On State Reduction in Linear Attention

Linear attention offers a computationally efficient yet expressive alternative to softmax attention. However, recent empirical results indicate that the hidden state of trained linear attention models often exhibits a low-rank structure, suggesting that these models underexploit their capacity in practice. To understand this phenomenon, we analyze how keys and values shape the rank of the recurrent state, providing a theoretical perspective on memory utilization in linear attention. In addition to these theoretical insights, we conjecture that low-rank states can be substantially reduced after pre-training. To this end, we propose a hardware-aware strategy to structurally prune the key and query matrices, reducing the state size while retaining compatibility with existing fast kernels. We adapt several existing pruning strategies to fit our framework and, building on our theoretical analysis, propose a robust structured pruning method based on a rank-revealing QR decomposition. Our empirical evaluations across model sizes show that the recurrent state of pretrained linear attention models can often be halved with only a modest increase in language modeling perplexity, while maintaining competitive performance on downstream tasks. The code for this project can be found at https://github.com/camail-official/LinearAttentionPruning.