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Is there a clear application to a practical, real world data set where TDA is the state of the art? Don’t get me wrong, I love maths and topology in particular but I’m trying to understand what are the advantages that these techniques bring to the table

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I’ve used algebraic topology for certain robotics tasks when statistical saftey wasn’t good enough and when trying to generalize techniques to learn on graphs.

The results are spectacular, but the application set is indeed very limited.

any references you can share? would love to read more about robotics applications of these approaches

This is a decent place to get started https://arxiv.org/pdf/math/0009118.pdf

You can extend some of the thinking in this paper to cover action spaces for deep RL models to really have some fun!

no, at least not for anything non-contrived.

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