Semi-inner-products in Banach Spaces with Applications to Regularized Learning, Sampling, and Sparse Approximation
Technical Report,01 May 2012,31 Dec 2015
University of Michigan - Ann Arbor Ann Arbor United States
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The goal of this project is to fully develop Banach space methods for kernel-based machine learning that extend the Hilbert space framework of regularized learning. We proposed to study Reproducing Kernel Banach Spaces RKBS by the semi-inner-product, develop the theory of vector-valued RKBS with applications of RKBS to manifold learning, study frames and Riesz bases for sequence spaces, and construct RKBS with the l1-norm known to enforce sparse solutions. We will also explore classification algorithms that are mathematically rigorous while rooted in human cognitive principles for categorization. Our execution plan include three specific topics Aims 1. Apply RKBS theory to Orlicz space, to perform convergence analysis, and to study Shannon sampling schemes 2. Work out vector-valued RKBS, and study s.i.p with l1 norm 3. Develop frames and Riesz bases for Banach spaces, and extend analysis and synthesis operators.
- Numerical Mathematics
- Theoretical Mathematics