Related papers: Metrics and stabilization in one parameter persist…
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…
Topological entanglement entropy is a topological invariant which can detect topological order of quantum many-body ground state. We assume an existence of such order parameter at finite temperature which is invariant under smooth…
Variational stability, in the sense of local good behavior of optimal values and solutions in problems of optimization under shifts in parameters, is important not only for validating model robustness in practical applications but also for…
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…
In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be…
In this paper, we introduce the persistence transformation, a novel methodology in Topological Data Analysis (TDA) for applications in time series data which can be obtained in various areas such as science, politics, economy, healthcare,…
A lower bound for the interleaving distance on persistence vector spaces is given in terms of rank invariants. This offers an alternative proof of the stability of rank invariants.
Probability metrics constitute an important tool in probability theory and statistics \cite{DKS91}, \cite{R91}, \cite{Z83} as they are specific metrics on spaces of random variables which, by satisfying an extra condition, concord well with…
The concept of fixed point plays a crucial role in various fields of applied mathematics. The aim of this paper is to establish the existence of a unique fixed point of some type of functions which satisfy a new contraction principle,…
Stability selection is a widely adopted resampling-based framework for high-dimensional variable selection. This paper seeks to broaden the use of an established stability estimator to evaluate the overall stability of the stability…
Nonlinear dynamical systems are complex and typically only simple systems can be analytically studied. In applications, these systems are usually defined with a set of tunable parameters and as the parameters are varied the system response…
Recently, many regularized procedures have been proposed for variable selection in linear regression, but their performance depends on the tuning parameter selection. Here a criterion for the tuning parameter selection is proposed, which…
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…
Predictive geometric models deliver excellent results for many Machine Learning use cases. Despite their undoubted performance, neural predictive algorithms can show unexpected degrees of instability and variance, particularly when applied…
We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy…
We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter…
We introduce two novel concepts, topological difference and topological correlation, that offer a new perspective on the discriminative power of multiparameter persistence. The former quantifies the discrepancy between multiparameter and…
In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a…
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…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…