# FastICA: what happens when we have more source than channel?

I am reading the original article proposing FastICA and I have a couple of question on point not covered in the article. Both answering me or providing a source will be very appreciated.

I want to use it in blind audio source separation context, so I am going to use that language.

Let's suppose that I have more source than channels.

$$x_1 = a_{11}s_1+a_{12}s_2 + a_{13}s_3+a_{14}s_4$$ $$x_1 = a_{21}s_1+a_{22}s_2 + a_{23}s_3+a_{24}s_4$$

Where $$x_1$$, and $$x_2$$ are our observed channels, $$s_i$$ the unknown sources and $$A = (a_{ij})$$ the unknown mixing matrix.

FastICA suppose that we have as many sources as channels, but does not check it in any way. If I am going to run the algorithm on my data, it would extract two sources and a 2x2 mixing matrix.

What should I expect from those sources? Are they going to be weighted mixing of my true sources? If yes: mixed how?

Thanks in advance, and sorry if I did some language mistake.

PS: I don't have a strong formation in Signal Processing, so if you can provide some source or small explanation when using specific language it would be appreciated.

– A_A
Jan 24, 2020 at 15:50

FastICA suppose that we have as many sources as channels, but does not check it in any way. If I am going to run the algorithm on my data, it would extract two sources and a 2x2 mixing matrix.

ICA will indeed provide a separation, along the lines of section 3 ("What is independence?") which would also require some elements of section 4.2("Measures of non-gaussianity"), 4.3("Minimisation of mutual information") and 4.4("Maximum likelihood estimation")

ICA is trying to express some $$x$$ (in your notation), as a mixture ($$A$$) of a collection of waveforms ($$s$$).

A better way of looking at this is that $$x$$ is the combined result of a number of processes that move from state to state with some probabiliy. By expressing $$x$$ in this way and at the same time requiring that you want to find independent components, you arrive at the concept of statistical independence.

You want the waveforms in $$s$$ to be as different to each other as possible but at the same time, when you sum them up (with some mixing weights determined by $$A$$) you want their sum to amount to $$x$$.

This is what ICA will do and it will do it whether you feed it audio samples or some random signals that are completely out of context.

What should I expect from those sources? Are they going to be weighted mixing of my true sources? If yes: mixed how?

This is a very interesting and at the same time difficult question to answer accurately.

You should expect that these "sources" will be "independent" to each other but not that they correspond to meaningful waveforms in an audio context.

A trivial example is this: I record 2 trumpets playing exactly the same score. Should I expect that they will be "separated" by ICA?

No.

ICA will try to find a minimal set of independent "components" as if these were the output of a number of independent processes. One possible decomposition might have one component sounding very closely to the score and the other being a totally random set of values. These two would be independent to each other and when mixed (at the right levels) would result to the recorded waveforms.

Don't expect necessarily "meaning" out of these independent components.

A better "result" can be obtained by attempting to discriminate between percussive sounds and the rest of the sounds in a recording. Again, this does not mean that it will give you the isolated drums as if you had a contact mic on the drum set and recorded only those sounds. This is because, percussive sounds have long(er) "silences" and their distributions tend to be non-gaussian (a key requirement for the independent components (section 4)).

Hope this helps.