SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this subsection, we present numerical results to illustrate the effectiveness and efficiency of our proposed REALM, namely, Algorithm 1. We choose  $\epsilon_{1} = 2\|C\|_{\infty}\epsilon_{2}$ ,  $\epsilon_{2} = 10^{-6}\max\{\|r\|_{\infty}, \|c\|_{\infty}\}$ ,  $\epsilon_{c} = 10^{-3}$ , and  $\eta_{1} = 20$ . To avoid possible numerical instability, we restrict the maximum number of updating  $\pi^{k}$  as 8. To prevent too small  $\eta_{k}$ , in our implementation, we update  $\eta_{k+1} = \min\{\gamma_{\eta}\eta_{k}, \eta_{\min}\}$  other than using (16), where  $\eta_{\min}$  is a preset positive number. We set  $\gamma_{\eta} = \gamma_{\epsilon} = 0.25$ . Moreover, we denote by REALM- $(\eta_{\min}, \gamma_{W})$  the REALM with particular parameters  $\eta_{\min}$  and  $\gamma_{W}$ . Note that choosing  $\gamma_{W} = 0$  means that we adopt a continuation technique to solve (11) with  $\pi^{k} \equiv \pi^{1}$  and  $\eta_{k} = \eta_{\min}$ , which is generally better than directly solving a single (11) with
Table 2: Average results of REALM for Dataset C.1, “a” and “b” stand for REALM-(0.0035, 0) and REALM-(0.007, 0.9), respectively.
data	\(\widehat{p}_{k}^{2}\)	nGrad	\(nS k_{exp}/nS k_{log}\)	time	iter
a	b	a	b	a	b	a	b	a	b
H5/JC	0.09270	0.10985	1081	675	23809/8267	9786/0	84.1	8.6	0.0/8.0	8.0/15.0
H/MV	0.06378	0.06424	1116	727	3864/13382	30997/0	175.7	14.9	0.0/8.0	8.0/15.0
H/RJ	0.21706	0.22607	547	788	3091/4062	11747/0	63.3	15.6	0.0/8.0	8.0/15.0
JC/MV	0.06270	0.06235	1324	711	12875/9727	17073/0	55.3	5.9	0.0/8.0	8.0/15.0
JC/O	0.04221	0.04277	1650	1100	4997/26612	35031/4500	204.7	46.7	0.0/8.0	8.0/15.0
MV/O	0.04181	0.03661	890	822	6101/5289	31651/0	57.0	13.3	0.0/8.0	8.0/15.0
AVG	0.11338	0.11461	863	787	7824/7364	20261/300	78.0	15.0	0.0/8.0	8.0/15.0
$\eta_{k} = \eta_{\mathrm{min}}$ . Subproblem (11) in REALM is solved by iRBBS. If  $\max \{\| r\|_{\infty},\| c\|_{\infty}\} \geq 500\eta_{k}$  or  $\| C(U^t) - \eta_k\log \pi^k\|_{\mathrm{var}}\geq 900\eta_k$ , we set  $\theta = 10$  and perform the Sinkhorn iteration (21); otherwise, we set  $\theta = 0.1$  and perform an equivalent formulation of Sinkhorn iteration (28).
The results over 20 runs on Dataset C.1 are reported in Table 2. In this table, the terms "nSklog" and "nSKexp" mean the total numbers of Sinkhorn iterations (21) and (28), respectively, the pair  $k_{1}/k$  in the column "iter" means that the corresponding algorithm stops at the  $k$ -iteration and updates the multiplier matrix  $k_{1}$  times. To save space, we only report instances where one method can return the value " $\widehat{\mathcal{P}}_k^2$ " larger than 1.005 times of the smaller one of the two  $\widehat{\mathcal{P}}_k^2$  values returned by the two methods. The better " $\widehat{\mathcal{P}}_k^2$ " is marked in bold. Besides, the average performance over all 15 instances is also kept in the "AVG" line.
