$$
\begin{array}{l} \| \underline {{\mathbf {X}}} - \underline {{\mathbf {X}}} ^ {c} \| _ {\mathrm {F}} \\ = \left\| \underline {{\mathbf {U}}} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \left(\underline {{\mathbf {V}}} ^ {c}\right) ^ {\top} \right\| _ {\mathrm {F}} \\ = \left\| \underline {{\mathbf {U}}} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} + \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \underline {{\mathbf {V}}} ^ {\top} \right. \\ + \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \left(\underline {{\mathbf {V}}} ^ {c}\right) ^ {\top} \| _ {\mathbf {F}} \\ \leq \left\| \underline {{\mathbf {U}}} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} \right\| _ {\mathrm {F}} + \left\| \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \underline {{\mathbf {V}}} ^ {\top} \right\| _ {\mathrm {F}} \\ + \left\| \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \left(\underline {{\mathbf {V}}} ^ {c}\right) ^ {\top} \right\| _ {\mathrm {F}} \\ \leq \left\| \underline {{\mathbf {U}}} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} \right\| _ {\mathrm {F}} + \left\| \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \underline {{\mathbf {V}}} ^ {\top} \right\| _ {\mathrm {F}} \\ + \left\| \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \underline {{\mathbf {V}}} ^ {\top} - \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} \left(\underline {{\mathbf {V}}} ^ {c}\right) ^ {\top} \right\| _ {\mathrm {F}} \\ \leq \left\| (\underline {{\mathbf {U}}} - \underline {{\mathbf {U}}} ^ {c}) * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} \right\| _ {\mathrm {F}} + \left\| \underline {{\mathbf {U}}} ^ {c} * _ {M} (\underline {{\mathbf {S}}} - \underline {{\mathbf {S}}} ^ {c}) * _ {M} \underline {{\mathbf {V}}} ^ {\top} \right\| _ {\mathrm {F}} + \left\| \underline {{\mathbf {U}}} ^ {c} * _ {M} \underline {{\mathbf {S}}} ^ {c} * _ {M} (\underline {{\mathbf {V}}} - \underline {{\mathbf {V}}} ^ {c}) ^ {\top} \right\| _ {\mathrm {F}}. \\ \end{array}
$$