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

# 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

# initial papers

## neuro style

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

## 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  such that the following norm turns H into a complete metric space: $f = \sqrt{}$
• 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.