Related papers: A Theory of Sub-Barcodes
We define a simple, explicit map sending a morphism $f:M \rightarrow N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this…
We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be…
The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special…
Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski,…
In a nutshell, submodular functions encode an intuitive notion of diminishing returns. As a result, submodularity appears in many important machine learning tasks such as feature selection and data summarization. Although there has been a…
The persistence barcode (equivalently, the persistence diagram), which can be obtained from the interval decomposition of a persistence module, plays a pivotal role in applications of persistent homology. For multi-parameter persistent…
We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects…
When filtering a topological space by a single parameter, the theory of quiver representations provides a complete framework for decomposing the resulting persistence module to obtain its barcode. This is achieved by interpreting the…
If M is a model category and Z is an object of M, then there are model category structures on the category of objects of M over Z and the category of objects of M under Z under which a map is a cofibration, fibration, or weak equivalence if…
When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P} \rightarrow \mathcal{C}$ from a category theory perspective. This generalizes the standard…
The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e. a…
A theory of modules over posets is developed to define computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology with multiple real parameters. To replace…
Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. Subsystem codes generalize all major quantum error protection schemes, and therefore are especially versatile. This…
We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is…
In this note we recall the relations between the barcodes in level and sub-level persistence and make precise their relation with the Morse-Novikov complex of a Morse real- or angle-valued map. The results in this papers are implicit in my…
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…
Stabilizer codes are a powerful method for implementing fault-tolerant quantum memory and in the case of topological codes, they form useful models for topological phases of matter. In this paper, we discuss the theory of stabilizer codes…
In this paper, we investigate the problem of classifying feature vectors with mutually independent but non-identically distributed elements. First, we show the importance of this problem. Next, we propose a classifier and derive an…
Barcodes form a complete set of invariants for interval decomposable persistence modules and are an important summary in topological data analysis. The set of barcodes is equipped with a canonical one-parameter family of metrics, the…
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…