4,273 research outputs found
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
Perturbative Quantum Field Theory on Random Trees
In this paper we start a systematic study of quantum field theory on random
trees. Using precise probability estimates on their Galton-Watson branches and
a multiscale analysis, we establish the general power counting of averaged
Feynman amplitudes and check that they behave indeed as living on an effective
space of dimension 4/3, the spectral dimension of random trees. In the `just
renormalizable' case we prove convergence of the averaged amplitude of any
completely convergent graph, and establish the basic localization and
subtraction estimates required for perturbative renormalization. Possible
consequences for an SYK-like model on random trees are briefly discussed.Comment: 44 page
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