10 research outputs found
Computational and Theoretical Issues of Multiparameter Persistent Homology for Data Analysis
The basic goal of topological data analysis is to apply topology-based descriptors
to understand and describe the shape of data. In this context, homology is one of
the most relevant topological descriptors, well-appreciated for its discrete nature,
computability and dimension independence. A further development is provided
by persistent homology, which allows to track homological features along a oneparameter
increasing sequence of spaces. Multiparameter persistent homology, also
called multipersistent homology, is an extension of the theory of persistent homology
motivated by the need of analyzing data naturally described by several parameters,
such as vector-valued functions. Multipersistent homology presents several issues in
terms of feasibility of computations over real-sized data and theoretical challenges
in the evaluation of possible descriptors. The focus of this thesis is in the interplay
between persistent homology theory and discrete Morse Theory. Discrete Morse
theory provides methods for reducing the computational cost of homology and persistent
homology by considering the discrete Morse complex generated by the discrete
Morse gradient in place of the original complex. The work of this thesis addresses
the problem of computing multipersistent homology, to make such tool usable in real
application domains. This requires both computational optimizations towards the
applications to real-world data, and theoretical insights for finding and interpreting
suitable descriptors. Our computational contribution consists in proposing a new
Morse-inspired and fully discrete preprocessing algorithm. We show the feasibility
of our preprocessing over real datasets, and evaluate the impact of the proposed
algorithm as a preprocessing for computing multipersistent homology. A theoretical
contribution of this thesis consists in proposing a new notion of optimality for such
a preprocessing in the multiparameter context. We show that the proposed notion
generalizes an already known optimality notion from the one-parameter case. Under
this definition, we show that the algorithm we propose as a preprocessing is optimal
in low dimensional domains. In the last part of the thesis, we consider preliminary
applications of the proposed algorithm in the context of topology-based multivariate
visualization by tracking critical features generated by a discrete gradient field compatible
with the multiple scalar fields under study. We discuss (dis)similarities of such
critical features with the state-of-the-art techniques in topology-based multivariate
data visualization
Parallel decomposition of persistence modules through interval bases
We introduce an algorithm to decompose any finite-type persistence module
with coefficients in a field into what we call an {em interval basis}. This
construction yields both the standard persistence pairs of Topological Data
Analysis (TDA), as well as a special set of generators inducing the interval
decomposition of the Structure theorem. The computation of this basis can be
distributed over the steps in the persistence module. This construction works
for general persistence modules on a field , not necessarily
deriving from persistent homology. We subsequently provide a parallel algorithm
to build a persistent homology module over by leveraging the Hodge
decomposition, thus providing new motivation to explore the interplay between
TDA and the Hodge Laplacian
Parallel decomposition of persistence modules through interval bases
We introduce an algorithm to decompose any finite-type persistence module
with coefficients in a field into what we call an {\em interval basis}. This
construction yields both the standard persistence pairs of Topological Data
Analysis (TDA), as well as a special set of generators inducing the interval
decomposition of the Structure theorem. The computation of this basis can be
distributed over the steps in the persistence module. This construction works
for general persistence modules on a field , not necessarily
deriving from persistent homology. We subsequently provide a parallel algorithm
to build a persistent homology module over by leveraging the Hodge
decomposition, thus providing new motivation to explore the interplay between
TDA and the Hodge Laplacian.Comment: 37 pages, 6 figure
Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology
The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires
the detection and interpretation of the possible interdependence among data components. To this end, in this paper, we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical
cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables
of persistence modules from above and below, via the number of critical cells. Our last result shows that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2
Persistent homology: a topological tool for higher-interaction systems
3nonenoneVaccarino, Francesco; Fugacci, Ulderico; Scaramuccia, SaraVaccarino, Francesco; Fugacci, Ulderico; Scaramuccia, Sar
Computing multiparameter persistent homology through a discrete Morse-based approach
Persistent homology allows for tracking topological features, like loops, holes and their higher-dimensional analogues, along a single-parameter family of nested shapes. Computing descriptors for complex data characterized by multiple parameters is becoming a major challenging task in several applications, including physics, chemistry, medicine, and geography. Multiparameter persistent homology generalizes persistent homology to allow for the exploration and analysis of shapes endowed with multiple filtering functions. Still, computational constraints prevent multiparameter persistent homology to be a feasible tool for analyzing large size data sets. We consider discrete Morse theory as a strategy to reduce the computation of multiparameter persistent homology by working on a reduced dataset. We propose a new preprocessing algorithm, well suited for parallel and distributed implementations, and we provide the first evaluation of the impact of multiparameter persistent homology on computations