Research map / Learning theory and algorithms / Kernels and regression theory

Extending Mercer's expansion to indefinite and asymmetric kernels

ICLR 2025 · Learning theory and algorithms › Kernels and regression theory · approach: RKHS kernel methods · 0 citations (OpenAlex)

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Extends Mercer's expansion to general continuous indefinite and asymmetric kernels, giving counterexamples where the expansion fails to converge pointwise, absolutely, or uniformly, proving pointwise a.e., unconditional a.e., and almost uniform convergence for continuous kernels of uniform bounded variation, plus singular value decay bounds and a pseudo-skeleton algorithm.

Problem

Mercer's expansion is classically only guaranteed for continuous symmetric positive definite kernels; its convergence properties for continuous indefinite and asymmetric kernels are subtle and lack rigorous theory.

Approach

Constructs explicit continuous symmetric indefinite kernels whose Mercer expansions fail to converge pointwise, absolutely, or uniformly, and proves that continuous kernels of uniform bounded variation (in each variable separately) have singular value expansions converging pointwise a.e., unconditionally a.e., and almost uniformly, using singular value decay σ_n = O(n^{-1}) derived from Legendre e

Key results

• Constructs continuous symmetric indefinite kernels whose Mercer expansion does not converge pointwise, converges pointwise but not uniformly, or converges pointwise but not absolutely.
• Proves continuous kernels of uniform bounded variation have Mercer expansions converging pointwise a.e., unconditionally a.e., and almost uniformly, with singular values sigma_n = O(n^{-1}).
• Shows smoother kernels have singular value bound sigma_{2k} = O(k^{-(r+1)}) for r-times differentiable kernels; the algorithm computes tanh(100xy+1) with R=96 pivots.

Surprising findings

• A continuous symmetric indefinite kernel can have a Mercer expansion that does not converge pointwise, even though the kernel is continuous.
• Continuity alone is not enough to guarantee a pointwise convergent Mercer expansion for general continuous kernels.

Limitations

• Uniform absolute and uniform convergence for bounded variation kernels is not proven.
• Singular value bounds are conjectured, not proven, to be asymptotically tight.
• Analysis restricted to closed subsets of the real line.

Experimental setup

Metrics: L2 approximation error singular value decay

Baselines: Hammerstein's uniform convergence result Smithies' decay result

Builds on: Mercer's theorem Eckart-Young-Mirsky theorem Rademacher-Menchov theorem

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