Single-photon emission computed tomography images of murine tumors are interpreted as the values of functions on a three-dimensional domain. Motivated by Morse theory, the local maxima of the tumor image functions are analyzed. This analysis captures tumor heterogeneity that cannot be identified with standard measures. Utilizing decreasing sequences of uptake values to filter the images, a modi...

Persistent homology appears as a fundamental tool in Topological Data Analysis. It studies the evolution of k−dimensional holes along a sequence of simplicial complexes (i.e. a filtration). The set of intervals representing birth and death times of k−dimensional holes along such sequence is called the persistence barcode. k−dimensional holes with short lifetimes are informally considered to be ...

We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which use...

The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensio...

In this thesis, we explore techniques in statistics and persistent homology, which detect features among data sets such as graphs, triangulations and point cloud. We accompany our theorems with algorithms and experiments, to demonstrate their effectiveness in practice. We start with the derivation of graph scan statistics, a measure useful to assess the statistical significance of a subgraph in...

Abstract We use methods from computational algebraic topology to study functional brain networks in which nodes represent regions and weighted edges encode the similarity of magnetic resonance imaging (fMRI) time series each region. With these tools, allow one characterize topological invariants such as loops high-dimensional data, we are able gain understanding low-dimensional structures a way...

This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent homological features in the diagrams for the different coefficient fields. This computation allows us to infer the prime divisors of the torsion coefficients of the...

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants suc...

We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is generally seen as a procedure which starts with a filtered complex and ends with a persistence diagram. This procedure is stable (at least to certain types of perturbations of the input). This justifies the use of the...

Persistent homology is a common technique in topological data analysis providing geometrical and information about the sample space. All this information, known as features, summarized persistence diagrams, main interest identifying most persisting ones since they correspond to Betti number values. Given randomness inherent sampling process, complex structure of space where diagrams take values...