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【期刊论文】Reproducing kernel Banach spaces with the norm
Applied and Computational Harmonic Analysis,2013,34(1):96-116
2013年01月01日
Targeting at sparse learning, we construct Banach spaces of functions on an input space X with the following properties: (1) possesses an norm in the sense that is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on and are representable through a bilinear form with a kernel function; and (3) regularized learning schemes on satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.
Reproducing kernel Banach spaces Sparse learning Lasso Basis pursuit Regularization The representer theorem The Brownian bridge kernel The exponential kernel
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【期刊论文】Vector-valued reproducing kernel Banach spaces with applications to multi-task learning
Journal of Complexity,2013,29(2):195-215
2013年04月01日
Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBSs). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RKBSs. The theory is then applied to multi-task machine learning. Especially, the representer theorem and characterization equations for the minimizer of regularized learning schemes in vector-valued RKBSs are established.
Vector-valued reproducing kernel Banach spaces Feature maps Regularized learning The representer theorem Characterization equations
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【期刊论文】Refinement of Operator-valued Reproducing Kernels
Journal of Machine Learning Research,-0001,13(4):91−136
-1年11月30日
This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given kernel as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representations of operator-valued reproducing kernels. Concrete examples of refining translation invariant and finite Hilbert-Schmidt operator-valued reproducing kernels are provided. Other examples include refinement of Hessian of scalar-valued translation-invariant kernels and transformation kernels. Existence and properties of operator-valued reproducing kernels preserved during the refinement process are also investigated.
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【期刊论文】Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products
Applied and Computational Harmonic Analysis,2011,31(1):1-25
2011年07月01日
Frames in a Banach space were defined as a sequence in its dual space ⁎ in some recent references. We propose to define them as a collection of elements in by making use of semi-inner products. Classical theory on frames and Riesz bases is generalized under this new perspective. We then aim at establishing the Shannon sampling theorem in Banach spaces. The existence of such expansions in translation invariant reproducing kernel Hilbert and Banach spaces is discussed.
Frames Riesz bases Bessel sequences Riesz–Fischer sequences Banach spaces Semi-inner products Duality mappings Shannonʼs sampling expansions Reproducing kernel Banach spaces Reproducing kernel Hilbert spaces Gaussian kernels
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【期刊论文】Reproducing Kernel Banach Spaces for Machine Learning
The Journal of Machine Learning Research,-0001,10():2741-2775
-1年11月30日
We introduce the notion of reproducing kernel Banach spaces (RKBS) and study special semi-inner-product RKBS by making use of semi-inner-products and the duality mapping. Properties of an RKBS and its reproducing kernel are investigated. As applications, we develop in the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, support vector machines, and kernel principal component analysis. In particular, existence, uniqueness and representer theorems are established.
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