Mathematics in Independent Component Analysis

1.1. Introduction 5
auditory
cortex
auditory
cortex 2
word
detection
decision
(a) cocktail party problem (b) linear mixing model
(c) neural cocktail party
Figure 1.2: Cocktail party problem: (a) a linear superposition of the speakers is recorded at
each microphone. This can be written as the mixing model x(t) = As(t) (1.1) with speaker
voices s(t) and activity x(t) at the microphones (b). Possible applications lie in neuroscience:
given multiple activity recordings of the human brain, the goal is to identify the underlying
hidden sources that make up the total activity (c).
A typical application of BSS lies in the cocktail party problem: at a cocktail party, a set
of microphones records the conversations of the guests. Each microphone records a linear superposition
of the conversations, and at each microphone, a slightly different superposition is
recorded depending on the position, see figure 1.2. In the following we will see that given some
rather weak assumptions on the conversations themselves such as independence of the various
talks, it is then possible to recover the original sources and the mixing matrix (which encodes
the position of the speakers) using only the signals recorded at the microphones. Note that in
real-world situations the nice linear mixing situation deteriorates due to noise, convolutions and
nonlinearities.
t=1
t=2
t=3
t=4