Abstract

Independent Component Analysis (ICA) is a widely used statistical technique for decomposing multivariate signals (mixtures) into their underlying (non-Gaussian) independent components. Mathematically, ICA models observations Y ∈ Rd, d ≥ 2, as a mixture of unobserved independent source components X ∈ Rd, via an unknown nonsingular mixing matrix A, namely, Y = AX. The goal of ICA is to estimate the unmixing matrix B = A−1 based on IID data Yi = AXi, i = 1, . . . , n; to separate the mixed signals into the underlying independent components.
Our work proposes a novel distance-based approach for estimating B. The estimation is performed by minimizing the distance ρw between the joint empirical distribution function of X1, . . . ,Xn, and the marginal empirical distribution functions of the coordinates. We establish that ρw is asymptotically a U-statistic and derive its theoretical properties. Further, we analyze the empirical process to derive an estimation strategy for B. We devise two separate methods to construct confidence intervals. The first one uses the U-statistic and a specialized weight function that makes it independent of the distributions of the sources. The second method relies on the principal components of the empirical process. To make this approach computationally feasible, we propose a Gradient Descent Algorithm (GDA) to compute the estimate, and demonstrate its effectiveness by comparing it to the prevalent FastICA method (cf. Hyvarinen and Oja (2000)).
This work contributes both theoretically and numerically to the field of ICA by introducing a new approach to the estimation problem as well as providing a practical algorithmic implementation procedure.