The Geometry of Categorical and Hierarchical Concepts in Large Language Models

Very nice. Well-written. Feels "natural."

Besides helping with interpretability, my immediate thought is that maybe we could pretrain models faster by adding regularization terms in the objective function that induce representations of distinct categories to be in subspaces that are orthogonal to each other, and representations of subcategories to be in orthogonal subspaces that can form polytopes. The data necessary for doing so is readily available: Wordnet synsets. Induce representations of synsets to be orthogonal to each other and representations of hierarchically related synsets to be arranged in polytopes. There's already some evidence that we can leverage Wordnet synsets to pretrain some models faster. Take a look at https://news.ycombinator.com/item?id=40160728 for example.

Thank you for sharing this on HN.

> We find a remarkably simple structure: simple categorical concepts are represented as simplices, hierarchically related concepts are orthogonal in a sense we make precise, and (in consequence) complex concepts are represented as polytopes constructed from direct sums of simplices, reflecting the hierarchical structure.

It's satisfying that the structure is precisely the one you would hope for.

Beautiful paper, relatively well written and accessible, too.

I think everyone _knew_ in some sense that the structure of categorial information about vectors must be hierarchical and have this general kind of structure, but they managed to formalize that intuition into just a few theorems that seem sort of inevitable only in retrospect.

Wow. This seems really important, because LLMs have been such black boxes.

Is this result useful only for basic concepts backed by huge numbers of cases in the training data, or is it more general than that?

Comments?

There's been a decent amount of work using simplicial complexes and related ideas to generalize graph neural networks, e.g. [0], [1]. If LLMs obey a similar geometry, it could be a promising direction for multimodal models and more principled RAGs with better inductive biases.

[0] https://arxiv.org/pdf/2010.03633

[1] https://arxiv.org/pdf/2012.06333

GitHub repo at

https://github.com/KihoPark/LLM_Categorical_Hierarchical_Rep...

Well, if concepts turn out to be simplicial (or cellular) complexes maybe philosophy can be made into applied algebraic topology.

reminds me of the anthropic's recent work on identifying the neuron sets that correlate to various semantic concepts in Claude: https://news.ycombinator.com/item?id=40429540 "Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet"