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

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

Testing the sensibility of the manifold coordinates

Testing the sensibility of the manifold coordinates

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

Some consecutive frames:

Well-lit faces are turning left, poorly-lit faces are turning left.

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.

Sparse neighbor graphs result in high 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 manifoldMake a better picture to illustrate that it’s more of a band than a path

Traversing the manifoldFix oval so that it encricles the middle three rows of faces instead

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 found.

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

End

<josh going nuts>