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- 1. manifold learningwith applications to object recognitionAdvanced PerceptionDavid R. Thompson
- 2. agenda1. why learn manifolds?2. Isomap3. LLE4. applications
- 3. types of manifoldsexhaust Sir Waltermanifold Synnot Manifold 1849-1928 low-D surface embedded in high-D space
- 4. manifold learningFind a low-D basis fordescribing high-D data.X → X S.T.dim(X) << dim(X)uncovers the intrinsicdimensionality(invertible)
- 5. manifolds in visionplenoptic function / motion / occlusion
- 6. manifolds in vision appearance variationimages from hormel corp.
- 7. manifolds in vision deformationimages from www.golfswingphotos.com
- 8. why do manifold learning?1. data compression2. “curse of dimensionality”3. de-noising4. visualization5. reasonable distance metrics *
- 9. reasonable distance metrics
- 10. reasonable distance metrics
- 11. reasonable distance metrics ?
- 12. reasonable distance metrics ? linear interpolation
- 13. reasonable distance metrics ? manifold interpolation
- 14. agenda1. why learn manifolds?2. Isomap3. LLE4. applications
- 15. IsomapFor n data points, and a distance matrix D, i Dij = j...we can construct a m-dimensional space topreserve inter-point distances by using the topeigenvectors of D scaled by their eigenvalues. yi= [ λ1v1i , λ2v2i , ... , λmvmi ]
- 16. Isomap Infer a distance matrix using distances along the manifold.
- 17. Isomap1. Build a sparse graph with K-nearest neighborsDg =(distance matrix issparse)
- 18. Isomap2. Infer other interpoint distances by findingshortest paths on the graph (Dijkstrasalgorithm).Dg =
- 19. Isomap3. Build a low-D embedded space to bestpreserve the complete distance matrix.Error function: inner product distances in new inner product coordinate distances in system graph L2 normSolution – set points Y to top eigenvectors of Dg
- 20. Isomapshortest-distance on a graph is easy tocompute
- 21. Isomap results: hands
- 22. Isomap: pro and con- preserves global structure- few free parameters- sensitive to noise, noise edges- computationally expensive (densematrix eigen-reduction)
- 23. Locally Linear Embedding Find a mapping to preserve local linear relationships between neighbors
- 24. Locally Linear Embedding
- 25. LLE: Two key steps1. Find weight matrix W of linearcoefficients:Enforce sum-to-one constraint with theLagrange Multiplier:
- 26. LLE: Two key steps2. Find projected vectors Y to minimizereconstruction errormust solve for whole datasetsimultaneously
- 27. LLE: Two key stepsWe add constraints to preventmultiple / degenerate solutions:
- 28. LLE: Two key stepscost function becomes:the optimal embedded coordinates aregiven by bottom m+1 eigenvectors ofthe matrix M
- 29. LLE: Resultpreserves localtopology PCA LLE
- 30. LLE: pro and con- no local minima, one free parameter- incremental & fast- simple linear algebra operations- can distort global structure
- 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. No Free Lunchthe “curvier” yourmanifold, the denser yourdata must be bad OK!
- 33. conclusionsManifold learning is a key tool in yourobject recognition toolboxA formal framework for many differentad-hoc object recognition techniques

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