Understanding deep convolutional networks

By Stéphane Mallat. 2016

687 citations recorded by the source corpus.

Deep convolutional networks provide state-of-the-art classifications and regressions results over many high-dimensional problems. We review their architecture, which scatters data with a cascade of linear filter weights and nonlinearities. A mathematical framework is introduced to analyse their properties. Computations of invariants involve multiscale contractions with wavelets, the linearization of hierarchical symmetries and sparse separations. Applications are discussed.

Subfields: Artificial Intelligence, Computer Vision and Pattern Recognition

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Citations in the corpus

What makes ImageNet good for transfer learning? (2016, neutral): “Recently, a mathematical framework for analyzing CNN features was provided by [28].”
Scattering Networks for Hybrid Representation Learning (2018, background): “Recent works [12], [27], [36] have suggested that deep networks could build an approximation of the group of symmetries of a classification task and apply transformations along the orbits of this group, like convolutions.”
Adversarial Perturbations Against Deep Neural Networks for Malware Classification (2016, background): “On the one hand, convolutional neural networks work well on input containing translation invariant properties, which can be found in images [13].”
Safety Verification of Deep Neural Networks (2016, background): “It would be interesting to see if the notions of regularity suggested in [24] permit a symbolic approach, and whether an abstraction reﬁnement framework can be formulated to improve the scalability and computational performance.”
Black-Box vs. White-Box: Understanding Their Advantages and Weaknesses From a Practical Point of View (2019, background): “layer, at least one hidden layer, and an output layer [35].”
Geometric Deep Learning: Going beyond Euclidean data (2016, background): “Most tasks studied in computer vision are not only translation invariant/equivariant, but also stable with respect to local deformations [57], [18].”
Excessive Invariance Causes Adversarial Vulnerability (2018, background): “The role of invariance has been studied extensively in the context of deep neural networks (Bruna & Mallat, 2013; Mallat, 2016; Achille & Soatto, 2018; Behrmann et al., 2018; Cadena et al., 2018).”
i-RevNet: Deep Invertible Networks (2018, background): “The duality between invariance and separation of the classes is discussed in Mallat (2016).”
An empirical study on evaluation metrics of generative adversarial networks (2018, background): “This is not that surprising as convolutional networks are well know for being invariant to certain transformations (Mallat, 2016).”
Scaling the Scattering Transform: Deep Hybrid Networks (2017, neutral): “Recent works [25, 7, 17] have suggested that deep networks could build an approximation of the group of symmetries of a classification task and apply transformations along the orbits of this group, like convolutions.”
A General Theory of Equivariant CNNs on Homogeneous Spaces (2018, neutral): “Other related work includes [24–31].”
Optical Flow Estimation Using a Spatial Pyramid Network (2016, background): “l. This could vastly reduce the size of the network and the number of parameters that need to be learned. In the spatial domain, for example, there is work relating the ﬁlters of convnets to wavelets [28]. Note that pyramids have well-known limitations for dealing with large motions [7, 36]. In part”
Deep Learning for Mortgage Risk* (2016, background): “7See Mallat (2016), Telgarsky (2016), Eldan & Shamir (2016), Bengio & LeCun (2007), and Montufar, Pascanu, Cho & Bengio (2014).”
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].”
Understanding the Representation and Computation of Multilayer Perceptrons: A Case Study in Speech Recognition (2017, methodology): “…methods and techniques for visualization (Zeiler & Fergus, 2013; J. Yosinski & Lipson, 2016), as well as studies of instability in the face of adversarial examples (C. Szegedy & Fergus, 2013; A. Nguyen & Clune, 2015) and contraction and separation properties of these models (Mallat, 2016).”