Microsoft:

"Our algorithm first reconstructs the 3D input camera path as well as dense, per-frame proxy geometries. We then optimize a novel camera path for the output video (shown in red) that is smooth and passes near the input cameras while ensuring that the virtual camera looks in directions that can be rendered well from the input."

Instagram:

"... Smartphones didn’t have nearly enough power to replicate video-editing software, but they did have built-in gyroscopes. On a smartphone, instead of using power-hungry algorithms to model the camera’s movement, he could measure it directly. And he could funnel those measurements through a simpler algorithm that could map one frame to the next, giving the illusion that the camera was being held steady"

What it looks like the Instagram app is doing is selecting and stabilizing individual frames instead of the full 3D scene reconstruction that the MSFT paper described. EG there's no pop in and out of details like the MSFT demos and there's still a small but noticeable judder in some of the videos.