Gaby Contardo (Flatiron) and I had our weekly today on our geometric data-analysis methods for finding and characterizing gaps or valleys in point clouds. We are starting at 2D data, which is the simplest case, and it is interesting and we have nice results. But scaling up to more dimensions is hard. For one, there is the curse of dimensionality, such that anything that relies on, or approximates, density estimation gets hard fast. And for another, the kinds of geometric structures or options for gaps blows up combinatorically (or faster than linearly anyway) with the number of dimensions. Do we have to enumerate the possibilities and track them all? Or are there more clever things? We don't yet have answers, even for 3D, let alone 6D!
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