标题： 光谱变换核回归 显示英文标题

标题： Spectrally Transformed Kernel Regression

摘要： 未标记数据是现代机器学习中的关键组成部分。一般来说，未标记数据的作用是施加一种平滑性，通常是从基础核函数中编码的相似性信息中获取的，例如 \$\epsilon$-邻域核或图的邻接矩阵。本文重新审视了光谱变换核回归（STKR）的经典思想，并提供了一类新的通用且可扩展的STKR估计器，能够利用未标记的数据。直观上，通过光谱变换，STKR利用了未标记数据可以提供额外信息的数据分布。首先，我们展示了STKR是一种原则性强且通用的方法，通过刻画一种“目标平滑性”的普适类型，并证明了任何足够平滑的函数都可以被STKR学习到。其次，我们为归纳设置和一般变换函数提供了可扩展的STKR实现，而先前的工作大多局限于传递设置。第三，我们为两种情况推导了统计保证：已知多项式变换的STKR以及未知变换时使用核主成分分析（PCA）的STKR。总体而言，我们认为这项工作有助于加深我们对如何处理未标记数据的理解，其通用性使得它更容易启发新方法。 显示英文摘要

摘要： Unlabeled data is a key component of modern machine learning. In general, the role of unlabeled data is to impose a form of smoothness, usually from the similarity information encoded in a base kernel, such as the $\epsilon$-neighbor kernel or the adjacency matrix of a graph. This work revisits the classical idea of spectrally transformed kernel regression (STKR), and provides a new class of general and scalable STKR estimators able to leverage unlabeled data. Intuitively, via spectral transformation, STKR exploits the data distribution for which unlabeled data can provide additional information. First, we show that STKR is a principled and general approach, by characterizing a universal type of "target smoothness", and proving that any sufficiently smooth function can be learned by STKR. Second, we provide scalable STKR implementations for the inductive setting and a general transformation function, while prior work is mostly limited to the transductive setting. Third, we derive statistical guarantees for two scenarios: STKR with a known polynomial transformation, and STKR with kernel PCA when the transformation is unknown. Overall, we believe that this work helps deepen our understanding of how to work with unlabeled data, and its generality makes it easier to inspire new methods.