Over the last decade, an increasing amount of data has become available for data analysts to understand. Datasets containing books, images, networks, or other types of data have been studied. A recent group of methods proposes to analyze samples in datasets based on a description of their shape. This group of methods is often referred to as Topological Data Analysis (TDA). In this thesis, an extension to the most commonly used TDA method, called Persistent Homology (PH), is proposed. PH only describes topological features, while this extension additionally allows for the description of geometric properties. The new information is obtained via the persistent Laplacian, a recently proposed operator that encodes the topological information of persistent homology in its kernel and geometric information in its non-zero spectrum. The persistent Laplacian contains a lot of information and extracting the relevant parts has not yet been standardized. In this thesis, a new operator, the persistent multiplicity operator, is proposed. The new operator summarizes the information of the persistent Laplacian such that it can easily be extracted and used to extend PH. This allows the many previously studied methods based on persistent homology to additionally describe geometric properties, as opposed to only topological features. For the multiplicity operator, the trace is analyzed and the features captured by it are discussed. Besides analyzing the sum of the eigenvalues, it is argued that individual eigenvalues could contain more information. However, these are deemed hard to understand. Therefore, an adjusted multiplicity operator is proposed that contains separately interpretable eigenvalues. Finally, the operators are used to classify handwritten digits from the MNIST dataset and to make statistical tests that can detect different generation processes of artificially made cross sections of crystalline structures.

Neural network is an active research field which involves many different (unsolved) issues, for example, different types of configuration of the network architectures, training strategies, etc. Amongst these active issues, the choice of loss (or cost) functions plays an important role in how a neural network model is to be optimized (trained) and how the model will perform after the training. Given the choice of measurement criteria, loss functions measure how far an estimated output is from its true value. And the measurement criteria can change depending on the task in hand and the goal to be met. The objective of this project is to understand the role of different loss functions and to evaluate the dependence of the performance on the loss functions using the language prediction problem.