SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathcal {K} (\boldsymbol {x}, \boldsymbol {y}) = \left\{ \begin{array}{l l} \max  \left\{\rho - \| \boldsymbol {x} - \boldsymbol {y} \| _ {1}, 0 \right\}, & \angle \boldsymbol {x} \boldsymbol {y} \geq 0 \\ \max  \left\{0. 2 \rho - \| \boldsymbol {x} - \boldsymbol {y} \| _ {1}, 0 \right\}, & \angle \boldsymbol {x} \boldsymbol {y} <   0 \end{array} , \right. \tag {9}
$$
where  $\angle xy$  is the angle between  $x$ ,  $y$  on the x-y plane. The asymmetric kernel is composed of two truncated  $\ell_1$  kernels [31], which measures asymmetric similarity based on angles between  $x$  and  $y$ .
Figure 3: Dimension reduction results of six methods (DM, ADM, KPCA, ME, MME, and MagDM) on the Möbius strip which is endowed with the asymmetric kernel (9) and colored by the angle.  $\rho = 5$  and  $q = 0.09$ .
The dimension reduction results are illustrated in Fig. 3. DM and KPCA fail to distinguish the asymmetric geometry, as they only learn the symmetric part of (9). Although the ADM method contains the color drift after reducing the dimension of data, there is still a sharp turn on the green part of the data, which is not desired. The ME method distinguishes the color drift only on the phase of the first eigenvector, while the MME method fails to provide clear directed information. Notably, the MagDM method provides a clear and smooth result along both two phase axes.
5.3 Trophic networks
Figure 4: The dimension reduction of the Florida network is shown using six methods (DM, ADM, KPCA, ME, MME, and MagDM), with  $q = 0.045$ . Florida is composed of four categories consisting of different elements: Producers (green nodes), Consumers (red and brown nodes), Organic Matters (blue and turquoise blue nodes), and one Decomposer (purple node). The low-dimensional embeddings using the different methods are visually compared.
To further illustrate the concept, we have chosen two specific real-world trophic networks from the Pajek datasets<sup>1</sup>: the Mondego network [32], which records trophic exchanges at the Mondego estuary, and the Florida network [33], which records trophic exchanges in Florida Bay during the wet season.
Based on the roles of the nodes in ecosystems, we classify them into different categories, as depicted in Appendix F.4. The different categories in the trophic networks consist of Producers, which generate their own food through photosynthesis or chemosynthesis (green nodes); Consumers, consisting of both low-level consumers (brown nodes) and high-level consumers (red nodes) that feed on other organisms for energy; and Organic Matters (blue and turquoise blue nodes). As for the Florida network, there is also a Decomposer that break down dead or decaying organic matter (purple node). The adjacency matrix  $W$  denotes the intensity of trophic exchanges and the asymmetric kernel matrix
<sup>1</sup> http://vlado.fmf.uni-lj.si/pub/networks/data/bio/foodweb/foodweb.htm
for the trophic networks are defined by,
$$
\boldsymbol {K} _ {i j} = \log_ {2} (\boldsymbol {W} _ {i j} + 1).
$$
The dimension reduction results for two trophic networks are presented in Figs. 4 and 5. For the Florida network, MagDM outperforms other methods, as shown in Fig. 4. All DM, ADM, and KPCA methods fail to differentiate between different categories. On the phases results of the remaining three methods, ME and MME are unable to extract clear asymmetric information, while MagDM almost distinguishes between producers and consumers. For better illustration, the real and imaginary parts of the first two non-trivial eigenvectors of ME, MME, and MagDM are visually compared. It is observed that all three methods fail to distinguish categories on the first eigenvector axis, but on the second eigenvector axis, MagDM not only differentiates the categories but also tightly clusters high-level consumers together, resulting in a clear classification structure. Similar results are obtained for the Mondego network. These findings demonstrate that MagDM is a promising approach for dealing with asymmetric proximity in trophic networks.
