On using affine sketches for multiple-response dynamic graph regression | The Journal of Supercomputing Skip to main content
Springer Nature Link
Log in

On using affine sketches for multiple-response dynamic graph regression

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

In the multiple-response dynamic graph regression problem, given a \(n\times d\) matrix embedding \({\varvec{M}}\) of a graph G and a \(n \times d'\) response matrix \({\varvec{B}}\), we want to update the solution of \(argmin_{\varvec{X}} ||{\varvec{M}} \cdot {\varvec{X}} - {\varvec{B}}||_F^2\), after update operations in the graph. In this paper, we present new theoretical results for the update time of this problem. More precisely, we show that using an affine embedding defined as subsampled randomized Hadamard transform and after an edge insertion or an edge deletion, a \(1\pm \epsilon \) approximation to the optimal solution can be updated in \( O\left( d' \epsilon ^{-2}\log ^2 n \right) \) time. Moreover, using an affine embedding defined as CountSketch and after a node insertion or a node deletion or an edge insertion or an edge deletion, an approximate solution can be updated in \(O\left( d' \epsilon ^{-2} \log ^6 \epsilon ^{-1} \right) \) time. To the best of our knowledge, these are the best theoretical results obtained for the update time of approximate multiple-response dynamic graph regression.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Note that there are other notions of embeddings in the field of machine learning with graphs, such as embeddings in frequent pattern discovery [26, 27], that are different from affine (and subspace) embeddings used in this paper.

References

  1. Kovac Arne, Smith Andrew DAC (2011) Nonparametric regression on a graph. J Comput Gr Stat 20(2):432–447

    Article  MathSciNet  Google Scholar 

  2. Chehreghani MH (2019) On the theory of dynamic graph regression problem. CoRR, abs/1903.10699

  3. Mostafa Haghir Chehreghani (2021) Sublinear update time randomized algorithms for dynamic graph regression. Appl Math Comput 410:126434

    MathSciNet  MATH  Google Scholar 

  4. Petros D, Mahoney Michael W, Muthukrishnan S, Tamás Sarlós (2011) Faster least squares approximation. Numerische Mathematik 117(2):219–249

    Article  MathSciNet  MATH  Google Scholar 

  5. Clarkson Kenneth L, Woodruff David P (2017) Low-rank approximation and regression in input sparsity time. J ACM 63(6):54

    MathSciNet  MATH  Google Scholar 

  6. Kleinberg Jon M, Tardos Éva (2002) Approximation algorithms for classification problems with pairwise relationships: metric labeling and markov random fields. J ACM 49(5):616–639

    Article  MathSciNet  MATH  Google Scholar 

  7. Herbster M, Pontil M, Wainer L (2005) Online learning over graphs. In: Luc De Raedt and Stefan Wrobel, editors, Machine Learning, Proceedings of the Twenty-Second International Conference (ICML 2005), Bonn, Germany, August 7-11, 2005, volume 119 of ACM International Conference Proceeding Series, pages 305–312. ACM

  8. Herbster M, Lever G, Pontil M (2008) Online prediction on large diameter graphs. In: Daphne Koller, Dale Schuurmans, Yoshua Bengio, and Léon Bottou, editors, Advances in Neural Information Processing Systems 21, Proceedings of the Twenty-Second Annual Conference on Neural Information Processing Systems, Vancouver, British Columbia, Canada, December 8-11, 2008, pages 649–656. Curran Associates, Inc.,

  9. Herbster Mark, Lever Guy (2009) Predicting the labelling of a graph via minimum \(p\)-seminorm interpolation. In: COLT 2009 - The 22nd Conference on Learning Theory, Montreal, Quebec, Canada, June 18-21, 2009

  10. Herbster Mark, Pasteris Stephen, Pontil Massimiliano (2015) Predicting a switching sequence of graph labelings. J Mach Learn Res 16:2003–2022

    MathSciNet  MATH  Google Scholar 

  11. Culp Mark, Michailidis George, Johnson Kjell (2009) On multi-view learning with additive models. Annals Appl Stat 3(1):292–318

    Article  MathSciNet  MATH  Google Scholar 

  12. Velickovic P, Cucurull G, Casanova A, Romero A, Liò P, Bengio Y (2018) Graph attention networks. In: 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings. OpenReview.net

  13. Mostafa Haghir Chehreghani (2022) Half a decade of graph convolutional networks. Nat Mach Intell 4(3):192–193

    Article  MATH  Google Scholar 

  14. Wang H, Leskovec J (2022) Combining graph convolutional neural networks and label propagation. ACM Trans Inf Syst 40(4):73

    Article  Google Scholar 

  15. Chehreghani Mostafa Haghir (2021) Dynamical algorithms for data mining and machine learning over dynamic graphs. WIREs Data Mining Knowl Discov 11(2):e1393

    MATH  Google Scholar 

  16. Dhanjal Charanpal, Gaudel Romaric, Clémençon Stéphan (2014) Efficient eigen-updating for spectral graph clustering. Neurocomputing 131:440–452

