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

2. 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

3. 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

4. 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

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

6. T ANGENT S PACE R ECONSTRUCTION

7. T ANGENT S PACE E STIMATION

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

9. Q UADRATIC R EGRESSION ON H IGHER C OMPONENTS

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

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

12. T HANK Y OU