Related papers: Removal and Contraction Operations in $n$D General…
0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…
This work is concerned with a representation of shapes that disentangles fine, local and possibly repeating geometry, from global, coarse structures. Achieving such disentanglement leads to two unrelated advantages: i) a significant…
We consider different ways to control the magnification in self-organizing maps (SOM) and neural gas (NG). Starting from early approaches of magnification control in vector quantization, we then concentrate on different approaches for SOM…
A unicellular collection on a surface is a collection of curves whose complement is a single disk. There is a natural surgery operation on unicellular collections, endowing the set of such with a graph structure where the edge relation is…
An important problem in computational topology is to calculate the homology of a space from samples. In this work, we develop a statistical approach to this problem by calculating the expected rank of an induced map on homology from a…
Deep neural networks (DNNs) require very large amounts of computation both for training and for inference when deployed in the field. A common approach to implementing DNNs is to recast the most computationally expensive operations as…
Convolutional neural networks (CNNs) are a standard tool for computer vision tasks such as image classification. However, typical model architectures may result in the loss of topological information. In specific domains such as…
This paper presents new methods for analyzing and evaluating generalized plans that can solve broad classes of related planning problems. Although synthesis and learning of generalized plans has been a longstanding goal in AI, it remains…
A new approach to the study of Generalized Graphs as semantic data structures using machine learning techniques is presented. We show how vector representations maintaining semantic characteristics of the original data can be obtained from…
These notes present an approach to obtaining monoid operations which are compatible with a given family of mappings in the sense that the mappings become left translations in the monoid. This can be applied to various situations such as the…
We prove that counting copies of any graph $F$ in another graph $G$ can be achieved using basic matrix operations on the adjacency matrix of $G$. Moreover, the resulting algorithm is competitive for medium-sized $F$: our algorithm recovers…
Neural implicit representations have shown substantial improvements in efficiently storing 3D data, when compared to conventional formats. However, the focus of existing work has mainly been on storage and subsequent reconstruction. In this…
The computation of triangular decompositions are based on two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations relying on modular…
Following the success of Word2Vec embeddings, graph embeddings (GEs) have gained substantial traction. GEs are commonly generated and evaluated extrinsically on downstream applications, but intrinsic evaluations of the original graph…
Image segmentation is to extract meaningful objects from a given image. For degraded images due to occlusions, obscurities or noises, the accuracy of the segmentation result can be severely affected. To alleviate this problem, prior…
This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their…
Despite there being significant work on developing spectral, and metric embedding based approximation algorithms for hypergraph generalizations of conductance, little is known regarding the approximability of hypergraph partitioning…
We present algorithms to compute the vector space of homomorphisms Hom(X,Y) between finitely generated representations of the partially ordered set Z^d. Our results generalise to any partially ordered set. Our main theoretical contribution…
A new method is given for computing generators of the homology groups with integer coefficients for any finite $T_0$-space. An important role in this method is played by irreducible cycles which are defined here and give rise to continuous…
We study the problem of distance-preserving graph compression for weighted paths and trees. The problem entails a weighted graph $G = (V, E)$ with non-negative weights, and a subset of edges $E^{\prime} \subset E$ which needs to be removed…