用线性最小二乘法精准估算渐近展开中的未知参数，还能保证收敛速度。
Learning Asymptotics with Convergence-Rate Guarantees using Linear Least Squares
- 基于滑动最小二乘与Tikhonov正则化，优化求解渐近参数
- 严格证明收敛条件与收敛速率，确保结果可靠性
- 适用于组合数学中的渐近计数，补充传统方法不足
我们提出一个新的研究领域——渐近学习理论（ALT），将优化与渐近分析相结合。该理论提供了一种统一方法，用于通过优化理论计算已知渐近展开中未知的常数或参数。本文聚焦于一类广泛适用的渐近形式，研究了两种强大的数值方法：滑动线性最小二乘法（sLLSQ）和滑动Tikhonov线性最小二乘法（sT-LLSQ）。对这两种方法，我们严格证明了渐近估计，给出了收敛至正确参数值的充分条件及收敛速率保证。尽管优势显著，但两者在某些情况下仍存在收敛缓慢甚至反常发散的问题。此外，我们在解析组合学领域展示了基础应用，该领域利用复分析研究离散结构的渐近计数。所提方法补充了比值法及其变体等现有手段。数值实验验证了理论结果。最后，文章讨论了ALT领域的若干潜在研究方向。
原文摘要 · Abstract (English)
We introduce a new research area that is called Asymptotics Learning Theory (ALT) and combines optimization with asymptotic analysis. In particular, ALT provides a unified approach for computing unknown constants/parameters in proven asymptotic expansions using optimization theory. In this paper, we focus on a general asymptotic form which includes a broad class of asymptotics. Furthermore, we study two powerful numerical methods, namely, sliding Linear Least Squares (sLLSQ) and sliding Tikhonov Linear Least Squares (sT-LLSQ). For these techniques we rigorously prove asymptotic estimates that lead to sufficient conditions for convergence (to the correct values of unknown parameters) and convergence-rate guarantees. Despite their strengths, both methods have also limitations, e.g., slow convergence---or even, counterintuitively, divergence---in some cases. Moreover, we present fundamental applications in analytic combinatorics, a beautiful field of mathematics that deals with asymptotic enumeration of discrete structures using complex analysis. The proposed techniques complement existing approaches, such as the ratio method and its variants. Numerical examples also verify the theoretical results. Finally, we discuss interesting research directions in ALT.
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