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What Happens When the Prior Is Wrong?

The BG-MMSE denoiser is Bayes-optimal when the prior is correctly specified — i.e., the scene is truly Bernoulli-Gaussian with the assumed parameters $(\rho, \sigma_c^2)$ . In practice, the prior is always wrong: the true scene may have a different sparsity, a different amplitude distribution, or structure that no parametric prior captures (extended targets, spatial correlations).

How robust is OAMP to such mismatch? Can we still use state evolution to predict performance under a mismatched denoiser? This section answers both questions.

Definition:
Mismatched State Evolution

Let $p_0$ be the true prior and $\tilde{p}_0$ be the assumed prior used to design the denoiser $\tilde{\eta}_t$ . The mismatched state evolution for OAMP is:

$v_1^t = F(v_2^{t-1}; \mathbf{A}),$

$v_2^t = \mathbb{E}_{p_0}\bigl[|\tilde{\eta}_t(C_0 + \sqrt{v_1^t}\,Z) - C_0|^2\bigr],$

where the expectation is over $C_0 \sim p_0$ (the true prior) and $Z \sim \mathcal{CN}(0, 1)$ . The LMMSE step is unchanged (it does not depend on the prior).

The key point: state evolution is still valid under mismatch. It tracks the actual algorithm, not the intended algorithm. But the MSE is degraded because $\tilde{\eta}_t$ is not optimal for $p_0$ .

Mismatched state evolution enables offline analysis of robustness: sweep over possible true priors $p_0$ while keeping the denoiser fixed, and plot the resulting MSE.

Theorem: MSE Degradation Under Prior Mismatch

Let $v_2^*$ be the fixed-point MSE of OAMP with the matched (Bayes-optimal) denoiser for the true prior $p_0$ , and let $\tilde{v}_2^*$ be the fixed-point MSE with the mismatched denoiser $\tilde{\eta}$ designed for $\tilde{p}_0$ . Then

$\tilde{v}_2^* \geq v_2^*,$

with equality if and only if $\tilde{\eta}$ equals the Bayes-optimal denoiser for $p_0$ almost everywhere.

Moreover, the MSE degradation is bounded by

$\tilde{v}_2^* - v_2^* \leq \sup_{v_1 \geq 0} \bigl[ \text{mse}(\tilde{\eta}, v_1; p_0) - \text{mmse}(p_0, v_1)\bigr] \cdot \bigl(1 - \frac{\partial F}{\partial v_2} \big|_{v_2^*}\bigr)^{-1},$

where the supremum is over noise levels, and the second factor accounts for the amplification through the LMMSE step.

The mismatched denoiser is suboptimal at each iteration — it removes less noise than the Bayes-optimal denoiser. This excess MSE feeds back into the LMMSE step, which sees a larger prior variance, producing a noisier output, creating a vicious cycle. The bound quantifies how much the per-step suboptimality is amplified by the iterative loop.

Proof

Per-step MSE comparison

At any noise level $v_1$ , the MMSE denoiser minimizes the MSE by definition: $\text{mse}(\tilde{\eta}, v_1; p_0) \geq \text{mmse}(p_0, v_1)$ . The gap $\Delta(v_1) = \text{mse}(\tilde{\eta}, v_1; p_0) - \text{mmse}(p_0, v_1) \geq 0$ is the per-step excess MSE.

Fixed-point shift

The state evolution map with the mismatched denoiser is $\tilde{g}(v_2) = F^{-1}(\text{mse}(\tilde{\eta}, F(v_2)))$ . Since $\tilde{g}(v_2) \geq g(v_2)$ (the matched map), the fixed point shifts up: $\tilde{v}_2^* \geq v_2^*$ .

Bounding the shift via sensitivity analysis

By the implicit function theorem applied to the fixed-point equation, the shift is bounded by the maximum per-step gap divided by the stability margin $1 - \frac{\partial g}{\partial v_2}\big|_{v_2^*}$ . $\blacksquare$

Example: Sparsity Mismatch in BG-MMSE

The true scene is Bernoulli-Gaussian with sparsity $\rho_0 = 0.10$ and variance $\sigma_c^2 = 1$ . The denoiser uses an assumed sparsity $\tilde{\rho}$ . Compute the OAMP fixed-point NMSE via mismatched state evolution for $\tilde{\rho} \in \{0.01, 0.05, 0.10, 0.15, 0.20, 0.30\}$ .

Parameters: $\delta = 0.4$ , $\text{SNR} = 25\,\text{dB}$ , $N_1 = N_2 = 32$ (Kronecker sensing with partial DFT).

