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Background:

In Euclidean space, a simple and easy to compute distance metric between two vectors $\mathbf{u}$ and $\mathbf{v}$, is the cosine similarity $ \mathbf{u} \cdot \mathbf{v} / |\mathbf{u}| |\mathbf{v}| $. On an intuitive level, it's nice since measures the angle between the two vectors.

Question:

I have two geodesics on a hypersphere $\mathbf{a}, \mathbf{b}$. Since they are on a hypersphere, the geodesics take the form of a great circle. Is there an easy to compute distance metric that can compare the similarity between two great circles in high-dimensional space?

Current Attempts:

The geodesics that I'm using are found using Slerp, thus they are realized by a finite set of points. Hence I'm approximating the true geodesic by a series of small line segments. My distance measure is then the average cosine similarity for the two paths by taking equal steps along the parametrization.

Hooked
  • 6,785
  • How about the angle between the two geodesics at their point(s) of intersection? – Amitai Yuval Jun 25 '15 at 14:40
  • @AmitaiYuval there is absolutely no guarantee that they intersect, in fact most of them don't. – Hooked Jun 25 '15 at 14:46
  • Of course, in high dimension they need not intersect. How about the usual distance between two (compact) disjoint sets then? – Amitai Yuval Jun 25 '15 at 14:51
  • @AmitaiYuval maybe I misunderstood your comment, but I'm not interested in the distance between the two sets, rather I'm looking to see if the two paths moving along the hypercoordinates in the same way. I know that this is vague (hence the question), but imagine a normal two sphere: you can parameterize it by spherical coordinates $\phi, \psi$. Two arcs would be 100% similar if the change along the parametrized paths would invoke approximately the same change in these coordinates. – Hooked Jun 25 '15 at 14:57

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