Interpolating distributions on S1

Let f be a discrete distribution on the unit circle, defined on a domain ℙ = {pi}i=1k, p i ∈ S1, thus f = {f i}i=1k for f i = f(pi) On another set of points {qj}j=1k, q j ∈ S1, called observation set, we define the corresponding to f set of covariances

k 1-∑ --→ --→ T Σ [f ](qj) = k (qjpi)(qjpi) f (pi). i=1

We consider two discrete distributions f1 and f2 on the unit circle, defined on the domain ℙ with corresponding set of covariances

Σs (qj) = Σ[fs](qj), s = 1,2.
Any similarity invariant h introduces a pseudo-distance between F’s
k ∑ dh(f1,f2) = h (Σ1(qj),Σ2(qj)). j=1
It is pseudo because the triangular inequality may eventually fail.

Next, we propose an interpolation criterion based on the pseudo-distance dh. For a α ∈ [0, 1], define

H (f;α ) = (1 - α )dh(f,f1) + αdh(f,f2)
and
fˆα = argmin H (f ;α). f
It turns out that ˆf α is continuous in α at all points where H(f; α) has a well separated minimum.

Similarity invariants h for which H(f; α) is convex functional is of particular interest for they can be easily optimized and for which the interpolation problem has a unique solution. For example, the invariant

-1 -1 hloglik(A, B) = tr(AB ) - ln(det(AB )) - 2
also known as normal log-likelihood, gives us a nice convex interpolation problem.

The following applet implements the proposed interpolation method for hloglik. The solution fα is found by gradient descent algortithm. The user can specify the number of iterations and gradient update step size. We report the minimum value achieved H(fˆ ,α) as well as the convergence curve of the gradient descent for diagnostic. When the optimal solution is achieved, this curve is decreasing and smoothly converges to the horizontal. Otherwise one may want to adjust the step size and the number of iterations.

What we show are the initial two distributions f1 and f2 in blue and red, the linear interpolation (1 - α)f1 + αf2 in gray and our solution ˆ f α in green. The inner circle shows the distributions domain of size k that is uniform on S1 and the observation points qj as black dots.

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