Incremental inference on higher-order probabilistic graphical models applied to constraint satisfaction problems

Thesis (PhD)--Stellenbosch University, 2022.ENGLISH ABSTRACT: Probabilistic graphical models (PGMs) are used extensively in the probabilistic reasoning domain. They are powerful tools for solving systems of complex relationships over a variety of probability distributions, such as medical and fault diagnosis, predictive modelling, object recognition, localisation and mapping, speech recognition, and language processing [5, 6, 7, 8, 9, 10, 11]. Furthermore, constraint satisfaction problems (CSPs) can be formulated as PGMs and solved with PGM inference techniques. However, the prevalent literature on PGMs shows that suboptimal PGM structures are primarily used in practice and a suboptimal formulation for constraint satisfaction PGMs. This dissertation aimed to improve the PGM literature through accessible algorithms and tools for improved PGM structures and inference procedures, specifically focusing on constraint satisfaction. To this end, this dissertation presents three published contributions to the current literature: a comparative study to compare cluster graph topologies to the prevalent factor graphs [1], an application of cluster graphs in land cover classification in the field of cartography [2], and a comprehensive integration of various aspects required to formulate CSPs as PGMs and an algorithm to solve this formulation for problems too complex for traditional PGM tools [3]. First, we present a means of formulating and solving graph colouring problems with probabilistic graphical models. In contrast to the prevailing literature that mostly uses factor graph configurations, we approach it from a cluster graph perspective, using the general-purpose cluster graph construction algorithm, LTRIP. Our experiments indicate a significant advantage for preferring cluster graphs over factor graphs, both in terms of accuracy as well as computational efficiency. Secondly, we use these tools to solve a practical problem: land cover classification. This process is complex due to measuring errors, inefficient algorithms, and low-quality data. We proposed a PGM approach to boost geospatial classifications from different sources and consider the effects of spatial distribution and inter-class dependencies (similarly to graph colouring). Our PGM tools were shown to be robust and were able to produce a diverse, feasible, and spatially-consistent land cover classification even in areas of incomplete and conflicting evidence. Lastly, in our third publication, we investigated and improved the PGM structures used for constraint satisfaction. It is known that tree-structured PGMs always result in an exact solution [12, p355], but is usually impractical for interesting problems due to exponential blow-up. We, therefore, developed the “purge-and merge” algorithm to incrementally approximate a tree-structured PGM. This algorithm iteratively nudges a malleable graph structure towards a tree structure by selectively merging factors. The merging process is designed to avoid exponential blow-up through sparse data structures from which redundancy is purged as the algorithm progresses. This algorithm is tested on constraint satisfaction puzzles such as Sudoku, Fill-a-pix, and Kakuro and manages to outperform other PGM-based approaches reported in the literature [13, 14, 15]. Overall, the research reported in this dissertation contributed to developing a more optimised approach for higher order probabilistic graphical models. Further studies should concentrate on applying purge-and-merge on problems closer to probabilistic reasoning than constraint satisfaction and report its effectiveness in that domain.AFRIKAANSE OPSOMMING: Grafiese waarskynlikheidsmodelle (PGM) word wyd gebruik vir komplekse waarskynlikheidsprobleme. Dit is kragtige gereedskap om sisteme van komplekse verhoudings oor ‘n versameling waarskynlikheidsverspreidings op te los, soos die mediese en foutdiagnoses, voorspellingsmodelle, objekherkenning, lokalisering en kartering, spraakherkenning en taalprosessering [5, 6, 7, 8, 9, 10, 11]. Voorts kan beperkingvoldoeningsprobleme (CSP) as PGM’s geformuleer word en met PGM gevolgtrekkingtegnieke opgelos word. Die heersende literatuur oor PGM’s toon egter dat sub-optimale PGM-strukture hoofsaaklik in die praktyk gebruik word en ‘n sub-optimale PGM-formulering vir CSP’s. Die doel met die verhandeling is om die PGM-literatuur deur toeganklike algoritmes en gereedskap vir verbeterde PGM-strukture en gevolgtrekking-prosedures te verbeter deur op CSP toepassings te fokus. Na aanleiding hiervan voeg die verhandeling drie gepubliseerde bydraes by die huidige literatuur: ‘n vergelykende studie om bundelgrafieke tot die heersende faktorgrafieke te vergelyk [1], ‘n praktiese toepassing vir die gebruik van bundelgrafieke in “land-cover”- klassifikasie in die kartografieveld [2] en ‘n omvattende integrasie van verskeie aspekte om CSP’s as PGM’s te formuleer en ‘n algoritme vir die formulering van probleme te kompleks vir tradisionele PGM-gereedskap [3] Eerstens bied ons ‘n wyse van formulering en die oplos van grafiekkleurprobleme met PGM’s. In teenstelling met die huidige literatuur wat meestal faktorgrafieke gebruik, benader ons dit van ‘n bundelgrafiek-perspektief deur die gebruik van die automatiese bundelgrafiekkonstruksie-algoritme, LTRIP. Ons eksperimente toon ‘n beduidende voorkeur vir bundelgrafieke teenoor faktorgrafieke, wat akku raatheid asook berekende doeltreffendheid betref. Tweedens gebruik ons die gereedskap om ‘n praktiese probleem op te los: “landcover”-klassifikasie. Die proses is kompleks weens metingsfoute, ondoeltreffende algoritmes en lae-gehalte data. Ons stel ‘n PGM-benadering voor om die georuimtelike klassifikasies van verskillende bronne te versterk, asook die uitwerking van ruimtelike verspreiding en interklas-afhanklikhede (soortgelyk aan grafiekkleurprobleme). Ons PGM-gereedskap is robuus en kon ‘n diverse, uitvoerbare en ruimtelik-konsekwente “land-cover”-klassifikasie selfs in gebiede van onvoltooide en konflikterende inligting bewys. Ten slotte het ons in ons derde publikasie die PGM-strukture vir CSP’s ondersoek en verbeter. Dit is bekend dat boomstrukture altyd tot ‘n eksakte oplossing lei [12, p355], maar is weens eksponensiële uitbreiding gewoonlik onprakties vir interessante probleme. Ons het gevolglik die algoritme, purge-and-merge, ontwikkel om inkrementeel ‘n boomstruktuur na te doen. Die algoritme hervorm ‘n bundelgrafiek stapsgewys in ‘n boomstruktuur deur faktore selektief te “merge”. Die saamsmeltproses is ontwerp om eksponensiële uitbreiding te vermy deur van yl datastrukture gebruik te maak waarvan die waarskeinlikheidsruimte ge-“purge” word namate die algoritme vorder. Die algoritme is getoets op CSP-speletjies soos Sudoku, Fill-a-pix en Kakuro en oortref ander PGM-gegronde benaderings waaroor in die literatuur verslag gedoen word [13, 14, 15]. In die geheel gesien, het die navorsing bygedra tot die ontwikkeling van ‘n meer geoptimaliseerde benadering vir hoër-orde PGM’s. Verdere studies behoort te fokus op die toepassing van purge-and-merge op probleme nader aan waarskynlikheidsredenasie-probleme as aan CSP’s en moet sy effektiwiteit in daar die domein rapporteer.Doctora

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