SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We have incorporated quantitative experiments to measure the performance of the proposed method (MagDM) in comparison to two other methods (ME and MME). In the first network, we set the forward flow probability to 0.5 and the backward flow probability to  $\mathrm{P} = 0 / 0.2 / 1$ . We then cluster the low-dimensional embeddings of the three methods using the k-means algorithm with  $\mathrm{k} = 3$ . To evaluate performance for the clustering results, we consider two internal evaluation metrics: the silhouette coefficient (SC) and the Davies-Bouldin index (DB), as well as two external evaluation metrics: the adjusted Rand index (ARI) and the normalized mutual information (NMI). These metrics assess the quality of clustering.
The results of the quantitative experiments evaluating the performance of the proposed MagDM are presented in Tab. Ape.1 of the attached PDF file. It can be seen that MagDM achieves higher scores in the metrics. Particularly when  $\mathrm{P} = 1$ , where interconnections are more complicated, MagDM significantly outperforms other methods, which highlights the superior effectiveness and robustness of MagDM compared to other methods.
Table Ape.1: Quantitative experiments on the first network in Section 5.1. The symbol  $\uparrow/\downarrow$  indicates that higher/ lower scores correspond to better clustering results.
Evaluation metrics	MagDM	ME	MME
P=0	SC↑	0.8583±0.0461	0.8444±0.0251	0.8616±0.0694
DB↓	0.3647±0.0610	0.3918±0.0709	0.3525±0.0587
ARI↑	0.9365±0.0581	0.9085±0.1051	0.9285±0.0586
NMI↑	0.9617±0.0624	0.9227±0.0529	0.9548±0.0481
P=0.2	SC↑	0.8839±0.0408	0.8701±0.0168	0.8788±0.0447
DB↓	0.3277±0.0717	0.3343±0.0625	0.3012±0.0805
ARI↑	0.8940±0.0718	0.8863±0.0756	0.86770.0626
NMI↑	0.9106±0.1115	0.8608±0.0762	0.8259±0.0871
P=1	SC↑	0.8206±0.0423	0.7987±0.0342	0.7979±0.0354
DB↓	0.3924±0.0396	0.4311±0.0848	0.4269±0.0777
ARI↑	0.9266±0.0888	0.2555±0.1818	-0.0030±0.0091
NMI↑	0.9484±0.0667	0.2124±0.1716	0.0074±0.0061
H Experiments regarding the importance of the parameter  $t$
We have conducted additional experiments on MagDM, focusing on diffusion time  $t \in \{1,2,3,4,5,10\}$ . The results are presented in Fig. Ape.6, which can be found in the attached PDF file under the General Response section.
In Fig. Ape.6, we illustrate the clusters that evolve from different values of  $t$  ( $t \in \{1, 2, 3, 4, 5, 10\}$ ). At  $t = 1$ , the local geometry becomes apparent with four clusters visible on the real and imaginary axes. Due to the presence of asymmetric connections, the diffusion distance between samples within the same group is small, while the diffusion distance between samples in different groups is large. Consequently, even with reduced dimensions, we can clearly distinguish the four clusters in the diffusion space. As the diffusion process progresses, the diffusion distance between different groups decreases. At  $t = 4$ , the groups start to connect with each other, forming a single structure. Finally, at  $t = 10$ , the four groups merge to create a single super-cluster structure, with very small diffusion distances between points. Interestingly, the clustering on the phase axes remains clear and preserved throughout the diffusion time. This observation demonstrates the strong ability of MagDM to capture asymmetric information during the diffusion process. The dynamic behavior we observe underscores the effectiveness of representing and visualizing the data.
I Discussion of the diffusion framework
The MagDM framework, proposed for asymmetric scenarios, employs a mathematical technique to transform a pair of real-valued asymmetric similarities into complex-valued and conjugate symmetric similarities. By leveraging the concept of random walks on graphs, the framework computes the diffusion distance  $D^{t}(x,y)$ . This distance is a functional weighted  $L^2$  distance between complex-valued proximities, enabling a more flexible and adaptive measure of similarity.  $D^{t}(x,y)$  will be
Figure Ape.6: Clusters evolving from  $t \in \{1, 2, 3, 4, 5, 10\}$  on the four groups colored by red, green, blue, and orange, with  $q = 1/3$ . For better illustration, the real and imaginary parts of the first two non-trivial eigenvectors are reported, in addition to the phases.
small if two samples are similar in the complex-valued plane and vice versa. Consequently, it becomes capable of capturing both local and global patterns within the data.
In this framework, the diffusion kernel, denoted as  $\rho(x,y)$ , represents the transition probabilities of the random walks. It signifies the likelihood of transitioning from one data point to another within the network. In summary, the proposed diffusion distance within the MagDM framework facilitates a geometric interpretation of asymmetric similarity. It encompasses the network structure and takes into account the asymmetric relationships between data points.
J The comparison with [4]
The MagDM is not a simple derivation of diffusion map for  $U(1)$ -connection graphs because there are three key differences between MagDM and the graph connection Laplacian (GCL) approach.
