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A Unique Inverse Decomposition of Positive Definite Matrices under Linear Constraints

Yan Dolinsky, Or Zuk

TL;DR

The paper introduces a unique inverse-based decomposition of a symmetric positive definite matrix $A$ into $A=(B^{*})^{-1}+C^{*}$ with $C^{*}\in S$ and $B^{*}\in S^{\perp}$, proving global existence, uniqueness, and smooth dependence under the nondegeneracy condition $S\cap\mathbb{S}^n_{+}=\{0\}$. It recasts the problem as a strictly convex log-determinant optimization, derives a dual formulation, and establishes a suite of structural properties including symmetry inheritance and stability. It then develops feasibility-preserving Newton-CG algorithms with detailed derivative formulas and complexity analyses, including extensions to primal, dual, and group-invariant settings. Finally, it demonstrates relevance to exponential utility maximization in Gaussian finance models, linking the decomposition to pricing, hedging, and information-structure constraints, with numerical studies on mixed fractional Brownian motion. Overall, the framework provides a rigorous, structure-exploiting approach to structured covariance/precision-matrix decomposition with practical computational strategies and finance applications.

Abstract

We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse component is required to belong to the orthogonal complement of that subspace with respect to the trace inner product. Under a sharp nondegeneracy condition on the subspace, we show that every positive definite matrix admits a \emph{unique} decomposition of this form. This decomposition admits a variational characterization as the unique minimizer of a strictly convex log-determinant optimization problem, which in turn yields a natural dual formulation that can be efficiently exploited computationally. We derive several properties, including the stability of the decomposition. We further develop feasibility-preserving Newton-type algorithms with provable convergence guarantees and analyze their per-iteration complexity in terms of algebraic properties of the decomposed matrix and the underlying subspace. Finally, we show that the proposed decomposition arises naturally in exponential utility maximization, a central problem in mathematical finance.

A Unique Inverse Decomposition of Positive Definite Matrices under Linear Constraints

TL;DR

The paper introduces a unique inverse-based decomposition of a symmetric positive definite matrix into with and , proving global existence, uniqueness, and smooth dependence under the nondegeneracy condition . It recasts the problem as a strictly convex log-determinant optimization, derives a dual formulation, and establishes a suite of structural properties including symmetry inheritance and stability. It then develops feasibility-preserving Newton-CG algorithms with detailed derivative formulas and complexity analyses, including extensions to primal, dual, and group-invariant settings. Finally, it demonstrates relevance to exponential utility maximization in Gaussian finance models, linking the decomposition to pricing, hedging, and information-structure constraints, with numerical studies on mixed fractional Brownian motion. Overall, the framework provides a rigorous, structure-exploiting approach to structured covariance/precision-matrix decomposition with practical computational strategies and finance applications.

Abstract

We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse component is required to belong to the orthogonal complement of that subspace with respect to the trace inner product. Under a sharp nondegeneracy condition on the subspace, we show that every positive definite matrix admits a \emph{unique} decomposition of this form. This decomposition admits a variational characterization as the unique minimizer of a strictly convex log-determinant optimization problem, which in turn yields a natural dual formulation that can be efficiently exploited computationally. We derive several properties, including the stability of the decomposition. We further develop feasibility-preserving Newton-type algorithms with provable convergence guarantees and analyze their per-iteration complexity in terms of algebraic properties of the decomposed matrix and the underlying subspace. Finally, we show that the proposed decomposition arises naturally in exponential utility maximization, a central problem in mathematical finance.
Paper Structure (19 sections, 3 theorems, 64 equations, 2 figures, 2 tables, 3 algorithms)

This paper contains 19 sections, 3 theorems, 64 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Decomposition Theorem
  3. A dual variational formulation
  4. Primal formulation.
  5. Dual formulation.
  6. Properties of the decomposition
  7. Algorithms for Computing the Decomposition
  8. Derivatives
  9. Newton-CG with feasibility-preserving backtracking
  10. Convergence.
  11. Per-iteration cost and algorithmic considerations
  12. Newton-CG for the dual problem
  13. Parameterization and derivatives.
  14. Group-invariant optimization
  15. Complexity.
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $A\in\mathbb{S}^n_{++}$. Let $S\subset\mathbb{S}^n$ be a linear subspace, and let the orthogonal complement with respect to the trace inner product $\langle X,Y\rangle:=\mathop{\mathrm{tr}}\nolimits(XY)$. Assume that $S$ contains no nonzero positive semidefinite matrix, that is, Then there exists a unique pair $(B^\ast \in \mathbb{S}^n_{++} \cap S^{\perp}, C^\ast \in S)$ such that

Figures (2)

  • Figure 1: Optimal investment value $v_N^*$ as a function of the Hurst parameter $\mathcal{H}$ in $[0.5,1)$ for three information structures: full information (red), Markovian strategies (blue), and observation of $S$ only (green).
  • Figure 1: Block-permutation invariant decomposition (Example III), with $n=25$ (five blocks of size five). Top left: input matrix $A$. Top right: constraint matrix $C\in S$, block-constant on symmetric block pairs; red stars indicate the active block pairs. Bottom left: precision matrix $B\in S^\perp$. Bottom right: covariance matrix $B^{-1}$, exhibiting exact zeros on block pairs for which one block is always active in $C$ (red circles).

Theorems & Definitions (9)

  • Theorem 2.1: General structured decomposition
  • Lemma 2.2
  • Proof 1
  • Proof 2
  • Remark 2.3
  • Remark 3.1
  • Proposition 4.1
  • Remark 4.2
  • Proof 3