Dimension reduction techniques for multi-dimensional numerical integrations based on Fourier-cosine series expansion

A wide range of practical problems involve computing multi-dimensional integrations. However, in most cases, it is hard to find analytical solutions to these multi-dimensional integrations. Their numerical solutions always suffer from the `curse of dimension', which means the computational complexity grows exponentially with respect to the...

All-at-once optimization for kernel machines with canonical polyadic decompositions: Enabling large scale learning for kernel machines

This thesis studies the Canonical Polyadic Decomposition (CPD) constrained kernel machine for large scale learning, i.e. learning with a large number of samples. The kernel machine optimization problem is solved in the primal space, such that the complexity of the problem scales linearly in the number of samples as opposed to scaling cubically...

Tensor decomposition for Independent Component Analysis: Through implicit cumulant tensor manipulation

Blind Source Separation (BSS), the separation of latent source components from observed mixtures, is relevant to many fields of expertise such as neuro-imaging, economics and machine learning. Reliable estimates of the sources can be obtained through diagonalization of the cumulant tensor, i.e., a fourth-order symmetric multi-linear array...

Solving multivariate expectations using dimension-reduced fourier-cosine series expansion and its application in finance

The computation of multivariate expectations is a common task in various fields related to probability theory. This thesis aims to develop a generic and efficient solver for multivariate expectation problems, with a focus on its application in the field of quantitative finance, specifically for the quantification of Counterparty Credit Risk (CCR...

Uncertainty quantification for tensor network constrained kernel machines: A frequentist and Bayesian approach

This research aims at quantifying the uncertainty in the predictions of tensor network constrained kernel machines, focusing on the Canonical Polyadic Decomposition (CPD) constrained kernel machine. Constraining the parameters in the kernel machine optimization problem to be a CPD results in a linear computational complexity in the number of...

A novel cosine network for pricing European and barrier options under stochastic volatility models

This thesis presents a comprehensive exploration of the rough Heston model as a means to enhance financial derivative pricing and calibration in the context of the complex behavior of market volatility. Recognizing the limitations of classical models, such as the Black-Scholes and the standard Heston model, which assume constant or mean...