One-Bit Phase Retrieval: Optimal Rates and Efficient Algorithms
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| Format: | Preprint |
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2024
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| _version_ | 1866914204721610752 |
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| author | Chen, Junren Yuan, Ming |
| author_facet | Chen, Junren Yuan, Ming |
| contents | In this paper, we study the sample complexity and develop efficient optimal algorithms for 1-bit phase retrieval: recovering a signal $\mathbf{x}\in\mathbb{R}^n$ from $m$ phaseless bits $\{\mathrm{sign}(|\mathbf{a}_i^\top\mathbf{x}|-τ)\}_{i=1}^m$ generated by standard Gaussian $\mathbf{a}_i$s. By investigating a phaseless version of random hyperplane tessellation, we show that (constrained) hamming distance minimization uniformly recovers all unstructured signals with Euclidean norm bounded away from zero and infinity to the error $\mathcal{O}((n/m)\log(m/n))$, and $\mathcal{O}((k/m)\log(mn/k^2))$ when restricting to $k$-sparse signals. Both error rates are shown to be information-theoretically optimal, up to a logarithmic factor. Intriguingly, the optimal rate for sparse recovery matches that of 1-bit compressed sensing, suggesting that the phase information is non-essential for 1-bit compressed sensing. We also develop efficient algorithms for 1-bit (sparse) phase retrieval that can achieve these error rates. Specifically, we prove that (thresholded) gradient descent with respect to the one-sided $\ell_1$-loss, when initialized via spectral methods, converges linearly and attains the near optimal reconstruction error, with sample complexity $\mathcal{O}(n)$ for unstructured signals and $\mathcal{O}(k^2\log(n)\log^2(m/k))$ for $k$-sparse signals. Our proof is based upon the observation that a certain local (restricted) approximate invertibility condition is respected by Gaussian measurements. Our results establish the major findings of (memoryless) 1-bit compressed sensing in a phaseless setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04733 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | One-Bit Phase Retrieval: Optimal Rates and Efficient Algorithms Chen, Junren Yuan, Ming Information Theory In this paper, we study the sample complexity and develop efficient optimal algorithms for 1-bit phase retrieval: recovering a signal $\mathbf{x}\in\mathbb{R}^n$ from $m$ phaseless bits $\{\mathrm{sign}(|\mathbf{a}_i^\top\mathbf{x}|-τ)\}_{i=1}^m$ generated by standard Gaussian $\mathbf{a}_i$s. By investigating a phaseless version of random hyperplane tessellation, we show that (constrained) hamming distance minimization uniformly recovers all unstructured signals with Euclidean norm bounded away from zero and infinity to the error $\mathcal{O}((n/m)\log(m/n))$, and $\mathcal{O}((k/m)\log(mn/k^2))$ when restricting to $k$-sparse signals. Both error rates are shown to be information-theoretically optimal, up to a logarithmic factor. Intriguingly, the optimal rate for sparse recovery matches that of 1-bit compressed sensing, suggesting that the phase information is non-essential for 1-bit compressed sensing. We also develop efficient algorithms for 1-bit (sparse) phase retrieval that can achieve these error rates. Specifically, we prove that (thresholded) gradient descent with respect to the one-sided $\ell_1$-loss, when initialized via spectral methods, converges linearly and attains the near optimal reconstruction error, with sample complexity $\mathcal{O}(n)$ for unstructured signals and $\mathcal{O}(k^2\log(n)\log^2(m/k))$ for $k$-sparse signals. Our proof is based upon the observation that a certain local (restricted) approximate invertibility condition is respected by Gaussian measurements. Our results establish the major findings of (memoryless) 1-bit compressed sensing in a phaseless setting. |
| title | One-Bit Phase Retrieval: Optimal Rates and Efficient Algorithms |
| topic | Information Theory |
| url | https://arxiv.org/abs/2405.04733 |