Beyond Perturbation Magnitude: Direction-Dependent Responses in Multimodal Geometric Representations

Abstract

Geometric alignment scores based on Gram determinants provide a compact way to model higher-order consistency among modalities, yet how such scores respond to modality degradation is poorly understood. This paper asks whether the response of a multimodal geometric score is determined primarily by the magnitude of the perturbation-induced displacement. Using frozen cohorts from MSR-VTT (N=878) and DiDeMo (N=980), we apply controlled video blur and audio noise and analyze the response in the relational geometry on which the score is defined. Displacement magnitude explains at most 15% of the out-of-sample variance in the absolute response, and magnitude-matched pairs respond systematically differently, so scalar magnitude does not organize the response. The closed-form first-order expansion of the Gramian volume yields the Directional Geometric Response (DGR): the projection of the displacement onto the local volume gradient, which jointly captures the clean operating point, displacement magnitude, and displacement direction. The absolute first-order DGR term explains the observed response with out-of-sample R^2 of 0.838-0.969, matched-magnitude ranking accuracies of 0.864-0.963, and response-sign accuracies of 0.909-0.989, whereas the tested direction-free alternatives remain weak or unstable under the corresponding evaluation protocols. A pre-specified gain-normalization candidate, V/(g_V+eps), fails its predictability and clean-order gates. DGR uses the observed degraded-state displacement and is therefore an explanatory quantity, not a deployment-time predictor: geometric response depends on where the representation operates, how far degradation moves the relational geometry, and in which direction it moves.