Graph　convolutional　networks　(GCNs)　have　been　attracting　widespread　attentions　due　to　their　encouraging　performance　and　powerful　generalizations.　However,　few　work　provide　a　general　view　to　interpret　various　GCNs　and　guide　GCNs’　designs.　In　this　paper,　by　revisiting　the　original　GCN,　we　induce　an　interpretable　regularizer-centerd　optimization　framework,　in　which　by　building　appropriate　regularizers　we　can　interpret　most　GCNs,　such　as　APPNP,　JKNet,　DAGNN,　and　GNN-LF/HF.　Further,　under　the　proposed　framework,　we　devise　a　dual-regularizer　graph　convolutional　network　(dubbed　tsGCN1)　to　capture　topological　and　semantic　structures　from　graph　data.　Since　the　derived　learning　rule　for　tsGCN　contains　an　inverse　of　a　large　matrix　and　thus　is　time-consuming,　we　leverage　the　Woodbury　matrix　identity　and　low-rank　approximation　tricks　to　successfully　decrease　the　high　computational　complexity　of　computing　infinite-order　graph　convolutions.　Extensive　experiments　on　eight　public　datasets　demonstrate　that　tsGCN　achieves　superior　performance　against　quite　a　few　state-of-the-art　competitors　w.r.t.　classification　tasks.　Copyright　©　2023,　Association　for　the　Advancement　of　Artificial　Intelligence　(www.aaai.org).　All　rights　reserved.