Isabel Papadimitriou

Neural Collapse predicts that balanced one-hot classification pushes model representations to be equally far from each other; a symmetric configuration that depends only on the output label and ignores any semantic similarity in the inputs. This creates a puzzle: next-token prediction language models are trained predominantly (as context length increases) with one-hot labels: the same context is very unlikely to appear twice in training with different labels. However, they clearly learn latent structural features. That is, despite the one-hot training regime, a language model's contextual embeddings represent the fact that the next word in "Mary broke the ____" is likely to be filled by tokens in the latent classes of a) medium-sized, b) rigid, c) inanimate nouns. How does gradient descent find such categorical semantic structure when co-occurrence statistics collapse to one-hot sparsity, eliminating any shared next-tokens among different contexts? To investigate this tension we identify three synthetic controlled settings where inputs have latent semantic factors but are mapped to distinct one-hot labels. We find that semantic geometry emerges early in training, and that representations cluster by shared attributes despite receiving no explicit supervision to do so. This structure is transient: with sufficient capacity and time, the model eventually reaches the predicted symmetric state where all representations are equally separated. We study this phase transition through Gram matrix analysis and propose a preliminary modification to the commonly used unconstrained features model to capture the emerged semantic geometry.

Large language models work by manipulating the geometry of input embedding vectors over multiple layers. Here, we ask: how are the input vocabulary representations of language models structured, and how and when does this structure evolve over training? To answer this question, we use representational similarity analysis, running a suite of experiments that correlate the geometric structure of the input embeddings and output embeddings of two open-source models (Pythia 12B and OLMo 7B) with semantic, syntactic, and frequency-based metrics over the course of training. Our key findings are as follows: 1) During training, the vocabulary embedding geometry quickly converges to its training-final correlations with a suite of semantic and syntactic features; 2) Embeddings of high-frequency and function words (e.g., “the,” “of”) converge to their final vectors faster than lexical and low-frequency words, which retain some alignment with the bias in their random initializations. These findings help map the dynamic trajectory by which input embeddings organize around linguistic structure, revealing distinct roles for word frequency and function. Our findings motivate a deeper study of how the evolution of vocabulary geometry may facilitate specific capability gains during model training.