4,323 research outputs found
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
Geometric deep learning: going beyond Euclidean data
Many scientific fields study data with an underlying structure that is a
non-Euclidean space. Some examples include social networks in computational
social sciences, sensor networks in communications, functional networks in
brain imaging, regulatory networks in genetics, and meshed surfaces in computer
graphics. In many applications, such geometric data are large and complex (in
the case of social networks, on the scale of billions), and are natural targets
for machine learning techniques. In particular, we would like to use deep
neural networks, which have recently proven to be powerful tools for a broad
range of problems from computer vision, natural language processing, and audio
analysis. However, these tools have been most successful on data with an
underlying Euclidean or grid-like structure, and in cases where the invariances
of these structures are built into networks used to model them. Geometric deep
learning is an umbrella term for emerging techniques attempting to generalize
(structured) deep neural models to non-Euclidean domains such as graphs and
manifolds. The purpose of this paper is to overview different examples of
geometric deep learning problems and present available solutions, key
difficulties, applications, and future research directions in this nascent
field
Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry
In Arakelov theory a completion of an arithmetic surface is achieved by
enlarging the group of divisors by formal linear combinations of the ``closed
fibers at infinity''. Manin described the dual graph of any such closed fiber
in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody
endowed with a Schottky uniformization. In this paper we consider arithmetic
surfaces over the ring of integers in a number field, with fibers of genus
. We use Connes' theory of spectral triples to relate the hyperbolic
geometry of the handlebody to Deninger's Archimedean cohomology and the
cohomology of the cone of the local monodromy at arithmetic infinity as
introduced by the first author of this paper.Comment: 68 pages, 10pt LaTeX, xy-pic (v2: to appear in Selecta Mathematica
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