Gabriel Mindlin

Ximena Fernández, Eugenio Borghini, Gabriel Mindlin et al.

We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with a sample metric known as Fermat distance. We prove that such metric space converges almost surely to the manifold itself endowed with an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This fact implies the convergence of the associated persistence diagrams. The use of this intrinsic distance when computing persistent homology presents advantageous properties such as robustness to the presence of outliers in the input data and less sensitiveness to the particular embedding of the underlying manifold in the ambient space. We use these ideas to propose and implement a method for pattern recognition and anomaly detection in time series, which is evaluated in applications to real data.

Diego E Shalom, Mariano Sigman, Gabriel Mindlin et al.

Language change involves the competition between alternative linguistic forms (1). The spontaneous evolution of these forms typically results in monotonic growths or decays (2, 3) like in winner-take-all attractor behaviors. In the case of the Spanish past subjunctive, the spontaneous evolution of its two competing forms (ended in -ra and -se) was perturbed by the appearance of the Royal Spanish Academy in 1713, which enforced the spelling of both forms as perfectly interchangeable variants (4), at a moment in which the -ra form was dominant (5). Time series extracted from a massive corpus of books (6) reveal that this regulation in fact produced a transient renewed interest for the old form -se which, once faded, left the -ra again as the dominant form up to the present day. We show that time series are successfully explained by a two-dimensional linear model that integrates an imitative and a novelty component. The model reveals that the temporal scale over which collective attention fades is in inverse proportion to the verb frequency. The integration of the two basic mechanisms of imitation and attention to novelty allows to understand diverse competing objects, with lifetimes that range from hours for memes and news (7, 8) to decades for verbs, suggesting the existence of a general mechanism underlying cultural evolution.