Algebraic Representation of Probability Distributions

We show that the use of techniques from algebra and algebraic geometry can be highly beneficial for tackling machine learning problems, where the set of desired solutions can be described in terms of approximate polynomial equations. Namely, they allow one to directly solve learning problems using algebraic operations thus avoiding iterative optimization which may converge to local minima. In this spirit, we suggest a novel representation for probability distributions in terms of elements in the polynomial ring derived from estimated cumulants. Demonstrating the versatility of our framework, we present an algorithm for finding the linear subspace on which estimated probability distributions are stationary w.r.t. to chosen cumulants of arbitrary degree and for arbitrary numbers of dimensions. Apart from showing interesting theoretical proof techniques, we illustrate that the algebraic approach is significantly more accurate than optimization-based methods in numerical simulations.