# Measuring trainable degrees of freedom in materials graph neural networks: a random-subspace intrinsic dimension analysis

arXiv: https://arxiv.org/abs/2609.36084 · HTML: https://mlsanity.com/paper/2609.36084/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 30 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [Learning Efficient Convolutional Networks through Network Slimming](https://mlsanity.com/paper/1708.06519/) (2017)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Graph neural networks for materials science and chemistry](https://mlsanity.com/paper/2208.09481/) (2022)
- [Periodic Graph Transformers for Crystal Material Property Prediction](https://mlsanity.com/paper/2209.11807/) (2022)
- [Structure-based out-of-distribution (OOD) materials property prediction: a benchmark study](https://mlsanity.com/paper/2401.08032/) (2024)
- [ET-AL: Entropy-Targeted Active Learning for Bias Mitigation in Materials Data](https://mlsanity.com/paper/2211.07881/) (2022)

## Abstract (arXiv)

Final predictive accuracy is the standard basis for comparing graph neural networks (GNNs) in materials-property prediction, but it does not show how strongly performance depends on access to trainable parameter-space directions. Here, we introduce trainable-degree dependence as a complementary characterization of materials GNN learning. Using random-subspace intrinsic-dimension analysis, we train CGCNN, ALIGNN, and DimeNet++ in randomly oriented parameter subspaces across six prediction tasks and measure how performance recovers as independent trainable degrees of freedom are restored. The resulting recovery curves separate endpoint accuracy from the trainable-dimensional demand required to recover it. They reveal distinctions that final errors alone miss: metallic classification and log-bulk-modulus regression recover near-reference performance from small fractional subspaces, formation-energy and band-gap prediction show stronger architecture dependence, and phonon prediction is most sensitive to dimensional restriction. Dataset-size sweeps show that band-gap models require larger fractional subspaces as training data grows, whereas formation-energy and bulk-modulus responses are more stable. A width sweep shows that fractional thresholds can remain stable while absolute threshold dimensions increase with model size. Random-subspace analysis therefore provides a targeted stress test for how materials GNNs use their optimization space.