A symmetric positive semidefinite function k(x,y) on an index set that represents the covariance function of a second‑order stochastic process or the Mercer kernel of a positive integral operator; for any finite collection {x_i} the matrix [k(x_i,x_j)] is positive semidefinite.

Principle

Positive definiteness characterizes covariance: k is a covariance kernel iff all finite Gram matrices are positive semidefinite, which yields a reproducing kernel Hilbert space (RKHS) and an integral operator with nonnegative spectrum.

Demonstration

The Gaussian radial basis kernel k(x,y)=σ^2 exp(−||x−y||^2/(2ℓ^2)) is a covariance kernel of a stationary Gaussian process and defines a compact, positive integral operator on L2 of a bounded domain with Mercer expansion in orthonormal eigenfunctions.

Misapplication

Using a symmetric similarity measure that is not positive semidefinite as if it were a covariance kernel, leading to Gram matrices with negative eigenvalues and impossible variances or ill‑posed Gaussian process inferences.

Consequence

A true covariance kernel defines the covariance structure of Gaussian process priors, induces an RKHS of functions for interpolation and regularization, and ensures that associated integral operators have nonnegative spectra enabling spectral decompositions.

Reversal

A general symmetric similarity or distance function lacking positive definiteness cannot serve as a covariance kernel; it may still be useful for heuristics but fails to guarantee probabilistic consistency or an RKHS structure.

Boundary

Applies to functions symmetric in (x,y) and positive semidefinite on finite samples; excludes kernels that depend on non‑symmetric cross‑terms, signed covariances without semidefiniteness, or objects requiring noncommutative covariance notions (operator‑valued kernels).

Semantic Tension

Tension exists between covariance kernels used for probabilistic modeling (requiring PSD) and heuristic similarity measures in machine learning that prioritize discrimination or locality without PSD guarantees.

Synthesis

A covariance kernel is a symmetric positive semidefinite function whose finite Gram matrices are PSD; it encodes second‑order dependence, yields RKHS structure, and underlies Gaussian process covariance operators and spectral decompositions.