Su, JiahaoNeural networks are general-purpose function approximators. Given a problem, engineers or scientists select a hypothesis space of functions with specific properties by designing the network architecture. However, mainstream designs are often ad-hoc, which could suffer from numerous undesired properties. Most prominently, the network architectures are gigantic, where most parameters are redundant while consuming computational resources. Furthermore, the learned networks are sensitive to adversarial perturbation and tend to underestimate the predictive uncertainty. We aim to understand and address these problems using spectral methods --- while these undesired properties are hard to interpret from network parameters in the original domain, we could establish their relationship when we represent the parameters in a spectral domain. These relationships allow us to design networks with certified properties via the spectral representation of parameters.enDissertationElectrical engineeringdeep learningneural networkssignal processingspectral methods