提出新复杂度度量，解释学习核谱算法为何泛化更好
Alignment-Sensitive Minimax Rates for Spectral Algorithms with Learned Kernels
- 引入有效跨度维数（ESD），融合信号、谱与噪声，衡量学习核的复杂性
- 当ESD≤K时，最小最大风险为σ²K，揭示泛化误差上限
- 证明梯度流可降低ESD，连接自适应特征学习与泛化提升
我们研究从数据中学习核的谱算法。提出有效跨度维数（ESD），一种依赖于信号、谱和噪声水平σ²的对齐敏感复杂度度量，适用于任意核与信号，无需特征衰减或源条件假设。我们证明：对于ESD不超过K的序列模型，最小最大过风险为σ²K。进一步分析过参数化梯度流，证明其能降低ESD。这一发现建立了自适应特征学习与谱算法泛化性能提升之间的理论联系。我们将ESD框架扩展至线性模型和再生核希尔伯特空间（RKHS）回归，并通过数值实验验证理论。该框架为超越传统固定核理论的泛化提供了新视角。
原文摘要 · Abstract (English)
We study spectral algorithms in the setting where kernels are learned from data. We introduce the effective span dimension (ESD), an alignment-sensitive complexity measure that depends jointly on the signal, spectrum, and noise level $σ^2$. The ESD is well-defined for arbitrary kernels and signals without requiring eigen-decay conditions or source conditions. We prove that for sequence models whose ESD is at most $K$, the minimax excess risk scales as $σ^2 K$. Furthermore, we analyze over-parameterized gradient flow and prove that it can reduce the ESD. This finding establishes a connection between adaptive feature learning and provable improvements in generalization of spectral algorithms. We demonstrate the generality of the ESD framework by extending it to linear models and RKHS regression, and we support the theory with numerical experiments. This framework provides a novel perspective on generalization beyond traditional fixed-kernel theories.
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