A Linear-Time Construction on $\mathbb{S}^2$ Achieving a Sublinear Energy Gap on Smale's Seventh Problem
Rishabh Kabra, Wim Dehaen, Andrea Karlova, Mark Hamilton, Joao Carreira, Maks Ovsjanikov, Jonathan T. Barron
原文摘要
Smale's seventh problem asks for a polynomial-time algorithm that places $N$ points on the unit sphere $\mathbb{S}^2$ with logarithmic energy (the sum of $-\log$ of all pairwise distances) within $O(\log N)$ of the minimum. The minimal energy has an asymptotic expansion with explicit terms of order $N^2$ and $N\log N$ and a linear term whose constant is unknown. Brauchart, Hardin, and Saff conjectured this constant; the resulting expansion, $E_{BHS}(N)$, is known to be an upper bound on the minimum, up to $o(N)$. All previously analyzed constructions on $\mathbb{S}^2$ have energy exceeding $E_{BHS}(N)$ by an amount that grows at least linearly in $N$, i.e. $Ω(N)$, far from the $O(\log N)$ accuracy sought by Smale. We present TerracedZonal, a deterministic construction that generates $N$ points in $Θ(N)$ time and memory. Points are placed on latitude rings whose populations are held constant across bands ("terraces") chosen by a one-dimensional dynamic program, so that adjacent rings within each terrace lock into a staggered triangular lattice. Under a locality assumption, we prove that the energy exceeds $E_{BHS}(N)$ by only $Θ(N^{2/3})$, with an explicit constant; exact evaluation up to $N = 10^9$ confirms this rate. Under the same assumption, its distance from the true minimum is sublinear if the conjecture holds. To our knowledge, this is the first algorithmic construction with a sublinear excess over $E_{BHS}(N)$ of explicit order (proven under the locality assumption), a necessary step toward Smale's seventh problem. With one parameter changed, the same algorithm also gives an equal-weight integration rule on $\mathbb{S}^2$ that runs just as fast and whose error decays at an empirical rate of $N^{-4}$ on smooth functions, versus $N^{-3/2}$ for every previous fast construction we tested.
cs.CG · math.MG