Flip Probabilities for Random Projections of Θ-separated Vectors

We give the probability that two vectors in d-dimensional Euclidean space m,n ∈ R which are separated in R by an angle θ ∈ [0, π/2] have angular separation θR > π/2 following random projection into a k-dimensional subspace of R, k < d. This probability, which we call the ‘flip probability’, has several interesting properties: It is polynomial of order k in θ; it is independent of the original dimensionality d depending only on the projection dimension k and the original separation θ of the vectors; it recovers the existing result for flip probability under random projection when k = 1 as a special case, and has a geometric interpretation as the quotient of the surface area of a hyperspherical cap by the area of the corresponding hypersphere which is a natural generalisation of the k = 1 case. We also prove the useful fact that, for all k ∈ N, the flip probability when projecting to dimension k is greater than the flip probability when projecting to dimension k + 1. 1. Statement of Theorem and Proof Theorem 1 (Flip Probability). Let n, m ∈ R with angular separation θ ∈ [0, π/2]. Let R ∈ Mk×d, k < d, be a random projection matrix with entries rij iid ∼ N (0, 1/d) and let Rn, Rm ∈ R be the projections of n, m into R with angular separation θR. (1) The ‘flip probability’ PrR[θR > π/2|θ] = PrR[(Rn)Rm < 0|nm > 0] is given by: PrR[(Rn) Rm < 0|nm > 0] = Γ(k) (Γ(k/2))2 ∫ ψ 0 z(k−2)/2 (1 + z)k dz (1.1) where ψ = (1− cos(θ))/(1 + cos(θ)). (2) The expression above can be shown to be of the form of the quotient of the surface area of a hyperspherical cap subtending an 1991 Mathematics Subject Classification. Primary 60D05; Secondary 15B52.