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The geometry of data: the missing metric tensor and the Stein score [Part II]
Note: This is a continuation of the previous post: Thoughts on Riemannian metrics and its connection with diffusion/score matching [Part I], so if you haven't read it yet, please consider reading as I won't be re-introducing in depth the concepts (e.g., the two scores) that I described there already. This article became a bit long,
While the tangent space provides a linear approximation of the manifold at a point, it still doesn’t allow us to be able to define lengths, vectors or angles between them (we still cannot calculate an inner product), for this we will need the metric tensor, which we will talk about it below. This metric tensor is basically the core of information geometry, which is dealing with distribution parameters as the point \(p\), allowing you to compute inner products, distances, lengths, geodesics, angles in the statistical manifold. Note now how the path coming from the left is bending to take advantage of the curvature at the bottom of the plot and then we can see that the geodesic starts to be “attracted” by the regions of high data density.
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