Hubness, the tendency for a few points to be among the nearest neighbours of a disproportionate number of other points, commonly arises when applying standard distance measures to high-dimensional data, often negatively impacting distance-based analysis.As autoregressive large language models (LLMs) operate on high-dimensional representations, we ask whether they are also affected by hubness.We first prove that the only large-scale representation comparison operation performed by LLMs, namely that between context and unembedding vectors to determine continuation probabilities, is not characterized by the concentration of distances phenomenon that typically causes the appearance of nuisance hubness.We then empirically show that this comparison still leads to a high degree of hubness, but the hubs in this case do not constitute a disturbance.They are rather the result of context-modulated frequent tokens often appearing in the pool of likely candidates for next token prediction.However, when other distances are used to compare LLM representations, we do not have the same theoretical guarantees, and, indeed, we see nuisance hubs appear.There are two main takeaways.First, hubness, while omnipresent in high-dimensional spaces, is not a negative property that needs to be mitigated when LLMs are being used for next token prediction.Second, when comparing representations from LLMs using Euclidean or cosine distance, there is a high risk of nuisance hubs and practitioners should use mitigation techniques if relevant.

When and why representations learned by different deep neural networks are similar is an active research topic. We choose to address these questions from the perspective of identifiability theory, which suggests that a measure of representational similarity should be invariant to transformations that leave the model distribution unchanged. Focusing on a model family which includes several popular pre-training approaches, e.g., autoregressive language models, we explore when models which generate distributions that are close have similar representations. We prove that a small Kullback--Leibler divergence between the model distributions does not guarantee that the corresponding representations are similar. This has the important corollary that models with near-maximum data likelihood can still learn dissimilar representations---a phenomenon mirrored in our experiments with models trained on CIFAR-10. We then define a distributional distance for which closeness implies representational similarity, and in synthetic experiments, we find that wider networks learn distributions which are closer with respect to our distance and have more similar representations. Our results thus clarify the link between closeness in distribution and representational similarity.

Diffusion models may be viewed as hierarchical variational autoencoders (VAEs) with two improvements: parameter sharing for the conditionals in the generative process and efficient computation of the loss as independent terms over the hierarchy. We consider two changes to the diffusion model that retain these advantages while adding flexibility to the model. Firstly, we introduce a data and depth-dependent mean function in the diffusion process, which leads to a modified diffusion loss. Our proposed framework, DiffEnc, achieves a statistically significant improvement in likelihood on CIFAR-10. Secondly, we let the ratio of the noise variance of the reverse encoder process and the generative process be a free weight parameter rather than being fixed to one. This leads to theoretical insights: For a finite depth hierarchy, the evidence lower bound (ELBO) can be used as an objective for a weighted diffusion loss approach and for optimizing the noise schedule specifically for inference. For the infinite-depth hierarchy, on the other hand, the weight parameter has to be one to have a well-defined ELBO.