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Fast projection onto the simplex and the \(\pmb {l}_\mathbf {1}\) ball

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Abstract

A new algorithm is proposed to project, exactly and in finite time, a vector of arbitrary size onto a simplex or an \(l_1\)-norm ball. It can be viewed as a Gauss–Seidel-like variant of Michelot’s variable fixing algorithm; that is, the threshold used to fix the variables is updated after each element is read, instead of waiting for a full reading pass over the list of non-fixed elements. This algorithm is empirically demonstrated to be faster than existing methods.

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Fig. 1
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Notes

  1. Actually, Algorithm 4 is an improvement of Michelot’s algorithm, with the test “\(>\)” instead of “\(\ge \)” at step 2.1. This modification has been proposed in [13, Sect. 5.7]. Algorithm 4 is also the same algorithm as in [9].

References

  1. Bentley, J.L., McIlroy, M.D.: Engineering a sort function. Softw. Pract. Exp. 23(11), 1249–1265 (1993)

    Article  Google Scholar 

  2. Bioucas-Dias, J.M., Plaza, A., Dobigeon, N., Parente, M., Du, Q., Gader, P., Chanussot, J.: Hyperspectral unmixing overview: geometrical, statistical, and sparse regression-based approaches. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 5(2), 354–379 (2012)

    Article  Google Scholar 

  3. Blondel, M., Fujino, A., Ueda, N.: Large-scale multiclass support vector machine training via Euclidean projection onto the simplex. In: Proceedings of the 22th International Conference on Pattern Recognition (ICPR), pp. 1289–1294 (2014)

  4. Blum, M., Floyd, R.W., Pratt, V.R., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. J. Comput. Syst. Sci. 7(4), 448–461 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brodie, J., Daubechies, I., Mol, C.D., Giannone, D., Loris, I.: Sparse and stable Markowitz portfolios. Proc. Natl. Acad. Sci. 106(30), 12267–12272 (2009)

    Article  MATH  Google Scholar 

  6. Cominetti, R., Mascarenhas, W.F., Silva, P.J.S.: A Newton’s method for the continuous quadratic knapsack problem. Math. Program. Comp. 6, 151–169 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duchi, J., Shalev-Shwartz, S., Singer, Y., Chandra, T.: Efficient projections onto the \(\ell _1\)-ball for learning in high dimensions. In: Proceedings of the 25th International Conference on Machine learning (ICML) (2008)

  8. Fadili, J., Peyré, G.: Total variation projection with first order schemes. IEEE Trans. Image Process. 20(3), 657–669 (2011)

    Article  MathSciNet  Google Scholar 

  9. Gong, P., Gai, K., Zhang, C.: Efficient Euclidean projections via piecewise root finding and its application in gradient projection. Neurocomputing 74, 2754–2766 (2011)

    Article  Google Scholar 

  10. Held, M., Wolfe, P., Crowder, H.: Validation of subgradient optimization. Math. Program. 6, 62–88 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kiwiel, K.C.: On Floyd and Rivest’s SELECT algorithm. Theor. Comput. Sci. 347, 214–238 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kiwiel, K.C.: Breakpoint searching algorithms for the continuous quadratic knapsack problem. Math. Program. Ser. A 112, 473–491 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kiwiel, K.C.: Variable fixing algorithms for the continuous quadratic knapsack problem. J. Optim. Theory Appl. 136, 445–458 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Knuth, D.E.: The Art of Computer Programming, vol. 2, 3rd edn, p. 232. Addison-Wesley, Boston (1998)

    Google Scholar 

  15. Lellmann, J., Kappes, J.H., Yuan, J., Becker, F., Schnörr, C.: Convex multi-class image labeling by simplex-constrained total variation. In: Proceedings of Scale Space and Variational Methods in Computer Vision (SSVM), vol. 5567, pp. 150–162 (2009)

  16. Liu, J., Ye, J.: Efficient Euclidean projections in linear time. In: Proceedings of the 26th International Conference Machine Learning (ICML) (2009)

  17. Michelot, C.: A finite algorithm for finding the projection of a point onto the canonical simplex of \({\mathbb{R}}^n\). J. Optim. Theory Appl. 50(1), 195–200 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Patriksson, M.: A survey on the continuous nonlinear resource allocation problem. Eur. J. Oper. Res. 185, 1–46 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Patriksson, M., Strömberg, C.: Algorithms for the continuous nonlinear resource allocation problem—new implementations and numerical studies. Eur. J. Oper. Res. 243(3), 703–722 (2015)

    Article  MathSciNet  Google Scholar 

  20. van den Berg, E., Friedlander, M.P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31, 890–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Grenoble Alpes, GIPSA-Lab, 38000, Grenoble, France

    Laurent Condat

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  1. Laurent Condat
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Correspondence to Laurent Condat.

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This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025).

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Condat, L. Fast projection onto the simplex and the \(\pmb {l}_\mathbf {1}\) ball. Math. Program. 158, 575–585 (2016). https://doi.org/10.1007/s10107-015-0946-6

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