# Structured sampling and fast reconstruction of smooth graph signals

arXiv: https://arxiv.org/abs/1705.02202 · HTML: https://mlsanity.com/paper/1705.02202/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2017.

## Built on by (most cited first, 3 of 3)

- [Adaptive Gaussian Processes on Graphs via Spectral Graph Wavelets](https://mlsanity.com/paper/2110.12752/) (2021)
- [Efficient Signed Graph Sampling via Balancing & Gershgorin Disc Perfect Alignment](https://mlsanity.com/paper/2208.08726/) (2022)
- [Graph Unfolding and Sampling for Transitory Video Keyframe Selection via Gershgorin Disc Alignment](https://mlsanity.com/paper/2408.01859/) (2024)

## Abstract (arXiv)

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 k-bandlimited signals is driven by a quantity called the group graph cumulative coherence. For some optimised sampling distributions, we show that sampling O(klog(k)) groups is always sufficient to stably embed the set of k-bandlimited signals but that this number can be smaller -- down to O(log(k)) -- depending on the structure of the groups of nodes. Fast methods to approximate these sampling distributions are detailed. Second, we consider k-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.