This report is concerned with power-weighted weight functionals associated with a minimal graph spanning a random sample of n points from a general multivariate Lebesgue density f over [0, 1]. It is known that under broad conditions, when the functional applies power exponent γ ∈ (1, d) to the graph edge lengths, the log of the functional normalized by n(d−γ)/d is a strongly consistent estimato...

The convergence of the discrete graph Laplacian to the continuous manifold Laplacian in the limit of sample size N → ∞ while the kernel bandwidth ε → 0, is the justification for the success of Laplacian based algorithms in machine learning, such as dimensionality reduction, semi-supervised learning and spectral clustering. In this paper we improve the convergence rate of the variance term recen...

We study the approach to obtaining least squares solutions to systems of linear algebraic equations over networks by using distributed algorithms. Each node has access to one of the linear equations and holds a dynamic state. The aim for the node states is to reach a consensus as a least squares solution of the linear equations by exchanging their states with neighbors over an underlying intera...

We consider partially asynchronous parallel iteration of a fixed nonnegative matrix with stationary ergodic interprocessor communication delays. We study the iteration via a random graph describing the interprocessor influences. Our major result is an invariant description of the rates of convergence of arbitrary sequences of individual processor-time values. In the course of proving this resul...

We construct harmonic diffeomorphisms from the complex plane C onto any Hadamard surface M whose curvature is bounded above by a negative constant. For that, we prove a JenkinsSerrin type theorem for minimal graphs in M × R over domains of M bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains converging to an entire minimal graph in M...

In this paper we give a complete characterization of the convergence of stationary vector subdivision schemes and the regularity of the associated limit function. These results extend and complete our earlier work on vector subdivision and its use in the construction of multiwavelets.

Kramer and Bruckner defined the following transformation on a weighted graph on n vertices. The transformation replaces the weight at every vertex by either the minimum or the maximum of the weights in its closed neighborhood, the choice being that extremum which is closer in value to the original weight. They showed that the system always converges after a finite number of iterations of the tr...

Graph partitioning problems have a wide range of applications in machine learning. This work analyzes convergence conditions for a class of diffuse interface algorithm [A.L. Bertozzi and A. Flenner, Multiscale Modeling & Simulation, 10(3):109

A general framework for solving combinatorial optimization problems heuristically by the Ant System approach is developed. The framework is based on the concept of a construction graph, a graph assigned to an instance of the optimization problem under consideration, encoding feasible solutions by walks. It is shown that under certain conditions, the solutions generated in each iteration of this...

Computing the k dominant eigenvalues and eigenvectors of massive graphs is a key operation in numerous machine learning applications; however, popular solvers suffer from slow convergence, especially when k is reasonably large. In this paper, we propose and analyze a novel multi-scale spectral decomposition method (MSEIGS), which first clusters the graph into smaller clusters whose spectral dec...