AMP Implementation

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This is the correct answer because it captures the core concept.

Common Mistake: Common Mistake in Section 1

Mistake:

Overlooking a critical implementation detail.

Correction:

Always verify results against known benchmarks and theoretical predictions.

Key Term 1

Core concept from section 1 of chapter 42.

Definition:

Approximate Message Passing (AMP)

AMP iterates:

r(t)=y−Ax^(t)+x^(t)M∑iη′(ri(t−1))\mathbf{r}^{(t)} = \mathbf{y} - \mathbf{A}\hat{\mathbf{x}}^{(t)} + \frac{\hat{\mathbf{x}}^{(t)}}{M}\sum_i \eta'(r_i^{(t-1)}) x^(t+1)=η(AHr(t)+x^(t))\hat{\mathbf{x}}^{(t+1)} = \eta(\mathbf{A}^H\mathbf{r}^{(t)} + \hat{\mathbf{x}}^{(t)})

The Onsager correction term x^(t)M∑iη′\frac{\hat{\mathbf{x}}^{(t)}}{M}\sum_i \eta' is crucial for convergence.

Definition:

Orthogonal AMP (OAMP)

OAMP uses orthogonal projection to ensure divergence-free iterations for non-i.i.d. matrices:

r(t)=W(y−Ax^(t))+x^(t)\mathbf{r}^{(t)} = \mathbf{W}(\mathbf{y} - \mathbf{A}\hat{\mathbf{x}}^{(t)}) + \hat{\mathbf{x}}^{(t)}

where W\mathbf{W} is designed so that the effective noise is white.

Definition:

Vector AMP (VAMP)

VAMP alternates between two denoisers in a symmetric fashion, converging for a broader class of matrices than AMP.

Definition:

State Evolution

State evolution tracks the MSE of AMP across iterations:

τ(t+1)=σ2+1δMSE(η,τ(t))\tau^{(t+1)} = \sigma^2 + \frac{1}{\delta}\text{MSE}(\eta, \tau^{(t)})

where δ=M/N\delta = M/N is the measurement ratio.

Prerequisites & NotationOAMP / VAMP Implementation