相关论文: The Theory of the Interleaving Distance on Multidi…
Unsupervised embeddings are fundamental to numerous machine learning applications, yet their evaluation remains a challenging task. Traditional assessment methods often rely on extrinsic variables, such as performance in downstream tasks,…
We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on…
This paper establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance.…
We investigate the validity of the "Einstein relations" in the general setting of unimodular random networks. These are equalities relating scaling exponents: $d_w = d_f + \tilde{\zeta}$ and $d_s = 2 d_f/d_w$, where $d_w$ is the walk…
We study dispersion properties of layered metal-dielectric media having different layers thicknesses ratios. Plotting dispersion diagrams and isofrequency contours, we find that strong nonlocality is inherent property of such periodic…
The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of…
Multiparameter persistence modules come up naturally in topological data analysis and topological robotics. Given a metric graph $(X,\delta)$, the second configuration space of $(X,\delta)$ with proximity parameters (for example, the…
Standard multidimensional scaling takes as input a dissimilarity matrix of general term $\delta _{ij}$ which is a numerical value. In this paper we input $\delta _{ij}=[\underline{\delta _{ij}},\overline{\delta _{ij}}]$ where…
Distances have a ubiquitous role in persistent homology, from the direct comparison of homological representations of data to the definition and optimization of invariants. In this article we introduce a family of parametrized pseudometrics…
This paper considers the problem of nonlinear dimensionality reduction. Unlike existing methods, such as LLE, ISOMAP, which attempt to unfold the true manifold in the low dimensional space, our algorithm tries to preserve the nonlinear…
We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems.…
Motivated by the need to relate the biparameter persistence module induced by a pair of scalar functions with the monoparameter persistence modules induced by each function separately, we introduce a construction that defines a kind of…
Let $X$ be a real analytic manifold endowed with a distance satisfying suitable properties and let $\mathbf{k}$ be a field. In [PS20], the authors construct a pseudo-distance on the derived category of sheaves of $\mathbf{k}$-modules on…
In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of $4$d Chern-Simons theory with defects, which provided a…
Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a…
For any $\varepsilon \in (0,+\infty)$, consider the metric spaces $\mathbb{R} \times [0,\varepsilon]$ in the Euclidean plane named layers or strips. B. Baslaugh in 1998 found the minimal width $\varepsilon \in (0,1)$ of a layer such that…
The persistence theory has been employed by several authors in order to study persistence properties of dynamical systems generated by ordinary differential equations or maps across diverse disciplines. In this note, the author discusses a…
We believe three ingredients are needed for further progress in persistence and its use: invariants not relying on decomposition theorems to go beyond 1-dimension, outcomes suitable for statistical analysis and a setup adopted for…
The notion of the ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the…
We propose an algorithmic framework for dataset normalization in data augmentation pipelines that preserves topological stability under non-uniform scaling transformations. Given a finite metric space \( X \subset \mathbb{R}^n \) with…