3
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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? :^) $\endgroup$ – tm1212 Jun 18 '18 at 14:03
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.