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Recent advances in diffusion and flow matching models have highlighted a shift in the preferred prediction target -- moving from noise ($\varepsilon$) and velocity (v) to direct data (x) prediction -- particularly in high-dimensional settings. However, a formal explanation of why the optimal target depends on the specific properties of the data remains elusive. In this work, we provide a theoretical framework based on a generalized prediction formulation $u=kx-(1-k)n$, where $x$ represents clean data and $n$ denotes noise. This formulation accommodates arbitrary targets, of which $\varepsilon$-, v-, and x-prediction are special cases. We derive the analytical relationship between data's geometry and the optimal prediction target, offering a rigorous justification for why x-prediction becomes superior when the ambient dimension significantly exceeds the data's intrinsic dimension. Furthermore, our controlled analysis reveals how time conditioning and loss variations systematically modulate this optimal objective. While our theory identifies dimensionality as the governing factor for the optimal prediction target, the intrinsic dimension of manifold-bound data is typically intractable to estimate in practice. To bridge this gap, we propose k-Diff, a framework that employs a data-driven approach to learn the optimal prediction parameter k directly from data, bypassing the need for explicit dimension estimation. Extensive experiments in both latent-space and pixel-space image generation demonstrate that k-Diff consistently outperforms fixed-target baselines across varying architectures and data scales, providing a principled and automated approach to enhancing generative performance.

Language-guided UAV navigation has largely focused on long-horizon route planning, leaving short-range, object-relative flight maneuvers underexplored. To address this gap, we formulate Flying-on-a-Word (Flow), a task requiring a UAV to convert concise natural-language instruction, egocentric visual observation, and current state dynamics into continuous flight actions. Flow encompasses both primitive motion commands and complex object-interactive behaviors, assessing motion intent comprehension and spatial context grounding. To enable policy learning and closed-loop evaluation, we introduce UAV-Flow, a comprehensive dataset comprising 30,692 expert-piloted real-world trajectories across eight distinct motion categories, alongside 10,109 simulation trajectories in the UAV-Flow-Sim environment equipped with task-specific completion criteria. Furthermore, we develop FlowVLA, which undergoes imitation pretraining on mixed real-sim demonstrations and online post-training via Tail-Aware GRPO (TGRPO). By leveraging group reward distributions, TGRPO effectively reinforces high-reward action sequences. Evaluated on 273 held-out UAV-Flow-Sim episodes, pretrained FlowVLA achieves a mean success rate of 75.58%, outperforming the strongest baseline by 6.72 percentage points, which TGRPO post-training further elevates to 93.16% with superior sample efficiency. Physical UAV experiments demonstrate robust execution of object-relative commands in both indoor and outdoor real-world scenes. Together, Flow establishes a novel benchmark and learning framework for language-conditioned flight maneuvers.