Title: Outlier-Robust Phase Retrieval in Nearly-Linear Time
Venue: NeurIPS 2025
Accepted: no
Keywords: high-dimensional robust statistics

learning theory

phase retrieval
Abstract: Phase retrieval is a fundamental problem in signal processing, where the goal is to recover a (complex-valued) signal from phaseless intensity measurements. In this paper, we propose and study the (real-valued) outlier-robust phase retrieval problem. Specifically, we seek to recover a vector $x \in \mathbb{R}^d$ from $n$ intensity measurements $y_i = (a_i^\top x)^2$, where a small fraction of the $(a_i,y_i)$ pairs are adversarially corrupted. Our main result is a near-sample-optimal and nearly-linear-time algorithm that provably recovers the ground-truth vector. Our algorithm first solves a lightweight convex program to find an initial point close to the ground truth, and then runs a robust version of gradient descent to achieve exact recovery. Our approach is conceptually simple and provides a framework for developing robust algorithms for other non-convex optimization problems.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a computationally efficient, nearly linear-time algorithm for solving the real-valued robust phase recovery problem, which aims to reconstruct a signal from phaseless measurements corrupted by adversarial outliers. The proposed method consists of two key stages: (1) a robust spectral initialization via a lightweight convex program to obtain a rough estimate near the ground truth, followed by (2) a refinement step using a robust gradient descent variant to achieve exact recovery within specified accuracy. The authors demonstrate that their approach is both near-sample-optimal and computationally efficient, with potential applicability to other non-convex optimization problems. While the paper provides theoretical analysis of the algorithm's superiority, some concerns were raised about insufficient mathematical rigor (e.g., unproven lemmas) and lack of numerical simulations, casting doubt on whether it currently meets NeurIPS publication standards. The framework is noted for its conceptual simplicity and generalizability, but may require additional theoretical and empirical validation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: Compared with existing works, the main innovations of this article are as follows:\n1. Phase retrieval is a well-studied non-convex problem, and analyzing the performance of robust algorithms on it remains crucial.\n2. This article proposes a new low-complexity algorithm for robust phase recovery through robust spectral initialization and robust gradient descent.\n3. This article demonstrates that the proposed algorithm can approximately solve the robust phase recovery problem within a linear time of the input size, under any given accuracy condition. \n\nWeaknesses:\n1. The paper does not explicitly point out any difficulties in terms of theory or technology that the theoretical analysis and algorithm derivation of the proposed algorithm compared with the existing works.\n2. This paper lacks numerical simulation experiments, which prevents it from demonstrating the superiority of the proposed algorithm compared to other algorithms.\n3. The descriptions of some lemmas in this paper are rather vague, for example:\n3.1. The description of the parameter $\\epsilon$ in Lemma 3.1 as \"sufficiently small\"\n3.2. The description of the parameter $\\delta$ in Lemma 4.1 is insufficient\n3.3. The constant corresponding to $O\\left(\\left\\|z-x\\right\\|_{2}^{2}\\right)$ in Lemma 5.1 is not given\n4. The proofs of some theorems in this article are not rigorous, for example:\n4.1. In the proof of Lemma 3.1, on line 254 of the formula, the subtraction of two $O(\\delta)$ terms results in $0$, which is incorrect\n4.2. In the proof of Lemma 5.2, on line 300, the second inequality selects $\\epsilon(c)=O(c^{2})$, which may not satisfy the condition $\\epsilon(c)=\\epsilon_{3}/2$\n\nQUESTIONS:\n1. Why not conduct numerical simulation to compare with the existing algorithms and verify the impact of different attack strategies on the proposed algorithm?\n2. Why does the article extensively use Big O notation instead of accurately writing out the specific constants?\n3. Why is the step size of gradient descent in Algorithm 2 set to $1/300$?\n4. In line 127, \"the corrupted samples can only add directions to Yw but cannot remove directions.\" What is the meaning of \"direction\"? Why do corrupted samples add directions to Y_{w}?