Okay, so thanks, Leon and Daniel for the nice introductions.

So just to reiterate, please feel free to jump in and ask a question at any time during

the talk.

I have a lot of stuff to cover here, but so if nobody has questions, I'll just be sort

of flying through everything.

So please, please jump in anytime.

So the talk is going to be about various parts of my work recently on random walks and PDEs

and graph-based learning.

It's a collaboration with a lot of people, including my PhD student, Brendan Cook, and

several other researchers at the University of Minnesota and other places, and supported

by the NSF and the Alfred P. Sloan Foundation.

So just for some background, the talk is mainly on graph-based learning.

And so throughout the talk, our graph will have vertices script X, which you can think

of as points in Euclidean space, and then some non-negative edge weights attached to

those points.

So that's just a matrix W. So for each X and Y, there's an edge weight Wxy.

It could be zero, which means there's no edge between those points.

And the idea is when the edge weight is larger, the points X and Y are more similar.

Here's an example of a K nearest neighbor graph that I built.

So it's just a very simple toy data set with sort of two clusters that are connected by

a thin tube.

And so you just take each point, connect to its K nearest neighbors.

And you can sort of start to think about how a graph can encode important properties of

your data set.

If you're thinking about random walks, let's say each cluster in this graph is sort of

very good at trapping a random walker, in the sense that if I release a random walker

from the red cluster, it's going to take a long time to find this narrow passageway between

the two clusters and land on the other side.

So this is sort of one of the reasons why random walks or diffusion processes and PDEs

on graphs are important things in problems like classification, clustering, things like

this.

And so these are the kind of things we like to do on graphs, grouping similar data points

together, clustering or semi-supervised learning, which is a problem of propagating labels on

a graph.

You can think of semi-supervised learning with very few labels as kind of like clustering,

where somebody tells you one or two of the labels.

So they say, you know, these two points over here are blue, these two points over there

are red, and then I want you to label the rest of the points as blue or red.

So for a very concrete example of something you can do, this is basically a toy example

now, is the MNIST data set of handwritten digits from zero to nine.

There are 70,000 of these digits.

Each is a 28 by 28 pixel image.

And so here's 100 of them in this grid here.

You could just naively think of each image as a flattened vector in R784.

You can define geometric weights on the graph, which means for any two images, x and y, the

weight between them is a non-negative kernel eta applied to the distance between x and

y divided by a length scale epsilon.

Oftentimes, this is a Gaussian, let's say eta.

And you know, for most real data sets, having the same bandwidth epsilon for the entire

graph is not so practical because you have very dense or very sparse regions.