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The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions with geometry and analysis. In particular, we present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, we discuss topological function theory which provides a new insight on oscillation of functions. The material should be accessible to readers with a basic background in algebraic and differential topology.
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Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules, including many of those that have been previously studied, and describe the relationships between them. In the cases where these classes are sets, interleaving distance induces a topology. We undertake a systematic study the resulting topological spaces and their basic topological properties.
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional scaling, while providing a firm theoretical ground. Many modern machine learning algorithms, however, are based on kernels. This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis. PIFs can be calculated and compared in linear time and have many beneficial properties, such as the availability of a kernel-based similarity measure. We demonstrate their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data.
We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram correspond to the local maxima and minima, respectively, of the k-th persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between k-th graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the k-th graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams.
The classical persistence algorithm virtually computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological data analysis. Its input is a simplicial filtration defined over the integers $mathbb{Z}$ giving rise to a $1$-parameter persistence module. It has been recognized that multi-parameter version of persistence modules given by simplicial filtrations over $d$-dimensional integer grids $mathbb{Z}^d$ is equally or perhaps more important in data science applications. However, in the multi-parameter setting, one of the main challenges is that topological summaries based on algebraic structure such as decompositions and bottleneck distances cannot be as efficiently computed as in the $1$-parameter case because there is no known extension of the persistence algorithm to multi-parameter persistence modules. We present an efficient algorithm to compute the unique decomposition of a finitely presented persistence module $M$ defined over the multiparameter $mathbb{Z}^d$.The algorithm first assumes that the module is presented with a set of $N$ generators and relations that are emph{distinctly graded}. Based on a generalized matrix reduction technique it runs in $O(N^{2omega+1})$ time where $omega<2.373$ is the exponent for matrix multiplication. This is much better than the well known algorithm called Meataxe which runs in $tilde{O}(N^{6(d+1)})$ time on such an input. In practice, persistence modules are usually induced by simplicial filtrations. With such an input consisting of $n$ simplices, our algorithm runs in $O(n^{2omega+1})$ time for $d=2$ and in $O(n^{d(2omega + 1)})$ time for $d>2$.
The notion of persistence partial matching, as a generalization of partial matchings between persistence modules, is introduced. We study how to obtain a persistence partial matching $mathcal{G}_f$, and a partial matching $mathcal{M}_f$, induced by a morphism $f$ between persistence modules, both being linear with respect to direct sums of morphisms. Some of their properties are also provided, including their stability after a perturbation of the morphism $f$, and their relationship with other induced partial matchings already defined in TDA.
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