From Table 2, we can observe that REALM-(0.007, 0.9) can not only return better solutions than REALM-(0.0035, 0) but also is about  $5.2\mathrm{x}$  faster. On average, REALM-(0.007, 0.9) updates the multiplier matrix 8 times in 15 total iterations, which shows that updating the multiplier matrix does help. The reasons why REALM with updating the multiplier matrix outperforms REALM without updating the multiplier matrix in terms of solution quality and speed are as follows. First, updating the multiplier matrix in REALM can keep the solution quality even using a larger  $\eta_{k}$ . Second, solving the subproblem with a larger  $\eta_{k}$  is always easier, which enables that REALM-(0.07, 0.9) computes less  $\operatorname{grad}_U\mathcal{L}_{\eta_k}(\mathbf{x}^t,\pi^k)$  and performs less Sinkhorn iterations (21) which involves computing the log-sum-exp function  $\log \sum_{i}\exp (x_{i} / \eta_{k}) = x_{\max} / \eta_{k} + \log \sum_{i}\exp ((x_{i} - x_{\max}) / \eta_{k})$  for small  $\eta_{k}$ .
5 Concluding Remarks
In this paper, we considered the computation of the PRW distance. By reformulating this problem as an optimization problem over the Cartesian product of the Stiefel manifold and the Euclidean space with additional nonlinear inequality constraints, we proposed a method called REALM. The convergence of REALM was also established. To solve the subproblem in REALM efficiently, we developed a practical iRBBS method with convergence and iteration complexity guarantees, wherein the Riemannian BB stepsize (based on the inexact Riemannian gradient information) and Sinkhorn iterations are employed. The complexity of iRBBS to attain an  $\epsilon$ -stationary point of the original PRW distance problem matches the best known iteration complexity result. Numerical results showed that, compared with the state-of-the-art methods, our proposed REALM and iRBBS methods have advantages in solution quality and speed.
Moreover, our proposed REALM and iRBBS can also be extended to solve some important minimax problems over the Riemannian manifolds arising from machine learning, such as the fair PCA problem [45] and the projection robust Wasserstein barycenters [27], etc.
Lastly, a limitation of our work is that we did not establish the positive lower bound of  $\eta_{k}$  in REALM, despite the fact that REALM performs well in practice and can avoid too small  $\eta_{k}$  in many cases. We shall investigate the conditions under which it is possible to establish a lower bound of  $\eta_{k}$  in REALM. This can be achievable by extending the analysis and conditions in [20] to the Riemannian case.
Acknowledgments The work of Bo Jiang was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 11971239, Grant 12371314, and in part by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province under Grant 21KJA110002. The work of Ya-Feng Liu was supported in part by NSFC under Grant 11991020, Grant 11991021, and Grant 12288201.
The appendices are organized as follows. In Appendix A, we present more details on our proposed REALM in Section 2. Some proofs in Section 3 are given in Appendix B. Some details on the dataset and implementation in Section 4 are shown in Appendix C. Finally, we present additional numerical results to evaluate the performance of our proposed methods to compute the PRW distance (1) in Appendix D.
A More Details on Our Proposed REALM in Section 2
In this section, we provide more materials on REALM in Section 2, including the first-order necessary conditions of our reformulation (6), relations between different  $(\epsilon_{1},\epsilon_{2})$ -stationary points of the PRW distance problem (1), the connections of the approximate stationary points of problems (11) and (1), and the convergence analysis of Algorithm 1.
