Diffusions for global optimizations
SIAM Journal on Control and Optimization
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
What is the goal of sensory coding?
Neural Computation
Independent component analysis, a new concept?
Signal Processing - Special issue on higher order statistics
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
On Advances in Statistical Modeling of Natural Images
Journal of Mathematical Imaging and Vision
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Tools for application-driven linear dimension reduction
Neurocomputing
Optimal linear representations of images for object recognition
CVPR'03 Proceedings of the 2003 IEEE computer society conference on Computer vision and pattern recognition
A Bayesian approach to geometric subspace estimation
IEEE Transactions on Signal Processing
Fast and robust fixed-point algorithms for independent component analysis
IEEE Transactions on Neural Networks
A theory for learning based on rigid bodies dynamics
IEEE Transactions on Neural Networks
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Problems involving high-dimensional data, such as pattern recognition, image analysis, and gene clustering, often require a preliminary step of dimension reduction before or during statistical analysis. If one restricts to a linear technique for dimension reduction, the remaining issue is the choice of the projection. This choice can be dictated by desire to maximize certain statistical criteria, including variance, kurtosis, sparseness, and entropy, of the projected data. Motivations for such criteria comes from past empirical studies of statistics of natural and urban images. We present a geometric framework for finding projections that are optimal for obtaining certain desired statistical properties. Our approach is to define an objective function on spaces of orthogonal linear projections--Stiefel and Grassmann manifolds, and to use gradient techniques to optimize that function. This construction uses the geometries of these manifolds to perform the optimization. Experimental results are presented to demonstrate these ideas for natural and facial images.