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Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods
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Derives exact asymptotic MMSE and a spectral estimator (via Bayes-AMP and interpolation) for rank-1 matrix denoising with doubly heteroscedastic noise, achieving the weak recovery threshold and Bayes-optimal error for one-sided heteroscedasticity.
Problem
Estimating the singular vectors of a rank-1 signal observed under noise with both column and row covariance correlations (doubly heteroscedastic noise), where prior methods (whitening, singular value shrinkage) lack exact error characterization and are suboptimal.
Approach
Incorporates the noise covariances into the signal priors so the model reduces to whitened signals with i.i.d. Gaussian noise, computes the free energy via a Guerra-style interpolation, and derives a pre-processed matrix A* whose top singular vectors (with a correction) are a spectral estimator matching the Bayes-AMP fixed point.
Key results
- Spectral estimator attains the weak recovery threshold under the criticality condition and sigma_2^* < 1, with theory matching simulations.
- For one-sided heteroscedasticity (Xi = I), the estimator's left-vector correlation eta_u equals sqrt(q_u*), achieving Bayes-optimal MMSE for u* u*^T.
- Numerical experiments (n=4000, d=2000) show the spectral estimator outperforms HeteroPCA, OptShrink, ScreeNOT and Leeb-type methods, especially at low SNR.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed spectral estimator instead of Leeb–Romanov, OptShrink, ScreeNOT, or HeteroPCA | Rank-1 matrix denoising with one-sided heteroscedastic noise · One-sided heteroscedastic noise, especially low SNR | normalized signal correlation | Helped |
| Use the proposed spectral estimator instead of Leeb, OptShrink, or ScreeNOT | Rank-1 matrix denoising with doubly heteroscedastic noise · Doubly heteroscedastic noise, especially low SNR | normalized signal correlation | Helped |
| Use the proposed spectral estimator | Estimating the left signal when noise is heteroscedastic only on the other side and priors are Gaussian · One-sided heteroscedastic noise | matrix MSE for estimating u*u*^T | Helped |
| Use whitening, Xi^{-1/2} A Sigma^{-1/2}, in place of the proposed Bayes-AMP preprocessing, Xi^{-1} A Sigma^{-1} (ablation) | Bayes-AMP for doubly heteroscedastic matrix denoising · At least one of Xi, Sigma is not a multiple of the identity | weak recovery threshold | Hurt |
Surprising findings
- Whitening-based AMP (operating on Xi^{-1/2} A Sigma^{-1/2}) yields a strictly suboptimal weak recovery threshold when either covariance is not a multiple of identity.
- Even when Xi = I, the estimator of u* uses the column covariance Sigma non-trivially, so the two estimation tasks cannot be decoupled.
Limitations
- Optimality for both signals is proven only for one-sided heteroscedasticity; general doubly heteroscedastic case is not Bayes-optimal.
- Technical condition sigma_2^* < 1 is verified only numerically, not analytically.
- Results assume Gaussian priors and known covariances.
Experimental setup
Benchmarks: synthetic doubly heteroscedastic spiked model
Metrics: normalized correlation with u* normalized correlation with v* matrix MSE for u*v*^T
Baselines: HeteroPCA OptShrink ScreeNOT Leeb-Romanov Leeb
Builds on: Guerra interpolation approximate message passing HeteroPCA OptShrink
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