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A knowledgeable user graciously submitted this edit for the topology tag, but the scope of the content is such that I wanted to bring it to the community for discussion:


USAGE GUIDANCE Topology is the study of geometric constructs and whether they can or cannot arise from other geometric constructs solely from stretching, contracting, bending, or twisting. In the context of AI, the topology correlates closely with the categorical capabilities of the system.


TAG WIKI Topology is the study of geometric constructs as to whether they can or cannot arise from other geometric constructs solely from stretching, contracting, bending, or twisting. See the Miriam-Webster online dictionary definition of topology below.

In the context of AI, the topology correlates closely with the categorical capabilities of the system.

Correct Use

The feedback of signals from the output of a network or a layer to a previous point in signal flow is a topological feature involving splitting and joining which matches the use of the term in mechanics, mathematics, semantic web analysis, and IT network provisioning.

Gating and Attention organelles are topological.

The signal paths that create the adversarial balance between generative and discriminative networks in the GAN design create a topology that is unique to collaborative circuit pairing.

Misuse

Although the term topology has been used in relation to the number of activation elements in a neural network layer, there is an undeniable semantic flaw with that usage:

Neural networks form a geometry of discrete elements without meaningful size attributes. Activations and input attenuation parameters cannot be stretched, contracted, bent, or twisted meaningfully. Moving a vertex with the edges still attached in the graphic representation also lacks functional meaning.

The only possible meaning of morphing a neural network layer is to change the quantity of its array elements, which means that the artificial neuron counts in the layers CANNOT be part of a network's topology if the word topology is to delineate anything at all.

Only Possible Logical Conclusion

If topology is to be applied to AI design, only features that cannot be changed solely by modifying the dimensionality of one or more arrays in the design can qualify as topological features.


The Miriam-Webster online dictionary definition of Topology

a :
    1. a branch of mathematics concerned with those properties of geometric configurations (such as point sets) which are unaltered by elastic deformations (such as a stretching or a twisting) that are homeomorphisms
    2. the set of all open subsets of a topological space

b : configuration — topology of a molecule topology of a magnetic field
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I strongly recommend against using this tag info, for the following reasons:

  1. Tag info should be easily understandable, provide clear information that tells a user whether or not to use that tag / whether or not it's relevant for their question.
  2. Tag info should be unambiguous, correct, and not be up for debate. The information in there should be "generally agreed upon" by people familiar with the relevant field(s) to be true.

I don't think either of these points are satisfied here.


For the first point, try reading through that text once, from top to bottom. Do you feel like you're now well aware of when the tag is or isn't applicable, what it's about? I certainly don't. I have the following concerns here:

  • Usage guidance doesn't really tell us for which questions it should or shouldn't be used. It starts out with a bunch of fancy words that don't tell me anything about its relation to AI. It ends by telling us that "topology" is supposedly "closely related" to something, but still don't know what it is.
  • Again, the main text / tag wiki doesn't provide clear information either. Again, lots of fancy words, but not much real information (definitely not clear information).
  • The tag wiki gives some examples of things that are "features of topology" or are "topological", but we still don't have a clear description of what it's supposed to be.

For the second point, I have the following concern:

  • It dives straight into "Correct Use" and "Misuse" headers, which is already hinting at the definition being up for debate, having multiple uses, being potentially ambiguous or not generally agreed upon. More importantly, as someone familiar with Neural Networks this might be plain incorrect according to my experiences. I say "might be" rather than "is" because the text is so incomprehensibly complicated that I can't tell for sure what it's actually saying.

In general, in AI, when people are talking about "topology" in the context of Neural Networks, it's used to describe the "architecture" of the Network; how many layers, what types of layers, how large is each layer, what activation functions do we put in between, where are the connections (typically a feature of layer type). That's basically it, and that can be explained very clearly in language that can be understood easily. Some sources:

Many more similar sources can be found through a google search for "Neural Network topology".

That is specifically in the context of Neural Networks. This will likely be the most common natural usage of the tag on this site. However, as also mentioned in the proposed tag wiki, topology is also a completely different field of mathematics. And that field of mathematics may sometimes be relevant in a completely different manner in the field of AI. So, "topology" can be ambiguous, and probably should not be restricted only to the usage in the context of Neural Networks.


As a final concern, I am wondering what a header saying "Only Possible Logical Conclusion" is doing in a tag wiki. That's a header I'd expect in an opinion piece, or maybe as an overly-sensationalized header after a mathematical proof. This is not a header that belongs in a neutral, informative Tag Wiki.


Now, given the use of language, I know immediately precisely which person proposed that tag wiki edit. For context, I think it's important to note that I've previously had a long discussion with this user concerning terminology, which can be read here.

Note that, in that discussion, it becomes very clear that this particular user is knowingly and actively trying to push the usage of new terminology that he personally believes to be "better" than commonly-used terminology across the entire field. That is fine, he can do that if he likes, even on this site by e.g. asking questions like "Wouldn't X be a better term instead of Y because reasons Z?" But this should not be done through tag wikis. Tag wikis should be consistent with language used commonly across the field, otherwise every single non-expert user visiting the site (and maybe even expert users) is going to be confused.

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    $\begingroup$ Dennis, thanks for commenting! I agree with you here, but I'm still struggling with how to define our topology tag. (Finally decided to ask a question on the general site so the experts can weigh in.) $\endgroup$ – DukeZhou Aug 8 '18 at 20:32
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Disclaimer: NNs are not my area, and I only have a general grasp of General Topology. I come to AI via combinatorics and combinatorial games which influences my perspective.

When researching the field of topology, I noticed it did not deal with certain aspects of geometrical game boards, which are more in line with the "proto-topology" pioneered by Euler with the Bridges of Königsberg. (For instance, you can add or remove playable cells from a Chess or Go board, which alters the structure in terms of number of connections (the network topology. From this standpoint, topological alterations to game boards potentially break game solutions, which may have an impact on AI performance/strength.)

Also wondering how similar the network topology usage is re: NNs

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