Simplicial Complex Representation Learning

Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d simplicial complexes, has witnessed a great attention and success recently. Due to the additional complexity higher dimensional simplicial hold, there has not been enough effort to extend representation learning to these objects especially when it comes to entire-simplicial complex representation learning. In this work, we propose a method for simplicial complex-level representation learning that embeds a simplicial complex to a universal embedding space in a way that complex-tocomplex proximity is preserved. Our method utilizes a simplex-level embedding induced by a pre-trained simplicial autoencoder to learn an entire simplicial complex representation. To the best of our knowledge, this work presents the first method for learning simplicial complex-level representation.

[1]  Nir Ailon,et al.  Deep Metric Learning Using Triplet Network , 2014, SIMBAD.

[2]  Paul Rosen,et al.  Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology , 2017, 2018 IEEE Pacific Visualization Symposium (PacificVis).

[3]  Claudio Gutierrez,et al.  Survey of graph database models , 2008, CSUR.

[4]  Yang Liu,et al.  graph2vec: Learning Distributed Representations of Graphs , 2017, ArXiv.

[5]  Jure Leskovec,et al.  Hierarchical Graph Representation Learning with Differentiable Pooling , 2018, NeurIPS.

[6]  Jure Leskovec,et al.  node2vec: Scalable Feature Learning for Networks , 2016, KDD.

[7]  R. Ho.  Algebraic Topology , 2022 .

[8]  Yizhou Sun,et al.  Unsupervised Inductive Whole-Graph Embedding by Preserving Graph Proximity , 2019, ArXiv.

[9]  Pietro Lio,et al.  Deep Graph Mapper: Seeing Graphs Through the Neural Lens , 2020, Frontiers in Big Data.

[10]  Jeffrey Dean,et al.  Efficient Estimation of Word Representations in Vector Space , 2013, ICLR.

[11]  Alexander J. Smola,et al.  Distributed large-scale natural graph factorization , 2013, WWW.

[12]  Guido Mont'ufar,et al.  Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks , 2021, ICML.

[13]  Keenan Crane,et al.  Geodesics in heat: A new approach to computing distance based on heat flow , 2012, TOGS.

[14]  Steven Skiena,et al.  DeepWalk: online learning of social representations , 2014, KDD.

[15]  Jure Leskovec,et al.  Representation Learning on Graphs: Methods and Applications , 2017, IEEE Data Eng. Bull..

[16]  Celia Hacker.  K-simplex2vec: a Simplicial Extension of Node2vec , 2020, ArXiv.

[17]  Edouard Pineau,et al.  A Simple Baseline Algorithm for Graph Classification , 2018, ArXiv.

[18]  Giovanni Petri,et al.  Simplex2Vec embeddings for community detection in simplicial complexes , 2019, ArXiv.

[19]  Kurt Mehlhorn,et al.  Weisfeiler-Lehman Graph Kernels , 2011, J. Mach. Learn. Res..

[20]  Mikhail Belkin,et al.  Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering , 2001, NIPS.

[21]  Jian Pei,et al.  A Survey on Network Embedding , 2017, IEEE Transactions on Knowledge and Data Engineering.

[22]  Santiago Segarra,et al.  Principled Simplicial Neural Networks for Trajectory Prediction , 2021, ICML.

[23]  Hisashi Koga,et al.  GL2vec: Graph Embedding Enriched by Line Graphs with Edge Features , 2019, ICONIP.

[24]  E. Munch,et al.  Fast and Scalable Complex Network Descriptor Using PageRank and Persistent Homology , 2020, 2020 International Conference on Intelligent Data Science Technologies and Applications (IDSTA).

[25]  Glenn M. Fung,et al.  Simplicial 2-Complex Convolutional Neural Nets. , 2020, 2012.06010.

[26]  Gard Spreemann,et al.  Simplicial Neural Networks , 2020, ArXiv.

[27]  Max Welling,et al.  Semi-Supervised Classification with Graph Convolutional Networks , 2016, ICLR.

[28]  Austin R. Benson,et al.  Random Walks on Simplicial Complexes and the normalized Hodge Laplacian , 2018, SIAM Rev..

[29]  M. Hajij,et al.  Cell Complex Neural Networks , 2020, ArXiv.

[30]  Emmanuel Müller,et al.  NetLSD: Hearing the Shape of a Graph , 2018, KDD.

[31]  Jure Leskovec,et al.  Inductive Representation Learning on Large Graphs , 2017, NIPS.

[32]  Jan Eric Lenssen,et al.  Fast Graph Representation Learning with PyTorch Geometric , 2019, ArXiv.

[33]  Paul Rosen,et al.  Mesh Learning Using Persistent Homology on the Laplacian Eigenfunctions , 2019, ArXiv.