Given $n \times n$ positive semi-definite matrix $\boldsymbol{A}$ and $n \times 1$ vector $\boldsymbol{b}$. The goal is to find a set of $K$ vectors $\boldsymbol{x}_1,\dots,\boldsymbol{x}_K$ (each being $n \times 1$), with smallest possible value for $K$, such that these vectors satisfy moment constraints: \begin{equation} \sum_{k=1}^K \boldsymbol{x}_k = \boldsymbol{b} \quad,\quad \sum_{k=1}^K \boldsymbol{x}_k \boldsymbol{x}_k^T = \boldsymbol{A} \,. \end{equation} Any thoughts on how to obtain $\boldsymbol{x}_1,\dots,\boldsymbol{x}_K$?
Thank you!