A.1 First-Order Necessary Conditions of Problem (6)
Lemma A.1 (First-order necessary conditions of problem (6)). Given  $\bar{\mathbf{x}}\in \mathcal{M}$  and  $\bar{y}\in \mathbb{R}$ , suppose that  $(\bar{\mathbf{x}},\bar{y})$  is a local minimizer of problem (6), then  $(\bar{\mathbf{x}}, - \varphi (\bar{\mathbf{x}})_{\mathrm{min}})$  is a stationary point of problem (6), namely, there exists  $\bar{\pi}\in \Pi (r,c)$  such that
$$
\operatorname {P r o j} _ {\mathrm {T} _ {\bar {U}} \mathcal {U}} (- 2 V _ {\bar {\pi}} \bar {U}) = 0, \quad \langle \bar {\pi}, Z (\bar {\mathbf {x}}) \rangle = 0, \tag {29}
$$
where  $Z(\bar{\mathbf{x}})\in \mathbb{R}^{n\times n}$  is defined as
$$
Z (\bar {\mathbf {x}}) _ {i j} = \varphi (\bar {\mathbf {x}}) _ {i j} - \varphi (\bar {\mathbf {x}}) _ {\min } \tag {30}
$$
Proof. Since  $(\bar{\mathbf{x}},\bar{y})$  is a local minimizer of problem (6), there must hold  $\bar{y} = -\varphi (\bar{\mathbf{x}})_{\mathrm{min}} = \max_{ij}\{-\varphi (\bar{\mathbf{x}})_{ij}\}$ . Moreover, such  $\bar{\mathbf{x}}$  is also a local minimizer of problem  $\min_{\mathbf{x}\in \mathcal{M}}h(\mathbf{x})$  with  $h(\mathbf{x}) = r^{\mathsf{T}}\alpha +c^{\mathsf{T}}\beta -\varphi (\mathbf{x})_{\mathrm{min}}$ . For fixed  $\alpha$  and  $\beta$ , it is easy to see that  $-\varphi (\mathbf{x})_{ij} + \| C\|_{\infty}\| U\|_{\mathsf{F}}^{2}$  is convex with respect to  $U$  since  $\| M_{ij}\| _2 = C_{ij}\leq \| C\|_{\infty}$ . We thus know that the function
$$
- \varphi (\bar {\mathbf {x}}) _ {\mathrm {m i n}} + \| C \| _ {\infty} \| U \| _ {\mathsf {F}} ^ {2} = \max _ {i j} \left\{- \varphi (\mathbf {x}) _ {i j} + \| C \| _ {\infty} \| U \| _ {\mathsf {F}} ^ {2} \right\}
$$
is convex with respect to  $U$ , which means that the function  $-\varphi(\bar{\mathbf{x}})_{\mathrm{min}}$  and thus  $h(\mathbf{x})$  is  $\|C\|_{\infty}$ -weakly convex with respect to  $U$  [48, Proposition 4.3]. Let  $\mathcal{A}(\bar{\mathbf{x}}) = \{(i,j) \in [n] \times [n] \mid \varphi(\bar{\mathbf{x}})_{ij} = \varphi(\bar{\mathbf{x}})_{\mathrm{min}}\}$ . By [48, Proposition 4.6], we have
$$
\partial h (\bar {\mathbf {x}}) = (r, c, 0) + \operatorname {c o n v} \left\{\left(- e _ {i}, - e _ {j}, - 2 M _ {i j} \bar {U}\right) \mid (i, j) \in \mathcal {A} (\bar {\mathbf {x}}) \right\},
$$
where  $e_i$  is the  $i$ -th standard unit vector in  $\mathbb{R}^n$ . Moreover, by [53, Theorem 4.1], [53, Theorem 5.1], and  $\mathrm{T}_{\bar{\mathbf{x}}}\mathcal{M} = \mathbb{R}^n\times \mathbb{R}^n\times \mathrm{T}_{\bar{\mathbf{x}}}\mathcal{U}$ , there must hold that  $0\in \mathrm{Proj}_{\mathrm{T}_{\bar{\mathbf{x}}}\mathcal{M}}\partial h(\bar{\mathbf{x}})$ . Putting all the above things together shows that there exists  $\bar{\pi}\in \mathbb{R}_+^{n\times n}$  with  $\bar{\pi}_{ij} = 0$  for all  $(i,j)\notin \mathcal{A}(\bar{\mathbf{x}})$  and  $\sum_{(i,j)\in \mathcal{A}(\bar{\mathbf{x}})}\bar{\pi}_{i,j} = 1$  such that  $\bar{\pi}\in \Pi (r,c)$  and (29) hold. The proof is completed.
A.2 Explanations on Remark 2.2
From [28, Section B], we know that the  $(\epsilon_1,\epsilon_2)$ -stationary point in [28, Definition 3.1] is stronger than that in [34, Definition 2.7]. To show that our Definition 2.1 of the  $(\epsilon_1,\epsilon_2)$ -stationary point is stronger than both definitions, we only need to verify that  $\langle \tilde{\pi},Z(\tilde{\mathbf{x}})\rangle \leq \epsilon_2$  can imply
$$
\langle \tilde {\pi}, C (\tilde {U}) \rangle - \min  _ {\pi \in \Pi (r, c)} \langle \pi , C (\tilde {U}) \rangle \leq \epsilon_ {2},
$$
which is clear since  $\langle \tilde{\pi}, C(\tilde{U}) \rangle - \min_{\pi \in \Pi(r, c)} \langle \pi, C(\tilde{U}) \rangle \leq \langle \tilde{\pi}, Z(\tilde{\mathbf{x}}) \rangle$ . This inequality comes from the fact that problem (5) is a dual formulation of  $\min_{\pi \in \Pi(r, c)} \langle \pi, C(U) \rangle$  for fixed  $U$  and the strong duality theorem.
