5 research outputs found
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
Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment
Graph sampling set selection, where a subset of nodes are chosen to collect
samples to reconstruct a smooth graph signal, is a fundamental problem in graph
signal processing (GSP). Previous works employ an unbiased least-squares (LS)
signal reconstruction scheme and select samples via expensive extreme
eigenvector computation. Instead, we assume a biased graph Laplacian
regularization (GLR) based scheme that solves a system of linear equations for
reconstruction. We then choose samples to minimize the condition number of the
coefficient matrix---specifically, maximize the smallest eigenvalue
. Circumventing explicit eigenvalue computation, we maximize
instead the lower bound of , designated by the smallest
left-end of all Gershgorin discs of the matrix. To achieve this efficiently, we
first convert the optimization to a dual problem, where we minimize the number
of samples needed to align all Gershgorin disc left-ends at a chosen
lower-bound target . Algebraically, the dual problem amounts to optimizing
two disc operations: i) shifting of disc centers due to sampling, and ii)
scaling of disc radii due to a similarity transformation of the matrix. We
further reinterpret the dual as an intuitive disc coverage problem bearing
strong resemblance to the famous NP-hard set cover (SC) problem. The
reinterpretation enables us to derive a fast approximation scheme from a known
SC error-bounded approximation algorithm. We find an appropriate target
efficiently via binary search. Extensive simulation experiments show that our
disc-based sampling algorithm runs substantially faster than existing sampling
schemes and outperforms other eigen-decomposition-free sampling schemes in
reconstruction error.Comment: Very fast deterministic graph sampling set selection algorithm
without explicit eigen-decompositio