发表机构
National Technical University of Athens（雅典国立技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入渐近学习理论（ALT），结合优化与渐近分析，聚焦一般渐近形式，研究滑动线性最小二乘法等两种数值方法，证明其渐近估计、收敛条件及速率保证，展示在解析组合学中的应用，验证理论结果并讨论研究方向。
AI 中文摘要
我们引入了一个名为渐近学习理论（ALT）的新研究领域，它将优化与渐近分析相结合。ALT为利用优化理论计算已证渐近展开中的未知常数/参数提供了统一方法。本文聚焦于一种包含广泛渐近性的一般渐近形式，研究了滑动线性最小二乘法（sLLSQ）和滑动蒂霍诺夫线性最小二乘法（sT - LLSQ）两种数值方法，严格证明了渐近估计，给出收敛条件和收敛速率保证。两种方法虽有优势但也有局限，如有时收敛慢甚至发散。我们还展示了在解析组合学中的基础应用，数值例子验证了理论结果，最后讨论了ALT有趣的研究方向。
英文摘要
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.
Comments62 pages, 7 tables, 5 figures; additional references
