Real-time Outdoor Localization Using Radio Maps: A Deep Learning Approach

This paper deals with the problem of localization in a cellular network in a dense urban scenario. Global Navigation Satellite Systems typically perform poorly in urban environments, where the likelihood of line-of-sight conditions between the devices and the satellites is low, and thus alternative localization methods are required for good accuracy. We present a deep learning method for localization, based merely on pathloss, which does not require any increase in computation complexity at the user devices with respect to the device standard operations, unlike methods that rely on time of arrival or angle of arrival information. In a wireless network, user devices scan the base station beacon slots and identify the few strongest base station signals for handover and user-base station association purposes. In the proposed method, the user to be localized simply reports such received signal strengths to a central processing unit, which may be located in the cloud. For each base station we have good approximation of the pathloss at every location in a dense grid in the map. This approximation is provided by RadioUNet, a deep learning-based simulator of pathloss functions in urban environment, that we have previously proposed and published. Using the estimated pathloss radio maps of all base stations and the corresponding reported signal strengths, the proposed deep learning algorithm can extract a very accurate localization of the user. The proposed method, called LocUNet, enjoys high robustness to inaccuracies in the estimated radio maps. We demonstrate this by numerical experiments, which obtain state-of-the-art results.

The Modern Mathematics of Deep Learning

We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.

Approximation spaces of deep neural networks

We study the expressivity of deep neural networks. Measuring a networks complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. Using results from classical approximation theory, we show that this class can be endowed with a (quasi)-norm that makes it a linear function space, called approximation space. We establish that allowing the networks to have certain types of skip connections does not change the resulting approximation spaces. We also discuss the role of the networks nonlinearity (also known as activation function) on the resulting spaces, as well as the role of depth. For the popular ReLU nonlinearity and its powers, we relate the newly constructed spaces to classical Besov spaces. The established embeddings highlight that some functions of very low Besov smoothness can nevertheless be well approximated by neural networks, if these networks are sufficiently deep.

A Haar Wavelet-Based Perceptual Similarity Index for Image Quality Assessment

In most practical situations, the compression or transmission of images and videos creates distortions that will eventually be perceived by a human observer. Vice versa, image and video restoration techniques, such as inpainting or denoising, aim to enhance the quality of experience of human viewers. Correctly assessing the similarity between an image and an undistorted reference image as subjectively experienced by a human viewer can thus lead to significant improvements in any transmission, compression, or restoration system. This paper introduces the Haar wavelet-based perceptual similarity index (HaarPSI), a novel and computationally inexpensive similarity measure for full reference image quality assessment. The HaarPSI utilizes the coefficients obtained from a Haar wavelet decomposition to assess local similarities between two images, as well as the relative importance of image areas. The consistency of the HaarPSI with the human quality of experience was validated on four large benchmark databases containing thousands of differently distorted images. On these databases, the HaarPSI achieves higher correlations with human opinion scores than state-of-the-art full reference similarity measures like the structural similarity index (SSIM), the feature similarity index (FSIM), and the visual saliency-based index (VSI). Along with the simple computational structure and the short execution time, these experimental results suggest a high applicability of the HaarPSI in real world tasks.

Efficient Resolution of Anisotropic Structures

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in $L_2$. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For frequent query problems, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings.

Robust Dimension Reduction, Fusion Frames, and Grassmannian Packings

We consider estimating a random vector from its noisy projections onto low dimensional subspaces constituting a fusion frame. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first determine the minimum mean-squared error (MSE) in linearly estimating the random vector of interest from its fusion frame projections, in the presence of white noise. We show that MSE assumes its minimum value when the fusion frame is tight. We then analyze the robustness of the constructed linear minimum MSE (LMMSE) estimator to erasures of the fusion frame subspaces. We prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures. We call such fusion frames equi-distance tight fusion frames, and prove that the chordal distance between subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for construction of equi-distance tight fusion frames.