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A Organization of the Appendix
The appendix includes the missing proofs, detailed discussions of some argument in the main body and more numerical experiments. We organize the appendix as follows:
- The proof of infeasibility condition (Theorem 3.2) is provided in Section B.
- Explanations on conditions derived in Theorem 3.2 are included in Section C.
- The proof of properties of the proposed model (r)LogSpecT (Proposition 3.4 & 3.6) is given in Section D and some additional properties are discussed.
- The truncated Hausdorff distance based proof details of Theorem 4.1 and Corollary 4.4 are given in Section E.
- Details of L-ADMM and its convergence analysis are in Section F.
Additional experiments and discussions on synthetic and real data are included in Section G.
B Proof of Theorem 3.2
Since the linear system (4) has no solution, we know from Farkas' lemma that the following system has solutions:
$$
\left\{ \begin{array}{l} {\left[ \boldsymbol {I} _ {m - 1} \quad \mathbf {0} _ {\frac {(m - 1) (m - 2)}{2}} \right] \boldsymbol {B} ^ {\top} \boldsymbol {A} _ {n} ^ {\top} \boldsymbol {x} <   \mathbf {0} _ {(m - 1) \times 1},} \\ {\left[ \mathbf {0} _ {\frac {(m - 1) (m - 2)}{2} \times (m - 1)} \quad \boldsymbol {I} _ {\frac {(m - 1) (m - 2)}{2}} \right] \boldsymbol {B} ^ {\top} \boldsymbol {A} _ {n} ^ {\top} \boldsymbol {x} \leq \mathbf {0} _ {\frac {(m - 1) (m - 2)}{2} \times 1}.} \end{array} \right. \tag {11}
$$
Let  $\pmb{x}^{*}\in \mathbb{R}^{m^{2}}$  be a solution to (11). Denote  $\pmb{x}_{+}\coloneqq \max \{\pmb{x}^{*},\pmb {0}\}$ $\pmb{x}_{-}\coloneqq \max \{-\pmb{x}^{*},\pmb {0}\}$  . Then, there exists  $c\in (0,1]$  such that
$$
\pmb {B} ^ {\top} \pmb {A} _ {n} ^ {\top} (\pmb {x} _ {+} - \pmb {x} _ {-}) + c \mathbf {1} _ {m ^ {2}} ^ {\top} (\pmb {x} _ {+} + \pmb {x} _ {-}) [ \mathbf {1} _ {m - 1}; \mathbf {0} _ {\frac {(m - 1) (m - 2)}{2}} ] \leq \mathbf {0}.
$$
Define  $y \coloneqq -\mathbf{1}_{m^2}^\top (\pmb{x}_+ + \pmb{x}_-), z \coloneqq c\mathbf{1}_{m^2}^\top (\pmb{x}_+ + \pmb{x}_-)$  and set  $\bar{\delta} = c$ . For all  $\delta \in [0, \bar{\delta})$ ,  $(\pmb{x}_+, \pmb{x}_-, y, z)$  is a solution to the following linear system:
$$
\left\{ \begin{array}{c} \boldsymbol {B} ^ {\top} \boldsymbol {A} _ {n} ^ {\top} (\boldsymbol {x} _ {+} - \boldsymbol {x} _ {-}) + z [ \mathbf {1} _ {m - 1}; \mathbf {0} _ {\frac {(m - 1) (m - 2)}{2}} ] \leq \mathbf {0}, \\ \quad \mathbf {1} _ {m ^ {2}} ^ {\top} (\boldsymbol {x} _ {+} + \boldsymbol {x} _ {-}) + y \leq 0, \\ \quad \delta y + z > 0, \\ \quad \boldsymbol {x} _ {+}, \boldsymbol {x} _ {-}, - y \geq \mathbf {0}. \end{array} \right.
