Hi all,
Previously I asked a question that currently has no satisfactory answer Least sum squares given constraints on subcomponents
It comes from an engineering problem. I was thinking to formulate it differently and hope that someone becomes interested and/or know how to solve it.
By formulating differently, I will have the following system:
$\mathbf{D} \mathbf{R} \mathbf{\theta} = \mathbf{0}_{2N \times 1}$
D is a $2N \times 6N$ block diagonal matrix that contains unknowns
$\mathbf{D} = diag(\mathbf{x}^T – a_1 \mathbf{z}^T, \mathbf{y}^T – b_1 \mathbf{z}^T, \dots, \mathbf{x}^T – a_N \mathbf{z}^T, \mathbf{x}^T – b_N \mathbf{z}^T)$
where $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are 3×1 orthogonal unit vectors that we need to find (ie., $\mathbf{x}^T \mathbf{x} = 1, \mathbf{x}^T \mathbf{y} = 0, \dots$). $a_i, b_i$ are known parameters.
R is a $6N \times M$ matrix that contains only numerical entries (measured and computed). $\mathbf{\theta}$ is a vector of M other unknown parameters.
By doing so, I isolated unknowns in two separate matrices (vectors). However, it is still not trivial.
I tried to solve this, but there is no obvious way. One way is that I tried to find $\mathbf{x}, \mathbf{y}, \mathbf{z}$ such that $\mathbf{D} \mathbf{R}$ is rank-deficient. I'm not sure if this can be solved, either exactly or in least-squared, in closed-form or numerically. Any idea, discussion is appreciated.
edit: I was not clear. If I set determinant of any MxM sub-matrix of $\mathbf{D} \mathbf{R}$ to 0 and express it as a function of elements in $\mathbf{x}, \mathbf{y}, \mathbf{z}$, then I have a number of homogeneous polynomials. That's why the title comes.
edit2: M is much smaller than 2N.
Best Answer
One suggests that we can have a good prior for $\theta$ (but not for $\mathbf{D}$). Then, he proposed that we fix $\theta$, find $\mathbf{R}$ for least square $|| \mathbf{D} \mathbf{R} \theta||$ given the constraints above; then fix $\mathbf{R}$ and find $\theta$ for least square $|| \mathbf{D} \mathbf{R} \theta|| $ given $||\theta|| = 1$. Then keep repeating (fix $\theta$ then $\mathbf{R}$) until they converges.
The latter least square is easy, and the former, I believe there is a closed form or numerical solution.
But my question is if such numerical scheme works? Is there any proof for it? What is the name of this method? I could not recall this numerical scheme.
Please enlighten.