Homework 3

Question 1

Part a

The dimentionality of this plot is 2. This is because after increasing dimensionality after 2 does'nt result in significant decrease in residual variance i.e. the elbow appears at 2.

Part b

2 dimensional Manifold of system.
Plot with some superimposed images showing the variation.

Part c

From the 2-D isomap below, we can clearly see that :-
- images having almost same value of Theta₁(red rod is in same orientation) are lying in radially outward direction of torus.
- images having almost same value of Theta₂(blue rod is in same orientation) are lying in tangential direction of torus

Part d

3-D embedding of points is below

Part e

Using Neural Network of 6 Neurons. Converging at 200 epochs. Time Taken 1:36 minutes (Map from angles to SinesCosines)

Using Neural Network of 6 Neurons. Converging at 271 epochs. Time taken 2:34 minutes. (Map from image to SinesCosines)

As we can see from the above graphs, approach one converges faster.
The path from start ie 1.png to 161.png as embedded on 2-D isomap gives

Part f

For the first obstacle, red rot cannot rotate completely
- therefore the taurus opens up and hence to reach destination it has to traverse through the opposite direction.

Part g

For the second obstacle, red rod can still rotate, therefore only some values of theta2 are cut off for certain theta1
- the path can still pass through the ramaining unsliced toroid