Sampling-based Nyström Approximation and Kernel Quadrature
Authors: Satoshi Hayakawa, Harald Oberhauser, Terry Lyons
ICML 2023 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Finally, we discuss their application to convex kernel quadrature and give novel theoretical guarantees as well as numerical observations. See Table 1 for a summary of our quantitative results. Section 5, we discuss how our bounds yields new theories and methods for the recent random kernel quadrature construction, which enables us to explain the empirical performance as well as to build some strong candidates whose performance is assessed by numerical experiments. 5.2. Numerical Examples. Figure 1 shows the results for (d, r) = (1, 1), (2, 1), (3, 3) with N = n2 and n = 4, 8, 16, 32, 64, 128. From Figure 1(a, b), we can see that our methods indeed recover (and perform slightly better than) the rate of k H from a contaminated sample Z. |
| Researcher Affiliation | Academia | Satoshi Hayakawa 1 Harald Oberhauser 1 Terry Lyons 1 1Mathematical Institute, University of Oxford, Oxford, United Kingdom. Correspondence to: Satoshi Hayakawa <hayakawa@maths.ox.ac.uk>. |
| Pseudocode | No | The paper does not contain structured pseudocode or algorithm blocks. |
| Open Source Code | Yes | Code is available at the nystrom folder in https://github.com/ satoshi-hayakawa/kernel-quadrature. |
| Open Datasets | Yes | Periodic Sobolev Spaces. The class of RKHS we use is called periodic Sobolev spaces of functions on X = [0, 1] (a.k.a. Korobov spaces), and given by the following kernel for a positive integer r: kr(x, y) = 1 + ( 1)r 1(2π)2r (2r)! B2r(|x y|), where B2r is the 2r-th Bernoulli polynomial (Wahba, 1990; Bach, 2017). |
| Dataset Splits | No | The paper does not provide explicit training/test/validation dataset splits by percentages or sample counts, nor does it refer to predefined splits with citations for reproducibility. |
| Hardware Specification | Yes | All the experiments were conducted on a Mac Book Pro with Apple M1 Max chip and 32GB unified memory. |
| Software Dependencies | No | The paper states that code is available but does not provide specific version numbers for software dependencies or libraries used to run the experiments. |
| Experiment Setup | No | The paper outlines the setup for numerical examples, including how samples Y and Z are generated and the types of kernel quadrature rules compared, but it does not specify concrete hyperparameter values or detailed system-level training configurations typically found in experimental setups. |