A Graph-Matching Formulation of the Interleaving Distance between Merge Trees

In this work we study the interleaving distance between merge trees from a combinatorial point of view. We use a particular type of matching between trees to obtain another formulation of the distance. With such formulation, we tackle the problem of approximating the interleaving distance by solving linear integer optimization problems in a recursive and dynamical fashion. We implement the algorithm and compare its outputs with another approximation procedure presented by other authors. We believe that further research in this direction could lead to faster algorithms to compute the distance and novel theoretical developments on the topic

A Locally Stable Edit Distance for Merge Trees

In this paper we define a novel edit distance for merge trees. Then we consider the metric space obtained and study the properties of such space obtaining completeness, compactness results and local approximations of such space by means of euclidean spaces. We also present results about its geodesic structure, with particular attention to objects called Frech\'et Means

A Locally Stable Edit Distance for Functions Defined on Merge Trees

In this work we define a novel metric structure to work with functions defined on merge trees. The metric introduced possesses some stability properties and can be computed with a dynamical integer linear programming approach. We showcase its feasibility and the effectiveness of the whole framework with simulated data sets. Using functions defined on merge trees proves to be very effective in situation where other topological data analysis tools, like persistence diagrams, can not be meaningfully employed

Wasserstein Principal Component Analysis for Circular Measures

We consider the 2-Wasserstein space of probability measures supported on the unit-circle, and propose a framework for Principal Component Analysis (PCA) for data living in such a space. We build on a detailed investigation of the optimal transportation problem for measures on the unit-circle which might be of independent interest. In particular, we derive an expression for optimal transport maps in (almost) closed form and propose an alternative definition of the tangent space at an absolutely continuous probability measure, together with the associated exponential and logarithmic maps. PCA is performed by mapping data on the tangent space at the Wasserstein barycentre, which we approximate via an iterative scheme, and for which we establish a sufficient a posteriori condition to assess its convergence. Our methodology is illustrated on several simulated scenarios and a real data analysis of measurements of optical nerve thickness

Projected Statistical Methods for Distributional Data on the Real Line with the Wasserstein Metric

We present a novel class of projected methods, to perform statistical analysis on a data set of probability distributions on the real line, with the 2-Wasserstein metric. We focus in particular on Principal Component Analysis (PCA) and regression. To define these models, we exploit a representation of the Wasserstein space closely related to its weak Riemannian structure, by mapping the data to a suitable linear space and using a metric projection operator to constrain the results in the Wasserstein space. By carefully choosing the tangent point, we are able to derive fast empirical methods, exploiting a constrained B-spline approximation. As a byproduct of our approach, we are also able to derive faster routines for previous work on PCA for distributions. By means of simulation studies, we compare our approaches to previously proposed methods, showing that our projected PCA has similar performance for a fraction of the computational cost and that the projected regression is extremely flexible even under misspecification. Several theoretical properties of the models are investigated and asymptotic consistency is proven. Two real world applications to Covid-19 mortality in the US and wind speed forecasting are discussed

Functional Data Representation with Merge Trees

In this paper we face the problem of representation of functional data with the tools of algebraic topology. We represent functions by means of merge trees and this representation is compared with that offered by persistence diagrams. We show that these two tree structures, although not equivalent, are both invariant under homeomorphic re-parametrizations of the functions they represent, thus allowing for a statistical analysis which is indifferent to functional misalignment. We employ a novel metric for merge trees and we prove a few theoretical results related to its specific implementation when merge trees represent functions. To showcase the good properties of our topological approach to functional data analysis, we first go through a few examples using data generated {\em in silico} and employed to illustrate and compare the different representations provided by merge trees and persistence diagrams, and then we test it on the Aneurisk65 dataset replicating, from our different perspective, the supervised classification analysis which contributed to make this dataset a benchmark for methods dealing with misaligned functional data

Sliced Max-Flow on Circular Mapper Graphs

In this work we develop a novel global invariant of circle-parametrized mapper graphs in order to analyse periodic sets, which often arise in materials science applications. This invariant describes a flow in a graph, slicing it with the fibers of the associated map onto the circle

Being on time in magnetic reconnection

The role of magnetic reconnection on the evolution of the Kelvin-Helmholtz instability is investigated in a plasma configuration with a velocity shear field. It is shown that the rate at which the large-scale dynamics drives the formation of steep current sheets, leading to the onset of secondary magnetic reconnection instabilities, and the rate at which magnetic reconnection occurs compete in shaping the final state of the plasma configuration. These conclusions are reached within a two-fluid plasma description on the basis of a series of two-dimensional numerical simulations. Special attention is given to the role of the Hall term. In these simulations, the boundary conditions, the symmetry of the initial configuration and the simulation box size have been optimized in order not to affect the evolution of the system artificially

Radiation Reaction Effects on Electron Nonlinear Dynamics and Ion Acceleration in Laser-solid Interaction

Radiation Reaction (RR) effects in the interaction of an ultra-intense laser pulse with a thin plasma foil are investigated analytically and by two-dimensional (2D3P) Particle-In-Cell (PIC) simulations. It is found that the radiation reaction force leads to a significant electron cooling and to an increased spatial bunching of both electrons and ions. A fully relativistic kinetic equation including RR effects is discussed and it is shown that RR leads to a contraction of the available phase space volume. The results of our PIC simulations are in qualitative agreement with the predictions of the kinetic theory

Data analysis with merge trees

Today’s data are increasingly complex and classical statistical techniques need growingly more refined mathematical tools to be able to model and investigate them. Paradigmatic situations are represented by data which need to be considered up to some kind of trans- formation and all those circumstances in which the analyst finds himself in the need of defining a general concept of shape. Topological Data Analysis (TDA) is a field which is fundamentally contributing to such challenges by extracting topological information from data with a plethora of interpretable and computationally accessible pipelines. We con- tribute to this field by developing a series of novel tools, techniques and applications to work with a particular topological summary called merge tree. To analyze sets of merge trees we introduce a novel metric structure along with an algorithm to compute it, define a framework to compare different functions defined on merge trees and investigate the metric space obtained with the aforementioned metric. Different geometric and topolog- ical properties of the space of merge trees are established, with the aim of obtaining a deeper understanding of such trees. To showcase the effectiveness of the proposed metric, we develop an application in the field of Functional Data Analysis, working with functions up to homeomorphic reparametrization, and in the field of radiomics, where each patient is represented via a clustering dendrogram