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Trace of Multi-variable Matrix Functions and its Application to Functions of Graph Spectrum

Subhrajit Bhattacharya

TL;DR

The paper develops a general multivariable matrix-function framework by extending scalar functions to tuples of matrices via joint eigen-decompositions and tensor products, and analyzes how the trace operator preserves monotonicity and convexity. It proves that for $M_l$ Hermitian (or diagonalizable) and suitable regularity of $f$, the trace $\mathrm{Tr}(f((M_l)_{l=1}^m))$ inherits the monotonicity and convexity properties of $f$, with explicit formulas such as $\mathrm{Tr}(f((M_l)_{l=1}^m)) = \sum_{j_1,\dots, j_m} f((\lambda_{l j_l})_{l=1}^m)$. The authors derive concrete results for monomials, polynomials, and general $\mathcal{C}^m$ functions, and specialize to the diagonal case $L$ to obtain criteria for monotone and convex behavior. They then apply the theory to the spectrum of the weighted graph Laplacian, showing that spectrum-based objectives defined as sums of multivariate functions of eigenvalues are monotone or convex in edge weights when $f$ is increasing or convex, respectively. This provides a principled tool for spectral optimization on graphs and other matrix-analytic settings.

Abstract

Matrix extension of a scalar function of a single variable is well-studied in literature. Of particular interest is the trace of such functions. It is known that for diagonalizable matrices, $M$, the function $g(M) = \text{Tr}(f(M)) = \sum_{j=1}^n f(μ_j)$ (where $\{μ_j\}_{j=1,2,\cdots,n}$ are the eigenvalues of $M$) inherits the monotonocity and convexity properties of $f$ (i.e., for $g$ to be convex, $f$ need not be operator convex -- convexity is sufficient). In this paper we formalize the idea of matrix extension of a function of multiple variables, study the monotonicity and convexity properties of the trace, and thus show that a function of form $g(M) = \sum_{j_1=1}^n \sum_{j_2=1}^n \cdots \sum_{j_m=1}^n f(μ_{j_1}, μ_{j_2},\cdots, μ_{j_m})$ also inherits the monotonocity and convexity properties of the multi-variable function, $f$. We apply these results to functions of the spectrum of the weighted Laplacian matrix of undirected, simple graphs.

Trace of Multi-variable Matrix Functions and its Application to Functions of Graph Spectrum

TL;DR

The paper develops a general multivariable matrix-function framework by extending scalar functions to tuples of matrices via joint eigen-decompositions and tensor products, and analyzes how the trace operator preserves monotonicity and convexity. It proves that for Hermitian (or diagonalizable) and suitable regularity of , the trace inherits the monotonicity and convexity properties of , with explicit formulas such as . The authors derive concrete results for monomials, polynomials, and general functions, and specialize to the diagonal case to obtain criteria for monotone and convex behavior. They then apply the theory to the spectrum of the weighted graph Laplacian, showing that spectrum-based objectives defined as sums of multivariate functions of eigenvalues are monotone or convex in edge weights when is increasing or convex, respectively. This provides a principled tool for spectral optimization on graphs and other matrix-analytic settings.

Abstract

Matrix extension of a scalar function of a single variable is well-studied in literature. Of particular interest is the trace of such functions. It is known that for diagonalizable matrices, , the function (where are the eigenvalues of ) inherits the monotonocity and convexity properties of (i.e., for to be convex, need not be operator convex -- convexity is sufficient). In this paper we formalize the idea of matrix extension of a function of multiple variables, study the monotonicity and convexity properties of the trace, and thus show that a function of form also inherits the monotonocity and convexity properties of the multi-variable function, . We apply these results to functions of the spectrum of the weighted Laplacian matrix of undirected, simple graphs.
Paper Structure (10 sections, 12 theorems, 17 equations)

This paper contains 10 sections, 12 theorems, 17 equations.

Table of Contents

  1. Introduction: Matrix Extension of a Scalar Function of a Single Variable
  2. Trace of $f(M)$
  3. Hermitian Matrices
  4. Multi-variable Functions
  5. Introduction: Function of Two Variables
  6. General Case: Multi-Variable Functions
  7. Trace of $f\left((M_l)_{l=1}^m\right)$
  8. Hermitian Matrices
  9. Evaluation on the Diagonal
  10. Application to Function of Graph Spectrum

Key Result

Proposition 1.1

Consider real-eigenvalued diagonaliable matrix, $M\in \mathcal{M}_{\text{diag}^n}$, parametrized by a real parameter $t$ such that the elements of $M(t)$ are $\mathcal{C}^1$ in $t$. Let $f:[a,b]\rightarrow\mathbb{R}$ be a function that is $\mathcal{C}^1$ in $[a,b]$. Then, for all $t$ such that $M(t)\in\mathcal{M}_{\text{diag}^n}([a,b])$, and $\left\|\frac{d M(t)}{dt}\right\|, \|(P(t))^{-1}\|$ are

Theorems & Definitions (31)

  • Definition 1
  • Definition 2: Partial Order based on Positive Semi-definiteness of Difference
  • Definition 3: Extension of a Scalar Function to Diagonalizable Matrices bhatia
  • Example 1.1
  • Proposition 1.1: Derivative of Trace of a Differentiable function of $M$ carlen
  • Corollary 1.2: Monotonic function carlen
  • Example 1.2
  • Lemma 1.3
  • Definition 4: Operator Monotone and Operator Convex Functions bhatia
  • Proposition 1.4: Monotonicity and Convexity Properties of Trace of a Function carlen
  • ...and 21 more