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27thSEPTEMBER2017

READING

Jayanth Koushik. Understanding Convolutional Neural Networks. CoRR abs/1605.09081 (2016)

Koushik, based on the work of Mallat [][], illustrates the connection between convolutional neural networks (at least the kind of convolutional neural networks considering single-channel convolutions) and the scattering transform by Mallat []. Essentially, considering a convolutional neural network, computing

$x_J(u, k_j) = \rho(\rho(\ldots \rho(x \ast W_{1})\ast \ldots )\ast W_J)$

When unrolling the individual layers ($\ast$ is the discrete convolution), Koushik argues that this can equivalently be expressed using wavelets. Specifically, there exists a sequence $\lambda_1,\ldots,\lambda_m$ with

$x_J(u,k_J) = S_J[p]x(u) = (U[p] x \ast \phi_J)(u) = (\rho(\rho(\ldots \rho(x\ast \psi_{\lambda_1})\ast \ldots )\ast\psi_{\lambda_m})(u)$

where $\phi_J$ is an averaging filter and $\psi_{\lambda_i}$ are suitably chosen wavelet filters. Unfortunately, examples and illustrations are missing such that it is quite hard to extract any useful conclusions from Koushik's paper. For details, I refer to [] and [].

  • [] Stéphane Mallat. Group invariant scattering. Communications on Pure and Applied Mathematics, 65(10):1331–1398, 2012.
  • [] Stéphane Mallat. Understanding deep convolutional networks. arXiv preprint arXiv:1601.04920, 2016.

What is your opinion on the summarized work? Or do you know related work that is of interest? Let me know your thoughts in the comments below or get in touch with me: