Independent Component Analysis (ICA) is another linear algorithm that identifies a new basis on which to represent the original data, but pursues a different objective to PCA.

ICA emerged in signal processing, and the problem it aims to solve is called blind source separation. It is typically framed as the cocktail party problem, in which a given number of guests are speaking at the same time so that a single microphone would record overlapping signals. ICA assumes there are as many different microphones as there are speakers, each placed at different locations so as to record a different mix of the signals. ICA then aims to recover the individual signals from the different recordings.

In other words, there are n original signals and an unknown square mixing matrix A that produces an n-dimensional set of m observations, so that:

The goal is to find the matrix W=A^-1 that untangles the mixed signals to recover the sources.

The ability to uniquely determine the matrix W hinges on the non-Gaussian distribution of the data. Otherwise, W could be rotated arbitrarily given the multivariate normal distribution's symmetry under rotation.

Furthermore, ICA assumes the mixed signal is the sum of its components and is unable to identify Gaussian components because their sum is also normally distributed.