[Solved] MATH5473 Homework 6-Manifold Learning

1. Order the faces: The following dataset contains 33 faces of the same person (Y R^1129233) in different angles, https://yao-lab.github.io/data/face.mat

You may create a data matrix X R^np where n = 33,p = 112 92 = 10304 (e.g.

X=reshape(Y,[10304,33]); in matlab).

• Explore the MDS-embedding of the 33 faces on top two eigenvectors: order the facesaccording to the top 1st eigenvector and visualize your results with figures.
• Explore the ISOMAP-embedding of the 33 faces on the k = 5 nearest neighbor graph and compare it against the MDS results. Note: you may try Tenenbaums Matlab code https://yao-lab.github.io/data/isomapII.m
• Explore the LLE-embedding of the 33 faces on the k = 5 nearest neighbor graph and compare it against ISOMAP. Note: you may try the following Matlab code https://yao-lab.github.io/data/lle.m

2. Manifold Learning: The following codes by Todd Wittman contain major manifold learning algorithms talked on class.

http://math.stanford.edu/~yuany/course/data/mani.m

Precisely, eight algorithms are implemented in the codes: MDS, PCA, ISOMAP, LLE, Hessian Eigenmap, Laplacian Eigenmap, Diffusion Map, and LTSA. The following nine examples are given to compare these methods,

• Swiss roll;
• Swiss hole;
• Corner Planes;
• Punctured Sphere;
• Twin Peaks;
• 3D Clusters;
• Toroidal Helix;
• Gaussian;
• Occluded Disks.

1

Homework 6. Manifold Learning 2

Run the codes for each of the nine examples, and analyze the phenomena you observed.

*Moreover if possible, play with t-SNE using scikit-learn manifold package:

http://scikit-learn.org/stable/modules/generated/sklearn.manifold.TSNE.html or any other implementations collected at

http://lvdmaaten.github.io/tsne/

3. Nystrom method: In class, we have shown that every manifold learning algorithm can be regarded as Kernel PCA on graphs: (1) given N data points, define a neighborhood graph with N nodes for data points; (2) construct a positive semidefinite kernel K; (3) pursue spectral decomposition of K to find the embedding (using top or bottom eigenvectors).

However, this approach might suffer from the expensive computational cost in spectral decomposition of K if N is large and K is non-sparse, e.g. ISOMAP and MDS.

To overcome this hurdle, Nystrom method leads us to a scalable approach to compute eigenvectors of low rank matrices. Suppose that an N-by-N positive semidefinite matrix

0 admits the following block partition

. (1)

where A is an n-by-n block. Assume that A has the spectral decomposition A = UU^T,

= diag(_i) (_{1 2 k} > _k+1 = = 0) and U = [u₁,,u_n] satisfies U^TU = I.

• Assume that K = XX^T for some X = [X₁;X₂] R^Nk with the block X₁ R^nk.

Show that X₁ and X₂ can be decided by:

1/2

X₁ = U_kk , (2)

X2 = BTUkk 1/2, (3)

where U_k = [u₁,,u_k] consists of those k columns of U corresponding to top k eigenvalues _i (i = 1,,k).

• Show that for general 0, one can construct an approximation from (2) and (3),

. (4)

where , and denoting the Moore-

Penrose (pseudo-) inverse of A. Therefore kK Kk_F = kC B^TABk_F. Here the matrix C B^TAB =: K/A is called the (generalized) Schur Complement of A in K.

• Explore Nystrom method on the Swiss-Roll dataset (http://yao-lab.github.io/data/ mat contains 3D-data X; http://yao-lab.github.io/data/swissroll. m is the matlab code) with ISOMAP. To construct the block A, you may choose either of the following:

n random data points;

Homework 6. Manifold Learning 3

*n landmarks as minimax k-centers (https://yao-lab.github.io/data/kcenter. m);

Some references can be found at:

[dVT04] Vin de Silva and J. B. Tenenbaum, Sparse multidimensional scaling using landmark points, 2004, downloadable at http://pages.pomona.edu/~vds04747/ public/papers/landmarks.pdf;

[P05] John C. Platt, FastMap, MetricMap, and Landmark MDS are all Nystrom Algorithms, 2005, downloadable at http://research.microsoft.com/en-us/um/people/ jplatt/nystrom2.pdf.

• *Assume that A is invertible, show that

det(K) = det(A) det(K/A),

• *Assume that A is invertible, show that

rank(K) = rank(A) + rank(K/A).

• *Can you extend the identities in (c) and (d) to the case of noninvertible A? A good reference can be found at,

[Q81] Diane V. Quellette, Schur Complements and Statistics, Linear Algebra and Its Applications, 36:187-295, 1981. http://www.sciencedirect.com/science/ article/pii/0024379581902329