56,853 research outputs found
Computing Persistent Homology within Coq/SSReflect
Persistent homology is one of the most active branches of Computational
Algebraic Topology with applications in several contexts such as optical
character recognition or analysis of point cloud data. In this paper, we report
on the formal development of certified programs to compute persistent Betti
numbers, an instrumental tool of persistent homology, using the Coq proof
assistant together with the SSReflect extension. To this aim it has been
necessary to formalize the underlying mathematical theory of these algorithms.
This is another example showing that interactive theorem provers have reached a
point where they are mature enough to tackle the formalization of nontrivial
mathematical theories
Convergence between Categorical Representations of Reeb Space and Mapper
The Reeb space, which generalizes the notion of a Reeb graph, is one of the
few tools in topological data analysis and visualization suitable for the study
of multivariate scientific datasets. First introduced by Edelsbrunner et al.,
it compresses the components of the level sets of a multivariate mapping and
obtains a summary representation of their relationships. A related construction
called mapper, and a special case of the mapper construction called the Joint
Contour Net have been shown to be effective in visual analytics. Mapper and JCN
are intuitively regarded as discrete approximations of the Reeb space, however
without formal proofs or approximation guarantees. An open question has been
proposed by Dey et al. as to whether the mapper construction converges to the
Reeb space in the limit.
In this paper, we are interested in developing the theoretical understanding
of the relationship between the Reeb space and its discrete approximations to
support its use in practical data analysis. Using tools from category theory,
we formally prove the convergence between the Reeb space and mapper in terms of
an interleaving distance between their categorical representations. Given a
sequence of refined discretizations, we prove that these approximations
converge to the Reeb space in the interleaving distance; this also helps to
quantify the approximation quality of the discretization at a fixed resolution
Computable decision making on the reals and other spaces via partiality and nondeterminism
Though many safety-critical software systems use floating point to represent
real-world input and output, programmers usually have idealized versions in
mind that compute with real numbers. Significant deviations from the ideal can
cause errors and jeopardize safety. Some programming systems implement exact
real arithmetic, which resolves this matter but complicates others, such as
decision making. In these systems, it is impossible to compute (total and
deterministic) discrete decisions based on connected spaces such as
. We present programming-language semantics based on constructive
topology with variants allowing nondeterminism and/or partiality. Either
nondeterminism or partiality suffices to allow computable decision making on
connected spaces such as . We then introduce pattern matching on
spaces, a language construct for creating programs on spaces, generalizing
pattern matching in functional programming, where patterns need not represent
decidable predicates and also may overlap or be inexhaustive, giving rise to
nondeterminism or partiality, respectively. Nondeterminism and/or partiality
also yield formal logics for constructing approximate decision procedures. We
implemented these constructs in the Marshall language for exact real
arithmetic.Comment: This is an extended version of a paper due to appear in the
proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS) in
July 201
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