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Question in brief

Let $a$ and $b$ be unit vectors in $\mathbb{R}^d$. Let $f$ be the $1-step$ transition function of a random walk on the $d$ dimensional unit sphere. I am interested in evaluating $\mathrm{E}[\exp(a^T f(b))]$ and one of things I want to know is any transition function/distribution exist where this quantity can be computed in closed form? The transition function should not be "trivial" e.g. it should be dependent on the input and it should not be identity.

More Details

I will also love to know which papers/textbooks deal with this problem? If there is no way to get an exact form, then what methods have people used to study the behavior of $\mathrm{E}[\exp(a^Tf(b))]$ as a function of $a$ and $b$? Is it possible to compute this or approximate this when $f$ is a random orthonormal matrix from the Haar measure?

What I did

I tried to evaluate this quantity in two "simple" cases but both of them turned out to be intractable. In the first case, I assumed $f$ to be a random householder reflection, so I assumed that the walk proceeded by first sampling a vector $x$ from $U_{S^{d-1}}$, i.e. the uniform distribution over $S^{d-1}$. Then $f = I - 2xx^T$. But finding the marginal of a single projection of $U_{S^{d-1}}$ is itself difficult, and I could not figure out the value of $\mathrm{E}[\exp( (a^Tx)(b^Tx))]$ which requires a joint distribution of two projections of $x$.

The second case I tried was with random givens rotations. I chose the indices $i,j$ uniformly randomly from the $\binom d2$ possible pairs and then chose the rotation angle $\theta$ from a distribution that was symmetric about $\pi$. But this formulation boils down to the following computation $\mathrm{E}_{\theta}[\exp(\alpha \cos(\theta) + \beta \sin(\theta))]$. I couldn't think of any distribution over $\theta$ such that this expression could be evaluated.

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Just found out that $\int_0^{2\pi} \exp(\alpha \cos\theta + \beta \sin\theta) = 2\pi I_0(\alpha^2 + \beta^2)$. So random givens rotations seem to be amenable to analysis.

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