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Review
. 2016 Apr 13;374(2065):20150203.
doi: 10.1098/rsta.2015.0203.

Understanding deep convolutional networks

Stéphane Mallat  1
Affiliations
Review

Understanding deep convolutional networks

Stéphane Mallat. Philos Trans A Math Phys Eng Sci. .

Abstract

Deep convolutional networks provide state-of-the-art classifications and regressions results over many high-dimensional problems. We review their architecture, which scatters data with a cascade of linear filter weights and nonlinearities. A mathematical framework is introduced to analyse their properties. Computations of invariants involve multiscale contractions with wavelets, the linearization of hierarchical symmetries and sparse separations. Applications are discussed.

Keywords: deep convolutional neural networks; learning; wavelets.

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Figures

Figure 1.
Figure 1.
Wavelet transform of an image x(u), computed with a cascade of convolutions with filters over J=4 scales and K=4 orientations. The low-pass and K=4 band-pass filters are shown on the first arrows. (Online version in colour.)
Figure 2.
Figure 2.
A convolution network iteratively computes each layer xj by transforming the previous layer xj−1, with a linear operator Wj and a pointwise nonlinearity ρ. (Online version in colour.)
Figure 3.
Figure 3.
First row: original images. Second row: realization of a Gaussian process with same second covariance moments. Third row: reconstructions from first- and second-order scattering coefficients.
Figure 4.
Figure 4.
A multiscale hierarchical networks computes convolutions along the fibres of a parallel transport. It is defined by a group Gj of symmetries acting on the index set Pj of a layer xj. Filter weights are transported along fibres. (Online version in colour.)
See this image and copyright information in PMC

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