Tensor dimensionality reduction and applications

Dimensionality reduction is a fundamental idea in data science and machine learning. Tensor is ubiquitous in modern data science due to its representation power for complex data settings. In this thesis, we study to efficiently reduce the size of tensor-structured data while preserving the essential information.

We generalize classical reduction methods for vectors, matrices, such as random projections and low-rank decompositions, to be suitable for higher-order tensors. Numerical experiments show the proposed reduction algorithms result in significant storage savings and computation speed-ups. As a trade-off, from the theoretical perspective, the approximation errors between the original and reduced data are rigorously analyzed. Finally, these tensor dimension reduction techniques find usages in solving inverse problems, anomaly detection and financial portfolio allocation.