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I need to find $x$ that minimizes the cost function $\|y-Ax\|_p$ when $p$ is close to $0$, subject to the constraint $\|x\|_2=1$ where $x$ and $y$ are vectors in $\mathbb{R}^n$ and $A$ is an $n\times n$ matrix of full rank such that $A^TA$ is diagonal with strictly positive entries and $\operatorname{Tr}(A^TA)=N$. The matrix $A$ and the vector $y$ are both known. The elements of $A$ are in $\mathbb{R}$. Do note that $A$ itself is not diagonal. Specifically (if it helps), $A=\begin{pmatrix} B& C\\ -C& B \end{pmatrix}$ where $B$ and $C$ are of size $n/2 \times n/2$ and are both symmetric matrices.

The problem is non-convex. Any thoughts on solving the expression with $A$ as an identity or diagonal matrix are more than welcome as well

The notation $\|z\|_p$ highlights the $L_p$ norm and is $\|z\|_p=\sum|z_i|^p$ where $z_i$ are the elements of $z$. For $p<1$, $\|\cdot\|_p$ is not a norm in the actual definition of the word as it defies the triangle inequality.

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