62 research outputs found
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
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
Chunk Reduction for Multi-Parameter Persistent Homology
The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among all the complexes quasi-isomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation
Discrete Morse theory for computing cellular sheaf cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology,
greatly generalizing persistent homology. We develop an algorithm for
simplifying the computation of cellular sheaf cohomology via (discrete)
Morse-theoretic techniques. As a consequence, we derive efficient techniques
for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.
Delaunay Bifiltrations of Functions on Point Clouds
The Delaunay filtration of a point cloud is a central tool of computational topology. Its use is justified
by the topological equivalence of and the offset
(i.e., union-of-balls) filtration of . Given a function , we introduce a Delaunay bifiltration
that satisfies an analogous topological
equivalence, ensuring that topologically
encodes the offset filtrations of all sublevel sets of , as well as the
topological relations between them. is of size
, which for odd matches the worst-case
size of . Adapting the Bowyer-Watson algorithm for
computing Delaunay triangulations, we give a simple, practical algorithm to
compute in time . Our implementation, based on CGAL, computes
with modest overhead compared to computing
, and handles tens of thousands of points in
within seconds.Comment: 28 pages, 7 figures, 8 tables. To appear in the proceedings of SODA2
Computing Multipersistence by Means of Spectral Systems
In their original setting, both spectral sequences and persistent homology are algebraic topology tools defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. Recently, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are related, generalizing results valid in the case of filtrations over Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a new invariant providing information on multifiltrations and applications of our algorithms to spaces of infinite type
Chunk Reduction for Multi-Parameter Persistent Homology
The extension of persistent homology to multi-parameter setups is an
algorithmic challenge. Since most computation tasks scale badly with the size
of the input complex, an important pre-processing step consists of simplifying
the input while maintaining the homological information. We present an
algorithm that drastically reduces the size of an input. Our approach is an
extension of the chunk algorithm for persistent homology (Bauer et al.,
Topological Methods in Data Analysis and Visualization III, 2014). We show that
our construction produces the smallest multi-filtered chain complex among all
the complexes quasi-isomorphic to the input, improving on the guarantees of
previous work in the context of discrete Morse theory. Our algorithm also
offers an immediate parallelization scheme in shared memory. Already its
sequential version compares favorably with existing simplification schemes, as
we show by experimental evaluation
- …