Context: The main purpose of this thesis is to build an automatic handwritten digit recognition method for the recognition of connected handwritten digit strings. To accomplish the recognition task, first, the digits were segmented into individual digits. Then, a digit recognition module is employed to classify each segmented digit completing the handwritten digit string recognition task. In this study, different machine learning methods, which are SVM, ANN and CNN architectures are used to achieve high performance on the digit string recognition problem. In these methods, images of digit strings are trained with the SVM, ANN and CNN model with HOG feature vectors and Deep learning methods structure by sliding a fixed size window through the images labeling each sub-image as a part of a digit or not. After the completion of the segmentation, to achieve the complete recognition of handwritten digits.Objective: The main purpose of this thesis is to find out the recognition performance of the methods. In order to analyze the performance of the methods, data is needed to be used for training using machine learning methods. Then digit data is tested on the desired machine learning technique. In this thesis, the following methods are performed: Implementation of HOG Feature extraction method with SVM Implementation of HOG Feature extraction method with ANN Implementation of Deep Learning methods with CNN Methods: This research will be carried out using two methods. The first research method is the ¨Literature Review¨ and the second ¨Experiment¨. Initially, a literature review is conducted to get a clear knowledge on the algorithms and techniques which will be used to answer the first research question i.e., to know which type of data is required for the machine learning methods and the data analysis is performed. Later on, with the knowledge of RQ1, Experimentation is conducted to answer the RQ2, RQ3, RQ4. Quantitative data is used to perform the experimentation because qualitative data which obtains from case-study and survey cannot be used for this experiment method as it contains non-numerical data. In this research, an experiment is conducted to find the best suitable machine learning method from the existing methods. As mentioned above in the objectives, an experiment is conducted using SVM, ANN, and CNN. By considering the results obtained from the experiment, a comparison is made on the metrics considered which results in CNN as the best method suitable for Documents Images. Results: Compare the results for SVM, ANN with HOG Feature extraction and the CNN method by using segmented results. Based on the Experiment results it is found that SVM and ANN have some drawbacks like low accuracy and low performance in the recognition of documented images. So, the other method i.e., CNN has greater performance with high accuracy. The following are the results of the recognition rates of each method. SVM performance - 39% ANN performance - 37% CNN performance - 71%. Conclusion: This research concentrates on providing an efficient method for recognition of automatic handwritten digits recognition. Here a sample training data is treated with existing machine learning and deep learning methods like SVM, ANN, and CNN. By the results obtained from the experimentation, it clearly is shown that the CNN method is much efficient with 71% performance when compared to ANN and SVM methods. Keywords: Handwritten Digit Recognition, Handwritten Digit Segmentation, Handwritten Digit Classification, Machine Learning Methods, Deep Learning, Image processing on document images, Support Vector Machine, Conventional Neural Networks, Artificial Neural Networks

In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.