SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We consider the measure  $\min_{0\leq k\leq K}(\| \pi^{k + 1} - \pi^k\| _F + \| \alpha^{k + 1} - \alpha^k\| _2 + \| \beta^{k + 1} - \beta^k\| _2)$  as the stationary measure, and it is observed that the convergence rate of our algorithm is  $\mathcal{O}\left(\frac{1}{\sqrt{K}}\right)$ . Similarly, the Frank-Wolfe algorithm for PGW also exhibits a convergence rate of  $\mathcal{O}\left(\frac{1}{\sqrt{K}}\right)$ . The literature on UGW does not provide a discussion on the convergence rate of alternate Sinkhorn minimization for UGW. In each iteration of PGW, UGW, and RGW, the computation of  $\mathcal{L}(D,\bar{D})\otimes \pi^k$  is a crucial step. According to [31], the complexity of this computation is  $\mathcal{O}(n^{2}m + m^{2}n)$ . Additionally, PGW involves utilizing the network simplex algorithm to solve a linear programming problem as a subroutine, which has a complexity of  $\mathcal{O}((n^2 m + m^2 n)\log^2 (n + m))$ . On the other hand, both UGW and RGW utilize the sinkhorn algorithm to solve an entropic unbalanced optimal transport problem. The complexity of the sinkhorn algorithm for unbalanced OT is  $\mathcal{O}((n^2 +m^2) / (\varepsilon \log (\varepsilon)))$  for computing an  $\varepsilon$ -approximation.
D.1 Additional Experiment Results on Subgraph Alignment
Source codes of all baselines used in this paper:
- FW [17]: https://github.com/PythonOT/POT
- SpecGW [15]: https://github.com/trneedham/Spectral-Gromov-Wasserstein
- eBPG [17]: https://github.com/PythonOT/POT
- BAPG [24]: https://github.com/squareRoot3/Gromov-Wasserstein-for-Graph
- UGW [35]: https://github.com/thibsej/unbalanced_gromov_wasserstein
- PGW [10, 17]: https://github.com/PythonOT/POT
- srGW: [42]: https://github.com/cedricvincentcuaz/srGW
- RGWD: [25]: https://github.com/cxxszz/rgdl
Results in Figure 4 The data utilized to create Figure 4 is provided in Table 2 and Table 3.
Table 2: Comparison of the average matching accuracy (%) and wall-clock time (seconds) on subgraph alignment of  $30\%$  subgraph and  $20\%$  subgraph.
30% subgraph	20% subgraph
Method	Synthetic	Proteins	Enzymes	Synthetic	Proteins	Enzymes
Acc	Time	Acc	Time	Acc	Time	Acc	Time	Acc	Time	Acc	Time
FW	2.22	17.06	12.96	14.83	12.08	5.37	2.24	6.65	10.83	11.34	9.53	4.92
SpecGW	1.38	2.24	10.64	11.54	9.41	3.74	1.71	2.21	10.78	10.15	8.35	3.21
eBPG	0.65	0.49	8.12	1022.02	3.84	476.83	1.17	0.42	7.23	545.50	2.66	94.78
BPG	1.86	17.53	17.89	86.85	17.69	52.89	1.64	11.66	12.99	55.47	14.35	32.89
BAPG	2.94	35.90	18.79	36.02	16.85	10.88	3.80	23.29	14.07	23.92	11.22	8.38
srGW	3.17	86.38	22.75	89.14	27.45	41.18	5.49	88.89	18.38	17.72	23.13	17.11
RGWD	1.94	933.25	16.90	3674.98	16.34	3322.16	1.94	933.25	16.90	3674.98	16.34	3322.16
UGW	35.15	168.03	14.32	10298	10.91	5552.27	4.48	251.41	11.75	7813.96	10.40	4019.62
PGW	2.06	339.23	11.68	507.11	11.77	174.26	1.90	227.87	9.34	365.88	7.97	165.27
RGW	52.35	679.00	30.17	947.48	37.12	538.04	11.58	229.05	23.51	546.15	25.39	879.93
Table 3: Comparison of the average matching accuracy (%) and wall-clock time (seconds) on subgraph alignment of  $40\%$  subgraph.
40% subgraph
Method	Synthetic	Proteins	Enzymes
Acc	Time	Acc	Time	Acc	Time
FW	1.84	17.96	15.34	19.64	14.22	6.36
SpecGW	1.72	3.25	11.21	12.17	9.59	3.88
eBPG	0.38	0.51	12.16	1628.38	9.96	943.49
BPG	3.41	18.61	24.31	108.10	24.58	62.81
BAPG	7.61	22.55	23.78	36.81	24.82	11.13
srGW	2.45	120.12	22.58	74.58	27.02	32.14
RGWD	3.48	930.84	22.73	4490.60	22.63	3205.03
UGW	79.61	960.04	21.22	11398	22.26	5589.73
PGW	2.17	483.10	11.95	491.64	9.51	182.58
RGW	90.79	662.15	38.94	769.25	48.11	291.74
Selection of Stepsize  $t_k$ ,  $c_k$ , and  $r_k$  In the subgraph alignment task, we have used constant values for the stepsizes  $t_k$ ,  $c_k$  and  $r_k$ . We have conducted a sensitivity analysis for these parameters, and the details are summarized in Tables 4 and 5. Specifically, Table 4 reveals that RGW achieves its highest accuracy with  $t$  in the range of 0.01 to 0.05, allowing us to select  $t = 0.01$  as the default. Table 5 further indicates that accuracy is not significantly affected by variations in  $c$ , leading us to set  $c = 0.1$  as the default.
