A growing body of work on KV-cache compression aims to reduce the memory requirements of autoregressive generation. These methods compress the stored context while approximately preserving attention computation, enabling generation with reduced memory consumption. Almost tight theoretical results on the worst case space complexity of attention have recently been obtained, but they are achieved by instances where queries can be perfectly aligned with keys. This directly contradicts the observed anisotropy of the key/query geometry in real transformers. In particular, it has been shown empirically that key/query cosine similarities are small even for most correlated pairs, i.e., near-perfect alignment of queries with keys does not happen in practice. In this paper, we study the space complexity of attention approximation under the assumption that key/query cosine similarities belong to a small interval around zero, i.e., a query cannot align too much with any key. We give a nearly space optimal algorithm and a corresponding space lower bound. Surprisingly, the distribution of key/query cosine similarities in our lower bound exhibits the anisotropy property empirically established and studied in the literature. Additionally, we show that the simple uniform sampling algorithm is nearly optimal for a typical range of parameters, supporting its surprising effectiveness on real datasets.

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