Group Invariant Scattering

By Stéphane Mallat. 2011

1,123 citations recorded by the source corpus.

This paper constructs translation‐invariant operators on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$, which are Lipschitz‐continuous to the action of diffeomorphisms. A scattering propagator is a path‐ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz‐continuous to the action of C2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high‐order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L2(G), where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ and on L2(SO(d)) defines a translation‐ and rotation‐invariant scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$. © 2012 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Radiology, Nuclear Medicine and Imaging, Statistical and Nonlinear Physics, Representation & Self-Supervised Learning

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Citations in the corpus

Scattering Networks for Hybrid Representation Learning (2018, methodology): “The construction of this network has strong mathematical foundations [35], meaning it is well understood, relies on few parameters, and is stable to a large class of geometric transformations.”
Representation Learning: A Review and New Perspectives (2012, background): “Alternatively, one can also use explicit knowledge of the expected invariants expressed mathematically to define transformations that are robust to a known family of input deformations, using the so-called scattering operators [139], [40], which can be computed in a way interestingly analogous to de”
Robust Large Margin Deep Neural Networks (2016, background): “Consequences for very deep networks-It is instructive to observe the result of [15] that suggests that the GE of DNNs with ReLUs behaves as provided that the energy of the training samples is bounded.”
Geometric Deep Learning: Going beyond Euclidean data (2016, background): “Moreover, if one specifies the convolutional tensors to be complex wavelet decomposition operators and uses complex modulus as pointwise nonlinearities, one can provably obtain stability to local deformations [17].”
i-RevNet: Deep Invertible Networks (2018, background): “An illustration of this framework on the Euclidean group is given by the scattering transform (Mallat, 2012), which builds invariance to small translations while being recoverable to a certain extent.”
Invariant Scattering Convolution Networks (2012, methodology): “It is computed with a ﬁlter bank of dilated and rotated wavelets having no orthogonality property.”
Understanding deep convolutional networks (2016, background): “It defines a nonlinear scattering transform, introduced in [9], where wavelets have a separation and linearization role.”
Manitest: Are classifiers really invariant? (2015, methodology): “This result is in agreement with the theoretical evidence [5, 24] showing that scattering classiﬁers are invariant to deformations.”
On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups (2018, background): “Another architecture that achieves invariance by stacking equivariant layers followed by a ﬁnal invariant one is that of scattering networks (Mallat, 2012).”
Deep roto-translation scattering for object classification (2014, background): “ decision but depends upon the choice of geometric invariant as we shall see. It has been proved mathematically that translation invariance and linearization of deformations is obtained with wavelets [20]. They separate the image information along multiple scales and orientations. Cascading wavelet ”
N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials (2018, background): “One particularly attractive approach is the scattering networks framework of Mallat and coworkers, which, at least in the limit of an infinite number of neural network layers, provides a representation of functions that is both globally invariant with respect to symmetries and Lipschitz with respect”
On Invariance and Selectivity in Representation Learning (2015, background): “In particular, the results in [10,28] develop invariant and stable representations within a signal processing framework.”
Scaling the Scattering Transform: Deep Hybrid Networks (2017, positive): “A major improvement over SIFT can be found in the scattering transform [24, 6, 33], which is a type of deep convolutional network, which permits to retain discriminative information normally discarded by methods like SIFT while introducing geometric invariances and stability.”
Wavelet Scattering Regression of Quantum Chemical Energies (2016, background): “In the following, we concentrate on the variability along u and thus only compute scattering coefficients along the translation variable u ∈ R(3) [31].”
Super-Resolution with Deep Convolutional Sufficient Statistics (2015, background): “Scattering networks (Bruna & Mallat (2013); Mallat (2012)) are obtained by cascading complex wavelet decompositions with complex modulus.”
Building a Regular Decision Boundary with Deep Networks (2017, background): “Geometric variabilities can be successfully handled by prior representations such as scattering transforms [23, 4, 20, 19, 25].”
On the Inductive Bias of Neural Tangent Kernels (2019, methodology): “In contrast to the kernels studied in [11], which for instance cover the limiting kernels that arise from training only the last layer of a ReLU CNN, we find that the obtained NTK kernel mappings for the ReLU activation lack a desired Lipschitz property which is needed for stability to deformations ”
Deep Convolutional Neural Networks for Image Classification: A Comprehensive Review (2017, background): “To this end, Mallat (2012) proved translation invariance and deformation stability of the features extracted by scattering convolutional networks.”