Related papers: A Theory of Sub-Barcodes

We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…

This text is based on lectures given by authors in summer 2015. It contains an introduction to the theory of limits over the category of presentations, with examples of different well-known functors like homology or derived functors of…

We are concerned with the problem of designing large families of subsets over a common labeled ground set that have small pairwise intersections and the property that the maximum discrepancy of the label values within each of the sets is…

This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric…

We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we…

We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use "algebraic" characterizations of fibrations to produce factorizations…

Graphcodes were recently introduced as a technique to employ two-parameter persistence modules in machine learning tasks (Kerber and Russold, NeurIPS 2024). We show in this work that a compressed version of graphcodes yields a description…

The output of persistent homology is an algebraic object called a persistence module. This object admits a decomposition into a direct sum of interval persistence modules described entirely by the barcode invariant. In this paper we…

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain…

Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In…

We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of promonoidal category. Since the theory of promonoidal categories has not been developed…

Weak superimposed codes are combinatorial structures related closely to generalized cover-free families, superimposed codes, and disjunct matrices in that they are only required to satisfy similar but less stringent conditions. This class…

A barcode is a finite multiset of intervals on the real line, $B = \{ (b_i, d_i)\}_{i=1}^n$. Barcodes are important objects in topological data analysis, where they serve as summaries of the persistent homology groups of a filtration. The…

Solving optimization tasks based on functions and losses with a topological flavor is a very active, growing field of research in data science and Topological Data Analysis, with applications in non-convex optimization, statistics and…

Submodularity is a fundamental phenomenon in combinatorial optimization. Submodular functions occur in a variety of combinatorial settings such as coverage problems, cut problems, welfare maximization, and many more. Therefore, a lot of…

This paper provides new constructions and lower bounds for subspace codes, using Ferrers diagram rank-metric codes from matchings of the complete graph and pending blocks. We present different constructions for constant dimension codes with…

We introduce the factor complex of a neural code, and show how intervals and maximal codewords are captured by the combinatorics of factor complexes. We use these results to obtain algebraic and combinatorial characterizations of…

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…

Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit…

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of…