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Let $\pi \in \mathbb{R}^k$, and I want to solve the following:

$$\min_{x,G} || G\pi - x||_2^2$$ such that $G \in \mathbb{R}^{k \times k}$ and $x_i > 0$ and $\sum_{i=1}^k x_i = 1$.

(I know there is a solution in which the minimizer gives value 0).

When is the solution going to be unique?

More verbally: for a given vector $\pi$, when is there going to be a single linear transformation that transforms $\pi$ into a point in the probability simplex (without any coordinate being 0)?

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There is a linear transformation which maps your $\pi$ to $(1/n,\dots,1/n)$; many, in fact. (Assuming $\pi\neq0$, of course; if $\pi=0$, there is no $G$) –  Mariano Suárez-Alvarez Feb 14 '13 at 2:15
There are no restrictions on $G$? –  Dominique Feb 14 '13 at 2:16

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