7 research outputs found
Intertwining wavelets or Multiresolution analysis on graphs through random forests
We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure
Exponential moments of self-intersection local times of stable random walks in subcritical dimensions
Let be an -stable random walk with values in
. Let be its local time. For ,
not necessarily integer, is the so-called -fold
self- intersection local time of the random walk. When , we
derive precise logarithmic asymptotics of the probability for
all scales r_t \gg \E(I_t). Our result extends previous works by Chen, Li and
Rosen 2005, Becker and K\"onig 2010, and Laurent 2012