A.3 Proof of Theorem 2.4
Proof. We first show that (13a) is true. First, by [12, Lemma 3.2], we have  $\hat{\pi} \in \Pi(r,c)$  and
$$
\left\| \hat {\pi} - \phi_ {\eta_ {k}} \left(\tilde {\mathbf {x}}, \pi^ {k}\right) \right\| _ {1} \leq \left\| \phi_ {\eta_ {k}} \left(\tilde {\mathbf {x}}, \pi^ {k}\right) \mathbf {1} - r \right\| _ {1} + \left\| \phi_ {\eta_ {k}} \left(\tilde {\mathbf {x}}, \pi^ {k}\right) ^ {\top} \mathbf {1} - c \right\| _ {1} = \mathrm {e} _ {\eta_ {k}} ^ {2} \left(\tilde {\mathbf {x}}, \pi^ {k}\right) \leq \epsilon_ {2}. \tag {31}
$$
Second, by the triangular inequality and the non-expansive property of the projection operator, we have
$$
\begin{array}{l} \| \operatorname {P r o j} _ {\mathrm {T} _ {U} \mathcal {U}} (A _ {1}) \| _ {\mathsf {F}} \leq \| \operatorname {P r o j} _ {\mathrm {T} _ {U} \mathcal {U}} (A _ {1}) - \operatorname {P r o j} _ {\mathrm {T} _ {U} \mathcal {U}} (A _ {2}) \| _ {\mathsf {F}} + \| \operatorname {P r o j} _ {\mathrm {T} _ {U} \mathcal {U}} (A _ {1}) \| _ {\mathsf {F}} \\ \leq \| A _ {1} - A _ {2} \| _ {\mathsf {F}} + \left\| \operatorname {P r o j} _ {\mathrm {T} _ {U} \mathcal {U}} (A _ {1}) \right\| _ {\mathsf {F}}, \\ \end{array}
$$
where  $A_{1}, A_{2} \in \mathbb{R}^{d \times k}$ . Hence, we have
$$
\left\| \operatorname {P r o j} _ {\mathrm {T} _ {U} \mathcal {U}} \left(- 2 V _ {\hat {\pi}} \tilde {U}\right) \right\| _ {\mathsf {F}} \leq 2 \| \left(V _ {\hat {\pi}} - V _ {\phi_ {\eta_ {k}} \left(\tilde {\mathbf {x}}, \pi^ {k}\right)}\right) \tilde {U} \| _ {\mathsf {F}} + \epsilon_ {1} \leq 2 \| V _ {\hat {\pi}} - V _ {\phi_ {\eta_ {k}} \left(\tilde {\mathbf {x}}, \pi^ {k}\right)} \| _ {\mathsf {F}} + \epsilon_ {1}, \tag {32}
$$
where the first inequality uses  $\| \mathrm{Proj}_{\mathrm{T}_{\tilde{U}}\mathcal{U}}(-2V_{\phi_{\eta_k}(\tilde{\mathbf{x}},\pi^k)}\tilde{U})\|_{\mathsf{F}}\leq \epsilon_1$  and the second inequality is due to the fact that  $\| A\tilde{U}\|_{\mathsf{F}}\leq \| A\|_{\mathsf{F}}$  for any matrix  $A\in \mathbb{R}^{d\times d}$ . Moreover, observe that
$$
\| V _ {\hat {\pi}} - V _ {\phi_ {\eta_ {k}} (\tilde {\mathbf {x}}, \pi^ {k})} \| _ {\mathsf {F}} = \left\| \sum_ {i j} (\hat {\pi} _ {i j} - \phi_ {\eta_ {k}} (\tilde {\mathbf {x}}, \pi^ {k}) _ {i j}) M _ {i j} \right\| _ {\mathsf {F}} \leq \| C \| _ {\infty} \| \hat {\pi} - \phi_ {\eta_ {k}} (\tilde {\mathbf {x}}, \pi^ {k}) \| _ {1}.
$$
Combining the above assertions with (31) and (32) yields (13a).