Definition of the similarity matrices. MagDM focuses on asymmetric similarity between samples and calculates the  $U(1)$  transporter for all pairs of samples. On the other hand, GCL focuses on symmetric cases and calculates the orthogonal  $O(d)$  transporter for graph edges. While there is an isomorphism between  $U(1)$  and  $SO(2)$ , they differ in implementation. MagDM measures scalar
similarity between two samples, denoted as  $\mathbf{x}$  and  $\mathbf{y}$ , while the similarity between  $\mathbf{x}$  and  $\mathbf{y}$  in GCL is a  $d * d$  block. The corresponding Gram matrix in MagDM, denoted by  $\rho$ , has a size of  $N * N$ , while the corresponding Gram matrix in GCL, denoted by  $S$ , is  $Nd * Nd$ .
Definition of diffusion distance. In MagDM, the diffusion distance is defined as  $[D^t (x,y)]^2 = \int_X(\rho^t (x,u) - \rho^t (y,u))^2\mu (du)$ . It measures the number of paths of length t connecting x and y in the complex-valued plane. The diffusion distance is calculated by the sum of squares of the difference between  $\rho^t (x,\bullet)$  and  $\rho^t (y,\bullet)$ . In GCL, diffusion distance has a different form:  $[D^{t}(x,y)]^{2} = Tr(S^{t}(x,y)S^{t}(x,y)^{\top})$ , which measures the agreement between transporters and the number of paths of length t connecting x and y. Based on these two definitions, it can be observed that  $U(1)$ -connection graphs exhibit different diffusion distances in MagDM and GCL.
Definition of the diffusion map. Due to the differences in diffusion distances, the diffusion maps of the two methods also differ. In MagDM, eigenvalues and eigenvectors are defined as  $\{\lambda_i\}_{i=1}^N, \{\phi_i\}_{i=1}^N$ , respectively. The diffusion map is defined as  $\Phi^t(x) = (\lambda_1^t\phi_1(x), \lambda_2^t\phi_2(x), \dots, \lambda_N^t\phi_N(x))$ . For vector diffusion map, its eigenvalues and eigenvectors are defined as  $\{\eta_i\}_{i=1}^{Nd}, \{\psi_i\}_{i=1}^{Nd}$ , respectively. Its diffusion map is defined by  $\Phi^t(x) = ((\eta_i\eta_j)^t < \psi_i(x), \psi(x) >)_{i,j=1}^{Nd}$ . These two diffusion maps have different forms from each other. Overall, MagDM is not a derivation to a notion of diffusion map for  $U(1)$ -connection graphs.
K Eigenvalues and robustness
Indeed,  $H$  and  $I - H$  can be transferred to each other when considering both positive and negative eigenvalues. However, it should be noted that in [18, 19], the negative eigenvalues are truncated, which means that  $H$  and  $I - H$  are not identical in this regard. We focus on the k eigenvalues that include both positive and negative eigenvalues with the largest absolute magnitudes. These top k eigenvalues correspond to the slowest diffusion processes or the global structures in the data. It is also believed that the larger magnitude of eigenvalues corresponds to better discriminative information, as discussed in [37]. One notable advantage of this approach is its robustness when confronted with perturbed asymmetric similarity. Even if the similarity measure is distorted, we can still obtain the principal components using this method. However, there is no specific reference to a related study because most previous work has mainly focused on positive semi-definite matrices. When dealing with asymmetry, it becomes necessary to consider negative eigenvalues. This aspect is worth exploring in future research. Currently, we can observe this phenomenon experimentally in Fig. 1 of the manuscript. In the case where  $\mathrm{P} = 0$ , it represents a situation where there are only links in one direction, making it an asymmetric but relatively easy scenario. When  $\mathrm{P} = 1 / 0.8$ , the situation becomes more complicated and backward flow is perturbed by the forward, MagDM effectively distinguishes between the three groups while ME/MME struggles to capture the information, highlighting stronger robustness and effectiveness of MagDM.
In the experiments involving asymmetric similarity, it is indeed possible for the largest eigenvalues to have different signs. Here, we present the counts of eigenvalues for  $\phi_{1}$  and  $\phi_{2}$  that exhibit both positive signs  $(++)$  and opposite signs  $(+-)$  among 100 trials for the network generated by the forward and backward flows (Experiment 1). The forward flow probability is fixed at 0.5, while the backward flow probability  $P$  is selected from the following set:  $\{0, 0.3, 0.4, 0.5, 0.6, 0.65, 0.675, 0.7, 0.8, 1\}$ .
Table Ape.2: The counts of signs for different P.
P	0	0.3	0.4	0.5	0.6	0.65	0.675	0.7	0.8	1
++	100	100	96	90	81	57	16	2	0	0
+-	0	0	4	10	19	43	84	98	100	100
In Tab. Ape.2, it can be observed that as P increases, the number of opposite signs also increases. Particularly, when P is larger than 0.675, opposite signs occur more frequently. We have discovered that the principal components align with both positive and negative eigenvalues, highlighting the superior performance of MagDM. These results further substantiate the robustness of MagDM.