Κατανεμημένη τανυστική αποδόμηση στη μηχανική μάθηση για πλήρως ή μερικώς παρατηρούμενα πολυδιάστατα δεδομένα μεγάλου όγκου

In this dissertation, we propose a basic algorithm (GenProxSGD) for calculating a CANDECOMP/PARAFAC (CP) decomposition from partially observed data, which is based on the proximal operator. This algorithm deals with the decomposition’s optimization problem by solving local optimization problems, in the sense that in each iteration a proximal to the previous point is calculated that leads to the minimum of the objective function. Furthermore, we propose two distributed algorithms (ParallelProxSGD and StrProxSGD) that are based on the proximal operator and GenProxSGD for calculating a CP decomposition from data with missing entries. These algorithms are suitable for big data. These algorithms are assessed on big data and by experiments we show that possess great scaling properties in respect to different parameters like dimensionality, decomposition rank, speed-up etc. Subsequently, we propose a classification algorithm for Multiple Instance Learning (MIL) called TensMIL. In TensMIL we propose a new representation of instances by a tensor of order 3 and a method for instance-level features extraction by a CP decomposition, from fully or partially observed data. TensMIL is developed in two phases. In the first phase, we calculate the responses of the instances by a non-linear regression and in the second phase the distributions of the instances’ responses, within each bag, are calculated and a QDA classifier is trained. By TensMIL we solved two problems that are of totally different nature: (1) classification of histopathological colour images in benign and malignant and (2) estimation of the frailty index of elderly people by using multidimensional monitoring physiological signals of all day activities, that comprise signals from accelerometers, electrocardiogram (ECG) and breath monitoring. In both situations we assessed our algorithm in fully and partially (90% missing values) observed data. Our experiments showed, that our method achieved comparable performance when compared with state-of-the-art MIL algorithms. Furthermore, we extended and improved TensMIL by adding non-negativity constraints in the CP decomposition and by adding a phase of instance selection based on the confidence intervals of the responses of the non-linear regression. This new algorithm was assessed in the problem of the classification of natural images of wild animals, in the dataset Tiger, Fox, Elephant, a benchmark dataset in the literature of Multiple Instance Learning. By experiments, we showed that TensMIL2 has better performance than other state-of-the-art Multiple Instance Learning algorithms, as well as from Multiple Instance Learning algorithms that are based on deep learning algorithms. We assessed TensMIL2 on fully and partially observed data (90% missing values). By experiments, we showed that, in the majority of the cases investigated, TensMIL2, when applied to partially observed data, had a better or comparable performance we compared with state-of-the-art algorithms applied on fully observed data. Furthermore, we propose different techniques for instance selection for Multiple Instance Learning Algorithms. We propose a technique based on the information entropy measure and a technique based on the quality of clustering that is incorporated in JC2MIL algorithm, which was not equipped with an instance selection phase in the first place. Finally, we propose a new process for extracting multi-scale instances and features by a series of CP decompositions. The assessment of these techniques was done by using as reference algorithm JC2MIL. By experiments, we showed that in the majority of the investigated settings, the proposed techniques for instance selection have increased the classification performance of JC2MIL.