  We propose a novel algebraic framework for treating probability distributions
represented by their cumulants such as the mean and covariance matrix. As an
example, we consider the unsupervised learning problem of finding the subspace
on which several probability distributions agree. Instead of minimizing an
objective function involving the estimated cumulants, we show that by treating
the cumulants as elements of the polynomial ring we can directly solve the
problem, at a lower computational cost and with higher accuracy. Moreover, the
algebraic viewpoint on probability distributions allows us to invoke the theory
of Algebraic Geometry, which we demonstrate in a compact proof for an
identifiability criterion.
