Foundations of Data Science

Exercise 10.18 Repeat the above exercise but instead of adding edges to form cliques,
use each block to form a G(100,p) graph. For how small a p can you recover the blocks?
What if you add G(1,000,q) to the graph for some small value of q.
Exercise 10.19 Construct an n × m matrix A where each of the m columns is a 0-1
indicator vector with approximately 1/4 entries being 1. Then B = AA T is a symmetric
matrix that can be viewed as the adjacency matrix of an n vertex graph. Some edges will
have weight greater than one. The graph consists of a number of possibly over lapping
cliques. Your task given B is to find the cliques by the following technique of finding a
0-1 vector in the column space of B by the following linear program for finding b and x.
subject to
b = argmin||b|| 1
Bx = b
b 1 = 1
Then subtract bb T from B and repeat.
0 ≤ b i ≤ 1 2 ≤ i ≤ n
Exercise 10.20 Construct an example of a matrix A satisfying the following conditions
1. The columns of A are 0-1 vectors where the support of no two columns overlap by
50% or more.
2. No column’s support is totally within the support of another column.
3. The minimum 1-norm vector in the column space of A is not a 0-1 vector.
Exercise 10.21 Let M = L+R where L is a low rank matrix corrupted by a sparse noise
matrix R. Why can we not recover L from M if R is low rank or if L is sparse?
Exercise 10.22
1. Suppose for a univariate convex function f and a finite interval D, |f ′′ (x)| ≤ δ|f ′ (x)|
for every x. Then, what is a good step size to choose for gradient descent? Derive a
bound on the number of steps needed to get an approximate minimum of f in terms
of as few parameters as possible.
2. Generalize the statement and proof to convex functions of d variables.
Exercise 10.23 Prove that the maximum of a convex function over a polytope is attained
at one of its vertices.
Exercise 10.24 Prove Lemma 10.11.
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