Exact Inference Algorithms and Their Optimization in Bayesian Clustering

Clustering is a central task in computational statistics. Its aim is to divide observed data into groups of items, based on the similarity of their features. Among various approaches to clustering, Bayesian model-based clustering has recently gained popularity. Many existing works are based on stochastic sampling methods. This work is concerned with exact, exponential-time algorithms for the Bayesian model-based clustering task. In particular, we consider the exact computation of two summary statistics: the number of clusters, and pairwise incidence of items in the same cluster. We present an implemented algorithm for computing these statistics substantially faster than would be achieved by direct enumeration of the possible partitions. The method is practically applicable to data sets of up to approximately 25 items. We apply a variant of the exact inference method into graphical models where a given variable may have up to four parent variables. The parent variables can then have up to 16 value combinations, and the task is to cluster them and find combinations that lead to similar conditional probability tables. Further contributions of this work are related to number theory. We show that a novel combination of addition chains and additive bases provides the optimal arrangement of multiplications, when the task is to use repeated multiplication starting from a given number or entity, but only a certain kind of function of the successive powers is required. This arrangement speeds up the computation of the posterior distribution for the number of clusters. The same arrangement method can be applied to other multiplicative tasks, for example, in matrix multiplication. We also present new algorithmic results related to finding extremal additive bases. Before this work, the extremal additive bases were known up to length 23. We have computed them up to length 24 in the unrestricted case, and up to length 41 in the restricted case.Klusterointi on keskeinen laskennallisen tilastotieteen menetelmä. Klusteroinnissa samankaltaisia havaintoja ryhmitellään yhteen. Eri klusterointimenetelmiä on lukuisia; viime aikoina on yleistynyt bayesiläinen mallipohjainen klusterointi. Siihen on yleensä sovellettu stokastisia menetelmiä: kokeillaan useita erilaisia tapoja klusteroida samat havainnot, osin satunnaisesti, ja näin saadaan laskennallinen arvio oikeasta klusterointiratkaisusta. Tässä työssä sen sijaan tutkitaan bayesiläistä mallipohjaista klusterointia eksakteilla algoritmeilla, joiden aikavaativuus on eksponentiaalinen. Niillä voidaan laskea tarkka todennäköisyysjakauma kahdelle oikeaa klusterointiratkaisua kuvaavalle tunnusluvulle: klusterien lukumäärälle sekä sille, mitkä havaintoparit kuuluvat samaan klusteriin. Työssä toteutettu algoritmi laskee nämä tunnusluvut huomattavasti nopeammin kuin jos yksinkertaisesti käytäisiin läpi kaikki mahdolliset klusterointiratkaisut. Käytännössä menetelmää voi käyttää enintään noin 25 havainnon klusterointiin. Algoritmin muunnelmaa sovelletaan graafisiin todennäköisyysmalleihin, joissa kullakin muuttujalla voi olla enintään neljä vanhempaa. Vanhempien mahdolliset arvot muodostavat siten enintään 16 erilaista yhdistelmää. Tehtävänä on klusteroida näitä yhdistelmiä lapsimuuttujan ehdollisen todennäköisyyden perusteella. Työ sisältää myös lukuteoreettisia tuloksia. Työssä osoitetaan, että lisäysjonot ja additiiviset kannat voidaan yhdistää siten, että saadaan optimaalinen tapa järjestää suoritettavat kertolaskut, kun kertolaskuja toistetaan annetusta lähtöarvosta alkaen ja näin syntyvistä potensseista tarvitaan vain tietynlainen funktio. Järjestämällä laskutoimitukset tämän ratkaisun mukaisesti pystytään vähentämään edellämainitussa klusterointitehtävässä tarvittavaa laskentatyötä. Samaa menetelmää voi käyttää myös muissa ketjutetuissa kertolaskuissa, esimerkiksi matriisikertolaskussa. Lopuksi työssä esitetään lukuteorian algoritmisia tuloksia, jotka liittyvät additiivisten kantojen etsintään. Aiemmin on laskettu kaikki ekstremaaliset additiiviset kannat, joiden pituus on enintään 23. Tässä työssä kyseiset additiiviset kannat on laskettu pituuteen 24 asti ilman rajoituksia, ja pituuteen 41 asti, kun oletetaan additiivisen kannan olevan rajoitettu

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Efficient Methods for Finding Optimal Convolutional Self-Doubly Orthogonal Codes

