在紧阿贝尔群上学习卷积算子，给出理论保证与频域解释。
Learning convolution operators on compact Abelian groups
- 基于正则化方法，用岭回归估计卷积核。
- 在有限样本下提供误差上界，证明了方法有效性。
- 揭示正则性假设的频域局部化新含义，适合理论研究者。
我们研究紧阿贝尔群相关卷积算子的学习问题。采用基于正则化的方案，在卷积核属于平移不变希尔伯特空间的自然正则条件下，分析了自然的岭回归（RR）估计器。结合现有岭回归结果，以有限样本界刻画了估计器的准确性。有趣的是，经典岭回归中的正则性假设，在空间/频率局部化意义上具有新颖且自然的解释。理论结果通过数值模拟得以验证。
原文摘要 · Abstract (English)
We consider the problem of learning convolution operators associated to compact Abelian groups. We study a regularization-based approach and provide corresponding learning guarantees under natural regularity conditions on the convolution kernel. More precisely, we assume the convolution kernel is a function in a translation invariant Hilbert space and analyze a natural ridge regression (RR) estimator. Building on existing results for RR, we characterize the accuracy of the estimator in terms of finite sample bounds. Interestingly, regularity assumptions which are classical in the analysis of RR, have a novel and natural interpretation in terms of space/frequency localization. Theoretical results are illustrated by numerical simulations.
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