The sum of squared logarithms inequality in arbitrary dimensions

Lev Borisov; Patrizio Neff; Suvrit Sra; Christian Thiel

doi:10.1016/j.laa.2016.06.026

The sum of squared logarithms inequality in arbitrary dimensions

Lev Borisov, Patrizio Neff, Suvrit Sra, Christian Thiel

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x,y∈Rⁿ whose elementary symmetric polynomials satisfy e_k(x)≤e_k(y) (for 1≤k<n) and e_n(x)=e_n(y), the inequality ∑_i(log⁡x_i)²≤∑_i(log⁡y_i)² holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f:M⊆Cⁿ→R with f(z)=∑_i(log⁡z_i)² has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. This property leads to our proof. We conclude by providing applications and wider connections of the SSLI.

Original language	English
Pages (from-to)	124-146
Number of pages	23
Journal	Linear Algebra and Its Applications
Volume	528
DOIs	https://doi.org/10.1016/j.laa.2016.06.026
State	Published - 1 Sep 2017
Externally published	Yes

Keywords

Algebraic geometry
Elementary symmetric polynomials
Fundamental theorem of algebra
Geodesics
Hencky energy
Logarithmic strain tensor
Matrix analysis
Polynomials
Positive definite matrices

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10.1016/j.laa.2016.06.026

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title = "The sum of squared logarithms inequality in arbitrary dimensions",

abstract = "We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x,y∈Rn whose elementary symmetric polynomials satisfy ek(x)≤ek(y) (for 1≤kn(x)=en(y), the inequality ∑i(log⁡xi)2≤∑i(log⁡yi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f:M⊆Cn→R with f(z)=∑i(log⁡zi)2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. This property leads to our proof. We conclude by providing applications and wider connections of the SSLI.",

keywords = "Algebraic geometry, Elementary symmetric polynomials, Fundamental theorem of algebra, Geodesics, Hencky energy, Logarithmic strain tensor, Matrix analysis, Polynomials, Positive definite matrices",

author = "Lev Borisov and Patrizio Neff and Suvrit Sra and Christian Thiel",

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TY - JOUR

T1 - The sum of squared logarithms inequality in arbitrary dimensions

AU - Borisov, Lev

AU - Neff, Patrizio

AU - Sra, Suvrit

AU - Thiel, Christian

PY - 2017/9/1

Y1 - 2017/9/1

N2 - We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x,y∈Rn whose elementary symmetric polynomials satisfy ek(x)≤ek(y) (for 1≤kn(x)=en(y), the inequality ∑i(log⁡xi)2≤∑i(log⁡yi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f:M⊆Cn→R with f(z)=∑i(log⁡zi)2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. This property leads to our proof. We conclude by providing applications and wider connections of the SSLI.

AB - We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x,y∈Rn whose elementary symmetric polynomials satisfy ek(x)≤ek(y) (for 1≤kn(x)=en(y), the inequality ∑i(log⁡xi)2≤∑i(log⁡yi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f:M⊆Cn→R with f(z)=∑i(log⁡zi)2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. This property leads to our proof. We conclude by providing applications and wider connections of the SSLI.

KW - Algebraic geometry

KW - Elementary symmetric polynomials

KW - Fundamental theorem of algebra

KW - Geodesics

KW - Hencky energy

KW - Logarithmic strain tensor

KW - Matrix analysis

KW - Polynomials

KW - Positive definite matrices

UR - http://www.scopus.com/inward/record.url?scp=84996618712&partnerID=8YFLogxK

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DO - 10.1016/j.laa.2016.06.026

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SN - 0024-3795

VL - 528

SP - 124

EP - 146

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

The sum of squared logarithms inequality in arbitrary dimensions

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