UC Davis

This dissertation comprises two research projects focused on inverse problems in graph models, each addressing a specific application. The central theme is inverse graph problems, specifically the challenge of recovering unknown network parameters that define the underlying network structure, using only nodal measurements. In both projects, we provide theoretical guarantees for parameter recovery under certain measurement conditions.

The first project, detailed in the first chapter, presents theoretical results on the inverse problem for power grids. We demonstrate that, given full access to phasor measurements and power injection data at all nodes, it is possible to recover sparse nodal susceptance based on the simplified AC power flow model. This recovery is achievable as long as the number of measurements exceeds a specified lower bound, a result that has not been extensively explored in the context of AC power grids.

The second project, presented in the second and third chapters, addresses the inverse problem of parameter learning in black box neural networks. Specifically, we investigate a layer-by-layer approach for learning parameters in single-hidden-layer feedforward neural networks, focusing on networks with rectified/monotone and binary activation functions. In addition to providing theoretical guarantees for parameter recovery, we also explore algorithm design and test various algorithms on small-scale networks.