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For brevity, collect all vectors $ \left\{\boldsymbol{a}_{i}\right\}_{i=1}^{m} $ in the $ m \times n $ matrix $ \boldsymbol{A}:=\left[\begin{array}{lll}\boldsymbol{a}_{1} & \cdots & \boldsymbol{a}_{m}\end{array}\right]^{\mathcal{T}} $, and all amplitudes $ \left\{\psi_{i}\right\}_{i=1}^{m} $ to form the vector $ \boldsymbol{\psi}:=\left[\begin{array}{lll}\psi_{1} & \cdots & \psi_{m}\end{array}\right]^{\mathcal{T}} $. One can rewrite the amplitudebased cost function in matrix-vector representation as \[ \underset{z \in \mathbb{R}^{n}}{\operatorname{minimize}} \ell(z):=\frac{1}{m} \sum_{i=1}^{m} \ell_{i}(z)=\frac{1}{2 m}\|\boldsymbol{\psi}-|\boldsymbol{A} \boldsymbol{z}|\|^{2} \] where $ \ell_{i}(z):=\frac{1}{2}\left(\psi_{i}-\left|\boldsymbol{a}_{i}^{\mathcal{T}} z\right|\right)^{2} $ with the superscript $ { }^{\mathcal{T}}\left({ }^{\mathcal{H}}\right) $ denoting (Hermitian) transpose; and with a slight abuse of notation, $ |\boldsymbol{A} z|:=\left[\left|\boldsymbol{a}_{1}^{\mathcal{T}} z\right| \cdots\left|\boldsymbol{a}_{m}^{\mathcal{T}} z\right|\right]^{\mathcal{T}} $. Apart from being

          For brevity, collect all vectors \( \left\{\boldsymbol{a}_{i}\right\}_{i=1}^{m} \) in the \( m \times n \) matrix \( \boldsymbol{A}:=\left[\begin{array}{lll}\boldsymbol{a}_{1} & \cdots & \boldsymbol{a}_{m}\end{array}\right]^{\mathcal{T}} \), and all amplitudes \( \left\{\psi_{i}\right\}_{i=1}^{m} \) to form the vector \( \boldsymbol{\psi}:=\left[\begin{array}{lll}\psi_{1} & \cdots & \psi_{m}\end{array}\right]^{\mathcal{T}} \). One can rewrite the amplitudebased cost function in matrix-vector representation as
\[
\underset{z \in \mathbb{R}^{n}}{\operatorname{minimize}} \ell(z):=\frac{1}{m} \sum_{i=1}^{m} \ell_{i}(z)=\frac{1}{2 m}\|\boldsymbol{\psi}-|\boldsymbol{A} \boldsymbol{z}|\|^{2}
\]
where \( \ell_{i}(z):=\frac{1}{2}\left(\psi_{i}-\left|\boldsymbol{a}_{i}^{\mathcal{T}} z\right|\right)^{2} \) with the superscript \( { }^{\mathcal{T}}\left({ }^{\mathcal{H}}\right) \) denoting (Hermitian) transpose; and with a slight abuse of notation, \( |\boldsymbol{A} z|:=\left[\left|\boldsymbol{a}_{1}^{\mathcal{T}} z\right| \cdots\left|\boldsymbol{a}_{m}^{\mathcal{T}} z\right|\right]^{\mathcal{T}} \). Apart from being

$For brevity, collect all vectors {ai}i=1^m in the m × n matrix A:=[ a1 ⋯ am ]^𝒯, and all amplitudes {ψi}i=1^m to form the vector ψ:=[ ψ1 ⋯ ψm ]^𝒯. One can rewrite the amplitudebased cost function in matrix-vector representation as z ∈ℝ^nminimizeℓ(z):=(1)/(m)∑i=1^mℓi(z)=(1)/(2 m)ψ-|Az|^2 where ℓi(z):=(1)/(2)(ψi-|ai^𝒯 z|)^2 with the superscript ^𝒯(^ℋ) denoting (Hermitian) transpose; and with a slight abuse of notation, |A z|:=[|a1^𝒯 z| ⋯|am^𝒯 z|]^𝒯. Apart from being$

Added by Vanessa B.

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