구형 포장 상한을 위한 AI 기반 모델‑기반 탐색
📝 Abstract
Sphere packing, Hilbert’s eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension n = 8, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions 4-16, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, modelbased search can make tangible progress on mathematically rigid, evaluationlimited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.
💡 Analysis
Sphere packing, Hilbert’s eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension n = 8, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions 4-16, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, modelbased search can make tangible progress on mathematically rigid, evaluationlimited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.
📄 Content
Artificial intelligence tools have shown an exceptional ability to solve mathematical problems from a wide spectrum of difficulty levels [1,2,14]. One such tool is AlphaEvolve [13,15], a system that has effectively utilised LLMs to tackle problems that are hard to solve but easy to evaluate (i.e., allow for fast evaluation of candidate solutions). These problems, known in theoretical computer science as NP-complete, involve tasks where determining an exact solution is difficult, yet verifying the quality of any proposed solution is typically straightforward.
However, NP-complete problems represent only a subset of a larger class known as NPhard problems, where even verifying candidate solutions can be complex and (frustratingly) slow. One prominent example in such a class is that of sphere packing, which aims to fill the Euclidean space R n with spheres of the same or different volumes to maximise their spatial coverage. Despite its seemingly simple and abstract formulation, this problem has seen widespread applications, e.g., in biology [8], chemistry [12], and medicine [19], to name a few. Finding optimal sphere-packing densities is hard! In fact, it is much harder than local setups like kissing number problems, as sphere packing requires understanding infinitely large configurations, not just a single neighbourhood. The arrangement of spheres in one region must remain consistent with placements arbitrarily far away, creating long-range constraints that do not appear in local problems. As the dimension grows, these global dependencies only become more complex, making density calculations even more difficult.
Interestingly, the roots of the sphere packing problem go back over two millennia, to Euclid’s exploration of space and volume around 300 BC. Yet despite centuries of mathematical progress, the problem remains unsolved in most dimensions. To date, optimal arrangements of identical unit-volume spheres have been discovered only for n=2 [9], 3 [10], 8 [18], and 24 [6]. The eight-dimensional case, proved by Maryna Viazovska and recognised with the 2022 Fields Medal, highlights the enduring significance of sphere packing in contemporary mathematics.
For dimensions where optimal arrangements are still unknown, researchers have focused on deriving rigorous upper and lower bounds on the maximum packing density, which are available for all dimensions [3]. Importantly, improving upper bounds for packing problems has proven to be a powerful strategy for identifying optimal sphere configurations. Notably, Cohn et al. [4] improved the upper bounds in dimensions n = 8 and n = 24, showing densities close to those of the E 8 root lattice and the Leech lattice, respectively. This work provided compelling evidence that these lattices might indeed be optimal and guided subsequent efforts to prove it. Building on this foundation, Viazovska later proved the optimality of the E 8 lattice by further refining the upper bound using a distinct method, which was subsequently extended to the Leech lattice case. Thus, improvements in upper bounds not only constrain possible configurations but can also illuminate the path toward exact optimality proofs.
Among other approaches, the three-point method proposed by Cohn et al. [5] attains state-of-the-art results by solving carefully designed semidefinite programming (SDP) problems, where the optimal objective value of those SDPs directly provides an upper bound on the maximum packing density. Each such SDP instance requires two main components: 1) a set of continuous geometric parameters in 2048-bit precision (approximately 617 digits in decimal representation) and 2) a set of specifically crafted constraints formulated in terms of matrix polynomials. The search for these parameters and the corresponding matrix polynomial representations has become a hallmark of recent progress in this area. Different parameterisations yield distinct SDP formulations, each defining a new optimisation landscape whose solutions may yield tighter upper bounds on packing density.
Given that the discovery of improved sphere packing bounds can be formulated as a search over continuous parameters and polynomial constraint representations, a natural question arises: Can AI-assisted methods accelerate this search and uncover new stateof-the-art results on such long-standing mathematical problems? At first glance, recent breakthroughs such as AlphaEvolve [17] and AlphaGeometry [2] suggest so, showcasing the promise of AI in mathematical discovery. However, these systems rely primarily on modelfree and evolutionary search strategies, where learning proceeds through extensive trial and error, i.e., evaluating vast numbers of candidates to gradually refine a policy or population. Such approaches thrive in domains where solutions are hard to find but cheap to evaluate. In contrast, our setting lies at the opposite extreme: evaluating a single candidate, by solving its corresponding semidefinite program, can t
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