of \[Alg1\] is based on the Kurdyka-Lojasiewicz (KL) property in [@Kurdyk1998; @Lojasiewicz1995]. It is in general difficult to directly verify whether a given function $f$ satisfies the KL property by the definition. Instead, we can verify this through several special types of functions which are proven to satisfy the KL property, such as analytic functions and semi-algebraic functions (e.g. [@H.Attouch2010; @C.Bao2016; @J.Bolte2014; @Y.Xu2013]).

For the notational simplicity, we introduce $\z=\left(\bsv,\bc,\bsW\right)$ and $$\begin{aligned}
P(\z)=\f{1}{2}\left\|\bmP_{\MM}\bsv-\bsf\right\|_2^2+\f{\mu}{2}\left\|\bsW\left(\La\bsv\right)-\bc\right\|_2^2.\end{aligned}$$ Recall that $\mC\simeq\DD_{R}^{N^2}$ as in \[ConstraintSet\], and we introduce $$\begin{aligned}
\mD=\left\{\bsW\in\C^{M_2N^2\times N^2}:\bsW^*\bsW=\bsI_{N^2\times N^2}\right\}.\end{aligned}$$ Using this notation, we reformulate \[ProposedCSMRIModel\] as $$\begin{aligned}
\label{ObjectModi}
\min_{\z}~H(\z):=P(\z)+\gamma\left\|\bc\right\|_0+\imath_{\mC}(\bsv)+\imath_{\mD}(\bsW),\end{aligned}$$ where $\imath_{\mA}$ is the indicator function of a set $\mA$: $\imath_{\mA}(\z)=0$ if $\z\in\mA$, and $\infty$ otherwise.

It is not hard to verify that the indicator functions are analytic, and $\left\|\cdot\right\|_0$ is a semi-algebraic function. Since the remaining term $P(\z)$ is merely a polynomial, we can conclude that the function $H(\z)$ in \[ObjectModi\] satisfies the KL-property. Under this formulation, we can present the following result on the global convergence of \[Alg1\].

\[Th1\] Let $H(\z)$ be defined as \[ObjectModi\]. Then the sequence $\{\z_n:n\in\N\}$ generated by \[Alg1\] is globally convergent, and its limit is a critical point of $H$. Moreover, we have $$\begin{aligned}
\label{FiniteLength}
\sum_{n=0}^{\infty}\left\|\z_{n+1}-\z_n\right\|_2<\infty,\end{aligned}$$ i.e. the 