A generalized preimage for the digital analytical hyperplane recognition

International audienceA new digital hyperplane recognition method is presented. This algorithm allows the recognition of digital analytical hyperplanes, such as Naive, Standard and Supercover ones. The principle is to incrementally compute in a dual space the generalized preimage of the ball set corresponding to a given hypervoxel set according to the chosen digitization model. Each point in this preimage corresponds to a Euclidean hyperplane the digitization of which contains all given hypervoxels. An advantage of the generalized preimage is that it does not depend on the hypervoxel locations. Moreover, the proposed recognition algorithm does not require the hypervoxels to be connected or ordered in any way

Local non-planarity of three dimensional surfaces for an invertible reconstruction: k-cuspal cells

International audienceThis paper addresses the problem of the maximal recognition of hyperplanes for an invertible reconstruction of 3D discrete objects. k- cuspal cells are introduced as a three dimensional extension of discrete cusps defined by R.Breton. With k-cuspal cells local non planarity on discrete surfaces can be identified in a very straightforward way

Two discrete-continuous operations based on the scaling transform

International audienceIn this paper we study the relationship between the Euclidean and the discrete world thru two operations based on the Euclidean scaling function: the discrete smooth scaling and the discrete based geometrical simplification

Discrete-Euclidean operations

International audienceIn this paper we study the relationship between the Euclidean and the discrete space. We study discrete operations based on Euclidean functions: the discrete smooth scaling and the discrete-continuous rotation. Conversely, we study Euclidean oper- ations based on discrete functions: the discrete based simplification, the Euclidean- discrete union and the Euclidean-discrete co-refinement. These operations operate partly in the discrete and partly in the continuous space. Especially for the discrete smooth scaling operation, we provide error bounds when different such operations are chained

Machine learning for Internet of Things data analysis: A survey

Rapid developments in hardware, software, and communication technologies have allowed the emergence of Internet-connected sensory devices that provide observation and data measurement from the physical world. By 2020, it is estimated that the total number of Internet-connected devices being used will be between 25 and 50 billion. As the numbers grow and technologies become more mature, the volume of data published will increase. Internet-connected devices technology, referred to as Internet of Things (IoT), continues to extend the current Internet by providing connectivity and interaction between the physical and cyber worlds. In addition to increased volume, the IoT generates Big Data characterized by velocity in terms of time and location dependency, with a variety of multiple modalities and varying data quality. Intelligent processing and analysis of this Big Data is the key to developing smart IoT applications. This article assesses the different machine learning methods that deal with the challenges in IoT data by considering smart cities as the main use case. The key contribution of this study is presentation of a taxonomy of machine learning algorithms explaining how different techniques are applied to the data in order to extract higher level information. The potential and challenges of machine learning for IoT data analytics will also be discussed. A use case of applying Support Vector Machine (SVM) on Aarhus Smart City traffic data is presented for a more detailed exploration.Comment: Digital Communications and Networks (2017

Topological Expressiveness of Neural Networks: Topology of Learning

Dissertation presented as the partial requirement for obtaining a Master's degree in Data Science and Advanced AnalyticsGiven a neural network, how many di erent problems can it solve? This important and open question in deep learning is usually referred to as the problem of the expressive power of a neural network. Previous research has tackled this issue through statistical and geometrical methods. This work proposes a new method based on a topological perspective. Topology is the eld of mathematics aimed at describing spaces and functions through robust characterizing features. Topological Data Analysis is the young eld developed to extract topological insight from data. We rst show that topological features of the decision boundary describe the closest notion of the intrinsic complexity of a classi cation problem. These topological features divide classi cation problems into several equivalence classes. Linear-separability is an example of such a class. We establish the topological expressive power of a network architecture as the number of di erent topological classes it is able to express. Being a novel work in a young research eld, most of the thesis is devoted to developing this perspective and creating the tools required. The main objective of this thesis is to tackle neural network’s understanding in general and architecture selection in particular, through a novel approach. Our results show that topological expressiveness has a complex correlation with many features in a neural network’s architecture depending weakly on the total number of parameters. Some of our results recapitulate previous research on geometrical properties, while others are unique to this novel topological point of view, sometimes challenging previous research.Quantos problemas di erentes consegue uma dada rede neuronal resolver? Esta pergunta aberta é central no ramo de aprendizagem profunda e conhecida como o poder expressivo de uma rede neuronal. Tentativas anteriores em resolver este problema zeram-no usando métodos estatísticos ou geométricos. Este trabalho apresenta um novo método baseado numa prespectiva topologica. Topologia é o ramo de matemática responsável por descrever espaços e transformações com base em caracteristicas fundamentais. Topological Data Analysis (Análise Topológica de Dados) é o recente ramo de investigação desenvolvido para extrair conhecimento Topológico de dados. Começamos por mostrar que uma caraterização topológica da barreira de decisão é a noção mais próxima da complexidade de um problema de classi cação. Estas caracteristicas topoólicas dividas os problemas de classi cação em diversas classes de equivalência. O conjunto de problemas separaveis por uma reta são um exemplo de uma destas classes. Establecemos que a expressividade topológica de uma architectura neuronal é equivalente a quantas destas classes consegue resolver. Dado que é um método novo num ramo de investigação recente, grande parte desta tese foca-se em desenvolver esta perspectiva e em criar as ferramentas necessárias para o seu estudo. O objectivo desta dissertação é, apartir de uma abordagem original, enfrentar a falta de compreensão de redes neuronais no geral e, em particular, informar a escolha de arquitecturas. Os resultados obtidos mostram que a expressividade topológica tem correlações complexas com diversos elementos da arquitectura de uma rede, mostrando uma depêndencia ténue no número total de parametros. Alguns resultados seguem a mesma linha que a investigação gemétrica anterior, outros são únicos à perspectiva apresentada e complementando resultados anteriores

