arXiv:2507.11688v4 Announce Type: replace Abstract: Contemporary large models often exhibit behaviors suggesting the presence of low-level primitives that compose into modules with richer functionality, but these fundamental building blocks remain poorly understood. We investigate this compositional structure in linear layers by asking: can we identify/synthesize linear transformations from a minimal set of geometric primitives? Using Clifford algebra, we show that linear layers can be expressed as compositions of bivectors -- geometric objects encoding oriented planes -- and introduce a diffe
The explosion of large AI models, while powerful, highlights a fundamental lack of understanding regarding their internal workings, making research into foundational composition crucial for future advancements.
Understanding and synthesizing linear transformations from fundamental geometric primitives could lead to more efficient, interpretable, and architecturally novel AI models, moving beyond current black-box approaches.
This research suggests a new mathematical framework (Clifford algebra and bivectors) for designing and understanding linear layers in AI, potentially displacing less geometrically intuitive methods.
- · AI researchers
- · Deep learning architects
- · Hardware developers (for novel AI architectures)
- · Developers reliant solely on empirical AI tuning
- · AI architectures that cannot leverage geometric primitives
Improved theoretical understanding of neural network functionality and component behaviors.
Development of fundamentally new, more efficient, and interpretable AI model architectures inspired by geometric composition.
Accelerated progress towards general AI through a deeper grasp of how intelligence emerges from basic computational elements.
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Read at arXiv cs.LG