Résumé: Au cours des dernières années, la hausse sans précédent du nombre d'ultrabooks et d'appareils mobiles s'est accompagnée d'un besoin toujours croissant d'accès aux technologies permettant des communications sans-fil fiables et à haut débit. Pour atténuer ou éliminer les erreurs induites par les interférences et le bruit dans les canaux de communication, il est important de développer des systèmes de codage efficaces pour la correction d'erreurs. En effet, lors de communications de données numériques sur un canal ayant un faible rapport signal sur bruit, ces codes permettent de conserver un taux d'erreur faible tout en augmentant le débit des transmissions et/ou en diminuant la puissance d'émission requise. Ceci contribue grandement à améliorer l'efficacité énergétique de ces dispositifs électroniques sans-fil et, ainsi, à prolonger leur autonomie. Dans cette thèse par articles, nous présentons un algorithme de recherche efficace pour trouver deux types de codes correcteurs d'erreur: les codes convolutionnels doublement orthogonaux (CDO) et les codes convolutionnels doublement orthogonaux simplifiés (S-CDO). En effet, ces codes sont utilisés dans un système de contrôle d'erreurs ayant un décodage à seuil itératif différent de la procédure de décodage Turbo classique, puisqu'il ne nécessite aucun entrelaceur, ni à l'encodage, ni aux étapes de décodage. Néanmoins, son processus de décodage à seuil nécessite que ces codes convolutionnels systématiques satisfassent des propriétés dites de « double orthogonalité », allant au-delà des conditions requises par les codes « simplement orthogonaux », bien connus et habituellement utilisés lors d'un décodage à seuil non-itératif. Afin de pouvoir construire des codecs à haute performance et à faible latence avec ces codes, il est important de minimiser leur longueur de contrainte ou « span » pour un nombre J de connexions donné. Bien que trouver des codes CDO et S-CDO ne soit pas difficile, déterminer les codes ayant un span minimal (dit optimal) pour un ordre J donné est mathématiquement très complexe. En effet, la construction directe de codes CDO / S-CDO à span court/optimal reste un problème ouvert et qui est soupçonné d'être NP-complet. Cette thèse présente un total de trois articles: deux articles publiés dans IEEE Transactions on Communications et un article soumis au journal IEEE Transactions on Parallel and Distributed Systems . Dans ces articles, nous décrivons un nouvel algorithme de recherche parallèle, efficace et implicitement-exhaustif pour trouver des codes CDO et S-CDO systématiques, à taux R=1/2 et ayant un span plus court, voire minimal, c.à.d. optimal. Comparé à l'algorithme de recherche implicitement-exhaustif de référence, l'algorithme de recherche à haute performance proposé reste exhaustif mais fournit un facteur d'accélération très important, supérieur à 16300 pour les codes CDO (J=7) et supérieur à 6300 pour les codes S-CDO (J=8).----------Abstract: In recent years, the rise of ultrabooks and mobile devices has been accompanied by an ever increasing need for reliable high-bandwidth wireless communications. To mitigate or eliminate the errors that are invariably introduced due to noise and interference in the communication channels, it is important to develop efficient error-correcting coding schemes. Indeed, these codes may be used to preserve the error performance while allowing the data-rate of digital communications to be increased and the transmission power at lower signal-to-noise ratios to be reduced, thereby improving the overall power efficiency of these devices. In this manuscript-based thesis, we present an efficient search algorithm for finding optimal/short-span Convolutional Self-Doubly Orthogonal (CDO) codes and Simplified Convolutional Self-Doubly Orthogonal (S-CDO) codes. These error-correcting codes are employed in an iterative error-control coding scheme that differs from the classical Turbo code procedure, as it does not require any interleaver, neither at the encoding nor at the decoding stages. However, its iterative threshold decoding procedure requires that these systematic convolutional codes satisfy some “double orthogonality properties”, beyond those of the well-known orthogonal codes used in the usual non-iterative threshold decoding. In order to build high-performance, low-latency codecs with these codes, it is important to minimize the constraint length, also called “span”, for a given number J of generator connections. Although finding CDO/S-CDO codes is not difficult, determining the optimal/short-span codes for a given order J is computationally very challenging. The direct construction of optimal or shortest-span CDO and S-CDO codes has so far eluded analysis, and the search for these codes is believed to be an NP-complete problem. The thesis presents a total of three articles: two articles that were published in IEEE Transactions on Communications , and one article that was submitted for publication to IEEE Transactions on Parallel and Distributed Systems . In these articles, we describe a novel efficient and parallel implicitly-exhaustive search algorithm for finding rate R=1/2 systematic optimal/short-span CDO and S-CDO codes. The high-performance search algorithm is still exhaustive in nature, yet it provides an impressive speedup that is larger than 16300 (CDO, J=7) and 6300 (S-CDO, J=8) over the reference implicitly-exhaustive search algorithm, and larger than 2000 (CDO, J=17) over the fastest known CDO validation function used in high-performance pseudo-random search algorithms

