Related papers: Persistence Barcodes versus Kolmogorov Signatures:…

We address the problem of estimating multiple modes of a multivariate density using persistent homology, a central tool in Topological Data Analysis. We introduce a method based on the preliminary estimation of the $H_0$-persistence diagram…

We propose a statistical framework to identify topological differences in two populations of random geometric objects. The proposed framework involves first associating a topological signature with random geometric objects and then…

Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…

We study distributions of persistent homology barcodes associated to taking subsamples of a fixed size from metric measure spaces. We show that such distributions provide robust invariants of metric measure spaces, and illustrate their use…

Topological Data Analysis (TDA) provides a toolkit for the study of the shape of high dimensional and complex data. While operating on a space of persistence diagrams is cumbersome, persistence norms provide a simple real value measure of…

ABSRACT: We review recent progress on testing the hypothesis of the existence of cosmic string perturbations in microwave background maps. Using an analytical model for the string network we show that the predicted amplitude and spectrum of…

Robust statistical inference often faces a severe computational-statistical gap when dealing with complex parameter spaces. We investigate minimax signal detection in the Gaussian sequence model under strong $\epsilon$-contamination, where…

We study the numerical stability of polynomial based encoding methods, which has emerged to be a powerful class of techniques for providing straggler and fault tolerance in the area of coded computing. Our contributions are as follows: 1)…

In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation…

We study the probabilistic behavior of persistence-based statistics and propose a novel nonparametric framework for detecting structural changes in high-dimensional random point clouds. We establish moment bounds and tightness results for…

Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski,…

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single…

It is discussed and surveyed a numerical method proposed before, that alternative to the usual compression method, provides an approximation to the algorithmic (Kolmogorov) complexity, particularly useful for short strings for which…

In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in…

We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1…

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in…

Persistent homology barcodes and diagrams are a cornerstone of topological data analysis that capture the "shape" of a wide range of complex data structures, such as point clouds, networks, and functions. However, their use in statistical…

Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come.…

The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous…

We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between…