Solution

Mismatched SE computation

For each $\tilde{\rho}$ , we compute the BG-MMSE denoiser with assumed sparsity $\tilde{\rho}$ and evaluate its MSE under the true prior with $\rho_0 = 0.10$ :

$\tilde{\rho}$	NMSE (dB)	Gap from optimal
0.01	$-14.3$	$8.8$ dB
0.05	$-20.5$	$2.6$ dB
0.10	$-23.1$	$0.0$ dB (matched)
0.15	$-22.4$	$0.7$ dB
0.20	$-21.2$	$1.9$ dB
0.30	$-19.0$	$4.1$ dB

Asymmetry of mismatch

Underestimating the sparsity ( $\tilde{\rho} < \rho_0$ ) is much more damaging than overestimating it. At $\tilde{\rho} = 0.01$ , the denoiser aggressively kills components, missing many true scatterers. At $\tilde{\rho} = 0.30$ , it is too conservative (preserves noise), but the penalty is milder.

Practical guideline

When the true sparsity is uncertain, it is safer to overestimate $\rho$ by a factor of 1.5–2x. Alternatively, use EM-GAMP (Chapter 19) to learn $\rho$ and $\sigma_c^2$ from the data.

OAMP Performance Under Prior Mismatch

Explore how OAMP's NMSE degrades when the assumed sparsity or signal variance differs from the truth. The dashed line shows the Bayes-optimal NMSE.

Parameters

True sparsity

\rho_0

0.1

True signal std

\sigma_c

1

\delta

0.4

SNR (dB)25

Mismatch dimension

Minimax and Robust Denoisers

When no reliable prior information is available, one can use a minimax denoiser that minimizes the worst-case MSE over a class of signals (e.g., the $\ell_2$ ball of radius $R$ ). The minimax denoiser for AWGN is the James-Stein estimator:

$\eta_{\text{JS}}(\mathbf{r}) = \Bigl(1 - \frac{(N-2)v} {\|\mathbf{r}\|^2}\Bigr)_+\,\mathbf{r},$

which shrinks toward zero by a data-dependent factor.

Minimax denoisers are conservative: they sacrifice performance when the prior is favorable in exchange for robustness when the prior is unfavorable. In RF imaging, this tradeoff is usually not worthwhile — the scene statistics are often well-characterized from training data, and a learned denoiser outperforms minimax by a large margin.

🔧Engineering Note

Online Parameter Learning via EM

Rather than choosing prior parameters offline, they can be learned from the measurements using an EM (expectation- maximization) approach interleaved with the OAMP iterations:

E-step: Run one OAMP iteration with current parameters $(\tilde{\rho}, \tilde{\sigma}_c^2)$ .
M-step: Update $\tilde{\rho}$ and $\tilde{\sigma}_c^2$ using the posterior statistics from the denoiser output.

This is essentially EM-GAMP (Chapter 19) applied to the OAMP framework. It converges in 5–10 outer EM iterations for typical RF imaging problems, adding minimal overhead.

Practical Constraints

•
EM converges to a local maximum of the marginal likelihood; multiple initializations may be needed
•
For very low SNR or very high undersampling, the EM landscape may have spurious local optima

Common Mistake: Signal Variance Mismatch Can Be Worse Than Sparsity Mismatch

Mistake:

Carefully tuning the sparsity parameter $\rho$ but using a default signal variance $\sigma_c^2 = 1$ without checking whether it matches the actual signal power.

Correction:

The BG-MMSE denoiser depends on both $\rho$ and $\sigma_c^2$ . A 10x error in $\sigma_c^2$ can cause 5–8 dB of MSE degradation, comparable to a 5x error in $\rho$ . Always estimate both parameters, either from training data or via EM within OAMP.

Quick Check

Mismatched state evolution for OAMP:

Still accurately predicts the empirical MSE of the mismatched algorithm

Overestimates the MSE because it assumes worst-case mismatch

Is invalid because the orthogonality condition requires a matched prior

Correction:

Still accurately predicts the empirical MSE of the mismatched algorithm

State evolution tracks the actual algorithm behavior, regardless of whether the denoiser is optimal. It predicts the MSE of whatever denoiser is used.

Key Takeaway

OAMP is moderately robust to prior mismatch: a 2x error in the sparsity parameter costs about 1–2 dB, while a 10x error can cost 5–8 dB. Underestimating sparsity is more harmful than overestimating it. Mismatched state evolution remains valid and enables offline robustness analysis. When prior parameters are uncertain, EM-based learning or learned denoisers provide the most robust reconstructions.

Mismatch Analysis

What Happens When the Prior Is Wrong?

Definition: Mismatched State Evolution

Theorem: MSE Degradation Under Prior Mismatch

Per-step MSE comparison

Fixed-point shift

Bounding the shift via sensitivity analysis

Example: Sparsity Mismatch in BG-MMSE

Mismatched SE computation

Asymmetry of mismatch

Practical guideline

OAMP Performance Under Prior Mismatch

Parameters

Minimax and Robust Denoisers

Online Parameter Learning via EM

Common Mistake: Signal Variance Mismatch Can Be Worse Than Sparsity Mismatch

Quick Check

Key Takeaway

Definition:
Mismatched State Evolution