Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations

The success of deep convolutional architectures is often attributed in part to their ability to learn multiscale and invariant representations of natural signals. However, a precise study of these properties and how they affect learning guarantees is still missing. In this paper, we consider deep convolutional representations of signals; we study their invariance to translations and to more general groups of transformations, their stability to the action of diffeomorphisms, and their ability to preserve signal information. This analysis is carried by introducing a multilayer kernel based on convolutional kernel networks and by studying the geometry induced by the kernel mapping. We then characterize the corresponding reproducing kernel Hilbert space (RKHS), showing that it contains a large class of convolutional neural networks with homogeneous activation functions. This analysis allows us to separate data representation from learning, and to provide a canonical measure of model complexity, the RKHS norm, which controls both stability and generalization of any learned model. In addition to models in the constructed RKHS, our stability analysis also applies to convolutional networks with generic activations such as rectified linear units, and we discuss its relationship with recent generalization bounds based on spectral norms

Neurogenesis Deep Learning

Neural machine learning methods, such as deep neural networks (DNN), have achieved remarkable success in a number of complex data processing tasks. These methods have arguably had their strongest impact on tasks such as image and audio processing - data processing domains in which humans have long held clear advantages over conventional algorithms. In contrast to biological neural systems, which are capable of learning continuously, deep artificial networks have a limited ability for incorporating new information in an already trained network. As a result, methods for continuous learning are potentially highly impactful in enabling the application of deep networks to dynamic data sets. Here, inspired by the process of adult neurogenesis in the hippocampus, we explore the potential for adding new neurons to deep layers of artificial neural networks in order to facilitate their acquisition of novel information while preserving previously trained data representations. Our results on the MNIST handwritten digit dataset and the NIST SD 19 dataset, which includes lower and upper case letters and digits, demonstrate that neurogenesis is well suited for addressing the stability-plasticity dilemma that has long challenged adaptive machine learning algorithms.Comment: 8 pages, 8 figures, Accepted to 2017 International Joint Conference on Neural Networks (IJCNN 2017

Deep Convolutional Neural Networks Based on Semi-Discrete Frames

Deep convolutional neural networks have led to breakthrough results in practical feature extraction applications. The mathematical analysis of these networks was pioneered by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on identical semi-discrete wavelet frames in each network layer, and proved translation-invariance as well as deformation stability of the resulting feature extractor. The purpose of this paper is to develop Mallat's theory further by allowing for different and, most importantly, general semi-discrete frames (such as, e.g., Gabor frames, wavelets, curvelets, shearlets, ridgelets) in distinct network layers. This allows to extract wider classes of features than point singularities resolved by the wavelet transform. Our generalized feature extractor is proven to be translation-invariant, and we develop deformation stability results for a larger class of deformations than those considered by Mallat. For Mallat's wavelet-based feature extractor, we get rid of a number of technical conditions. The mathematical engine behind our results is continuous frame theory, which allows us to completely detach the invariance and deformation stability proofs from the particular algebraic structure of the underlying frames.Comment: Proc. of IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, June 2015, to appea