215 research outputs found
Cellular spanning trees and Laplacians of cubical complexes
We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell
complex in terms of the eigenvalues of its cellular Laplacian operators,
generalizing a previous result for simplicial complexes. As an application, we
obtain explicit formulas for spanning tree enumerators and Laplacian
eigenvalues of cubes; the latter are integers. We prove a weighted version of
the eigenvalue formula, providing evidence for a conjecture on weighted
enumeration of cubical spanning trees. We introduce a cubical analogue of
shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of
shifted cubical complexes, in particular, these eigenvalues are also integers.
Finally, we recover Adin's enumeration of spanning trees of a complete colorful
simplicial complex from the cellular Matrix-Tree Theorem together with a result
of Kook, Reiner and Stanton.Comment: 24 pages, revised version, to appear in Advances in Applied
Mathematic
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Towards quantum advantage via topological data analysis
Even after decades of quantum computing development, examples of generally
useful quantum algorithms with exponential speedups over classical counterparts
are scarce. Recent progress in quantum algorithms for linear-algebra positioned
quantum machine learning (QML) as a potential source of such useful exponential
improvements. Yet, in an unexpected development, a recent series of
"dequantization" results has equally rapidly removed the promise of exponential
speedups for several QML algorithms. This raises the critical question whether
exponential speedups of other linear-algebraic QML algorithms persist. In this
paper, we study the quantum-algorithmic methods behind the algorithm for
topological data analysis of Lloyd, Garnerone and Zanardi through this lens. We
provide evidence that the problem solved by this algorithm is classically
intractable by showing that its natural generalization is as hard as simulating
the one clean qubit model -- which is widely believed to require
superpolynomial time on a classical computer -- and is thus very likely immune
to dequantizations. Based on this result, we provide a number of new quantum
algorithms for problems such as rank estimation and complex network analysis,
along with complexity-theoretic evidence for their classical intractability.
Furthermore, we analyze the suitability of the proposed quantum algorithms for
near-term implementations. Our results provide a number of useful applications
for full-blown, and restricted quantum computers with a guaranteed exponential
speedup over classical methods, recovering some of the potential for
linear-algebraic QML to become one of quantum computing's killer applications.Comment: 29 pages, 3 figures. New results added and improved expositio
Combinatorial and Hodge Laplacians: Similarity and Difference
As key subjects in spectral geometry and combinatorial graph theory
respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian
share similarities in revealing the topological dimension and geometric shape
of data and in their realization of diffusion and minimization of harmonic
measures. It is believed that they also both associate with vector calculus,
through the gradient, curl, and divergence, as argued in the popular usage of
"Hodge Laplacians on graphs" in the literature. Nevertheless, these Laplacians
are intrinsically different in their domains of definitions and applicability
to specific data formats, hindering any in-depth comparison of the two
approaches.
To facilitate the comparison and bridge the gap between the combinatorial
Laplacian and Hodge Laplacian for the discretization of continuous manifolds
with boundary, we further introduce Boundary-Induced Graph (BIG) Laplacians
using tools from Discrete Exterior Calculus (DEC). BIG Laplacians are defined
on discrete domains with appropriate boundary conditions to characterize the
topology and shape of data. The similarities and differences of the
combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are then examined.
Through an Eulerian representation of 3D domains as level-set functions on
regular grids, we show experimentally the conditions for the convergence of BIG
Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes.Comment: 26 page
Efficient 1-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences
We present efficient algorithms for solving systems of linear equations in 1-Laplacians of well-shaped simplicial complexes. 1-Laplacians, or higher-dimensional Laplacians, generalize graph Laplacians to higher-dimensional simplicial complexes and play a key role in computational topology and topological data analysis. Previously, nearly-linear time solvers were developed for simplicial complexes with known collapsing sequences and bounded Betti numbers, such as those triangulating a three-ball in R3 (Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA’2014], Black, Maxwell, Nayyeri, and Winkelman [SODA’2022], Black and Nayyeri [ICALP’2022]). Furthermore, Nested Dissection provides quadratic time solvers for more general systems with nonzero structures representing well-shaped simplicial complexes embedded in R3. We generalize the specialized solvers for 1-Laplacians to simplicial complexes with additional geometric structures but without collapsing sequences and bounded Betti numbers, and we improve the runtime of Nested Dissection. We focus on simplicial complexes that meet two conditions: (1) each individual simplex has a bounded aspect ratio, and (2) they can be divided into “disjoint” and balanced regions with well-shaped interiors and boundaries. Our solvers draw inspiration from the Incomplete Nested Dissection for stiffness matrices of well-shaped trusses (Kyng, Peng, Schwieterman, and Zhang [STOC’2018]).ISSN:1868-896
Weighted Tree-Numbers of Matroid Complexes
International audienceWe give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s -invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes.Nous présentons une nouvelle formule pour les nombres d’arbres pondérés de grande dimension des matroïdes complexes. Cette formule est dérivée du résultat que le spectre des Laplaciens combinatoires pondérés des matrides complexes sont des polynômes à plusieurs variables. Dans la formule, le ;-invariant de Crapo apparaît comme étant le facteur clé reliant les Laplaciens combinatoires pondérés et les nombres d’arbres pondérés des matroïdes complexes
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