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.
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Akian, M., Gaubert, S., Hochart, A.: Minimax representation of nonexpansive functions and application to zero-sum recursive games. J. Convex Anal., to appear
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)
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)
Basseville, M.: Distance measures for signal processing and pattern recognition. Signal Process. 18, 349–369 (1989)
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)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
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)
Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)
Bercher, J.-F.: Some properties of generalized Fisher information in the context of nonextensive thermostatistics. Physica A 392, 3140–3154 (2013)
Berlinet, A., Vajda, I.: Selection rules based on divergences. Statistics 45, 479–495 (2011)
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
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)
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)
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)
Bouchitté, G., Fragalà, I., Lucardesi, I., Seppecher, P.: Optimal thin torsion rods and Cheeger sets. SIAM J. Math. Anal. 44, 483–512 (2012)
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)
Bougeard, M.L., Caquineau, C.D.: Parallel proximal decomposition algorithms for robust estimation. Ann. Oper. Res. 90, 247–270 (1999)
Brasco, L., Buttazzo, G., Santambrogio, F.: A Benamou-Brenier approach to branched transport. SIAM J. Math. Anal. 43, 1023–1040 (2011)
Briceño-Arias, L.M., Kalise, D., Silva, F.J.: Proximal methods for stationary mean field games with local couplings, arXiv:1608.07701v1.pdf(2016)
Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. A86, 595–614 (1999)
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)
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)
Choquet, G.: Topologie. Masson, Paris (1964). (English translation: Topology. Academic Press, New York, 1966)
Cohen, M.L.: The Fisher information and convexity. IEEE Trans. Inform. Theory 14, 591–592 (1968)
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
Csiszár, I.: Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2, 299–318 (1967)
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)
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)
Fisher, R.A.: Theory of statistical estimation. Proc. Cambridge. Philos. Soc. 22, 700–725 (1925)
Fitschen, J.H., Laus, F., Steidl, G.: Transport between RGB images motivated by dynamic optimal transport. J. Math. Imaging Vis. 56, 409–429 (2016)
Frieden, B.R., Gatenby, R.A. (eds.): Exploratory Data Analysis Using Fisher Information. Springer, New York (2007)
Hartley, R.I., Zisserman, A.: Multiple view geometry in computer vision, 2nd ed. Cambridge University Press (2003)
Hijazi, H., Bonami, P., Cornuéjols, G., Ouorou, A.: Mixed-integer nonlinear programs featuring “on/off” constraints. Comput. Optim. Appl. 52, 537–558 (2012)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, New York (1993)
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)
Huber, P.J.: Robust estimation of a location parameter. Ann. Stat. 35, 73–101 (1964)
Huber, P.J., Ronchetti, E.M.: Robust Statistics, 2nd ed. Wiley, New York (2009)
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)
Lambert-Lacroix, S., Zwald, L.: Robust regression through the Huber’s criterion and adaptive lasso penalty. Electron. J. Stat. 5, 1015–1053 (2011)
Lambert-Lacroix, S., Zwald, L.: The adaptive BerHu penalty in robust regression. J. Nonparametr. Stat. 28, 487–514 (2016)
Laurent, P.-J.: Approximation et Optimisation. Hermann, Paris (1972)
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)
Lemaréchal, C.: Personnal communication
Liese, F., Vajda, I.: On divergences and informations in statistics and information theory. IEEE Trans. Inform. Theory 52, 4394–4412 (2006)
Lions, P.-L., Toscani, G.: A strengthened central limit theorem for smooth densities. J. Funct. Anal. 129, 148–167 (1995)
Micchelli, C.A., Morales, J.M., Pontil, M.: Regularizers for structured sparsity. Adv. Comput. Math. 38, 455–489 (2013)
Moehle, N., Boyd, S.: A perspective-based convex relaxation for switched-affine optimal control. Systems Control Lett. 86, 34–40 (2015)
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)
Ndiaye, E., Fercoq, O., Gramfort, A., Leclère, V., Salmon, J.: Efficient smoothed concomitant lasso estimation for high dimensional regression, arXiv:1606.02702v1.pdf (2016)
Nesterov, Yu., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Philadelphia (1994)
Nikolova, M., Ng, M.K.: Analysis of half-quadratic minimization methods for signal and image recovery. SIAM J. Sci. Comput. 27, 937–966 (2005)
Noll, D.: Reconstruction with noisy data: An approach via eigenvalue optimization. SIAM J. Optim. 8, 82–104 (1998)
Owen, A.B.: A robust hybrid of lasso and ridge regression. Contemp. Math. 443, 59–71 (2007)
Papadakis, N., Peyré, G., Oudet, E.: Optimal transport with proximal splitting. SIAM J. Imaging Sci. 7, 212–238 (2014)
Pardo, L.: Statistical Inference Based on Divergence Measures. Chapman and Hall/CRC, Boca Raton (2006)
Rey, W.J.J.: Introduction to Robust and Quasi-Robust Statistical Methods. Springer, Berlin (1983)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Toscani, G.: A strengthened entropy power inequality for log-concave densities. IEEE Trans. Inform. Theory 61, 6550–6559 (2015)
Vapnik, V.N.: The Nature of Statistical Learning Theory, 2nd ed. Springer, New York (2000)
Villani, C.: Fisher information estimates for Boltzmann’s collision operator. J. Math. Pures. Appl. 77, 821–837 (1998)
Zach, C., Pollefeys, M.: Practical methods for convex multi-view reconstruction. Lecture Notes Comput. Sci. 6314, 354–367 (2010)
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The work of P. L. Combettes was partially supported by the CNRS MASTODONS project under grant 2016TABASCO.
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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
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DOI: https://doi.org/10.1007/s11228-017-0407-x