When you think about how large language models (LLMs) represent meaning, it helps to imagine the data not as scattered points in a vast high-dimensional space, but as forming a much more structured shape—a manifold. A manifold is a mathematical concept describing a space that, while possibly very complex and curved globally, looks like ordinary, flat Euclidean space when you zoom in close enough. In other words, a manifold is a surface or shape that locally resembles regular space, but can have rich, twisted, or curved structure when viewed as a whole.

In the context of LLMs, the high-dimensional vectors that encode words, sentences, or even concepts are not distributed randomly throughout the entire latent space. Instead, these representations tend to cluster and organize along lower-dimensional manifolds embedded within the high-dimensional space. This means that, although the model has access to thousands of dimensions, the meaningful data it learns to represent often lies on a much smaller, structured subset—a manifold whose geometry reflects the relationships and constraints present in natural language.

To build geometric intuition, consider how manifolds capture both local and global structure. Locally, the manifold is smooth and can be approximated by a flat plane, making it possible to use familiar geometric tools to analyze small neighborhoods of points. This is why LLMs can capture subtle, nuanced distinctions between similar words or phrases—nearby points on the manifold represent semantically related tokens. Globally, however, the manifold may have curvature, twists, or folds, allowing it to wrap complex semantic relationships into the latent space. Curvature is what enables the model to encode more abstract connections, such as analogies or hierarchical relationships, that would be difficult to represent in a purely flat space.

By modeling language representations as lying on manifolds, you can better understand how LLMs organize meaning: local neighborhoods reflect fine-grained semantic similarity, while the global shape of the manifold encodes broader linguistic structure. This perspective is essential for analyzing how LLMs generalize, interpolate, and capture semantic relationships.

Key Insights

• Manifolds provide a framework for understanding how high-dimensional representations can have low-dimensional, structured organization;
• Local structure on the manifold enables LLMs to capture fine-grained semantic similarity;
• Global curvature and topology of the manifold encode complex, abstract relationships;
• Recognizing manifold structure is critical for interpreting, visualizing, and improving LLM representations.