Unlabeled Sensing: Reconstruction Algorithm and Theoretical Guarantees

It often happens that we are interested in reconstructing an unknown signal from partial measurements. Also, it is typically assumed that the location (temporal or spatial) of the samples is known and that the only distortion present in the observations is due to additive measurement noise. However, there are some applications where such location information is lost. In this paper, we consider the situation in which the order of noisy samples out of a linear measurement system is missing. Previous work on this topic has only considered the noiseless case and exhaustive search combinatorial algorithms. We propose a much more efficient algorithm based on a geometrical viewpoint of the problem. We also study the uniqueness of the solution under different choices of the sampling matrix and its robustness to noise for the case of two-dimensional signals. Finally we provide simulation results to confirm the theoretical findings of the paper.