Chapter 3 Geometry of convex functions - Meboo Publishing ...

244 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
whose gradient (D.2.3)
∇ X ‖XA − I‖ 2 F = 2 ( XAA T − A T) = 0 (573)
vanishes when
XAA T = A T (574)
When A is fat full-rank, then AA T is invertible, X ⋆ = A T (AA T ) −1 is the
pseudoinverse A † , and AA † =I . Otherwise, we can make AA T invertible
by adding a positively scaled identity, for any A∈ R m×n
X = A T (AA T + t I) −1 (575)
Invertibility is guaranteed for any finite positive value of t by (1427). Then
matrix X becomes the pseudoinverse X → A † X ⋆ in the limit t → 0 + .
Minimizing instead ‖AX − I‖ 2 F yields the second flavor in (1829).
3.7.0.0.3 Example. Hyperplane, line, described by affine function.
Consider the real affine function of vector variable, (confer Figure 71)
f(x) : R p → R = a T x + b (576)
whose domain is R p and whose gradient ∇f(x)=a is a constant vector
(independent of x). This function describes the real line R (its range), and
it describes a nonvertical [195,B.1.2] hyperplane ∂H in the space R p × R
for any particular vector a (confer2.4.2);
{[
∂H =
having nonzero normal
x
a T x + b
η =
[ a
−1
] }
| x∈ R p ⊂ R p ×R (577)
]
∈ R p ×R (578)