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Simplicial Convolutional Filters

Published 27 Jan 2022 in eess.SP, cs.LG, cs.SI, math.AT, and math.SP | (2201.11720v3)

Abstract: We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted as generalizations of graphs that account for nodes, edges, triangular faces etc. To process such signals, we develop simplicial convolutional filters defined as matrix polynomials of the lower and upper Hodge Laplacians. First, we study the properties of these filters and show that they are linear and shift-invariant, as well as permutation and orientation equivariant. These filters can also be implemented in a distributed fashion with a low computational complexity, as they involve only (multiple rounds of) simplicial shifting between upper and lower adjacent simplices. Second, focusing on edge-flows, we study the frequency responses of these filters and examine how we can use the Hodge-decomposition to delineate gradient, curl and harmonic frequencies. We discuss how these frequencies correspond to the lower- and the upper-adjacent couplings and the kernel of the Hodge Laplacian, respectively, and can be tuned independently by our filter designs. Third, we study different procedures for designing simplicial convolutional filters and discuss their relative advantages. Finally, we corroborate our simplicial filters in several applications: to extract different frequency components of a simplicial signal, to denoise edge flows, and to analyze financial markets and traffic networks.

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References (39)
  1. M. Yang, E. Isufi, M. T. Schaub, and G. Leus, “Finite impulse response filters for simplicial complexes,” in 2021 29th European Signal Processing Conference (EUSIPCO), 2021, pp. 2005–2009.
  2. D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst, “The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains,” IEEE Signal Processing Magazine, vol. 30, no. 3, pp. 83–98, 2013.
  3. A. Ortega, P. Frossard, J. Kovačević, J. M. Moura, and P. Vandergheynst, “Graph signal processing: Overview, challenges, and applications,” Proceedings of the IEEE, vol. 106, no. 5, pp. 808–828, 2018.
  4. A. Sandryhaila and J. M. Moura, “Discrete signal processing on graphs,” IEEE Transactions on Signal Processing, vol. 61, no. 7, pp. 1644–1656, 2013.
  5. ——, “Discrete signal processing on graphs: Frequency analysis,” IEEE Transactions on Signal Processing, vol. 62, no. 12, pp. 3042–3054, 2014.
  6. W. Huang, T. A. Bolton, J. D. Medaglia, D. S. Bassett, A. Ribeiro, and D. Van De Ville, “A graph signal processing perspective on functional brain imaging,” Proceedings of the IEEE, vol. 106, no. 5, pp. 868–885, 2018.
  7. K. K. Leung, W. A. Massey, and W. Whitt, “Traffic models for wireless communication networks,” IEEE Journal on selected areas in Communications, vol. 12, no. 8, pp. 1353–1364, 1994.
  8. M. T. Schaub and S. Segarra, “Flow smoothing and denoising: Graph signal processing in the edge-space,” in 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP).   IEEE, 2018, pp. 735–739.
  9. X. Jiang, L.-H. Lim, Y. Yao, and Y. Ye, “Statistical ranking and combinatorial hodge theory,” Mathematical Programming, vol. 127, no. 1, pp. 203–244, 2011.
  10. O. Candogan, I. Menache, A. Ozdaglar, and P. A. Parrilo, “Flows and decompositions of games: Harmonic and potential games,” Mathematics of Operations Research, vol. 36, no. 3, pp. 474–503, 2011.
  11. T. Gebhart, X. Fu, and R. J. Funk, “Go with the flow? a large-scale analysis of health care delivery networks in the united states using hodge theory,” in 2021 IEEE International Conference on Big Data (Big Data).   IEEE, 2021, pp. 3812–3823.
  12. A. Mock and I. Volić, “Political structures and the topology of simplicial complexes,” Mathematical Social Sciences, vol. 114, pp. 39–57, 2021.
  13. W. Huang and A. Ribeiro, “Metrics in the space of high order networks,” IEEE Transactions on Signal Processing, vol. 64, no. 3, pp. 615–629, 2015.
  14. C. Bick, E. Gross, H. A. Harrington, and M. T. Schaub, “What are higher-order networks?” arXiv preprint arXiv:2104.11329, 2021.
  15. A. Patania, G. Petri, and F. Vaccarino, “The shape of collaborations,” EPJ Data Science, vol. 6, pp. 1–16, 2017.
  16. L.-H. Lim, “Hodge laplacians on graphs,” SIAM Review, vol. 62, no. 3, pp. 685–715, 2020.
