Interpolating Distributions on the Sphere, S2

Here we implement interpolation of probability mass functions (pmf) on the unit 2-sphere. The underlying theory is exposed here.

The user may choose a family of distributions, Normal or Fisher-von-Mises, sample size, k, and similarity invariant. Three choices for invariant are available

hloglik(X ) = tr(X ) - log(det(X )) - 2,
htrsq(X ) = tr2(X - I2),
- 12 htrsqinv(X ) = tr (X - X ) .

The optimization problem we try to solve is the following. For a given α in[0, 1], distribution support set {pi}i=1k and observation set {q j}j=1k, find

∑k ˆfh = argminf {(1 - α)γh (G Σ [f ],G Σ1 )|qj + αγh(G Σ [f ],G Σ2)|qj}, j=1
where sample covariances are Σs(qj) = 1 k ∑ i=1k(--→ qjpi)(--→ qjpi)T f s(pi), s = 1, 2.

The optimization is based on gradient discent and three parameters need to be specified, the number of iterations, gradient step size and γ. We report the convergence curve, which serves as an indicator for the success of the descent, and the values of H(f; α) for the linear combination ˆf lin of f1 and f2 and the found ˆfh.

In the display window the user sees, from top left to botton right, f1, f2, flin and ˆfh. By dragging on the display area, one can simultaneously rotate the four spheres and visually compare the results.

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