Related papers: Lifting Cocycles: From Heuristic to Theory
Dimensionality reduction is a crucial technique in data analysis, as it allows for the efficient visualization and understanding of high-dimensional datasets. The circular coordinate is one of the topological data analysis techniques…
We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such…
Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for…
Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes…
A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood equations. We…
Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide…
Let $V$ be a complete discrete valuation ring with residue field $\mathbb{F}$. We define a cyclic homology theory for algebras over $\mathbb{F}$, by lifting them to free algebras over $V$, which we enlarge to tube algebras and complete…
We describe a method to obtain spherical parameterizations of arbitrary data through the use of persistent cohomology and variational optimization. We begin by computing the second-degree persistent cohomology of the filtered Vietoris-Rips…
A recent result of ours [GM] shows that all Hopf algebra liftings of a given diagram in the sense of Andruskiewitsch and Schneider are cocycle deformations of each other. Here we develop a "non-abelian" cohomology theory, which gives a…
We present a distributed algorithm to compute the first homology of a simplicial complex. Such algorithms are very useful in topological analysis of sensor networks, such as its coverage properties. We employ spanning trees to compute a…
Symmetry is the essential element of lifted inference that has recently demon- strated the possibility to perform very efficient inference in highly-connected, but symmetric probabilistic models models. This raises the question, whether…
The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseux-valued ``lift'' of this point in the…
Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We…
A key goal in the design of probabilistic inference algorithms is identifying and exploiting properties of the distribution that make inference tractable. Lifted inference algorithms identify symmetry as a property that enables efficient…
The compression of geometric structures is a relatively new field of data compression. Since about 1995, several articles have dealt with the coding of meshes, using for most of them the following approach: the vertices of the mesh are…
Many complex networks, ranging from social to biological systems, exhibit structural patterns consistent with an underlying hyperbolic geometry. Revealing the dimensionality of this latent space can disentangle the structural complexity of…
Topological Data Analysis has grown in popularity in recent years as a way to apply tools from algebraic topology to large data sets. One of the main tools in topological data analysis is persistent homology. This paper uses undergraduate…
Natural and man-made transport webs are frequently dominated by dense sets of nested cycles. The architecture of these networks, as defined by the topology and edge weights, determines how efficiently the networks perform their function.…
In this expository paper we present some ideas of algebraic topology (more precisely, of homology theory) in a language accessible to non-specialists in the area. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is…
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in…