$$
Again, from Farkas' lemma, this implies that the following linear system does not have a solution:
$$
\left\{ \begin{array}{c} A _ {n} B s + t \mathbf {1} _ {m ^ {2}} \geq \mathbf {0}, \\ A _ {n} B s - t \mathbf {1} _ {m ^ {2}} \leq \mathbf {0}, \\ t \leq \delta , \\ \left[ \mathbf {1} _ {m - 1} \quad \mathbf {0} _ {\frac {(m - 1) (m - 2)}{2}} \right] s = 1, \end{array} \right. \tag {12}
$$
where  $s \in \mathbb{R}^{m(m - 1) / 2}$  and  $t \in \mathbb{R}$ . Since (12) is equivalent to:
$$
\left\{ \begin{array}{c} \| C _ {n} \boldsymbol {S} - \boldsymbol {S} C _ {n} \| _ {\infty , \infty} \leq \delta , \\ (\boldsymbol {S} \mathbf {1}) _ {1} = 1, \\ \boldsymbol {S} \in \boldsymbol {S}, \end{array} \right. \tag {13}
$$
the above argument indicates that (13) does not have a solution. Suppose rSpecT has a feasible solution  $S'$ , then
$$
\| \boldsymbol {C} _ {n} \boldsymbol {S} ^ {\prime} - \boldsymbol {S} ^ {\prime} \boldsymbol {C} _ {n} \| _ {\infty , \infty} \leq \| \boldsymbol {C} _ {n} \boldsymbol {S} ^ {\prime} - \boldsymbol {S} ^ {\prime} \boldsymbol {C} _ {n} \| _ {F} \leq \delta .
$$
Hence,  $S'$  is also a solution to (13). However, (13) does not have a solution. We can conclude that rSpecT is infeasible in this case.
C Explanations on Sufficient Conditions in Theorem 3.2
We elaborate more on the infeasibility condition that  $A_{n}B$  has full column rank. An application of the condition is Example 3.1. Specifically, we know that in this case,
$$
\boldsymbol {B} = \left( \begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \end{array} \right) \quad \text {a n d} \quad \boldsymbol {A} _ {n} = \left( \begin{array}{c c c c} 0 & h _ {1 2} & - h _ {1 2} & 0 \\ h _ {1 2} & h _ {2 2} - h _ {1 1} & 0 & - h _ {1 2} \\ - h _ {1 2} & 0 & h _ {1 1} - h _ {2 2} & h _ {1 2} \\ 0 & - h _ {1 2} & h _ {1 2} & 0 \end{array} \right).
$$
$$
\boldsymbol {A} _ {n} \boldsymbol {B} = \left( \begin{array}{c} 0 \\ h _ {2 2} - h _ {1 1} \\ h _ {1 1} - h _ {2 2} \\ 0 \end{array} \right).
$$
Hence, when  $h_{11} \neq h_{22}$ ,  $A_n B$  has full column rank. This means that when  $\delta$  is small enough (from Example 3.1 we know  $\tilde{\delta} = \sqrt{2} |h_{11} - h_{22}|$ ), the model rSpecT is infeasible.
