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Sketching for Geometric Problems (Invited Talk)

Author David P. Woodruff

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David P. Woodruff

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David P. Woodruff. Sketching for Geometric Problems (Invited Talk). In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 1:1-1:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ESA.2017.1

Abstract

In this invited talk at the European Symposium on Algorithms (ESA), 2017, I will discuss a tool called sketching, which is a form of data dimensionality reduction, and its applications to several problems in high dimensional geometry. In particular, I will show how to obtain the fastest possible algorithms for fundamental problems such as projection onto a flat, and also study generalizations of projection onto more complicated objects such as the union of flats or subspaces. Some of these problems are just least squares regression problems, with many applications in machine learning, numerical linear algebra, and optimization. I will also discuss low rank approximation, with applications to clustering. Finally I will mention a number of other applications of sketching in machine learning, numerical linear algebra, and optimization.

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Keywords
  • dimensionality reduction
  • low rank approximation
  • projection
  • regression
  • sketching

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References

  1. Haim Avron, Kenneth L. Clarkson, and David P. Woodruff. Sharper bounds for regression and low-rank approximation with regularization. In RANDOM, 2017. Google Scholar
  2. Maria-Florina Balcan, Vandana Kanchanapally, Yingyu Liang, and David Woodruff. Improved distributed principal component analysis. In Advances in Neural Information Processing Systems (NIPS), 2014. URL: https://arxiv.org/pdf/1408.5823.
  3. Maria-Florina Balcan, Yingyu Liang, Le Song, David Woodruff, and Bo Xie. Communication efficient distributed kernel principal component analysis. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pages 725-734. ACM, 2016. URL: https://arxiv.org/pdf/1503.06858.
  4. Jean Bourgain, Sjoerd Dirksen, and Jelani Nelson. Toward a unified theory of sparse dimensionality reduction in euclidean space. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 499-508, 2015. Google Scholar
  5. Christos Boutsidis and David P. Woodruff. Optimal CUR matrix decompositions. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC), pages 353-362. ACM, https://arxiv.org/pdf/1405.7910, 2014.
  6. Christos Boutsidis, David P. Woodruff, and Peilin Zhong. Optimal principal component analysis in distributed and streaming models. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 236-249. ACM, 2016. URL: https://arxiv.org/pdf/1504.06729.
  7. Kenneth L. Clarkson and David P. Woodruff. Low rank approximation and regression in input sparsity time. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 81-90, 2013. URL: https://arxiv.org/pdf/1207.6365.
  8. Kenneth .L Clarkson and David P. Woodruff. Input sparsity and hardness for robust subspace approximation. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 310-329. IEEE, 2015. URL: https://arxiv.org/pdf/1510.06073.
  9. Kenneth L. Clarkson and David P. Woodruff. Low-rank PSD approximation in input-sparsity time. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2061-2072, 2017. Google Scholar
  10. Michael B. Cohen, Sam Elder, Cameron Musco, Christopher Musco, and Madalina Persu. Dimensionality reduction for k-means clustering and low rank approximation. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (STOC), pages 163-172. ACM, 2015. URL: https://arxiv.org/pdf/1410.6801.
  11. Michael B. Cohen, Jelani Nelson, and David P. Woodruff. Optimal approximate matrix product in terms of stable rank. In Proceedings of the 43rd International Colloquium on Automata, Languages and Programming (ICALP), Rome, Italy, July 12-15, 2016, volume 55 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://arxiv.org/pdf/1507.02268, URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.11.
  12. Dan Feldman, Melanie Schmidt, and Christian Sohler. Turning big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1434-1453, 2013. Google Scholar
  13. Ravindran Kannan, Santosh S Vempala, and David P. Woodruff. Principal component analysis and higher correlations for distributed data. In Proceedings of The 27th Conference on Learning Theory (COLT), pages 1040-1057, 2014. Google Scholar
  14. Xingguo Li and David P. Woodruff. Near optimal sketching of low-rank tensor regression, 2017. Manuscript. Google Scholar
  15. Xiangrui Meng and Michael W. Mahoney. Low-distortion subspace embeddings in input-sparsity time and applications to robust linear regression. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 91-100. ACM, 2013. URL: https://arxiv.org/pdf/1210.3135.
  16. Cameron Musco and David P. Woodruff. Sublinear time low-rank approximation of positive semidefinite matrices. CoRR, abs/1704.03371, 2017. Google Scholar
  17. Jelani Nelson and Huy L. Nguyên. OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 117-126. IEEE, 2013. URL: https://arxiv.org/pdf/1211.1002.
  18. Ilya Razenshteyn, Zhao Song, and David P. Woodruff. Weighted low rank approximations with provable guarantees. In Proceedings of the 48th Annual Symposium on the Theory of Computing (STOC), 2016. Google Scholar
  19. Tamás Sarlós. Improved approximation algorithms for large matrices via random projections. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 143-152, 2006. Google Scholar
  20. Zhao Song, David P. Woodruff, and Peilin Zhong. Low rank approximation with entrywise 𝓁₁-norm error. In Proceedings of the 49th Annual Symposium on the Theory of Computing (STOC). ACM, 2017. URL: https://arxiv.org/pdf/1611.00898.
  21. David P. Woodruff. Sketching as a tool for numerical linear algebra. Foundations and Trends in Theoretical Computer Science, 10(1-2):1-157, 2014. Google Scholar
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