Dimensionality reduction is a long standing challenging problem in the fields of statistical learning, pattern recognition and computer vision. Numerous algorithms have been proposed and studied in the past decades. In this dissertation we address several challenging problems emerged in recent studies of dimensionality reduction. We first explore the dimensionality reduction method for semi-supervised classification via the idea of mixed label propagation in which we attempt to find the best one dimensional embedding of the data in which data points in different classes can be well separated and the class labels are obtained by simply thresholding the one dimensional representation. In the next, we explore the dimensionality reduction methods for non-vector data representations. We first look into the problem in which a datum is represented by a matrix. We give a convex formulation to the problem of dimensionality reduction for matrices and developed an efficient approximating algorithm to solve the associated semi-definite programming problem. In the last, we studied the problem ofdimensionality reduction with even more challenging data representation, i.e., each datum is described as a different numberof unordered vectors. We convert the problem into the functional data dimensionality reduction and study it in the context of large scale image retrieval.
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