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Computer examples Tenenbaum, de Silva, Langford “A Global Geometric Framework for Nonlinear Dimensionality Reduction”

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Statue face database 698 64x64 grayscale images 2 mins, 12 secs on a ~600 (?) MHz PIII

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The computed manifold

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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

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Right Left

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

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Lit on left Lit on right

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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

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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.

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Testing the sensibility of the manifold coordinates Geodesic distance approximator can’t be perfect in the face of sparse data

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Testing the sensibility of the manifold coordinates The test expected this face:

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Testing the sensibility of the manifold coordinates …to be a bit more left-facing than this face:

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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.

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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.

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Traversing the manifold

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D = 20 (Half the range of the “up-down” dimension)

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Traversing the manifold (D = 30)

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Traversing the manifold D = 40 (using 80% of the faces)

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Traversing the manifold D = 50 (using 98% of the faces)

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Comparison to LLE Run both algorithms on 100 of the statue faces (64 x 64 pixels) Isomap LLE

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Comparison to LLE Running time for 100 64x64 images: LLE: 5 secs Isomap: 1.39 secs

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Comparison to LLE The collapsing-to-primary-dimension-test:

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

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Comparison to LLE The horizontal manifold traversal test (7 frames)

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

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

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Weakness under translation Uniform translation, no overlap Input images: Output images:

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Weakness under translation Uniform translation, 1-column overlap Input images: Output images:

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Weakness under translation Uniform translation, 1-column overlap

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Weakness under translation Uniform translation, with a skip

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Weakness under translation Isomap with k = 1 (like before) (Original) (Reconstruction)

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Weakness under translation Isomap with k = 2 (Original) (Reconstruction)

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Overestimating k Isomap with k = 2

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