Thanks for your clear explanation, understanding the topology of the space seems fascinating. If it’s a vector space, I would assume its topology is simple, but I can see why you would be interested in the subspaces of it where meaningful information might actually be stored. I imagine that since topology is the most abstract form of geometry, the topological structure would represent some of the most abstract and general ideas the neural network thinks about.
Yeah! I think distance, direction, and position (not topology) are at least locally important in semantic spaces, if not globally important, but continuity and connectedness (yes topology) are probably important for understanding the different semantic regions, especially since so much of what neural nets seem to do is warping the spaces in a way that wouldn’t change anything about them from a topological perspective!
subspaces of it where meaningful information might actually be stored
At least for vanilla networks, the input can be embedded into higher dimensions or projected into lower dimensions, so you’re only ever really throwing away information, which I think is an interesting perspective for when thinking about the idea that meaningful information would be stored in different subspaces. It feels to me more like specific kinds of data points (inputs) which had specific locations in the input distribution would, if you projected their activation for some layer into some subspace, tell you something about that input. But whatever it tells you was in the semantic topology of the input distribution, it just needed to be transformed geometrically before you could do a simple projection to a subspace to see it.
Thanks for your clear explanation, understanding the topology of the space seems fascinating. If it’s a vector space, I would assume its topology is simple, but I can see why you would be interested in the subspaces of it where meaningful information might actually be stored. I imagine that since topology is the most abstract form of geometry, the topological structure would represent some of the most abstract and general ideas the neural network thinks about.
Yeah! I think distance, direction, and position (not topology) are at least locally important in semantic spaces, if not globally important, but continuity and connectedness (yes topology) are probably important for understanding the different semantic regions, especially since so much of what neural nets seem to do is warping the spaces in a way that wouldn’t change anything about them from a topological perspective!
At least for vanilla networks, the input can be embedded into higher dimensions or projected into lower dimensions, so you’re only ever really throwing away information, which I think is an interesting perspective for when thinking about the idea that meaningful information would be stored in different subspaces. It feels to me more like specific kinds of data points (inputs) which had specific locations in the input distribution would, if you projected their activation for some layer into some subspace, tell you something about that input. But whatever it tells you was in the semantic topology of the input distribution, it just needed to be transformed geometrically before you could do a simple projection to a subspace to see it.