Fast Summation of Radial Kernels via QMC Slicing
Authors: Johannes Hertrich, Tim Jahn, Michael Quellmalz
ICLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Numerical examples demonstrate that our QMC-slicing approach significantly outperforms existing methods like (QMC-)random Fourier features, orthogonal Fourier features or non-QMC slicing on standard test datasets. |
| Researcher Affiliation | Academia | 1 Universit e Paris Dauphine-PSL and UCL, EMAIL 2 Technische Universit at Berlin, EMAIL |
| Pseudocode | No | The paper describes methods and processes in text, such as in Section 2 (SLICING OF RADIAL KERNELS) and Section 4 (NUMERICAL EXAMPLES), but does not contain any structured pseudocode or algorithm blocks. |
| Open Source Code | Yes | The code for the numerical examples is provided online2. 2available at https://github.com/johertrich/fastsum_QMC_slicingAdditionally, we provide a general Py Torch implementation for fast kernel summations via slicing and non-equispaced fast Fourier transforms4. 4available at https://github.com/johertrich/simple_torch_NFFT |
| Open Datasets | Yes | We evaluate the kernel sums on Letters dataset (d = 16, Slate, 1991), MNIST (reduced to d = 20 dimensions via PCA, Le Cun et al., 1998) and Fashion MNIST (reduced to d = 30 dimensions via PCA, Xiao et al., 2017) |
| Dataset Splits | No | We evaluate the kernel sums on Letters dataset (d = 16, Slate, 1991), MNIST (reduced to d = 20 dimensions via PCA, Le Cun et al., 1998) and Fashion MNIST (reduced to d = 30 dimensions via PCA, Xiao et al., 2017), where (x1, ..., x N) and (y1, ..., y M) constitute the whole dataset and the weights (w1, ..., w N) are set to 1. In particular, we have M = N = 20000 for the Letters dataset and M = N = 60000 for MNIST and Fashion MNIST.The paper utilizes the entire datasets for its kernel summation and gradient flow experiments, rather than splitting them into distinct training, validation, or test sets. |
| Hardware Specification | Yes | on an NVIDIA RTX 4090 GPU.We benchmark the computation times on a single thread of an AMD Ryzen Threadripper 7960X CPU |
| Software Dependencies | No | The paper mentions using Julia and Python for implementation, and references packages like SciPy, Sobol.jl, Py Keops, Py Torch, and NFFT.jl, but does not specify the version numbers for these software dependencies (e.g., 'Python 3.8' or 'PyTorch 1.9'). |
| Experiment Setup | Yes | The parameters σ, α and β are chosen by the median rule... where m is the median of all considered input norms x of the basis functions F and γ is some scaling factor which we set to γ { 1 2, 1, 2}.We set the threshold T to 0.3 for the Gauss kernel, to 0.2 for the Mat ern kernel and to 0.1 for the Laplace kernel.In our experiments, we choose Nft = 128 for the Gauss, Nft = 512 for the Mat ern and Nft = 1024 for the Laplace kernel.We run the MMD flow for 50000 steps with step size τ = 1, where we compute the function Gy by (Monte Carlo) Slicing and by QMC Slicing with P = 1000 projections.add a momentum parameter m = 0.9. |