Local differential privacy for physical sensor data and sparse recovery

In this work, we exploit the ill-posedness of linear inverse problems to design algorithms to release differentially private data or measurements of the physical system. We discuss the spectral requirements on a matrix such that only a small amount of noise is needed to achieve privacy and contrast this with the ill-conditionedness. We then instantiate our framework with several diffusion operators and explore recovery via constrained minimisation. Our work indicates that it is possible to produce locally private sensor measurements that both keep the exact locations of the heat sources private and permit recovery of the "general geographic vicinity" of the sources.