Next, we show that (13b) is also true. By the Cauchy-Schwarz inequality and (31), we have
$$
\begin{array}{l} \langle \hat {\pi}, Z (\tilde {\mathbf {x}}) \rangle = \left\langle \hat {\pi} - \phi_ {\eta_ {k}} \left(\tilde {\mathbf {x}}, \pi^ {k}\right), Z (\tilde {\mathbf {x}}) \right\rangle + \left\langle \phi_ {\eta_ {k}} \left(\tilde {\mathbf {x}}, \pi^ {k}\right), Z (\tilde {\mathbf {x}}) \right\rangle \\ \leq \left\| Z (\tilde {\mathbf {x}}) \right\| _ {\infty} \epsilon_ {2} + \left\langle \phi_ {\eta_ {k}} (\tilde {\mathbf {x}}, \pi^ {k}), Z (\tilde {\mathbf {x}}) \right\rangle \\ \leq \left(\| \tilde {\alpha} \| _ {\operatorname {v a r}} + \| \tilde {\beta} \| _ {\operatorname {v a r}} + \| C \| _ {\infty}\right) \epsilon_ {2} + \left\langle \phi_ {\eta_ {k}} \left(\tilde {\mathbf {x}}, \pi^ {k}\right), Z (\tilde {\mathbf {x}}) \right\rangle , \tag {33} \\ \end{array}
$$
where the second inequality uses (30), the definition of  $\varphi (\mathbf{x})$  in (5), and
$$
\langle M _ {i j}, U U ^ {\mathsf {T}} \rangle = [ C (U) ] _ {i j} \leq C _ {i j} \leq \| C \| _ {\infty} \quad \forall U \in \mathcal {U}. \tag {34}
$$
The remaining is to bound  $\langle \phi_{\eta_k}(\tilde{\mathbf{x}},\pi^k),Z(\tilde{\mathbf{x}})\rangle$ . By  $\| \zeta_{\eta_k}(\tilde{\mathbf{x}},\pi^k)\| _1 = 1$ , (7), and (12), we have  $\varphi (\tilde{\mathbf{x}})_{ij} = \eta_k(\log \pi_{ij}^k -\log \phi_{\eta_k}(\tilde{\mathbf{x}},\pi^k)_{ij})$  and  $\varphi (\tilde{\mathbf{x}})_{\mathrm{min}}\geq \eta_k(\log \pi^k)_{\mathrm{min}}$ . Again with (30), we have
$$
\begin{array}{l} \left\langle \phi_ {\eta_ {k}} (\tilde {\mathbf {x}}, \pi^ {k}), Z (\tilde {\mathbf {x}}) \right\rangle = \left\langle \phi_ {\eta_ {k}} (\tilde {\mathbf {x}}, \pi^ {k}), \varphi (\tilde {\mathbf {x}}) \right\rangle - \varphi (\tilde {\mathbf {x}}) _ {\min } \\ \leq \eta_ {k} \left(\left\langle \log \pi^ {k}, \phi_ {\eta_ {k}} (\tilde {\mathbf {x}}, \pi^ {k}) \right\rangle + H (\phi_ {\eta_ {k}} (\tilde {\mathbf {x}}, \pi^ {k})) - (\log \pi^ {k}) _ {\min}\right), \\ \end{array}
$$
which, together with  $H(\phi_{\eta_k}(\tilde{\mathbf{x}}, \pi^k)) \leq 2 \log n$  and  $\langle \log \pi^k, \phi_{\eta_k}(\tilde{\mathbf{x}}, \pi^k) \rangle \leq (\log \pi^k)_{\max}$ , further implies  $\langle \phi_{\eta_k}(\tilde{\mathbf{x}}, \pi^k), Z(\tilde{\mathbf{x}}) \rangle \leq (2 \log n + \| \log \pi^k \|_{\mathrm{var}}) \eta_k$ . This, together with (33), yields (13b). The proof is completed.
A.4 Proof of Theorem 2.7
We first use the requirements in (14) to show that  $r^{\top}\alpha^{k} \leq r^{\top}\alpha^{0}$  for all  $k \geq 0$ .