Figure 5: The dimension reduction of the Mondego network is shown using six methods (DM, ADM, KPCA, ME, MME, and MagDM), with  $q = 0.04$ . Mondego is composed of three categories consisting of different elements: Producers (green nodes), Consumers (red and brown nodes), and Organic Matters (blue and turquoise blue nodes). The low-dimensional embeddings using the different methods are visually compared.
In this paper, we have presented a novel framework named MagDM for representing data endowed with asymmetric kernels using the magnetic transform within the context of diffusion maps. We introduced the integral operator with the proposed magnetic transform kernel and investigated several
of its properties, including compactness, spectral decomposition, and spectral radius. Moreover, we proposed a diffusion distance that embeds asymmetric kernels into the diffusion map in complex-valued function spaces, allowing the identification and analysis of asymmetric geometries and patterns in data. This could lead to new insights into the structure and properties of asymmetric kernels, potentially leading to new applications or algorithmic improvements. Experiments demonstrated the effectiveness and robustness of MagDM on three synthetic datasets and two trophic networks.
There are several interesting avenues for future work in the study of asymmetric kernels. In terms of methodology, exploring variants of other dimension reduction schemes for asymmetric kernels such as MDS [34] and Isomap [35] would be appealing. In terms of theory, studying geometric harmonics of functions with magnetic transform kernels could provide a deeper understanding of the intrinsic and extrinsic geometries in data. In terms of applications, using MagDM as feature mappings for classification and regression tasks could be particularly useful for problems involving asymmetric proximity, such as social networks and commodity trading between economies.
Broader impacts. The proposed MagDM method described in this paper is not expected to have significantly different impacts on society compared to other dimension reduction techniques.
This work was partially supported by National Natural Science Foundation of China (62376155), Research Program of Shanghai Municipal Science and Technology Committee (22511105600), and Shanghai Municipal Science and Technology Major Project (2021SHZDX0102).
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A The proof of Proposition 1
Proof. Because  $\mathcal{K}$  is a Hilbert-Schmidt kernel, we have  $\iint_{X\times X}\left|\mathcal{K}(\boldsymbol {x},\boldsymbol {y})\right|^2\mathrm{d}\mu (\boldsymbol {x})\mathrm{d}\mu (\boldsymbol {y}) < \infty$  . Then,
$$
\begin{array}{l} \iint_ {X \times X} | \mathcal {H} ^ {(q)} (\boldsymbol {x}, \boldsymbol {y}) | ^ {2} \mathrm {d} \mu (\boldsymbol {x}) \mathrm {d} \mu (\boldsymbol {y}) = \int_ {X} \int_ {X} | \frac {\mathcal {K} (\boldsymbol {x} , \boldsymbol {y}) + \mathcal {K} (\boldsymbol {y} , \boldsymbol {x})}{2} | ^ {2} \mathrm {d} \mu (\boldsymbol {x}) \mathrm {d} \mu (\boldsymbol {y}) \\ \leq \int_ {X} \int_ {X} \frac {1}{2} \left(\left| \mathcal {K} (\boldsymbol {x}, \boldsymbol {y}) \right| ^ {2} + \left| \mathcal {K} (\boldsymbol {y}, \boldsymbol {x}) \right| ^ {2}\right) d \mu (\boldsymbol {x}) d \mu (\boldsymbol {y}) \\ = \int_ {X} \int_ {X} | \mathcal {K} (\boldsymbol {x}, \boldsymbol {y}) | ^ {2} d \mu (\boldsymbol {x}) d \mu (\boldsymbol {y}) <   \infty . \\ \end{array}
$$
It holds that the symbol “ $\leq$ ” in the third row of the above equation because of the Cauchy-Schwarz inequality. The Cauchy-Schwarz inequality states that  $\forall a_i, b_i \in \mathbb{R}, i = 1, \dots, k$ , then  $\left(\sum_{i=1}^{k} a_i b_i\right)^2 \leq \left(\sum_{i=1}^{k} a_i^2\right)\left(\sum_{i=1}^{k} b_i^2\right)$ . Then, we have  $\left|\frac{\mathcal{K}(\boldsymbol{x}, \boldsymbol{y})}{2} + \frac{\mathcal{K}(\boldsymbol{y}, \boldsymbol{x})}{2}\right|^2 \leq \left(\frac{1}{4} + \frac{1}{4}\right) \cdot \left(\left|\mathcal{K}(\boldsymbol{x}, \boldsymbol{y})\right|^2 + \left|\mathcal{K}(\boldsymbol{y}, \boldsymbol{x})\right|^2\right)$ , and measure  $\mu$  is non-negative. Thus, symbol “ $\leq$ ” in the third row holds. We finish the proof.
