Related papers: Representing Vineyard Modules
Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$…
We study pointwise free and finitely-generated persistence modules over a principal ideal domain, indexed by a (possibly infinite) totally-ordered poset category. We show that such persistence modules admit interval decompositions if and…
The representation of graphs is commonly based on the adjacency matrix concept. This formulation is the foundation of most algebraic and computational approaches to graph processing. The advent of deep learning language models offers a wide…
The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of a matroid. In this paper we indicate the role that tableaux can play in…
A foundational question in the theory of linear compartmental models is how to assess whether a model is structurally identifiable -- that is, whether parameter values can be inferred from noiseless data -- directly from the combinatorics…
We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter…
Multiparameter persistence module can capture more topological differences across data instances compared to using a single parameter, where the well-studied matching distance investigates the distance along a straight line in the…
We introduce a general technique to create an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by…
A flat-injective presentation of a multiparameter persistence module $M$ characterizes $M$ as the image of a morphism from a flat to an injective persistence module. Like flat or injective presentations, flat-injective presentations can be…
Time-delay mappings constructed using neural networks have proven successful in performing nonlinear system identification; however, because of their discrete nature, their use in bifurcation analysis of continuous-time systems is limited.…
Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram…
Value Iteration Networks (VINs) are effective differentiable path planning modules that can be used by agents to perform navigation while still maintaining end-to-end differentiability of the entire architecture. Despite their…
Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior…
We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets,…
Hierarchical structure and repetition are prevalent in graphs originating from nature or engineering. These patterns can be represented by a class of parametric-structure graphs, which are defined by templates that generate structure by way…
The $\gamma$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules…
We present a methodology for object-modular reasoning about hybrid system code using VCC, a deductive verifier for concurrent C code. We define in VCC an explicit time model, in which the passage of time must respect the invariants of…
Let $ m , n \in \mathbb{N}$, $D$ be a division ring, and $M_{m \times n}(D)$ denote the bimodule of all $m \times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m \times n}(D)$ in terms of left row…
One of the most powerful techniques to study protein structures is to look for recurrent fragments (also called substructures or spatial motifs), then use them as patterns to characterize the proteins under study. An emergent trend consists…
Molecular graphs generally contain subgraphs (known as groups) that are identifiable and significant in composition, functionality, geometry, etc. Flat latent representations (node embeddings or graph embeddings) fail to represent, and…