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Exponential Spectral Pursuit: An Effective Initialization Method for Sparse Phase Retrieval
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Proposes exponential spectral pursuit (ESP), a hyperparameter-free spectral initialization for sparse phase retrieval that achieves a δ-neighborhood estimate with sampling complexity m = Ω(||x||²/|x_max|² · k log n), removing the log³n factor of the truncated power method.
Problem
Sparse phase retrieval: reconstructing a k-sparse n-dimensional signal from magnitude-only measurements; practical algorithms need Ω(k² log n) samples because the initialization stage dominates sampling complexity, creating a gap with the information-theoretic bound Ω(k log n).
Approach
Replaces the heavy-tailed fourth-power spectrum used by truncated power and similar methods with an exponential-weighted spectrum exp(-y²/λ²) that has sub-exponential tails, enabling direct concentration bounds; ESP then does diagonal search, top-k support selection on the chosen column, and eigenvector estimation on the restricted matrix.
Key results
- ESP attains δ-neighborhood initialization with m = Ω(||x||²/|x_max|² k log n), eliminating the log³n term present in the truncated power method's bound.
- At a 30% sampling ratio (n=1000, k=10), ESP recovers about 40% of support indices versus 24% for TP, 18% for SPARTA and 9% for TP-UD.
- ESP outperforms TP clearly when s = 1/√k, and in the phase transition plot has a higher success rate than TP for small sparsity relative to m (e.g., m=1000, k=30).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ESP's exponential-spectrum diagonal search and support recovery instead of TP | Sparse phase retrieval initialization · Sparse Gaussian, k=10, sampling ratio 30% | fraction of recovered support: 24% -> 40% | Helped |
| Use ESP's exponential-spectrum diagonal search and support recovery instead of SPARTA | Sparse phase retrieval initialization · Sparse Gaussian, k=10, sampling ratio 30% | fraction of recovered support: 18% -> 40% | Helped |
| Optimize TP's truncation hyper-parameters rather than use η1=0.9, η2=1.1 | TP · Sparse Gaussian, k=10, sampling ratio 30% | fraction of recovered support: 9% -> 24% | Helped |
| Use ESP instead of TP | Sparse phase retrieval initialization · Sparse 0-1 signal, k=10 | relative error | Mixed |
| Use ESP instead of TP | Sparse phase retrieval initialization · Phase transition, n=1000, m=1,000, k=30 | successful recovery rate | Helped |
| Use ESP instead of TP | Sparse phase retrieval initialization · Phase transition, n=1000, m=1,000, k=10 | successful recovery rate | No clear effect |
Surprising findings
- For sparse 0-1 signals, ESP's advantage over TP narrows and is comparable to TP at low sampling ratios, since equal-magnitude nonzeros shrink the support gap.
Limitations
- Bound depends on the ratio ||x||²/|x_max|², so it is only better than TP when the signal is sufficiently sparse (k < s log^4 n).
- Advantage over TP is less pronounced for sparse 0-1 signals with equal-magnitude nonzeros at low sampling ratios.
- Relies on proxy spectrum analysis; the expectation of the actual spectrum is not derived exactly.
Experimental setup
Benchmarks: synthetic sparse phase retrieval simulations
Metrics: relative error fraction of recovered support successful recovery rate
Baselines: SPARTA CoPRAM TP TP-UD Hadamard WF
Builds on: truncated power method Hadamard wirtinger flow SPARTA Gao & Xu spectral analysis
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