  We describe a method for unmixing mixtures of freely independent random
variables in a manner analogous to the independent component analysis (ICA)
based method for unmixing independent random variables from their additive
mixtures. Random matrices play the role of free random variables in this
context so the method we develop, which we call Free component analysis (FCA),
unmixes matrices from additive mixtures of matrices. Thus, while the mixing
model is standard, the novelty and difference in unmixing performance comes
from the introduction of a new statistical criteria, derived from free
probability theory, that quantify freeness analogous to how kurtosis and
entropy quantify independence. We describe the theory, the various algorithms,
and compare FCA to vanilla ICA which does not account for spatial or temporal
structure. We highlight why the statistical criteria make FCA also vanilla
despite its matricial underpinnings and show that FCA performs comparably to,
and sometimes better than, (vanilla) ICA in every application, such as image
and speech unmixing, where ICA has been known to succeed. Our computational
experiments suggest that not-so-random matrices, such as images and short time
fourier transform matrix of waveforms are (closer to being) freer "in the wild"
than we might have theoretically expected.