Turning the map : construction and deconstruction of Galilee in Luke-Acts

Masterthesis Nieuwe Testamen

Applications of Spin Glasses across Disciplines: From Complex Systems to Quantum Computing and Algorithm Development

The main subjects of this dissertation are spin glass applications in other disciplines and spin glass algorithms. Spin glasses are magnetic systems with disorder and frustration, and the essential physics of spin glasses lies not in the details of their microscopic interactions but rather in the competition between quenched ferromagnetic and antiferromagnetic interactions. Concepts that arose in the study of spin glasses have led to applications in areas as diverse as computer science, biology, and finance, as well as a variety of others. In the first part of this dissertation I study the equilibrium and non-equilibrium properties of Boolean decision problems with competing interactions on scale-free networks in an external bias (a magnetic field). First, I perform finite-temperature Monte Carlo simulations in a field to test the robustness of the spin-glass phase and I show that the system has a spin-glass phase in a field, i.e., it exhibits a de Almeida–Thouless line. Then I study avalanche distributions when the system is driven by a field at zero temperature to test whether the system displays self-organized criticality. The numerical results suggest that avalanches (damage) can spread across the entire system with nonzero probability when the decay exponent of the interaction degree is less than or equal to 2, i.e., that Boolean decision problems on scale-free networks with competing interactions can be fragile when the system is not in thermal equilibrium. In the second part of this dissertation I discuss the best-case performance of quantum annealers on native spin-glass benchmarks, i.e., how chaos can affect success probabilities. We perform classical parallel-tempering Monte Carlo simulations of the archetypal benchmark problem, an Ising spin glass, on the native chip topology. Using realistic uncorrelated noise models for the D-Wave Two quantum annealer, I study the best-case resilience, or the probability that the ground-state configuration is not affected by random fields and random-bond fluctuations found on the chip. We compute the upper-bound success probabilities for different instance classes based on these simple error models, and I present strategies for developing robust and hard benchmark instances. In the third part of this dissertation I present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by several orders of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits (improved thermalization and achievement of more equiprobable sampling of ground states) of the isoenergetic cluster moves in two and three space dimensions, as well as in the nonplanar Chimera topology found in the D-Wave quantum annealing machine. Finally, I study the thermodynamic properties of the two-dimensional Edwards-Anderson Ising spin-glass model on a square lattice using the tensor renormalization group method based on a higher-order singular-value decomposition. Our estimates of the partition function without a high precision data type lead to negative values at very low temperatures, thus illustrating that the method can not be applied to frustrated magnetic systems

Envisioning the Future of Cyber Security in Post-Quantum Era: A Survey on PQ Standardization, Applications, Challenges and Opportunities

The rise of quantum computers exposes vulnerabilities in current public key cryptographic protocols, necessitating the development of secure post-quantum (PQ) schemes. Hence, we conduct a comprehensive study on various PQ approaches, covering the constructional design, structural vulnerabilities, and offer security assessments, implementation evaluations, and a particular focus on side-channel attacks. We analyze global standardization processes, evaluate their metrics in relation to real-world applications, and primarily focus on standardized PQ schemes, selected additional signature competition candidates, and PQ-secure cutting-edge schemes beyond standardization. Finally, we present visions and potential future directions for a seamless transition to the PQ era

GEOGRAPHICAL SYSTEMS IN THE FIRST CENTURY BC: POSIDONIUS’ F 49 E ̶ K AND VITRUVIUS’ ON ARCHITECTURE VI 1. 3 ̶ 13

The article analyses innovative ethno-geographical systems of the first century BC. During Hellenistic times, the science of geography made use of increasingly advanced mathematical and astronomical skills to ensure a scientific basis for the cartographical project; however, this geographical research apparently disregarded the natural and human environments. There is a paradigm change in the referred century. The Stoic Posidonius focuses on the concept of zones found in the early philosophers and finds a compromise between the ‘scientific’ and the ‘descriptive’ geographies. Likewise, Vitruvius conveys a geographical system which associates climatic, somatic, and psychic features