Related papers: Deformations, Derived Categories, and Multiparamet…
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and…
Motivated both by theoretical and practical considerations in topological data analysis, we generalize the $p$-Wasserstein distance on barcodes to multiparameter persistence modules. For each $p\in [1,\infty]$, we in fact introduce two such…
Persistent homology is a popular technique in topological data analysis that tracks the lifespans of homological features in a nested sequence of spaces. This data is typically presented in a multi-set called a persistence diagram or a…
A complete representation of 3D objects requires characterizing the space of deformations in an interpretable manner, from articulations of a single instance to changes in shape across categories. In this work, we improve on a prior…
This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy…
The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…
This is the second of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. This paper outlines how the framework can assist in the development of homotopy…
We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is…
The relationship between overparameterization, stability, and generalization remains incompletely understood in the setting of discontinuous classifiers. We address this gap by establishing a generalization bound for finite function classes…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets,…
In this work, we establish a deformation-based framework for learning solution mappings of PDEs defined on varying domains. The union of functions defined on varying domains can be identified as a metric space according to the deformation,…
The interleaving distance, although originally developed for persistent homology, has been generalized to measure the distance between functors modeled on many posets or even small categories. Existing theories require that such a poset…
The homology of configuration spaces of point-particles in manifolds has been studied intensively since the 1970s; in particular it is known to be stable if the underlying manifold is connected and open. Closely related to configuration…
In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution…
Popular network models such as the mixed membership and standard stochastic block model are known to exhibit distinct geometric structure when embedded into $\mathbb{R}^{d}$ using spectral methods. The resulting point cloud concentrates…
Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…
The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from…
Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira, after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in…
The stability of topological persistence is one of the fundamental issues in topological data analysis. Numerous methods have been proposed to address the stability of persistent modules or persistence diagrams. Recently, the concept of…