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Motivated by the increasing availability of low- and mixed-precision arithmetic on modern hardware, we develop mixed-precision variants of Lloyd's algorithm for k-means clustering. The main ingredient is a family of mixed-precision kernels…
Let $N$ be a normal subgroup of a group $G$. An $N$-module $Q$ is $G$-stable provided that $Q$ is equivalent to the twist $Q^g$ of $Q$ by $g$, for every $g\in G$. If the action of $N$ on $Q$ extends to an action of $G$ on $Q$, $Q$ is…
The concept of edit distance, which dates back to the 1960s in the context of comparing word strings, has since found numerous applications with various adaptations in computer science, computational biology, and applied topology. By…
Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto $\mathbb{R}$.…
Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams…
Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other…
In this paper we formulate and prove a general theorem of stability of exactness properties under the pro-completion, which unifies several such theorems in the literature and gives many more. The theorem depends on a formal approach to…
We apply poset cocalculus, a functor calculus framework for functors out of a poset, to study the problem of decomposing multipersistence modules into simpler components. We both prove new results in this topic and offer a new perspective…
In this article, we introduce a new parameterized family of topological descriptors, taking the form of candidate decompositions, for multi-parameter persistence modules, and we identify a subfamily of these descriptors, that we call…
We consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the…
For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ of $\mathbf{P}$, we give a formula for the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the…
We investigate a linear diffusion equation incorporating historical effects, characterised by a finite non-negative Borel measure on \((0, \mathfrak T]\). This approach accommodates both distributed memory and discrete delays within a…
The interleaving distance, although originally developed for persistent homology, has been generalized to measure the distance between functors modeled on many posets or even small categories. Existing theories require that such a poset…
Persistent homology is a way of determining the topological properties of a data set. It is well known that each persistence module admits the structure of a representation of a finite totally ordered set. In previous work, the authors…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to…
Spatial perception aims to estimate camera motion and scene structure from visual observations, a problem traditionally addressed through geometric modeling and physical consistency constraints. Recent learning-based methods have…
In this paper, we first study the Gorenstein projective/flat dimension of complexes of modules. The relation between the Gorenstein projective/flat dimension for complexes and that for modules are investigated. Then we study Tate, stable…
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that this construction can produce…
Distances on merge trees facilitate visual comparison of collections of scalar fields. Two desirable properties for these distances to exhibit are 1) the ability to discern between scalar fields which other, less complex topological…