Connections between quadratic transform and schur complement revealed
Connections Between Quadratic Transform for Fractional Programming and Schur Complement
Information Theory
Summary
This paper reveals a deep link between two mathematical techniques used in optimization and matrix analysis: the quadratic transform and the Schur complement. The authors show these methods are closely related, providing new ways to understand and generalize both. They apply this insight to a problem in information theory involving noisy communication channels, especially when certain matrices are singular and traditional methods struggle. This work offers a fresh perspective that simplifies complex calculations in such cases.
What this means in practice
- •For communications engineers: Handle singular noise covariance in Gaussian broadcast channel capacity calculations using generalized quadratic transform.
- •For optimization algorithm developers: Design improved fractional programming solvers that accommodate generalized matrix inverses by leveraging the connection with Schur complement.
A theory result. No direct application yet.
Authors
Kaiming Shen, Kareem M. Attiah, Yannan Chen, Wei Yu
Abstract
This paper shows that there are intimate connections between the quadratic transform technique for solving fractional programming (FP) problems and the Schur-complement technique in matrix analysis. We demonstrate that the quadratic transform technique is related to two aspects of the Schur complement: (i) the linear matrix inequality (LMI) condition for positive semidefiniteness and (ii) the matrix determinant formula. Specifically, we establish that the quadratic transform and the Schur-complement LMI condition imply each other. This connection allows us to provide new interpretations of the auxiliary variable in the quadratic transform, and it allows us to rederive the Schur-complement determinant formula. Furthermore, this connection leads to generalizations of the quadratic transform in FP and the Schur-complement LMI that can accommodate generalized matrix inverse. As an application in information theory, we apply the generalized FP framework to the least-favorable-noise minimax formulation of the Gaussian vector broadcast channel sum capacity problem. When the least-favorable noise covariance is singular, matrix-inverse-based Karush-Kuhn-Tucker (KKT) analysis would require a careful analysis of the input and output spaces of the channel. We show using generalized FP that an auxiliary-variable representation of the singular matrix fraction directly yields the reciprocal multiple-access channel and recovers the uplink-downlink duality relation for sum capacity.