Error Bounds for Learning with Vector-Valued Random Features
Authors: Samuel Lanthaler, Nicholas H. Nelsen
NeurIPS 2023 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | To study how our theory holds up in practice, we numerically implement the vector-valued RF-RR algorithm on a benchmark operator learning dataset. Figure 2a shows the decay of the relative squared test error as M increases (with λ 1/M) for fixed N. |
| Researcher Affiliation | Academia | Samuel Lanthaler California Institute of Technology slanth@caltech.edu Nicholas H. Nelsen California Institute of Technology nnelsen@caltech.edu |
| Pseudocode | No | The paper contains mathematical proofs and derivations, but no pseudocode or algorithm blocks are provided. |
| Open Source Code | Yes | All code used to produce the numerical results and figures in this paper are available at https://github.com/nickhnelsen/error-bounds-for-vv RF. |
| Open Datasets | Yes | The data {(un, G(un))}N n=1 is noise-free, the {un} are iid Gaussian random fields, and G : L2(T; R) L2(T; R) is a nonlinear operator defined as the time one flow map of the viscous Burgers equation on the torus T. The initial conditions un ν are sampled iid from a centered Matérn-like Gaussian process according to [23, Sect. 6.3, p. 32]. |
| Dataset Splits | No | The paper mentions a test set of size N=500 and a training input set, but does not specify the size of the training set or the splitting methodology for the training/validation/test sets, nor does it mention a validation set. |
| Hardware Specification | No | No specific hardware details (e.g., GPU/CPU models, memory) used for running experiments are provided. |
| Software Dependencies | No | The paper mentions the use of ELU, but does not specify any software names with version numbers for libraries, frameworks, or programming languages used in the experiments. |
| Experiment Setup | Yes | The random features are defined in a similar way to the Fourier Space RFs in [32, Sect. 3.1, p. 15]: φ(u(0); θ) = 2.6 ELU F 1{1(|k| kmax)χk (Fu(0))k (Fθ)k}k Z and θ µ , where µ is also a centered Matérn Gaussian measure with covariance operator 1.82( d2 dx2 +152 Id) 3. In the above display, F maps a function to its Fourier series coefficients, and F 1 expresses a Fourier coefficient sequence as a function expanded in the Fourier basis. The filter χ is given by [32, Eqn. 3.6, p. 15] with δ = 0.32 and β = 0.1. We take kmax = 64. In Figures 2a and 3a, the regularization factor is chosen as λ = 7 10 4/M and as λ = 3 10 6/ N in Figures 2b and 3b. |