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Estimating Intrinsic Dimension

Estimating Intrinsic Dimension. Justin Eberhardt UMD, Mathematics and Statistics Advisor: Dr. Kang James. Outline. Introduction Nearest Neighborhood Estimators Regression Estimator Maximum Likelihood Estimator Revised Maximum Likelihood Estimator Comparison Summary. 2.

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Estimating Intrinsic Dimension

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  1. Estimating Intrinsic Dimension Justin Eberhardt UMD, Mathematics and Statistics Advisor: Dr. Kang James

  2. Outline • Introduction • Nearest Neighborhood Estimators • Regression Estimator • Maximum Likelihood Estimator • Revised Maximum Likelihood Estimator • Comparison • Summary 2

  3. Intrinsic Dimension Definition • The least number of parameters required to generate a dataset • Minimum number of dimensions that describes a dataset without significant loss of feature 3

  4. z x y Ex 1: Intrinsic Dimension Flatten (Unroll) y x Int Dim = 2 4

  5. Ex 2: Intrinsic Dimension 1 28 56 28 X 28 One Image: 784 Dimensional

  6. No Loop Top & Bottom Loop Ex 2: Intrinsic Dimension [Isomap Project, J. Tenenbaum & J. Langford, Stanford] Int Dim = 2 6

  7. Applications • Biometrics • Facial Recognition, Fingerprints, Iris • Genetics 7

  8. Why do we need to reduce dimensionality? • Low dimensional datasets are more efficient • Not even supercomputers can handle very high-dimensional matrices • Data in 1,2 and 3 dimensions can be visualized 8

  9. Ex: Facial Recognition in MN • 5 Million People • 2 Images per Person (Front and Profile) • 1028 X 1028 Pixels per Image (1 Megapixel) • Total Memory Required: • n = 5,000,000 • p = (2)(1028)(1028)= 2.11 Million Dimensions • Matrix Size: (5 x 106)(2.11 x 106) = 10 billion cells • Memory: 2(10 x 1012) = 20 x 1012 = 20 Terabytes

  10. Intrinsic Dimension Estimators Objective: To find a simple formula that uses nearest neighbor (NN) information to quickly estimate intrinsic dimension 10

  11. Intrinsic Dimension Estimators Project Description: Through simulation, we will compare the effectiveness of three proposed NN intrinsic dimension estimators. 11

  12. Intrinsic Dimension Estimators Note: Traditional methods for estimating Intrinsic Dimension, such as PCA, fail on non-linear manifolds. 12

  13. Intrinsic Dimension Estimators Nearest-Neighbor Methods • Regression Estimator K. Pettis, T. Bailey, A. Jain & R. Dubes, 1979 • Maximum Likelihood Estimator E. Levina, & P. Bickel, 2005 D. MacKay and Z. Ghahramani, 2005 13

  14. Distance Matrix The distance from x2to x3 Di,j: Euclidean distance from xi to xj 14

  15. Nearest Neighbor Matrix The distance between x2 and the kth NN to x2 Ti,k: Euclidean distance between xi and the kth NN to xi 15

  16. Notation • m: Intrinsic Dimension • p: Dimension of the Raw Dataset • n: Number of Observations • f(x): density pdf for observation x • Tx,k or Tk: distance from observation x to kth NN • N(t,x): # obs within dist t of observation x 16

  17. N(t,x) = 3 t Notation p = 2 m = 1 N = 12 t2 x t1 t3 17

  18. NN Regression Estimator Density of Distance to kth NN (Single Observation, appx as Poisson) 1 Expected Distance to kth NN (Single Observation) 2a Sample-Averaged Distance to kth NN 2b Expected Distance to Sample-Averaged kth NN 3

  19. Trinomial Distribution Binomial Distribution Regression Estimator Distance to Kth NN pdf • Assumptions • f(x) is constant • n is large • f(x)Vt is small 19

  20. Regression Estimator Approximate as Poisson Expected distance to Kth NN

  21. Gk,m Cn Estimate m using simple linear regression 21

  22. Ex: Swiss Roll Dataset m=0.49 22

  23. Datasets Gaussian Sphere Raw Dim = 3 Int Dim = 3 Swiss Roll Raw Dim = 3 Int Dim = 2 Dbl Swiss Roll Raw Dim = 3 Int Dim = 2 Faces: Raw Dimension = 4096, Int Dim ~ 3 to 5 23

  24. ResultsRegression Estimator ~ 3.0 ~ 2.0 ~ 2.0 ~ 3.5 FACES K = N / 100 24

  25. NN Maximum Likelihood Estimator Counting Process Binomial (appx as Poisson) 1 Joint Counting Probability Joint Occurrence Density 2 Log-likelihood Function 3 4

  26. Maximum Likelihood Estimator N(t,x) = # Counts within Distance t of x # Counts btw Distance r and s is BIN 26

  27. Maximum Likelihood Estimator

  28. Joint pdf of Distances to K NN 28

  29. Log-Likelihood Function 29

  30. E. Levina & P. Bickel Averaging over N observations Averaging inverses over N observations (Using MLE) D. MacKay & Z. Ghahramani 30

  31. ResultsMLE Estimator (Revised MacKay & Ghahramani) ~ 3.0 ~ 2.0 ~ 2.1 ~ 3.5 FACES K = N / 100 31

  32. Comparison 32

  33. Comparison 33

  34. Comparison 34

  35. Comparison 35

  36. Comparison 36

  37. Comparison 37

  38. Isomap 38

  39. Summary • The regression and revised MLE estimators share similar characteristics when intrinsic dimension is small • As intrinsic dimension increases, the estimators become more dependent on K • Distribution type does not appear to be highly influential when the intrinsic dimension is small 39

  40. Thank You! • Dr. Kang James & Dr. Barry James • Dr. Steve Trogdon

  41. Example Swiss Roll Data Int Dim = 2

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