Skip to main content
Log in

Perspective Functions: Properties, Constructions, and Examples

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

Many functions encountered in applied mathematics and in statistical data analysis can be expressed in terms of perspective functions. One of the earliest examples is the Fisher information, which appeared in statistics in the 1920s. We analyze various algebraic and convex-analytical properties of perspective functions and provide general schemes to construct lower semicontinuous convex functions from them. Several new examples are presented and existing instances are featured as special cases.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akian, M., Gaubert, S., Hochart, A.: Minimax representation of nonexpansive functions and application to zero-sum recursive games. J. Convex Anal., to appear

  2. Ali, S.M., Silvey, S.D.: A general class of coefficients of divergence of one distribution from another. J. Roy. Statist. Soc. B28, 131–142 (1966)

    MathSciNet  MATH  Google Scholar 

  3. Alibert, J.J., Bouchitté, G., Fragalà, I., Lucardesi, I.: A nonstandard free boundary problem arising in the shape optimization of thin torsion rods. Interfaces Free Bound. 15, 95–119 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Basseville, M.: Distance measures for signal processing and pattern recognition. Signal Process. 18, 349–369 (1989)

    Article  MathSciNet  Google Scholar 

  5. Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Comm. Contemp. Math. 3, 615–647 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  MATH  Google Scholar 

  7. Ben-Tal, A., Ben-Israel, A., Teboulle, M.: Certainty equivalents and information measures: Duality and extremal principles. J. Math. Anal. Appl. 157, 211–236 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bercher, J.-F.: Some properties of generalized Fisher information in the context of nonextensive thermostatistics. Physica A 392, 3140–3154 (2013)

    Article  MathSciNet  Google Scholar 

  10. Berlinet, A., Vajda, I.: Selection rules based on divergences. Statistics 45, 479–495 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bien, J., Gaynanova, I., Lederer, J., Müller, C.L.: Non-convex global minimization and false discovery rate control for the TREX. J. Comp. Graph. Stat., to appear

  12. Boekee, D.E.: An Extension of the Fisher Information Measure. In: Csiszár, I., Elias, P. (eds.) Topics in Information Theory, János Bolyai Mathematical Society, vol. 16, pp 113–123. North-Holland, Keszthely (1977)

    Google Scholar 

  13. Borwein, J.M., Lewis, A.S., Limber, M.N., Noll, D.: Maximum entropy reconstruction using derivative information, part 2: Computational results. Numer. Math. 69, 243–256 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Borwein, J.M., Lewis, A.S., Noll, D.: Maximum entropy reconstruction using derivative information, part 1: Fisher information and convex duality. Math. Oper. Res. 21, 442–468 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bouchitté, G., Fragalà, I., Lucardesi, I., Seppecher, P.: Optimal thin torsion rods and Cheeger sets. SIAM J. Math. Anal. 44, 483–512 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bougeard, M.L.: Connection between some statistical estimation criteria, lower-C2 functions and Moreau-Yosida approximates. In: Bulletin International Statistical Institute, 47th session, contributed papers, vol. 1, 159–160 (1989)

  17. Bougeard, M.L., Caquineau, C.D.: Parallel proximal decomposition algorithms for robust estimation. Ann. Oper. Res. 90, 247–270 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brasco, L., Buttazzo, G., Santambrogio, F.: A Benamou-Brenier approach to branched transport. SIAM J. Math. Anal. 43, 1023–1040 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Briceño-Arias, L.M., Kalise, D., Silva, F.J.: Proximal methods for stationary mean field games with local couplings, arXiv:1608.07701v1.pdf(2016)

  20. Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. A86, 595–614 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chaux, C., Combettes, P.L., Pesquet, J.-C., Wajs, V.R.: A variational formulation for frame-based inverse problems. Inverse Problems 23, 1495–1518 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, J.-S.: The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem. J. Global Optim. 36, 565–580 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Choquet, G.: Topologie. Masson, Paris (1964). (English translation: Topology. Academic Press, New York, 1966)

    MATH  Google Scholar 

  24. Cohen, M.L.: The Fisher information and convexity. IEEE Trans. Inform. Theory 14, 591–592 (1968)

    Article  Google Scholar 

  25. Combettes, P.L., Müller, C. L.: Perspective functions: Proximal calculus and applications in high-dimensional statistics. J. Math. Anal. Appl., published online 2016-12-15

  26. Csiszár, I.: Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2, 299–318 (1967)

    MathSciNet  MATH  Google Scholar 

  27. Dacorogna, B., Maréchal, P.: The role of perspective functions in convexity, polyconvexity, rank-one convexity and separate convexity. J. Convex Anal. 15, 271–284 (2008)

    MathSciNet  MATH  Google Scholar 

  28. Elad, M., Matalon, B., Zibulevsky, M.: Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization. Appl. Comput. Harmon. Anal. 23, 346–367 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fisher, R.A.: Theory of statistical estimation. Proc. Cambridge. Philos. Soc. 22, 700–725 (1925)

