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R ECONSTRUCTION ON S MOOTH M ANIFOLDS Bhuwan Dhingra Dual Degree Student Electrical Engineering IIT Kanpur

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I NTRODUCTION A lot of high-dimensional datasets lie on or near a smooth low-dimensional manifold Ex: Disk Images Each image lies on a 2-dimensional manifold in a 100x100x3 space

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D IMENSION R EDUCTION Non-Linear methods (ISOMAP, LLE etc.) find a low-dimensional embedding of the sampled points on the manifold Out-of-Sample reconstruction – construct the high-dimensional images for new test points, given the training point images

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L OCAL L INEAR I NTERPOLATION Specifically, can we improve over simple linear interpolation in a local neighborhood on the manifold? Linear interpolation: Find k -nearest neighbors of new point Minimize Reconstruction

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T ANGENT S PACE P ARAMETERIZATION (T YAGI, V URAL AND F ROSSARD )

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T ANGENT S PACE R ECONSTRUCTION

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T ANGENT S PACE E STIMATION

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L INEAR R EGRESSION ON T ANGENT S PACE Red circle shows the interpolated test point on the tangent space

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Q UADRATIC R EGRESSION ON H IGHER C OMPONENTS

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T ANGENT S PACE V L INEAR R ECONSTRUCTION Actual Data Tangent Space Linear

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V IDEO F RAME I NTERPOLATION Foreman video sequence ISOMAP used to embed into m = 1 dimensional space

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T HANK Y OU

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