Nieves Atienza

A new topological entropy-based approach for measuring similarities among piecewise linear functions

In this paper we present a novel methodology based on a topological entropy, the so-called persistent entropy, for addressing the comparison between discrete piecewise linear functions. The comparison is certified by the stability theorem for persistent entropy that is presented here. The theorem is used in the implementation of a new algorithm. The algorithm transforms a discrete piecewise linear function into a filtered simplicial complex that is analyzed via persistent homology and persistent entropy. Persistent entropy is used as a discriminant feature for solving the supervised classification problem of real long-length noisy signals of DC electrical motors. The quality of classification is stated in terms of the area under receiver operating characteristic curve (AUC=93.87%). HighlightsDefinition of a new entropy from the persistent barcode.Proof of the stability theorem for the persistent entropy.Development of a new entropy-based methodology for studying piecewise linear function.Development of a new entropy-based algorithm for the classification of real signals

Separating Topological Noise from Features Using Persistent Entropy

In this paper, we derive a simple method for separating topological noise from topological features using a novel measure for comparing persistence barcodes called persistent entropy.

Consistency of maximum likelihood estimators in finite mixture models of the union of °W-type families

ABSTRACT In finite mixture models, maximum likelihood estimators have good properties, such as efficiency, consistency, and asymptotic normality under some uniform integrability assumptions on the mixture and its derivatives up to the third order. These conditions are frequently not easy to check because complex computations on bounding a lot of derivatives are involved. We give results implying these conditions for a new class of families of distributions, 𝒲-type families, which make it easier to check the conditions in many cases. Many useful and known families of distributions such as Weibull, Generalized Gamma, Log-gamma, inverse Log-gamma, inverse Gaussian, and all of the exponential families are 𝒲-type families. Hence, these results have broad applications.

Persistent entropy for separating topological features from noise in vietoris-rips complexes

Persistent homology studies the evolution of k-dimensional holes along a nested sequence of simplicial complexes (called a filtration). The set of bars (i.e. intervals) representing birth and death times of k-dimensional holes along such sequence is called the persistence barcode. k-Dimensional holes with short lifetimes are informally considered to be “topological noise”, and those with long lifetimes are considered to be “topological features” associated to the filtration. Persistent entropy is defined as the Shannon entropy of the persistence barcode of the filtration. In this paper we present new important properties of persistent entropy of Vietoris-Rips filtrations. Later, using these properties, we derive a simple method for separating topological noise from features in Vietoris-Rips filtrations.

Computing balanced islands in two colored point sets in the plane

Abstract Let S be a set of n points in general position in the plane, r of which are red and b of which are blue. In this paper we present algorithms to find convex sets containing a balanced number of red and blue points. We provide an O ( n 4 ) time algorithm that for a given α ∈ [ 0 , 1 2 ] finds a convex set containing exactly ⌈ α r ⌉ red points and exactly ⌈ α b ⌉ blue points of S. If ⌈ α r ⌉ + ⌈ α b ⌉ is not much larger than 1 3 n , we improve the running time to O ( n log ⁡ n ) . We also provide an O ( n 2 log ⁡ n ) time algorithm to find a convex set containing exactly ⌈ r + 1 2 ⌉ red points and exactly ⌈ b + 1 2 ⌉ blue points of S, and show that balanced islands with more points do not always exist.

A new entropy based summary function for topological data analysis

Abstract Topological data analysis (TDA) aims to obtain useful information from data sets using topological concepts. In particular, it may help to infer from finite sample when a configuration space is a manifold. So far, there is no automatic process to decide the main topological features of a given sampled manifold. In this article, we present an entropy-based summary function which may help to decide the most relevant Betti numbers from finite samples of a given manifold.

Cover contact graphs

We study problems that arise in the context of covering certain geometric objects (so-called seeds, e.g., points or disks) by a set of other geometric objects (a so-called cover, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds).