  In this paper, we focus on the nonlinear least squares:
$\mbox{min}_{\mathbf{x} \in \mathbb{H}^d}\| |A\mathbf{x}|-\mathbf{b}\|$ where
$A\in \mathbb{H}^{m\times d}$, $\mathbf{b} \in \mathbb{R}^m$ with $\mathbb{H}
\in \{\mathbb{R},\mathbb{C} \}$ and consider the uniqueness and stability of
solutions. Such problem arises, for instance, in phase retrieval and absolute
value rectification neural networks. For the case where
$\mathbf{b}=|A\mathbf{x}_0|$ for some $\mathbf{x}_0\in \mathbb{H}^d$, many
results have been developed to characterize the uniqueness and stability of
solutions. However, for the case where $\mathbf{b} \neq |A\mathbf{x}_0| $ for
any $\mathbf{x}_0\in \mathbb{H}^d$, there is no existing result for it to the
best of our knowledge. In this paper, we first focus on the uniqueness of
solutions and show for any matrix $A\in \mathbb{H}^{m \times d}$ there always
exists a vector $\mathbf{b} \in \mathbb{R}^m$ such that the solution is not
unique. But, in real case, such ``bad'' vectors $\mathbf{b}$ are negligible,
namely, if $\mathbf{b} \in \mathbb{R}_{+}^m$ does not lie in some measure zero
set, then the solution is unique. We also present some conditions under which
the solution is unique. For the stability of solutions, we prove that the
solution is never uniformly stable. But if we restrict the vectors $\mathbf{b}$
to any convex set then it is stable.
