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

Towards End-to-End Acoustic Localization using Deep Learning: from Audio Signal to Source Position Coordinates

This paper presents a novel approach for indoor acoustic source localization using microphone arrays and based on a Convolutional Neural Network (CNN). The proposed solution is, to the best of our knowledge, the first published work in which the CNN is designed to directly estimate the three dimensional position of an acoustic source, using the raw audio signal as the input information avoiding the use of hand crafted audio features. Given the limited amount of available localization data, we propose in this paper a training strategy based on two steps. We first train our network using semi-synthetic data, generated from close talk speech recordings, and where we simulate the time delays and distortion suffered in the signal that propagates from the source to the array of microphones. We then fine tune this network using a small amount of real data. Our experimental results show that this strategy is able to produce networks that significantly improve existing localization methods based on \textit{SRP-PHAT} strategies. In addition, our experiments show that our CNN method exhibits better resistance against varying gender of the speaker and different window sizes compared with the other methods.Comment: 18 pages, 3 figures, 8 table

Accelerating Eulerian Fluid Simulation With Convolutional Networks

Efficient simulation of the Navier-Stokes equations for fluid flow is a long standing problem in applied mathematics, for which state-of-the-art methods require large compute resources. In this work, we propose a data-driven approach that leverages the approximation power of deep-learning with the precision of standard solvers to obtain fast and highly realistic simulations. Our method solves the incompressible Euler equations using the standard operator splitting method, in which a large sparse linear system with many free parameters must be solved. We use a Convolutional Network with a highly tailored architecture, trained using a novel unsupervised learning framework to solve the linear system. We present real-time 2D and 3D simulations that outperform recently proposed data-driven methods; the obtained results are realistic and show good generalization properties.Comment: Significant revisio