[Paper Review] Level sets and non Gaussian integrals of positively homogeneous functions

Jean B. Lasserre|arXiv (Cornell University)|Oct 30, 2011

Advanced Optimization Algorithms Research11 references4 citations

TL;DR

This paper establishes a convexity property for the volume of sublevel sets of positively homogeneous functions and derives a surprising connection between non-Gaussian integrals and level set volumes via action-independence. It shows that integrating a homogeneous function $ h $ over $ \{\mathbf{x} : g(\mathbf{x}) \leq y\} $ is proportional to the integral of $ h\exp(-g) $ over $ \mathbb{R}^n $, with a constant depending only on homogeneity degrees and the function $ \phi $. The key result is a convexity of volume as a function of polynomial coefficients, enabling new convex optimization schemes for minimum-volume containment problems.

ABSTRACT

We investigate various properties of the sublevel set $\{x \,:\,g(x)\leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(-g)$ on the whole space $R^n$ (a non Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of $g$. In fact, whenever $h$ is nonnegative, the functional $\int ϕ(g(x))h(x)dx$ is a convex function of $g$ for a large class of functions $ϕ:R_+ o R$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set $\{x \,:\,g(x)\leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.

Motivation & Objective

• To investigate the properties of sublevel sets $ \{\mathbf{x} : g(\mathbf{x}) \leq 1\} $ and integrals of $ h $ over these sets when $ g $ and $ h $ are positively homogeneous functions.
• To establish a functional relationship between the integral of $ h $ over sublevel sets and the non-Gaussian integral $ \int_{\mathbb{R}^n} h\exp(-g)\,d\mathbf{x} $, showing action-independence for a broad class of functions $ \phi $.
• To prove that the volume of the sublevel set $ \{\mathbf{x} : g(\mathbf{x}) \leq 1\} $ is a convex function of the coefficients of $ g $ when $ g $ is a nonnegative homogeneous polynomial.
• To develop numerical approximation schemes for computing the volume or integrating $ h $ over sublevel sets, and to solve the minimum-volume sublevel set containment problem via convex optimization.
• To show that for sums of squares of homogeneous polynomials, non-Gaussian integrals exhibit a Gaussian-like behavior, and to identify critical points of a specific functional.

Proposed method

• Derives a general action-independence identity: $ \int_{\mathbb{R}^n} \phi(g(\mathbf{x}))h(\mathbf{x})\,d\mathbf{x} = C(\phi,d,p) \cdot \int_{\mathbb{R}^n} h(\mathbf{x})\exp(-g(\mathbf{x}))\,d\mathbf{x} $, where $ C(\phi,d,p) $ depends only on $ \phi $, degree $ d $, and homogeneity degree $ p $ of $ h $.
• Uses Euler's identity and scaling arguments to prove that the integral $ \int_{\{g(\mathbf{x}) \leq y\}} h(\mathbf{x})\,d\mathbf{x} $ scales as $ y^{(n+p)/d} $, leading to the explicit formula involving the Gamma function.
• Applies Fatou’s Lemma and weak-* compactness in the space of Borel measures to prove convergence of sequences of polynomials and establish optimality in convex optimization problems.
• Proposes two convergent numerical schemes for solving the minimum-volume sublevel set containment problem, based on semidefinite programming and moment relaxation techniques.
• Characterizes the dual cone of $ C_d(\mathbf{K}) $, the set of nonnegative homogeneous polynomials of degree $ d $ on a compact set $ \mathbf{K} $, as the set of moment sequences of finite Borel measures on $ \mathbf{K} $.
• Uses the structure of the dual cone to prove that the optimal solution of the volume minimization problem is attained in the limit of a sequence of feasible polynomials.

Experimental results

Research questions

• RQ1 Can the integral of a positively homogeneous function $ h $ over the sublevel set $ \{\mathbf{x} : g(\mathbf{x}) \leq y\} $ be expressed in terms of the non-Gaussian integral $ \int_{\mathbb{R}^n} h(\mathbf{x})\exp(-g(\mathbf{x}))\,d\mathbf{x} $?
• RQ2 Is the volume of the sublevel set $ \{\mathbf{x} : g(\mathbf{x}) \leq 1\} $ a convex function of the coefficients of $ g $ when $ g $ is a nonnegative homogeneous polynomial?
• RQ3 Can the problem of finding the minimum-volume sublevel set containing a given compact set $ \mathbf{K} $ be formulated as a convex optimization problem?
• RQ4 Do non-Gaussian integrals of homogeneous polynomials that are sums of squares exhibit a Gaussian-like behavior?
• RQ5 What is the dual cone of the set of nonnegative homogeneous polynomials of degree $ d $ on a compact set $ \mathbf{K} $, and how does it relate to moment sequences of measures?

Key findings

• The integral $ \int_{\{g(\mathbf{x}) \leq y\}} h(\mathbf{x})\,d\mathbf{x} $ is proportional to $ \int_{\mathbb{R}^n} h(\mathbf{x})\exp(-g(\mathbf{x}))\,d\mathbf{x} $, with the proportionality constant depending only on the homogeneity degrees of $ g $ and $ h $, and the function $ \phi $.
• For a nonnegative homogeneous polynomial $ g $ of degree $ d $, the volume of $ \{\mathbf{x} : g(\mathbf{x}) \leq 1\} $ is a convex function of the coefficients of $ g $.
• The functional $ \int \phi(g(\mathbf{x}))h(\mathbf{x})\,d\mathbf{x} $ is convex in $ g $ for a large class of functions $ \phi: \mathbb{R}_+ \to \mathbb{R} $, when $ h $ is nonnegative and $ g $ is a nonnegative homogeneous polynomial.
• The minimum-volume sublevel set containing a given compact set $ \mathbf{K} $ is a convex optimization problem, and two convergent numerical schemes are proposed to solve it.
• The dual cone of $ C_d(\mathbf{K}) $, the set of nonnegative homogeneous polynomials of degree $ d $ on a compact set $ \mathbf{K} $, is the set of moment sequences of finite Borel measures on $ \mathbf{K} $.
• For homogeneous polynomials that are sums of squares, the non-Gaussian integral $ \int_{\mathbb{R}^n} \exp(-g(\mathbf{x}))\,d\mathbf{x} $ exhibits a Gaussian-like behavior, and critical points of a specific functional are identified.