Complex KDA: Understanding and Enhancing the Expressivity of Kimi Delta Attention

Julien Siems, Riccardo Grazzi, Korbinian Pöppel, Jaisidh Singh, Arber Zela +6

Linear RNNs based on the delta-rule enable efficient sequence modeling, but their linear updates with a low-rank correction constrain their expressivity. Prior work has shown that composing two delta-rule transitions in a single recurrent update can model a 2D rotation, but this increases the rank and the cost of the updates compared to a single transition. We show that Kimi Delta Attention (KDA) can realize 2D rotations by combining a single delta-rule transformation with a second reflection supplied by its channel-wise gate. This requires extending the parameter ranges of KDA by combining two existing range extensions: allowing gates in $[-1,1]$ and the delta-rule coefficient $β$ in $[0,2]$. We call the resulting model Complex KDA (CKDA). It preserves KDA's stability and efficiency, with transitions that remain diagonal-plus-rank-one and non-expansive, while reaching the state-tracking expressivity of DeltaProduct$_2$. We characterize the expressivity of CKDA and prove that every orthogonal diagonal-plus-rank-one matrix is exactly a CKDA transition matrix. A single CKDA layer can track every finite group isomorphic to a subgroup of $\mathrm{SO}(3)$, and many state-tracking…