# On Transferring Transferability: Towards a Theory for Size Generalization

arXiv: https://arxiv.org/abs/2505.23599 · HTML: https://mlsanity.com/paper/2505.23599/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2025.

## Built on by (most cited first, 10 of 10)

- [Any-Dimensional Invariant Universality](https://mlsanity.com/paper/2605.23156/) (2026)
- [Size-Transferability of Graph Transformers with Convolutional Positional Encodings](https://mlsanity.com/paper/2602.15239/) (2026)
- [On the Expressive Power of GNNs to Solve Linear SDPs](https://mlsanity.com/paper/2604.27786/) (2026)
- [Any-Dimensional Learning by Sampling](https://mlsanity.com/paper/2607.07680/) (2026)
- [Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs](https://mlsanity.com/paper/2602.10031/) (2026)
- [Learning to Approximate Uniform Facility Location via Graph Neural Networks](https://mlsanity.com/paper/2602.13155/) (2026)
- [Warm-starting PDE solvers with any-dimensional machine learning](https://mlsanity.com/paper/2609.38916/) (2026)
- [Transferable Graph Metanetworks](https://mlsanity.com/paper/2610.00420/) (2026)
- [The Expressive Power of Random-Feature MPNNs Across Graph Sizes](https://mlsanity.com/paper/2610.12809/) (2026)
- [Dimension-Free Scaling Laws for Size Generalization: An Exponential Separation](https://mlsanity.com/paper/2610.19279/) (2026)

## Abstract (arXiv)

Many modern learning tasks require models that can take inputs of varying sizes. Consequently, dimension-independent architectures have been proposed for domains where the inputs are graphs, sets, and point clouds. Recent work on graph neural networks has explored whether a model trained on low-dimensional data can transfer its performance to higher-dimensional inputs. We extend this body of work by introducing a general framework for transferability across dimensions. We show that transferability corresponds precisely to continuity in a limit space formed by identifying small problem instances with equivalent large ones. This identification is driven by the data and the learning task. We instantiate our framework on existing architectures, and implement the necessary changes to ensure their transferability. Finally, we provide design principles for designing new transferable models. Numerical experiments support our findings.