- The paper introduces a novel algorithm that leverages structured prompts to map token embedding spaces up to homeomorphism.
- It employs response analysis to compute local dimensions and curvature, revealing the stratified manifold nature of token embeddings.
- Experiments on Llemma-7B confirm the method's efficacy, with implications for improved LLM interpretability and architectural optimization.
Probing the Space of Tokens in LLMs
Introduction
The paper "Probing the topology of the space of tokens with structured prompts" (2503.15421) presents a novel methodology leveraging structured prompts to uncover the topology of token embeddings in LLMs. It transcends the limitations of prior work by enabling topological exploration without direct access to token embeddings. Through a mathematically grounded approach, this method captures the token subspace up to homeomorphism, enabling the analysis of semantic structures within the latent space.
Method Overview
The approach involves crafting structured prompts, which when fed into an LLM, elicit responses that reflect the underlying topological arrangement of tokens. The premise is built on the idea that tokens, embedded in a latent space, form a stratified manifold rather than a simple manifold. By employing an algorithmic probing technique, it is possible to reconstruct the token subspace's geometry using only the model's outputs.
Algorithm Details
The proposed method (Algorithm \ref{alg:token_prompting}) is designed to prompt the LLM in a way that reveals its token subspace. This involves querying the model with structured inputs and analyzing the resulting outputs to infer topological features. The algorithm calculates the coordinates of tokens within an embedded space, translating these coordinates into meaningful topological structures by examining local dimensions and curvature properties.
Key steps include:
- Structured Prompt Creation: Using specific sequences to clear and reset the context window, ensuring consistent probe conditions.
- Response Collection: Generating sequences and measuring probabilities for token predictions across a defined number of positions.
- Topological Analysis: Interpreting these measurements to construct a topological map that mirrors the manifold structure of token embeddings.
Theoretical Justification
The algorithm's validity is supported by Theorem \ref{thm:autoregressive_embedding}, which assures that the structured prompts can be used to map token subspaces to their latent representations effectively under certain conditions. The theoretical framework leverages concepts from transverse intersections and residual subsets within smooth manifold theory, providing a solid proof for the embedding capability of nonlinear autoregressive processes, of which LLMs are a subclass.
Experiments and Results
Experiments with the {\tt Llemma-7B} model demonstrate the efficacy of this approach. The method successfully recovers the token subspace's dimensional structure, aligning closely with known embeddings. This involves estimating token dimensions and analyzing their distribution, revealing the stratified manifold nature of text embeddings.
Implementation Challenges
Computational complexity arose as a challenge, particularly concerning data collection and dimensional estimation processes. Advanced sampling techniques and thorough probabilistic analysis were employed to mitigate these challenges, allowing efficient processing on standard hardware platforms.
Implications and Future Work
The implications of this research are profound, as it not only facilitates a deeper understanding of LLM internals but also opens avenues for optimizing model architectures for better semantic comprehension. Future work may focus on refining the probing technique to improve dimensionality estimation accuracy and extend applicability to more complex models, exploring the impact on model performance and interpretability.
Conclusion
The paper provides a substantial advancement in the analysis of token spaces in LLMs, demonstrating how structured prompts can unveil hidden geometric and topological structures. This method paves the way for deeper investigations into semantic embeddings, offering a robust framework for future explorations of AI-driven natural language understanding.