D Proofs of Properties of (r)LogSpecT
D.1 Proof of Proposition 3.4
Since the constraint set  $S$  is a cone, it follows that for all  $\gamma > 0$ ,  $\gamma S = S$ . Then, we know that
$$
\begin{array}{l} \operatorname {O p t} (\boldsymbol {C}, \alpha) = \underset {\boldsymbol {S} \in \mathcal {S}, \boldsymbol {C S} = \boldsymbol {S C}} {\operatorname {a r g m i n}} \| \boldsymbol {S} \| _ {1, 1} - \alpha \boldsymbol {1} ^ {\top} \log (\boldsymbol {S} \boldsymbol {1}) \\ = \gamma \cdot \operatorname * {a r g m i n} _ {\gamma \boldsymbol {S} \in \mathcal {S}, \boldsymbol {C} \gamma \boldsymbol {S} = \gamma \boldsymbol {S C}} \| \gamma \boldsymbol {S} \| _ {1, 1} - \alpha \boldsymbol {1} ^ {\top} \log (\gamma \boldsymbol {S} \boldsymbol {1}) \\ = \gamma \cdot \operatorname * {a r g m i n} _ {\boldsymbol {S} \in \frac {1}{\gamma} \boldsymbol {\mathcal {S}}, \boldsymbol {C} \boldsymbol {S} = \boldsymbol {S} \boldsymbol {C}} \gamma \| \boldsymbol {S} \| _ {1, 1} - \alpha \boldsymbol {1} ^ {\top} \log (\boldsymbol {S} \boldsymbol {1}) \\ = \gamma \cdot \operatorname * {a r g m i n} _ {\boldsymbol {S} \in \mathcal {S}, \boldsymbol {C S} = \boldsymbol {S C}} \| \boldsymbol {S} \| _ {1, 1} - \frac {\alpha}{\gamma} \mathbf {1} ^ {\top} \log (\boldsymbol {S} \mathbf {1}) \\ = \gamma \operatorname {O p t} (C, \alpha / \gamma), \\ \end{array}
$$
where the third equality is from the basic calculus rule of the logarithm function. Set  $\gamma = \alpha$  and then  $\mathrm{Opt}(\pmb {C},\alpha) = \alpha \mathrm{Opt}(\pmb {C},1)$ , which completes the proof.
D.2 Proof of Proposition 3.6
The proof will be conducted by constructing a feasible solution for rLogSpecT. Recall that  $A_{n} = I \otimes C_{n} - C_{n} \otimes I$  and the matrix  $B \in \mathbb{R}^{m^2 \times m(m-1)/2}$  that maps a non-negative vector to the vectorization of a valid adjacency matrix. Let  $S = \min \left\{ \frac{\delta}{\|A_{n}B s\|_{2}}, 1 \right\} \cdot \mathrm{mat}(B s)$  with  $s \in \mathbb{R}^{(m-1)m/2}$  being a non-negative vector, where  $\mathrm{mat}(\cdot)$  is the matricization operator. Note that
$$
\operatorname {v e c} \left(\boldsymbol {C} _ {n} \boldsymbol {S} - \boldsymbol {S C} _ {n}\right) = (\boldsymbol {I} \otimes \boldsymbol {C} _ {n} - \boldsymbol {C} _ {n} \otimes \boldsymbol {I}) \operatorname {v e c} (\boldsymbol {S}) = \boldsymbol {A} _ {n} \operatorname {v e c} (\boldsymbol {S}).
$$
Then, we know that
$$
\left\| \boldsymbol {C} _ {n} \boldsymbol {S} - \boldsymbol {S C} _ {n} \right\| _ {F} = \left\| \operatorname {v e c} \left(\boldsymbol {C} _ {n} \boldsymbol {S} - \boldsymbol {S C} _ {n}\right) \right\| _ {2} = \min  \left\{\frac {\delta}{\left\| \boldsymbol {A} _ {n} \boldsymbol {B} \boldsymbol {s} \right\| _ {2}}, 1 \right\} \cdot \left\| \boldsymbol {A} _ {n} \boldsymbol {B} \boldsymbol {s} \right\| _ {2} \leq \delta .
$$
Thus, the given  $S$  is a feasible solution for rLogSpecT and it completes the proof.
D.3 Properties of optimal solutions and values of (r)LogSpecT
In this section, we further discuss some properties of the optimal solutions/value of the proposed models, which are useful for deriving the recovery guarantee. More specifically, we obtain an upper bound on the optimal solutions (which may not be unique) independent of the sample size  $n$  and the inaccuracy parameter  $\delta_{n}$ . Also, a lower bound of optimal values follows.
