v2009.01.01 - Convex Optimization

2.3. HULLS 61
In case k = N , the Fantope is identity matrix I . More generally, the set
{UU T | U ∈ R N×k , U T U = I} (84)
comprises the extreme points (2.6.0.0.1) of its convex hull. By (1367), each
and every extreme point UU T has only k nonzero eigenvalues λ and they all
equal 1 ; id est, λ(UU T ) 1:k = λ(U T U) = 1. So the Frobenius norm of each
and every extreme point equals the same constant
‖UU T ‖ 2 F = k (85)
Each extreme point simultaneously lies on the boundary of the positive
semidefinite cone (when k < N ,2.9) and on the boundary of a hypersphere
of dimension k(N −
√k(1 k + 1) and radius − k ) centered at k
I along
2 2 N N
the ray (base 0) through the identity matrix I in isomorphic vector space
R N(N+1)/2 (2.2.2.1).
Figure 18 illustrates extreme points (84) comprising the boundary of a
Fantope, the boundary of a disc corresponding to k = 1, N = 2 ; but that
circumscription does not hold in higher dimension. (2.9.2.5)
2.3.2.0.2 Example. Convex hull of rank-1 matrices.
From (83), in Example 2.3.2.0.1, we learn that the convex hull of normalized
symmetric rank-1 matrices is a slice of the positive semidefinite cone.
In2.9.2.4 we find the convex hull of all symmetric rank-1 matrices to be
the entire positive semidefinite cone.
In the present example we abandon symmetry; instead posing, what is
the convex hull of bounded nonsymmetric rank-1 matrices:
conv{uv T | ‖uv T ‖ ≤ 1, u∈ R m , v ∈ R n } = {X ∈ R m×n | ∑ i
σ(X) i ≤ 1} (86)
where σ(X) is a vector of singular values. Since ‖uv T ‖= ‖u‖‖v‖ (1496),
norm of each vector constituting a dyad uv T in the hull is effectively
bounded above by 1.
Proof. (⇐) Suppose ∑ σ(X) i ≤ 1. As inA.6, define a singular
value decomposition: X = UΣV T where U = [u 1 ... u min{m,n} ]∈ R m×min{m,n} ,
V = [v 1 ... v min{m,n} ]∈ R n×min{m,n} , and whose sum of singular values is
∑ σ(X)i = tr Σ = κ ≤ 1. Then we may write X = ∑ σ i
κ
√ κui
√ κv
T
i which is a
convex combination of dyads each of whose norm does not exceed 1. (Srebro)