Understanding Smoothness of Vector Gaussian Processes on Product Spaces
Authors: Emilio Porcu, Ana Paula Peron, Eugenio Massa, Xavier Emery
TMLR 2024 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | This paper challenges the problem of quantifying smoothness of matrix-valued continuous kernels that are associated with mean-square continuous vector Gaussian processes defined over non-Euclidean product manifolds. After noting that a constructive RKHS approach is unsuitable for this specific task, we proceed through the analysis of spectral properties. Specifically, we find a spectral representation to quantify smoothness through Sobolev spaces that are adapted to certain measure spaces of product measures obtained through the tensor product of Haar measures with multivariate Gaussian measures. Our results allow to measure smoothness in a simple way, and open for the study of foundational properties of certain machine learning techniques over product spaces. |
| Researcher Affiliation | Academia | Emilio Porcu EMAIL Department of Mathematics, College of Computing and Mathematical Sciences Khalifa University, Abu Dhabi & ADIA Lab, Abu Dhabi. Ana Paula Peron EMAIL Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos SP, Brazil. Eugenio Massa EMAIL Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos SP, Brazil. Xavier Emery EMAIL Department of Mining Engineering, Universidad de Chile & Advanced Mining Technology Center, Universidad de Chile |
| Pseudocode | No | The paper describes a 'Route to Smoothness' with numbered steps (1. Consider the measure space in Equation (9). 2. Provide an orthonormal basis. 3. Provide a suitable Karhunen-Loève expansion. 4. Define a proper Sobolev space. 5. Quantify smoothness of the kernel K.), but these are descriptive text points and not structured pseudocode or an algorithm block. |
| Open Source Code | No | The paper does not provide any statements about releasing code, nor does it include links to source code repositories or mention code in supplementary materials. The paper is theoretical in nature. |
| Open Datasets | No | The paper is theoretical and does not present experimental results that would involve the use or release of datasets. Therefore, no information about publicly available or open datasets is provided. |
| Dataset Splits | No | The paper is theoretical and does not involve experimental evaluation using datasets. Consequently, there is no mention of training/test/validation dataset splits. |
| Hardware Specification | No | The paper is theoretical and focuses on mathematical proofs and definitions. It does not describe any experiments that would require specific hardware. Therefore, no hardware specifications are mentioned. |
| Software Dependencies | No | The paper is a theoretical work focusing on mathematical concepts. It does not describe any computational experiments or implementations that would require specific software dependencies or versions. |
| Experiment Setup | No | The paper is theoretical and does not present any empirical experiments. As such, there are no details provided regarding experimental setup, hyperparameters, or system-level training settings. |