With the first two requirements in (14), we have  $\mathcal{L}_{\eta_k}(\mathbf{x}^k,\pi^k) = 2r^\top \alpha^k$ . By the fact  $\| \zeta_{\eta_1}(\mathbf{x}^0,\pi^1)\| _1 = 1$  with  $\pi^1 = \mathbf{1}\mathbf{1}^\top$ , we know from (7) that  $\varphi (\mathbf{x}^0)_{ij}\geq 0$ . Due to the update rules of  $\pi^k$  and  $\eta_{k}$ , we know that  $0 < \pi_{ij}^{k}\leq 1$  and  $0 < \eta_{k}\leq \eta_{1}$  for all  $k\geq 1$ . Therefore, we have from (7) that
$$
[ \zeta_ {\eta_ {k}} (\mathbf {x} ^ {0}, \boldsymbol {\pi} ^ {k}) ] _ {i j} = \pi_ {i j} ^ {k} \exp \left(- \frac {\varphi (\mathbf {x} ^ {0}) _ {i j}}{\eta_ {k}}\right) \leq \exp \left(- \frac {\varphi (\mathbf {x} ^ {0}) _ {i j}}{\eta_ {1}}\right) = [ \zeta_ {\eta_ {1}} (\mathbf {x} ^ {0}, \boldsymbol {\pi} ^ {1}) ] _ {i j},
$$
which further implies  $\| \zeta_{\eta_k}(\mathbf{x}^0,\pi^k)\| _1\leq \| \zeta_{\eta_1}(\mathbf{x}^0,\pi^1)\| _1 = 1$  . With  $r^{\top}\alpha^{0} = c^{\top}\beta^{0}$  , we know  $\mathcal{L}_{\eta_k}(\mathbf{x}^0,\pi^k)\leq 2r^\top \alpha^0$  . By the third requirement in (14), we further have
$$
r ^ {\top} \alpha^ {k} \leq r ^ {\top} \alpha^ {0}, \quad \forall k \geq 0. \tag {35}
$$
Besides, since  $\mathbf{x}^k$  is an  $(\epsilon_{k,1},\epsilon_{k,2})$ -stationary point of (11), we have
$$
\left\| \operatorname {P r o j} _ {\mathrm {T} _ {U ^ {k}} \mathcal {U}} \left(- 2 V _ {\tilde {\pi} ^ {k + 1}} U ^ {k}\right) \right\| _ {\mathsf {F}} \leq \epsilon_ {k, 1}, \quad \| r - \tilde {\pi} ^ {k + 1} \mathbf {1} \| _ {1} + \| c - \left(\tilde {\pi} ^ {k + 1}\right) ^ {\mathsf {T}} \mathbf {1} \| _ {1} \leq \epsilon_ {k, 2}. \tag {36}
$$
We next consider two cases, which is motivated by the proofs of Theorem 3.1 in [13].
Case (i).  $\eta_{k}$  is bounded below by some threshold value  $\underline{\eta} > 0$ . Due to the update rule of  $\eta_{k+1}$ , we can see that (16) is invoked only finite times. Besides, due to  $0 < \pi_{ij}^{k} \leq 1$ , without loss of generality,
we assume  $\eta_k \equiv \eta$  for all  $k \geq 2$  and  $\lim_{k \to \infty} \pi_{ij}^k = \pi_{ij}^\infty$ . Meanwhile, due to the update rule of  $\pi^k$ , we have  $\pi_{ij}^k = \tilde{\pi}_{ij}^k$  for  $k \geq 2$  and
$$
\left| \min  \left\{\underline {{\eta}} \pi_ {i j} ^ {k + 1}, \varphi \left(\mathbf {x} ^ {k}\right) _ {i j} \right\}\right|\rightarrow 0 \quad \forall (i, j) \in [ n ] \times [ n ]. \tag {37}
$$
We now show that  $\{\mathbf{x}^k\}$  has at least a limit point. By (37), we must have  $\varphi (\mathbf{x}^k)_{ij}\to 0$  for each  $(i,j)\notin \mathcal{A}(\pi^{\infty})\coloneqq \{(i,j)\in [n]\times [n]\mid \pi_{ij}^{\infty} = 0\}$  and there must exist  $K_{1} > 0$  such that for all  $k\geq K_1$ , there holds  $\varphi (\mathbf{x}^k)_{ij}\geq -1$  for each  $(i,j)\in \mathcal{A}(\pi^{\infty})$ . Recalling the definition of  $\varphi (\mathbf{x})$  in (5), then we can conclude that there exists  $K_{2} > 0$  such that for all  $k\geq K_{2}$ , there holds that
$$
\alpha_ {i} ^ {k} + \beta_ {j} ^ {k} + \left\langle M _ {i j}, U ^ {k} \left(U ^ {k}\right) ^ {\mathsf {T}} \right\rangle \geq - 1 \quad \forall (i, j) \in [ n ] \times [ n ].
