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In monocular visual-inertial navigation, it is desirable to initialize the system as quickly and robustly as possible. A state-of-the-art initialization method typically constructs a linear system to find a closed-form solution using the image features and inertial measurements and then refines the states with a nonlinear optimization. These methods generally require a few seconds of data, which however can be expedited (less than a second) by adding constraints from a robust but only up-to-scale monocular depth network in the nonlinear optimization. To further accelerate this process, in this work, we leverage the scale-less depth measurements instead in the linear initialization step that is performed prior to the nonlinear one, which only requires a single depth image for the first frame. Importantly, we show that the typical estimation of all feature states independently in the closed-form solution can be modeled as estimating only the scale and bias parameters of the learned depth map. As such, our formulation enables building a smaller minimal problem than the state of the art, which can be seamlessly integrated into RANSAC for robust estimation. Experiments show that our method has state-of-the-art initialization performance in simulation as well as on popular real-world datasets (TUM-VI, and EuRoC MAV). For the TUM-VI dataset in simulation as well as real-world, we demonstrate the superior initialization performance with only a 0.3 s window of data, which is the smallest ever reported, and validate that our method can initialize more often, robustly, and accurately in different challenging scenarios.