Related papers: Computing the Tropical Abel--Jacobi Transform and …
Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper we compute the Euclidean distance degree and polar degrees…
Hyperbolicity is a graph parameter related to how much a graph resembles a tree with respect to distances. Its computation is challenging as the main approaches consist in scanning all quadruples of the graph or using fast matrix…
Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world. The "space of networks", however, has a complex structure that cannot be adequately described…
Recently, many graph matching methods that incorporate pairwise constraint and that can be formulated as a quadratic assignment problem (QAP) have been proposed. Although these methods demonstrate promising results for the graph matching…
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.
Torus networks of moderate degree have been widely used in the supercomputer industry. Tori are superb when used for executing applications that require near-neighbor communications. Nevertheless, they are not so good when dealing with…
Graph analytics can lead to better quantitative understanding and control of complex networks, but traditional methods suffer from high computational cost and excessive memory requirements associated with the high-dimensionality and…
Comparing two geometric graphs embedded in space is important in the field of transportation network analysis. Given street maps of the same city collected from different sources, researchers often need to know how and where they differ.…
This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large.…
Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display…
Graph is an important data representation which appears in a wide diversity of real-world scenarios. Effective graph analytics provides users a deeper understanding of what is behind the data, and thus can benefit a lot of useful…
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 a novel formulation of the distance. With such formulation, we tackle the…
We view molecular optimization as a graph-to-graph translation problem. The goal is to learn to map from one molecular graph to another with better properties based on an available corpus of paired molecules. Since molecules can be…
We present local biplots, a an extension of the classic principal components biplot to multi-dimensional scaling. Noticing that principal components biplots have an interpretation as the Jacobian of a map from data space to the principal…
Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs…
Molecular graphs generally contain subgraphs (known as groups) that are identifiable and significant in composition, functionality, geometry, etc. Flat latent representations (node embeddings or graph embeddings) fail to represent, and…
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means…
\textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we…
Tropical algebraic geometry offers new tools for elimination theory and implicitization. We determine the tropicalization of the image of a subvariety of an algebraic torus under any homomorphism from that torus to another torus.
Tropical geometry has recently found several applications in the analysis of neural networks with piecewise linear activation functions. This paper presents a new look at the problem of tropical polynomial division and its application to…