HOMEWORK 6

Problem 1. (4pt total) Manifold learning:

(a) (2pt) Write 1/2 page report each on PCA, kernel PCA and Local linear

embedding. You can use web resources but don’t simply cut and paste.

(b) (2pt) Please refer to http://www.mathworks.com/help/stats/multidimensional￾scaling.html

This example shows how to construct a map of 10 US cities based on the pairwise

distances between those cities. The pairwise distance matrix between cities is given

by: cities = {’Atl’,’Chi’,’Den’,’Hou’,’LA’,’Mia’,’NYC’,’SF’,’Sea’,’WDC’}; D = [ 0 587

1212 701 1936 604 748 2139 2182 543; 587 0 920 940 1745 1188 713 1858 1737 597;

1212 920 0 879 831 1726 1631 949 1021 1494; 701 940 879 0 1374 968 1420 1645

1891 1220; 1936 1745 831 1374 0 2339 2451 347 959 2300; 604 1188 1726 968 2339 0

1092 2594 2734 923; 748 713 1631 1420 2451 1092 0 2571 2408 205; 2139 1858 949

1645 347 2594 2571 0 678 2442; 2182 1737 1021 1891 959 2734 2408 678 0 2329; 543

597 1494 1220 2300 923 205 2442 2329 0];

.

Generate 2D locations of the cities using MDS and overlay the map of the US

on the found locations to show that the estimated locations are indeed correct.

Problem 2. Groups (8pt total):

(1) (2pt) Prove that the set of matrix similarity transformations (A ∼ KK 1AK

as discussed in the class) forms a group.

(2) (2pt) Prove that the set of rotations form the special orthogonal group

under multiplication SO(m) for m = 2.

(3) (2pt) Prove that the set of isotropic scalings forms a group under multipli￾cation.

(4) (2pt) Prove that the set of translations in two dimensions forms a group

under addition.

Problem 3. (8pt) Manifolds

(1) (2pt) Please define smooth manifold, tangent space.

(2) (2pt) What is a Riemannian manifold?

(3) (2pt) What is exponential mapping and logarithmic mapping

(4) (2pt) What is equivalence relationship and Quotient Manifold?