Geometry-Aware Attention with a Shared Second-Order Feature Matrix for Lightweight Image Super-Resolution

Existing lightweight single-image super-resolution (SISR) methods often rely on standard dot-product attention when explicit token-to-token similarity is modeled. Although the query and key projections are learned, the similarity in the projected coordinates is still evaluated by the Euclidean inner product. In this work, we propose GAASR, a lightweight SISR network that augments window-based attention with a shared, data-conditioned second-order feature matrix. For each attention layer, an uncentered second-order descriptor is pooled across the windows and heads of the current forward pass and combined with a learnable base matrix. The resulting symmetric matrix is shared within that layer and is used in two complementary ways: to linearly transform query features and to precondition the token-aligned query–key residual through a linear solve. A trace-derived scalar then controls the contribution of the residual correction before the standard dot-product attention is evaluated. This design introduces only small feature-dimension-dependent overhead and preserves the dominant quadratic dependence of window attention on the number of tokens. The proposed mechanism is formulated using symmetric matrix transformations and linear-system solving while retaining the standard dot-product attention structure. Experiments on multiple standard SISR benchmarks show consistent reconstruction gains under comparable parameter counts and reported computational budgets, with noticeable improvements on texture-rich and high-frequency regions.