[Paper Review] Discrepancy, separation and Riesz energy of finite point sets on compact connected Riemannian manifolds

Paul Leopardi|arXiv (Cornell University)|Mar 26, 2014

Mathematical Approximation and Integration21 references3 citations

TL;DR

This paper establishes that on a smooth, compact, connected Riemannian manifold, any admissible sequence of finite point sets that is both asymptotically equidistributed and well-separated exhibits Riesz $ s $-energy converging to the continuous energy double integral, with convergence rate governed by geodesic ball discrepancy. The result generalizes prior findings on the sphere to general Riemannian manifolds using discrepancy-based estimates from Blümlinger (1988).

ABSTRACT

On a smooth compact connected $d$-dimensional Riemannian manifold $M$, if $0 < s < d$ then an asymptotically equidistributed sequence of finite subsets of $M$ that is also well-separated yields a sequence of Riesz $s$-energies that converges to the energy double integral, with a rate of convergence depending on the geodesic ball discrepancy. This generalizes a known result for the sphere.

Motivation & Objective

• To extend known results on Riesz energy convergence from the unit sphere to general compact connected Riemannian manifolds.
• To establish a quantitative convergence rate for Riesz $ s $-energy of finite point sets based on geodesic ball discrepancy.
• To formalize the interplay between equidistribution, separation, and energy minimization on Riemannian manifolds.
• To provide a framework for analyzing energy convergence using discrepancy theory in a Riemannian geometric setting.
• To bridge the gap between discrete energy minimization and continuous energy integrals on manifolds using geometric and measure-theoretic tools.

Proposed method

• Define the normalized Riesz $ s $-energy of a finite point set $ X $ on a $ d $-dimensional Riemannian manifold $ M $, using the geodesic distance as the kernel.
• Introduce the normalized ball discrepancy $ ilde{rak{D}}(X) $ as a measure of how uniformly $ X $ is distributed relative to the volume measure $ \sigma_M $.
• Use Blümlinger's discrepancy estimates (1988) to bound the difference between discrete and continuous energy integrals via discrepancy norms.
• Establish that for $ 0 < s < d $, the Riesz $ s $-energy of $ X $ converges to the double integral of the Riesz kernel over $ M \times M $, with rate depending on $ \tilde{\frak{D}}(X) $.
• Define admissible sequences as those with both discrepancy $ \tilde{\frak{D}}(X_\ell) < \delta(N_\ell) $ and separation $ \operatorname{dist}(x,y) > \Delta(N_\ell) $, where $ \delta, \Delta \to 0 $ as $ N_\ell \to \infty $.
• Leverage the compactness and smoothness of $ M $ to ensure uniform bounds on the Riesz kernel and its integrability, enabling the use of discrepancy-based error estimates.

Experimental results

Research questions

• RQ1 Can the convergence of Riesz $ s $-energy for finite point sets on the sphere be generalized to arbitrary compact connected Riemannian manifolds?
• RQ2 What is the dependence of the Riesz $ s $-energy convergence rate on the geometric distribution of the point set?
• RQ3 How do discrepancy and separation properties of a point set influence its Riesz energy on a Riemannian manifold?
• RQ4 Is there a general discrepancy-based error estimate that bounds the difference between discrete and continuous Riesz energy on manifolds?
• RQ5 Can the methods used for spherical codes be adapted to prove energy convergence on general Riemannian manifolds with geodesic distance kernels?

Key findings

• For any smooth, compact, connected $ d $-dimensional Riemannian manifold $ M $, and for $ 0 < s < d $, the Riesz $ s $-energy of an admissible sequence of $ M $-codes converges to the continuous energy double integral.
• The rate of convergence of the Riesz $ s $-energy is bounded by a constant multiple of the normalized ball discrepancy $ \tilde{\frak{D}}(X) $, which quantifies the uniformity of point distribution.
• Well-separated sequences with separation $ \Delta(N) = \Omega(N^{-1/d}) $ exist on such manifolds, and this order is optimal due to volume packing constraints.
• The result generalizes prior work on the sphere by showing that the same convergence mechanism holds in the broader Riemannian setting using geodesic distance.
• The proof relies on adapting Blümlinger’s discrepancy estimates to the Riemannian context, establishing a quantitative link between distribution uniformity and energy convergence.
• The framework excludes diagonal terms in the energy sum, ensuring finite discrete energy even for $ s < d $, which is essential for convergence analysis.