###### tags: `one-offs` `expository`

# Gaussian Smoothing

**Overview**: In this note, I describe a couple of nice structural features of the Gaussian smoothing technique.

## Shape Properties and Gaussian Smoothing

There are a few elementary facts about Gaussian smoothing which I think are very neat, and hence worth sharing. To be clear, by 'Gaussian smoothing', what I mean is that one starts off with some sensible function $f:\mathbf{R}^{d}\to\mathbf{R}$, picks a $\sigma\geqslant0$, and defines

\begin{align}
f_{\sigma}\left(x\right):=\mathbf{E}\left[f\left(x+\sigma G\right)\right],
\end{align}

where $G \sim \mathcal{N} \left(0,I_{d}\right)$.

A first fact is that for $\sigma>0$, the function $f_{\sigma}$ is automatically smooth in the classical sense, i.e. it has derivatives of all orders. Actually, this is a fairly common reason for even working with such smoothings in the first place. Moreover, as $\sigma$ increases, one obtains increasingly good quantitative bounds on the size of those derivatives.

At the other end of the spectrum, for $\sigma \approx 0$, it is quite tractable to show that $f_{\sigma}$ is close to the original $f$ in many relevant senses. see e.g. the developments in [this paper](https://econpapers.repec.org/paper/corlouvco/2011001.htm) for some help along this path.

Another aspect which is quite convenient is that Gaussian smoothing preserves many structural properties of $f$. I mean this in the sense that if $f$ is { positive, bounded, monotone, convex, ... }, then so is $f_{\sigma}$. This can be quite useful, and is also not too difficult to prove directly.

There are other interesting properties of Gaussian smoothing, but for now, I prefer to keep it short and simple, and so I will leave it at this for now.