This dissertation introduces a novel model for solving the L2 support vector machine dubbed Direct L2 Support Vector Machine (DL2 SVM). DL2 SVM represents a new classification model that transforms the SVM's underlying quadratic programming problem into a system of linear equations with nonnegativity constraints. The devised system of linear equations has a symmetric positive definite matrix and a solution vector has to be nonnegative. Furthermore, this dissertation introduces a novel algorithm dubbed Non-Negative Iterative Single Data Algorithm (NN ISDA) which solves the underlying DL2 SVM's constrained system of equations. This solver shows significant speedup compared to several other state-of-the-art algorithms. The training time improvement is achieved at no cost, in other words, the accuracy is kept at the same level. All the experiments that support this claim were conducted on various datasets within the strict double cross-validation scheme. DL2 SVM solved with NN ISDA has faster training time on both medium and large datasets. In addition to a comprehensive DL2 SVM model we introduce and derive its three variants. Three different solvers for the DL2's system of linear equations with nonnegativity constraints were implemented, presented and compared in this dissertation.

In this thesis, we propose a novel hyper-rectangle cover theory which provides a new approach to analyzing mathematical problems with nonnegativity constraints on variables. In this theory, two fundamental concepts, cover order and cover length, are introduced and studied in details. In the same manner as determining the rank of a matrix, we construct a specific e ́chelon form of the matrix to obtain the cover order of a given matrix efficiently and effectively. We discuss various structures of the e ́chelon form for some special cases in detail. Based on the structure and properties of the constructed e ́chelon form, the concepts of non-negatively linear independence and non-negatively linear dependence are developed. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with nonnegativity constraints on variables for both homogeneous and non-homogeneous cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with nonnegativity constraints on variables, including linear programming problems and non-negative least squares (NNLS) problems. For linear programming problem, we study the three possible behaviors of the solutions for it through hyper-rectangle cover theory, and show that a series of feasible solutions for the problem with the zero-cover e ́chelon form structure. On the other hand, we develop a method to obtain the cover length of the covered variable. In the process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem. / Thesis / Doctor of Philosophy (PhD)

De nombreux domaines applicatifs conduisent à résoudre des problèmes inverses où le signal ou l'image à reconstruire est à la fois parcimonieux et positif. Si la structure de certains algorithmes de reconstruction parcimonieuse s'adapte directement pour traiter les contraintes de positivité, il n'en va pas de même des algorithmes gloutons orthogonaux comme OMP et OLS. Leur extension positive pose des problèmes d'implémentation car les sous-problèmes de moindres carrés positifs à résoudre ne possèdent pas de solution explicite. Dans la littérature, les algorithmes gloutons positifs (NNOG, pour “Non-Negative Orthogonal Greedy algorithms”) sont souvent considérés comme lents, et les implémentations récemment proposées exploitent des schémas récursifs approchés pour compenser cette lenteur. Dans ce manuscrit, les algorithmes NNOG sont vus comme des heuristiques pour résoudre le problème de minimisation L0 sous contrainte de positivité. La première contribution est de montrer que ce problème est NP-difficile. Deuxièmement, nous dressons un panorama unifié des algorithmes NNOG et proposons une implémentation exacte et rapide basée sur la méthode des contraintes actives avec démarrage à chaud pour résoudre les sous-problèmes de moindres carrés positifs. Cette implémentation réduit considérablement le coût des algorithmes NNOG et s'avère avantageuse par rapport aux schémas approximatifs existants. La troisième contribution consiste en une analyse de reconstruction exacte en K étapes du support d'une représentation K-parcimonieuse par les algorithmes NNOG lorsque la cohérence mutuelle du dictionnaire est inférieure à 1/(2K-1). C'est la première analyse de ce type. / Non-negative sparse approximation arises in many applications fields such as biomedical engineering, fluid mechanics, astrophysics, and remote sensing. Some classical sparse algorithms can be straightforwardly adapted to deal with non-negativity constraints. On the contrary, the non-negative extension of orthogonal greedy algorithms is a challenging issue since the unconstrained least square subproblems are replaced by non-negative least squares subproblems which do not have closed-form solutions. In the literature, non-negative orthogonal greedy (NNOG) algorithms are often considered to be slow. Moreover, some recent works exploit approximate schemes to derive efficient recursive implementations. In this thesis, NNOG algorithms are introduced as heuristic solvers dedicated to L0 minimization under non-negativity constraints. It is first shown that the latter L0 minimization problem is NP-hard. The second contribution is to propose a unified framework on NNOG algorithms together with an exact and fast implementation, where the non-negative least-square subproblems are solved using the active-set algorithm with warm start initialisation. The proposed implementation significantly reduces the cost of NNOG algorithms and appears to be more advantageous than existing approximate schemes. The third contribution consists of a unified K-step exact support recovery analysis of NNOG algorithms when the mutual coherence of the dictionary is lower than 1/(2K-1). This is the first analysis of this kind.