Le démélange de sources pour la pollution de l'air peut être formulé comme un problème de NMF en décomposant la matrice d'observation X en le produit de deux matrices non négatives G et F, respectivement la matrice de contributions et de profils. Généralement, les données chimiques sont entâchées d'une part de données aberrantes. En dépit de l'intérêt de la communauté pour les méthodes de NMF, elles souffrent d'un manque de robustesse à un faible nombre de données aberrantes et aux conditions initiales et elles fournissent habituellement de multiples minimas. En conséquence, cette thèse est orientée d'une part vers les méthodes de NMF robustes et d'autre part vers les NMF informées qui utilisent une connaissance experte particulière. Deux types de connaissances sont introduites dans la matrice de profil F. La première hypothèse est la connaissance exacte de certaines composantes de la matrice F tandis que la deuxième information utilise la propriété de somme-à-1 de chaque ligne de la matrice F. Une paramétrisation qui tient compte de ces deux informations est développée et des règles de mise à jour dans le sous-espace des contraintes sont proposées. L'application cible qui consiste à identifier les sources de particules dans l'air dans la région côtière du nord de la France montre la pertinence des méthodes proposées. Dans la série d'expériences menées sur des données synthétiques et réelles, l'effet et la pertinence des différentes informations sont mises en évidence et rendent les résultats de factorisation plus fiables. / Source apportionment for air pollution may be formulated as a NMF problem by decomposing the data matrix X into a matrix product of two factors G and F, respectively the contribution matrix and the profile matrix. Usually, chemical data are corrupted with a significant proportion of abnormal data. Despite the interest for the community for NMF methods, they suffer from a lack of robustness to a few abnormal data and to initial conditions and they generally provide multiple minima. To this end, this thesis is oriented on one hand towards robust NMF methods and on the other hand on informed NMF by using some specific prior knowledge. Two types of knowlodge are introduced on the profile matrix F. The first assumption is the exact knowledge on some of flexible components of matrix F and the second hypothesis is the sum-to-1 constraint on each row of the matrix F. A parametrization able to deal with both information is developed and update rules are proposed in the space of constraints at each iteration. These formulations have been appliede to two kind of robust cost functions, namely, the weighted Huber cost function and the weighted αβ divergence. The target application-namely, identify the sources of particulate matter in the air in the coastal area of northern France - shows relevance of the proposed methods. In the numerous experiments conducted on both synthetic and real data, the effect and the relevance of the different information is highlighted to make the factorization results more reliable.

A recommender system is a popular tool used by companies to increase customer satisfaction and to increase revenue. Collaborative filtering and content-based filtering are the two most common approaches when implementing a recommender system, where the former provides recommendations based on user behaviour, and the latter uses the characteristics of the items that are recommended. The aim of the study was to develop and compare different recommender system approaches, for both book and author recommendations and their ability to predict user ratings of an e-book application. The evaluation of the models was done by measuring root mean square error (RMSE) and mean absolute error (MAE). Two pure models were developed, one based on collaborative filtering and one based on content-based filtering. Also, three different hybrid models using a combination of the two pure approaches were developed and compared to the pure models. The study also explored how aggregation of book data to author level could be used to implement an author recommender system. The results showed that the aggregated author data was more difficult to predict. However, it was difficult to draw any conclusions of the performance on author data due to the data aggregation. Although it was clear that it was possible to derive author recommendations based on data from books. The study also showed that the collaborative filtering model performed better than the content-based filtering model according to RMSE but not according to MAE. The lowest RMSE and MAE, however, were achieved by combining the two approaches in a hybrid model.

