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We study the problem of distinguishing between two distributions on a metric space; i.e., given metric measure spaces $({mathbb X}, d, mu_1)$ and $({mathbb X}, d, mu_2)$, we are interested in the problem of determining from finite data whether or not $mu_1$ is $mu_2$. The key is to use pairwise distances between observations and, employing a reconstruction theorem of Gromov, we can perform such a test using a two sample Kolmogorov--Smirnov test. A real analysis using phylogenetic trees and flu data is presented.
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An augmented metric space is a metric space $(X, d_X)$ equipped with a function $f_X: X to mathbb{R}$. This type of data arises commonly in practice, e.g, a point cloud $X$ in $mathbb{R}^d$ where each point $xin X$ has a density function value $f_X(x)$ associated to it. An augmented metric space $(X, d_X, f_X)$ naturally gives rise to a 2-parameter filtration $mathcal{K}$. However, the resulting 2-parameter persistent homology $mathrm{H}_{bullet}(mathcal{K})$ could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode $mathrm{H}_0(mathcal{K})$. Specifically, if $n = |X|$, the elder-rule-staircode consists of $n$ number of staircase-like blocks in the plane. We show that if $mathrm{H}_0(mathcal{K})$ is interval decomposable, then the barcode of $mathrm{H}_0(mathcal{K})$ is equal to the elder-rule-staircode. Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), and the graded Betti numbers of $mathrm{H}_0(mathcal{K})$ can all be efficiently computed once the elder-rule-staircode is given. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in $O(n^2log n)$ time, which can be improved to $O(n^2alpha(n))$ if $X$ is from a fixed dimensional Euclidean space $mathbb{R}^d$, where $alpha(n)$ is the inverse Ackermann function.
Distribution function is essential in statistical inference, and connected with samples to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties. This connection creates a paradigm for statistical inference. However, existing distribution functions are defined in Euclidean spaces and no longer convenient to use in rapidly evolving data objects of complex nature. It is imperative to develop the concept of distribution function in a more general space to meet emerging needs. Note that the linearity allows us to use hypercubes to define the distribution function in a Euclidean space, but without the linearity in a metric space, we must work with the metric to investigate the probability measure. We introduce a class of metric distribution functions through the metric between random objects and a fixed location in metric spaces. We overcome this challenging step by proving the correspondence theorem and the Glivenko-Cantelli theorem for metric distribution functions in metric spaces that lie the foundation for conducting rational statistical inference for metric space-valued data. Then, we develop homogeneity test and mutual independence test for non-Euclidean random objects, and present comprehensive empirical evidence to support the performance of our proposed methods.
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter spatiotemporal filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov-Hausdorff distance which permits metrizing the collection of all DMSs. On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well known interleaving distance. We also study the computational complexity associated to both choices of d.
In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its optimality properties and goodness of fit. Further, we present a notion of MDS on infinite metric measure spaces that generalizes these optimality properties. As a consequence we can study the MDS embeddings of the geodesic circle $S^1$ into $mathbb{R}^m$ for all $m$, and ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into $mathbb{R}^m$. Finally, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space $X$, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of $X$?
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