# MATH5473 A Mathematical Introduction to Data Science Homework 6. Manifold Learning

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1. Order the faces: The following dataset contains 33 faces of the same person (Y ∈ R
112×92×33)
in different angles,
https://yao-lab.github.io/data/face.mat
You may create a data matrix X ∈ R
n×p where n = 33, p = 112 × 92 = 10304 (e.g.
X=reshape(Y,[10304,33])’; in matlab).
(a) Explore the MDS-embedding of the 33 faces on top two eigenvectors: order the faces
according to the top 1st eigenvector and visualize your results with figures.
(b) 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 Tenenbaum’s Matlab code
https://yao-lab.github.io/data/isomapII.m
(c) 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,
(a) Swiss roll;
(b) Swiss hole;
(c) Corner Planes;
(d) Punctured Sphere;
(e) Twin Peaks;
(f) 3D Clusters;
(g) Toroidal Helix;
(h) Gaussian;
(i) 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. Nystr¨om 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, Nystr¨om method leads us to a scalable approach to compute
eigenvectors of low rank matrices. Suppose that an N-by-N positive semidefinite matrix
K  0 admits the following block partition
K =

A B
BT C

. (1)
where A is an n-by-n block. Assume that A has the spectral decomposition A = UΛU
T
,
Λ = diag(λi) (λ1 ≥ λ2 ≥ . . . λk > λk+1 = . . . = 0) and U = [u1, . . . , un] satisfies U
TU = I.
(a) Assume that K = XXT
for some X = [X1; X2] ∈ R
N×k with the block X1 ∈ R
n×k
.
Show that X1 and X2 can be decided by:
X1 = UkΛ
1/2
k
, (2)
X2 = B
TUkΛ
−1/2
k
, (3)
where Uk = [u1, . . . , uk] consists of those k columns of U corresponding to top k eigenvalues λi (i = 1, . . . , k).
(b) Show that for general K  0, one can construct an approximation from (2) and (3),
Kˆ =

A B
BT Cˆ

. (4)
where A = X1XT
1
, B = X1XT
2
, and Cˆ = X2XT
2 = BT A†B, A† denoting the MoorePenrose (pseudo-) inverse of A. Therefore kKˆ − KkF = kC − BT A†BkF . Here the
matrix C − BT A†B =: K/A is called the (generalized) Schur Complement of A in K.
(c) Explore Nystr¨om method on the Swiss-Roll dataset (http://yao-lab.github.io/data/
swiss_roll_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 Nystr¨om Algorithms”, 2005, downloadable at http://research.microsoft.com/en-us/um/people/
jplatt/nystrom2.pdf.
(d) *Assume that A is invertible, show that
det(K) = det(A) · det(K/A),
(e) *Assume that A is invertible, show that
rank(K) = rank(A) + rank(K/A).
(f) *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