Linear attention reduces the quadratic sequence-length complexity of softmax attention by maintaining fixed-size recurrent states. However, these states scale quadratically with the representation dimension and require repeated data transfers during decoding, creating a substantial GPU memory bottleneck. Recent studies have revealed significant low-rank redundancy in recurrent states, suggesting opportunities for more efficient representations. We introduce SPARE (State Projection for Accelerated Recurrent Execution), a training-free method that exploits this redundancy through fixed low-dimensional subspaces, heterogeneous rank allocation, and lightweight write compensation. By performing recurrent reads and updates directly on compact states, SPARE reduces memory traffic without online decomposition or dense-state reconstruction. SPARE achieves 2.55x full-model decoding speedup, 7.49x lower recurrent-kernel DRAM traffic, and 3.31x lower peak GPU memory, while preserving better associative-retrieval accuracy than structured pruning at matched memory budgets.

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