TITLE: Confusion related to curse of dimensionality in k nearest neighbor
QUESTION [11 upvotes]: I have this confusion related to curse of dimensionality in k nearest neighbor search.
It says that as the number of dimensions are higher I need to cover more space to get the same number of training examples. I didn't get it what is it trying to show and how does it occur. Any clarifications?

REPLY [2 votes]: Lets say we have a p-dimensional unit cube representing our data. (where each dimension/feature corresponds to an edge of the cube).
Lets say we try to use the K-nearest neighbor classifier to predict the output for test data based on the output values of inputs that are close to the test input. So let us explore a fraction 'r' of the unit volume of the p-dimensional cube. So the volume of the cube that we are trying to explore is 1/r. So the length of the edge of cube is [math]r^{1/p}[math].
So say we have 100 features in our dataset (p=100 and )if we want to explore 1%(0.01) of the space of cube while doing nearest neighbour,  the length of the edge of the cube is [math]0.1^{1/100}[/math] = 0.977 or we need to explore 97% of each feature to get information about 1% of the whole data. This is such a huge information to explore and takes a lot of time and space complexity. This is what we mean intuitively by curse of dimensionality, where the actual data to explore is bigger than what we think we need to explore.