Related papers: Removal and Contraction Operations in $n$D General…
Cartograms depict geographic regions with areas proportional to quantitative data. However, when created using density-equalizing map projections, cartograms may exhibit invalid topologies if boundary polygons are drawn using only a finite…
Graph coarsening reduces the size of a graph while preserving certain properties. Most existing methods preserve either spectral or spatial characteristics. Recent research has shown that preserving topological features helps maintain the…
We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the "invariants by differentiation" property, and can hence be…
We describe a graph reduction operation, generalizing three graph reduction operations related to gene assembly in ciliates. The graph formalization of gene assembly considers three reduction rules, called the positive rule, double rule,…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
Stratified digraphs are popular models for feedforward neural networks. However, computation of their path homologies has been limited to low dimensions due to high computational complexity. A recursive algorithm is proposed to compute…
Self-organizing maps (SOM) are widely used for their topology preservation property: neighboring input vectors are quantified (or classified) either on the same location or on neighbor ones on a predefined grid. SOM are also widely used for…
We present our ongoing work on understanding the limitations of graph convolutional networks (GCNs) as well as our work on generalizations of graph convolutions for representing more complex node attribute dependencies. Based on an analysis…
Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
In this paper, three Computational Topology methods (namely effective homology, persistent homology and discrete vector fields) are mixed together to produce algorithms for homological digital image processing. The algorithms have been…
For matrices with displacement structure, basic operations like multiplication, inversion, and linear system solving can all be expressed in terms of the following task: evaluate the product $\mathsf{A}\mathsf{B}$, where $\mathsf{A}$ is a…
Tensor network contraction is central to problems ranging from many-body physics to computer science. We describe how to approximate tensor network contraction through bond compression on arbitrary graphs. In particular, we introduce a…
To perform visual data exploration, many dimensionality reduction methods have been developed. These tools allow data analysts to represent multidimensional data in a 2D or 3D space, while preserving as much relevant information as…
We present an innovative algorithm that simplifies the topology of a cross-field. Our algorithm works through macro-operations that allow us editing the graph of separatrices, which is extracted from a cross-field, while maintaining it…
We contribute to the theory for minimal liftings of cut-generating functions. In particular, we give three operations that preserve the so-called covering property of certain structured cut-generating functions. This has the consequence of…
A new geometric procedure to construct symplectic methods for constrained mechanical systems is developed in this paper. The definition of a map coming from the notion of retraction maps allows to adapt the continuous problem to the…
In this paper, we study the construction and transformation of two-dimensional persistent graphs. Persistence is a generalization to directed graphs of the undirected notion of rigidity. In the context of moving autonomous agent formations,…
Many computational algorithms applied to geometry operate on discrete representations of shape. It is sometimes necessary to first simplify, or coarsen, representations found in modern datasets for practicable or expedited processing. The…
In this article, we show that the algorithm of maintaining expander decompositions in graphs undergoing edge deletions directly by removing sparse cuts repeatedly can be made efficient. Formally, for an $m$-edge undirected graph $G$, we say…