A Generalized Representer Theorem

By Bernhard Schölkopf, Ralf Herbrich, Alex Smola. 2001

1,972 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “(3.44) Put differently, any solution f̂ can be expressed in terms of the kernel mean embedding of P1, . . . ,Pn. Note that if we restrict M1+ to contain only Dirac measures δx, then the solution becomes f̂ = ∑n i=1 αik(xi, ·), i.e., the classical representer theorem (Schölkopf et al. 2001).”
Security Evaluation of Support Vector Machines in Adversarial Environments (2014, background): “1 This is an instance of the Representer Theorem which states that solutions to a large class of regularized ERM problems lie in the span of the training data [60].”
The hardness of conditional independence testing and the generalised covariance measure (2018, background): “nce for Continuous Random Variables, 2004. EURANDOM-report 2004-049. [6]T. B. Berrett and R. J. Samworth. Nonparametric independence testing via mutual information. ArXiv e-prints (1711.06642), 2017. [7]Thomas B Berrett, Yi Wang, Rina Foygel Barber, and Richard J Samworth. The conditional permutatio”
Kernel Mean Shrinkage Estimators (2014, background): “By representer theorem (Schölkopf et al., 2001), any function g ∈ H that is a minimizer of (20) lies in a subspace spanned by {k(·, x1), .”
Density Estimation in Infinite Dimensional Exponential Families (2013, methodology): “In Theorem 4 (proved in Section 8.5), we provide an alternate expression for f λ,n as a solution of a simple ﬁnite-dimensional linear system (see (7) and (8)), using the ideas in the proof of representer theorem [24, 31].”
Learning from Distributions via Support Measure Machines (2012, background): “Furthermore, if we restrict P to a class of Dirac measures δx on X and consider the training set {(δxi , yi)}mi=1, the functional (3) reduces to the usual regularization functional (Schölkopf et al. 2001) and the solution reduces to f = ∑m i=1 αik(xi, ·).”
Kernels for Vector-Valued Functions: a Review (2011, neutral): “The above result is the well known representer theorem originally proved in [51] (see also [88] and [26] for recent results and further references).”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, background): “, 2000), it is well known that kernel-based algorithms (for classification/regression) generally invoke the representer theorem (Kimeldorf and Wahba, 1970; Schölkopf et al., 2001) and learn a function in H that has the representation, f := ∑”
Kernel Continual Learning (2021, methodology): “Based on the Representer theorem (Schölkopf et al., 2001), we have:”
A Comprehensive Survey on Transfer Learning (2019, background): “In light of the representer theorem [82], the parameter θ can be defined as θ = ∑n i=1 αiΦ(xi), and thus we have”
A la Carte - Learning Fast Kernels (2014, methodology): “Instead, one uses the representer theorem [Kimeldorf and Wahba, 1970, Sch¨olkopf et al., 2001] to show that when solving regularized risk minimization problems, the optimal solution f ( x ) = (cid:104) w, φ ( x ) (cid:105) can be found as linear combination of kernel functions: While this expansion…”
Empirical Risk Minimization under Fairness Constraints (2018, background): “By the representer theorem (Schölkopf et al., 2001), the solution to Problem (16) is a linear combination of the feature vectors φ(x1), .”
Robust kernel density estimation (2008, neutral): “Similar results are known for supervised kernel methods (Schölkopf et al., 2001).”
Biological Sequence Modeling with Convolutional Kernel Networks (2017, positive): “Conveniently, the representer theorem [33] provides that even though f may live in a space of infinite dimension, the optimal solution f? of (1) may be parametrized by n scalar values and (1) is equivalent to an optimization problem in Rn, which is convex if L is convex.”
Kernels for Multi--task Learning (2004, background): “This result improves upon [9] where it is proven only for a global minimizer.”
Fast and simple gradient-based optimization for semi-supervised support vector machines (2014, methodology): “using ||f ||2H = ∑n i=1 ∑n j=1 cicjk(xi,xj) [31].”
Learning to Personalize from Observational Data (2016, methodology): “By expressing the hinge with constraints and using the representer theorem [23] and the kernel Gram matrix Kij = K(Xi, Xj), we can write this problem as a convex quadratic program:”