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Principle Component Analysis

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Re: Principle Component Analysis

#21
post #19
post #8

Earlier quoted context omitted.

It's often true that you can figure out what the first handful (say, 3 to 6) PCA components mean, in a large problem. The first is usually the mean of the quantities. It is typical in practice to compute PCA by using the SVD of the data itself; if you subtract the mean first, then of course it will not appear as the first PCA component. In matlab, this is literally a one-liner using the svd of the original data -- no…

FYI, Gabor-like filters pop out from doing ICA (i.e. Independent Components Analysis), not PCA. While PCA looks for orthogonal vectors onto which the data's projection is normally-distributed (among other properties), ICA, roughly speaking, looks for a set of orthogonal vectors onto which the data's projection has maximal kurtosis (among other properties). It is the kurtosis-maximization of ICA that tends to produce…

Thanks for the reminder. I was thinking of this 1991 paper, which I ran into a long time ago:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.1...

They used a (linear) "neural network" with gradient descent training that implemented PCA (kind of an iterative graham-schmidt process), and got Gabor-like filters. I think a lot of people have done similar experiments, with varying results.

Re: Principle Component Analysis

#22
post #21
post #19

Earlier quoted context omitted.

FYI, Gabor-like filters pop out from doing ICA (i.e. Independent Components Analysis), not PCA. While PCA looks for orthogonal vectors onto which the data's projection is normally-distributed (among other properties), ICA, roughly speaking, looks for a set of orthogonal vectors onto which the data's projection has maximal kurtosis (among other properties). It is the kurtosis-maximization of ICA that tends to produce…

Thanks for the reminder. I was thinking of this 1991 paper, which I ran into a long time ago: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.1... They used a (linear) "neural network" with gradient descent training that implemented PCA (kind of an iterative graham-schmidt process), and got Gabor-like filters. I think a lot of people have done similar experiments, with varying results.

I hadn't seen that paper before; thanks for the reference. I read through it and saw that they were reweighting the sampled image patches with a Gaussian mask prior to learning, which explains how they got Gabor-like filters. The masking effectively forced the learned filters to have localized receptive fields, while locality/nonlocality is generally one of the (visually) clearer differences between filters learned with ICA/PCA.

In other words, the Gaussian-modulated part of Gaussian-modulated sinusoids was built into their learning process, rather than appearing as an emergent property. I also chuckled a bit when they described how computing eigenvectors for 4096x4096 matrices was "beyond reasonable computation".

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