Sampling methods for the Nyström method
The Journal of Machine Learning Research
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In dimensionality reduction, a set of points in 1Rd is mapped into III, with the target dimension k smaller than the original dimension d, while distances between all pairs of points are approximately preserved. Currently popular methods for achieving this involve random projection, or choosing a linear mapping (a k × d matrix) front a distribution that is independent of the input points. Applying the mapping (chosen according to this distribution) is shown to give the desired property with at least constant probability. The contributions in this thesis are twofold. First, we provide a framework for designing such distributions. Second, we derive efficient random projection algorithms using this framework. Our results achieve performance exceeding other existing approaches. When the target dimension is significantly smaller than the original dimension we gain significant improvement by designing efficient algorithms for applying certain linear algebraic transforms. To demonstrate practicality, we supplement the theoretical derivations with experimental results.