Independent component analysis (ICA) is an important method for blind source separation and unsupervised learning. Recently, the method has been extended to overcomplete situation where the number of sources is greater than the number of receivers. Comparing complete ICA and overcomplete ICA in existing
literature, one can notice that complete ICA does not assume noise
in observations, and the sources can be explicitly solved from the
receivers, whereas the overcomplete ICA in general assumes noise in observations and the sources are implicitly solved by gradient type algorithms. In this paper, we present an explicit null-space
representation for overcomplete ICA in the noiseless situation based on singular value decomposition (SVD), and develop an algorithm for estimating mixing matrix and recovering the sources. The null-space representation makes the connection between complete ICA
and overcomplete ICA more apparent, and leads to a mathematical
explanation of lateral inhibition in the context of overcomplete
linear model. It also appears to work well on the experimental examples. Moreover, it can be extended to the situation where there is noise in observations, and may lead to more efficient algorithms in this situation.