Correlation sketching for ordered data

Methods based on order statistics are often used in finance, quality control, data and signal processing, especially when signals of interest are immersed in impulsive noise. These allow to include rank information by increasing the dimension of the problem. In large dimension problems, we are usually required to know only the second order statistics. In this article we use a rank-one quadratic measurement model based on sketches to estimate the correlation matrix for ordered data. Furthermore, we exploit this matrix's structure to design a convex relaxation optimization problem to recover the matrix. This reconstruction takes a number of measurements proportional to the original size of the problem (without ordering). We provide simulations to show the reconstruction performance of the proposed scheme, and the robustness of this estimation when uniform noise is present.