4 research outputs found
Interpolation of Sparse Graph Signals by Sequential Adaptive Thresholds
This paper considers the problem of interpolating signals defined on graphs.
A major presumption considered by many previous approaches to this problem has
been lowpass/ band-limitedness of the underlying graph signal. However,
inspired by the findings on sparse signal reconstruction, we consider the graph
signal to be rather sparse/compressible in the Graph Fourier Transform (GFT)
domain and propose the Iterative Method with Adaptive Thresholding for Graph
Interpolation (IMATGI) algorithm for sparsity promoting interpolation of the
underlying graph signal.We analytically prove convergence of the proposed
algorithm. We also demonstrate efficient performance of the proposed IMATGI
algorithm in reconstructing randomly generated sparse graph signals. Finally,
we consider the widely desirable application of recommendation systems and show
by simulations that IMATGI outperforms state-of-the-art algorithms on the
benchmark datasets in this application.Comment: 12th International Conference on Sampling Theory and Applications
(SAMPTA 2017
Structured sampling and fast reconstruction of smooth graph signals
This work concerns sampling of smooth signals on arbitrary graphs. We first
study a structured sampling strategy for such smooth graph signals that
consists of a random selection of few pre-defined groups of nodes. The number
of groups to sample to stably embed the set of -bandlimited signals is
driven by a quantity called the \emph{group} graph cumulative coherence. For
some optimised sampling distributions, we show that sampling
groups is always sufficient to stably embed the set of -bandlimited signals
but that this number can be smaller -- down to -- depending on the
structure of the groups of nodes. Fast methods to approximate these sampling
distributions are detailed. Second, we consider -bandlimited signals that
are nearly piecewise constant over pre-defined groups of nodes. We show that it
is possible to speed up the reconstruction of such signals by reducing
drastically the dimension of the vectors to reconstruct. When combined with the
proposed structured sampling procedure, we prove that the method provides
stable and accurate reconstruction of the original signal. Finally, we present
numerical experiments that illustrate our theoretical results and, as an
example, show how to combine these methods for interactive object segmentation
in an image using superpixels
Random sampling of bandlimited signals on graphs
We study the problem of sampling k-bandlimited signals on graphs. We propose
two sampling strategies that consist in selecting a small subset of nodes at
random. The first strategy is non-adaptive, i.e., independent of the graph
structure, and its performance depends on a parameter called the graph
coherence. On the contrary, the second strategy is adaptive but yields optimal
results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure
an accurate and stable recovery of all k-bandlimited signals. This second
strategy is based on a careful choice of the sampling distribution, which can
be estimated quickly. Then, we propose a computationally efficient decoder to
reconstruct k-bandlimited signals from their samples. We prove that it yields
accurate reconstructions and that it is also stable to noise. Finally, we
conduct several experiments to test these techniques