揭示无限维空间中无条件收敛的七种等价条件，为算法稳定性提供理论支撑。
Algorithmic Stability in Infinite Dimensions: Characterizing Unconditional Convergence in Banach Spaces
- 提出七种等价条件统一刻画无条件收敛
- 证明排列不变性与系数截断在算法中的有效性
- 适合研究算法稳定性和泛函分析交叉领域的学者
无限维空间中条件收敛、无条件收敛和绝对收敛的区别对计算算法具有根本影响。尽管在有限维空间三者等价，但Dvoretzky-Rogers定理表明它们在一般Banach空间中严格分离。本文提出一个综合性刻画定理，将七种等价条件统一：排列不变性、网收敛性、子级数检验、符号稳定性、有界乘子性质以及弱一致收敛性。这些理论结果直接指导算法稳定性分析，规范随机梯度下降中梯度累积的排列不变性，并支持基于框架的信号处理中的系数阈值化。本工作连接经典泛函分析与现代计算实践，为顺序无关且数值鲁棒的求和过程提供严格基础。
原文摘要 · Abstract (English)
The distinction between conditional, unconditional, and absolute convergence in infinite-dimensional spaces has fundamental implications for computational algorithms. While these concepts coincide in finite dimensions, the Dvoretzky-Rogers theorem establishes their strict separation in general Banach spaces. We present a comprehensive characterization theorem unifying seven equivalent conditions for unconditional convergence: permutation invariance, net convergence, subseries tests, sign stability, bounded multiplier properties, and weak uniform convergence. These theoretical results directly inform algorithmic stability analysis, governing permutation invariance in gradient accumulation for Stochastic Gradient Descent and justifying coefficient thresholding in frame-based signal processing. Our work bridges classical functional analysis with contemporary computational practice, providing rigorous foundations for order-independent and numerically robust summation processes.
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