# The sum of squared logarithms inequality in arbitrary dimensions

@article{Borisov2015TheSO, title={The sum of squared logarithms inequality in arbitrary dimensions}, author={Lev Borisov and Patrizio Neff and Suvrit Sra and Christian Thiel}, journal={PAMM}, year={2015}, volume={16} }

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 ≤ k < n) and en(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…

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## References

SHOWING 1-10 OF 48 REFERENCES

### A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm

- Mathematics
- 2013

The unitary polar factor Q = Up in the polar decomposition of Z = Up H is the minimizer over unitary matrices Q for both kLog(QZ)k 2 and its Hermitian part ksym * (Log(QZ))k 2 over both R and C for…

### The minimization of matrix logarithms - on a fundamental property of the unitary polar factor

- Mathematics
- 2013

### Symmetric polynomials in information theory: entropy and subentropy

- MathematicsArXiv
- 2014

It is seen that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory.

### On the sum of squared logarithms inequality and related inequalities

- Mathematics
- 2014

We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors…

### Sum of squared logarithms - an inequality relating positive definite matrices and their matrix logarithm

- Mathematics
- 2013

AbstractLet y1,y2,y3,a1,a2,a3∈(0,∞) be such that y1y2y3=a1a2a3 and
y1+y2+y3≥a1+a2+a3,y1y2+y2y3+y1y3≥a1a2+a2a3+a1a3.
Then
…

### Towards a geometrical interpretation of quantum-information compression (6 pages)

- Mathematics
- 2004

Let S be the von Neumann entropy of a finite ensemble E of pure quantum states. We show that S may be naturally viewed as a function of a set of geometrical volumes in Hilbert space defined by the…

### On the generalised sum of squared logarithms inequality

- Mathematics
- 2014

Assume n≥2$n\geq2$. Consider the elementary symmetric polynomials ek(y1,y2,…,yn)$e_{k}(y_{1},y_{2},\ldots, y_{n})$ and denote by E0,E1,…,En−1$E_{0},E_{1},\ldots,E_{n-1}$ the elementary symmetric…

### Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices

- MathematicsSIAM J. Matrix Anal. Appl.
- 2006

This work defines the Log‐Euclidean mean from a Riemannian point of view, based on a lie group structure which is compatible with the usual algebraic properties of this matrix space and a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure.