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John Zarka

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4 papers

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ICLR Conference 2022 Conference Paper

Phase Collapse in Neural Networks

Florentin Guth
John Zarka
Stéphane Mallat

Deep convolutional classifiers linearly separate image classes and improve accuracy as depth increases. They progressively reduce the spatial dimension whereas the number of channels grows with depth. Spatial variability is therefore transformed into variability along channels. A fundamental challenge is to understand the role of non-linearities together with convolutional filters in this transformation. ReLUs with biases are often interpreted as thresholding operators that improve discrimination through sparsity. This paper demonstrates that it is a different mechanism called \emph{phase collapse} which eliminates spatial variability while linearly separating classes. We show that collapsing the phases of complex wavelet coefficients is sufficient to reach the classification accuracy of ResNets of similar depths. However, replacing the phase collapses with thresholding operators that enforce sparsity considerably degrades the performance. We explain these numerical results by showing that the iteration of phase collapses progressively improves separation of classes, as opposed to thresholding non-linearities.

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ICLR Conference 2021 Conference Paper

Separation and Concentration in Deep Networks

John Zarka
Florentin Guth
Stéphane Mallat

Numerical experiments demonstrate that deep neural network classifiers progressively separate class distributions around their mean, achieving linear separability on the training set, and increasing the Fisher discriminant ratio. We explain this mechanism with two types of operators. We prove that a rectifier without biases applied to sign-invariant tight frames can separate class means and increase Fisher ratios. On the opposite, a soft-thresholding on tight frames can reduce within-class variabilities while preserving class means. Variance reduction bounds are proved for Gaussian mixture models. For image classification, we show that separation of class means can be achieved with rectified wavelet tight frames that are not learned. It defines a scattering transform. Learning $1 \times 1$ convolutional tight frames along scattering channels and applying a soft-thresholding reduces within-class variabilities. The resulting scattering network reaches the classification accuracy of ResNet-18 on CIFAR-10 and ImageNet, with fewer layers and no learned biases.

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ICLR Conference 2020 Conference Paper

Deep Network Classification by Scattering and Homotopy Dictionary Learning

John Zarka
Louis Thiry
Tomás Angles
Stéphane Mallat

We introduce a sparse scattering deep convolutional neural network, which provides a simple model to analyze properties of deep representation learning for classification. Learning a single dictionary matrix with a classifier yields a higher classification accuracy than AlexNet over the ImageNet 2012 dataset. The network first applies a scattering transform that linearizes variabilities due to geometric transformations such as translations and small deformations. A sparse $\ell^1$ dictionary coding reduces intra-class variability while preserving class separation through projections over unions of linear spaces. It is implemented in a deep convolutional network with a homotopy algorithm having an exponential convergence. A convergence proof is given in a general framework that includes ALISTA. Classification results are analyzed on ImageNet.

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JMLR Journal 2020 Journal Article

Kymatio: Scattering Transforms in Python

Mathieu Andreux
Tomás Angles
Georgios Exarchakis
Roberto Leonarduzzi
Gaspar Rochette
Louis Thiry
John Zarka
Stéphane Mallat

The wavelet scattering transform is an invariant and stable signal representation suitable for many signal processing and machine learning applications. We present the Kymatio software package, an easy-to-use, high-performance Python implementation of the scattering transform in 1D, 2D, and 3D that is compatible with modern deep learning frameworks, including PyTorch and TensorFlow/Keras. The transforms are implemented on both CPUs and GPUs, the latter offering a significant speedup over the former. The package also has a small memory footprint. Source code, documentation, and examples are available under a BSD license at https://www.kymat.io. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2020. ( edit, beta )

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