# Semi-Supervised Classification via Hypergraph Convolutional Extreme Learning Machine  
[PERSON]  
1 School of Computer Science, China University of Geosciences, Wuhan 430078, China; [EMAIL_ADDRESS] (Z.L. ); [EMAIL_ADDRESS] (Z.Z. ); [EMAIL_ADDRESS] (Y.C. ); [EMAIL_ADDRESS] (Y.M.) [PERSON]  
1 School of Computer Science, China University of Geosciences, Wuhan 430078, China; [EMAIL_ADDRESS] (Z.L. ); [EMAIL_ADDRESS] (Z.Z. ); [EMAIL_ADDRESS] (Y.C. ); [EMAIL_ADDRESS] (Y.M.) [PERSON]  
1 School of Computer Science, China University of Geosciences, Wuhan 430078, China; [EMAIL_ADDRESS] (Z.L. ); [EMAIL_ADDRESS] (Z.Z. ); [EMAIL_ADDRESS] (Y.C. ); [EMAIL_ADDRESS] (Y.M.) [PERSON]  
1 School of Computer Science, China University of Geosciences, Wuhan 430078, China; [EMAIL_ADDRESS] (Z.L. ); [EMAIL_ADDRESS] (Z.Z. ); [EMAIL_ADDRESS] (Y.C. ); [EMAIL_ADDRESS] (Y.M.) [PERSON]  
2 Guangqi Key Laboratory of Marine Disaster in the Beibu Gulf, Beibu Gulf University, Qinzhou 535011, China 2  
samples are significantly limited, the auxiliary information of unlabeled data is surprisingly helpful to improve generalization ability [19; 20]. To this end, Semi-supervised ELM (SS-ELM) [21] has drawn increasing attention. SS-ELM adopts a frequently used scheme, i.e., graph Laplacian regularization, to incorporate manifold structure. To combine multiple graphs into SS-ELM, [PERSON] et al. [22] introduced an adaptive multiple graph regularized SS-ELM approach. These efforts have demonstrated that structured information is beneficial for ELM. Nonetheless, the simple graph can only reveal the pairwise relationship of data points, thus ignoring the high-order relationship. Moreover, these methods basically work in the Euclidean domain, leading to indirect and insufficient use of structured information. Recently, Graph Neural Networks (GNNs) [23] is emerging as a powerful technique for generalizing neural networks on graph-structured data. Technically, GNNs capture the dependence of graphs via message passing between the nodes of graphs [24]. Graph Convolutional Network (GCN) [25] is one of the most popular GNN models for node-level tasks [26] and graph-level tasks [27]. It has been proven that GCN is equivalent to a fixed low-pass filter [28; 29] followed by a linear classifier. Thus, it often suffers from the over-smoothing problem when model depth increased. More recently, some works have attempted to extending GCN into hypergraph scenarios [30; 31] so that the high-order relationship of data can be embedded. Despite considerable success, current GCNs highly rely on arduous model update via gradient descent. Depending on the observation that GCN is equivalent to a multilayer perceptron multiplied by an adjacent matrix, [32], [PERSON] et al. proposed a Graph Convolutional ELM (GCELM) for semi-supervised learning. As a pioneer work, GCELM follows the same pipeline as ELM but extends it into the non-Euclidean domain, enabling ELM to deal with graph-structured data directly. However, GCELM mainly focuses on the graph embedding of pairwise connections, resulting in failure to capture intraclass variations between data points. This motivates us to further develop an enhanced version of HGCELM. We refer to the method as Hypergraph Convolutional Extreme Learning Machine (HGCELM). Specifically, HGCELM contains a random hypergraph convolutional layer that produces hypergraph embeddings, and a hypergraph regression layer with a closed-form solution. To sum up, the main contributions of this paper are two-fold:  
1. We propose a simple but effective hypergraph convolutional ELM, i.e., HGCELM, for semi-supervised classification. The HGCELM method not only inherits all the advantages from ELM but enables ELM to model the high-order relationship of data. The successful attempt signifies that structured information, especially high-order relationships, among data is important for ELM, which offers an alternative orientation for ELM representation learning. 2. We have shown that the traditional ELMs are the special cases of HGCELM on the Euclidian data.