These dots were sampled from a 2-component Gaussian mixture and then put through a smooth invertible warp. There aren’t any clusters, but the concept is still present and recoverable from the data (altho too hard for our visual cortex to recover in this particular case).
The shortest rule to describe this scatter is that each dot is an independent draw from a mixture of two modes. You have to specify where the two modes are, and you can guess decently well which mode each dot is from. Thru this you’ve rediscovered color without needing clustering.
It looks like x and y are the dimensions of your thingspace here. If you had a different thingspace, would you still be able to recover the same concept?
Yup! This is a very weird space to call a ‘thingspace’ but most transformations of it (anything that’s at least approximately injective, for example) will preserve the same concept.
Here’s what that same distribution I used above looks like if you plot the closed-form pushforward density analytically. In this picture it’s easier for the visual cortex to pick up on the patterns (although it would still be nontrivial for a human to figure out what should be colored red and what should be colored blue if you erased the colors).
Nope! Here’s an example:
These dots were sampled from a 2-component Gaussian mixture and then put through a smooth invertible warp. There aren’t any clusters, but the concept is still present and recoverable from the data (altho too hard for our visual cortex to recover in this particular case).
The shortest rule to describe this scatter is that each dot is an independent draw from a mixture of two modes. You have to specify where the two modes are, and you can guess decently well which mode each dot is from. Thru this you’ve rediscovered color without needing clustering.
It looks like x and y are the dimensions of your thingspace here. If you had a different thingspace, would you still be able to recover the same concept?
Yup! This is a very weird space to call a ‘thingspace’ but most transformations of it (anything that’s at least approximately injective, for example) will preserve the same concept.
Here’s what that same distribution I used above looks like if you plot the closed-form pushforward density analytically. In this picture it’s easier for the visual cortex to pick up on the patterns (although it would still be nontrivial for a human to figure out what should be colored red and what should be colored blue if you erased the colors).