    Article  Google Scholar 

  17. Yao Y, Holder LB (2014) Scalable svm-based classification in dynamic graphs. In: Ravi Kumar, Hannu Toivonen, Jian Pei, Joshua Zhexue Huang, and Xindong Wu, editors, 2014 IEEE International Conference on Data Mining, ICDM 2014, Shenzhen, China, December 14-17, 2014, pages 650–659. IEEE Computer Society

  18. Ailon N, Liberty E (2008) Fast dimension reduction using rademacher series on dual BCH codes. In: Shang-Hua Teng, editor. Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008

  19. Pareja A, Domeniconi G, Chen J, Ma T, Suzumura T, Kanezashi H, Kaler T, Schardl TB, Leiserson CE (2020) Evolvegcn: Evolving graph convolutional networks for dynamic graphs. In: The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020, pages 5363–5370. AAAI Press

  20. You J, Du T, Leskovec J (2022) ROLAND: graph learning framework for dynamic graphs. In: Aidong Zhang and Huzefa Rangwala, (eds.), KDD ’22: The 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Washington, DC, USA, August 14 - 18, 2022, pages 2358–2366. ACM

  21. Qin T, Liu T-Y, Zhang X-D, Wang D-S, Li H (2008) Global ranking using continuous conditional random fields. In: Daphne Koller, Dale Schuurmans, Yoshua Bengio, and Léon Bottou, (eds.), Advances in Neural Information Processing Systems 21, Proceedings of the Twenty-Second Annual Conference on Neural Information Processing Systems, Vancouver, British Columbia, Canada, December 8-11, 2008, pages 1281–1288. Curran Associates, Inc

  22. Sohn K-A, Kim S (2012) Joint estimation of structured sparsity and output structure in multiple-output regression via inverse-covariance regularization. In: Neil D. Lawrence and Mark A. Girolami, (eds.), Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, AISTATS 2012, La Palma, Canary Islands, Spain, April 21-23, 2012, volume 22 of JMLR Proceedings, pages 1081–1089. JMLR.org,

  23. Wytock M, Kolter JZ (2013) Sparse gaussian conditional random fields: Algorithms, theory, and application to energy forecasting. In: Proceedings of the 30th International Conference on Machine Learning, ICML 2013, Atlanta, GA, USA, 16-21 June 2013

  24. Han C, Cao XH, Stanojevic M, Ghalwash MF, Obradovic Z (2019) Temporal graph regression via structure-aware intrinsic representation learning. In: Tanya Y. Berger-Wolf and Nitesh V. Chawla, (eds.), Proceedings of the 2019 SIAM International Conference on Data Mining, SDM 2019, Calgary, Alberta, Canada, May 2-4, 2019, pages 360–368. SIAM

  25. Peng Hao, Wang Hongfei, Bowen Du, Bhuiyan Md Zakirul Alam, Ma Hongyuan, Liu Jianwei, Wang Lihong, Yang Zeyu, Linfeng Du, Wang Senzhang, Philip SYu (2020) Spatial temporal incidence dynamic graph neural networks for traffic flow forecasting. Inf Sci 521:277–290

    Article  Google Scholar 

  26. Mostafa Haghir Chehreghani and Maurice Bruynooghe (2016) Mining rooted ordered trees under subtree homeomorphism. Data Min Knowl Discov 30(5):1249–1272

    Article  MathSciNet  MATH  Google Scholar 

  27. Haghir Chehreghani Mostafa, Talel Abdessalem, Albert Bifet, Meriem Bouzbila (2020) Sampling informative patterns from large single networks. Future Gener Comput Syst 106:653–658

    Article  Google Scholar 

  28. Boutsidis Christos, Gittens Alex (2013) Improved matrix algorithms via the subsampled randomized hadamard transform. SIAM J Matrix Anal Appl 34(3):1301–1340

    Article  MathSciNet  MATH  Google Scholar 

  29. Greville TNE (1960) Some applications of the pseudoinverse of a matrix. SIAM Rev 2:15–22

    Article  MathSciNet  MATH  Google Scholar 

  30. Meyer Jr Carl D (1973) Generalized inversion of modified matrices. SIAM J Appl Math 24(3):315–323

    Article  MathSciNet  MATH  Google Scholar 

  31. van den Brand Jan (2020) Unifying matrix data structures: Simplifying and speeding up iterative algorithms. CoRR, arXiv:abs/2010.13888

  32. Meng X, Mahoney MW (2013) Low-distortion subspace embeddings in input-sparsity time and applications to robust linear regression. In: Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, (eds.), Symposium on Theory of Computing Conference, STOC’13, Palo Alto, CA, USA, June 1-4, 2013, pages 91–100. ACM

  33. Nelson J, Nguyen HL (2013) OSNAP: faster numerical linear algebra algorithms via sparser subspace embeddings. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 117–126. IEEE Computer Society

Download references

Author information

Authors and Affiliations

  1. Department of Computer Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

    Mostafa Haghir Chehreghani

Authors
  1. Mostafa Haghir Chehreghani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mostafa Haghir Chehreghani.

Ethics declarations

Conflict of interest

None.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Haghir Chehreghani, M. On using affine sketches for multiple-response dynamic graph regression. J Supercomput 79, 5139–5153 (2023). https://doi.org/10.1007/s11227-022-04865-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-022-04865-x

Keywords

Search

Navigation