Table 4: The performance of RGW with different stepsize  $t$  on three subgraph alignment databases.
RGW	Synthetic	Proteins-1	Enzymes
Acc	Time	Acc	Time	Acc	Time
t = 0.01	94.64	541.50	53.07	567.09	63.23	139.82
t = 0.05	90.49	288.25	53.37	1030.15	63.69	304.11
t = 0.1	87.00	233.65	53.69	713.07	62.38	506.83
t = 0.5	87.09	779.84	51.56	1797.13	60.07	822.23
t = 1.0	71.10	491.93	50.21	3071.27	58.31	1166.88
Experiment Results of Normalized Degree as Marginal Distribution In addition to employing the uniform distribution as node distribution, we also explore the use of normalized degrees as node distribution. The results presented in Table 6 confirm that RGW surpasses other methods in terms of accuracy when utilizing normalized degrees as marginal distributions.
Experiment Results of Adding Noise to Query Graph We conducted an experiment by adding  $10\%$  pseudo edges to the subgraphs, and the results can be found in Table 7. These findings
Table 5: The performance of RGW with different stepsize  $c$  on three subgraph alignment databases.
RGW	Synthetic	Proteins-1	Enzymes
Acc	Time	Acc	Time	Acc	Time
c = 0.01	90.39	481.06	53.17	459.43	63.06	163.45
c = 0.05	90.39	967.98	53.12	919.18	62.86	327.44
c = 0.1	90.39	1460.94	53.37	1374.43	63.69	490.91
c = 0.5	90.34	1969.18	53.35	1822.70	63.36	650.25
c = 1.0	90.28	2488.17	53.43	2262.07	63.53	807.57
Table 6: Subgraph alignment results (Acc.) of  $50\%$  subgraph of compared GW-based methods using normalized degree.
Method	Synthetic	Proteins-1	Proteins-2	Enzymes
FW	2.96	14.82	42.16	15.90
SpecGW	1.57	8.92	43.10	12.07
eBPG	5.27	13.77	31.90	13.51
BPG	12.52	20.88	57.77	29.42
BAPG	74.39	24.65	63.26	31.92
srGW	4.22	13.67	12.39	23.05
RGWD	13.54	25.92	57.30	28.12
UGW	99.56	25.51	62.92	39.71
PGW	3.97	11.59	37.79	13.08
RGW	99.61	50.61	66.09	63.59
demonstrate that RGW significantly outperforms other methods on the Enzymes and Proteins datasets.
Table 7: Subgraph alignment results (Mean ± Std.) of 50% subgraph in 5 independent trials over different random seeds in the noise generating process.
Method	Synthetic	Proteins-1	Enzymes
FW	2.38±0.27	10.21±1.22	12.58±12.58
SpecGW	1.79±0.26	9.91±0.56	10.00±0.81
eBPG	6.84±1.89	15.63±0.73	14.31±0.77
BPG	17.71±2.23	20.96±1.57	22.29±1.19
BAPG	43.39 ± 4.84	23.08±1.12	24.42±1.54
srGW	1.75±0.18	13.34±0.56	20.03±0.73
RGWD	17.30±1.90	22.70±0.33	24.07±0.44
UGW	81.24±2.34	22.99±0.16	26.68±1.54
PGW	1.67 ± 0.43	8.19±0.92	7.33±0.45
RGW	88.79 ± 1.59	38.88±1.31	49.01±0.99
D.2 Additional Experiment Results on Partial Shape Correspondence
Convergence of BPALM The convergence results for the proposed BPALM using various step sizes are presented in Figure 7 for the toy example discussed in Section 4.1.
Additional Experiment Result on TOSCA Dataset Additional experiment results on TOSCA Dataset are shown in Figure 8, 9, and 10.
Figure 7: Convergence result of toy example with different stepsizes
Figure 8: (a): 3D shape geometry of the source and target; (b)-(e): visualization of ground truth, initial point obtained from the partial functional map, and the matching results of PGW and RGW.
(a) Shape Geometry
(b) Ground Truth
(c) Partial Functional Map
(d) Partial GW
(e) Robust GW
Figure 9: (a): 3D shape geometry of the source and target; (b)-(e): visualization of ground truth, initial point obtained from the partial functional map, and the matching results of PGW and RGW.
(c) Partial Functional Map
Figure 10: (a): 3D shape geometry of the source and target; (b)-(e): visualization of ground truth, initial point obtained from the partial functional map, and the matching results of PGW and RGW.
(e) Robust GW