Recommender systems are an integral part of almost all modern e-commerce companies. They contribute significantly to the overall customer satisfaction by helping the user discover new and relevant items, which consequently leads to higher sales and stronger customer retention. It is, therefore, not surprising that large e-commerce shops like Amazon or streaming platforms like Netflix and Spotify even use multiple recommender systems to further increase user engagement. Finding the most relevant items for each user is a difficult task that is critically dependent on the available user feedback information. However, most users typically interact with products only through noisy implicit feedback, such as clicks or purchases, rather than providing explicit information about their preferences, such as product ratings. This usually makes large amounts of behavioural user data necessary to infer accurate user preferences. One popular approach to make the most use of both forms of feedback is called collaborative filtering. Here, the main idea is to compare individual user behaviour with the behaviour of all known users. Although there are many different collaborative filtering techniques, matrix factorization models are among the most successful ones. In contrast, while neural networks are nowadays the state-of-the-art method for tasks such as image recognition or natural language processing, they are still not very popular for collaborative filtering tasks. Therefore, the main focus of this thesis is the derivation of multiple wide neural network architectures to mimic and extend matrix factorization models for various collaborative filtering problems and to gain insights into the connection between these models. The basics of the proposed architecture are wide and shallow feedforward neural networks, which will be established for rating prediction tasks on explicit feedback datasets. These networks consist of large input and output layers, which allow them to capture user and item representation similar to matrix factorization models. By deriving all weight updates and comparing the structure of both models, it is proven that a simplified version of the proposed network can mimic common matrix factorization models: a result that has not been shown, as far as we know, in this form before. Additionally, various extensions are thoroughly evaluated. The new findings of this evaluation can also easily be transferred to other matrix factorization models. This neural network architecture can be extended to be used for personalized ranking tasks on implicit feedback datasets. For these problems, it is necessary to rank products according to individual preferences using only the provided implicit feedback. One of the most successful and influential approaches for personalized ranking tasks is Bayesian Personalized Ranking, which attempts to learn pairwise item rankings and can also be used in combination with matrix factorization models. It is shown, how the introduction of an additional ranking layer forces the network to learn pairwise item rankings. In addition, similarities between this novel neural network architecture and a matrix factorization model trained with Bayesian Personalized Ranking are proven. To the best of our knowledge, this is the first time that these connections have been shown. The state-of-the-art performance of this network is demonstrated in a detailed evaluation. The most comprehensive feedback datasets consist of a mixture of explicit as well as implicit feedback information. Here, the goal is to predict if a user will like an item, similar to rating prediction tasks, even if this user has never given any explicit feedback at all: a problem, that has not been covered by the collaborative filtering literature yet. The network to solve this task is composed out of two networks: one for the explicit and one for the implicit feedback. Additional item features are learned using the implicit feedback, which capture all information necessary to rank items. Afterwards, these features are used to improve the explicit feedback prediction. Both parts of this combined network have different optimization goals, are trained simultaneously and, therefore, influence each other. A detailed evaluation shows that this approach is helpful to improve the network's overall predictive performance especially for ranking metrics.

We report on some experience with a new version of the well known Gröbner algorithm with factorization and constraint inequalities, implemented in our REDUCE package CALI, [12]. We discuss some of its details and present run time comparisons with other existing implementations on well splitting examples.

An n × n matrix factorization of a polynomial f is a pair of n × n matrices (P, Q) such that PQ = f In, where In is the n × n identity matrix. In this dissertation, we study matrix factorizations of an arbitrary element in a given unital ring. This study is motivated on the one hand by the construction of the unit object in the bicategory LGK of Landau-Ginzburg models (of great utility in quantum physics) whose 1−cells are matrix factorizations of polynomials over a commutative ring K, and on the other hand by the existing tensor product of matrix factorizations b⊗. We observe that the pair of n × n matrices that appear in the matrix factorization of an element in a unital ring is not unique. Next, we propose a new operation on matrix factorizations denoted e⊗ which is such that if X is a matrix factorization of an element f in a unital ring (e.g. the power series ring K[[x1, ..., xr]] f) and Y is a matrix factorization of an element g in a unital ring (e.g. g ∈ K[[y1, ..., ys]]), then Xe⊗Y is a matrix factorization of f g in a certain unital ring (e.g. in case f ∈ K[[x1, ..., xr]] and g ∈ K[[y1, ..., ys]], then f g ∈ K[[x1, ..., xr , y1, ..., ys]]). e⊗ is called the multiplicative tensor product of X and Y. After proving that this product is bifunctorial, many of its properties are also stated and proved. Furthermore, if MF(1) denotes the category of matrix factorizations of the constant power series 1, we define the concept of one-step connected category and prove that there is a one-step connected subcategory of (MF(1),e⊗) which is semi-unital semi-monoidal. We also define the concept of right pseudo-monoidal category which generalizes the notion of monoidal category and we prove that (MF(1),e⊗) is an example of this concept. Furthermore, we define a summand-reducible polynomial to be one that can be written in the form f = t1 + · · · + ts + g11 · · · g1m1 + · · · + gl1 · · · glml under some specified conditions where each tk is a monomial and each gji is a sum of monomials. We then use b⊗ and e⊗ to improve the standard method for matrix factorization of polynomials on this class and we prove that if pji is the number of monomials in gji, then there is an improved version of the standard method for factoring f which produces factorizations of size 2 Qm1 i=1 p1i+···+ Qml i=1 pli−( Pm1 i=1 p1i+···+ Pml i=1 pli) times smaller than the size one would normally obtain with the standard method. Moreover, details are given to elucidate the intricate construction of the unit object of LGK. Thereafter, a proof of the naturality of the right and left unit maps of LGK with respect to 2−morphisms is presented. We also prove that there is no direct inverse for these (right and left) unit maps, thereby justifying the fact that their inverses are found only up to homotopy. Finally, some properties of matrix factorizations are exploited to state and prove a necessary condition to obtain a Morita context in LGK.

