On the use of sparsity for recovering discrete probability distributions from their moments

We address the problem of determining the probability distribution of a discrete random variable from its moments, using a sparsity-based approach. If the random variable can take at most K different values from a potential set of M ≫ K values, then its moments can be represented as linear measurements of a if-sparse probabilities vector, where the measurement matrix is a fat Vandermonde matrix. With this measurement matrix, Compressed Sensing theory asserts that if at least the 2K -1 first moments are available, a unique K-sparse solution exists, but is generally not attainable via ℓ₁ minimization (since other, non-sparse solutions with the same ℓ₁ norm may exist). Using the concept of neighborly poly-topes, we show that if (and only if) the first 2K moments are known, then the solution is always unique, and is therefore attainable via (degenerate) ℓ₁ minimization.