Prerequisites & Notation

Before You Begin

This chapter develops the computational machinery that turns the abstract inverse problem $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ into a tractable numerical pipeline. We assume familiarity with the following material.

Kronecker products and the $\text{vec}$ operator(Review telecom-ch01-s07)
Self-check: Can you evaluate $(\mathbf{A} \otimes \mathbf{B})\text{vec}(\mathbf{X})$ without forming the Kronecker product explicitly?
Proximal operators, ADMM, and primal-dual splitting(Review telecom-ch03)
Self-check: Can you state the ADMM update equations and explain the role of the augmented Lagrangian parameter?
The RF imaging forward model and sensing operator(Review rfi-ch02)
Self-check: Can you write out $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ and explain each term?
Regularization and inverse problem well-posedness(Review rfi-ch02)
Self-check: Can you explain why regularization is necessary for underdetermined linear systems?
Basic Python/NumPy and familiarity with array broadcasting
Self-check: Can you reshape and transpose multidimensional arrays without confusion?

Notation for This Chapter

Symbols introduced or used prominently in this chapter. See also the global notation table in the front matter.

Symbol	Meaning	Introduced
$\mathbf{A}$	Sensing / measurement matrix	ch02
$\mathbf{A}_{1}, \mathbf{A}_{2}$	Kronecker factors of the sensing matrix: $\mathbf{A} = \mathbf{A}_{1} \otimes \mathbf{A}_{2}$	s01
$\mathbf{c}$	Discretized reflectivity vector	ch02
$\mathbf{y}$	Observation vector	ch02
$\mathbf{w}$	Noise vector	ch02
$\otimes$	Kronecker product	s01
$\text{vec}(\cdot)$	Column-major vectorization of a matrix	s01
$r^{(t)}$	Primal residual at iteration $t$	s04
$s^{(t)}$	Dual residual at iteration $t$	s04
$\nabla f$	Gradient of $f$	s03
$\mathbf{J}_f$	Jacobian matrix of $f$	s03

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