The Kernel Trick for Distances

By Bernhard Schölkopf. 2000

713 citations recorded by the source corpus.

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Citations in the corpus

DISCO Nets : DISsimilarity COefficients Networks (2016, methodology): “0;x n) G(z k;x n)jj 2 i (7) This speciﬁc case of our objective function is related to the Maximum Mean Discrepancy method (MMD) of Gretton et al. [7, 8]. Indeed as shown in Proposition 3 of Schölkopf [23], the function k: YY! R deﬁned as k(y;y0) = jjy y0jj 2 for 0 <2 is a conditionally positive d”
On the Schoenberg Transformations in Data Analysis: Theory and Illustrations (2010, background): “…the object features themselves) corresponds an equally eﬃcient “distance trick”, stating that Euclidean distances themselves (and not their underlying coordinates) permit to express all the real quantities of interest, as in (5), (6), or Section 5; see also Sch¨olkopf (2000) and Williams (2002).”
Open-Ended Learning Leads to Generally Capable Agents (2021, positive): “With this construction, the norm of each vector is constant ‖ג(·)‖2 = 𝑛 · 𝑁𝜙 and thus we can compute a ג-kernel (Scholkopf, 2001): Kג (G𝑖,G 𝑗) ∝ −‖G𝑖 − G 𝑗‖2𝔊 This way we can utilise any kernelised method (such as Kernel PCA (Schölkopf et al., 1998)) to analyse the Game Space without…”
Quick Shift and Kernel Methods for Mode Seeking (2008, methodology): “) kernel K [18].”
Generalized RBF feature maps for Efficient Detection (2010, background): “Recall that, for any PD kernel K(x,y), the equation D2(x,y) = K(x,x)+K(y,y)−2K(x,y) (11) defines a corresponding squared metric [15].”
Tikhonov, Ivanov and Morozov regularization for support vector machine learning (2016, unknown): “…solution function h(x) can be written as: h(x) = w · φ(x) + b = n∑ i=1 piφ(xi ) · φ(x) + b = n∑ i=1 pi K (xi , x) + b (12) wherewemade use of thewell-knownkernel trick (Berlinet andThomas-Agnan 2004;Vapnik 1998; Schölkopf 2001; Shawe-Taylor and Cristianini 2004) and K (·, ·) is the kernel function.”
Rule-Based Learning Systems for Support Vector Machines (2006, methodology): “Moreover, it was already recognized in [21] that the SVM relies only on distances in the Reproducing Kernel Hilbert Space.”
Feature space interpretation of SVMs with indefinite kernels (2005, background): “These kernels are known to produce convenient convex optimization problems for SVMs [13].”
Dimension Reduction: A Guided Tour (2010, background): “Defining the projection matrix P c ≡ (1 − ec′), for any c ∈ Rm such that e′c = 1, then for any CND matrix D, the matrix −P cDP ′c is positive semidefinite (and hence a dot product matrix) [10, 79] (note that P c is not necessarily symmetric).”
Secrets of GrabCut and Kernel K-Means (2015, background): “Large bandwidth and basic K-means: Consider Gaussian kernels of large bandwidth σ approaching the data range.”
A Study on Sigmoid Kernels for SVM and the Training of non-PSD Kernels by SMO-type Methods (2005, methodology): “For example, K = [ 1 0 0 − 1 ] is CPSD as for any α 1 + α 2 = 0, α T Kα = 0.”
Gesture recognition for human-robot collaboration: A review (2017, methodology): “Some papers also introduced Derivative Dynamic Time Warping (DDTW) as an extension to DTW (Keogh and Pazzani, 2001; Rautaray and Agrawal, 2015).”
The Greedy Miser: Learning under Test-time Budgets (2012, methodology): “Typically, the mapping h can be performed implicitly through the kernel-trick (Sch ¨ olkopf, 2001) or explicitly through, for example, the boosting-trick (Friedman, 2001; Rosset et al., 2004; Chapelle et al., 2010).”
Support Measure Data Description for group anomaly detection (2015, unknown): “7, pp. 215-219 [22] and Sec.”
Hilbertian Metrics and Positive Definite Kernels on Probability Measures (2005, methodology): “Nevertheless this class is of great interest since Schölkopf showed in [11] that all translation invariant kernel methods can also use the bigger class of cpd kernels.”