  A prototypical blind signal separation problem is the so-called cocktail
party problem, with n people talking simultaneously and n different microphones
within a room. The goal is to recover each speech signal from the microphone
inputs. Mathematically this can be modeled by assuming that we are given
samples from an n-dimensional random variable X=AS, where S is a vector whose
coordinates are independent random variables corresponding to each speaker. The
objective is to recover the matrix A^{-1} given random samples from X. A range
of techniques collectively known as Independent Component Analysis (ICA) have
been proposed to address this problem in the signal processing and machine
learning literature. Many of these techniques are based on using the kurtosis
or other cumulants to recover the components.
  In this paper we propose a new algorithm for solving the blind signal
separation problem in the presence of additive Gaussian noise, when we are
given samples from X=AS+\eta, where \eta is drawn from an unknown, not
necessarily spherical n-dimensional Gaussian distribution. Our approach is
based on a method for decorrelating a sample with additive Gaussian noise under
the assumption that the underlying distribution is a linear transformation of a
distribution with independent components. Our decorrelation routine is based on
the properties of cumulant tensors and can be combined with any standard
cumulant-based method for ICA to get an algorithm that is provably robust in
the presence of Gaussian noise. We derive polynomial bounds for the sample
complexity and error propagation of our method.
