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# Manifold learning with application to object recognition

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# Manifold learning with application to object recognition

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### Manifold learning with application to object recognition

1. 1. manifold learning with applications to object recognition Advanced Perception David R. Thompson
2. 2. agenda 1. why learn manifolds? 2. Isomap 3. LLE 4. applications
3. 3. types of manifolds exhaust Sir Walter manifold Synnot Manifold 1849-1928 low-D surface embedded in high-D space
4. 4. manifold learning Find a low-D basis for describing high-D data. X → X' S.T. dim(X') << dim(X) uncovers the intrinsic dimensionality (invertible)
5. 5. manifolds in vision plenoptic function / motion / occlusion
6. 6. manifolds in vision appearance variation images from hormel corp.
7. 7. manifolds in vision deformation images from www.golfswingphotos.com
8. 8. why do manifold learning? 1. data compression 2. “curse of dimensionality” 3. de-noising 4. visualization 5. reasonable distance metrics *
9. 9. reasonable distance metrics
10. 10. reasonable distance metrics
11. 11. reasonable distance metrics ?
12. 12. reasonable distance metrics ? linear interpolation
13. 13. reasonable distance metrics ? manifold interpolation
14. 14. agenda 1. why learn manifolds? 2. Isomap 3. LLE 4. applications
15. 15. Isomap For n data points, and a distance matrix D, i Dij = j ...we can construct a m-dimensional space to preserve inter-point distances by using the top eigenvectors of D scaled by their eigenvalues. yi= [ λ1v1i , λ2v2i , ... , λmvmi ]
16. 16. Isomap Infer a distance matrix using distances along the manifold.
17. 17. Isomap 1. Build a sparse graph with K-nearest neighbors Dg = (distance matrix is sparse)
18. 18. Isomap 2. Infer other interpoint distances by finding shortest paths on the graph (Dijkstra's algorithm). Dg =
19. 19. Isomap 3. Build a low-D embedded space to best preserve the complete distance matrix. Error function: inner product distances in new inner product coordinate distances in system graph L2 norm Solution – set points Y to top eigenvectors of Dg
20. 20. Isomap shortest-distance on a graph is easy to compute
21. 21. Isomap results: hands
22. 22. Isomap: pro and con - preserves global structure - few free parameters - sensitive to noise, noise edges - computationally expensive (dense matrix eigen-reduction)
23. 23. Locally Linear Embedding Find a mapping to preserve local linear relationships between neighbors
24. 24. Locally Linear Embedding
25. 25. LLE: Two key steps 1. Find weight matrix W of linear coefficients: Enforce sum-to-one constraint with the Lagrange Multiplier:
26. 26. LLE: Two key steps 2. Find projected vectors Y to minimize reconstruction error must solve for whole dataset simultaneously
27. 27. LLE: Two key steps We add constraints to prevent multiple / degenerate solutions:
28. 28. LLE: Two key steps cost function becomes: the optimal embedded coordinates are given by bottom m+1 eigenvectors of the matrix M
29. 29. LLE: Result preserves local topology PCA LLE
30. 30. LLE: pro and con - no local minima, one free parameter - incremental & fast - simple linear algebra operations - can distort global structure
31. 31. Others you may encounter ● Laplacian Eigenmaps (Belkin 2001) ● spectral method similar to LLE ● better preserves clusters in data ● Kernel PCA ● Kohonen Self-Organizing Map (Kohonen, 1990) ● iterative algorithm fits a network of pre- defined connectivity ● simple, fast for on-line learning ● local minima ● lacking theoretical justification
32. 32. No Free Lunch the “curvier” your manifold, the denser your data must be bad OK!
33. 33. conclusions Manifold learning is a key tool in your object recognition toolbox A formal framework for many different ad-hoc object recognition techniques