[Paper Review] Orlicz addition for measures and an optimization problem for the $f$-divergence

Shaoxiong Hou, Deping Ye|arXiv (Cornell University)|Jun 6, 2016

Point processes and geometric inequalities22 references3 citations

TL;DR

This paper introduces the Orlicz addition of measures and provides a functional interpretation of the $f$-divergence as the first-order variation of total mass under linear Orlicz addition. It establishes a dual functional Orlicz-Brunn-Minkowski inequality and proves that the dual functional Orlicz affine and geominimal surface areas for measures attain their extrema at Gaussian measures under convexity conditions on the function $\phi$, yielding new functional affine isoperimetric inequalities.

ABSTRACT

In this paper, the Orlicz addition of measures is proposed and an interpretation of the $f$-divergence is provided based on a linear Orlicz addition of two measures. Fundamental inequalities, such as, a dual functional Orlicz-Brunn-Minkowski inequality, are established. We also investigate an optimization problem for the $f$-divergence and establish functional affine isoperimetric inequalities for the dual functional Orlicz affine and geominimal surface areas of measures.

Motivation & Objective

• To develop a functional analogue of the dual Orlicz-Brunn-Minkowski theory for measures by introducing Orlicz addition for measures.
• To interpret the $f$-divergence as the first-order variation of the total mass under linear Orlicz addition of two measures.
• To investigate an optimization problem for the $f$-divergence and define dual functional Orlicz affine and geominimal surface areas for measures.
• To establish functional affine isoperimetric inequalities for these new surface area functionals.
• To show that the extremal values of these functionals are attained at Gaussian measures under appropriate convexity and monotonicity conditions on the function $\phi$.

Proposed method

• Define the Orlicz addition of nonnegative measurable functions $p_1, \dots, p_m \in \mathscr{F}$ via an implicit equation involving a function $\varphi \in \Phi_m$ that is strictly increasing in each component and satisfies $\varphi(\mathbf{0}) = 0$.
• Extend the Orlicz addition to measures by defining the total mass of the resulting measure as the $L^1$-norm of the Orlicz sum of the density functions.
• Use the $f$-divergence $D_f(P,Q) = \int_\Omega f(p/q) q \, d\mu$ as the core functional tool, interpreting it as a first-order variation under linear Orlicz addition.
• Establish the dual functional Orlicz-Brunn-Minkowski inequality by showing it is equivalent to Jensen’s inequality for integrals.
• Define the dual functional Orlicz affine surface area $\widetilde{\Omega}_\phi^{orlicz}(P)$ as an infimum of $f$-divergences over a class of measures $\mathscr{D}$, and similarly define the geominimal surface area $\widetilde{G}_\phi^{orlicz}(P)$.
• Prove that for log-concave measures $P$, the functionals $\widetilde{\Omega}_\phi^{orlicz}(P)$ and $\widetilde{G}_\phi^{orlicz}(P)$ attain their maximum at the Gaussian measure $\gamma_n \circ c_1$ when $\phi \in \Phi$.

Experimental results

Research questions

• RQ1 How can the $f$-divergence be interpreted as a first-order variation under a linear Orlicz addition of two measures?
• RQ2 What is the functional analogue of the dual Orlicz-Brunn-Minkowski theory for measures, and how does it relate to the classical Brunn-Minkowski inequality?
• RQ3 Under what conditions do the dual functional Orlicz affine and geominimal surface areas for measures achieve their extrema?
• RQ4 Is the dual functional Orlicz-Brunn-Minkowski inequality equivalent to Jensen’s inequality for integrals?
• RQ5 Can the dual functional Orlicz surface areas be minimized or maximized, and if so, at which measures do they attain their extrema?

Key findings

• The dual functional Orlicz-Brunn-Minkowski inequality is equivalent to Jensen’s inequality for integrals, establishing a deep link between information theory and functional analysis.
• For $\phi \in \Phi$, the dual functional Orlicz affine surface area $\widetilde{\Omega}_\phi^{orlicz}(P)$ is bounded above by $\phi(c_1^n) c_1^{-n} \mu(\gamma_n)$, with equality when $P = \gamma_n \circ c_1$.
• The dual functional Orlicz geominimal surface area $\widetilde{G}_\phi^{orlicz}(P)$ attains its maximum at the Gaussian measure $\gamma_n \circ c_1$ among all log-concave measures $P \in \mathscr{D}$.
• When $\phi$ is strictly convex but not in $\Phi$, the upper bound $\phi(c_1^n) c_1^{-n} \mu(\gamma_n)$ still holds for $\widetilde{\Omega}_\phi^{orlicz}(P)$, with $c_1 > 0$ as defined in Theorem 15.
• The functional affine isoperimetric inequalities for the dual Orlicz surface areas are sharp and are attained precisely at Gaussian measures under the given conditions.
• The proposed Orlicz addition operation generalizes classical $L_p$-addition and provides a flexible framework for studying geometric and information-theoretic functionals under nonlinear, non-homogeneous operations.