Proposition D.1. The following statements hold:
- For an optimal solution  $S^{*}$  (resp.  $S_{n}^{*}$ ) to LogSpecT (resp. rLogSpecT with any given sample size  $n$ ), it follows that
$$
\| \boldsymbol {S} ^ {*} \| _ {1, 1} = \alpha m a n d \| \boldsymbol {S} _ {n} ^ {*} \| _ {1, 1} \leq \alpha m, \forall \delta_ {n} > 0.
$$
- If  $\delta_n \geq 2\alpha m \| C_n - C_\infty \|$ , then
$$
\alpha m (1 - \log \alpha) \leq f _ {n} ^ {*} \leq f ^ {*}, \forall n,
$$
where  $f^{*}$  (resp.  $f_{n}^{*}$ ) denotes the optimal value of LogSpecT (resp. rLogSpecT).
For the first statement, let us consider the Karush-Kuhn-Tucker (KKT) conditions of LogSpecT and rLogSpecT. Since the LogSpecT is a convex problem and Slater's condition holds, the KKT conditions are necessary and sufficient for the optimality, i.e., there exists  $(\Lambda_1,\Lambda_2)\in \mathbb{R}^{m\times m}\times \mathcal{N}_S(S^*)$  such that
$$
\left\{ \begin{array}{l} \nabla_ {\boldsymbol {S}} \left(\left\| \boldsymbol {S} ^ {*} \right\| _ {1, 1} - \alpha \mathbf {1} ^ {\top} \log \left(\boldsymbol {S} ^ {*} \mathbf {1}\right)\right) + C _ {\infty} \boldsymbol {\Lambda} _ {1} - \boldsymbol {\Lambda} _ {1} C _ {\infty} + \boldsymbol {\Lambda} _ {2} = \mathbf {0}, \\ \boldsymbol {C} _ {\infty} \boldsymbol {S} ^ {*} = \boldsymbol {S} ^ {*} \boldsymbol {C} _ {\infty}, \\ \boldsymbol {S} ^ {*} \in \mathcal {S}, \end{array} \right. \tag {14}
$$
where  $\mathcal{N}_{\mathcal{S}}(S^{*})\coloneqq \{N\in \mathbb{R}^{m\times m}:\sup_{X\in \mathcal{S}}\langle X - S^{*},N\rangle \leq 0\}$  is the normal cone of  $\mathcal{S}$  at  $S^{*}$ , and  $\nabla \| S^{*}\|_{1,1}$  is well-defined since  $\| \cdot \|_{1,1} = \langle \cdot ,\mathbf{11}^{\top}\rangle$  at  $S^{*}\geq 0$ , which is differentiable. Taking further calculation gives that
$$
\nabla \| \boldsymbol {S} ^ {*} \| _ {1, 1} = \mathbf {1} \mathbf {1} ^ {\top}, \quad (\nabla_ {\boldsymbol {S}} \mathbf {1} ^ {\top} \log (\boldsymbol {S} ^ {*} \mathbf {1})) _ {i j} = \frac {1}{(\boldsymbol {S} ^ {*} \mathbf {1}) _ {i}}.
$$
Combining this with (14) by taking inner product of both sides with  $S^{*}$ , we obtain that
$$
\sum_ {i, j} \left(\boldsymbol {S} ^ {*}\right) _ {i j} - \alpha \sum_ {i, j} \frac {\left(\boldsymbol {S} ^ {*}\right) _ {i j}}{\left(\boldsymbol {S} ^ {*} \mathbf {1}\right) _ {i}} + \left\langle \boldsymbol {\Lambda} _ {1}, \boldsymbol {C} _ {\infty} \boldsymbol {S} ^ {*} - \boldsymbol {S} ^ {*} \boldsymbol {C} _ {\infty} \right\rangle + \left\langle \boldsymbol {\Lambda} _ {2}, \boldsymbol {S} ^ {*} \right\rangle = 0. \tag {15}
$$
From the structure of  $S$  and the fact that  $\Lambda_2\in \mathcal{N}_S(S^*)$ , one has that  $\langle \Lambda_2,S^*\rangle = 0$ . Also, note that  $C_{\infty}S^{*} = S^{*}C_{\infty}$ . Hence, the equation (15) can be simplified as the desired result:
$$
\| \boldsymbol {S} ^ {*} \| _ {1, 1} = \sum_ {i, j} (\boldsymbol {S} ^ {*}) _ {i j} = \alpha \sum_ {i, j} \frac {(\boldsymbol {S} ^ {*}) _ {i j}}{(\boldsymbol {S} ^ {*} \mathbf {1}) _ {i}} = \alpha \sum_ {i = 1} ^ {m} \sum_ {j = 1} ^ {m} \frac {(\boldsymbol {S} ^ {*}) _ {i j}}{(\boldsymbol {S} ^ {*} \mathbf {1}) _ {i}} = \alpha m.
