The transitivity of correlation (or of the inner product on the sphere) is a reformulation of the triangle inequality for spheres with the Riemannian metric (and also for the surface of the unit ball in a real Hilbert space).
Let d be the Riemannian metric on the sphere Sn−1={x∈Rn:∥x∥=1}. The value d(x,y) is just the length of the shortest path ℓ:[0,1]→Sn−1 with ℓ(0)=x,ℓ(1)=y. If x,y∈Sn−1, then d(x,y)>∥x−y∥ since the path ℓ cannot be a straight line segment in Rn. The metric d can also be defined by d(x,y)=cos−1(⟨x,y⟩). From the triangle inequality, if x,y,z∈Sn−1, we know that cos−1(⟨x,z)⟩=d(x,z)≤d(x,y)+d(y,z)=cos−1⟨x,y⟩+cos−1⟨y,z⟩. Therefore, by applying the cosine function (which is order reversing) to this inequality, we have ⟨x,z⟩=cos(cos−1(⟨x,z⟩))≥cos(cos−1⟨x,y⟩+cos−1⟨y,z⟩)
The transitivity of correlation (or of the inner product on the sphere) is a reformulation of the triangle inequality for spheres with the Riemannian metric (and also for the surface of the unit ball in a real Hilbert space).
Let d be the Riemannian metric on the sphere Sn−1={x∈Rn:∥x∥=1}. The value d(x,y) is just the length of the shortest path ℓ:[0,1]→Sn−1 with ℓ(0)=x,ℓ(1)=y. If x,y∈Sn−1, then d(x,y)>∥x−y∥ since the path ℓ cannot be a straight line segment in Rn. The metric d can also be defined by d(x,y)=cos−1(⟨x,y⟩). From the triangle inequality, if x,y,z∈Sn−1, we know that cos−1(⟨x,z)⟩=d(x,z)≤d(x,y)+d(y,z)=cos−1⟨x,y⟩+cos−1⟨y,z⟩. Therefore, by applying the cosine function (which is order reversing) to this inequality, we have ⟨x,z⟩=cos(cos−1(⟨x,z⟩))≥cos(cos−1⟨x,y⟩+cos−1⟨y,z⟩)
=cos(cos−1(⟨x,y⟩)cos(cos−1(⟨y,z⟩)+sin(cos−1(⟨x,y⟩)sin(cos−1(⟨y,z⟩)
=⟨x,y⟩⋅⟨y,z⟩−√1−⟨x,y⟩2⋅√1−⟨y,z⟩2.
For the reverse inequality, we apply cosine to the inequality d(x,z)≥d(x,y)−d(y,z) to obtain
⟨x,z⟩≤⟨x,y⟩⋅⟨y,z⟩+√1−⟨x,y⟩2⋅√1−⟨y,z⟩2.