arXiv subjects

John C. Urschel

Publications and source records attributed to John C. Urschel.

8 recordsLinked to original sources

arXiv2020

Uniform Error Estimates for the Lanczos Method

The Lanczos method is one of the most powerful and fundamental techniques for solving an extremal symmetric eigenvalue problem. Convergence-based error estimates depend heavily on the eigenvalue gap. In practice, this gap is often relatively small, resulting in significant overestimates of error. One way to avoid this issue is through the use of uniform error estimates, namely, bounds that depend only on the dimension of the matrix and the number of iterations. In this work, we prove explicit upper and lower uniform error estimates for the Lanczos method. These lower bounds, paired with numerical results, imply that the maximum error of $m$ iterations of the Lanczos method over all $n \times n$ symmetric matrices does indeed depend on the dimension $n$ in practice. The improved bounds for extremal eigenvalues translate immediately to error estimates for the condition number of a symmetric positive definite matrix. In addition, we prove more specific results for matrices that possess some level of eigenvalue regularity or whose eigenvalues converge to some limiting empirical spectral distribution. Through numerical experiments, we show that the theoretical estimates of this paper do apply to practical computations for reasonably sized matrices.

math.NA↗

arXiv2020

Discrete Trace Theorems and Energy Minimizing Spring Embeddings of Planar Graphs

Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their neighbors, then this results in a unique embedding, and this embedding is planar. It also follows fairly quickly that this embedding minimizes the sum of squared edge lengths, conditional on the embedding of the outer face. However, it is not at all clear how to embed this outer face. We consider the minimization problem of embedding this outer face, up to some normalization, so that the sum of squared edge lengths is minimized. In this work, we show the connection between this optimization problem and the Schur complement of the graph Laplacian with respect to the interior vertices. We prove a number of discrete trace theorems, and, using these new results, show the spectral equivalence of this Schur complement with the boundary Laplacian to the one-half power for a large class of graphs. Using this result, we give theoretical guarantees for this optimization problem, which motivates an algorithm to embed the outer face of a spring embedding.

math.CO↗

arXiv2020

Testing Gap k-planarity is NP-complete

For all $k \geq 1$, we show that deciding whether a graph is $k$-planar is NP-complete, extending the well-known fact that deciding 1-planarity is NP-complete. Furthermore, we show that the gap version of this decision problem is NP-complete. In particular, given a graph with local crossing number either at most $k\ge 1$ or at least $2k$, we show that it is NP-complete to decide whether the local crossing number is at most $k$ or at least $2k$. This algorithmic lower bound proves the non-existence of a $(2-ε)$-approximation algorithm for any fixed $k \ge 1$. In addition, we analyze the sometimes competing relationship between the local crossing number (maximum number of crossings per edge) and crossing number (total number of crossings) of a drawing. We present results regarding the non-existence of drawings that simultaneously approximately minimize both the local crossing number and crossing number of a graph.

math.CO↗

arXiv2020

Regarding two conjectures on clique and biclique partitions

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures -- one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.

math.CO↗

arXiv2018

Constructing Frequency Domains on Graphs in Near-Linear Time

Analysis of big data has become an increasingly relevant area of research, with data often represented on discrete networks both constructed and organic. While for structured domains, there exist intuitive definitions of signals and frequencies, the definitions are much less obvious for data sets associated with a given network. Often, the eigenvectors of an induced graph Laplacian are used to construct an orthogonal set of low-frequency vectors. For larger graphs, however, the computational cost of creating such structures becomes untenable, and the quality of the approximation is adequate only for signals near the span of the set. We propose a construction of a full basis of frequencies with computational complexity that is near-linear in time and linear in storage. Using this frequency domain, we can compress data sets on unstructured graphs more robustly and accurately than spectral-based constructions.

math.NA↗

arXiv2016

On the Approximation of Laplacian Eigenvalues in Graph Disaggregation

Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian.

math.NA↗

arXiv2015

On the Maximal Error of Spectral Approximation of Graph Bisection

Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class of graphs, we prove that the standard spectral graph bisection can produce bisections that are far from optimal. In particular, we show that the maximum error in the spectral approximation of the optimal bisection (partition sizes exactly equal) cut for such graphs is bounded below by a constant multiple of the order of the graph squared.

math.NA↗

arXiv2014

A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.

math.NA↗