Structural and Computational Existence Results for Multidimensional Subshifts

Symbolic dynamics is a branch of mathematics that studies the structure of infinite sequences of symbols, or in the multidimensional case, infinite grids of symbols. Classes of such sequences and grids defined by collections of forbidden patterns are called subshifts, and subshifts of finite type are defined by finitely many forbidden patterns. The simplest examples of multidimensional subshifts are sets of Wang tilings, infinite arrangements of square tiles with colored edges, where adjacent edges must have the same color. Multidimensional symbolic dynamics has strong connections to computability theory, since most of the basic properties of subshifts cannot be recognized by computer programs, but are instead characterized by some higher-level notion of computability. This dissertation focuses on the structure of multidimensional subshifts, and the ways in which it relates to their computational properties. In the first part, we study the subpattern posets and Cantor-Bendixson ranks of countable subshifts of finite type, which can be seen as measures of their structural complexity. We show, by explicitly constructing subshifts with the desired properties, that both notions are essentially restricted only by computability conditions. In the second part of the dissertation, we study different methods of defining (classes of ) multidimensional subshifts, and how they relate to each other and existing methods. We present definitions that use monadic second-order logic, a more restricted kind of logical quantification called quantifier extension, and multi-headed finite state machines. Two of the definitions give rise to hierarchies of subshift classes, which are a priori infinite, but which we show to collapse into finitely many levels. The quantifier extension provides insight to the somewhat mysterious class of multidimensional sofic subshifts, since we prove a characterization for the class of subshifts that can extend a sofic subshift into a nonsofic one.Symbolidynamiikka on matematiikan ala, joka tutkii äärettömän pituisten symbolijonojen ominaisuuksia, tai moniulotteisessa tapauksessa äärettömän laajoja symbolihiloja. Siirtoavaruudet ovat tällaisten jonojen tai hilojen kokoelmia, jotka on määritelty kieltämällä jokin joukko äärellisen kokoisia kuvioita, ja äärellisen tyypin siirtoavaruudet saadaan kieltämällä vain äärellisen monta kuviota. Wangin tiilitykset ovat yksinkertaisin esimerkki moniulotteisista siirtoavaruuksista. Ne ovat värillisistä neliöistä muodostettuja tiilityksiä, joissa kaikkien vierekkäisten sivujen on oltava samanvärisiä. Moniulotteinen symbolidynamiikka on vahvasti yhteydessä laskettavuuden teoriaan, sillä monia siirtoavaruuksien perusominaisuuksia ei ole mahdollista tunnistaa tietokoneohjelmilla, vaan korkeamman tason laskennallisilla malleilla. Väitöskirjassani tutkin moniulotteisten siirtoavaruuksien rakennetta ja sen suhdetta niiden laskennallisiin ominaisuuksiin. Ensimmäisessä osassa keskityn tiettyihin äärellisen tyypin siirtoavaruuksien rakenteellisiin ominaisuuksiin: äärellisten kuvioiden muodostamaan järjestykseen ja Cantor-Bendixsonin astelukuun. Halutunlaisia siirtoavaruuksia rakentamalla osoitan, että molemmat ominaisuudet ovat olennaisesti laskennallisten ehtojen rajoittamia. Väitöskirjan toisessa osassa tutkin erilaisia tapoja määritellä moniulotteisia siirtoavaruuksia, sekä sitä, miten nämä tavat vertautuvat toisiinsa ja tunnettuihin siirtoavaruuksien luokkiin. Käsittelen määritelmiä, jotka perustuvat toisen kertaluvun logiikkaan, kvanttorilaajennukseksi kutsuttuun rajoitettuun loogiseen kvantifiointiin, sekä monipäisiin äärellisiin automaatteihin. Näistä kolmesta määritelmästä kahteen liittyy erilliset siirtoavaruuksien hierarkiat, joiden todistan romahtavan äärellisen korkuisiksi. Kvanttorilaajennuksen tutkimus valottaa myös niin kutsuttujen sofisten siirtoavaruuksien rakennetta, jota ei vielä tunneta hyvin: kyseisessä luvussa selvitän tarkasti, mitkä siirtoavaruudet voivat laajentaa sofisen avaruuden ei-sofiseksi.Siirretty Doriast

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4