Norm$\times$Direction: Restoring the Missing Query Norm in Vision Linear Attention

Linear attention mitigates the quadratic complexity of softmax attention but suffers from a critical loss of expressiveness. We identify two primary causes: (1) The normalization operation cancels the query norm, which breaks the correlation between a query's norm and the spikiness (entropy) of the attention distribution as in softmax attention. (2) Standard techniques for enforcing non-negativity cause destructive information loss by nullifying valid inner-product interactions. To address these challenges, we introduce **NaLaFormer**, a novel linear attention mechanism built upon a norm$\times$direction (ND) decomposition of the query and key vectors. We leverage each component to solve a distinct problem: The *query norm* is injected into our kernel to create a query-norm-aware map that restores the attention distribution's spikiness. The *direction vectors* are processed by a geometric, cosine-based similarity metric that guarantees non-negativity while preserving the rich, fine-grained information of the inner product. We validate NaLaFormer through a comprehensive multi-modal evaluation, where it sets new state-of-the-art benchmarks for linear attention. Our model achieves up to a 7.5\% accuracy gain on ImageNet-1K and a 4.7\% mIoU improvement on ADE20K over comparable baselines. It demonstrates profound efficiency, reducing peak memory by a transformative 92.3\% in token-intensive super-resolution tasks (70K+ tokens). NaLaFormer's versatility is further confirmed as it surpasses strong baselines like Mamba on common-sense reasoning and sets a new state-of-the-art on the Long Range Arena (LRA) benchmark. Source code can be found in the supplementary materials.