Invariant Scattering Convolution Networks Hasanov Vagif 2014 06

Problem statement • In the context of image classification, consider the framework of kernel

Transformations Rigid: • Translation • Rotation • Scaling We want our representations to be

Transformations (2) Non-rigid: • Deformations due to different writing styles • Stretching • Bending

The proposed solution • A deep convolution network which cascades wavelet transforms and modulus

Previous methods to achieve translation invariant representations • Registration method • Fourier transform modulus

Instability of Fourier transform modulus

Instability of registration method For any choice of anchor point proven that it can

Wavelets A wavelet is a wave-like oscillation with an amplitude that begins at zero,

Wavelet family (2) Source: Stephane Mallat, NIPS 2012

Incorporating translation invariance • We can prove that is stable to deformations the same

Reconstruction Proven in: I. Waldspurger, S. Mallat “Recovering the phase of a complex wavelet

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Invariant Scattering Convolution Networks Hasanov Vagif 2014. 06. 19

Problem statement • In the context of image classification, consider the framework of kernel classifiers, where metrics are defined as a Euclidean distance applied on a representation Φ(x) of signals x. • We have significant variability within image classes, so we need to construct representations that are invariant to irrelevant deformations, and continuous to relevant ones.

Transformations Rigid: • Translation • Rotation • Scaling We want our representations to be invariant to rigid transformations.

Transformations (2) Non-rigid: • Deformations due to different writing styles • Stretching • Bending We want our representations to be stable to non-rigid deformations. A small deformation of x to x’ should correspond to a small Euclidean distance between Φ(x) and Φ(x’).

Transformations of handwritten digits

The proposed solution • A deep convolution network which cascades wavelet transforms and modulus operators.

Translation Invariance

Previous methods to achieve translation invariant representations • Registration method • Fourier transform modulus

Definition of stability to deformations

Instability of Fourier transform modulus

Instability of registration method For any choice of anchor point proven that it can be

Wavelets A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. Image source: http: //upload. wikimedia. org/wikipedia/commons/2/23/Wavelet_-_Morlet. png Text source: en. wikipedia. org/wiki/Wavelet

Morlet Wavelet

Wavelet family

Wavelet family (2) Source: Stephane Mallat, NIPS 2012

Incorporating translation invariance • We can prove that is stable to deformations the same way that wavelets are, is stable, and is invariant to translations. Proven in: J. Bruna, “Operators commuting with diffeomorphisms”, CMAP, Tech. Report, Ecole Polytechnique, 2012.

Reconstruction Proven in: I. Waldspurger, S. Mallat “Recovering the phase of a complex wavelet transform”, CMAP Tech. Report, Ecole Polytechnique, 2012.

Scattering transform

Scattering network