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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?