# Approximating Uniform Distribution with Mixture of Gaussians

Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$.

Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$: $$\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n \subset T$$

I would like to approximate continuous uniform distribution $u(\boldsymbol{x})$ over $T$ with mixture of Gaussians $g(\boldsymbol{x})$, having general, not necessarily diagonal covariances, and for which means of components are given by the set $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$, that is: $$g(\boldsymbol{x}) = \sum_{i=1}^{n} \pi_{i} \mathcal{N}(\boldsymbol{x} \,|\, \boldsymbol{\mu}_{i},\Sigma_{i})$$

We can assume that, informally speaking, means $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ are "equispaced" over $T$ or "nice" enough such that good approximation exists.

Any ideas on this?

The way I want to attack this is by minimizing some sort of computationally tractable divergence (Kullback–Leibler divergence, for example) between $u$ and $g$.

I'm curious if something similar was attempted before (searching the web didn't yield anything)?

Or could you point me to the relevant literature on this? Basically, any suggestions would be of help.

Motivation: this problem arises as an inverse problem in radiotherapy treatment planning. $T$ is considered to be a tumor and needs to receive radiation dose that is uniformly distributed over $T$. Dose spillage outside $T$ is undesirable. In this case the problem of designing such treatment can be reduced to estimating values of weights $\; \pi_1,\, \dots,\, \pi_n\;$ and covariance matrices $\; \Sigma_1,\, \dots,\, \Sigma_n$.

• A major problem with this suggestion is that the Normal components are distributed over $\mathbb{R}^3$ rather than the support $T$. – Xi'an Mar 18 '17 at 12:33
• @Xi'an The context suggests the intention is to truncate these Gaussians to $T$, but indeed this deserves clarification from the OP, especially in light of the "dose spillage remark." That suggests a different abstract formulation of the problem, to wit: find $\pi_i$ and $\Sigma_i$ for which $g|_T$ stays within acceptable bounds while some norm of $g$ restricted to the complement of $T$ does not exceed a given threshold. It's difficult to see why uniformity per se should be the objective. It's surprising, too, that $n$ and the $\mu_i$ are prescribed and not free variables. – whuber Mar 18 '17 at 13:14
• @whuber, No formulation, as stated, is to approximate $u$ with $g$, or more precisely find $\text{argmin}_{\left\{\pi_i \right\}_{i=1}^n,\, \left\{\Sigma_i \right\}_{i=1}^n} {d\left( u, g \right)}$, where $d(\, \cdot \,,\, \cdot \, )$ is some divergence between $u$ and $g$ over all of $\mathbb{R}^3$, that also needs to be determined. – aberdysh Mar 18 '17 at 16:09
• @whuber, If allowing $n$ and $\mu_i$ to vary freely makes solving problem (formulated as above) easier, then that is also fine – aberdysh Mar 18 '17 at 16:16
• Uniform per se, comes from the nature of the problem, where we want to deliver uniformly distributed radiation dose inside the tumor $T$, while minimizing exposure to radiation of healthy tissues outside the tumor. – aberdysh Mar 18 '17 at 16:21