Multiplicative gating is widely used in neural architectures and has recently been applied to attention layers to improve performance and training stability in large language models. Despite the success of gated attention, the mathematical implications of gated attention mechanisms remain poorly understood. We study attention through the geometry of its representations by modeling outputs as mean parameters of Gaussian distributions and analyzing the induced Fisher--Rao geometry. We show that ungated attention operator is restricted to intrinsically flat statistical manifolds due to its affine structure, while multiplicative gating enables non-flat geometries, including positively curved manifolds that are unattainable in the ungated setting. These results establish a geometric expressivity gap between ungated and gated attention. Empirically, we show that gated models exhibit higher representation curvature and improved performance on tasks requiring nonlinear decision boundaries whereas they provide no consistent advantage on tasks with linear decision boundaries. Furthermore, we identify a structured regime in which curvature accumulates under composition, yielding a systematic depth amplification effect.

We consider the problem of localizing a set of microphones together with a set of external acoustic events (e.g., hand claps), emitted at un- known times and unknown locations. We propose a solution that ap- proximates this problem under a far field approximation defined in the calculus of affine geometry, and that relies on singular value decompo- sition (SVD) to recover the affine structure of the problem. We then define low-dimensional optimization techniques for embedding the solu- tion into Euclidean geometry, and further techniques for recovering the locations and emission times of the acoustic events. The approach is use- ful for the calibration of ad-hoc microphone arrays and sensor networks.