Fisher Bingham (Isotropic) case
For my current research I am trying to find the parameters of the product of 2 ASG (anisotropic spherical Gaussians).
A common spherical Gaussian formula (SG) is:
$ae^{\lambda(\mu\cdot v - 1)}$
Where $\mu$ is the lobe of the distribution the sphere:
i.e in the following (figure is a wip) image it would be the vector pointing to the center of the white region:
[![enter image description here][1]][1]
The problem I am trying to solve is, find the the new lobe and $\lambda$ of the resulting product. With this formula it's very simple:
$a_1e^{\lambda_1(\mu_1\cdot v - 1)} \cdot a_2e^{\lambda_2(\mu_2\cdot v - 1)}$
$= a_1a_2e^{\lambda_1(\mu_1\cdot v - 1) + \lambda_2(\mu_2\cdot v - 1)}$
$= a_1a_2e^{\lambda_1\mu_1\cdot v - \lambda_1 + \lambda_2\mu_2\cdot v - \lambda_2}$
Let $u = \frac{\lambda_1\mu_1 + \lambda_2\mu_2}{|\lambda_1\mu_1 + \lambda_2\mu_2|}$. Note $u$ will be well defined as long as the inner angle between the 2 vectors is less than $\pi$. We get:
$= a_1a_2e^{|\lambda_1\mu_1 + \lambda_2\mu_2|u\cdot v - \lambda_1 - \lambda_2}$
$= a_1a_2e^{|\lambda_1\mu_1 + \lambda_2\mu_2|(u\cdot v - 1) - \lambda_1 - \lambda_2 + |\lambda_1\mu_1 + \lambda_2\mu_2|}$
Let's introduce $C$ to eat away all the constant factors and we get:
$= Ce^{|\lambda_1\mu_1 + \lambda_2\mu_2|(u\cdot v - 1)}$
Which is a new SG with lobe $u$ and sharpness $|\lambda_1\mu_1 + \lambda_2\mu_2|$
Anisotropic Case (Xu et al)
However this distribution is isotropic. I am interested in the ASG distribution introduced by Xu et al.
$c\max(0,v\cdot z)e^{-\lambda(v\cdot x)^2 - \mu(v\cdot y)^2}$
In this case $c, \lambda, \mu$ are constants and $x,y,z$ form an orthonormal basis of $\mathbb{R}^3$
I have been trying to find an explicit representation of the parameters for this distribution as well but I am stuck. The first thing is, I know this representation exists because given 2 ASG's:
You get this as their product:
(I had to greatly intensify the color otherwise it looks black)
I did multiple experiments and it seems that for every pair of Gaussians there is a well defined new lobe.
I don't seem to find resources on this topic anywhere.
What I have so far:
Let's focus on the non exponential term $V\cdot Z$. Let's assume we have 2 ASG with respective lobes $Z_1, Z_2$.
We are searching for a solution with respect to $Z'$ to:
$(V \cdot Z_1)(V\cdot Z_2) = V\cdot Z'$
So we get:
$V^TZ_1V^TZ_2 = V^TZ'$
$\iff Z_1^TVV^TZ_2 = V^TZ'$ (by commutativity of the dot product)
$VV^T$ is a symmetric matrix (trivial to show), and when $v = <0,0,1>$
We get that the third component of $Z' = Z'_z = Z_{1z}Z_{2z}$
If we restrict $V$ to just be the standard basis of $\mathbb{R}^3$
We get that $Z'=<Z_{1x}Z_{2x},Z_{1y}Z_{2y},Z_{1z}Z_{2z}>$
The problem with that solution is A) it's not a unit vector so not on the surface of the sphere, and B) this doesn't work for values of $V$ other than the standard basis.
[1]: https://i.stack.imgur.com/uGE Eb.png