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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 l.jpg

Computer examples

Tenenbaum, de Silva, Langford

“A Global Geometric Framework for Nonlinear Dimensionality Reduction”


Statue face database l.jpg
Statue face database

  • 698 64x64 grayscale images

  • 2 mins, 12 secs on a ~600 (?) MHz PIII




Testing the sensibility of the manifold coordinates l.jpg
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


Right left l.jpg
Right  Left


Up down l.jpg
Up  Down

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


Lit on left lit on right l.jpg
Lit on left  Lit on right


Testing the sensibility of the manifold coordinates9 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Testing the sensibility of the manifold coordinates

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


Traversing the manifold l.jpg
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 l.jpg
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 l.jpg
Traversing the manifold

D = 20

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



Traversing the manifold21 l.jpg
Traversing the manifold

D = 40 (using 80% of the faces)


Traversing the manifold22 l.jpg
Traversing the manifold

D = 50 (using 98% of the faces)


Comparison to lle l.jpg
Comparison to LLE

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

Isomap

LLE


Comparison to lle24 l.jpg
Comparison to LLE

Running time for 100 64x64 images:

LLE: 5 secs

Isomap: 1.39 secs


Comparison to lle25 l.jpg
Comparison to LLE

The collapsing-to-primary-dimension-test:


Comparison to lle26 l.jpg
Comparison to LLE

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


Comparison to lle27 l.jpg
Comparison to LLE

The horizontal manifold traversal test (7 frames)


Comparison to lle28 l.jpg
Comparison to LLE

  • LLE: once manifold is computed, meaningful paths through it need to be searched for.


Weakness under translation l.jpg
Weakness under translation

  • Images with a common background and a single translating object will have a rough time with pixel differences.


Weakness under translation30 l.jpg
Weakness under translation

  • Uniform translation, no overlap

Input images:

Output images:


Weakness under translation31 l.jpg
Weakness under translation

  • Uniform translation, 1-column overlap

Input images:

Output images:


Weakness under translation32 l.jpg
Weakness under translation

  • Uniform translation, 1-column overlap


Weakness under translation33 l.jpg
Weakness under translation

  • Uniform translation, with a skip


Weakness under translation34 l.jpg
Weakness under translation

  • Isomap with k = 1 (like before)

(Original)

(Reconstruction)


Weakness under translation35 l.jpg
Weakness under translation

  • Isomap with k = 2

(Original)

(Reconstruction)


Overestimating k l.jpg
Overestimating k

  • Isomap with k = 2



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