Thesis Defence: Tensor-Variate Spatially Constrained Gaussian Mixture Models

Hanzhang Lu, supervised by Dr. Jeff Andrews and Dr. Xiaoping Shi, will defend their thesis titled “Tensor-Variate Spatially Constrained Gaussian Mixture Models” in partial fulfillment of the requirements for the degree of Master of Science in Mathematics.

An abstract for Hanzhang Lu’s thesis is included below.

Defences are open to all members of the campus community as well as the general public. Registration is not required for in-person defences.

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Abstract

This thesis presents a tensor-variate spatially constrained Gaussian mixture model that can be employed to perform model-based clustering while estimating spatial patterns in the data. As data becomes increasingly abundant and complex, the demand for effective clustering methodologies grows. Unsupervised learning techniques, particularly clustering, seek to uncover hidden patterns without labeled examples. Model-based clustering offers a mathematically rigorous approach, assigning probabilities for cluster membership, but it struggles with the curse of dimensionality. Our proposed model effectively captures positive spatial correlation in tensor-variate data that takes advantage of a spatial coordinate system through a linear covariance structure with sigmoid decay. By applying an appropriate decomposition, this highly constrained covariance structure offers an efficient way to model spatial information while maintaining a constant number of free parameters. Additionally, the factor analyzers model is utilized to model the dependence among different spatial systems for further dimensionality reduction. We present both simulation studies and applications using Raman
spectroscopy data to demonstrate the model.