Orlicz addition for measures and an optimization problem for the \(f\)-divergence. (English) Zbl 1436.52009
Summary: This paper provides a functional analogue of the recently initiated dual Orlicz-Brunn-Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz-Brunn-Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous \(f\)-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz-Brunn-Minkowski inequality. An optimization problem for the \(f\)-divergence is proposed, and related functional affine isoperimetric inequalities are established.
MSC:
| 52A41 | Convex functions and convex programs in convex geometry |
| 26B25 | Convexity of real functions of several variables, generalizations |
| 28A25 | Integration with respect to measures and other set functions |
| 60A10 | Probabilistic measure theory |
| 94A15 | Information theory (general) |
| 94A17 | Measures of information, entropy |
Keywords:
affine isoperimetric inequality; dual Brunn-Minkowski theory; \(f\)-divergence; optimization problem for the \(f\)-divergenceReferences:
| [1] | Ali, M. and Silvey, D., A general class of coefficients of divergence of one distribution from another. J. R. Statist. Soc. Ser. B28(1966), 131-142. · Zbl 0203.19902 |
| [2] | Artstein-Avidan, S., Klartag, B., and Milman, V., The Santaló point of a function, and a functional form of the Santaló inequality. Mathematika51(2004), 33-48. · Zbl 1121.52021 |
| [3] | Artstein-Avidan, S., Klartag, B., Schütt, C., and Werner, E., Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality. J. Funct. Anal.262(2012), 4181-4204. · Zbl 1255.52010 |
| [4] | Barron, A., Györfi, L., and Van Der Meulen, E., Distribution estimates consistent in total variation and two types of information divergence. IEEE Trans. Inform. Theory38(1990), 1437-1454. · Zbl 0765.62007 |
| [5] | Bhattacharyya, A., On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc.35(1943), 99-109. · Zbl 0063.00364 |
| [6] | Blaschke, W., Vorlesungen über Differentialgeometrie II. In: Affine Differentialgeometrie. Springer-Verlag, Berlin, 1923. · JFM 49.0499.01 |
| [7] | Caglar, U., Fradelizi, M., Guedon, O., Lehec, J., Schuett, C., and Werner, E., Functional versions of L_p-affine surface area and entropy inequalities. Int. Math. Res. Not. IMRN2016(2016), 1223-1250. · Zbl 1337.52008 |
| [8] | Caglar, U. and Werner, E., Divergence for s-concave and log concave functions. Adv. Math.257(2014), 219-247. · Zbl 1310.26016 |
| [9] | Caglar, U. and Ye, D., Affine isoperimetric inequalities in the functional Orlicz-Brunn-Minkowski theory. Adv. Appl. Math.81(2016), 78-114. · Zbl 1357.52001 |
| [10] | Cover, T. and Thomas, J., Elements of information theory. Second ed., Wiley-Interscience, Hoboken, NJ, 2006. · Zbl 1140.94001 |
| [11] | Csiszár, I., Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Publ. Math. Inst. Hungar. Acad. Sci. Ser. A8(1963), 84-108. · Zbl 0124.08703 |
| [12] | Csiszár, I., I-divergence geometry of probability distributions and minimization problems. Ann. Probab.3(1975), 146-158. · Zbl 0318.60013 |
| [13] | Fradelizi, M. and Meyer, M., Some functional forms of Blaschke-Santaló inequality. Math. Z.256(2007), 379-395. · Zbl 1128.52007 |
| [14] | Frank, N. and Nock, R., On the Chi square and higher-order Chi distances for approximating f-Divergences. IEEE Signal Process. Lett.21(2014), 10-13. |
| [15] | Gardner, R., A positive answer to the Busemann-Petty problem in three dimensions. Ann. of Math.140(1994), 435-447. · Zbl 0826.52010 |
| [16] | Gardner, R., The Brunn-Minkowski inequality. Bull. Amer. Math. Soc.39(2002), 355-405. · Zbl 1019.26008 |
| [17] | Gardner, R., Hug, D., and Weil, W., The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities. J. Differential Geom.97(2014), 427-476. · Zbl 1303.52002 |
| [18] | Gardner, R., Hug, D., Weil, W., and Ye, D., The dual Orlicz-Brunn-Minkowski theory. J. Math. Anal. Appl.430(2015), 810-829. · Zbl 1320.52008 |
