CS365: Homework 3

Group Submission

Priya Goyal (10535) and Gaganpreet Singh Kalra(10258)

Problem A

• Manifold dimensionality for the images
  

  Dimensionality of manifold is best explained at 2

  The residual variance at dimension 2 reduces sharply and for dimensionality higher than 2, the residual variance drops very little only by 0.04 in beginning and then becomes almost constant. The robot has 2 degree of freedoms and this is best captured at dimensionality 2 in the scree plot.

Problem B

• Lower dimensional 2-D embedding for the arm images (using Euclidean Distance) with some superimposed images
  

Problem C

• By looking at the 2-D embedding image above with some superimposed images, we observe that theta1 is varying from -pi to pi. By zooming into a part of data, for some small range of theta 1 value, theta2 varies from -pi to pi (red arm is fixed in the small range i.e. theta1 is almost fixed and theta2 is varying from -pi to pi). The global topolgy of robot is a torus i.e. S^1xS^1 and the observation indicates that the map has captured the topology of robot correctly.

Problem D

• Lower dimensional 3-D embedding for the arm images (using Euclidean Distance)
  

  The toplogy in 3-Dimension on zooming in looks like a TORUS which is what we actually expect.

Problem E

• The error obtained (nearly 0.4378 with learning rate 1.5, batchsize 150, epochs 250) when learning map from (v1,v2) using Feed Forward Neural Network less than the error obtained (nearly 0.7679 with learning rate 1.5, batchsize 150, epochs 250) when learning map directly from images to (q1,q2,q3,q4)
  The 1st approach converges faster because in second approach, each image is of very high dimension and due to the curse of dimensionality the following problems arise:
  1. Due to the large dimensionality, a highly non-linear function has to be mapped by the neural network and this also consumes a lot of computer memory.
  2. Due to the large number of network weights a vary large training set would be required to train the network.
  3. Due to large training data requirement, the genralization ability of the network is poor.
  
  The shortest path from image 0001.png to 00161.png is:

Problem F

• The isomap embedding of the images not colliding with the obstacle and the shortest path between image 00001.png and 00161.png is as shown in the figure below. The gaps in the graph correspond to the removed images that were colliding with the robot.(for more clarity compare with the image in part E).

Problem G

• The isomap embedding of the images not colliding with the obstacle and the shortest path between image 00001.png and 00161.png is as shown in the figure below. The magenta color in the graph correspond to the removed images that were colliding with the robot. The path in this case differs completely from the path in F. In first case, obstacle is closer to the robot so to to attain the configuration in image 161, it has to take the entirely different path (i.e. in the clockwise direction) whereas in 2nd case, the obstace is not very close to the robot and hence to attain the configuration in image 161, it can go in the counter-clockwise direction.