Signal Recovery from Unlabeled Samples

In this paper, we study the recovery of a signal from a collection of unlabeled and possibly noisy measurements via a measurement matrix with random i.i.d. Gaussian components. We call the measurements unlabeled since their order is missing, namely, it is not known a priori which elements of the resulting measurements correspond to which row of the measurement matrix. We focus on the special case of ordered measurements, where only a subset of the measurements is kept and the order of the taken measurements is preserved. We identify a natural duality between this problem and the traditional Compressed Sensing, where we show that the unknown support (location of nonzero elements) of a sparse signal in Compressed Sensing corresponds in a natural way to the unknown location of the measurements kept in unlabeled sensing. While in Compressed Sensing it is possible to recover a sparse signal from an under-determined set of linear equations (less equations than the dimension of the signal), successful recovery in unlabeled sensing requires taking more samples than the dimension of the signal. We develop a low-complexity alternating minimization algorithm to recover the initial signal from the set of its unlabeled samples. We also study the behavior of the proposed algorithm for different signal dimensions and number of measurements both theoretically and empirically via numerical simulations. The results are a reminiscent of the phase-transition similar to that occurring in Compressed Sensing.