Compression of Recurrent Neural Networks using Matrix Factorization

Compressing neural networks is a key step when deploying models for real-time or embedded applications. Factorizing the model's matrices using low-rank approximations is a promising method for achieving compression. While it is possible to set the rank before training, this approach is neither flexible nor optimal. In this work, we propose a post-training rank-selection method called Rank-Tuning that selects a different rank for each matrix. Used in combination with training adaptations, our method achieves high compression rates with no or little performance degradation. Our numerical experiments on signal processing tasks show that we can compress recurrent neural networks up to 14x with at most 1.4% relative performance reduction

Mathematics and Morphogenesis of the City: A Geometrical Approach

Cities are living organisms. They are out of equilibrium, open systems that never stop developing and sometimes die. The local geography can be compared to a shell constraining its development. In brief, a city's current layout is a step in a running morphogenesis process. Thus cities display a huge diversity of shapes and none of traditional models from random graphs, complex networks theory or stochastic geometry takes into account geometrical, functional and dynamical aspects of a city in the same framework. We present here a global mathematical model dedicated to cities that permits describing, manipulating and explaining cities' overall shape and layout of their street systems. This street-based framework conciliates the topological and geometrical sides of the problem. From the static analysis of several French towns (topology of first and second order, anisotropy, streets scaling) we make the hypothesis that the development of a city follows a logic of division / extension of space. We propose a dynamical model that mimics this logic and which from simple general rules and a few parameters succeeds in generating a large diversity of cities and in reproducing the general features the static analysis has pointed out.Comment: 13 pages, 13 figure

Promenade dans les cartes de villes - Phénoménologie mathématique et physique de la ville - une approche géométrique

We are interested in the phenomenology of cities by restricting them to the geometry of their street network. This study aims at being synthetic, functional and interdisciplinary. It follows the large work that has been performed from the early XXth century by townplanners, social scientists, geographers, statisticians, physicists. We try to demonstrate that the street - as a coherent alignment of street segments - can be considered as an essential elementary structure of the city. How much information is encoded in the street network? To what extent does it constraint the city use? How are the current urban layout and its evolution determined jointly by traffic axis and structuring elements? We present a mathematical and computational framework allowing to consider the map of a city as a geometric continuum associated to the topology of a planar graph. To this graph we juxtapose a hypergraph structure using the street geometry to obtain easily the notion of axis and a multi-scale representation of the city. In spite of an apparent shape diversity, we show that the street network of a city is subjected to general laws that leave hallmarks on a city map. We propose several morphogenesis models of the city, implementing the idea that the city's growth follows a structured extension / division of space logic and able to reproduce hallmarks observed on actual maps. The understanding of regulation mechanism of the city allows us to propose functional algorithms whose computational efficiencies are very interesting. We present an algorithm recovering streets from a collection of street segments; the notion of simplest centrality whose calculus on a map allows a hierarchical analysis of it, revealing for instance main traffic axis and ill-deserved area; a fast approximate algorithm to nd the shortest path between two random points; and a Spectral Clustering based algorithm to produce morphological segmentations of maps. We also work on the identi cation of random tessellation models to be substituted to an actual road network and to solve large optimization problems using statistical equivalence.Nous nous intéressons à la phénoménologie des villes en nous limitant à la géométrie induite par le squelette de leur réseau de rues. C'est une étude à volonté synthétique, fonctionnelle et interdisciplinaire qui vient s'ajouter aux travaux qui ont été menés à grande cadence depuis le début du XXème siècle par des urbanistes, sociologues, géographes, statisticiens, physiciens. Nous cherchons à montrer que la rue, en tant qu'alignement cohérent de segments de rues peut être considérée comme structure élémentaire de la ville. Quelle quantité d'information est donnée par la géométrie du réseau routier ? Dans quelle mesure contraint-il nos échanges ? Comment le paysage urbain actuel est-il déterminé par son évolution le long d'axes de circulation et d'éléments structurants ? Nous présentons un cadre mathématique permettant de considérer la carte d'une ville comme un continuum géométrique défi ni par la topologie d'un graphe planaire. Nous superposons à ce graphe une structure d'hypergraphe pour manipuler aisément la notion d'axes ainsi qu'une représentation multi-échelles de la ville. En dépit d'une grande diversité apparente de formes, nous montrons que le réseau de rues d'une ville se soumet à un certain nombre de lois générales qui laissent des traces sur le plan de la ville. Nous proposons des modèles de croissance et de morphogénèse de la ville, implé- mentant l'idée que l'évolution de la ville suit une logique d'extension / division structurée de l'espace et reproduisant les signatures observées sur les plans de villes réelles. La compréhension des mécanismes régulateurs de la ville nous permet de proposer des algorithmes fonctionnels dont le temps de calcul est très intéressant. Ainsi nous présentons un algorithme reconstituant les rues à partir de segments de rues ; la notion de centralité simple dont le calcul sur une carte permet une analyse hiérarchique de celle-ci, met en valeur les axes de trafic principaux et en évidence les zones mal desservies ; un algorithme permettant d'approximer rapidement le plus court chemin entre deux points aléatoires ; un algorithme prenant appui sur le Spectral Clustering pour produire des segmentations morphologiques de cartes et retravaillons l'identi cation de modèles de mosaïques aléatoires pour les substituer à un réseau urbain particulier dans la résolution par équivalents statistiques de grands problèmes d'optimisation

« Tentative de modélisation de la Morphogenèse du réseau des rues »

Maurizio Gribaudi Dir. et al.International audienc

Stochastic simulation of urban environments: Application to Path-loss in wireless systems

We are interested in the assessment of electromagnetic Path-Loss in complex environments. The Path-loss is the attenuation function $P$ of the electromagnetic power at a distance $d$ of an antenna. In free-space, $P(d) \propto 1/d^2$, in complex environments like cities, wave trajectory is altered by successive reflections and absorptions, the path-loss is not theoretically known and engineering rules postulate that $P(d) \simeq 1/d^{\gamma}, \, \gamma>2$.We place in a stochastic geometry context to answer the problem statistically. We present random models of 3D-city. These models reproduce main real cities' features, can be calibrated with simple mean formulae and can be fast simulated. For collections of random cities with the same mean morphology, we estimate by Monte-Carlo ray tracing techniques their attenuation maps. By averaging these maps, we show that the power expectancy actually follows a function $\sim 1/d^{\gamma}$ with $\gamma$ depending on the environment morphology

Stochastic simulation of urban environments: Application to Path-loss in wireless systems

International audienc

Advanced statistical methods applied to a simplifed assessment of population exposure induced by a LTE network

International audienceThis study presents a simplified methodology based on advanced statistical tools to evaluate the day-to-day global electromagnetic (EM) field exposure of a population taking into account the variability and uncertainties linked to propagation environment, information and communication technology usage, as well as EM fields from personal wireless devices and base stations. A sensitivity analysis was carried out in order to assess the influence of these parameters on EM field exposure. Results have highlighted the importance of received power density from base stations to the issue of global exposure induced by a macro LTE network for an entire population in an urban area