Flow and diffusion-based generative models achieve state-of-the-art quality, but require multiple forward passes at inference time, making them computationally costly. Consistency models overcome this by learning a flow map directly, enabling few-step generation and dramatically reducing the computation needed at test time. However, formal statistical guarantees for such one-step generators are scarce. Existing results either suffer from the curse of dimensionality or rely on impractical loss functions. We investigate whether one-step flow-based models can escape the curse of dimensionality under the assumption that the data distribution is generated by a low-dimensional surface perturbed with Gaussian noise. In this setup, we derive approximation and generalization error bounds for the flow map estimate that circumvent the curse of dimensionality. Our findings provide the first principled explanation for practical success of one-step flow-based generative models.

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