| [19] | Gardner, R. and Kiderlen, M., Operations between functions. Comm. Anal. Geom.26(2018), 787-855. · Zbl 1400.26027 |
| [20] | Gardner, R., Koldobski, A., and Schlumprecht, T., An analytic solution to the Busemann-Petty problem on sections of convex bodies. Ann. of Math.149(1999), 691-703. · Zbl 0937.52003 |
| [21] | Good, I., Maximum entropy for hypothesis formulation, especially for multidimensional contingency tables. Ann. Math. Statist.34(1963), 911-934. · Zbl 0143.40705 |
| [22] | Haberl, C. and Parapatits, L., The centro-affine Hadwiger theorem. J. Amer. Math. Soc.27(2014), 685-705. · Zbl 1319.52006 |
| [23] | Harremoes, P. and Topsoe, F., Inequalities between entropy and the index of coincidence derived from information diagrams. IEEE Trans. Inform. Theory47(2001), 2944-2960. · Zbl 1017.94005 |
| [24] | Ireland, C. and Kullback, S., Contingency tables with given marginals. Biometrika55(1968), 179-188. · Zbl 0155.26701 |
| [25] | Jaynes, E., Information theory and statistical mechanics. Phys. Rev.106(1957), 620-630. · Zbl 0084.43701 |
| [26] | Kullback, S. and Leibler, R., On information and sufficiency. Ann. Math. Statist.22(1951), 79-86. · Zbl 0042.38403 |
| [27] | Lehec, J., Partitions and functional Santaló inequalities. Arch. Math. (Basel)92(2009), 89-94. · Zbl 1168.39010 |
| [28] | Liese, F. and Vajda, I., On divergences and information in statistics and information theory. IEEE Trans. Inform. Theory52(2006), 4394-4412. · Zbl 1287.94025 |
| [29] | Ludwig, M., General affine surface areas. Adv. Math.224(2010), 2346-2360. · Zbl 1198.52004 |
| [30] | Ludwig, M. and Reitzner, M., A characterization of affine surface area. Adv. Math.147(1999), 138-172. · Zbl 0947.52003 |
| [31] | Ludwig, M. and Reitzner, M., A classification of SL (n) invariant valuations. Ann. of Math.172(2010), 1219-1267. · Zbl 1223.52007 |
| [32] | Lutwak, E., Intersection bodies and dual mixed volume. Adv. Math.71(1988), 232-261. · Zbl 0657.52002 |
| [33] | Lutwak, E., The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv. Math.118(1996), 244-294. · Zbl 0853.52005 |
| [34] | Morimoto, T., Markov processes and the H-theorem. J. Phys. Soc. Japan18(1963), 328-331. · Zbl 0156.23804 |
| [35] | Österreicher, F. and Vajda, I., A new class of metric divergences on probability spaces and its applicability in statistics. Ann. Inst. Statist. Math.55(2003), 639-653. · Zbl 1052.62002 |
| [36] | Petty, C. M., Geominimal surface area. Geom. Dedicata3(1974), 77-97. · Zbl 0285.52006 |
| [37] | Rényi, A., On measures of entropy and information. , Univ. California Press, Berkeley, CA, 1961, pp. 547-561. · Zbl 0106.33001 |
| [38] | Sanov, I., On the probability of large deviations of random magnitudes. (Russian), Mat. Sbornik42(1957), 11-44. |
| [39] | Schütt, C. and Werner, E., Surface bodies and p-affine surface area. Adv. Math.187(2004), 98-145. · Zbl 1089.52002 |
| [40] | Werner, E., Rényi divergence and L_p-affine surface area for convex bodies. Adv. Math.230(2012), 1040-1059. · Zbl 1251.52003 |
| [41] | Xi, D., Jin, H., and Leng, G., The Orlicz Brunn-Minkowski inequality. Adv. Math.260(2014), 350-374. · Zbl 1357.52004 |
| [42] | Ye, D., New Orlicz affine isoperimetric inequalities. J. Math. Anal. Appl.427(2015), 905-929. · Zbl 1316.52014 |
| [43] | Ye, D., L_p Geominimal surface areas and their inequalities. Int. Math. Res. Not. IMRN2015(2015), 2465-2498. · Zbl 1326.53086 |
| [44] | Ye, D., Dual Orlicz-Brunn-Minkowski theory: dual Orlicz L_𝜙 affine and geominimal surface areas. J. Math. Anal. Appl.443(2016), 352-371. · Zbl 1355.52003 |
| [45] | Zhang, G., A positive answer to the Busemann-Petty problem in four dimensions. Ann. of Math.149(1999), 535-543. · Zbl 0937.52004 |
| [46] | Zhu, B., Xing, S., and Ye, D., The dual Orlicz-Minkowski problem. J. Geom. Anal.28(2018), 3829-3855. · Zbl 1416.52009 |
| [47] | Zhu, B., Zhou, J., and Xu, W., Dual Orlicz-Brunn-Minkowski theory. Adv. Math.264(2014), 700-725. · Zbl 1307.52004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