\n5. Why do you compute w that minimizes the sum of the largest \"two\" eigenvalues of Y_{w}?\n\nLIMITATIONS:\nThis article is not only not clearly expressed in theory, but also lacks numerical simulation experiments.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis article proposes a two-stage algorithm for the real-valued phase retrieval problem, with a small fraction of measurement pair corrupted. The algorithm first estimates a high-quality initial point through a spectral-based convex optimization that adaptively weights corrupted sample matrices to approximate the ideal matrix. This initialization is then refined through corruption-resistant gradient descent using robust mean estimation. The paper claims its near-optimal sample complexity and near-linear runtime with theoretical guarantees.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**:\n\n1.\tPaper is well-written and provides a clear picture of the setting of the problem\n\n2.\tPhase retrieval is a non-convex problem with important applications, studying its robustness certainly is an important \n\n3.\tthe algorithm is weill supported by the theoretical analysis.\n\n**Weakness**\n\n1.  I do not believe the problem setting studied in this paper has genuine practical value. In phase retrieval, if noise is considered, then all pairs will be corrupted. If outliers are considered, they would correspond to measurements with very large noise levels. The setting proposed in the paper does not make sense in real-world scenarios.\n\n2. The analysis appears to be more incremental in nature compared to prior theoretical works on phase retrieval. Specifically, the main novelty of the spectral initialization step lies in assigning a nonnegative weight to each sample. For the gradient descent step, the problem is essentially reduced to the analysis of robust mean estimation algorithms.\n\n3. Although the authors claim it is a theoretical paper, the work essentially proposes an algorithm with theoretical justification. The technique in the theoretical derivations themselves have no novelty. Since the primary objective of the paper is to develop a robust algorithm, it is quite surprising that no experimental evaluation is provided to assess its performance or efficiency. For an algorithm study paper, the absence of empirical validation is unacceptable.\n\nThis paper is a resubmission of the work originally submitted to NeurIPS 2024. One of the main concerns previously raised was the lack of numerical experiments; however, this revision fails to address that issue entirely.\n\nQUESTIONS:\n1.\tThe main contribution claimed is the near-linear runtime. However, the analysis about lemma 4.1 and 5.2 shows that the running time depends on the reciprocal of epsilon and desired precision. Since they are rather small, it's unclear whether the method is more efficient over other alternatives. How its practical runtime compared with similar approaches (e.g., [3])?\n\nLIMITATIONS:\nNo limitation is discussed.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies phase retrieval under the strong contamination model, where an adversary can arbitrarily replace an $\\epsilon$-fraction of measurements $y_i = \\langle a_i, x\\rangle^2$ and the corresponding measurement vectors $a_i$. It proposes suitable modifications to the standard 2-step approach (spectral initialization followed by gradient descent) that lead to an outlier-robust algorithm with near-optimal runtime $\\tilde{O}(nd)$ and sample complexity $O(d \\cdot \\mathrm{polylog}(1 / \\Delta))$, where $\\Delta$ is the target estimation accuracy. The first stage -- robust spectral initilization -- reduces to finding per-sample weights $w_i$ such that\n\n$$\n\\widehat{Y}_{w} := \\sum_{i} w_i y_i a_i a_i^{\\top} \\approx I + 2 xx^{\\top}\n$$\n\nby means of solving a generalized packing SDP proposed in prior work. The latter stage -- robust gradient descent -- leverages the fact that it is sufficient to obtain an estimate of the gradient, $\\widehat{g}(x_k)$, satisfying the aiming inequality\n\n$$\n\\langle \\widehat{g}(x_k), x_k - x\\rangle \\gtrsim \\|x_k - x\\|^2\n$$\n\nin a neighborhood of the solution set.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The overall algorithm follows the standard 2-step scheme and is simple to state (although the solver for the generalized packing SDP might be complicated).\n- The runtime is near-optimal and the sample complexity is linear in $d$ for moderate estimation accuracies $\\Delta$.\n\nWeaknesses:\n- Real gaussian measurement vectors are somewhat unrealistic in phase retrieval.