Optimal Recovery in Noisy ICA

Independent Component Analysis (ICA) is a popular model for blind signal separation. The ICA model assumes that a number of independent source signals are linearly mixed to form the observed signal. Traditional ICA algorithms typically aim to recover the mixing matrix, the inverse of which can be applied to data in order to recover the latent independent signals. However, in the presence of noise, this demixing process is non-optimal for signal recovery as measured by signal-to-interference-plus-noise ratio (SINR), even if the mixing matrix is recovered exactly. This paper has two main contributions. First, we show how any solution to the mixing matrix reconstruction problem can be used to construct an SINR-optimal ICA demixing. The proposed method is optimal for any noise model and applies in the underdetermined setting when there are more source signals than observed signals. Second, we improve the recently proposed Gradient Iteration ICA algorithm to obtain provable and practical SINR optimal signal recovery for Gaussian noise with an arbitrary covariance matrix. We also simplify the original algorithm making the cumbersome quasi-orthogonalization step unnecessary, leading to improved computational performance.