Theoretical Results and Applications Related to Dimension Reduction

To overcome the curse of dimensionality, dimension reduction is important andnecessary for understanding the underlying phenomena in a variety of fields.Dimension reduction is the transformation of high-dimensional data into ameaningful representation in the low-dimensional space. It can be furtherclassified into feature selection and feature extraction. In this thesis, whichis composed of four projects, the first two focus on feature selection, and thelast two concentrate on feature extraction.The content of the thesis is as follows. The first project presents severalefficient methods for the sparse representation of a multiple measurementvector (MMV); some theoretical properties of the algorithms are also discussed.The second project introduces the NP-hardness problem for penalized likelihoodestimators, including penalized least squares estimators, penalized leastabsolute deviation regression and penalized support vector machines. The thirdproject focuses on the application of manifold learning in the analysis andprediction of 24-hour electricity price curves. The last project proposes a newhessian regularized nonlinear time-series model for prediction in time series.