$$
The KKT conditions of rLogSpecT indicate that there exist  $\lambda_1 \geq 0$ ,  $\Lambda_2 \in \mathcal{N}_S(S_n^*)$  and  $Q \in \partial \| C_n S_n^* - S_n^* C_n \|_F$  (i.e., the subgradient of the function  $S \mapsto \| C_n S - S C_n \|_F$  at  $S_n^*$ ) such that
$$
\left\{ \begin{array}{l} \nabla_ {\boldsymbol {S}} \left(\left\| \boldsymbol {S} _ {n} ^ {*} \right\| _ {1, 1} - \alpha \mathbf {1} ^ {\top} \log \left(\boldsymbol {S} _ {n} ^ {*} \mathbf {1}\right)\right) + \lambda_ {1} \boldsymbol {Q} + \boldsymbol {\Lambda} _ {2} = \mathbf {0}, \\ \lambda_ {1} \left(\left\| \boldsymbol {C} _ {n} \boldsymbol {S} _ {n} ^ {*} - \boldsymbol {S} _ {n} ^ {*} \boldsymbol {C} _ {n} \right\| _ {F} - \delta_ {n}\right) = 0, \\ \boldsymbol {S} _ {n} ^ {*} \in \mathcal {S}. \end{array} \right. \tag {16}
$$
Moreover, from the definition of the convex subdifferential we know that  $0 \geq \| C_n S_n^* - S_n^* C_n\|_F - \langle Q, S_n^* \rangle$ . Thus, after taking inner product of both sides of the equation (16) with  $S_n^*$ , it follows that:
$$
\begin{array}{l} 0 = \sum_ {i, j} \left(\boldsymbol {S} _ {n} ^ {*}\right) _ {i j} - \alpha m + \lambda_ {1} \langle \boldsymbol {Q}, \boldsymbol {S} _ {n} ^ {*} \rangle + \langle \boldsymbol {\Lambda} _ {2}, \boldsymbol {S} _ {n} ^ {*} \rangle \\ \geq \sum_ {i, j} \left(\boldsymbol {S} _ {n} ^ {*}\right) _ {i j} - \alpha m + \lambda_ {1} \left\| \boldsymbol {C} _ {n} \boldsymbol {S} _ {n} ^ {*} - \boldsymbol {S} _ {n} ^ {*} \boldsymbol {C} _ {n} \right\| _ {F} + \left\langle \boldsymbol {\Lambda} _ {2}, \boldsymbol {S} _ {n} ^ {*} \right\rangle \\ = \sum_ {i, j} \left(S _ {n} ^ {*}\right) _ {i j} - \alpha m + \lambda_ {1} \delta_ {n}, \\ \end{array}
$$
which implies that  $\sum_{i,j}(S_n^*)_{ij} \leq \alpha m - \lambda_1\delta_n \leq \alpha m$ . This completes the proof of the first statement.
