Statistical Topological Data Analysis - A Kernel Perspective
We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as real-world data.
Summary: topological data analysis is a field which studies the geometric "shape" of a data set, using methods from algebraic topology. A tool that has been proven useful is the so-called persistent homology together the persistence diagrams. One is particularly interested in computing these persistence diagrams as they give the "topological summary" of the data shape. However a statistical treatment of these persistent diagrams is rather computationally expensive and difficult. A way to circumnavigate this difficulty goes by embedding probability measure on the space of persistence diagrams into a a reproducing kernel Hilbert space, since Hilbert spaces are highly regular and have an abundance of available well-studied methods that largely simplify calculations. Such an embedding can be achieved if the corresponding kernel is "universal", via the so-called "mean map". The paper proves that a slight modification of the recently introduced "PSS kernel", called the "u-PSS kernel", has the property of being "universal" (with respect to the 1-Wasserstein metric) [prop 2]. For example, one can then test whether two different samples exhibit significantly different homological features, while avoiding the direct computation of persistence diagrams and reducing it to the evaluation of the corresponding kernel. Major negative points- The paper is generally hard to read for people not so familiar with algebraic topology or functional analysis. Despite the background chapter, several rather more advanced notions or results in functional analysis are assumed familiar and not explained in further detail. - Many relevant definitions, argumentations from other papers and calculations are rather sketchy or even omitted.- The core argumentation of the paper is not written down in a concise chronological "start-to-finish argument", rather the different pieces of the argument are spread out throughout the paper, randomly mixed in with different information of secondary importance which makes the reading at times rather confusing.- The actual contribution is the proof of proposition 2, which in turn just boils down to proving that (\mathcal{S},d_{W,1}) is a compact metric, due to Theorem 1. In my opinion, the actual contribution is not that significant. The author mainly put together several already-published definitions and results, to achieve his result without introducing anything novel.   Positive points- the field of topological data analysis is a rather new and exciting area with many interesting applications, and the result seems rather exciting. As algebraic topology is a rather powerful tool, however not frequently used due to its computational complexity, a result leading to implementations of reduced complexity can lead to interesting and very diverse implications. - the experiments /implementations illustrate well the generality of theoretical result, and the diversity of its applications.Final conclusion: 5/10 (weak rejection)