This paper will examine observables in the bc system, a two-dimensional free conformal field theory. We begin by encoding the bc system into the BV formalism following procedures of Costello and Gwilliam. This will allow us to construct the factorization algebra of observables for the bc system. The cohomology of the factorization algebra recovers the observables themselves. In cohomology, we will compute the commutation relations and factorization algebra structure maps for observables supported on disks and annuli. These structure maps will be used to prove the equivalence of the factorization algebra and vertex algebra structures for the bc system. This proof provides a rigorous derivation of the free fermionic vertex algebra starting from the action functional of the bc system. Using this equivalence, we will provide a dictionary to translate the action of the Virasoro algebra to the language of factorization algebras. Also in this paper, we examine the bc system in four-dimensions. We construct its factorization algebra and show that its observables are anti-commutative. Lastly, we prove that the global observables of the bc system are one-dimensional on a compact manifold of complex dimension one or two.

A latent factor model for count data is popularly applied when deconvoluting mixed signals in biological data as exemplified by sequencing data for transcriptome or microbiome studies. Due to the availability of pure samples such as single-cell transcriptome data, the estimators can enjoy much better accuracy by utilizing the extra information. However, such an advantage quickly disappears in the presence of excessive zeros. To correctly account for such a phenomenon, we propose a zero-inflated non-negative matrix factorization that models excessive zeros in both mixed and pure samples and derive an effective multiplicative parameter updating rule. In simulation studies, our method yields smaller bias comparing to other deconvolution methods. We applied our approach to gene expression from brain tissue as well as fecal microbiome datasets, illustrating the superior performance of the approach. Our method is implemented as a publicly available R-package, iNMF. In zero-inflated non-negative matrix factorization (iNMF) for the deconvolution of mixed signals of biological data, pure-samples play a significant role by solving the identifiability issue as well as improving the accuracy of estimates. One of the main issues of using single-cell data is that the identities(labels) of the cells are not given. Thus, it is crucial to sort these cells into their correct types computationally. We propose a nonlinear latent variable model that can be used for sorting pure-samples as well as grouping mixed-samples via deep neural networks. The computational difficulty will be handled by adopting a method known as variational autoencoding. While doing so, we keep the NMF structure in a decoder neural network, which makes the output of the network interpretable.

In this thesis a heuristic method for factoring semiprimes by multiagent depth-limited search of PG2N graphs is presented. An analysis of PG2N graph connectivity is used to generate heuristics for multiagent search. Further analysis is presented including the requirements on choosing prime numbers to generate 'hard' semiprimes; the lack of connectivity in PG1N graphs; the counts of spanning trees in PG2N graphs; the upper bound of a PG2N graph diameter and a conjecture on the frequency distribution of prime numbers on Hamming distance. We further demonstrated the feasibility of the HD2 breadth first search of PG2N graphs for factoring small semiprimes. We presented the performance of different multiagent search heuristics in PG2N graphs showing that the heuristic of most connected seedpick outperforms least connected or random connected seedpick heuristics on small PG2N graphs of size N

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第20737号 / 情博第651号 / 新制||情||112(附属図書館) / 京都大学大学院情報学研究科社会情報学専攻 / (主査)教授 吉川 正俊, 教授 石田 亨, 教授 美濃 導彦, 准教授 馬 強 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(人間健康科学) / 甲第21040号 / 人健博第56号 / 新制||人健||4(附属図書館) / 京都大学大学院医学研究科人間健康科学系専攻 / (主査)教授 三谷 章, 教授 澤本 伸克, 教授 髙橋 良輔 / 学位規則第4条第1項該当 / Doctor of Human Health Sciences / Kyoto University / DFAM