# How can a neural network distinguish a rotated 6 and 9 digits?

Rotated MNIST is a popular dataset for benchmarking models equivariant to rotations on $$\mathbb{R}^2$$, described by $$SO(2)$$ group or its discrete subgroups like $$\mathbb{Z}^{n}$$:

It consists of all digits from 0 to 9 rotated on an arbitrary angle from $$[0, 2 \pi)$$. However, what makes me a bit puzzled is that digits $$6$$ and $$9$$ seem to be confused by any learning algorithms, since from the view of human perception $$6$$ rotated by 180 degrees is equivalent to $$9$$ and vice versa.

The original paper in the description of Rotated MNIST doesn't comment on this point at all, which is strange, since it is a very natural question to ask.

In the paper Oriented Response Networks - authors plot embeddings of rotated digits projected via t-SNE on a 2d plane. There is a clear separation between all rotated versions of 6 and the rotated version of 9 for ORN.

I do not understand how it can be achieved? Probably, the networks understand much more in writing the digit, there are some subtle features, inaccessible to humans, but recognizable by powerful classifier?

I think by writing left to right people create clockwise and counterclockwise patterns in the rounded parts of their typography.

For example, I think it'd be pretty unusual to write a 9 like this -->

I bet you could also detect the original orientation of the 8s if you wanted. The circle that isn't continuous/smooth is most likely at the top.

• Great explanation and great examples, @pip.pip!! Using your insight, I think I can pick out the 6's and 9's in many examples in the post. (I think I can also pick out the top and bottom of the 8's.) Having said that, it seems like the separation between 6's and 9's in the paper is perhaps a little too good, no? Oct 26, 2022 at 4:16

"... since from the view of human perception 6 rotated by 180 degrees is equivalent to 9 and vice versa.".

Only with some typefaces, and almost never with handwritten text; someone would have to make an extra effort to make their text ambiguous when rotated (or mirrored).

A "6" has a bent tail, while a "9" tends to be straighter.

Similarly with a "4", tending to have a closed top and a flattened midline, while an "h" can never have a closed bottom and usually has a curved midline.

It's how the character is written in its natural orientation that determines its appearance, rotation (and scaling) has no effect on that.

With a 9-pin Dot Matrix Printer all bets are off, since the usual font was not designed to differentiate between rotated characters.

If anything, the fontographer cheated and rotated the characters whenever possible to avoid the effort of creating a new character from scratch.

Later fonts such as Data-70 were able to avoid rotation ambiguities at the cost of readability, and increased geekiness.