Topological Data Analysis - Persistent Homology

Hi,

I am not a mathematician, but I have noticed some recent papers on this seemingly new field, called Topological Data Analysis (see this relevant paper).

I have had an overview of the applications and it seems that when you have data points that were sampled from some source (e.g. an image), you can use Persistent Homology to visualize what these data looks like in higher dimensions. (this is my understanding).

I am still unsure what this really means. Will any data set have higher dimensional shape or geometry?
 

fresh_42

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This is build on thin ice. Topological here means the lack of scales, metrics and coordinates. But data are measures somehow which gives natural coordinates, even though the author says differently. With topology you also get lost of all analytical means, plus that a finite set of data points only allow trivial topologies, except some hypothesis are added.

I wouldn't take the paper very serious, i.e. a closer examination of these additional conditions is due. Topology is a rather new field of mathematics - only 100 years old - so people are still looking for applications outside mathematics. Of course this is a personal opinion, so let's wait and see.
 

TeethWhitener

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I’m by no means an expert, but I have noticed a significant increase in interest around the field of topological data analysis by a number of US funding agencies.

As I understand it, the basic technique is to take high-dimensional data and find lower-dimensional features that are persistent over several different length scales (according to some relevant metric). Features that are persistent are presumed to be related functionally in some way, where’s features that aren’t are generally disregarded as noise. I have no idea how useful the technique is, but I wanted to chime in to point out that it seems to have caught funders’ attention here in the US.
 

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