For the second statement, we first prove that  $v_{n}^{*}$  and  $v^{*}$  are larger than  $\alpha m(1 - \log \alpha)$ . Define the auxiliary function  $g: \mathbb{R} \to \mathbb{R}$  such that  $g(x) \coloneqq x - \alpha \log x$  for any  $x \in \mathbb{R}_{+}$ , whose minimum is attained at  $\alpha$ . Since for any  $S \in S$ ,
$$
f (\boldsymbol {S}) = \sum_ {i = 1} ^ {m} g \left(\sum_ {j = 1} ^ {m} S _ {i j}\right),
$$
where  $f$  is the objective in LogSpecT, it follows that
$$
f (\boldsymbol {S}) \geq \sum_ {i = 1} ^ {m} g (\alpha) = \alpha m (1 - \log \alpha).
$$
This implies that  $v_{n}^{*}$  and  $v^{*}$  are larger than  $\alpha m(1 - \log \alpha)$ . Next, we will show  $v_{n}^{*} \leq v^{*}$ . Consider any optimal solution  $S^{*}$  to LogSpecT. We show that it is feasible for rLogSpecT.
$$
\begin{array}{l} \left\| \boldsymbol {C} _ {n} \boldsymbol {S} ^ {*} - \boldsymbol {S} ^ {*} \boldsymbol {C} _ {n} \right\| = \left\| \boldsymbol {C} _ {n} \boldsymbol {S} ^ {*} - \boldsymbol {C} _ {\infty} \boldsymbol {S} ^ {*} + \boldsymbol {S} ^ {*} \boldsymbol {C} _ {\infty} - \boldsymbol {S} ^ {*} \boldsymbol {C} _ {n} \right\| \\ \leq 2 \left\| \boldsymbol {S} ^ {*} \right\| _ {1, 1} \left\| \boldsymbol {C} _ {n} - \boldsymbol {C} _ {\infty} \right\| \leq 2 \alpha m \left\| \boldsymbol {C} _ {n} - \boldsymbol {C} _ {\infty} \right\| \leq \delta_ {n}, \\ \end{array}
$$
where the equality comes from  $C_{\infty}S^{*} = S^{*}C_{\infty}$ , the first inequality comes from the fact that  $\| X\mathbf{Y}\| \leq \| \mathbf{X}\| _F\| \mathbf{Y}\| \leq \| \mathbf{X}\|_{1,1}\| \mathbf{Y}\|$ , the second one comes from the first statement and the last one is due to  $\delta_n\geq 2\alpha m\| C_n - C_\infty \|$ . Hence,  $S^*$  is feasible for rLogSpecT, which indicates that  $v_{n}^{*}\leq v^{*}$ . The proof is completed.
E Proof of Theorem 4.1 & Corollary 4.4
E.1 Truncated Hausdorff distance
In this section, we introduce an advanced technique in optimization that is efficient in analyzing the recovery guarantee of robust formulations. Before that, we introduce the concept of truncated Hausdorff distance between two sets.
Definition E.1 (Truncated Hausdorff Distance [29, 6.J]). For any  $\rho \geq 0$ , the truncated Hausdorff distance between two sets  $\mathcal{C}$  and  $\mathcal{D}$  is defined as:
$$
\hat {\mathrm {d}} _ {\rho} (\mathcal {C}, \mathcal {D}) = \max  \{\operatorname {d i s t} (\mathcal {C} \cap \mathbb {B} (\mathbf {0}, \rho), \mathcal {D}), \operatorname {d i s t} (\mathcal {D} \cap \mathbb {B} (\mathbf {0}, \rho), \mathcal {C}) \}.
$$
It turns out that the distance between the optimum of two minimization problems can be bounded with the truncated Hausdorff distance of the epigraphs under some conditions. The result is captured in the following lemma.
Lemma E.2 ([29, Theorem 6.56]). Let  $\rho \in [0,\infty)$ . Suppose that the extended-real-valued lower semicontinuous functions  $f,g:\mathbb{R}^n\to \overline{\mathbb{R}}$  satisfy
- inf  $f, \inf g \in [-\rho, \rho]$ ,
- argmin  $f$ , argmin  $g \subseteq \mathbb{B}(\mathbf{0}, \rho)$ .
