###### tags: `one-offs` `rejection sampling` `isoperimetry`

# Isoperimetric Surrogates for the Gaussian

**Overview**: In this note, I outline some calculations relating to some one-dimensional probability measures which are close in character to the standard Gaussian measure, but which retain certain tractabilities which may be of interest.

## Motivation

To begin with, I was interested in cooking up some probability measures in one dimension which qualitatively resemble the Gaussian measure (in terms of e.g. concentration, isoperimetry, etc.), but are a bit more { tractable / tangible / etc. } in terms of analytic PDF, CDF, etc.

Anyways, it is known that the isoperimetric profile (which I won't describe, for now) of the standard Gaussian satisfies

\begin{align}
I\left(p\right) &=\left( \varphi\circ\Phi^{-1} \right) \left(p\right) \\
&\sim p \cdot \left(2\cdot\log\left(\frac{1}{p}\right)\right)^{1/2} \quad \mathrm{as} \,p\to0^{+}.
\end{align}

Using some well-known relationships, one can then reverse-engineer probability measures with this form of isoperimetric profile. In particular, for symmetric, log-concave probability measures in dimension one with density $f$ and cumulative distribution function $F$, it is known that

\begin{align}
I=f\circ F^{-1}\leadsto F^{'}=I\circ F.
\end{align}

As such, taking $I\left(p\right) = p \cdot \left(2\cdot\log\left( \frac{1}{p}\right) \right)^{1/2}$ leads to

\begin{align}
\frac{\mathrm{d}F}{F\cdot\left(2\cdot\log\left(\frac{1}{F}\right)\right)^{1/2}} =\mathrm{d}x\quad\leadsto\quad F\left(x\right)=\exp\left(-\frac{1}{2}\left(x-\sqrt{2\cdot\log2}\right)^{2}\right)
\end{align}

for $x\leqslant0$, and hence by symmetry, $F\left(x\right)=1-\exp\left(-\frac{1}{2}\left(x+\sqrt{2\cdot\log2}\right)^{2}\right)$ for $x\geqslant0$.

## Generalising and Checking

This suggests considering symmetric probability measures satisfying

\begin{align}
\forall r \geqslant 0, \quad \mathbf{P}\left(\left|X\right|\geqslant r\right)=\exp\left(\frac{1}{2}a^{2}-\frac{1}{2}\left(a+r\right)^{2}\right)
\end{align}

for some $a > 0$. These measures have density

\begin{align}
\mu_{a}\left(\mathrm{d}x\right) &= f_{a}\left(x\right)\:\mathrm{d}x \\
f_{a}\left(x\right) &= \frac{1}{2} \cdot \left(a+\left|x\right|\right)\cdot\exp\left(\frac{1}{2}a^{2}-\frac{1}{2}\left(a+\left|x\right|\right)^{2}\right)
\end{align}

and cumulative distribution function

\begin{align}
F_{a}\left(x\right)=\begin{cases}
\frac{1}{2}\cdot\exp\left(\frac{1}{2}a^{2}-\frac{1}{2}\left(a+\left|x\right|\right)^{2}\right) & x\leqslant0\\
1-\frac{1}{2}\cdot\exp\left(\frac{1}{2}a^{2}-\frac{1}{2}\left(a+\left|x\right|\right)^{2}\right) & x\geqslant0
\end{cases}.
\end{align}

In particular, the isoperimetric profile can then be read off as (for $p\in\left[0,\frac{1}{2}\right]$)

\begin{align}
I_{a}\left(p\right) &=\left(f_{a}\circ F_{a}^{-1}\right)\left(p\right) \\
&=p\cdot\left(2\cdot\log\left(\frac{\alpha_{a}}{p}\right)\right)^{1/2}
\end{align}

with $\alpha_{a}=\frac{1}{2} \cdot \exp \left( \frac{1}{2}a^{2} \right)$.

So, having constructed these measures with a goal in mind, we ask the natural question: will this probability measure be qualitatively close to the Gaussian in all relevant ways?

## Log-Concavity

There is one important detail which turns out to be sensitive to the precise choice of $a$: the log-concavity of the measure. Many important geometric properties of probability measures are particularly easy to work with for log-concave measures, and so it is of interest to ensure this. It will turn out that $a=1$ is a critical value for the log-concavity of $\mu_{a}$. Note that just differentiating $\log f_{a}$ twice to check for concavity is a risky strategy, due to the non-smoothness of $f_a$. Still, write

\begin{align}
-\log f_{a}\left(x\right) &=-\log\left(\frac{1}{2}\cdot\left(a+\left|x\right|\right)\cdot\exp\left(\frac{1}{2}a^{2}-\frac{1}{2}\left(a+\left|x\right|\right)^{2}\right)\right) \\
&=-\log\left(1+\frac{\left|x\right|}{a}\right)+\frac{1}{2}\left(a+\left|x\right|\right)^{2}-\frac{1}{2}a^{2}-\log a+\log2.
\end{align}

Now, $x\mapsto-\log\left(1+\frac{\left|x\right|}{a}\right)$ is convex on either side of $x=0$, but not globally. However, one can show that $x\mapsto\frac{\left|x\right|}{a}-\log\left(1+\frac{\left|x\right|}{a}\right)$ **is** globally convex, and so we regroup terms to see that

\begin{align}
-\log f_{a}\left(x\right) =&\left(\frac{\left|x\right|}{a}-\log\left(1+\frac{\left|x\right|}{a}\right)\right)+\frac{1}{2}\left(\left(a-a^{-1}\right)+\left|x\right|\right)^{2} \\
&-\frac{1}{2}\left(a-a^{-1}\right)^{2}-\log a+\log2.
\end{align}

We have just claimed that $x\mapsto\frac{\left|x\right|}{a}-\log\left(1+\frac{\left|x\right|}{a}\right)$ is convex, and one can check directly that $x\mapsto\left(b+\left|x\right|\right)^{2}$ is convex precisely when $b\geqslant0$; the remaining terms are constant in $x$ and so the claim follows. For $a\in\left(0,1\right)$, one can check directly by inspection that $-\log f_{a}$ fails to be convex in a neighbourhood of the origin.

## Next Steps

We now know that for $a\geqslant1$, log-concavity (even $1$-strong log-concavity) holds for $\mu_{a}$, and so these are reasonable model probability measures with which to do some concrete calculations. Sampling is also straightforward by e.g. inversion, i.e. draw $h\sim\mathrm{Exp}\left(1\right)$, $s\sim\mathrm{Unif}\left(\left\{ \pm1\right\} \right)$, and set $x=s\cdot\left(\left(a^{2}+2\cdot h\right)^{1/2}-a\right)$.

Moving beyond this, I can think of a few interesting things to play around with; including:

1. Since simulation is straightforward, it is (at least conceptually) interesting to consider using $\mu_{a}$ for rejection sampling for Gaussians; how should one set $a$ optimally?
2. What are the qualitative differences between the $\mu_{a}$ in the limits $a\to0^{+}$, $a\to\infty$?
3. Transport maps between $\mu_{a}$ and the standard Gaussian measure ought to have reasonably good regularity (e.g. Lipschitz constants). For which values of $a$ are the regularity properties best?
4. Do there exist any modifications (e.g. exponential tiltings, mixtures, conjugate families, higher-dimensional analogues, ...) of the $\mu_{a}$ which are interesting or tractable?