4 ms·
The klein bottle is relatively uninteresting. The technique which identified it is the interesting part here. The linked paper explains a method to take a high-
by jamii 15y ago
The klein bottle is relatively uninteresting. The technique which identified it is the interesting part here. The linked paper explains a method to take a high-dimensional point cloud and compute a 'bar code' which encodes fundamental geometric features of the solid of which the cloud is an approximation. Its a way of visualizing high-dimensional data without using dimensional reduction.
- Drbble 15y agoDoes the point cloud approximate the shape, or does the shape approximate the point cloud? That distinction informed a recent discussion on the origin of the word "regression", which on the surface seems a weird term for curve-fitting, at least from my point of view.
- jamii 15y agoI suppose the idea is that there is an underlying shape which the point cloud approximates. The fitted shape is then a statistical guess at the underlying shape. In which case it would be nice to see a proof that as more points are added the fitted shape eventually converges to the underlying shape.
- yread 15y agofrom the article "The full data set consists of roughly 8,000,000 points in E9. By normalizing with respect to mean intensity and restricting attention to high-contrast images (those away from the origin), the data set is projected to a set of points M a topological seven-sphere S7 ⊂ E8" http://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf http://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf