Kernel methods for approximation and learning theory

在许多不同的情况下，我们在一些非结构化的点上得到信息，并被要求从数据中推断出一些具体的东西。在近似理论中，我们典型地试图开发一种算法，该算法将在给定有限信息集的情况下恢复某一集合中所有点的信息。然后我们给出近似误差的估计值。我们可以问其他问题。给定一定数量的信息和给定的一组函数，就误差而言，使用线性过程(如果我们近似函数组合，我们可以获得相同的近似值，就好像我们已经分别近似组合中的每个分量，并以相同的比例将它们组合在一起)，我们所能期望的最佳结果是什么。这样的考虑存在于n宽度的领域，知道是否有任何特定的算法实现了可能的最佳误差是很重要的。与n宽度的概念密切相关的是熵的概念，它处理覆盖一个固定集合所需的一定大小的集合的数量(例如，覆盖一个半径为r的圆需要多少个半径为1的圆)。在机器学习中，我们经常对获得良好估计的概率感兴趣。可能在我的集合中有非常少量的坏函数，而n宽度给出了最坏的情况。平均情况可能要好得多，所以合理答案的概率估计是有用的信息。当我们所拥有的信息位于某种光滑流形上时，我们打算研究这些问题，比如球面。我们想写一本研究专著，让在这一领域没有太多经验的人能够以一种自给自足的方式达到可以进行研究的阶段。这将需要从广泛的数学学科中收集理论。

In many different situations we are given information at some unstructured set of points, and asked to deduce something concrete from the data. In approximation theory we tyypically try to develop an algorithm which will recover the information at all points in a certain set, given a finite set of information. We then provide estimates for the error of approximation.We can ask other questions. Given a certain amount of information and a given set of functions, what is the best we can expect to do in terms of error using a linear process (if we approximate a combination of functions we obtain the same approximation as if we had individually approximated each component of the combination, and combine these in the same proportion). Such consideration lies in the field of n-widths, and it clearly of importance to know if any particular algorithm realises the best possible error. Closely related to the notion of n-width is that of entropy, which deals with the number of sets of a certain size it would take to cover a fixed set (for instance, how many circles of radius 1 does it take to cover a circle of radius r).In machine learning we are often interested in the probability of obtaining a good estimate. It maybe that there are a very small number of bad functions in my set, and n-width gives the worst case scenario. The average case might be much better, so an estimate of probability of a reasonable answer is useful information.We intend to investigate these issues when the information we have lies on some sort of smooth manifold, like a sphere. We would like to write a research monograph which will allow people who do not have much experience in this area to get to the stage where they could do research, and in a self-contained way. This will require the collection of theory from a wide range of subjects in mathematics.