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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

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

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

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

(D = 30)

Traversing the manifold

D = 40 (using 80% of the faces)

Traversing the manifold

D = 50 (using 98% of the faces)

Comparison to LLE

The collapsing-to-primary-dimension-test:

Comparison to LLE

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

Comparison to LLE

The 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, 1-column overlap

Weakness under translation

- Uniform translation, with a skip

Overestimating k

- Isomap with k = 2

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