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We study multilevel techniques, commonly used in PDE multigrid literature, to solve structured optimization problems. For a given hierarchy of levels, we formulate a coarse model that approximates the problem at each level and provides a…

The present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete-time multi-agent systems. In the systems considered here, both the topology of the interconnection graph and the weight of…

In this note we study some properties of topological entropy for noncompact non-metrizable spaces.

This paper provides an optimized cable path planning solution for a tree-topology network in an irregular 2D manifold in a 3D Euclidean space, with an application to the planning of submarine cable networks. Our solution method is based on…

We study hierarchical clusterings of metric spaces that change over time. This is a natural geometric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent…

We use Pesin theory to study possible equilibrium measures for piecewise monotone maps of the interval. The maps may have unbounded derivative.

Hierarchical networks are attracting a renewal interest for modelling the organization of a number of biological systems and for tackling the complexity of statistical mechanical models beyond mean-field limitations. Here we consider the…

Several structure-learning algorithms for staged trees, asymmetric extensions of Bayesian networks, have been proposed. However, these either do not scale efficiently as the number of variables considered increases, a priori restrict the…

Mutual learning of a pair of tree parity machines with continuous and discrete weight vectors is studied analytically. The analysis is based on a mapping procedure that maps the mutual learning in tree parity machines onto mutual learning…

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the…

Coarse geometry studies metric spaces on the large scale. Our goal here is to study dynamics from a coarse point of view. To this end we introduce a coarse version of topological entropy, suitable for unbounded metric spaces, consistent…

Certain notions of convergence of sequences functions such as pointwise convergence and (uniform) convergence on compact or bounded sets come from suitable topological function spaces; see [1]. Under certain conditions these topologies…

Several different measures for digraph width have appeared in the last few years. However, none of them shares all the "nice" properties of treewidth: First, being \emph{algorithmically useful} i.e. admitting polynomial-time algorithms for…

Monotonicity is a simple yet significant qualitative characteristic. We consider the problem of segmenting an array in up to K segments. We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric for…

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is…

Since its inception in the 1980s, ID3 has become one of the most successful and widely used algorithms for learning decision trees. However, its theoretical properties remain poorly understood. In this work, we introduce a novel metric of a…

We study polynomial optimization problems whose objective has a composition or tensor train structure. These polynomials can be evaluated as a sequence of maps, giving rise to intermediate variables (``states'') of dimension lower than the…

In many situations, the choice of an adequate similarity measure or metric on the feature space dramatically determines the performance of machine learning methods. Building automatically such measures is the specific purpose of…

We present an efficient algorithm for the construction of a holographic simple tree graph model that realizes a given entropy vector, subject to a specific ``chordality'' condition first introduced in arXiv:2412.18018. We further develop…

The problem of subgroups is ubiquitous in scientific research (ex. disease heterogeneity, spatial distributions in ecology...), and piecewise regression is one way to deal with this phenomenon. Morse-Smale regression offers a way to…