Then, it follows that
$$
\left| \inf  f - \inf  g \right| \leq \hat {\mathrm {d}} _ {\rho} (\operatorname {e p i} f, \operatorname {e p i} g). ^ {3} \tag {17}
$$
Suppose further that  $\varepsilon \geq 2\hat{\mathrm{d}}_{\rho}(\mathrm{epi}f,\mathrm{epi}g)$ , then one has
$$
\operatorname {d i s t} \left(\boldsymbol {x} _ {g} ^ {*}, \varepsilon - \operatorname {a r g m i n} f\right) \leq \hat {\mathrm {d}} _ {\rho} (\operatorname {e p i} f, \operatorname {e p i} g), \tag {18}
$$
where  $\varepsilon$ -argmin  $f$  is the  $\varepsilon$ -suboptimal solution set of  $f$  that is defined as  $\varepsilon$ -argmin  $f := \{\pmb{x} \in \operatorname{dom} f : f(\pmb{x}) \leq \inf f + \varepsilon\}$ , and  $\pmb{x}_g^*$  is a minimizer of  $g$ .
From the above lemma, we know that if two optimization problems are close enough (in the sense of truncated Hausdorff distance), then the optimum of them should be close to each other. Hence, in order to apply this result, we need to bound the truncated Hausdorff distance in an explicit way, which is solved by the following Kenmochi condition.
3For a function  $f:\mathbb{R}^n\to \overline{\mathbb{R}}$  , its epigraph is defined as epi  $f\coloneqq \{(x,y)\mid y\geq f(x)\}$
Lemma E.3 (Kenmochi Condition [29, Proposition 6.58]). Let  $\rho \in [0,\infty)$ . Then, for  $f,g:\mathbb{R}^n\to \overline{\mathbb{R}}$  with nonempty epigraphs, one has that
$$
\hat{\mathrm{d}}_{\rho}(\operatorname {epi}f,\operatorname {epi}g) = \inf \left\{\eta >0:\begin{array}{l}\inf_{\mathbb{B}(\boldsymbol {x},\eta)}g\leq \max \{f(\boldsymbol {x}), - \rho \} +\eta ,  \forall \boldsymbol {x}\in [f\leq \rho ]\cap \mathbb{B}(\boldsymbol {0},\rho)\\ \inf_{\mathbb{B}(\boldsymbol {x},\eta)}f\leq \max \{g(\boldsymbol {x}), - \rho \} +\eta ,  \forall \boldsymbol {x}\in [g\leq \rho ]\cap \mathbb{B}(\boldsymbol {0},\rho) \end{array} \right\} ,
$$
where  $[f\leq \rho ]\coloneqq \{\pmb {x}\in \mathbb{R}^n:f(\pmb {x})\leq \rho \}$
E.2 Proof of Theorem 4.1
Before presenting the proof, we first introduce the following lemma.
Lemma E.4 (Hoffman's Error Bound [12]). Consider the set  $\mathcal{S} \coloneqq \{\pmb{x} \in \mathbb{R}^n : \pmb{A}\pmb{x} \leq \pmb{b}\}$ . There exists  $C > 0$  such that for any  $\pmb{x} \in \mathbb{R}^n$ , one has
$$
\operatorname {d i s t} (\boldsymbol {x}, \mathcal {S}) \leq C \cdot \| (\boldsymbol {A} \boldsymbol {x} - \boldsymbol {b}) _ {+} \| _ {2}.
