As Dave Barry once wrote: studies show that 74% of Americans — that’s nearly half! — have no sense of percentages and basic fractions.

]]>1) we actually used persistent H_1 as well (Loosely: magine the artery trees thickening up within the skull and keep track of persistent H_1 during this process. Precisely: subsample the trees and get a set of points, and then run 1-dim Rips on this, and appeal to the gods of stability theorems to say you’re doing the right thing). We found even stronger age correlations there (as well as a significant gender different). As was the case with H_0, it was the middle-length-range bars that did the trick. I would like to loudly and clearly proclaim that we have no particular notions why this is so!

2) In general, I think a very productive analytical pipeline is: take populations of filtered objects with some sort of ground-truth labels, compute persistence diagrams, extract features (how? Lots of ways, all very different, and we need much more theory!), and then “do machine-learning.” The brain artery paper is one (small) example of this pipeline, I think. A bunch of other folks are exploring signal analysis applications where the filtered objects are signal snippets (more precisely, functions f: I \to R and you then filter I by sublevel sets of f). The features you get from zero-dim persistence on this seem to given really interesting conclusions, and the features are clearly very different than things you’d get from, say, fourier or wavelet analysis. I don’t way to say “better,” but I think “different” is clear, and I think using them in combination will bear a lot of fruit.

3) All that said, the worry about the Null Hypothesis is extremely valid! The paper by Bobrowksi/Kahle/Skraba, which is really great work, attacks the length of the longest bar for this very specific type of filtered object. On the other hand, many of the exciting TDA/ML combo-papers are finding stuff using shorter bar lengths and very different filtering paradigms, so the theory/practice gap is vast. Of course, that’s true of almost all engineering that gets done “in practice,” and so I’d say Noah’s second paragraph is very much on point.

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