Computer examples Tenenbaum de Silva Langford A Global

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

Statue face database • 698 64 x 64 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

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.