B The proof of Proposition 2
Proof. The associated kernel  $\mu(\boldsymbol{x}, \boldsymbol{y}, q)$  of the operator (5) is a Hermitian kernel, i.e.,  $\mu(\boldsymbol{x}, \boldsymbol{y}, q) = \overline{\mu(\boldsymbol{y}, \boldsymbol{x}, q)}$ , then the eigenvalues of (5) are real and the spectral radius is equal to the operator norm  $R(\rho) = \|T^{(q)}\|$  [36]. Let  $f \in L^2(X, \mu)$ , we have
$$
\begin{array}{l} \langle T ^ {(q)} f, f \rangle = \int_ {X} \int_ {X} \rho (\boldsymbol {x}, \boldsymbol {y}, q) f (\boldsymbol {y}) \overline {{f (\boldsymbol {x})}} \mathrm {d} \mu (\boldsymbol {x}) \mathrm {d} \mu (\boldsymbol {y}) \\ = \int_ {X} \int_ {X} \mathcal {H} ^ {(q)} (\boldsymbol {x}, \boldsymbol {y}) \frac {f (\boldsymbol {y})}{\sqrt {m (\boldsymbol {y})}} \frac {\overline {{f (\boldsymbol {x})}}}{\sqrt {m (\boldsymbol {x})}} \mathrm {d} \mu (\boldsymbol {x}) \mathrm {d} \mu (\boldsymbol {y}). \\ \end{array}
$$
If we apply the Cauchy-Schwarz inequality as follows,
$$
\begin{array}{l} \left| \int_ {X} \mathcal {H} ^ {(q)} (\boldsymbol {x}, \boldsymbol {y}) \frac {f (\boldsymbol {y})}{\sqrt {m (\boldsymbol {y})}} \mathrm {d} \mu (\boldsymbol {y}) \right| \leq \left(\int_ {X} \left| \mathcal {H} ^ {(q)} (\boldsymbol {x}, \boldsymbol {y}) \right| \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} \left( \right.\int_ {X} \left| \mathcal {H} ^ {(q)} (\boldsymbol {x}, \boldsymbol {y}) \right. ^ {\frac {1}{2}} \frac {f (\boldsymbol {y})}{\sqrt {m (\boldsymbol {y})}} \left. \right| ^ {2} \mathrm {d} \mu (\boldsymbol {y})\left. \right) ^ {\frac {1}{2}} \\ \leq \left(\int_ {X} | \mathcal {H} ^ {(q)} (\boldsymbol {x}, \boldsymbol {y}) | \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} \left(\int_ {X} | \mathcal {H} ^ {(q)} (\boldsymbol {x}, \boldsymbol {y}) | \frac {| f (\boldsymbol {y}) | ^ {2}}{m (\boldsymbol {y})} \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} \\ = \left(\int_ {X} S (\boldsymbol {x}, \boldsymbol {y}) \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} \left(\int_ {X} S (\boldsymbol {x}, \boldsymbol {y}) \frac {| f (\boldsymbol {y}) | ^ {2}}{m (\boldsymbol {y})} \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}}, \\ \end{array}
$$
where  $S(\pmb{x},\pmb{y}) = \left|\mathcal{H}^{(q)}(\pmb{x},\pmb{y})\right| = \frac{1}{2}\left|\mathcal{K}(\pmb{x},\pmb{y}) + \mathcal{K}(\pmb{y},\pmb{x})\right|$ . The asymmetric kernel  $\mathcal{K}$  is non-negative, thus  $S(\pmb{x},\pmb{y}) = \frac{1}{2}\bigl (\mathcal{K}(\pmb{x},\pmb{y}) + \mathcal{K}(\pmb{y},\pmb{x})\bigr)$ . Then we have
$$
\begin{array}{l} \left| \int_ {X} \mathcal {H} ^ {(q)} (\boldsymbol {x}, \boldsymbol {y}) \frac {f (\boldsymbol {y})}{\sqrt {m (\boldsymbol {y})}} \mathrm {d} \mu (\boldsymbol {y}) \right| \leq \left(\int_ {X} S (\boldsymbol {x}, \boldsymbol {y}) \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} \left(\int_ {X} S (\boldsymbol {x}, \boldsymbol {y}) \frac {| f (\boldsymbol {y}) | ^ {2}}{m (\boldsymbol {y})} \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} \\ = \sqrt {m (\boldsymbol {x})} \left(\int_ {X} S (\boldsymbol {x}, \boldsymbol {y}) \frac {\left| f (\boldsymbol {y}) \right| ^ {2}}{m (\boldsymbol {y})} \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}}. \\ \end{array}
$$
$$
\langle T ^ {(q)} f, f \rangle \leq \int_ {X} | f (\boldsymbol {x}) | \left(\int_ {X} S (\boldsymbol {x}, \boldsymbol {y}) \frac {| f (\boldsymbol {y}) | ^ {2}}{m (\boldsymbol {y})} \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} \mathrm {d} \mu (\boldsymbol {x}).
$$
We can apply the Cauchy-Schwarz inequality again,
$$
\begin{array}{l} \langle T ^ {(q)} f, f \rangle \leq \| f \| \left(\int_ {X} \int_ {X} S (\boldsymbol {x}, \boldsymbol {y}) \frac {\left| f (\boldsymbol {y}) \right| ^ {2}}{m (\boldsymbol {y})} \mathrm {d} \mu (\boldsymbol {y}) \mathrm {d} \mu (\boldsymbol {x})\right) ^ {\frac {1}{2}} \\ = \| f \| \left(\int_ {X} \frac {\left| f (\boldsymbol {y}) \right| ^ {2}}{m (\boldsymbol {y})} \left(\int_ {X} S (\boldsymbol {x}, \boldsymbol {y}) \mathrm {d} \mu (\boldsymbol {x})\right) \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} \\ = \| f \| \left(\int_ {X} \frac {\left| f (\boldsymbol {y}) \right| ^ {2}}{m (\boldsymbol {y})} m (\boldsymbol {y}) \mathrm {d} \mu (\boldsymbol {y})\right) ^ {\frac {1}{2}} = \| f \| ^ {2}. \\ \end{array}
$$
It can be noticed that the operator norm  $\| T^{(q)}\| = 1$ . Therefore, the spectral radius  $R(\rho) = 1$ . We finish the proof.
C Selection of the scaling parameter q.
As illustrated in Fig. Ape.1, we suggest choosing a period that encompasses the range of the skew-symmetric component, i.e.,  $\overline{a}$  should be less than  $\pi$ , where  $\overline{a}$  is defined as follows,
$$
\overline {{a}} := \sup  _ {\boldsymbol {x}, \boldsymbol {y} \in X} | \mathcal {K} (\boldsymbol {x}, \boldsymbol {y}) - \mathcal {K} (\boldsymbol {y}, \boldsymbol {x}) |,
$$
and the period of the phase function is  $T = \frac{2\pi}{2\pi q} = 1 / q$ . Thus, we have
$$
T = \frac {1}{q} > 2 \overline {{a}} \Rightarrow q <   \frac {1}{2 \overline {{a}}}.
$$
Figure Ape.1: A simple illustration for selecting  $q$ .
