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Block Sparse Bayesian Learning: A Diversified Scheme
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Proposes a Diversified Block Sparse Prior and the DivSBL algorithm (EM plus dual ascent) that learns block structure adaptively, with global and local optimality theory and experiments on synthetic signals, audio, and images.
Problem
Block sparse signal recovery from compressed measurements requires pre-defined block sizes and locations, and existing block sparse Bayesian methods are sensitive to these choices and prone to overfitting when intra-block correlation is estimated.
Approach
Allow per-element intra-block variances and per-block correlation matrices that are diversified across blocks via weak constraints (solved by dual ascent), so block size and location are learned automatically; BSBL and RVM arise as special cases.
Key results
- On synthetic homoscedastic signals DivSBL reaches NMSE 0.0094 and Corr 0.9955, versus 0.0132 and 0.9936 for the next best (BSBL).
- On 2D image reconstruction at sample rate 0.5, DivSBL improves on other methods by an average of 9.8% on the test images.
- On audio at sample rate 0.25 with SNR 10 to 50 dB, DivSBL achieves the lowest NMSE at every SNR level tested.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DivSBL instead of BSBL | Synthetic homoscedastic signals · Synthetic homoscedastic | NMSE: 0.0132±0.0069 -> 0.0094±0.0053 | Helped |
| Use DivSBL instead of BSBL | Synthetic heteroscedastic signals · Synthetic heteroscedastic | NMSE: 0.0245±0.0125 -> 0.0086±0.0041 | Helped |
| Use DivSBL instead of BSBL | AudioSet recovery · AudioSet, sample rate 0.25, SNR 10–45 | NMSE | Helped |
| Use DivSBL instead of BSBL | Image reconstruction · Eight test images | sqrt NMSE: 9.8% average improvement | Helped |
| Add diversified correlation using weak constraints (ablation) | DivSBL · Synthetic block-sparse recovery | NMSE | Helped |
| Add a strong shared-correlation constraint before diversification (ablation) | DivSBL · Synthetic block-sparse recovery | NMSE | Helped |
| Add Toeplitz correction to correlation matrices (ablation) | DivSBL · Synthetic block-sparse recovery | NMSE | Helped |
| Use one-step dual ascent instead of complete dual ascent (ablation) | DivSBL · Synthetic block-sparse recovery | NMSE versus CPU time | Helped |
| Remove constraints on block correlation matrices (ablation) | DivSBL · Synthetic block-sparse recovery | NMSE | Hurt |
| Vary the preset block size | DivSBL · Synthetic block-sparse recovery | NMSE | No clear effect |
| Increase the sample rate | DivSBL · AudioSet | NMSE: +1 dB NMSE enhancement | Mixed |
Limitations
- DivSBL is slower than heuristic and convex-solver baselines such as StructOMP, CVX and SPGL1.
- Experiments are limited to signal and image reconstruction; supervised tasks such as regression and classification are left for future work.
- Global optimality holds only in the noiseless limit beta to infinity, and the local-minima result only bounds sparsity.
Experimental setup
Datasets: AudioSet standard grayscale test images (DAMP toolbox, NLR experiments) synthetic homoscedastic and heteroscedastic block sparse signals
Benchmarks: synthetic block sparse signals AudioSet clips standard grayscale test images
Metrics: NMSE correlation (cosine similarity)
Baselines: BSBL PC-SBL SBL Group Lasso Group BPDN StructOMP
Builds on: Sparse Bayesian Learning BSBL Pattern-Coupled SBL Relevance Vector Machine
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