Prerequisites & Notation

This chapter assumes familiarity with the following topics. If any item feels unfamiliar, revisit the linked material first.

OAMP/VAMP algorithms, state evolution, and the orthogonality principle for non-i.i.d. sensing matrices (Chapter 17) (Review ch17)
Self-check: Can you write the OAMP linear estimation step and explain the divergence-free condition?
Sparse recovery algorithms: ISTA, FISTA, ADMM, and their convergence properties (Chapter 13) (Review ch13)
Self-check: Can you write one full iteration of ISTA and identify the proximal operator?
Kronecker structure of the RF sensing operator and efficient matrix-vector products (Chapter 8) (Review ch08)
Self-check: Can you explain why $\mathbf{A} = \mathbf{A}_1 \otimes \mathbf{A}_2$ reduces the cost of $\mathbf{A}\mathbf{c}$ from $O(N^2)$ to $O(N^{3/2})$ ?

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$\mathbf{x}^{(k)}$	Estimate at iteration/layer $k$	s01
$T$	Number of unrolled layers (OAMP iterations)	s01
$\\theta = \\{\\theta_t\\}_{t=1}^T$	Learnable parameters across all layers	s01
$\\mathcal{D}_{\\theta_t}$	Learnable ProxNet denoiser at layer $t$	s01
$\\sigma_t^2$	Effective noise variance at layer $t$ (state evolution)	s01
$\\mathbf{W}_{\\text{LE}}$	OAMP linear estimator (LMMSE matrix)	s01
$K$	Number of LISTA/ADMM unrolled layers	s02
$\\mathbf{W}_k$	Learnable linear transform at LISTA layer $k$	s02
$\\tau_k$	Learnable threshold/step-size at layer $k$	s02
$\\rho_k$	Learnable ADMM penalty parameter at layer $k$	s02
$\\boldsymbol{\\tau}_k$	Spatially-varying threshold map (hierarchical soft-thresholding)	s04
$\\otimes$	Kronecker product	s01