proposal for STAT946 projects Fall 2010

Project 1 : Sampling Landmarks Using a Convergence Approach

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, starting from the center of the data space, alternatively obtain landmarks that are as near as possible to the center and as far away as possible from the center, effectively removing each newly-sampled landmark by shrinking the distances matrix. During these steps, the closest point from the center and the furthest point from the center from the remaining data space gradually converge towards each other. This is because, during the steps, the distance between the nearest point from the center and the center will gradually increase while at the same time the distance between the furthest point from the center and the center will gradually decrease as the nearest point from the center and the furthest point from the center are effectively removed from the data space.

Before starting the procedure, we require the distances matrix containing the pair-wise distances between the data points. This matrix is constructed only once, and it is updated two times during each step.

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 point and the furthest point from it, 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.

Using 27-dimensional phoneme data, 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).

( NOTE: STILL BEING EDITED )