This content was created by GHION.AI . This content was uploaded by GHION.AI . This is a YouTube talk titled "Gunnar Carlsson: Using Math to Reveal the Shape of Data" (recorded February 10, 2014). The excerpt is a single-speaker lecture by Gunnar Carlsson introducing topology and its role in data analysis: topology as pattern recognition and the introduction of invariants to measure shape, the idea of compressing continuous shapes (e.g., approximating a circle by a polygon), and the transfer of topological techniques since ~2000 into the point-cloud/data setting (topological data analysis).
Chapters
So topology is the part of mathematics that deals with trying to describe and represent shape. In fact, it's actually a form of pattern recognition. So it was started in the 1700s by a Swiss mathematician named Leonard Euler, and it's actually thrived on the pure math side. What it does is it actually introduces invariants that allow you to measure shape. So measuring shape is sort of a— that's a funny-sounding concept because shape to me is kind of a— it's an ill-formed or a kind of a vague notion. And so the idea of measuring it with numbers, you know, is a little counterintuitive. But it turns out that there are ways of doing that, actually very, you know, very interesting and powerful ways of doing that. The second part of it though is— and I don't think it's usually talked about this way, but Topology is about compressing shape. It is about finding— if you think of a circle, for example, a circle is infinitely many points and infinitely many pairwise distances between those points. Now if you're willing to sacrifice a little bit of detail, like the exact curvature and so on, you can represent it by, say, a hexagon or an octagon, which is, say, 8 nodes and 8 edges, which can be represented in a single byte. I mean, so it's very, very simple. And that notion of trying to sort of combinatorially compress the notion of shape into something much more understandable is the second thing that topology is about. And so in this area, what happened starting in about the year 2000, a lot of people started to have this idea that we should say these techniques in topology, which are about pattern recognition and representing shapes and sort of getting at, getting really precise about what one means by shapes, should now be transported from the pure math world where you're dealing with things where you have complete information, you have all the points or a description in terms of equations, to something where you only have sampled information, which is really more like real life. And so that's what's been going on in the last 15 years, porting all those techniques from understanding shapes— and I mean here even higher-dimensional notions of shape, not just 2 and 3 dimensions— porting them into what we would call the point cloud world, which is data, which is where data lives.
