1 research outputs found
Graph Sampling for Covariance Estimation
In this paper the focus is on subsampling as well as reconstructing the
second-order statistics of signals residing on nodes of arbitrary undirected
graphs. Second-order stationary graph signals may be obtained by graph
filtering zero-mean white noise and they admit a well-defined power spectrum
whose shape is determined by the frequency response of the graph filter.
Estimating the graph power spectrum forms an important component of stationary
graph signal processing and related inference tasks such as Wiener prediction
or inpainting on graphs. The central result of this paper is that by sampling a
significantly smaller subset of vertices and using simple least squares, we can
reconstruct the second-order statistics of the graph signal from the subsampled
observations, and more importantly, without any spectral priors. To this end,
both a nonparametric approach as well as parametric approaches including moving
average and autoregressive models for the graph power spectrum are considered.
The results specialize for undirected circulant graphs in that the graph nodes
leading to the best compression rates are given by the so-called minimal sparse
rulers. A near-optimal greedy algorithm is developed to design the subsampling
scheme for the non-parametric and the moving average models, whereas a
particular subsampling scheme that allows linear estimation for the
autoregressive model is proposed. Numerical experiments on synthetic as well as
real datasets related to climatology and processing handwritten digits are
provided to demonstrate the developed theory.Comment: Under peer review for Jour. of Sel. Topics in Signal Proc. (special
issue on graph signal processing), Nov. 201