$$
Multiplying both sides of the above assertions by  $c_{j}$  and then summing the obtained inequality from  $j = 1$  to  $n$ , we have
$$
\alpha_ {i} ^ {k} \geq - 1 - \sum_ {j} c _ {j} \left\langle M _ {i j}, U ^ {k} \left(U ^ {k}\right) ^ {\top} \right\rangle - c ^ {\top} \beta^ {k} \geq - 1 - \| C \| _ {\infty} - r ^ {\top} \alpha^ {0}, \tag {38}
$$
where the second inequality is due to  $r^{\top}\alpha^{k} = c^{\top}\beta^{k}$ , (35), and (34). Combining (35) and (38), we further have
$$
- 1 - \| C \| _ {\infty} - r ^ {\mathsf {T}} \alpha^ {0} \leq \alpha_ {i} ^ {k} \leq r _ {i} ^ {- 1} \big ((1 - r _ {i}) (1 + \| C \| _ {\infty} + r ^ {\mathsf {T}} \alpha^ {0}) + r ^ {\mathsf {T}} \alpha^ {0} \big).
$$
Similarly, we can establish a similar bound for each  $\beta_j^k$ . Recalling that  $U^k$  is in a compact set, it is thus safe to say that the sequence  $\{\mathbf{x}^k\}$  has at least one limit point, denoted by  $\mathbf{x}^\infty = \lim_{k \in \mathcal{K}, k \to \infty} \mathbf{x}^k$ . Again with  $\lim_{k \to \infty} \pi_{ij}^k = \pi^\infty$ , we have from (37) that  $\min \{\underline{\eta} \pi_{ij}^\infty, \varphi(\mathbf{x}^\infty)_{ij}\} = 0$ , which, together with  $\pi_{ij}^\infty \geq 0$  and the fact
$$
0 \leq \langle \pi^ {\infty}, Z (\mathbf {x} ^ {\infty}) \rangle = \sum_ {i j} \pi_ {i j} ^ {\infty} \varphi (\mathbf {x} ^ {\infty}) _ {i j} - \varphi (\mathbf {x} ^ {\infty}) _ {\min },
$$
further implies  $\varphi (\mathbf{x}^{\infty})_{\mathrm{min}} = 0$  and  $\langle \pi^{\infty},Z(\mathbf{x}^{\infty})\rangle = 0$ . This shows that  $(\mathbf{x}^{\infty},0)$  is feasible. Moreover, letting  $k\in \mathcal{K}$  go to infinity in (36), we further have  $\pi^{\infty}\in \Pi (r,c)$  and  $\| \mathrm{Proj}_{\mathrm{T}_U\infty}\mathcal{U}(-2V_{\pi^\infty}U^\infty)\|_{\mathsf{F}} = 0$ . From Lemma A.1, we know that  $(\mathbf{x}^{\infty},0)$  is a stationary point of problem (6) and  $(\mathbf{x}^{\infty},\pi^{\infty})$ , namely,  $(\mathbf{x}^{\infty},\tilde{\pi}^{\infty})$ , is a stationary point of problem (1).
Case (ii). The sequence  $\{\eta_k\}$  is not bounded below by any positive number, namely,  $\lim_{k\to \infty}\eta_k = 0$ . By the updating rule, we know that  $\eta_{k}$  is updated via (16) infinitely many times. Hence, there must exist  $k_{1} < k_{2} < \dots$  such that  $\eta_{k_{\ell}}\rightarrow 0$  as  $\ell \rightarrow \infty$  and
$$
\eta_ {s} = \eta_ {k _ {\ell}} = \min  \left\{\gamma_ {\eta} \eta_ {k _ {\ell} - 1}, \varrho_ {k _ {\ell} - 1} / \| \log \pi^ {k _ {\ell} - 1} \| _ {\infty} \right\} \text {w i t h} k _ {\ell} \leq s <   k _ {\ell + 1} \text {a n d} \pi^ {k _ {\ell}} = \pi^ {k _ {\ell} - 1}.
