Dimensionality Reduction of Hyperspectral Imagery Using Random Projections

Hyperspectral imagery is often associated with high storage and transmission costs. Dimensionality reduction aims to reduce the time and space complexity of hyperspectral imagery by projecting data into a low-dimensional space such that all the important information in the data is preserved. Dimensionality-reduction methods based on transforms are widely used and give a data-dependent representation that is unfortunately costly to compute. Recently, there has been a growing interest in data-independent representations for dimensionality reduction; of particular prominence are random projections which are attractive due to their computational efficiency and simplicity of implementation. This dissertation concentrates on exploring the realm of computationally fast and efficient random projections by considering projections based on a random Hadamard matrix. These Hadamard-based projections are offered as an alternative to more widely used random projections based on dense Gaussian matrices. Such Hadamard matrices are then coupled with a fast singular value decomposition in order to implement a two-stage dimensionality reduction that marries the computational benefits of the data-independent random projection to the structure-capturing capability of the data-dependent singular value transform. Finally, random projections are applied in conjunction with nonnegative least squares to provide a computationally lightweight methodology for the well-known spectral-unmixing problem. Overall, it is seen that random projections offer a computationally efficient framework for dimensionality reduction that permits hyperspectral-analysis tasks such as unmixing and classification to be conducted in a lower-dimensional space without sacrificing analysis performance while reducing computational costs significantly

A geometric approach to archetypal analysis and non-negative matrix factorization

Archetypal analysis and non-negative matrix factorization (NMF) are staples in a statisticians toolbox for dimension reduction and exploratory data analysis. We describe a geometric approach to both NMF and archetypal analysis by interpreting both problems as finding extreme points of the data cloud. We also develop and analyze an efficient approach to finding extreme points in high dimensions. For modern massive datasets that are too large to fit on a single machine and must be stored in a distributed setting, our approach makes only a small number of passes over the data. In fact, it is possible to obtain the NMF or perform archetypal analysis with just two passes over the data.Comment: 36 pages, 13 figure

Algorithms and literate programs for weighted low-rank approximation with missing data

Linear models identification from data with missing values is posed as a weighted low-rank approximation problem with weights related to the missing values equal to zero. Alternating projections and variable projections methods for solving the resulting problem are outlined and implemented in a literate programming style, using Matlab/Octave's scripting language. The methods are evaluated on synthetic data and real data from the MovieLens data sets