Symmetric vector least squares solution

Question

I have the similar problem as the Symmetric linear least squares solution. The least square problem of mine is that I want to find $$ minimize || Ax-b ||^2, $$

$$ where A\in m\times n, $$

$$ x,b \in n\times 1 $$ However, in my problem the symmetry belongs to the vector x, i.e. x = [1 2 2 1] or x = [1 2 3 2 1]. Is there any closed form solution or optimization problem for finding a symmetric solution x ? Any algorithm, solution, paper or idea is welcome.

(edit 1) : I found a similar way to achieve it from the matlab implemented function and problem for designing a linear phase filter

(edit 2 ): @user1551 You are so awesome. I have another uncorrelated question, I am fascinated with the various least square these days, I found that gilbert strang's book and his corresponding opened course for digging out some advanced linear algebra. However, I found that your appreciated skilled can't be available from here. The question of mine is how you have this idea, i.e. symmetric constraint setting up in the vector by (I-R)x = 0. Furthermore, do you have any recommended reference or book for learning ? THX!

Answer 1

Let $R$ be the reversal matrix, the anti-diagonal matrix obtained by flipping the identity matrix left-right. The constraint that $x$ is “symmetric” can be reformulated as $(I-R)x=0$. Any $x$ satisfying this constraint must be in the form of $$ x=[I-(I-R)^+(I-R)]v=\frac{I+R}{2}v=Pv $$ where $P=\frac{I+R}{2}$. Your constrained minimisation problem can then be reformulated to the usual unconstrained least-squares problem of minimising $\|APv-b\|^2$. Its minimisers are therefore given by $v=(AP)^+b+\big[I-(AP)^+(AP)\big]u$ or $$ x=P(AP)^+b+P\big[I-(AP)^+(AP)\big]u, $$ where $u$ is an arbitrary vector. In particular, $x=P(AP)^+b$ is a solution (if you take $u=0$).