Graph-Based Signal Sampling with Adaptive Subspace Reconstruction for Spatially-Irregular Sensor Data

Choosing an appropriate frequency definition and norm is critical in graph signal sampling and reconstruction. Most previous works define frequencies based on the spectral properties of the graph and use the same frequency definition and $\ell_2$-norm for optimization for all sampling sets. Our previous work demonstrated that using a sampling set-adaptive norm and frequency definition can address challenges in classical bandlimited approximation, particularly with model mismatches and irregularly distributed data. In this work, we propose a method for selecting sampling sets tailored to the sampling set adaptive GFT-based interpolation. When the graph models the inverse covariance of the data, we show that this adaptive GFT enables localizing the bandlimited model mismatch error to high frequencies, and the spectral folding property allows us to track this error in reconstruction. Based on this, we propose a sampling set selection algorithm to minimize the worst-case bandlimited model mismatch error. We consider partitioning the sensors in a sensor network sampling a continuous spatial process as an application. Our experiments show that sampling and reconstruction using sampling set adaptive GFT significantly outperform methods that used fixed GFTs and bandwidth-based criterion.