# The usage of “variance” (vs. “standard deviation”) in the 2008 t-SNE paper

In the t-SNE paper from 2008 the term "variance" is mentioned on several occasions. For example on page 3:

For nearby datapoints, $p_{j|i}$ is relatively high, whereas for widely separated datapoints, $p_{j|i}$ will be almost infinitesimal (for reasonable values of the variance of the Gaussian, $\sigma_i$).

I'm a bit confused here as the variance is usually defined as ${\sigma_i}^2$. Instead, $\sigma_i$ would be the standard deviation.

Is there any mathematical reason for the authors to break with this naming convention or am I simply missing something here?

• I would guess that it's for pure aesthetics. If they followed proper convention they would have ${\sigma_i}^4$ in the very first equation of the paper, and who wants that? :^) – tm1212 Jun 18 '18 at 14:03

## 1 Answer

You are right. It is clear from Equation (1) that $\sigma_i$ denotes the standard deviation and not the variance. So the text should have said

(for reasonable values of the variance of the Gaussian, $\sigma^2_i$)

The same applies to five or six (!!) other occurrences of $\sigma_i$ in the text, e.g.

... where $\sigma_i$ is the variance of the Gaussian that is centered on datapoint $x_i$.

Somehow in all these sentences the authors mixed up the variance and the standard deviation (or alternatively, if they did intend for $\sigma_i$ to be the variance, which is rather unconventional, then they should not have used the square in Eq. (1)).

I assume this went unnoticed because it is nevertheless very clear what was meant throughout the text and all the formulas are unambiguous.