Researchers Decompose Linear Layers in Neural Networks Using Geometric Primitives
Researchers have developed a method to decompose linear transformations in neural networks into simpler geometric components called rotors, derived from Clifford algebra. The approach reduces the parameters needed from O(d²) to O(log² d) while maintaining performance comparable to existing techniques like block-Hadamard and low-rank approximations. This work provides insight into how fundamental building blocks compose into higher-level functions in large language models.
A new paper appearing in NeurIPS 2025 investigates the compositional structure of linear layers in large neural networks by expressing them as compositions of bivectors—geometric objects representing oriented planes. Using Clifford algebra, the researchers introduce a differentiable algorithm that decomposes linear transformations into products of rotors, achieving significant parameter efficiency. When applied to key, query, and value projections in transformer attention layers, the rotor-based approach matches the performance of established baseline methods while using substantially fewer parameters. The work addresses a fundamental question about how low-level geometric primitives compose into modules with richer functionality, offering an algebraic perspective on neural network architecture.
What different sources said
- arXiv cs.LGCenter
Composing Linear Layers from Irreducibles
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