Deep convolutional networks have spectacular applications to classification and regression (LeCun et al., 2015), but they are a black box which are hard to analyze mathematically because of their architecture Despite its simplicity, it applies to complex image classification and reaches a higher accuracy than AlexNet (Krizhevsky et al., 2012) over ImageNet ILSVRC2012. It is implemented with a deep convolutional network architecture. Dictionary learning for classification was introduced in Mairal et al. (2009) and implemented with deep A major issue is to compute the sparse code with a small network. We introduce a new architecture based on homotopy continuation, which leads to exponential convergence. The ALIST A (Liu et al., 2019) sparse code is incorporated in We explain the implementation and mathematical properties of each element of the sparse scattering network.

The wavelet scattering transform is an invariant 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. All transforms may be executed on a GPU (in addition to CPU), offering a considerable speed up over CPU implementations. The package also has a small memory footprint, resulting inefficient memory usage. The source code, documentation, and examples are available undera BSD license at https://www.kymat.io/

Generative Adversarial Nets (GANs) and Variational Auto-Encoders (VAEs) provide impressive image generations from Gaussian white noise, but the underlying mathematics are not well understood. We compute deep convolutional network generators by inverting a fixed embedding operator. Therefore, they do not require to be optimized with a discriminator or an encoder. The embedding is Lipschitz continuous to deformations so that generators transform linear interpolations between input white noise vectors into deformations between output images. This embedding is computed with a wavelet Scattering transform. Numerical experiments demonstrate that the resulting Scattering generators have similar properties as GANs or VAEs, without learning a discriminative network or an encoder.