We consider a general regularised interpolation problem for learning a parameter vector from data. The well-known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is at the core of kernel methods in machine learning as it makes the problem computationally tractable. Most literature deals only with sufficient conditions for representer theorems in Hilbert spaces and shows that the regulariser being norm-based is sufficient for the existence of a representer theorem. We prove necessary and sufficient conditions for the existence of representer theorems in reflexive Banach spaces and show that any regulariser has to be essentially norm-based for a representer theorem to exist. Moreover, we illustrate why in a sense reflexivity is the minimal requirement on the function space. We further show that if the learning relies on the linear representer theorem, then the solution is independent of the regulariser and in fact determined by the function space alone. This in particular shows the value of generalising Hilbert space learning theory to Banach spaces.

我们考虑从数据中学习参数向量的一般正则化插值问题。著名的代表定理说，在正则化器上的某些条件下，数据点的线性跨度中存在解。这是机器学习中核方法的核心，因为它使问题在计算上易于处理。大多数文献只涉及 Hilbert 空间中表示定理的充分条件，并表明基于范数的正则化器足以满足表示定理的存在。我们证明了自反 Banach 空间中表示定理存在的充分必要条件，并表明任何正则化器必须本质上是基于范数的，才能使表示定理存在。而且，我们说明了为什么在某种意义上自反性是对函数空间的最低要求。我们进一步表明，如果学习依赖于线性表示定理，那么解决方案独立于正则化器，实际上仅由函数空间决定。这尤其显示了将 Hilbert 空间学习理论推广到 Banach 空间的价值。