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In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare…
We compute the cohomology ring of a generalised type of configuration space of points in $\mathbb{R}^r$. This configuration space is indexed by a graph. In the case the graph is complete the result is known and it is due to Arnold and…
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…
We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomology…
This paper presents the main features of a system that aims to transform regular expressions into shorter equivalent expressions. The system is also capable of computing other operations useful for simplification, such as checking the…
In this paper, we prove the integrality conjecture for quotient stacks arising from weakly symmetric representations of reductive groups. Our main result is a decomposition of the cohomology of the stack into finite-dimensional components…
Topological Data Analysis (TDA) provides tools to describe the shape of data, but integrating topological features into deep learning pipelines remains challenging, especially when preserving local geometric structure rather than…
Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…
We improve upon pairwise annotation for active learning in coreference resolution, by asking annotators to identify mention antecedents if a presented mention pair is deemed not coreferent. This simple modification, when combined with a…
A general method for constructing simplicial complex from observed time series of dynamical systems based on the delay coordinate reconstruction procedure is presented. The obtained simplicial complex preserves all pertinent topological…
Persistent homology (PH) is a relatively new field in applied mathematics that studies the components and shapes of discrete data. In this work, we demonstrate that PH can be used as a universal framework to identify phases in spin models,…
Compressed inverted indices in use today are based on the idea of gap compression: documents pointers are stored in increasing order, and the gaps between successive document pointers are stored using suitable codes which represent smaller…
Execution of concurrent programs implies frequent switching between different thread contexts. This property perplexes analyzing and reasoning about concurrent programs. Trace simplification is a technique that aims at alleviating this…
Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on $R^{2n}$ ($C^{2n}) are investigated under suitable continuity restrictions on cochains. The first and second cohomology…
We study classes of Dynamic Programming (DP) algorithms which, due to their algebraic definitions, are closely related to coefficient extraction methods. DP algorithms can easily be modified to exploit sparseness in the DP table through…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
The purpose of this article is to relate coarse cohomology of metric spaces with a more computable cohomology. We introduce a notion of boundedly supported cohomology and prove that coarse cohomology of many spaces are isomorphic to the…
Tries are popular data structures for storing a set of strings, where common prefixes are represented by common root-to-node paths. Over fifty years of usage have produced many variants and implementations to overcome some of their…
Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…
Clearing is a simple but effective optimization for the standard algorithm of persistent homology (PH), which dramatically improves the speed and scalability of PH computations for Vietoris--Rips filtrations. Due to the quick growth of the…