For a matrix $\mathbf{X} \in \mathbb{R}^{n\times l}$, we have the following problem of representing vectors in $\mathbf{X}$ as a convex combination of other vectors excluding the vector itself:
$\min\Vert\mathbf{X-XC}\Vert_F^2 \;\; s.t.\;\; diag(\mathbf{C})=0, \;\mathbf{c}_i\geq 0, \;\mathrm{and}\; \Vert\mathbf{c}_i\Vert_1=1$
where $\mathbf{c}_i$, is a column of $\mathbf{C}$, and $diag()$ extracts the diagonal elements of a matrix.
1) Do we have a unique solution or not?
2) What is an efficient way to solve this quadratic program?