Topological Data Analysis and Machine Learning Theory

Perhaps the most important idea in applied algebraic topology is persistence. It is a response to the first difficulty that one encounters in attempting to assign topological invariants to statistical data sets: that the topology is not robust and has a sensitive dependence on the length scale at which the data set is being considered. The solution is to calculate the topology (specifically the homology) at all scales simultaneously, and to encode the relationship between the different scales in an algebraic invariant called the persistence diagram. The effective algorithm for doing so was published in 2000 by Edelsbrunner, Letscher and Zomorodian [2]. Topological data analysis would not be possible without this tool. Since then, persistence has been developed and understood quite extensively. Cohen-Steiner, Edelsbrunner and Harer [3] proved the important (and nontrivial) theorem that the persistence diagram is stable under perturbations of the initial data. Zomorodian and Carlsson [7] studied persistence algebraically, identifying the points of the persistence diagram with indecomposable summands of a module over the polynomial ring k[t], the monomial t representing a change of scale by a fixed increment. Generalising this approach using polynomial rings with two or more variables, they showed that the corresponding situation with two or more independent length scales is in some sense algebraically intractable, with no complete descriptive invariants available. Carlsson and de Silva [1] showed that persistence can be made to work over “non-monotone” parameters, in contrast to “monotone” parameters such as length scale; this is known as zigzag persistence. Bubenik and Scott [4] have described and studied persistence in terms of category theory.

也许应用代数拓扑中最重要的概念是持续性。它是对在尝试给统计数据集赋予拓扑不变量时遇到的第一个困难的一种回应：即拓扑不稳健，并且对考虑数据集的长度尺度有敏感的依赖性。解决方案是同时计算所有尺度下的拓扑（特别是同调），并将不同尺度之间的关系编码在一个称为持续性图的代数不变量中。2000年，埃德尔布鲁纳（Edelsbrunner）、莱舍尔（Letscher）和佐莫罗迪安（Zomorodian）发表了实现此操作的有效算法[2]。没有这个工具，拓扑数据分析是不可能的。从那时起，持续性得到了相当广泛的发展和理解。科恩 - 施泰纳（Cohen - Steiner）、埃德尔布鲁纳和哈勒（Harer）[3]证明了一个重要（且非平凡）的定理，即持续性图在初始数据的扰动下是稳定的。佐莫罗迪安和卡尔松（Carlsson）[7]从代数角度研究持续性，将持续性图中的点与多项式环\(k[t]\)上一个模的不可分解直和项等同起来，单项式\(t\)表示按固定增量改变尺度。通过使用具有两个或更多变量的多项式环推广这种方法，他们表明具有两个或更多独立长度尺度的相应情况在某种意义上在代数上是棘手的，没有可用的完整描述性不变量。卡尔松和德席尔瓦（de Silva）[1]表明，与长度尺度等“单调”参数相反，持续性可以在“非单调”参数上起作用；这被称为之字形持续性。布贝尼克（Bubenik）和斯科特（Scott）[4]从范畴论的角度描述和研究了持续性。