝐳 via the roots of a characteristic polynomial derived from the kernel of the Hankel matrix.
• The method leverages complex-valued directions 𝐳 = 𝐚 + i𝐛 to enhance root-finding accuracy and recover all vertex projections even when real projections fail due to numerical instability.
• A Pade approximation-based procedure is used to recover projections on complex planes or real axes, improving robustness over the original method.
• The approach assumes uniform density and simple polytopes (each vertex has d+1 incident edges), simplifying the geometric structure and moment equations.
• Theoretical foundations rely on moment formulas involving tangent cones and determinants of edge vectors at each vertex, linking moments to vertex geometry.

Experimental results

Research questions

• RQ1 Can the inverse moment problem for convex polytopes be efficiently implemented with high numerical precision using finite-precision arithmetic?
• RQ2 How does using complex projections improve the robustness and accuracy of vertex recovery compared to real projections?
• RQ3 What level of numerical precision is required to reliably reconstruct polytope vertices from moment data?
• RQ4 How sensitive is the algorithm to numerical errors in moment measurements, especially in floating-point arithmetic?
• RQ5 Can harmonic moments (harmonic polynomials) be used to reconstruct vertices of both convex and non-convex polytopes?

Key findings

• Using complex projections instead of real projections significantly improves the recovery of all vertex projections, especially when numerical precision is limited.
• With 25-bit precision in real arithmetic, the algorithm missed some vertex projections due to spurious root loss in the characteristic polynomial.
• The same problem was resolved with complex arithmetic at 25-bit precision, recovering all 8 projections with high accuracy.
• The method achieves near-exact reconstruction under exact rational arithmetic, confirming theoretical correctness.
• Float arithmetic reduces running time dramatically—e.g., from over 10 minutes to 7.10 seconds for 40 vertices—while maintaining low reconstruction error (order E-3).
• The required precision for stable reconstruction increases with polytope size: 75-bit precision was needed for 20 vertices, and 210 bits for 40 vertices, to maintain shape fidelity.