scattering transform

1.8. scattering transform#

some of the researchers involved
- edouard oyallan, joan bruna, stephan mallat, Helmut Bölcskei, max welling

benefits
- all filters are defined
- more interpretable
- more biophysically plausible
scattering transform - computes a translation invariant repr. by cascading wavelet transforms and modulus pooling operators, which average the amplitude of iterated wavelet coefficients

classification with scattering operators (bruna & mallat 2010)
recursive interferometric repr. (mallat 2010)
group invariant scattering (mallat 2012)
- introduces scat transform
Generic deep networks with wavelet scattering (oyallan et al. 2013)
Invariant Scattering Convolution Networks (bruna & mallat 2012)
- introduces the scattering transform implemented as a cnn
Deep scattering spectrum (anden & mallat 2013)

Deep roto-translation scattering for object classification (oyallan & mallat 2014)
- use 1x1 conv on top of scattering coefs (only 1 layer)
- can capture rounded figures
- can further impose robustness to rotation variability (although not full rotation invariance)
- Deep learning in the wavelet domain (cotter & kingbury, 2017)- each conv layer is replaced by scattering transform + 1x1 conv
Visualizing and improving scattering networks (cotter et al. 2017)
- add deconvnet to visualize
Scattering Networks for Hybrid Representation Learning (oyallon et al. 2018)
- using early layers scat is good enough
i-RevNet: Deep Invertible Networks (jacobsen et al. 2018)
Scaling the scattering transform: Deep hybrid networks (oyallon et al. 2017)
- use 1x1 convolutions to collapse accross channels
jacobsen_17 “Hierarchical Attribute CNNs”
- modularity
cheng_16 “Deep Haar scattering networks”
Deep Network Classification by Scattering and Homotopy Dictionary Learning (zarka et al. 2019) - scat followed by sparse coding then linear

Parameterized Wavelets for Convolutional Neural Networks (2020) - a discrete wavelet CNN
An End-to-End Multi-Level Wavelet Convolutional Neural Networks for heart diseases diagnosis (el bouny et al. 2020) - stationary wavelet CNN
Fully Learnable Deep Wavelet Transform for Unsupervised Monitoring of High-Frequency Time Series (michau et al. 2021)

Deep Convolutional Neural Networks Based on Semi-Discrete Frames (wiatowski et al. 2015)
- allowing for different and, most importantly, general semidiscrete frames (such as, e.g., Gabor frames, wavelets, curvelets, shearlets, ridgelets) in distinct network layers
- translation-invariant, and we develop deformation stability results
wiatoski_18 “A mathematical theory of deep convolutional neural networks for feature extraction”
- encompasses general convolutional transforms - general semi-discrete frames (including Weyl-Heisenberg filters, curvelets, shearlets, ridgelets, wavelets, and learned filters), general Lipschitz-continuous non-linearities (e.g., rectified linear units, shifted logistic sigmoids, hyperbolic tangents, and modulus functions), and general Lipschitz-continuous pooling operators emulating, e.g., sub-sampling and averaging
  - all of these elements can be different in different network layers.
- translation invariance result of vertical nature in the sense of the features becoming progressively more translation-invariant with increasing network depth
- deformation sensitivity bounds that apply to signal classes such as, e.g., band-limited functions, cartoon functions, and Lipschitz functions.
wiatowski_18 “Energy Propagation in Deep Convolutional Neural Networks” (wiatowski et al. 2017)

yu_06 “A Nanoengineering Approach to Regulate the Lateral Heterogeneity of Self-Assembled Monolayers”
- regulate heterogeneity of self-assembled monlayers
  - used nanografting + self-assembly chemistry
bu_10 nanografting - makes more homogenous morphology
fleming_09 “dendrimers”
- scanning tunneling microscopy - provides highest spatial res
- combat this for insulators
lin_12_moire
- prob moire effect with near-field scanning optical microscopy
chen_12_crystallization

$L^2$ function is a function $f: X \to \mathbb{R}$ that is square integrable: $$\vert f\vert ^2 = \int_X \vert f\vert ^2 d\mu$$ with respect to the measure $\mu$
- \[\vert f\vert $$ is its $L^2$-norm - **measure** = nonnegative real function from a delta-ring F such that $m(\empty) = 0$ and $m(A) = \sum_n m(A_n)$ - **Hilbert space** H: a vectors space with an innor product $$<f, g>$$ such that the following norm turns H into a complete metric space: $$\vert f\vert = \sqrt{<f, f>}\]
diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.