We can visualize single, two, and three dimensions using websites or imagination.

In the context of AI and, in particular, machine learning, AI researchers often have to deal with multi-dimensional random vectors.

Suppose if we consider a dataset of human faces, each image is a vector in higher dimensional space and needs to understand measures on them.

How do they imagine them?

I can only imagine with 3D and then approximating with higher dimensions. Is there any way to visualize in higher dimensions for research?

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    $\begingroup$ Stop viewing dimensions spatially and start viewing them as degrees of freedom. $\endgroup$ – chessprogrammer Oct 9 '20 at 2:44
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    $\begingroup$ You can visualise 4 dimensions with 3 dimensions and significant amounts of color, but you aren't going to get any further than that. Even with that, it's generally not worth it. It's just a fact of reality that an organism stemming from millions of years of evolution in a 3 dimensional world will never be able to truly visualise 4 or more geometric dimensions. $\endgroup$ – Recessive Oct 9 '20 at 3:46

The most I can visualize or perceive are 4 dimensions. Yes, 4, because I can also watch videos (which have 3 spatial dimensions and 1 temporal one). Remember Einstein's spacetime?

When dealing with $n$-dimensional spaces, for $n > 4$, I simply do not care about visualizing them in my head, but, as someone suggests, we can think of them as "degrees of freedom". Maybe something like a tesseract may be interesting to you, but that's not really useful to me, to be honest.

When dealing with the math that involves $n$-dimensional spaces or objects, you often do not have to visualize anything, but just have to apply the rules. For example, if you are multiplying multi-dimensional arrays, you just need to make sure that the external dimensions match, and stuff like that.

There are cases, when dealing e.g. with TensorFlow's tensors, where you can imagine that there are matrices for each of the elements at that coordinate of the tensor, but that's not very common.

In case you really want to visualize $n$-dimensional objects, you could first project them to $2$ or $3$ dimensions with some dimensionality reduction technique (e.g. t-SNE).

  • $\begingroup$ I've tried to visualize 4 spacial dimensions by concatenating regular grids. The math works, but hard to say it's a faithful visualization. I've also searched for a rendering of a grid of tesseracts, but a fruitless search so far. $\endgroup$ – DukeZhou Oct 15 '20 at 2:31

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