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Nonlinear Dimension Reduction: A Comparative Study of Semi-Definite Embedding vs. Local Linear Embedding

This study compares Semi-Definite Embedding (SDE) and Local Linear Embedding (LLE) for nonlinear dimension reduction. Methods, constraints, and objective functions of SDE are discussed, along with the performance differences between SDE and LLE. Experiments on various datasets show the strengths and weaknesses of each method.

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Nonlinear Dimension Reduction: A Comparative Study of Semi-Definite Embedding vs. Local Linear Embedding

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  1. Nonlinear Dimension Reduction: Semi-Definite Embedding vs. Local Linear Embedding Li Zhang and Lin Liao

  2. Outline • Nonlinear Dimension Reduction • Semi-Definite Embedding • Local Linear Embedding • Experiments

  3. Dimension Reduction • To understand images in terms of their basic modes of variability. • Unsupervised learning problem: Given N high dimensional input Xi RD, find a faithful one-to-one mapping to N low dimensional output Yi Rd and d<D. • Methods: • Linear methods (PCA, MDS): subspace • Nonlinear methods (SDE, LLE): manifold

  4. Semi-Definite Embedding Given input X=(X1,...,XN) and k • Find the k nearest neighbors for each input Xi • Formulate and solve a corresponding semi-definite programming problem; find optimal Gram matrix of output K=YTY • Extract approximately a low dimensional embedding Y from the eigenvectors and eigenvalues of Gram matrix K

  5. Semi-Definite Programming Maximize C·X Subject to AX=b matrix(X) is positive semi-definite where X is a vector with size n2, and matrix(X) is a n by n matrix reshaped from X

  6. Semi-Definite Programming • Constraints: • Maintain the distance between neighbors |Yi-Yj|2=|Xi-Xj|2for each pair of neighbor (i,j)  Kii+Kjj-Kij-Kji= Gii+Gjj-Gij-Gji where K=YTY,G=XTX • Constrain the output centered on the origin ΣYi=0  ΣKij=0 • K is positive semidefinite

  7. Semi-Definite Programming • Objective function • Maximize the sum of pairwise squared distance between outputs Σij|Yi-Yj|2 Tr(K)

  8. Semi-Definite Programming • Solve the best K using any SDP solver • CSDP (fast, stable) • SeDuMi (stable, slow) • SDPT3 (new, fastest, not well tested)

  9. Locally Linear Embedding

  10. Swiss Roll SDE, k=4 LLE, k=18 N=800

  11. LLE on Swiss Roll, varying K K=5 K=6 K=8 K=10

  12. LLE on Swiss Roll, varying K K=12 K=14 K=16 K=18

  13. LLE on Swiss Roll, varying K K=20 K=30 K=40 K=60

  14. Twos SDE, k=4 LLE, k=18 N=638

  15. Teapots SDE, k=4 LLE, k=12 N=400

  16. LLE on Teapot, varying N N=400 N=200 N=100 N=50

  17. Faces SDE, failed LLE, k=12 N=1900

  18. SDE versus LLE • Similar idea • First, compute neighborhoods in the input space • Second, construct a square matrix to characterize local relationship between input data. • Finally, compute low-dimension embedding using the eigenvectors of the matrix

  19. SDE versus LLE • Different performance • SDE: good quality, more robust to sparse samples, but optimization is slow and hard to scale to large data set • LLE: fast, scalable to large data set, but low quality when samples are sparse, due to locally linear assumption

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