scattering transform
Contents
1.7. scattering transform#
some papers involving the scattering transform and similar developments bringing structure to replace learned filters
some of the researchers involved
edouard oyallan, joan bruna, stephan mallat, Helmut Bölcskei, max welling
1.7.1. goals#
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
1.7.2. review-type#
Understanding deep convolutional networks (mallat 2016)
Mathematics of deep learning (vidal et al. 2017)
Geometric deep learning: going beyond euclidean data (bronstein et al. 2017)
1.7.3. initial papers#
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)
1.7.3.1. scat_conv#
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
1.7.3.2. neuro style#
1.7.3.3. papers by other groups#
cohen_16 “Group equivariant convolutional networks”
introduce G-convolutions which share more wieghts than normal conv
worrall_17 “Interpretable transformations with encoder-decoder networks”
look at interpretability
bietti_17 “Invariance and stability of deep convolutional representations”
theory paper
1.7.3.4. wavelet style transfer#
1.7.4. adaptive wavelet papers#
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)
1.7.4.1. helmut lab papers#
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”
1.7.5. nano papers#
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
1.7.5.1. l2 functions#
\(L^2\) function is a function \(f: X \to \mathbb{R}\) that is square integrable: \(|f|^2 = \int_X |f|^2 d\mu\) with respect to the measure \(\mu\)
\(|f|\) 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: \(|f| = \sqrt{<f, f>}\)
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.