Noise A 1-confidence interval for the persistence diagram consists of an estimate and such that The 1st Persistent Landscape (Bubenik, 2012) is the maximum contour of the triangles: We want a confidence band for .-Compute c, the 1-quantile of the bootstrapped .

We introduce a new algorithm for computing zigzag persistence, designed in the same spirit as the standard persistence algorithm. Our algorithm reduces a single matrix, maintains an explicit set of chains encoding the persistent homology of the current zigzag, and updates it under simplex insertions and removals. The total worst-case running time matches the usual cubic bound. A noticeable diff...

We define persistent homology groups over any set of spaces which have inclusions defined so that the underlying graph between the spaces is directed and acyclic. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of arbitrary families of topological spaces or...

s _______________________________________________________________________________________ Ulrich Bauer Title:Induced Matchings and the Algebraic Stability of persistence Barcode Abstract: We define a simple, explicit map sending a morphism f : M → 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...

We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cell...

We construct a locality preserving weight matrix for Laplacian eigenmaps algorithm used in dimension reduction. Our point cloud data is sampled from a low dimensional stratified space embedded in a higher dimension. Specifically, we use tools developed in local homology, persistence homology for kernel and cokernels to infer a weight matrix which captures neighborhood relations among points in ...

In persistent homology, the persistence barcode encodes pairs of simplices meaning birth and death of homology classes. Persistence barcodes depend on the ordering of the simplices (called a filter) of the given simplicial complex. In this paper, we define the notion of “minimal” barcodes in terms of entropy. Starting from a given filtration of a simplicial complex K, an algorithm for computing...

We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cell...

The theory of zigzag persistence is a substantial extension of persistent homology, and its development has enabled the investigation of several unexplored avenues in the area of topological data analysis. In this paper, we discuss three applications of zigzag persistence: topological bootstrapping, parameter thresholding, and the comparison of witness complexes.

Topological persistence methods provide a robust framework for analyzing large point cloud datasets topologically, and have been applied with great success towards homology computations on simplicial complexes. In this paper, we apply the persistence algorithm towards calculating a set of invariants related to the cup product structure on the cohomology ring for a space. These invariants expres...