  17. S. Barbarossa and S. Sardellitti, “Topological signal processing over simplicial complexes,” IEEE Transactions on Signal Processing, vol. 68, pp. 2992–3007, 2020.
  18. M. T. Schaub, Y. Zhu, J.-B. Seby, T. M. Roddenberry, and S. Segarra, “Signal processing on higher-order networks: Livin’on the edge… and beyond,” Signal Processing, vol. 187, p. 108149, 2021.
  19. L. Wasserman, “Topological data analysis,” Annual Review of Statistics and Its Application, vol. 5, pp. 501–532, 2018.
  20. J. Jia, M. T. Schaub, S. Segarra, and A. R. Benson, “Graph-based semi-supervised & active learning for edge flows,” in Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2019, pp. 761–771.
  21. T. M. Roddenberry, M. T. Schaub, and M. Hajij, “Signal processing on cell complexes,” in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).   IEEE, 2022, pp. 8852–8856.
  22. S. Sardellitti and S. Barbarossa, “Topological signal representation and processing over cell complexes,” arXiv preprint arXiv:2201.08993, 2022.
  23. S. Sardellitti, S. Barbarossa, and L. Testa, “Topological signal processing over cell complexes,” in 2021 55th Asilomar Conference on Signals, Systems, and Computers.   IEEE, 2021, pp. 1558–1562.
  24. M. Yang, E. Isufi, and G. Leus, “Simplicial convolutional neural networks,” in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).   IEEE, 2022, pp. 8847–8851.
  25. S. Ebli, M. Defferrard, and G. Spreemann, “Simplicial neural networks,” in NeurIPS 2020 Workshop on Topological Data Analysis and Beyond, 2020. [Online]. Available: https://openreview.net/forum?id=nPCt39DVIfk
  26. C. Bodnar, F. Frasca, Y. Wang, N. Otter, G. F. Montufar, P. Lio, and M. Bronstein, “Weisfeiler and lehman go topological: Message passing simplicial networks,” in International Conference on Machine Learning.   PMLR, 2021, pp. 1026–1037.
  27. C. Bodnar, F. Frasca, N. Otter, Y. Wang, P. Lio, G. F. Montufar, and M. Bronstein, “Weisfeiler and lehman go cellular: Cw networks,” Advances in Neural Information Processing Systems, vol. 34, pp. 2625–2640, 2021.
  28. E. Bunch, Q. You, G. Fung, and V. Singh, “Simplicial 2-complex convolutional neural networks,” in NeurIPS 2020 Workshop on Topological Data Analysis and Beyond, 2020. [Online]. Available: https://openreview.net/forum?id=TLbnsKrt6J-
  29. T. M. Roddenberry, N. Glaze, and S. Segarra, “Principled simplicial neural networks for trajectory prediction,” in International Conference on Machine Learning.   PMLR, 2021, pp. 9020–9029.
  30. T. M. Roddenberry and S. Segarra, “Hodgenet: Graph neural networks for edge data,” in 2019 53rd Asilomar Conference on Signals, Systems, and Computers.   IEEE, 2019, pp. 220–224.
  31. M. T. Schaub, A. R. Benson, P. Horn, G. Lippner, and A. Jadbabaie, “Random walks on simplicial complexes and the normalized hodge 1-laplacian,” SIAM Review, vol. 62, no. 2, pp. 353–391, 2020.
  32. T. M. Roddenberry, F. Frantzen, M. T. Schaub, and S. Segarra, “Hodgelets: Localized spectral representations of flows on simplicial complexes,” in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).   IEEE, 2022, pp. 5922–5926.
  33. D. I. Shuman, P. Vandergheynst, D. Kressner, and P. Frossard, “Distributed signal processing via chebyshev polynomial approximation,” IEEE Transactions on Signal and Information Processing over Networks, vol. 4, no. 4, pp. 736–751, 2018.
  34. G. L. Sleijpen and H. A. Van der Vorst, “A jacobi–davidson iteration method for linear eigenvalue problems,” SIAM review, vol. 42, no. 2, pp. 267–293, 2000.
  35. V. L. Druskin and L. A. Knizhnerman, “Two polynomial methods of calculating functions of symmetric matrices,” USSR Computational Mathematics and Mathematical Physics, vol. 29, no. 6, pp. 112–121, 1989.
  36. D. K. Hammond, P. Vandergheynst, and R. Gribonval, “Wavelets on graphs via spectral graph theory,” Applied and Computational Harmonic Analysis, vol. 30, no. 2, pp. 129–150, 2011.
  37. H. Youn, M. T. Gastner, and H. Jeong, “Price of anarchy in transportation networks: efficiency and optimality control,” Physical review letters, vol. 101, no. 12, p. 128701, 2008.
  38. M. T. Schaub, J. Lehmann, S. N. Yaliraki, and M. Barahona, “Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution,” Network Science, vol. 2, no. 1, pp. 66–89, 2014.
  39. B. Stabler, H. Bar-Gera, and E. Sall, “Transportation networks for research core team,” 2018.
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