Notice: The reproducibility variables underlying each score are classified using an automated LLM-based pipeline, validated against a manually labeled dataset. LLM-based classification introduces uncertainty and potential bias; scores should be interpreted as estimates. Full accuracy metrics and methodology are described in [1].
Extending Mercer's expansion to indefinite and asymmetric kernels
Authors: Sungwoo Jeong, Alex Townsend
ICLR 2025 | Venue PDF | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | In this paper we extend Mercer s expansion to continuous kernels, providing a rigorous theoretical underpinning for indefinite and asymmetric kernels. We generalize Mercer s expansion to a general continuous kernel, where the kernel may be indefinite or asymmetric. The following two theorems summarize our main theoretical results about Mercer s expansion for general continuous kernels (see Section 3). |
| Researcher Affiliation | Academia | Sungwoo Jeong Department of Mathematics Cornell University EMAIL Alex Townsend Department of Mathematics Cornell University EMAIL |
| Pseudocode | Yes | 5 COMPUTING MERCER S EXPANSION FOR GENERAL KERNELS. The procedure for computing the low-rank SVD of KR involves the following steps, which is essentially a fast way to compute a Mercer s expansion of a finite rank kernel: 1. Perform two QR decompositions of Φ(x) and Ψ(y), using a function analogue of Householder QR (Trefethen, 2010). We can write this QR decomposition as Φ(x) = Qleft(x)R1 and Ψ(y) = Qright(y)R2, where R1, R2 RR R and the columns of Qleft(x) and Qright(y) are orthonormal functions. 2. Compute the SVD of an R R matrix formed by R1CR 2 = UΣV . 3. Construct the final SVD-based approximation by combining the singular values and the orthonormalized functions to form: Pj=1 σjuj(x)vj(y), (10) where σ1, . . . , σR are the diagonal entries of Σ, uj(x) = PR s=1 Usj Qleft s (x), and vj(y) = PR s=1 Vsj Qright s (y). |
| Open Source Code | No | The paper does not provide an explicit statement or link indicating that the source code for the methodology described is publicly available. |
| Open Datasets | No | The paper presents theoretical work and does not conduct experiments requiring datasets. |
| Dataset Splits | No | The paper is theoretical and does not involve experiments with dataset splits. |
| Hardware Specification | No | The paper focuses on theoretical contributions and does not describe hardware used for experiments. |
| Software Dependencies | No | The paper mentions 'Chebfun' as an implementation for a part of the algorithm, but it does not specify software dependencies with version numbers for reproducing the methodology described in this paper. |
| Experiment Setup | No | The paper is theoretical and does not present an experimental setup with hyperparameters or training configurations. |