The results of applying the isomap on digits are shown below.The value of k was taken as 7 in each case.The tangent method of finding the distance between the points gives a better result than the euclidean distance.But an algorithm by Whang,Zhang and Feng seems to be promising for data which is noisy.In this particular case,the results given by the euclidean distance and the gaussian euclidean distance don't seem to differ much which may be beacuse the images are encoded in binary format and most of them are not smudged.
Isomap data with clusters of 1 and 7. Click here for larger image.
Isomap data with clusters of 4 and 9. Click here for larger image.
Isomap data with clusters of all digits. Click here for larger image.
Isomap data with some digits displayed. Click here for larger image.
The results displayed by applying isomap with the distances computed by the tangent method are displayed.
Isomap data with clusters of 1 and 7 displayed.
Isomap data with clusters of 4 and 9 displayed.
Isomap data with clusters of all digits displayed.
The dimensionality of these images as observerd from the residual curve is two.In all these images,the robot is standing at the same position and all other parts of the images are fixed except its left arm and left forearm.Both of these can sway in the horizontal direction. The residual error has been calculated for k=6,k=7 and k=8.The best curve is obtained for k=7.Plotting with epsilon instead of k does not seem to give the right result as the data is scattered in a large region.
Residual curve for k=6.
Click here for larger image.
Click here for the isomap.
Residual curve for k=7.
Click here for larger image.
Click here for the isomap
Residual curve for k=8.
Click here for larger image.
Click here for the isomap
The most likely dimensionality for the images of the arms is four as obsereved from the residual curve.This is evident as each link of each arm can move independently,giving the arms a total of four degrees of freedom.
From the residual curve,the dimensionality for this set of images appears to be one as opposed to the dimensionality of four as observed in B,2 above.In these images,only one of the upper arms appears to trnaslate freely,constraining the other arm to translate in a way so that the box moves in the horizontal direction since the coordinates of the center of the box are fixed with respect to the origin.Moreover,there is no free movement of the lower arms in the images as their translation in the horizontal direction is constrained by the motion of the upper arm.A dimension of one ensures that the task of moving the box horizontally is carried out successfully.