$$
For the sake of brevity, we denote
$$
\bar {f} _ {n} (\boldsymbol {S}) := \| \boldsymbol {S} \| _ {1, 1} - \alpha \mathbf {1} ^ {\top} \log (\boldsymbol {S} \mathbf {1}) + \iota_ {\mathbb {R} _ {-}} \left(\| \boldsymbol {C} _ {n} \boldsymbol {S} - \boldsymbol {S} \boldsymbol {C} _ {n} \| _ {F} - \delta_ {n}\right) + \iota_ {\mathcal {S}} (\boldsymbol {S}),
$$
$$
\bar {f} (\boldsymbol {S}) := \| \boldsymbol {S} \| _ {1, 1} - \alpha \mathbf {1} ^ {\top} \log (\boldsymbol {S} \mathbf {1}) + \iota_ {\{0 \}} \left(\left\| \boldsymbol {C} _ {\infty} \boldsymbol {S} - \boldsymbol {S} \boldsymbol {C} _ {\infty} \right\| _ {F}\right) + \iota_ {\mathcal {S}} (\boldsymbol {S}).
$$
Hence, the optimization problem LogSpecT (resp. rLogSpecT) is equivalent to inf  $\bar{f}$  (resp. inf  $\bar{f}_n$ ).
Now, we aim to use Lemma E.3 to bound  $\hat{\mathrm{d}}_{\rho}(\mathrm{epi}\bar{f},\mathrm{epi}\bar{f}_n)$ . Let  $S\in S\cap \mathbb{B}(\mathbf{0},\rho)$  satisfy
$$
\bar {f} (\boldsymbol {S}) \leq \rho \quad \text {a n d} \quad \boldsymbol {S C} _ {\infty} = \boldsymbol {C} _ {\infty} \boldsymbol {S}.
$$
$$
\| \boldsymbol {S} \boldsymbol {C} _ {n} - \boldsymbol {C} _ {n} \boldsymbol {S} \| _ {F} \leq 2 \| \boldsymbol {S} \| _ {F} \| \boldsymbol {C} _ {n} - \boldsymbol {C} _ {\infty} \| \leq 2 \rho \| \boldsymbol {C} _ {n} - \boldsymbol {C} _ {\infty} \| \leq \delta_ {n},
$$
and consequently  $S$  is in the domain of  $\bar{f}_n$ . Then, it follows that for any  $\eta > 0$ , we have
$$
\inf  _ {\mathbb {B} (\boldsymbol {S}, \eta)} \bar {f} _ {n} \leq \bar {f} _ {n} (\boldsymbol {S}) = \bar {f} (\boldsymbol {S}) \leq \max  \left\{\bar {f} (\boldsymbol {S}), - \rho \right\}, \quad \forall \boldsymbol {S} \in [ \bar {f} \leq \rho ] \cap \mathbb {B} (\boldsymbol {0}, \rho). \tag {19}
$$
Before verifying the reverse side of the Kenmochi condition, we first consider the non-emptiness of  $[\bar{f}_n\leq \rho ]\cap \mathbb{B}(\mathbf{0},\rho)$ . Since
$$
\delta_ {n} \geq 2 \rho \| \boldsymbol {C} _ {n} - \boldsymbol {C} _ {\infty} \| \geq 2 \alpha m \| \boldsymbol {C} _ {n} - \boldsymbol {C} _ {\infty} \|,
$$
it follows from Proposition D.1 that  $\| S_n^*\|_{1,1}\leq \alpha m\leq \rho$  and  $f_{n}^{*}\leq f^{*}\leq \rho$ , which implies that  $[\bar{f}_n\leq \rho ]\cap \mathbb{B}(\mathbf{0},\rho)$  is nonempty. Let  $S_{n}\in [\bar{f}_{n}\leq \rho ]\cap \mathbb{B}(\mathbf{0},\rho)$ . Then, one has that
$$
\boldsymbol {S} _ {n} \in \mathcal {S} \quad \text {a n d} \quad \| \boldsymbol {C} _ {n} \boldsymbol {S} _ {n} - \boldsymbol {S} _ {n} \boldsymbol {C} _ {n} \| _ {F} \leq \delta_ {n}.
$$