Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations

By J. Mercer. 1909

The present memoir is the outcome of an attempt to obtain the conditions under which a given symmetric and continuous function k ( s, t ) is definite, in the sense of Hilbert. At an early stage, however, it was found that the class of definite functions was too restricted to allow the determination of necessary and sufficient conditions in terms of the determinants of § 10. The discovery that this could be done for functions of positive or negative type, and the fact that almost all the theorems which are true of definite functions are, with slight modification, true of these, led finally to the abandonment of the original plan in favour of a discussion of the properties of functions belonging to the wider classes. The first part of the memoir is devoted to the definition of various terms employed, and to the re-statement of the consequences which follow from Hilbert’s theorem.

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Next, we give an important characterization of a continuous positive definite kernel k on a compact set, known as Mercer’s theorem (Mercer 1909).”
A Kernel Method for the Two-Sample-Problem (2006, background): “xi −z) − 1 n X i κ(yi −z) # 2 dz (15) = 1 m2 Xm i,j=1 k(xi −xj) + 1 n2 Xn i,j=1 k(yi −yj)− 2 mn Xm,n i,j=1 k(xi −yj), (16) where k(x−y) = R κ(x−z)κ(y−z)dz. By its deﬁnition k(x−y) is a Mercer kernel (Mercer, 1909), as it can be viewed as inner product between κ(x− z) and κ(y− z) on the domain X. A d”
Kernel methods in machine learning (2007, neutral): “Instead, researchers considered kernels satisfying the conditions of Mercer’s theorem (Mercer [99], see, e.g., Cristianini and Shawe-Taylor [37] and Vapnik [141]).”
A la Carte - Learning Fast Kernels (2014, background): “Finally, let k : X ×X → R be a Hilbert Schmidt kernel [Mercer, 1909].”
Wavelet support vector machine (2004, background): “The Mercer theorem (see [7]) gives the conditions that a dot product kernel must satisfy.”
Accurate interatomic force fields via machine learning with covariant kernels (2016, background): “In particular, a kernel is positive semidefinite if and only if it is a scalar product in some (possibly infinite dimensional) vector space [23,41].”
Nonparametric Bayesian Kernel Models (2007, background): “Kernel functionsk : X×X → R that are continuous and positive semi-definite on a compact space X are Mercer kernels for which the RKHS is characterized (Mercer, 1909; König, 1986) as”
A tutorial on support vector regression (2004, unknown): “Theorem 2 (Mercer 1909).”
Facilitating the application of Support Vector Regression by using a universal Pearson VII function based kernel (2006, background): “In general, any function satisfying the so-called Mercer’s theorem conditions [14] belongs to the class of valid kernel functions.”
Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks (2020, methodology): “By Mercer’s theorem (Mercer, 1909; Rasmussen & Williams, 2005) , the kernel admits a representation in terms of its M kernel eigenfunctions { φ ρ ( x ) } , (5) where ψ ρ ( x ) = λ ρ φ ( x ) is the feature map we will work with.”
Preventing Over-Fitting during Model Selection via Bayesian Regularisation of the Hyper-Parameters (2007, background): “…of the data onto the feature space, F , that is, The interpretation of an inner-product in a ﬁxed feature space is valid for any Mercer kernel (Mercer, 1909), for which the Gram matrix, is positive semi-deﬁnite, that is, The Gram matrix effectively encodes the spatial relationships between…”