Prerequisites & Notation

Before You Begin

This chapter is the algorithmic heart of the RIS optimization thread. We go into the internals of the three standard methods for the unit-modulus QCQP. Prior familiarity with semidefinite programming (Telecom Ch. 3) is most useful; Riemannian geometry is introduced here from the ground up.

  • Semidefinite programming: feasibility, duality, interior-point methods(Review ch03)

    Self-check: Can you state the SDP dual of min⁑tr(CX)\min \text{tr}(\mathbf{C}\mathbf{X}) s.t. Xβͺ°0,A(X)=b\mathbf{X} \succeq 0, \mathcal{A}(\mathbf{X}) = \mathbf{b}?

  • Rank-1 PSD matrices: X=xxH\mathbf{X} = \mathbf{x}\mathbf{x}^H

    Self-check: Why is X=xxHβͺ°0\mathbf{X} = \mathbf{x}\mathbf{x}^H \succeq 0 and rank(X)=1\text{rank}(\mathbf{X}) = 1?

  • Gaussian random vectors on the complex sphere

    Self-check: If z∼CN(0,X)\mathbf{z} \sim \mathcal{CN}(\mathbf{0}, \mathbf{X}) with X=xxH\mathbf{X} = \mathbf{x}\mathbf{x}^H, what is the distribution of z\mathbf{z}?

  • Gradient and Hessian calculus for functions on CN\mathbb{C}^N(Review ch01)

    Self-check: For f(Ο•)=∣aHΟ•βˆ£2f(\boldsymbol{\phi}) = |\mathbf{a}^H \boldsymbol{\phi}|^2, compute βˆ‡Ο•f\nabla_{\boldsymbol{\phi}} f.

  • Block coordinate descent convergence (Ch. 5)(Review ch05)

    Self-check: What regularity conditions guarantee BCD convergence to a stationary point?

Notation for This Chapter

Algorithm-specific notation. We focus on the unit-modulus quadratic problem throughout, so the basic RIS symbols carry over from Ch. 5 and are supplemented by SDR- and manifold-specific terms.

SymbolMeaningIntroduced
X∈C(N+1)Γ—(N+1)\mathbf{X} \in \mathbb{C}^{(N+1) \times (N+1)}Lifted PSD variable in SDR: X=Ο•~Ο•~H\mathbf{X} = \boldsymbol{\tilde{\phi}} \boldsymbol{\tilde{\phi}}^H when rank 1s02
M\mathcal{M}Complex unit-modulus manifold: {Ο•βˆˆCN:βˆ£Ο•n∣=1βˆ€n}\{\boldsymbol{\phi} \in \mathbb{C}^N : |\phi_n| = 1 \forall n\}s03
TΟ•MT_{\boldsymbol{\phi}} \mathcal{M}Tangent space at Ο•βˆˆM\boldsymbol{\phi} \in \mathcal{M}s03
βˆ‡Ο•Riemf\nabla^{\text{Riem}}_{\boldsymbol{\phi}} fRiemannian gradient: Euclidean gradient projected onto tangent spaces03
RΟ•(ΞΎ)R_{\boldsymbol{\phi}}(\boldsymbol{\xi})Retraction operator: maps tangent vector ΞΎ\boldsymbol{\xi} back to manifolds03
LLNumber of Gaussian randomization samples in SDRs02
A,B\mathbf{A}, \mathbf{B}Quadratic-form matrices defining the passive objective (Ο•~HAΟ•~/Ο•~HBΟ•~\boldsymbol{\tilde{\phi}}^H \mathbf{A} \boldsymbol{\tilde{\phi}} / \boldsymbol{\tilde{\phi}}^H \mathbf{B} \boldsymbol{\tilde{\phi}})s01
← Ch 5The Unit-Modulus QCQP