Intrinsic dimension

Imagine you have a bunch of toys, and you want to put them away in a toy chest. But the toy chest isn't very big, so you have to choose which toys to put in and which ones to leave out.

Some toys are big and take up a lot of space, like a toy car or a stuffed animal. Other toys are small and don't take up a lot of space, like a toy soldier or a puzzle piece.

Now, imagine the toy chest is like a graph, and the toys are like points on the graph. Intrinsic dimension is like the size of the toy chest, meaning how many toys you can fit in it. But instead of measuring size, we're measuring how much information we need to describe the points on the graph.

Some points on the graph might be really important and tell us a lot about the data, while others might not give us much information at all. For example, if we're looking at a graph of ice cream flavors that people like, the fact that someone likes vanilla might not be very informative because a lot of people like vanilla. But if someone likes a really weird flavor like garlic or pickles, that's much more informative and tells us a lot about that person's tastes.

So, intrinsic dimension is kind of like how many important pieces of information we need to describe the data in a graph. Just like how you have to choose which toys to put in the toy chest, we have to choose which points on the graph are really important and which ones are less important. By doing this, we can figure out how many important pieces of information we need to describe the data.