Recovering Data Permutation from Noisy Observations: The Linear Regime

This paper considers a noisy data structure recovery problem. The goal is to investigate the following question: Given a noisy data observation, according to which permutation was the original data sorted? The focus is on scenarios where data is generated according to an isotropic Gaussian distribution, and the perturbation consists of adding Gaussian noise with an arbitrary covariance matrix. This problem is posed within a hypothesis testing framework. The objective is to study the linear regime in which the optimal decoder has a polynomial complexity in the data size, and it declares the permutation by simply computing a linear function of the noisy observation. The main result of the paper is a complete characterization of the linear regime in terms of the noise covariance matrix. Specifically, it is shown that this matrix must have a very flat spectrum with at most three distinct eigenvalues to induce the linear regime. Several practically relevant implications of this result are discussed, and the error probability incurred by the decision criterion in the linear regime is also characterized. A core technical component consists of using linear algebraic and geometric tools, such as Steiner symmetrization.