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If I have a simplicial complex, and a discrete Morse function defined on the simplices, I can use persistent homology to produce a barcode which helps me distinguish "persistent" shape from noise. To quote H. Edelsbruner and J. Harer in Persistent homology — a survey (Contemporary Mathematics 453, AMS, 2008).

Persistent homology is an algebraic method for measuring topological features of shapes and of functions. Small size features are often categorized as noise and much work on scientific datasets is concerned with de-noising or smoothing images and other records of observation. But noise is in the eye of the beholder...

What if I chose my own Morse function? This may be necessary when looking at complex networks, such as social networks, and therein trying to de-noise the topology of their clique complexes. No Morse function is naturally available, since the simplices are higher-order structure (only the network edges have weights, and even this is not necessary for simple relation networks based on "actors" and "have met").

What conditions does such a function need to be directly relevant to this process? What are the main intuitive ideas which lead to a Morse function, defined on a simplicial complex, successful applicable in persistent homology?

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    $\begingroup$ I'm not sure what you're asking here. You are free to define "your own Morse function" on a given clique complex, but I have no idea what it means for your Morse function to be "relevant to this process". Which process? Regarding the last sentence, you can use a filtered version of discrete Morse theory to speed up persistent homology computation: link.springer.com/article/10.1007/s00454-013-9529-6. See also Uli Bauer's thesis: webdoc.sub.gwdg.de/diss/2011/bauer_u/bauer_u.pdf $\endgroup$ – Vidit Nanda Jul 3 '19 at 20:24
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    $\begingroup$ Ok, so I understand more now that the Morse theory helps you preprocess the complex. Then you do the persistent homology. Any Morse function will lead to a new complex which is homotopy equivalent to the unprocessed complex, is that right? $\endgroup$ – apkg Jul 3 '19 at 21:52
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    $\begingroup$ Ok I am reading the notes of your course on discrete Morse theory. The whole point is to reduce the size. $\endgroup$ – apkg Jul 3 '19 at 21:58

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