\n- The algorithm requires a fresh batch of $d \\log(1 / \\Delta)$ samples in each iteration.\n- I am not sure if the strong contamination model is the right way to study corruptions to the measurement vectors (other than, of course, adversarial tampering). For example, neither hardware noise nor miscalibration seem to require an adversary that is allowed to inspect all inputs before deciding which ones to corrupt (and how to corrupt them).\n\nQUESTIONS:\nMy main questions are the following:\n\n1. Appendix C shows that prior \"robust\" approaches -- focusing on corruption solely in the measurement space -- fail when measurement vectors can be corrupted. However, my understanding is that the argument focuses on the failure of spectral initialization. Is it possible that, given a sufficiently good initial estimate, prior methods (such as optimizing the robust $\\ell_1$ loss a-la Duchi-Ruan) succeed in recovering $x$? In my opinion, a formal argument ruling this out would considerably strengthen this paper.\n2. Is it possible to remove the sample-splitting requirement by following an approach similar to the SEVER algorithm (https://proceedings.mlr.press/v97/diakonikolas19a.html), wherein samples are used to certify first-order stationarity and discarded iteratively based on outlier scores? See, also, the recent work by Gao et al. (https://arxiv.org/abs/2412.11003).\n\nOther minor questions:\n- What is the relationship between $\\epsilon(\\delta)$ in Lemma 4.1 and the upper bound $\\epsilon_0$ on the contamination level?\n- How much of the overall analysis depends on Gaussianity? I suspect the robust gradient descent analysis can be easily extended.\n- Can you expand on how you anticipate attacking the sample-splitting requirement? I suspect that verifying stability conditions without fresh samples is very difficult in general settings.\n- Is there a weaker notion of adversary suitable for some of the corruptions you describe in the introduction (hardware noise + miscalibration)?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work tackles the challenge of recovering real-valued signals in the phase retrieval problem under adversarial corruption in possibly both sampling vectors and intensity measurements. The proposed method operates in two stages: \n1. It first uses a convex optimization procedure to obtain a corruption-resilient initialization. \n2. Then refines the estimate through gradient descent combined with a robust mean estimation technique. \n\nA key result of the analysis is that the convergence rate is independent of the corruption level.\n\nSTRENGTHS AND WEAKNESSES:\nThe authors address an interesting problem that is relevant to both the robust statistics and signal processing communities. However, the paper is quite dense and can be difficult to follow at times. Given the theoretical nature of the contributions, it is especially important to maintain clearer bookkeeping of the key variables involved in the theoretical guarantees.\n\nThe central claim of the paper is that the results are independent of the corruption level $\\epsilon$. For instance, in lines 93–99, the authors assert that $\\epsilon_0$ is an absolute constant, independent of both $n$ and $d$. However, this assertion does not appear to be fully justified. A closer examination of the statement of Lemma 2.3 suggests otherwise.\n\n1. It is evident that the statement of Lemma 2.3 does not hold with high probability when $\\epsilon$ is very small, for example, when $\\epsilon = 0$. This issue can be addressed by assuming $\\epsilon \\leq \\epsilon_0$, allowing one to replace $\\epsilon$ with $\\epsilon_0$. However, this introduces a practical challenge, as the value of $\\epsilon_0$ is not explicitly specified.\n2. Even setting aside practical considerations, for the statement of Lemma 2.3 to hold with high probability, it is necessary that $\\epsilon_0 = \\omega(\\frac{1}{n})$. This implies that $\\epsilon_0$ cannot be any absolute constant independent of $n$. Similar issues arise elsewhere - for example, in Lemma 5.2, where the choice of $m$ must depend on $\\epsilon_0$ for the Lemma statement to hold.   \n\nThe presentation of the results would benefit from explicitly retaining all occurrences of $\\epsilon$ (or $\\epsilon_0$) in the theoretical statements. Doing so would provide better context and help clarify the dependence of the guarantees on the corruption level.\n\nQUESTIONS:\nPlease see previous sections.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}