Editing Independent component analysis (section)

==== Reduced Mixing Problem ====
'''Independent component analysis''' ('''ICA''') addresses the problem of recovering a set of unobserved source signals <math>s_i = (s_{i1}, s_{i2}, \dots, s_{im})^T</math> from observed mixed signals <math>x_i = (x_{i1}, x_{i2}, \dots, x_{im})^T</math>, based on the linear mixing model:

<math>x_i = A\,s_i,</math>

where the <math>A</math> is an <math>m \times m</math> invertible matrix called the '''mixing matrix''', <math>s_i</math> represents the m‑dimensional vector containing the values of the sources at time <math>t_i</math>, and <math>x_i</math> is the corresponding vector of observed values at time <math>t_i</math>. The goal is to estimate both <math>A</math> and the source signals <math>\{s_i\}</math> solely from the observed data <math>\{x_i\}</math>.

After centering, the Gram matrix is computed as:
<math>
(X^*)^T X^* = Q\,D\,Q^T,
</math>
where D is a diagonal matrix with positive entries (assuming <math>X^*</math> has maximum rank), and Q is an orthogonal matrix.<ref name="Springer"/>
Writing the SVD of the mixing matrix <math>A = U \Sigma V^T</math> and comparing with <math>AA^T = U \Sigma^2 U^T</math> the mixing A has the form 
<math>
A = Q\,D^{1/2}\,V^T.
</math>
So, the normalized source values satisfy
<math>s_i^* = V\,y_i^*</math>, where <math>y_i^* = D^{-\tfrac12}Q^T x_i^*.</math>
Thus, ICA reduces to finding the orthogonal matrix <math>V</math>. This matrix can be computed using optimization techniques via projection pursuit methods (see [[#Projection pursuit|Projection Pursuit]]).<ref name="Springer"/>

Well-known algorithms for ICA include [[infomax]], [[FastICA]], [[JADE (ICA)|JADE]], and [[kernel-independent component analysis]], among others. In general, ICA cannot identify the actual number of source signals, a uniquely correct ordering of the source signals, nor the proper scaling (including sign) of the source signals.

ICA is important to [[blind signal separation]] and has many practical applications. It is closely related to (or even a special case of) the search for a [[factorial code]] of the data, i.e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector (loss-free coding), but the code components are statistically independent.