Suppose $X \in \mathbb{R}^{m\times n}$, i.e., there are $n$ columns. Some of columns of $X$ are known, some of columns of $X$ are unknown (variables). Suppose I have $$\min_{x,z} \|X-z\mathbf{1}^T\|^2_2$$
where $z\in \mathbb{R}^{m}, \mathbf{1}\in \mathbb{R}^n$. So the optimization variables are those unknown columns $x$ and $z$.
How to show that this can be written as a least-squares problem? And what is the (geometric) meaning of $z$?
Please advise. Thanks!
First vectorize $X$ and $z$ together in some way, for example $$\text{vec}(X,z) = [\text{vec}(X),\text{vec}(z)]^T$$I will assume this vectorization (first orders all elements of $X$ and then all elements of $z$). Then build $$A=[M_I,M_{-1^T}]$$ Where $M$ is matrix representation of "multiplied with" subscript. Here you can use Kronecker products to your help. Now you want to minimize $$\|A \text{vec}(X,z)\|_2^2$$ To encode the $X$ values being known, just add a term like this: $$\|C(\text{vec}(X,z)-p)\|_2^2$$ Where $C$ is diagonal matrix encoding which values are known (large positive values if known, $\epsilon>0$ otherwise), and $p$ vector contains those values.