什么时候有表示者定理？向量与矩阵正则化器
When is there a representer theorem? Vector versus matrix regularizers
摘要：
我们考虑一类通用的正则化方法，它基于线性测量学习参数向量。众所周知，如果正则化子是内积的非减函数，则学习向量是输入数据的线性组合。这个结果被称为{\em表示定理}，是机器学习中基于内核的方法的基础。本文证明了上述条件的必要性，从而完成了基于正则化的核方法的刻画。我们进一步将我们的分析扩展到学习矩阵的正则化方法，这是一个由多任务学习应用程序激发的问题。在这方面，我们研究了一个更一般的表示定理，它适用于更大的一类正则化。我们提供了一个必要和充分条件，这些类矩阵正则化，并突出了它们与一些具体的例子，具有实际意义。我们的分析使用了矩阵理论的基本原理，特别是矩阵非减函数的有用概念。
英文摘要：
We consider a general class of regularization methods which learn a vector of parameters on the basis of linear measurements. It is well known that if the regularizer is a nondecreasing function of the inner product then the learned vector is a linear combination of the input data. This result, known as the {\em representer theorem}, is at the basis of kernel-based methods in machine learning. In this paper, we prove the necessity of the above condition, thereby completing the characterization of kernel methods based on regularization. We further extend our analysis to regularization methods which learn a matrix, a problem which is motivated by the application to multi-task learning. In this context, we study a more general representer theorem, which holds for a larger class of regularizers. We provide a necessary and sufficient condition for these class of matrix regularizers and highlight them with some concrete examples of practical importance. Our analysis uses basic principles from matrix theory, especially the useful notion of matrix nondecreasing function.
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