    Article  MATH  Google Scholar 

  30. Fitschen, J.H., Laus, F., Steidl, G.: Transport between RGB images motivated by dynamic optimal transport. J. Math. Imaging Vis. 56, 409–429 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Frieden, B.R., Gatenby, R.A. (eds.): Exploratory Data Analysis Using Fisher Information. Springer, New York (2007)

  32. Hartley, R.I., Zisserman, A.: Multiple view geometry in computer vision, 2nd ed. Cambridge University Press (2003)

  33. Hijazi, H., Bonami, P., Cornuéjols, G., Ouorou, A.: Mixed-integer nonlinear programs featuring “on/off” constraints. Comput. Optim. Appl. 52, 537–558 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, New York (1993)

    MATH  Google Scholar 

  35. Hiriart-Urruty, J.-B., Martínez-Legaz, J.-E.: Convex solutions of a functional equation arising in information theory. J. Math. Anal Appl. 328, 1309–1320 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Huber, P.J.: Robust estimation of a location parameter. Ann. Stat. 35, 73–101 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  37. Huber, P.J., Ronchetti, E.M.: Robust Statistics, 2nd ed. Wiley, New York (2009)

    Book  MATH  Google Scholar 

  38. Jung, M.N., Kirches, C., Sager, S.: On perspective functions and vanishing constraints in mixed-integer nonlinear optimal control. In: Facets of Combinatorial Optimization, pp 387–417. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  39. Lambert-Lacroix, S., Zwald, L.: Robust regression through the Huber’s criterion and adaptive lasso penalty. Electron. J. Stat. 5, 1015–1053 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lambert-Lacroix, S., Zwald, L.: The adaptive BerHu penalty in robust regression. J. Nonparametr. Stat. 28, 487–514 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Laurent, P.-J.: Approximation et Optimisation. Hermann, Paris (1972)

    MATH  Google Scholar 

  42. Lederer, J., Mu̇ller, C. L.: Don’t fall for tuning parameters: Tuning-free variable selection in high dimensions with the TREX, Proc. Twenty-Ninth AAAI Conf. Artif. Intell., pp. 2729–2735. AAAI Press, Austin (2015)

    Google Scholar 

  43. Lemaréchal, C.: Personnal communication

  44. Liese, F., Vajda, I.: On divergences and informations in statistics and information theory. IEEE Trans. Inform. Theory 52, 4394–4412 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lions, P.-L., Toscani, G.: A strengthened central limit theorem for smooth densities. J. Funct. Anal. 129, 148–167 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  46. Micchelli, C.A., Morales, J.M., Pontil, M.: Regularizers for structured sparsity. Adv. Comput. Math. 38, 455–489 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Moehle, N., Boyd, S.: A perspective-based convex relaxation for switched-affine optimal control. Systems Control Lett. 86, 34–40 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. Paris Sér. A Math. 255, 2897–2899 (1962)

    MathSciNet  MATH  Google Scholar 

  49. Ndiaye, E., Fercoq, O., Gramfort, A., Leclère, V., Salmon, J.: Efficient smoothed concomitant lasso estimation for high dimensional regression, arXiv:1606.02702v1.pdf (2016)

  50. Nesterov, Yu., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Philadelphia (1994)

  51. Nikolova, M., Ng, M.K.: Analysis of half-quadratic minimization methods for signal and image recovery. SIAM J. Sci. Comput. 27, 937–966 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Noll, D.: Reconstruction with noisy data: An approach via eigenvalue optimization. SIAM J. Optim. 8, 82–104 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  53. Owen, A.B.: A robust hybrid of lasso and ridge regression. Contemp. Math. 443, 59–71 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. Papadakis, N., Peyré, G., Oudet, E.: Optimal transport with proximal splitting. SIAM J. Imaging Sci. 7, 212–238 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  55. Pardo, L.: Statistical Inference Based on Divergence Measures. Chapman and Hall/CRC, Boca Raton (2006)

    MATH  Google Scholar 

  56. Rey, W.J.J.: Introduction to Robust and Quasi-Robust Statistical Methods. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  57. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  58. Toscani, G.: A strengthened entropy power inequality for log-concave densities. IEEE Trans. Inform. Theory 61, 6550–6559 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. Vapnik, V.N.: The Nature of Statistical Learning Theory, 2nd ed. Springer, New York (2000)

    Book  MATH  Google Scholar 

  60. Villani, C.: Fisher information estimates for Boltzmann’s collision operator. J. Math. Pures. Appl. 77, 821–837 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zach, C., Pollefeys, M.: Practical methods for convex multi-view reconstruction. Lecture Notes Comput. Sci. 6314, 354–367 (2010)

    Article  Google Scholar 

Download references

Acknowledgments

The work of P. L. Combettes was partially supported by the CNRS MASTODONS project under grant 2016TABASCO.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Patrick L. Combettes.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Combettes, P.L. Perspective Functions: Properties, Constructions, and Examples. Set-Valued Var. Anal 26, 247–264 (2018). https://doi.org/10.1007/s11228-017-0407-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-017-0407-x

Keywords

Navigation