[Paper Review] Refinements of the trace inequality of Belmega, Lasaulce and Debbah

Shigeru Furuichi, Minghua Lin|arXiv (Cornell University)|Dec 31, 2009

Mathematical Inequalities and Applications2 references20 citations

TL;DR

This paper refines a matrix trace inequality by Belmega, Lasaulce, and Debbah using trace inequalities involving positive definite and positive semidefinite matrices. It establishes a tighter lower bound via a novel application of trace inequalities and Cauchy-Schwarz-type estimates, proving that the original trace expression is bounded below by the absolute value of a cross-term, and further improves the inequality by introducing a scaling factor of 4, ensuring non-negativity even when individual terms may be negative.

ABSTRACT

In this short paper, we show a certain matrix trace inequality and then give a refinement of the trace inequality proven by Belmega, Lasaulce and Debbah. In addition, we give an another improvement of their trace inequality.

Motivation & Objective

• To refine the trace inequality of Belmega, Lasaulce, and Debbah for positive definite and positive semidefinite matrices.
• To provide a new, simpler proof of the original inequality using trace properties and matrix decomposition.
• To establish a tighter lower bound by introducing an absolute value term involving cross-products of matrix differences.
• To improve the original inequality by scaling the second term by a factor of 4, ensuring non-negativity even when the term alone may be negative.

Proposed method

• Derives a new matrix trace inequality using the Cauchy-Schwarz-type trace inequality for Hermitian and positive semidefinite matrices.
• Applies Lemma 2.2, which compares traces of products involving inverses of matrix sums.
• Uses Lemma 2.3 and Theorem 2.4 to bound sums of traces from below by twice the absolute value of a cross-term.
• Employs matrix decomposition and trace symmetry to express the original inequality in a form amenable to refinement.
• Introduces a weighted trace inequality using parameters a and b, later set to 1 and 4, to strengthen the bound.
• Applies the refined inequality to the original expression, showing that the sum remains non-negative due to the dominance of the cross-term bound.

Experimental results

Research questions

• RQ1 Can the trace inequality of Belmega, Lasaulce, and Debbah be refined to provide a tighter lower bound than the original non-negativity condition?
• RQ2 Is there a way to express the trace difference in terms of an absolute value of a cross-term, improving the inequality's precision?
• RQ3 Can the inequality be strengthened by introducing a scaling factor to ensure non-negativity even when individual components are negative?
• RQ4 Does the use of weighted trace inequalities with parameters a and b lead to a more robust bound in matrix trace expressions?
• RQ5 Can the refined inequality be generalized to include a scaling parameter r, preserving non-negativity for all positive r?

Key findings

• The paper proves a refined trace inequality where the original expression is bounded below by the absolute value of the cross-term Tr[(C−D)(B+D)⁻¹(A−B)(A+C)⁻¹].
• The refined bound is tighter than the original inequality, as it explicitly accounts for the interaction between matrix differences and inverse sums.
• A new inequality is established with a scaling factor of 4 in front of the second trace term, ensuring non-negativity even when the term alone may be negative.
• The proof technique relies on a weighted trace inequality that generalizes the Cauchy-Schwarz approach to matrix traces.
• The refined inequality holds for all positive definite matrices A, B and positive semidefinite matrices C, D, with the bound being sharp in the sense of trace dominance.
• A generalization to a parameter r is derived, showing that the inequality remains valid when scaling the matrices A and B by r, with the inequality holding for all r > 0.