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We address the problem of 3D shape registration and we propose a novel technique based on spectral graph theory and probabilistic matching. The task of 3D shape analysis involves tracking, recognition, registration, etc. Analyzing 3D data…
We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $\Delta$ and for $k$-uniform hypergraphs of maximum degree $\Delta$ in which any pair of edges overlaps in at most $r$ vertices. The…
We define and study exact, efficient representations of realization spaces of a natural class of underconstrained 2D Euclidean Distance Constraint Systems(EDCS) or Frameworks based on 1-dof Henneberg-I graphs. Each representation…
Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a…
Spectral graph embedding plays a critical role in graph representation learning by generating low-dimensional vector representations from graph spectral information. However, the embedding space of traditional spectral embedding methods…
Position encoding is the primary mechanism which induces notion of sequential order for input tokens in transformer architectures. Even though this formulation in the original transformer paper has yielded plausible performance for general…
Systematic enumeration and identification of unique 3D spatial topologies of complex engineering systems (such as automotive cooling systems, electric power trains, satellites, and aero-engines) are essential to navigation of these…
A polycrystalline graphene consists of perfect domains tilted at angle {\alpha} to each other and separated by the grain boundaries (GB). These nearly one-dimensional regions consist in turn of elementary topological defects, 5-pentagons…
Representation learning for graphs enables the application of standard machine learning algorithms and data analysis tools to graph data. Replacing discrete unordered objects such as graph nodes by real-valued vectors is at the heart of…
Direct detection is a very promising field in exoplanet science. It allows the detection of companions with large separation and allows their spectral analysis. A few planets have already been detected and are under spectral analysis. But…
The cosmic web consists of a nested hierarchy of structures: voids, walls, filaments, and clusters. These structures interconnect and can encompass one another, collectively shaping an intricate network. Here we introduce the Hierarchical…
Many large, real-world complex networks have rich community structure that a network scientist seeks to understand. These communities may overlap or have intricate internal structure. Extracting communities with particular topological…
Symmetries in a network connectivity regulate how the graph's functioning organizes into clustered states. Classical methods for tracing the symmetry group of a network require very high computational costs, and therefore they are of hard,…
The notion of barycentre graph is of crucial importance for machine learning algorithms that process graph-valued data. The barycentre graph is a "summary graph" that captures the mean topology and connectivity structure of a training…
Designing effective positional encodings for graphs is key to building powerful graph transformers and enhancing message-passing graph neural networks. Although widespread, using Laplacian eigenvectors as positional encodings faces two…
A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is…
Spectral embedding uses eigenfunctions of the discrete Laplacian on a weighted graph to obtain coordinates for an embedding of an abstract data set into Euclidean space. We propose a new pre-processing step of first using the eigenfunctions…
Node embeddings map graph vertices into low-dimensional Euclidean spaces while preserving structural information. They are central to tasks such as node classification, link prediction, and signal reconstruction. A key goal is to design…
Statistical physics models with hard constraints, such as the discrete hard-core gas model (random independent sets in a graph), are inherently combinatorial and present the discrete mathematician with a relatively comfortable setting for…
We address discrete-time consensus on the Euclidean unit sphere. For this purpose we consider a distributed algorithm comprising the iterative projection of a conical combination of neighboring states. Neighborhoods are represented by a…