Mathematics in Independent Component Analysis

232 Chapter 16. Proc. ICA 2006, pages 917-925
1.2 Uniqueness theorem
Definition 1. X=AS with A∈Gl(d), S=(SN, SG) and SN∈ L2(Ω,R n ) is called an
n-decomposition of X if SN and SG are stochastically independent and SG is Gaussian.
In this case, X is said to be n-decomposable.
Hence an n-decomposition of X corresponds to the NGSA problem. If as before
A=(AN, AG), then the n-dimensional subvectorspace im(AN)⊂R d is called the non-
Gaussian subspace, and im(AG) the Gaussian subspace of the decomposition; here
im(A) denotes the image of the linear map A.
Definition 2. X is denoted to be minimally n-decomposable if X is not (n−1)-decomposable.
Then dime(X) := n is called the essential dimension of X.
For example, the essential dimension dime(X) is zero if and only if X is Gaussian,
whereas the essential dimension of a d-dimensional mutually independent Laplacian is
d. The following theorem is the main theoretical contribution of this work. It essentially
connects uniqueness of the dimension reduction model with minimality, and gives a
simple characterization for it.
Theorem 1 (Uniqueness of NGSA). Let n