3 ms·

Could you elaborate on the connection between Deep Learning and type theory, resp. topos theory that you allude to? It sounds fascinating. At the some time, I'm

Could you elaborate on the connection between Deep Learning and type theory, resp. topos theory that you allude to? It sounds fascinating. At the some time, I'm slighly surprised that I am not aware of results in this direction.

zmgsabst 5y ago
I think it mostly exists at the level of folk lore right now, with preliminary bridges. I’ve spoken some with people at, eg HoTT 2019 or Topos Institute speakers, but I think the state of publishing is that the other direction is being worked out fully — since it solves problems in the math community like accelerating proof engines via GPUs. (Maybe, I hope.) There’s three approaches to seeing the connection: 1. If the manifold hypothesis is true, then the semantic information between the original manifold and reduced manifold will be in the topology of that map — and your TT is a relabeling of the synthetic topology description. 2. For the convolution comment, take a TT statement, view it as a CuTT structure, and draw the construction diagram of that “shape”. Project that into a matrix encoding and… convolutions are right-side type division! Eg, finding a submatrix that matches “Line(…)” in your overall matrix. But it’s right-sided division since you can only remove the “innermost” factor in a product that way. (There’s also a left-handed, but it’s less relevant here.) 3. The paper on ML via hyper plane splits to create data coverings also suggests that what we really want to know is the topological relationships between kinds of data. This is furthered by, eg, “animal” in word2vec ideally being the centroid of the open ball covering animals, as they’re embedded.
- mjburgess 5y ago
  The manifold hypothesis is true when the measurement bases aren't orthogonal, and there's lots of them eg., when we measure 3D objects by their relative position to 100 objects in a room. That isnt a bridge principle between spaces which measure different properties. The low-D structure of any high-D dataset is only a "washing out" of spare measurements of the same properties. These arent different "property-spaces", ie., the low-D structure isnt a world one has bridged to via statistics (or grad desc, etc.). The properties the formal structure of text captures aren't the properties of objects.. they're something like the social-vocab-coincidental usage of language users in how they "orient their words" around topics. If we change our usage, models which find this structure, become invalidated. We language-users routinely change this structure, and robustly against misunderstanding, because we bridge to the relevant property spaces by being in the world.
  - VirusNewbie 5y ago
    >because we bridge to the relevant property spaces by being in the world. Isn't there an implicit conclusion here that the extra dimensions of 'world being' would actually make the manifold hypothesis true, it's just missing data in the training model? Not to say it's trivial to fix, that missing data might be derived laws of physics, smell, shared world models of a society etc. but none the less I fail to see how conceptually any of that couldn't be reduced to data.
    - mjburgess 5y ago
      If you have all possible ways of measuring the world, and all possible measurements you can produce all possible models. A subset of those models will be appear relevantly intelligent. But that isnt intelligence. Intelligence is the solution to the problem of those "all possibles" being imposssible. Namely, intelligence is how animal's conduct themselves in the face of not knowing what the data is.. they do this by conducting themselves and their bodies in their environment -- they dont focus on the problem in the AI/ML/etc. sense... In this way they dont make decisions based on "data" -- they move according to self-regulatory goals. The data which they encounter, in that movement, becomes useful for reasoning about the world. But its radically more accidental, partial, and speculative than ML requires.

zmgsabst 5y ago

I think it mostly exists at the level of folk lore right now, with preliminary bridges. I’ve spoken some with people at, eg HoTT 2019 or Topos Institute speakers, but I think the state of publishing is that the other direction is being worked out fully — since it solves problems in the math community like accelerating proof engines via GPUs. (Maybe, I hope.) There’s three approaches to seeing the connection: 1. If the manifold hypothesis is true, then the semantic information between the original manifold and reduced manifold will be in the topology of that map — and your TT is a relabeling of the synthetic topology description. 2. For the convolution comment, take a TT statement, view it as a CuTT structure, and draw the construction diagram of that “shape”. Project that into a matrix encoding and… convolutions are right-side type division! Eg, finding a submatrix that matches “Line(…)” in your overall matrix. But it’s right-sided division since you can only remove the “innermost” factor in a product that way. (There’s also a left-handed, but it’s less relevant here.) 3. The paper on ML via hyper plane splits to create data coverings also suggests that what we really want to know is the topological relationships between kinds of data. This is furthered by, eg, “animal” in word2vec ideally being the centroid of the open ball covering animals, as they’re embedded.

mjburgess 5y ago
The manifold hypothesis is true when the measurement bases aren't orthogonal, and there's lots of them eg., when we measure 3D objects by their relative position to 100 objects in a room. That isnt a bridge principle between spaces which measure different properties. The low-D structure of any high-D dataset is only a "washing out" of spare measurements of the same properties. These arent different "property-spaces", ie., the low-D structure isnt a world one has bridged to via statistics (or grad desc, etc.). The properties the formal structure of text captures aren't the properties of objects.. they're something like the social-vocab-coincidental usage of language users in how they "orient their words" around topics. If we change our usage, models which find this structure, become invalidated. We language-users routinely change this structure, and robustly against misunderstanding, because we bridge to the relevant property spaces by being in the world.
- VirusNewbie 5y ago
  >because we bridge to the relevant property spaces by being in the world. Isn't there an implicit conclusion here that the extra dimensions of 'world being' would actually make the manifold hypothesis true, it's just missing data in the training model? Not to say it's trivial to fix, that missing data might be derived laws of physics, smell, shared world models of a society etc. but none the less I fail to see how conceptually any of that couldn't be reduced to data.
  - mjburgess 5y ago
    If you have all possible ways of measuring the world, and all possible measurements you can produce all possible models. A subset of those models will be appear relevantly intelligent. But that isnt intelligence. Intelligence is the solution to the problem of those "all possibles" being imposssible. Namely, intelligence is how animal's conduct themselves in the face of not knowing what the data is.. they do this by conducting themselves and their bodies in their environment -- they dont focus on the problem in the AI/ML/etc. sense... In this way they dont make decisions based on "data" -- they move according to self-regulatory goals. The data which they encounter, in that movement, becomes useful for reasoning about the world. But its radically more accidental, partial, and speculative than ML requires.