The Living Thing / Notebooks : Sparse basis dictionary expansions

Regression with dictionaries of basis functions, with respect to which you expect your representation to be sparse; i.e. in the statistical case, basis-sparse regression. But even outside statistics, you wish simply to approximate some data compactly. My focus here is on the noisy-observation case, although the same results are recycled enough throughout the field.

Orthogonal matching pursuit, wavelet, curvelets, chirplets, framelets, shearlets, whateverlets, content-specific basis dictionaries, designed or learned. Mammals visual cortexes seem to do some thing like this.

If they are not intended to be sparsely coded, it’s probably a plain old linear basis expansion or a relative.

To discuss:

Shift-invariant bases

I would like to find a general way of doing this in a phase/shift-robust fashion, although doing this naively can be computationally expensive outside of certain convenient bases. It can be better if you know you have have constraints, say, a convex combination (e.g. mixtures) or positive definite bases. (e.g. in kernel methods)

One method is “Shift Invariant Sparse coding”, ( BlDa04) and there are various versions out there. (GRKN07 One way is to include multiple shifted copies of your atoms, another is to actually shift them in a separate optimisation stage. Both these get annoying in the time domain for various reasons.

Affine tight framelets (DHRS03) and their less computationally tractable, more flexible cousins, shearlets also sound interesting here. For reasons I do not yet understand I am told they can naturally be used on graphs and manifolds, not just lattices. I just saw Xiaosheng Zhuang present these (e.g. HaZZ16 and WaZh16, where WaZh16 demonstrates a “Fast Framelet Transform” which is computationally as attractive as the FFT, supposedly.)

I have some ideas I call learning gamelan.


This boils down to clever optimisation to make the calculations tractable.

To read

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