A Hub of Short Rows Inflates Intrinsic Dimension Estimation of Token Embeddings
TL;DR
A token-embedding table holds a hub of short rows near its origin, and it is shown that this cluster biases what nearest-neighbor intrinsic-dimension (ID) estimators report, and it is shown that normalizing the rows instead of removing them gives the same lower reading.
Abstract
A token-embedding table holds a hub of short rows near its origin, and we show that this cluster biases what nearest-neighbor intrinsic-dimension (ID) estimators report. Because of the concentration of measure, a token is closer to the central cluster than to any other token, so its first two neighbors are both hub rows at nearly the same distance. As a result, the ID estimators such as TwoNN return a dimension far above the real ID. Measured one token at a time, dimension is a heavy-tailed distribution. Measured on the full vocabulary, it grows with the model's parameter count. However, when we remove the hub, the heavy tail disappears and the measured dimension collapses to a narrow range for eleven models, from GPT-2 to models such as K3 and GLM-4.7. The hub acts as a switch: a few hundred rows are enough to fully inflate the estimate. We reproduced an experiment stating that the intrinsic dimension (ID) of Pythia's token-embedding table grows with the parameter count, from $27$ to $122$ between 160M and 12B parameters. We show that this result disappears when the hub is removed: the table then reads $10$ to $17$ at every size. The hub contains a subset of the population that under-trained-token detectors flag, but on Pythia the hub that we detected and removed as a whole was updated during training: what seem to characterize these rows is simply their length, not an absence of updates. Finally, we show that normalizing the rows instead of removing them gives the same lower reading.
