SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figure 5: Top row: The trajectory of the relative error for the local search method for the problem  $\mathcal{U}$  on 2-dimensional data. Bottom row: The trajectory of the relative error for the local search method for the problem  $\mathcal{N}_1$  on 2-dimensional data. The solid blue line indicates the mean convergence rate for 20 runs and the dashed red line indicates 1 standard deviation from the mean.
A.6 Evaluation on the MNIST dataset
In this section we present the performance of the proposed method on shapes originating from the MNIST dataset [8]. Even though the arabic numerals are not entirely unique to reflections (e.g. "2" and "5") and rotations (e.g. "6" and "9"), the ability to compute the exact Gromov-Wasserstein discrepancy is of interest. When the images doesn't contain the same number of points, evenly distributed points are taken from the image with more points. The maximum number of points was set to 400. Number of iterations are presented in Table 3 and the computational time is presented in Table 4, showing that the proposed method works as anticipated on the data set. Examples of correspondances of numerals are shown in Figure 6.
Table 3: Number of iterations to compute the distance between two numeral shapes to a relative error gap of  $10^{-8}$ . Numbers presented are median in the upper table and [min - max] in the lower table. The number of random tests were 10 for each combination.
0	1	2	3	4	5	6	7	8	9
0	92.5	55.5	98.5	75.5	115.5	106.5	80	81	90	90
1	21	46.5	36	70.5	50.5	47	41	50	49
2	52	53.5	64	58.5	60.5	57.5	58	43
3	40.5	75	52.5	72.5	46.5	56	56.5
4	84.5	57	63.5	65	73.5	61.5
5	50	53.5	48	64.5	49.5
6	44	45.5	67	56.5
7	48.5	64	47.5
8	62.5	55.5
9	52.5
0	1	2	3	4	5	6	7	8	9
0	67-134	39-80	67-223	51-139	86-184	69-131	62-97	63-143	73-102	53-163
1	10-39	32-78	30-64	49-98	36-61	35-88	27-69	43-59	27-93
2	29-77	36-78	37-75	44-88	47-95	38-72	43-86	36-83
3	29-67	31-116	47-60	42-89	27-77	48-85	27-87
4	50-107	40-111	40-111	52-92	47-108	50-80
5	31-74	45-88	43-87	54-94	36-97
6	26-80	36-88	54-123	32-94
7	24-76	52-75	30-87
8	40-82	42-71
9	29-65
Table 4: Computational time for the distance between two numeral shapes to a relative error gap of  $10^{-8}$ . Numbers presented are [median]s in the upper table and [min - max]s in the lower table. The number of random tests were 10 for each combination.
0	1	2	3	4	5	6	7	8	9
0	0.85	0.48	1.03	0.76	1.09	1.33	0.83	0.81	0.98	0.95
1	0.05	0.18	0.15	0.32	0.22	0.18	0.15	0.3	0.29
2	0.34	0.33	0.41	0.39	0.46	0.37	0.47	0.34
3	0.31	0.7	0.41	0.64	0.39	0.52	0.51
4	0.85	0.61	0.6	0.67	0.86	0.7
5	0.53	0.62	0.49	0.75	0.5
6	0.54	0.49	0.96	0.66
7	0.61	0.84	0.59
8	0.77	0.74
9	0.92
0	1	2	3	4	5	6	7	8	9
0	0.5-1.4	0.3-1	0.6-2.4	0.5-1.5	0.9-1.9	0.6-1.6	0.6-1.2	0.5-1.7	0.6-1.4	0.5-2.1
1	0-0.1	0.1-0.4	0.1-0.4	0.2-0.4	0.2-0.3	0.1-0.4	0.1-0.4	0.2-0.4	0.1-0.5
2	0.2-0.5	0.2-0.6	0.2-0.5	0.3-0.6	0.3-0.8	0.3-0.5	0.3-0.7	0.2-0.7
3	0.2-0.7	0.2-1.2	0.3-0.5	0.3-0.9	0.2-0.7	0.4-0.7	0.2-0.8
4	0.4-1	0.3-1	0.3-1.3	0.4-0.8	0.4-1.3	0.4-0.9
5	0.2-0.9	0.4-1	0.3-0.9	0.5-1.1	0.4-1.1
6	0.2-1	0.3-1	0.6-1.8	0.3-1.2
7	0.3-0.9	0.7-1.2	0.1-1.4
8	0.6-1.4	0.5-1.1
9	0.5-1.1
Figure 6: Left image: Matching of the numeral "6" to the numeral "9". Right image: Matching of the numeral "2" to the numeral "5". The colors represent which point in one numeral corresponds to another point in the other numeral.