Results 1 
3 of
3
Estimates for nonlinear harmonic measures on trees
"... Abstract. In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set D the value at the origin of the solution to u(x) = F ((x, 0),..., (x,m−1)) for every x ∈ Tm, a directed tree with m branches with initial datum f + χD. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set D the value at the origin of the solution to u(x) = F ((x, 0),..., (x,m−1)) for every x ∈ Tm, a directed tree with m branches with initial datum f + χD. Here F is an averaging operator on Rm, x is a vertex of a directed tree Tm with regular mbranching and (x, i) denotes a successor of that vertex for 0 ≤ i ≤ m − 1. We also provide a characterization of the subsets of the tree for which the unique continuation property holds. 1.
Algorithms for Lipschitz Learning on Graphs *
, 2015
"... Abstract We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for lar ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large p of pLaplacian regularization. We present an algorithm that computes a minimal Lipschitz extension in expected linear time, and an algorithm that computes an absolutely minimal Lipschitz extension in expected time O(mn). The latter algorithm has variants that seem to run much faster in practice. These extensions are particularly amenable to regularization: we can perform l 0 regularization on the given values in polynomial time and l 1 regularization on the initial function values and on graph edge weights in time O(m 3/2 ). Our definitions and algorithms naturally extend to directed graphs.