Mathematics in Independent Component Analysis

58 Chapter 2. Neural Computation 16:1827-1850, 2004
A New Concept for Separability Problems in BSS 1829
2 Separated and Linearly Separated Functions
Definition 1. A function f : R n → C is said to be separated, respectively,
linearly separated, if there exist one-dimensional functions g1,...,gn : R → C
such that f (x) = g1(x1) ···gn(xn) respectively f (x) = g1(x1) +···+gn(xn) for
all x ∈ R n .
Note that the functions gi are uniquely determined by f up to a scalar
factor, respectively, an additive constant. If f is linearly separated, then exp f
is separated. Obviously the density function of an independent random
vector is separated. For brevity, we often use the tensor product and write
f ≡ g1 ⊗···⊗gn for separated f , where for any functions h, k defined on a
set U, h ≡ k if h(x) = k(x) for all x ∈ U.
Separatedness can also be defined on any open parallelepiped (a1, b1) ×
···×(an, bn) ⊂ R n in the obvious way. We say that f is locally separated
at x ∈ R n if there exists an open parallelepiped U such that x ∈ U and
f |U is separated. If f is separated, then f is obviously everywhere locally
separated. The converse, however, does not necessarily hold, as shown in
Figure 1.
0.2
0.1
0
4
3
2
1
0
−1
−2
0
−1
−2
−3 −3
Figure 1: Density of a random vector S with a locally but not globally separated
density. Here, p S := cχ[−2,2]×[−2,0]∪[0,2]×[1,3] where χU denotes the function that is
1onU and 0 everywhere else. Obviously, p S is not separated globally, but is
separated if restricted to squares of length < 1. Plotted is a smoothed version of
p S .
1
2
3
4