Learning the Kernel Function via Regularization

By Charles A. Micchelli, Massimiliano Pontil. Journal of machine learning research, 2005

459 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/62e2d0cbe656a6f95dbf349d2a7158b948ce5820

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Citations in the corpus

Learning to Multitask (2018, background): “…due to the von Neumann’s trace inequality, then we can get where µ i ( · ) denotes the i th singular value of a matrix, the second inequality holds due to Lemma 26 in [22], and the equality holds when µ i ( Ω ) = µ i ( W ) where φ is the Lagrange multiplier corresponding to the equality constraint.”
More efficiency in multiple kernel learning (2007, methodology): “…block regularization is equal to the adaptive weighted 2-norm regularization, (which is a particular case of a more general equivalence proposed by Micchelli and Pontil (2005)) i.e : 2 L L JfkJ2 min =JfkJ(10) dk=0, dk=1 dk k k k For proving this equality, we use a classical Lagrangian…”
When is there a representer theorem? Vector versus matrix regularizers (2008, methodology): “We first recall a lemma from [Micchelli and Pontil, 2004, Sec.2] which states that (linear or not) representer theorems for interpolation lead to represent r theorems for regularization, under no conditions on the error function.”
Generalization bounds for learning the kernel (2009, methodology): “Then the general kernel learning scheme in [20, 29] can be cast as a two-layer minimization problem:”
A Spectral Regularization Framework for Multi-Task Structure Learning (2007, methodology): “Without loss of generality, we can assume as in [15] that Kt are invertible for all t ∈ INT .”
Algorithms for Learning Kernels Based on Centered Alignment (2012, background): “…data (Cristianini et al., 2001; Chapelle et al., 2002; Bousquet and Herrmann, 2002; Lanckriet et al., 2004; Jebara, 2004; Argyriou et al., 2005; Micchelli and Pontil, 2005; Lewis et al., 2006; Argyriou et al., 2006; Kim et al., 2006; Cortes et al., 2008; Sonnenburg et al., 2006; Srebro and…”
Efficient Rule Ensemble Learning using Hierarchical Kernels (2011, methodology): “With repeated application of lemma 26 in Micchelli & Pontil (2005), it can be shown that 5 : where .”
$\ell_{p}-\ell_{q}$ Penalty for Sparse Linear and Sparse Multiple Kernel Multitask Learning (2011, result): “The following proposition extends that of Michelli et al. [ 35 ] to mixed norm.”
Sparse Modeling for Image and Vision Processing (2014, background): “Note that similar variational formulations may be obtained for the squared quasi-norms [Micchelli and Pontil, 2005].”
L2 Regularization for Learning Kernels (2009, methodology): “The problem of learning kernels has been investigated in a number of recent publications including (Lanckriet et al., 2004; Micchelli & Pontil, 2005; Argyriou et al., 2005; Ar-gyriou et al., 2006; Srebro & Ben-David, 2006; Ong et al., 2005; Lewis et al., 2006; Zien & Ong, 2007; Jebara, 2004; Bach,…”
A multiple kernel support vector machine scheme for feature selection and rule extraction from gene expression data of cancer tissue (2007, background): “The multiple kernel learning considers the multiple kernels or parameterizations of kernels to replace the single fixed kernel [46].”