Inferring Rankings Using Constrained Sensing

We consider the problem of recovering a function over the space of permutations (or, the symmetric group) over $n$ elements from given partial information; the partial information we consider is related to the group theoretic Fourier Transform of the function. This problem naturally arises in several settings such as ranked elections, multi-object tracking, ranking systems, and recommendation systems. Inspired by the work of Donoho and Stark in the context of discrete-time functions, we focus on non-negative functions with a sparse support (support size $\ll$ domain size). Our recovery method is based on finding the sparsest solution (through $\ell_0$ optimization) that is consistent with the available information. As the main result, we derive sufficient conditions for functions that can be recovered exactly from partial information through $\ell_0$ optimization. Under a natural random model for the generation of functions, we quantify the recoverability conditions by deriving bounds on the sparsity (support size) for which the function satisfies the sufficient conditions with a high probability as $n \to \infty$. $\ell_0$ optimization is computationally hard. Therefore, the popular compressive sensing literature considers solving the convex relaxation, $\ell_1$ optimization, to find the sparsest solution. However, we show that $\ell_1$ optimization fails to recover a function (even with constant sparsity) generated using the random model with a high probability as $n \to \infty$. In order to overcome this problem, we propose a novel iterative algorithm for the recovery of functions that satisfy the sufficient conditions. Finally, using an Information Theoretic framework, we study necessary conditions for exact recovery to be possible.