Foundations of Data Science

Exercise 8.24 The Gaussian kernel maps points to a higher dimensional space. What is
this mapping?
Exercise 8.25 Agglomerative clustering requires that one calculate the distances between
all pairs of points. If the number of points is a million or more, then this is impractical.
One might try speeding up the agglomerative clustering algorithm by maintaining a 100
clusters at each unit of time. Start by randomly selecting a hundred points and place each
point in a cluster by itself. Each time a pair of clusters is merged randomly select one of
the remaining data points and create a new cluster containing that point. Suggest some
other alternatives.
Exercise 8.26 Let A be the adjacency matrix of an undirected graph. Let d(S, S) = A(S,S)
|S|
be the density of the subgraph induced by the set of vertices S. Prove that d (S, S) is the
average degree of a vertex in S. Recall that A(S, T ) = ∑
a ij
i∈S,j∈T
Exercise 8.27 Suppose A is a matrix with non negative entries. Show that A(S, T )/(|S||T |)
is maximized by the single edge with highest a ij . Recall that A(S, T ) = ∑
Exercise 8.28 Suppose A is a matrix with non negative entries and
a ij
i∈S,j∈T
σ 1 (A) = x T Ay = ∑ i,j
x i a ij y j , |x| = |y| = 1.
Zero out all x i less than 1/2 √ n and all y j less than 1/2 √ d. Show that the loss is no more
than 1 / 4 th of σ 1 (A).
Exercise 8.29 Consider other measures of density such as A(S,T ) for different values of
|S| ρ |T | ρ
ρ. Discuss the significance of the densest subgraph according to these measures.
Exercise 8.30 Let A be the adjacency matrix of an undirected graph. Let M be the
matrix whose ij th element is a ij − d id j
. Partition the vertices into two groups S and ¯S.
2m
Let s be the indicator vector for the set S and let ¯s be the indicator variable for ¯S. Then
s T Ms is the number of edges in S above the expected number given the degree distribution
and s T M ¯s is the number of edges from S to ¯S above the expected number given the degree
distribution. Prove that if s T Ms is positive s T M ¯s must be negative.
Exercise 8.31 Which of the three axioms, scale invariance, richness, and consistency
are satisfied by the following clustering algorithms.
1. k-means
2. Spectral Clustering.
Exercise 8.32 (Research Problem): What are good measures of density that are also
effectively computable? Is there empirical/theoretical evidence that some are better than
others?
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