A1 MNIST: Isomap on Euclidean distances

Clusters of 1s and 7s Blue: 7   Red:1 MATLAB Figure MATLAB Figure
Clusters of 4s and 9s Blue: 9   Red:4 MATLAB Figure MATLAB Figure
Clusters of all the digits Magenta:0, Red:1, Purple:2, Teal:3, Yellowish green:4, Pink:5, Black:6, blue:7, Green:8, Purplish Pink:9 MATLAB Figure MATLAB Figure

A2 MNIST: Isomap on Tangent distances

Clusters of 1s and 7s Blue: 7   Red:1 MATLAB Figure MATLAB Figure
Clusters of 4s and 9s Blue: 7   Red:1 MATLAB Figure MATLAB Figure
Clusters of all digits Magenta:0, Red:1, Purple:2, Teal:3, Yellowish green:4, Pink:5, Black:6, blue:7, Green:8, Purplish Pink:9 MATLAB Figure MATLAB Figure

A3 Handwritten Hindi Numerals

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B1 Nao: Isomap on Euclidean & Tangent distances

Residual Variance (Euclidean and Tangent Distances) Dimensionality: 3-4 The Variance doesn't decrease much beyong the knee at 3. This dimensionality may be attributed to 3 DOF of Nao's arm: 1 at the elbow and 2 at the shoulders. MATLAB Figure MATLAB Figure
3D plots (Euclidean and Tangent Distances) The Euclidean 3D plot reveals a possible 3D manifold of Nao's images. Open the MATLAB figures to have a 3D view. MATLAB Figure MATLAB Figure

B2 Random Arms: Isomap on Euclidean & Tangent distances

Residual Variance (Euclidean and Tangent Distances) Dimensionality: 4 The Residual Variance doesn't decrease much beyond the knee at 4th dimensionality. This may be attributed to four angles which uniquely determine an image of this set. MATLAB Figure MATLAB Figure
3D plot with images(Euclidean and Tangent Distances) While the Euclidean plot reveals a correspondance between coordinates and the arms' configurations, the Tangent plot doesn't help so much (tangent algo fails here due to uniform shades of black, blue and red). MATLAB Figure MATLAB Figure
3D plot without images Open the MATLAB figures to have a 3D view. MATLAB Figure MATLAB Figure

B3 Random Arms &: Finding angles

Angles [Theta1, Theta2, Theta3, Theta4]= [144.115, -131.774, 75.867, 91.786]

B4 Random Arms & Box: Isomap on Euclidean & Tangent distances

Residual Variance (Euclidean and Tangent) Dimensionality: 1 Although it appears from the Euclidean plot that dimensionality is 5 (the knee of the graph), but a deeper analysis shows that variance drops only by 0.009 from 1st to 5th dimensionality. Since the mere position of box determines all the four angles of the arms, the dimension reduces to two. Since the box moves only horizontally, the dimension further falls down to 1. MATLAB Figure MATLAB Figure
3D plots with images (Euclidean and Tangent) The Euclidean plot reveals a spiral structure, which is a 1D manifold, in direct testimony to the finding that dimensionality is 1. MATLAB Figure MATLAB Figure
3D plots without images (Euclidean and Tangent) Open the MATLAB figures to have a 3D view. MATLAB Figure MATLAB Figure