165 research outputs found
Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues
Let be the smallest number such that the adjacency matrix of any
undirected graph with vertices or more has at least nonpositive
eigenvalues. We show that is well-defined and prove that the values of
for are respectively. In addition, we
prove that for all , , in which
is the Ramsey number for and , and is the triangular
number. This implies new lower bounds for eigenvalues of Laplacian matrices:
the -th largest eigenvalue is bounded from below by the -th largest
degree, which generalizes some prior results.Comment: 23 pages, 12 figure
A nodal domain theorem and a higher-order Cheeger inequality for the graph -Laplacian
We consider the nonlinear graph -Laplacian and its set of eigenvalues and
associated eigenfunctions of this operator defined by a variational principle.
We prove a nodal domain theorem for the graph -Laplacian for any .
While for the bounds on the number of weak and strong nodal domains are
the same as for the linear graph Laplacian (), the behavior changes for
. We show that the bounds are tight for as the bounds are
attained by the eigenfunctions of the graph -Laplacian on two graphs.
Finally, using the properties of the nodal domains, we prove a higher-order
Cheeger inequality for the graph -Laplacian for . If the eigenfunction
associated to the -th variational eigenvalue of the graph -Laplacian has
exactly strong nodal domains, then the higher order Cheeger inequality
becomes tight as
Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods
In this paper, we study matrix scaling and balancing, which are fundamental
problems in scientific computing, with a long line of work on them that dates
back to the 1960s. We provide algorithms for both these problems that, ignoring
logarithmic factors involving the dimension of the input matrix and the size of
its entries, both run in time where is the amount of error we are willing to
tolerate. Here, represents the ratio between the largest and the
smallest entries of the optimal scalings. This implies that our algorithms run
in nearly-linear time whenever is quasi-polynomial, which includes, in
particular, the case of strictly positive matrices. We complement our results
by providing a separate algorithm that uses an interior-point method and runs
in time .
In order to establish these results, we develop a new second-order
optimization framework that enables us to treat both problems in a unified and
principled manner. This framework identifies a certain generalization of linear
system solving that we can use to efficiently minimize a broad class of
functions, which we call second-order robust. We then show that in the context
of the specific functions capturing matrix scaling and balancing, we can
leverage and generalize the work on Laplacian system solving to make the
algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201
A Multiscale Pyramid Transform for Graph Signals
Multiscale transforms designed to process analog and discrete-time signals
and images cannot be directly applied to analyze high-dimensional data residing
on the vertices of a weighted graph, as they do not capture the intrinsic
geometric structure of the underlying graph data domain. In this paper, we
adapt the Laplacian pyramid transform for signals on Euclidean domains so that
it can be used to analyze high-dimensional data residing on the vertices of a
weighted graph. Our approach is to study existing methods and develop new
methods for the four fundamental operations of graph downsampling, graph
reduction, and filtering and interpolation of signals on graphs. Equipped with
appropriate notions of these operations, we leverage the basic multiscale
constructs and intuitions from classical signal processing to generate a
transform that yields both a multiresolution of graphs and an associated
multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure
- …