D Algorithm
To apply MagDM, we require a dataset and an asymmetric kernel or an asymmetric Gram matrix, as well as the scaling parameter  $q$  and the desired accuracy. Algorithm 1 outlines the MagDM procedure.
Algorithm 1 MagDM for asymmetric kernels
Input: The Gram matrix  $\pmb{K}$  of dataset  $X$  endowed with an asymmetric kernel  $\mathcal{K}$ , the scaling parameter  $q$  and a preset accuracy  $\delta$ .
Output: The diffusion map  $\psi^{t,(q)}$  of  $X$ .
1: Calculate the Hermitian Gram matrix  $\pmb{H}$  of the asymmetric Gram matrix  $\pmb{K}$  by (3) and (4).
2: Calculate the  $t$ -powers kernel matrix  $\pmb{H}^t$
3: Run eigen-decomposition of  $\pmb{H}^t$  and denote its eigen-system as  $\{\lambda_n^{(q)},\phi_n^{(q)}\}$ .
4:  $s(\delta ,t)\gets \max \{n\in \mathbb{N}:|\lambda_n^{(q)}| > \delta |\lambda_1^{(q)}|\}$
5: Return the diffusion map  $\psi^{t,(q)}$  by (8).
Limitations. Researchers should note that the MagDM method proposed in this paper has some limitations. One such limitation is its dependence on the choice of asymmetric kernel functions, which can impact its performance. Additionally, MagDM may be computationally expensive for large datasets, as it requires  $\mathcal{O}(N^2)$  memory and  $\mathcal{O}(N^3)$  computational complexity to derive the spectral decomposition. However, this limitation can be addressed through the use of out-of-sample extensions, which are discussed in Appendix E.
E Out-of-sample extensions.
Out-of-sample extensions are useful in many applications where low-dimensional embeddings computed on the original dataset are extended to new data. The Nyström extension is a well-known technique used in the machine learning community to approximate the Gram matrix by a low-rank embedding. However, the out-of-sample extension of duffision maps for asymmetric kernels has not been studied before. Here, we present the corresponding Nyström-based extension for out-of-sample cases. As discussed earlier, the integral operator (5) is compact and self-adjoint, whose spectral decomposition is  $\{\lambda_n^{(q)},\phi_n^{(q)}\}$ . If  $\lambda_n^{(q)}\neq 0$ , the following identity holds for  $x\in X$ :
$$
\phi_ {n} ^ {(q)} (\pmb {x}) = \frac {T ^ {(q)}}{\lambda_ {n} ^ {(q)}} \phi_ {n} ^ {(q)} (\pmb {x}) = \int_ {X} \frac {\rho (\pmb {x} , \pmb {y} , q)}{\lambda_ {n} ^ {(q)}} \phi_ {n} ^ {(q)} (\pmb {y}) \mathrm {d} \mu (\pmb {y}).
$$
The Nyström extension extends the equation above to new data  $Z$  such that  $X \subseteq Z$  as follows,
$$
\phi_ {n} ^ {(q)} (\boldsymbol {z}) = \int_ {X} \frac {\rho (\boldsymbol {z} , \boldsymbol {y} , q)}{\lambda_ {n} ^ {(q)}} \phi_ {n} ^ {(q)} (\boldsymbol {y}) \mathrm {d} \mu (\boldsymbol {y}), \tag {a1}
$$
where  $\pmb{z} \in Z$  and  $\phi_n^{(q)}(\pmb{z}) = \sum_{\pmb{y} \in X} \frac{\rho(\pmb{z}, \pmb{y}, q)}{N \lambda_n^{(q)}} \phi_n^{(q)}(\pmb{y})$  is the empirical form of (a1) for  $X$ . This allows the eigenfunctions to be extended for new data, enabling the extension of MagDM (8) as follows,
$$
\psi^ {t, (q)} (\pmb {z}) = \sum_ {\pmb {y} \in X} \left[ \frac {\rho (\pmb {z} , \pmb {y} , q)}{N} \phi_ {1} ^ {(q)} (\pmb {y}), \quad \frac {\rho (\pmb {z} , \pmb {y} , q)}{N} \phi_ {2} ^ {(q)} (\pmb {y}), \quad \dots , \quad \frac {\rho (\pmb {z} , \pmb {y} , q)}{N} \phi_ {s (\delta , t)} ^ {(q)} (\pmb {y}) \right] ^ {\top}.
