New Theoretical Bounds Established for the Geometric Structure of ReLU Neural Networks
Researchers have proven new theoretical bounds on the connectivity structure of linear regions formed by ReLU neural networks in input space. ReLU networks partition their input space into polyhedral linear regions, but the geometry of how these regions fit together has been poorly understood beyond basic region-count bounds. The findings reveal surprising structural constraints on network complexity that hold regardless of network width or depth, with implications for understanding neural network behavior.
A study accepted for oral presentation at ICLR 2026 establishes new theoretical results about the discrete geometry of ReLU neural networks, which are known to define continuous piecewise-linear functions whose linear regions form polyhedral complexes in input space. The researchers analyzed the connectivity graph of these regions — where nodes represent regions and edges connect adjacent region pairs — and proved that the average degree of this graph is upper bounded by twice the input dimension, independent of network width and depth. Additionally, they show that the diameter of this connectivity graph has an upper bound that does not depend on input dimension, a non-trivial result given that the number of regions grows exponentially with input dimension. These bounds suggest that despite the combinatorial explosion in the number of regions as networks grow larger, the way those regions connect to one another remains geometrically constrained. The theoretical results were corroborated through experiments on both synthetic and real-world datasets, and code to reproduce the findings has been made publicly available.
What's missing
The study focuses exclusively on fully-connected architectures, leaving open whether analogous results hold for convolutional or other network types. The practical downstream implications of these geometric constraints for tasks such as generalization, robustness, or network design are not addressed.
What different sources said
- arXiv cs.LGCenter
Characterizing the Discrete Geometry of ReLU Networks
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