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Computer examples. Tenenbaum, de Silva, Langford “A Global Geometric Framework for Nonlinear Dimensionality Reduction”. Statue face database. 698 64x64 grayscale images 2 mins, 12 secs on a ~600 (?) MHz PIII. The computed manifold. The computed manifold.

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computer examples

Computer examples

Tenenbaum, de Silva, Langford

“A Global Geometric Framework for Nonlinear Dimensionality Reduction”

statue face database
Statue face database
  • 698 64x64 grayscale images
  • 2 mins, 12 secs on a ~600 (?) MHz PIII
testing the sensibility of the manifold coordinates
Testing the sensibility of the manifold coordinates
  • One test you could do:
  • Sort all faces according to first manifold coordinate (“left-right”)
  • View them in order
  • See if the face makes a monotonic progression from left to right
up down
Up  Down

Cleaner, since light variation is strictly azimuthal (consistent chin shadow)

testing the sensibility of the manifold coordinates9
Testing the sensibility of the manifold coordinates

Semantic consistency of a dimension value deteriorates between points that are far away on the manifold.

4 consecutive frames from right  left movie:

Well-lit faces are turning to the left with respect to each other

Dimly-lit faces also don’t turn right w.r.t each other

testing the sensibility of the manifold coordinates10
Testing the sensibility of the manifold coordinates

Semantic consistency of a dimension value deteriorates between points that are far away on the manifold.

Explanations:

Geodesic distance on the manifold is approximated by shortest-path distance in a neighbor graph.

 Sparsity in neighbor graphs result in distance error for points far away on the graph.

testing the sensibility of the manifold coordinates11
Testing the sensibility of the manifold coordinates

Geodesic distance approximator can’t be perfect in the face of sparse data

testing the sensibility of the manifold coordinates13
Testing the sensibility of the manifold coordinates

…to be a bit more left-facing than this face:

traversing the manifold
Traversing the manifold
  • Collapsing the manifold to one dimension isn’t the way to use it.
  • Try tracing one dimension while keeping the other dimensions from jumping around too much.
traversing the manifold15
Traversing the manifold

Algorithm used:

Sort images by “left-right” coord as before

Draw a smooth line through the manifold

Only add images that are within a certain manifold distance D from this line.

traversing the manifold19
Traversing the manifold

D = 20

(Half the range of the “up-down” dimension)

traversing the manifold21
Traversing the manifold

D = 40 (using 80% of the faces)

traversing the manifold22
Traversing the manifold

D = 50 (using 98% of the faces)

comparison to lle
Comparison to LLE

Run both algorithms on 100 of the statue faces (64 x 64 pixels)

Isomap

LLE

comparison to lle24
Comparison to LLE

Running time for 100 64x64 images:

LLE: 5 secs

Isomap: 1.39 secs

comparison to lle25
Comparison to LLE

The collapsing-to-primary-dimension-test:

comparison to lle26
Comparison to LLE

Uh… the collapsing-to-second-dimension-test

comparison to lle27
Comparison to LLE

The horizontal manifold traversal test (7 frames)

comparison to lle28
Comparison to LLE
  • LLE: once manifold is computed, meaningful paths through it need to be searched for.
weakness under translation
Weakness under translation
  • Images with a common background and a single translating object will have a rough time with pixel differences.
weakness under translation30
Weakness under translation
  • Uniform translation, no overlap

Input images:

Output images:

weakness under translation31
Weakness under translation
  • Uniform translation, 1-column overlap

Input images:

Output images:

weakness under translation32
Weakness under translation
  • Uniform translation, 1-column overlap
weakness under translation33
Weakness under translation
  • Uniform translation, with a skip
weakness under translation34
Weakness under translation
  • Isomap with k = 1 (like before)

(Original)

(Reconstruction)

weakness under translation35
Weakness under translation
  • Isomap with k = 2

(Original)

(Reconstruction)

overestimating k
Overestimating k
  • Isomap with k = 2
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