$$
F Descriptions and figures of datasets
In this section, we visualize the datasets using either expert knowledge or a force-directed layout. We hope that these visualizations will help readers gain a better understanding of the data.
F.1 The first artificial network
Fig. Ape.2(a) provides the running flow of the first artificial network and Fig. Ape.2(b) shows an example of the directed graphs generated with  $P = 0$ , where the asymmetric adjacency connection can be considered as an asymmetric kernel.
(a) The running flow of three groups.
(b) Graph using the expert knowledge positions.
Figure Ape.2: An illustration of the first artificial network. (a) The running flow of three groups A, B and C. The directed/asymmetric information is nested in the running flow. (b) An instance of a directed graph generated by the running flow with backward flow probability  $P = 0$ .
F.2 The second artificial network
The running flow of the second artificial network comprises four groups (A, B, C, and D). The structure of the flow is apparent, with groups A and D serving as out-come and in-come nodes, respectively, while groups B and C function as communicators. Groups B and C are two dense sets containing 20 nodes and Group A and D are a pair of datasets whose interconnections are much more than Group B and C.
(a) The running flow of four groups.
Figure Ape.3: An illustration of the second artificial network. (a) The running flow of three groups A, B, C and D. The directed/asymmetric information is nested in the running flow. (b) An instance of a directed graph generated by the running flow with Groups A and D playing a particular role (green and orange nodes) and Groups B and C playing a role of a communicator (red and blue nodes).
(b) Graph using a force-directed layout.
F.3 The Möbius strip
The Möbius strip dataset is a set of 300 points randomly distributed along the Möbius strip. The parametric form of the Möbius strip is defined by,
where  $0 \leq u \leq 2\pi$  and  $-0.5 \leq v \leq 0.5$ . The dataset is with a color drift in the counterclockwise direction on the x-y plane in Fig. Ape.4.
Figure Ape.4: Dataset with 300 random points in the Möbius strip.
F.4 Two trophic networks
We have chosen two specific trophic networks: the Mondego [32] and Florida [33] networks, which are part of the Hajek datasets. These networks have recorded the trophic exchanges at Mondego estuary and Florida bay during the wet season, respectively. Based on the roles of the nodes in these ecosystems, we have classified them into different categories, as shown in Appendix Ape.5. The green nodes, such as 2um Spherical Phytoplankt and Phytoplankton, are producers that generate their own food through photosynthesis or chemosynthesis. The brown and red nodes are low-level consumers like littorina and high-level consumers like bonefish and crocodiles that feed on other organisms for energy. Additionally, the purple node in the Florida network represents decomposers that break down dead or decaying organic matter. Finally, the blue and turquoise blue nodes correspond to the input, output, and organic matter of the ecosystem, respectively.
(a) Mondego.
(b) Florida.
Figure Ape.5: The trophic networks using force-direct layout. Nodes among this network are classified into several categories, The green nodes are producers that generate their own food through photosynthesis or chemosynthesis. The brown and red nodes are low-level consumers and high-level consumers that feed on other organisms for energy. Additionally, the purple node in the Florida network represents decomposers that break down dead or decaying organic matter. The blue and turquoise blue nodes correspond to the input, output, and organic matter of the ecosystem.
G Quantitative experiments