scattering transform

view markdown

---

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

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

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)

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)

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

neuro style

• https://arxiv.org/pdf/1809.10504.pdf

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

wavelet style transfer

• Photorealistic Style Transfer via Wavelet Transforms>

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)

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”

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

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.

reversible/invertible models

• williams inverting w/ autoregressive models

• arora reversibility
• iResNet