Let $X$ be a symmetric $n\times n$ matrix and $A$ be $n \times m$ matrix and . $A$ is not related to $X$. Suppose we know there is an approximation $X \approx ACA^t$ where $C$ is $m\times m$ matrix ($m < n$).
Can we efficiently find $C$ ? (Without solving large optimization problem)
Can we do this efficiently if we know $C$ is sparse or diagonal ?
From this question I know there is no general connection between a general matrix $C$ and the eigenvalues of $X$, but what if we know $C$ is diagonal ?
Thanks.
Suppose $A$ has rank $m$. Then $A^t A$ is invertible, and $$C \approx (A^t A)^{-1} A^t X A (A^t A)^{-1}$$
More generally, consider $\tilde{C} = A^+ X {A^+}^t$ where $A^+$ is the Moore-Penrose pseudoinverse of $A$. If $X = A C A^t$, then $$A \tilde{C} A^t = A A^+ A C A^t {A^+}^t A^t = A C A^t = X$$