arXiv subjects

Mauricio Toledo-Acosta

Publications and source records attributed to Mauricio Toledo-Acosta.

5 recordsLinked to original sources

arXiv2025

Hyperoctant Search Clustering: A Method for Clustering Data in High-Dimensional Hyperspheres

Clustering of high-dimensional data sets is a growing need in artificial intelligence, machine learning and pattern recognition. In this paper, we propose a new clustering method based on a combinatorial-topological approach applied to regions of space defined by signs of coordinates (hyperoctants). In high-dimensional spaces, this approach often reduces the size of the dataset while preserving sufficient topological features. According to a density criterion, the method builds clusters of data points based on the partitioning of a graph, whose vertices represent hyperoctants, and whose edges connect neighboring hyperoctants under the Levenshtein distance. We call this method HyperOctant Search Clustering. We prove some mathematical properties of the method. In order to as assess its performance, we choose the application of topic detection, which is an important task in text mining. Our results suggest that our method is more stable under variations of the main hyperparameter, and remarkably, it is not only a clustering method, but also a tool to explore the dataset from a topological perspective, as it directly provides information about the number of hyperoctants where there are data points. We also discuss the possible connections between our clustering method and other research fields.

cs.LG↗

arXiv2023

Representations of Solvable Subgroups of PSL(3,C)

In this paper, we give a complete description of the representations of all upper triangular complex Kleinian subgroups of PSL(3,C). In https://doi.org/10.1007/s00574-021-00254-9 we show that any solvable group is virtually triangularizable and can be constructed as the semidirect product of two layers of parabolic elements and two layers of loxodromic elements. There are five families of purely parabolic discrete groups of PSL(3,C) https://doi.org/10.1016/j.laa.2022.08.027, therefore, the parabolic part of any upper triangular group belongs to one of these five families. In this paper we study which loxodromic elements can be combined with the elements of the parabolic part of an upper triangular discrete subgroup of PSL(3,C) in each of these five cases. These parabolic elements impose strong restrictions on the type, and number, of loxodromic elements that can be present in the group. We show that up to conjugation, there are 16 types of upper triangular complex Kleinian groups in PSL(3,C) containing loxodromic elements. These results are a further step towards the completion of the study of elementary discrete subgroups of PSL(3,C).

math.DS↗

arXiv2020

The Dynamics of Solvable Subgroups of PSL(3,C)

In this work we study and provide a full description, up to a finite index subgroup, of the dynamics of solvable complex Kleinian subgroups of PSL(3,C). These groups havesimpledynamics, contrary to strongly irre-ducible groups. Because of this, we propose to define elementary subgroups of PSL(3,C) as solvable groups. We show that triangular groups can be de-composed in four layers, via the semi-direct product of fourtypes of elements,with parabolic elements in the inner most layers and loxodromic elements inthe outer layers. It is also shown that solvable groups, up toa finite index sub-group, act properly and discontinuously on the complement of either a line,two lines, a line and a point outside of the line, or a pencil oflines passingthrough a point. These results are another step towards the completion of thestudy of elementary subgroups of PSL(3,C)

math.DS↗

arXiv2020

Lattice Diversities

Diversities are a generalization of metric spaces, where instead of the non-negative function being defined on pairs of points, it is defined on arbitrary finite sets of points. Diversities have a well-developed theory. This includes the concept of a diversity tight span that extends the metric tight span in a natural way. Here we explore the generalization of diversities to lattices. Instead of defining diversities on finite subsets of a set we consider diversities defined on members of an arbitrary lattice (with a 0). We show that many of the basic properties of diversities continue to hold. However, the natural map from a lattice diversity to its tight span is not a lattice homomorphism, preventing the development of a complete tight span theory as in the metric and diversity cases.

math.MG↗

arXiv2019

Three Generalizations Regarding Limit Sets for Complex Kleinian Groups

We study three problems related to the limit sets of discrete subgroups of PSL(n+1,C). In Chapter 2, we study the dynamics of solvable discrete subgroups of PSL(n+1,C). We prove that solvable groups are virtually triangularizable and we provide a description of the all the possible Kulkarni limit sets of solvable subgroups of PSL(n+1,C). Finally, we give the representations of these groups. With this description, the full description of the dynamics of general discrete subgroups of PSL(n+1,C) will be almost complete. In Chapter 3, we propose a new definition for the concept of limit set for the action of a discrete subgroup of PSL(n+1,C), we call it the Frances limit set. In complex dimension n=2, the Kulkarni limit set seems to be the right notion of limit set. However, in dimension n>2, the Kulkarni limit set is difficult to compute and it is bigger than it needs to be. This new limit set is, in general, smaller than the Kulkarni limit set. Also, it is made up of projective subspaces of the same dimension. The action of a discrete subgroup of PSL(n+1,C) on the complement of its Frances limit set is proper and discontinuous, also, this limit set is purely dimensional and unstable under deformations. In Chapter 4, we propose a way to generalize Patterson-Sullivan measures to the complex setting. We consider the Kobayashi metric on the complement of the Kulkarni limit set of an irreducible subgroup of PSL(3,C). These domains are the complement of arrays of complex lines in general position. We parametrize the space of such arrays of lines and we prove that, if for some subgroup of PSL(3,C) the entropy volume is finite, then we can construct these measures. We also give some concrete ideas on how to guarantee that the entropy volume of the Kobayashi metric is finite for certain groups.

math.DS↗