arXiv subjects

Andrew Lyons

Publications and source records attributed to Andrew Lyons.

5 recordsLinked to original sources

arXiv2026

A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation

The algorithmic differentiation (AD) of mathematical functions can be interpreted as a sequence of vertex eliminations in an underlying directed acyclic graph. The problem of determining a minimum-cost elimination ordering, which we call Optimal Vertex Elimination, is NP-complete. Consequently, much effort has been devoted to the design of heuristics. Many of these heuristics are widely believed to perform well in practice, but this hypothesis has so far been difficult to test due to the lack of scalable exact methods. We design and engineer new integer programming formulations for Optimal Vertex Eliminatioin and for a related objective we call Minimum Edge Count. Our implementations scale to graphs one-to-two orders of magnitude larger than existing techniques, enabling the assembly of a corpus of medium-sized graphs for which optimal solutions are known. This corpus facilitates a study of existing heuristics, confirming that on real data popular methods achieve high quality solutions. We also make several theoretical contributions. We give a tight analysis of the forward and reverse modes of AD, and extend our techniques to provide a simple algorithm for Optimal Vertex Elimination with approximation ratio parameterized by the size of a minimum source-sink separator. On the complexity side, we give the first approximation lower bounds for both problems.

cs.DS↗

arXiv2026

Nodal Deficiency of Neumann Eigenfunctions on a Symmetric Dumbbell Domain

We study the nodal deficiency of pairs of Neumann eigenfunctions defined over symmetric dumbbell domains. As the width of the connecting neck shrinks, these eigenfunctions converge to Neumann eigenfunctions defined over the ends of the dumbbell, together with a one-dimensional Sturm-Liouville solution in the neck. In this limit, the corresponding eigenvalues become degenerate, with multiplicity two. The nodal deficiency, defined as the difference between the eigenvalue index and the nodal domain count, is known by the Courant nodal domain theorem to be nonnegative. We show that, for small neck widths, the nodal deficiencies of the dumbbell eigenfunctions are no smaller than the nodal deficiencies of the limiting eigenfunctions in the ends, and we provide conditions under which equality is achieved. As a consequence, we establish a criterion for identifying eigenfunctions of zero nodal deficiency for the dumbbell domain.

math.AP↗

arXiv2025

Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary

We study the eigenvalues and eigenfunctions of a differential operator that governs the asymptotic behavior of the unsupervised learning algorithm known as Locally Linear Embedding when a large data set is sampled from an interval or disc. In particular, the differential operator is of second order, mixed-type, and degenerates near the boundary. We show that a natural regularity condition on the eigenfunctions imposes a consistent boundary condition and use the Frobenius method to estimate pointwise behavior. We then determine the limiting sequence of eigenvalues analytically and compare them to numerical predictions. Finally, we propose a variational framework for determining eigenvalues on other compact manifolds.

math.AP↗

arXiv2023

Nodal Sets of Laplacian Eigenfunctions with an Eigenvalue of Multiplicity 2

We study the effects of a domain deformation to the nodal set of Laplacian eigenfunctions when the eigenvalue is degenerate. In particular, we study deformations of a rectangle that perturb one side and how they change the nodal sets corresponding to an eigenvalue of multiplicity 2. We establish geometric properties, such as number of nodal domains, presence of crossings, and boundary intersections, of nodal sets for a large class of boundary deformations and study how these properties change along each eigenvalue branch for small perturbations. We show that internal crossings of the nodal set break under generic deformations and obtain estimates on the location and regularity of the nodal sets on the perturbed rectangle.

math.AP↗

arXiv2011

Acyclic and Star Colorings of Cographs

An \emph{acyclic coloring} of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of \emph{star coloring} requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding an optimal acyclic and star coloring of a cograph. If the graph is given in the form of a cotree, the algorithm runs in O(n) time. We also show that the acyclic chromatic number, the star chromatic number, the treewidth plus one, and the pathwidth plus one are all equal for cographs.

cs.DS↗