Algebraic identifiability in independent component analysis mixture models

In this talk, we discuss independent component analysis (ICA) mixture models. These models consist of convex combinations of ICA densities. An ICA density is associated with a random vector whose coordinates become mutually independent after an invertible linear transformation. For instance, every Gaussian density is an ICA density, and therefore ICA mixture models are nonparametric extensions of the celebrated family of Gaussian mixtures.

We consider the method-of-moments approach to density estimation over an ICA mixture model: given samples arising from an unknown density in this model, how can we find the best estimate of that density using empirical averages computed from these samples? In particular, we establish when the defining parameters of a density in the ICA mixture model – including its mixing coefficients, linear transformations, and source moments – can be recovered, up to a finite class, from observed moments of order at most d. This property is called algebraic identifiability. For n-variate distributions and $d >= 3$, we prove algebraic identifiability for models with asymptotically up to $1/d! n^{d-2}$ mixture components, and show that this is an asymptotically sharp bound on the number of mixture components. We finish by presenting a method-of-moments algorithm for density estimation over these models.