Any-Dimensional Learning by Sampling
Abstract
Many machine learning models are defined for inputs of different sizes, such as point clouds containing different numbers of points, sequences of tokens of different lengths, and graphs on different numbers of nodes. Such models are trained on finitely many examples of necessarily limited sizes. How well do these models generalize from inputs of small size to larger inputs of size not seen during training? Furthermore, evaluating such models on large inputs is often expensive. How can we sketch large inputs to obtain smaller ones on which the model takes similar values? At the heart of both questions is the need to compare inputs of different sizes and to approximate large inputs by small ones. We present a unified approach to address these questions by using random sampling maps to compare inputs of different sizes. The sampling maps we consider are generalizations of sampling with replacement, random binning, and species sampling. We characterize the application domains in which each type of sampling is appropriate in terms of the symmetries and relations between problem instances of different sizes in the domain. Our framework yields explicit generalization and sketching rates for function classes continuous with respect to a chosen notion of sampling, encompassing large families of functions defined on sequences, graphs, and tensors of different sizes. Specific examples include moment polynomials on measures, homomorphism densities and numbers of graphs, permutation-invariant transformers, and graph neural networks.
Related
- Sampling with Autoregressive Clock Jitter: Infinite-Dimensional Stability Failure and Finite-Dimensional Conditioning
- Low Dimensional Sampling under Reconstructed Constraints
- Sharp dimensional analysis of midpoint methods for Langevin sampling
- Rapid reconstruction-free classification from Lissajous sparse sampling using deep learning
- Learning to Replace MCMC in Split-Gibbs Diffusion Posterior Sampling via Deep Unfolding
BibTeX
@misc{levin2026anydimensional,
title = {{Any-Dimensional Learning by Sampling}},
author = {Eitan Levin and Venkat Chandrasekaran},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.07680},
eprint = {2607.07680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.07680},
}
RIS
TY - UNPB
TI - Any-Dimensional Learning by Sampling
AU - Eitan Levin
AU - Venkat Chandrasekaran
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2607.07680
UR - https://arxiv.org/abs/2607.07680
AB - Many machine learning models are defined for inputs of different sizes, such as point clouds containing different numbers of points, sequences of tokens of different lengths, and graphs on different numbers of nodes. Such models are trained on finitely many examples of necessarily limited sizes. How well do these models generalize from inputs of small size to larger inputs of size not seen during training? Furthermore, evaluating such models on large inputs is often expensive. How can we sketch large inputs to obtain smaller ones on which the model takes similar values? At the heart of both questions is the need to compare inputs of different sizes and to approximate large inputs by small ones. We present a unified approach to address these questions by using random sampling maps to compare inputs of different sizes. The sampling maps we consider are generalizations of sampling with replacement, random binning, and species sampling. We characterize the application domains in which each type of sampling is appropriate in terms of the symmetries and relations between problem instances of different sizes in the domain. Our framework yields explicit generalization and sketching rates for function classes continuous with respect to a chosen notion of sampling, encompassing large families of functions defined on sequences, graphs, and tensors of different sizes. Specific examples include moment polynomials on measures, homomorphism densities and numbers of graphs, permutation-invariant transformers, and graph neural networks.
ER -