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Suppose I have a model M which outputs a three-dimensional tensor of size 3x3x3. I have another model N which outputs a one-dimensional tensor of size 27.

Train both models on some arbitrary objective which requires the knowledge of space. For example:

M: Output 1 in any corner, 1 in the exact opposite corner, and 0 elsewhere. (3x3x3)

N: Output 1 in the indices which would correspond to any corner and its opposite, if you rebuilt the 3x3x3 cube from the 1D (1x27) array. Output 0 elsewhere.

Now, this might not be a perfect objective to evaluate spatial-competency, but you get the idea.

Will model M be as proficient as model N? Why or why not?

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    $\begingroup$ So far the models are too trivial, in that there is no input, and only a single target output. The "models" are simply the output structures, with no deeper internal structure, and both can achieve their goal perfectly without any "perception" $\endgroup$ Commented Apr 4 at 18:28
  • $\begingroup$ If you use an MLP anyway, it will deal with flattened tensor (eventually reshape later). $\endgroup$
    – Lelouch
    Commented Apr 5 at 8:32
  • $\begingroup$ @NeilSlater I see. I understand that ML models having a "perception" of anything is a bit of a misconception, as they are just math models at the core. Perhaps a better way to ask it would be: Can an ML model reduce its loss just as effectively on flattened input/output versus non-flattened input/output? $\endgroup$
    – schmixi
    Commented Apr 7 at 16:37
  • $\begingroup$ @schmibbler On the final output layer, with no defined input, or completely equivalent output, then yes, there is no useful way to encode spatial knowledge, and nothing can be shown with one output structure that doesn't map immediately to the other. You have to define more of the problem and/or more of the model, and then we can usefully discuss whether one architecture has some advantage. $\endgroup$ Commented Apr 7 at 17:48

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