computer examples tenenbaum, de silva, langford “a global geometric framework for nonlinear...
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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
Testing the sensibility of the manifold coordinates
One test you could do:
1. Sort all faces according to first manifold coordinate (“left-right”)
2. View them in order
3. See if the face makes a monotonic progression from left to right
Right Left
Up Down
Cleaner, since light variation is strictly azimuthal (consistent chin shadow)
Lit on left Lit on right
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 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 coordinates
Geodesic distance approximator can’t be perfect in the face of sparse data
Testing the sensibility of the manifold coordinates
The test expected this face:
Testing the sensibility of the manifold coordinates
…to be a bit more left-facing than this face:
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 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 manifold
Traversing the manifold
Traversing the manifold
Traversing the manifold
D = 20
(Half the range of the “up-down” dimension)
Traversing the manifold
(D = 30)
Traversing the manifold
D = 40 (using 80% of the faces)
Traversing the manifold
D = 50 (using 98% of the faces)
Comparison to LLERun both algorithms on 100 of the statue faces (64 x 64 pixels)
Isomap LLE
Comparison to LLE
Running time for 100 64x64 images:
LLE: 5 secs
Isomap: 1.39 secs
Comparison to LLE
The collapsing-to-primary-dimension-test:
Comparison to LLE
Uh… the collapsing-to-second-dimension-test
Comparison to LLEThe horizontal manifold traversal test (7 frames)
Comparison to LLE
• LLE: once manifold is computed, meaningful paths through it need to be searched for.
Weakness under translation
• Images with a common background and a single translating object will have a rough time with pixel differences.
Weakness under translation
• Uniform translation, no overlap
Input images:
Output images:
Weakness under translation
• Uniform translation, 1-column overlap
Input images:
Output images:
Weakness under translation
• Uniform translation, 1-column overlap
Weakness under translation
• Uniform translation, with a skip
Weakness under translation
• Isomap with k = 1 (like before)
(Original)
(Reconstruction)
Weakness under translation
• Isomap with k = 2
(Original)
(Reconstruction)
Overestimating k
• Isomap with k = 2
End