Title:  Hypergraph Convolutional Networks via Equivalency  between  Hypergraphs and Undirected Graphs. Abstract: As a powerful tool for modeling the complex relationships, hypergraphs are gaining popularity from the graph learning community. However, commonly used algorithms in deep hypergraph learning were not specifically designed for hypergraphs with edge-dependent vertex weights (EDVWs). To fill this gap, we build the equivalency condition between EDVW-hypergraphs and undirected simple graphs, which enables utilizing existing undirected graph neural networks as subroutines to learn high-order interactions induced by EDVWs of hypergraphs. Specifically, we define a generalized hypergraph with vertex weights by proposing a unified random walk framework, under which we present the equivalency condition between generalized hypergraphs and undigraphs. Guided by the equivalency results, we propose a Generalized Hypergraph Convolutional Network (GHCN) architecture for deep hypergraph learning.  Furthermore, to improve the long-range interactions and alleviate the over-smoothing issue, we further propose the Simple Hypergraph Spectral Convolution (SHSC) model by constructing the Discounted Markov Diffusion Kernel from our random walk framework. Extensive experiments from various domains including social network analysis, visual objective classification, and protein fold classification demonstrate that the proposed approaches outperform state-of-the-art spectral methods with a large margin.