Actually I guess that’s kind of trivial (the belief state geometry should be tensored together). Maybe a more interesting question is what happens if you use a Markov chain to transduce it.
If you use a Markov chain to transduce another Markov chain, the belief state geometry should kind of resemble a tensor of the two Markov chains, but taking some dependencies into account.
However, let’s return to the case of tensoring two independent variables. If the neural network is asked to learn that, it will presumably shortcut by representing them as a direct sum.
Due to the dependencies, the direct sum representation doesn’t work if you are transducing it, and arguably ideally we’d like something like a tensor. But in practice, there may be a shortcut between the two, where the neural network learns some compressed representation that mixes the transducer and the base together.
(A useful mental picture for understanding why I care about this: take the base to be “the real world” and the transducer to be some person recording data from the real world into text. Understanding how the base and the transducer relate to the learned representation of the transduction tells you something about how much the neural network is learning the actual world.)
What do the fractals look like if you tensor two independent variables together?
Actually I guess that’s kind of trivial (the belief state geometry should be tensored together). Maybe a more interesting question is what happens if you use a Markov chain to transduce it.
I guess to expand:
If you use a Markov chain to transduce another Markov chain, the belief state geometry should kind of resemble a tensor of the two Markov chains, but taking some dependencies into account.
However, let’s return to the case of tensoring two independent variables. If the neural network is asked to learn that, it will presumably shortcut by representing them as a direct sum.
Due to the dependencies, the direct sum representation doesn’t work if you are transducing it, and arguably ideally we’d like something like a tensor. But in practice, there may be a shortcut between the two, where the neural network learns some compressed representation that mixes the transducer and the base together.
(A useful mental picture for understanding why I care about this: take the base to be “the real world” and the transducer to be some person recording data from the real world into text. Understanding how the base and the transducer relate to the learned representation of the transduction tells you something about how much the neural network is learning the actual world.)