$$
By (7) and the second assertion in (14), we have  $\pi_{ij}^{k_\ell}\exp (-\varphi (\mathbf{x}^{k_\ell})_{ij} / \eta_{k_\ell})\leq 1$ , which, together with  $0 < \pi_{ij}^{k_\ell}\leq 1$  and (16), implies
$$
\varphi \left(\mathbf {x} ^ {k _ {\ell}}\right) _ {i j} \geq - \eta_ {k _ {\ell}} \left| \log \pi_ {i j} ^ {k _ {\ell}} \right| \geq - \eta_ {k _ {\ell}} \| \log \pi^ {k _ {\ell} - 1} \| _ {\infty} \geq - \varrho_ {k _ {\ell} - 1}, \quad \forall \ell \geq 1. \tag {39}
$$
Using the same arguments as in the proof in Case (i), we can show that  $\{\mathbf{x}^k\}$  is bounded over  $\{k_1, k_2, \ldots\}$  and thus  $\{\mathbf{x}^k\}$  has at least a limit point. Without loss of generality, assume  $\lim_{\ell \to \infty} \mathbf{x}^{k_\ell} = \mathbf{x}^\infty$ . By (39) and  $\varrho_k \to 0$  in (16), we know that  $\varphi(\mathbf{x}^\infty) \geq 0$  and thus that  $(\mathbf{x}^\infty, 0)$  is feasible to problem (6).
Due to the compactness of  $\tilde{\pi}^k$  in (15), there must exist a subset  $\mathcal{K} \subseteq \{k_1, k_2, \ldots\}$  such that  $\lim_{k \in \mathcal{K}, k \to \infty} \tilde{\pi}_{ij}^k = \tilde{\pi}^\infty$ . Recalling (15), (7), and the second requirement in (14), we have  $\tilde{\pi}_{ij}^{k+1} = \pi_{ij}^k \exp(-\varphi(\mathbf{x}^k)_{ij} / \eta_k)$  for each  $k \geq 1$ . Let  $\mathcal{A}(\mathbf{x}^\infty) = \{(i,j) \in [n] \times [n] \mid \varphi(\mathbf{x}^\infty)_{ij} = 0\}$ . We claim  $\mathcal{A}(\mathbf{x}^\infty) \neq \emptyset$ . Otherwise, for every  $(i,j)$  we have  $\varphi(\mathbf{x}^\infty)_{ij} > 0$  and thus
$$
0 \leq \lim  _ {k \in \mathcal {K}, k \rightarrow \infty} \tilde {\pi} _ {i j} ^ {k + 1} \leq \lim  _ {k \in \mathcal {K}, k \rightarrow \infty} \exp \left(- \frac {\varphi \left(\mathbf {x} ^ {k}\right) _ {i j}}{\eta_ {k}}\right) = 0, \tag {40}
$$
where the equality uses  $\lim_{\ell \to \infty} \mathbf{x}^{k_\ell} = \mathbf{x}^\infty$ ,  $\mathcal{K} \subseteq \{k_1, k_2, \ldots\}$ , and  $\lim_{k \to \infty} \eta_k \to 0$ . This makes a contradiction with  $\|\tilde{\pi}^{k+1}\|_1 = 1$ . Moreover, (40) also further implies that  $\langle \tilde{\pi}^\infty, Z(\mathbf{x}^\infty) \rangle = 0$ . Finally, with (36), we further have  $\| \mathrm{Proj}_{\mathrm{T}_U^\infty \mathcal{U}}(-2V_{\tilde{\pi}^\infty} U^\infty) \|_{\mathsf{F}} = 0$  and  $\tilde{\pi}^\infty \in \Pi(r, c)$ . Putting the above things together, we know that that  $(\mathbf{x}^\infty, 0)$  is a stationary point of (6) and  $(\mathbf{x}^\infty, \tilde{\pi}^\infty)$  is a stationary point of problem (1). The proof is completed.
B Proofs in Section 3
B.1 Proof of Lemma 3.1
For notational simplicity, we define
$$
h (\pi , U) = \left\langle C (U) - \eta_ {k} \log \pi^ {k}, \pi \right\rangle - \eta_ {k} H (\pi). \tag {41}
$$