Flat Facts, Curved Beliefs: A Geometric Hypothesis for Transformer Cognition

Medium Article: https://medium.com/@jsmith0475/flat-facts-curved-beliefs-a-geometric-hypothesis-for-transformer-cognition-5ad6f850ebd5 The article, by Dr. Jerry A. Smith, proposes a geometric hypothesis for transformer cognition, suggesting that beliefs might operate within a curved, hyperbolic mathematical space, unlike factual information which likely resides in a flatter, Euclidean space. This theory attempts to explain why opposing concepts, like "love" and "hate," appear artificially close in traditional, flattened visualizations of transformer's internal representations. The author suggests that different "attention heads" within transformers may specialize in different geometries, with some handling stable facts in Euclidean space and others managing nuanced beliefs in hyperbolic space, which naturally accommodates hierarchies and divergent ideas. The text outlines potential experiments to test this hypothesis, such as measuring geodesic distances between beliefs in a hyperbolic model and analyzing the "tree-like" quality of attention head graphs. Ultimately, this perspective implies that transformers have independently discovered the need for varied geometries to fully represent the complexity of meaning, moving beyond the limitations of simply increasing Euclidean dimensions to accurately model human-like understanding.

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