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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 learningwith applications to object recognitionAdvanced PerceptionDavid R. Thompson
  2. 2. agenda1. why learn manifolds?2. Isomap3. LLE4. applications
  3. 3. types of manifoldsexhaust Sir Waltermanifold Synnot Manifold 1849-1928 low-D surface embedded in high-D space
  4. 4. manifold learningFind a low-D basis fordescribing high-D data.X → X S.T.dim(X) << dim(X)uncovers the intrinsicdimensionality(invertible)
  5. 5. manifolds in visionplenoptic function / motion / occlusion
  6. 6. manifolds in vision appearance variationimages from hormel corp.
  7. 7. manifolds in vision deformationimages from
  8. 8. why do manifold learning?1. data compression2. “curse of dimensionality”3. de-noising4. visualization5. 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. agenda1. why learn manifolds?2. Isomap3. LLE4. applications
  15. 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. 16. Isomap Infer a distance matrix using distances along the manifold.
  17. 17. Isomap1. Build a sparse graph with K-nearest neighborsDg =(distance matrix issparse)
  18. 18. Isomap2. Infer other interpoint distances by findingshortest paths on the graph (Dijkstrasalgorithm).Dg =
  19. 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. 20. Isomapshortest-distance on a graph is easy tocompute
  21. 21. Isomap results: hands
  22. 22. Isomap: pro and con- preserves global structure- few free parameters- sensitive to noise, noise edges- computationally expensive (densematrix 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 steps1. Find weight matrix W of linearcoefficients:Enforce sum-to-one constraint with theLagrange Multiplier:
  26. 26. LLE: Two key steps2. Find projected vectors Y to minimizereconstruction errormust solve for whole datasetsimultaneously
  27. 27. LLE: Two key stepsWe add constraints to preventmultiple / degenerate solutions:
  28. 28. LLE: Two key stepscost function becomes:the optimal embedded coordinates aregiven by bottom m+1 eigenvectors ofthe matrix M
  29. 29. LLE: Resultpreserves localtopology 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 Lunchthe “curvier” yourmanifold, the denser yourdata must be bad OK!
  33. 33. conclusionsManifold learning is a key tool in yourobject recognition toolboxA formal framework for many differentad-hoc object recognition techniques