Subspace Phase Retrieval

TL;DR

Subspace Phase Retrieval (SPR) tackles sparse phase retrieval by combining a correlation-based initialization that captures a large energy fraction of the true support with a non-initial iterative stage that refines the support and estimates the signal within a subspace. The method leverages Wirtinger calculus and a quartic objective to perform matching, estimation, and pruning, achieving recovery under sampling bounds close to the information-theoretic limit, $m = \Omega(k\log n)$, for signals with $|x_{\min}| = \Omega(\|\mathbf{x}\|/\sqrt{k})$. Theoretical guarantees (Theorems 1–4) show initialization energy capture, expectation-based exact recovery in the subproblem, finite-sample recovery with high probability, and a partitioning technique to handle dependence across steps. Empirical results demonstrate SPR’s competitive performance on 1D and 2D signals, including Fourier-based settings, highlighting its practical robustness and potential for bridging the computational-to-statistical gap in sparse phase retrieval.

Abstract

In recent years, phase retrieval has received much attention in statistics, applied mathematics and optical engineering. In this paper, we propose an efficient algorithm, termed Subspace Phase Retrieval (SPR), which can accurately recover an $n$-dimensional $k$-sparse complex-valued signal $\x$ given its $Ω(k^2\log n)$ magnitude-only Gaussian samples if the minimum nonzero entry of $\x$ satisfies $|x_{\min}| = Ω(\|\x\|/\sqrt{k})$. Furthermore, if the energy sum of the most significant $\sqrt{k}$ elements in $\x$ is comparable to $\|\x\|^2$, the SPR algorithm can exactly recover $\x$ with $Ω(k \log n)$ magnitude-only samples, which attains the information-theoretic sampling complexity for sparse phase retrieval. Numerical Experiments demonstrate that the proposed algorithm achieves the state-of-the-art reconstruction performance compared to existing ones.

Figures (8)

• Figure 1: An illustration for the separation of $S^0$ and $\mathop{\mathrm{supp}}\limits(\mathbf{x})$
• Figure 2: An illustrative explanation on Proposition \ref{['lm:elemenEzxtbound']}.
• Figure 3: Illustrative diagram of our proof structure for exact recovery via SPR.
• Figure 4: An illustrative example where $\bar{s}$ and $|x_{\min}|$ affect different parts of $\mathbf{x}$.
• Figure 5: The numerical solution is clustered around the expected global minimum.

Theorems & Definitions (39)

• Theorem 1: Initialization
• Theorem 2: Expectation case
• Theorem 3: Finite sampling case
• Theorem 4: Overall condition
• Lemma 1: Bernstein's inequality hdpBook
• proof
• Proposition 1
• proof
• Proposition 2: Geometric Property in ${\mathbb C}^\mathcal{T}$
• Proposition 3: Concentration of Gradient