A wavelet scattering network computes a translation invariant image representation which is stable to deformations and preserves high-frequency information for classification. It cascades wavelet transform convolutions with nonlinear modulus and averaging operators. The first network layer outputs SIFT-type descriptors, whereas the next layers provide complementary invariant information that improves classification. The mathematical analysis of wavelet scattering networks explains important properties of deep convolution networks for classification. A scattering representation of stationary processes incorporates higher order moments and can thus discriminate textures having the same Fourier power spectrum. State-of-the-art classification results are obtained for handwritten digits and texture discrimination, with a Gaussian kernel SVM and a generative PCA classifier.

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Invariant Scattering Convolution Networks

Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

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The Convenient Setting of Global Analysis

Acknowledgements Basic notation Introduction 1 General measure theory 2 Covering and differentiation 3 Invariant measures 4 Hausdorff measures and dimension 5 Other measures and dimensions 6 Density theorems for Hausdorff and packing measures 7 Lipschitz maps 8 Energies, capacities and subsets of finite measure 9 Orthogonal projections 10 Intersections with planes 11 Local structure of s-dimensional sets and measures 12 The Fourier transform and its applications 13 Intersections of general sets 14 Tangent measures and densities 15 Rectifiable sets and approximate tangent planes 16 Rectifiability, weak linear approximation and tangent measures 17 Rectifiability and densities 18 Rectifiability and orthogonal projections 19 Rectifiability and othogonal projections 19 Rectifiability and analytic capacity in the complex plane 20 Rectifiability and singular intervals References List of notation Index of terminology

Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability

A wavelet scattering network computes a translation invariant image representation, which is stable to deformations and preserves high frequency information for classification. It cascades wavelet transform convolutions with non-linear modulus and averaging operators. The first network layer outputs SIFT-type descriptors whereas the next layers provide complementary invariant information which improves classification. The mathematical analysis of wavelet scattering networks explains important properties of deep convolution networks for classification. 
A scattering representation of stationary processes incorporates higher order moments and can thus discriminate textures having the same Fourier power spectrum. State of the art classification results are obtained for handwritten digits and texture discrimination, using a Gaussian kernel SVM and a generative PCA classifier.

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

A new direct proof of convergence of cluster expansions for polymer (contour) models is given in an abstract setting. It does not rely on Kirkwood-Salsburg type equations or “combinatorics of trees.” A distinctive feature is that, at all steps, the considered clusters contain every polymer at most once.

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Cluster expansion for abstract polymer models

Geometry of measures in $\mathbf{R}^n$: Distribution, rectifiability, and densities

Differentiability of Lipschitz functions on Banach spaces

A general “Mountain Pass” principle that extends the theorem of Ambrosetti-Rabinowitz and which gives more information about the location of critical points, is established. This theorem also covers the problem of the “limiting case”, i.e. when “the separating mountain range has zero altitude”. It is also shown how this principle yields localized versions of recent results of Hofer and Pucci-Serrin concerning the structure of the critical set.

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A general mountain pass principle for locating and classifying critical points

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient mappings between Banach spaces. It is proved, in particular, that any Banach space which is a uniform quotient of L

 p
 , 1 < p < 
$ \infty $
 , is already isomorphic to a linear quotient of L

 p
 .

David Preiss

Papers

Cluster expansion for abstract polymer models

Geometry of measures in $\mathbf{R}^n$: Distribution, rectifiability, and densities

Differentiability of Lipschitz functions on Banach spaces

A general mountain pass principle for locating and classifying critical points

Affine Approximation of Lipschitz Functions and Nonlinear Quotients