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What's the impact of a specific filter $\phi$ when building the wavelet scattering transform? I mean, if I compute the MFCC choosing different windows, (Hamming, Blackman, etc.), the behaviour concerning time-warping deformations changes. For instance, I got the best result (a pretty linear variation of the distance) with a Chebyshev window while other windows return a much faster increase of the distance with the deformation. I guess that it depends on the side lobes of the filter (it performs better where lobes are lower) but some ideas on this point would be appreciated.

Thanks.

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The choice of $\phi$ affects:

  1. Time-shift invariance: slower decay in time will increase it
  2. Time-warp stability: slower decay in time will increase it mainly for deformations along time (but not only time; Ctrl +F "frequential averaging effect" here)
  3. Time-frequency resolution: anything other than Gaussian (or DPSS) will yield a suboptimal joint resolution. Namely, if we attain greater invariance, features are poorer resolved in time, and an additional order of scattering will be required to recover the lost information.
  4. Subsampling: $\hat\phi$ must decay sufficiently quickly for a given $T$ to allow ~lossless subsampling. Filters with suboptimal joint resolution will be less permissive on subsampling for any given $T$ (see "Subsampling").
  5. Energy conservation: a tight frame (with several desirable properties) requires compatibility of $\hat \phi$'s shape with that of $\hat \psi$'s, else the LP summation will either overflow or underflow, which might also require an additional low frequency wavelet to tile the full frequency axis. See "Energy analysis". For most purposes, besides the potential incomplete tiling, the effect is negligible.

Gaussian + Morlets are a natural choice for optimizing all 5 for general purpose, but if something beats Gaussian in MFCC, it can certainly work better in scattering also, since MFCC approximately subsets scattering in extracted features.

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