Phase Retrieval: Stability and Recovery Guarantees (1211.0872v1)

Abstract: We consider stability and uniqueness in real phase retrieval problems over general input sets. Specifically, we assume the data consists of noisy quadratic measurements of an unknown input x in R^n that lies in a general set T and study conditions under which x can be stably recovered from the measurements. In the noise-free setting we derive a general expression on the number of measurements needed to ensure that a unique solution can be found in a stable way, that depends on the set T through a natural complexity parameter. This parameter can be computed explicitly for many sets T of interest. For example, for k-sparse inputs we show that O(k\log(n/k)) measurements are needed, and when x can be any vector in R^n, O(n) measurements suffice. In the noisy case, we show that if one can find a value for which the empirical risk is bounded by a given, computable constant (that depends on the set T), then the error with respect to the true input is bounded above by an another, closely related complexity parameter of the set. By choosing an appropriate number N of measurements, this bound can be made arbitrarily small, and it decays at a rate faster than N^{-1/2+\delta} for any \delta>0. In particular, for k-sparse vectors stable recovery is possible from O(k\log(n/k)\log k) noisy measurements, and when x can be any vector in R^n, O(n \log n) noisy measurements suffice. We also show that the complexity parameter for the quadratic problem is the same as the one used for analyzing stability in linear measurements under very general conditions. Thus, no substantial price has to be paid in terms of stability if there is no knowledge of the phase.

• The paper introduces a complexity parameter showing that O(k log(n/k)) and O(n) measurements are required for stable recovery of k‐sparse and general signals, respectively.
• It demonstrates that recovery error decreases faster than N^(-1/2+δ), highlighting improved accuracy with more noisy measurements.
• The analysis equates quadratic measurement complexity to linear cases, indicating that the absence of phase information minimally affects stability.

An Analytical Exploration of Phase Retrieval: Stability and Recovery Guarantees

Phase retrieval is a pertinent problem in various applied fields, notably in optics and imaging techniques such as X-ray crystallography and electron microscopy. The paper "Phase Retrieval: Stability and Recovery Guarantees" by Yonina C. Eldar and Shahar Mendelson provides a comprehensive theoretical analysis of phase retrieval, focusing on stability and recovery from quadratic measurements in the presence of noise.

Stability and Uniqueness in Phase Retrieval

The authors investigate the conditions necessary for the stable recovery of an unknown input vector $x \in R^n$ from noisy quadratic measurements, with consideration of a general input set $T$. They derive a pivotal complexity parameter that quantifies the number of measurements necessary for stable recovery. This parameter is notable for how it scales with the sparsity of the input: for $k$-sparse inputs, stability requires $O(k\log(n/k))$ measurements, whereas for arbitrary inputs in $R^n$, $O(n)$ measurements suffice. This provides a compelling insight into the trade-off between sparsity and measurement requirements.

Noisy Measurements and Empirical Minimization

In the presence of noise, the analysis becomes more intricate. The paper outlines that if one can compute a bounded empirical risk, then the error in estimating the input $x$ from noisy measurements can also be bounded. Notably, the error diminishes faster than $N^{-1/2+\delta}$ for any $\delta > 0$, indicating that increasing the number of measurements rapidly improves the accuracy of the phase estimation. The paper demonstrates that recovery is feasible with $O(k\log(n/k)\log k)$ noisy measurements for $k$-sparse vectors and $O(n \log n)$ for general vectors.

Complexity Measures and Their Significance

A significant contribution of the paper is the introduction of the complexity parameter, derived from the geometry of the input set $T$. This parameter reflects the intertwined effects of input structure and measurement dimensions on stable recovery. By equating the complexity in quadratic measurements to that in linear ones under general conditions, the authors suggest that the absence of phase information is not substantially disadvantageous in terms of stability.

Implications and Future Directions

The implications of this paper are substantial, both theoretically and practically. It provides a rigorous foundation for assessing measurement requirements in phase retrieval applications, which is vital for designing effective algorithms in signal processing and related fields. Future research may extend these results to complex-valued systems, exploring the limits of efficiency and stability in broader settings.

Conclusion

Eldar and Mendelson's paper delivers insightful theoretical underpinnings for phase retrieval through a detailed examination of stability and recovery guarantees. The paper enhances understanding of how sparsity and measurement complexity interrelate and sets a benchmark for future explorations in both theoretical and applied research in phase retrieval. The analytical tools and estimates provided give a strong foundation for further exploration into more complex scenarios, such as those involving non-Gaussian measurements or varying noise platforms.