Introduction

Invariance, Stability, Discriminability

Originated in 2010 by Stéphane Mallat as “Recursive Interferometry” [Mal10], the scattering representation aims at providing robust mid-level features for signal-based machine learning.

It relies on the idea that an ideal representation for classification should verify the three following criteria:

  1. Invariance to translation: if the signal gets globally translated, the representation should not change.
  2. Stability to small elastic deformations: if the signal get deformed, the representation should be deformed in a proportional way.
  3. Discriminability: signals in different classes should be represented differently.

Below is an informal review of the capacities of different signal descriptors in terms of these criteria.

  Invariant Stable Discriminative
the signal itself     ★ ★ ★
moving average ★ ★ ★ ★ ★  
Fourier transform modulus ★ ★ ★   ★
wavelet transform modulus   ★ ★ ★ ★
constant-Q transform, MFCC ★ ★ ★ ★ ★
scattering transform ★ ★ ★ ★ ★ ★ ★

Invariance and stability can easily be achieved by averaging the signal over time, at the cost of a poor discriminability. Conversely, the modulus of the wavelet transform (called scalogram) is stable and discriminative, yet not invariant to translation. The classical constant-Q transform is equivalent to averaging the scalogram over uniform windows : that is a gain in invariance, yet a loss in discriminability. The rationale behing the scattering transform is to compensate for this averaging by recovering the fast variations in the scalogram as well. This is handled by means of another wavelet transform, so as to guarantee a good property of stability.

Multivariable scattering

The theoretical framework of the scattering transform is not limited to translations of one-dimensional signals. In fact, it can be formulated for any source of variability as long as it follows an algebraic structure of group [Mal12]. Rotation (for images) and frequency transposition (for sounds) are examples of these sources of variability.

What’s in scattering.m

The scattering.m MATLAB toolbox intends to provide a pipeline for multi-variable scattering that is very generic and customizable, yet remaining as seamless and efficient as possible.

  • Morlet wavelets and auditory Gammatone wavelets
  • Tunable quality factor, maximum scale, and number of filters per octave
  • Pay only for what you use: a fraction of scattering paths can be explicitly spared
  • Logarithmic compression of the wavelet scalogram is opt-in
  • FFT-based convolutions with subsampling in the Fourier domain
  • Efficient products in the Fourier domain with narrowband wavelets
  • Multi-variable scattering: all variables share the same framework
  • Automatic padding of the log-frequency axis
  • Multiple-support filter banks to avoid extraneous padding
  • Spiral scattering by reshaping the log-frequency axis on the fly
  • Long signals are processed by constant-sized chunks. Unchunking is automatic
  • Signal reconstruction from scattering coefficients (even over multiple variables)
  • Efficient formatting into feature vectors for classification
[Mal10]Mallat, S. Recursive Interferometric Representations. in European Signal Processing Conference 716–720 (2010).
[Mal12]Mallat, S. Group Invariant Scattering. Communications on Pure and Applied Mathematics, vol. 65, issue 10, pages 1331–1398 (2012).