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The problem of finding a vector $x$ which obeys a set of quadratic equations$|a_k^\top x|^2=y_k$, $k=1,\cdots,m$, plays an important role in manyapplications. In this paper we consider the case when both $x$ and $a_k$ arereal-valued vectors of length $n$. A new loss function is constructed for thisproblem, which combines the smooth quadratic loss function with an activationfunction. Under the Gaussian measurement model, we establish that with highprobability the target solution $x$ is the unique local minimizer (up to aglobal phase factor) of the new loss function provided $m\gtrsim n$. Moreover,the loss function always has a negative directional curvature around its saddlepoints.

Towards the optimal construction of a loss function without spurious local minima for solving quadratic equations