Let’s make a set of (more or less random) assumptions.

Suppose that embedding spaces could be modeled as manifolds—or, if that seems too optimistic, as chart atlases. Let’s imagine that these embedding space regions are clustered into manifolds, with smooth bridges linking them. Let’s go even further: suppose that semantically similar domains (e.g., physics, mathematics, coding) share underlying geometric or topological properties. Let's suppose that LLMs are probabilitistic functions that have learned to travel, or operate on those surfaces.

Intuitions

1) Embeddings as Projections
In natural language processing, embeddings are one of the most established strategies for representing tokens (words, subwords, etc.) in a vector space of dimension d. Each token is associated with a vector in ℝᵈ, encapsulating syntactic and semantic traits. If we think of each token as a “point” on a map, its position reflects its semantic relationships with others. Just as latitude and longitude define geographic coordinates, the components of the embedding encode linguistic meaning.

2) Semantic Proximity
Short distances in embedding space are interpreted as high conceptual similarity between tokens; large distances imply dissimilarity. This has proven useful in tasks like analogies or synonym detection. However, Euclidean distance in high-dimensional spaces is an oversimplification.

3) Dynamism During Training
As the LLM learns, the positions of these points are “redrawn.” This process is analogous to progressively refining a map as more accurate measurements become available.

4) Multi-Head Attention

A distinctive feature of the Transformer architecture—it can be conceived as a set of “mirrors” examining the context from different angles. Each head focuses on specific (semantically entangled) relations: syntactic dependencies, gender/number agreement, long-range semantic associations, etc.

Strong Intuition: The combination of views from multiple heads creates a “high-dimensional reconstruction” of the input. It's analogous to fusing multiple photographs of the same object to obtain a more faithful 3D model. The backpropagation that updates each head’s weights acts as a rotation of these “mirrors” or representational lenses, correcting biases or focusing errors.

How Does This Relate to LLM “Weird” Behaviors?

Transformers operate as complex, non-linear transformations applied to a set o...

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