You shouldn’t want to have a high-dimensional space. High-dimensional spaces are hard to work with, it’s just that they come up often. You basically obtain one when you look at an object or concept or what have you, then think of everything you could measure about it and measure that.
Misha’s answer is almost always the right one, but you technically can project points into a higher-dimensional space using a kernel function. This comes up in Support Vector Machines where you’re trying to separate two classes of data points by drawing a hyperplane between them. If your data isn’t linearly separable, projecting it into a higher-dimensional space can sometimes help.
But most of the time, what you want to so is just measure everything you can think of, and let those measurements be your dimensions. When looking at rubes and bleggs, measure things like redness, blueness, roundedness, furryness, whatever you can think of. Each of those is one dimension. Before you know it, you’ve got a high-dimensional featurespace. Good luck dealing with it.
How does one obtain a high-dimensional featurespace to begin with? Can one bootstrap from a one-dimensional space?
I can’t think of any satisfactory way to do this right now.
You shouldn’t want to have a high-dimensional space. High-dimensional spaces are hard to work with, it’s just that they come up often. You basically obtain one when you look at an object or concept or what have you, then think of everything you could measure about it and measure that.
Misha’s answer is almost always the right one, but you technically can project points into a higher-dimensional space using a kernel function. This comes up in Support Vector Machines where you’re trying to separate two classes of data points by drawing a hyperplane between them. If your data isn’t linearly separable, projecting it into a higher-dimensional space can sometimes help.
But most of the time, what you want to so is just measure everything you can think of, and let those measurements be your dimensions. When looking at rubes and bleggs, measure things like redness, blueness, roundedness, furryness, whatever you can think of. Each of those is one dimension. Before you know it, you’ve got a high-dimensional featurespace. Good luck dealing with it.