Lecture 8 Least-norm solutions of undetermined equations
EE263 Autumn 2007-08
Stephen Boyd
Lecture 8 Least-norm solutions of undetermined
equations
? least-norm solution of underdetermined equations ? minimum norm solutions via QR factorization ? derivation via Lagrange multipliers ? relation to regularized least-squares ? general norm minimization with equality constraints
8?1
Underdetermined linear equations
we consider y = Ax
where A Rm?n is fat (m < n), i.e.,
? there are more variables than equations ? x is underspecified, i.e., many choices of x lead to the same y
we'll assume that A is full rank (m), so for each y Rm, there is a solution set of all solutions has form
{ x | Ax = y } = { xp + z | z N (A) } where xp is any (`particular') solution, i.e., Axp = y
Least-norm solutions of undetermined equations
8?2
? z characterizes available choices in solution ? solution has dim N (A) = n - m `degrees of freedom' ? can choose z to satisfy other specs or optimize among solutions
Least-norm solutions of undetermined equations
8?3
Least-norm solution
one particular solution is xln = AT (AAT )-1y
(AAT is invertible since A full rank)
in fact, xln is the solution of y = Ax that minimizes x
i.e., xln is solution of optimization problem
minimize x subject to Ax = y
(with variable x Rn)
Least-norm solutions of undetermined equations
8?4
suppose Ax = y, so A(x - xln) = 0 and (x - xln)T xln = (x - xln)T AT (AAT )-1y = (A(x - xln))T (AAT )-1y =0
i.e., (x - xln) xln, so x 2 = xln + x - xln 2 = xln 2 + x - xln 2 xln 2
i.e., xln has smallest norm of any solution
Least-norm solutions of undetermined equations
8?5
{ x | Ax = y }
N (A) = { x | Ax = 0 } xln
? orthogonality condition: xln N (A)
? projection interpretation: xln is projection of 0 on solution set { x | Ax = y }
Least-norm solutions of undetermined equations
8?6
? A = AT (AAT )-1 is called the pseudo-inverse of full rank, fat A ? AT (AAT )-1 is a right inverse of A ? I - AT (AAT )-1A gives projection onto N (A)
cf. analogous formulas for full rank, skinny matrix A: ? A = (AT A)-1AT ? (AT A)-1AT is a left inverse of A ? A(AT A)-1AT gives projection onto R(A)
Least-norm solutions of undetermined equations
8?7
Least-norm solution via QR factorization
find QR factorization of AT , i.e., AT = QR, with ? Q Rn?m, QT Q = Im ? R Rm?m upper triangular, nonsingular
then ? xln = AT (AAT )-1y = QR-T y ? xln = R-T y
Least-norm solutions of undetermined equations
8?8
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