Using t-Distributed Stochastic Neighbor Embedding (t-SNE) to visualize the two-dimensional unfolding of the sophon

In the famous science fiction novel The Three-Body Problem, the alien, named Trisolarans, created a kind of supercomputers, sophon, to spy on the Earth and disrupt all of Earth’s particle accelerators.

The sophon is made from a single proton, which is first unfolded from its eleven-dimensional form to a two-dimensional form, then etched, and finally folded back into the eleven-dimensional form.

When in the high-dimensional form, a sophon is minuscule to a low-dimensional observer. But when unfolded into its two-dimensional form, a sophon is enormous to a human observer, who is in the three-dimensional space.

This kind of transformation is very familiar to us who study machine learning: basically, it’s dimension reduction, among which principal component analysis (PCA) is best-known.

However, the sophon’s unfolding cannot be implemented by PCA because PCA is a linear transformation and hence a compressive mapping, which means that a sophon will only be smaller in lower dimensional spaces after this transformation.

Thus, the sophon’s unfolding can only be a nonlinear one. Then, what is it? No one knows, but I find it interesting to use t-Distributed Stochastic Neighbor Embedding (t-SNE) to “imagine” the sophon’s unfolding.

The method t-SNE is invented by Maaten and Hinton in 2008 and has since become very successful. It has the advantage that points close in the high-dimensional space tend to be also close when reduced to the low-dimensional space.

For this purpose, I made the video at the beginning of this post. Firstly, the sophon is in its eleven-dimensional form and is like a small ball when projected into the two-dimensional space. Different parts of the sophon are colored differently; points with the same color are considered close in the eleven-dimensional space.

Then, the sophon starts to unfold. Points with the same color attract each other, while points with different colors push each other away. Sometimes, the points with the same color cannot move together because the path connecting them is “occupied” by points with different colors. In this case, the sophon needs to further enlarge its volume to create room for these points to move together.

At the end, the sophon breaks into four parts. In the video, I intentionally set a large distance (in the eleven-dimensional space) for points with different colors. It is reasonable to imagine that the four parts can be closer if I set a smaller distance.

In the novel, at the end, the entire solar system was reduced to a two-dimensional space. This inspires me to apply t-SNE on real-world objects, such as the Eiffel Tower. The two-dimensional Eiffel Tower must be a wonderful piece of art, I believe. Maybe the next project?

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Written on July 9, 2019