I am a student of Stéphane Mallat so I'm going to comment on the topic.
The idea behind the scattering transform is fairly straightforward, but it requires some dexterity to have a fast implementation. The algorithm is a cascade of two wavelet transforms intertwined with complex modulus nonlinearities. Each level of the cascade is then averaged by the scaling function phi of the corresponding wavelet filterbank.
From my point of view, the improvements needed to make it fast are, by decreasing order of importance:

wavelet transforms implemented as products in the Fourier domain
downsampling of subbands according to the critical resolution after each of the wavelet transforms, by truncation in the Fourier domain
leverage the fact that Gabor wavelets have narrow support to restrict the Fourier-domain product to its nonzero coefficients
if the resolution of the second wavelet is finer that the resolution of the first wavelet (j1 > j2 in Mallat's notations), the result will be negligible so the computation of these coefficients can be spared
for classification, the end user typically wants only a portion of the coefficients (j1 not to large, j2 not too small) so again, the corresponding computations could be spared
vectorizing the output of the low-pass filtering to avoid stacking all coefficients at the end, hence reducing the memory footprint

I have limited experience with LTFAT and, if I recall well, your paradigm is to apply pyramidal schemes when possible. So you will possibly want to replace improvements (2) and (3) by a dyadic downsampling algorithm. I would love to hear your advice on this.

I have implemented the scattering transform in the MATLAB toolbox "scattering.m" I use for my research: https://github.com/lostanlen/scattering.m
The joint time-frequency scattering transform (see Andén et al MLSP 2015) is also implemented. It computes a wavelet transform over the log-frequency axis as well as over the time axis, hence bearing resemblance with cepstral coefficients. Is it in the scope of LTFAT ?
It is released under MIT license so feel free to re-use my code and research ideas. I can help with the integration into this repo.

Add Mallat's scattering transform #50

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