proposal for STAT946 projects Fall 2010

Project 1 : Sampling Landmarks Recursively for Landmark MDS

By: Yongpeng Sun

Intuition:

When we have a large number of data points and we wish to sample a small portion of them as landmarks, using simple random sampling (SRS) does not guarantee good results. The reason is that, when the sample is very small, SRS sometimes results in a sample that is a small cluster within the data space that would not be representative of all of the data. We would like our landmarks to be as representative of the data set as possible. Thus, we can, using each landmark that we obtain, generate two new landmarks using that landmark; and these two newly-generated landmarks can be the nearest point and the furthest point from that landmark. The resulting landmarks obtained from this procedure should then be very representative of the entire data set.

For this procedure, we require the following items:

- he distances matrix containing the pair-wise distances between the data points. This matrix is constructed only once, and it is updated a number of times during the procedure of obtaining the landmarks.

- a queue containing landmarks that we obtain and add to it. For each point at the front of the queue, we use it to find two new landmarks, and then we delete it from the queue.

- a list containing the landmarks, to which we add all of the landmarks that we obtain.

The procedure for obtaining the landmarks can be described by the following steps:

Step 1: Find the point closet to the center, and add this point to the to-do queue and the landmarks. Update the distances matrix by removing the row and the column regarding this point.

Step 2: For the front-most point in the to-do queue, find the closest and furthest points, and add these two points to the to-do queue and the landmarks. Update the distances matrix by removing the row and the column regarding each of these two points.

Remove the front-most point from the to-do-queue.

Step 3: If the number of landmarks suffices, then stop. Otherwise, go back to step 2.

I would like to compare the efficiency of my method of sampling landmarks against the most common approach of sampling landmarks, which is by using simple random sampling (SRS).