Abstract:
This dissertation consists of two main contributions. The first contribution focuses on developing a new framework of kernelizing Mahalanobis distance learners. The new KPCA trick framework offers several practical advantages over the classical kernel trick framework, e.g. no mathematical formulas and no reprogramming are required for a kernel implementation, a way to speed up an algorithm is provided with no extra work, the framework avoids troublesome problems such as singularity. Rigorous representer theorems in countably-infinite dimensional spaces are given to validate our framework. Furthermore, unlike previous works which always apply brute force methods to select a kernel, we derive a kernel alignment formula based on quadratic programming which can efficiently construct an appropriate kernel for a given dataset. In the second contribution, we present a general framework of semi-supervised dimensionality reduction for manifold learning which naturally generalizes existing supervised and unsupervised learning frameworks which apply the spectral decomposition. Algorithms derived under our framework are able to employ both labeled and unlabeled examples and are able to handle complex problems where data form separate clusters of manifolds. Our framework offers simple views, explains relationships among existing frameworks and provides further extensions which can improve existing algorithms.
