636 research outputs found
Decision trees, monotone functions, and semimatroids
We define decision trees for monotone functions on a simplicial complex. We
define homology decidability of monotone functions, and show that various
monotone functions related to semimatroids are homology decidable. Homology
decidability is a generalization of semi-nonevasiveness, a notion due to
Jonsson. The motivating example is the complex of bipartite graphs, whose Betti
numbers are unknown in general.
We show that these monotone functions have optimum decision trees, from which
we can compute relative Betti numbers of related pairs of simplicial complexes.
Moreover, these relative Betti numbers are coefficients of evaluations of the
Tutte polynomial, and every semimatroid collapses onto its broken circuit
complex.Comment: 16 page
Avoiding the Global Sort: A Faster Contour Tree Algorithm
We revisit the classical problem of computing the \emph{contour tree} of a
scalar field , where is a
triangulated simplicial mesh in . The contour tree is a
fundamental topological structure that tracks the evolution of level sets of
and has numerous applications in data analysis and visualization.
All existing algorithms begin with a global sort of at least all critical
values of , which can require (roughly) time. Existing
lower bounds show that there are pathological instances where this sort is
required. We present the first algorithm whose time complexity depends on the
contour tree structure, and avoids the global sort for non-pathological inputs.
If denotes the set of critical points in , the running time is
roughly , where is the depth of in
the contour tree. This matches all existing upper bounds, but is a significant
improvement when the contour tree is short and fat. Specifically, our approach
ensures that any comparison made is between nodes in the same descending path
in the contour tree, allowing us to argue strong optimality properties of our
algorithm.
Our algorithm requires several novel ideas: partitioning in
well-behaved portions, a local growing procedure to iteratively build contour
trees, and the use of heavy path decompositions for the time complexity
analysis
Building Efficient and Compact Data Structures for Simplicial Complexes
The Simplex Tree (ST) is a recently introduced data structure that can
represent abstract simplicial complexes of any dimension and allows efficient
implementation of a large range of basic operations on simplicial complexes. In
this paper, we show how to optimally compress the Simplex Tree while retaining
its functionalities. In addition, we propose two new data structures called the
Maximal Simplex Tree (MxST) and the Simplex Array List (SAL). We analyze the
compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List
under various settings.Comment: An extended abstract appeared in the proceedings of SoCG 201
Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening
This work introduces a number of algebraic topology approaches, such as
multicomponent persistent homology, multi-level persistent homology and
electrostatic persistence for the representation, characterization, and
description of small molecules and biomolecular complexes. Multicomponent
persistent homology retains critical chemical and biological information during
the topological simplification of biomolecular geometric complexity.
Multi-level persistent homology enables a tailored topological description of
inter- and/or intra-molecular interactions of interest. Electrostatic
persistence incorporates partial charge information into topological
invariants. These topological methods are paired with Wasserstein distance to
characterize similarities between molecules and are further integrated with a
variety of machine learning algorithms, including k-nearest neighbors, ensemble
of trees, and deep convolutional neural networks, to manifest their descriptive
and predictive powers for chemical and biological problems. Extensive numerical
experiments involving more than 4,000 protein-ligand complexes from the PDBBind
database and near 100,000 ligands and decoys in the DUD database are performed
to test respectively the scoring power and the virtual screening power of the
proposed topological approaches. It is demonstrated that the present approaches
outperform the modern machine learning based methods in protein-ligand binding
affinity predictions and ligand-decoy discrimination
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