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 (Map from angles to SinesCosines)

• 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