PhaseLift: Convex Approach to Phase Retrieval

• PhaseLift is a convex phase retrieval method that lifts quadratic measurements into linear constraints using rank-one PSD matrices.
• It reformulates the nonconvex phase retrieval problem into a trace-minimization semidefinite program, ensuring exact and robust recovery under various noise models.
• The approach extends to applications in coded diffraction, quantum tomography, and integrated photonics, demonstrating stability even with adversarial noise.

Searching arXiv for recent and foundational PhaseLift papers to ground the article. arxiv_search.query({"search_query":"all:PhaseLift", "start":0, "max_results":10, "sort_by":"relevance", "sort_order":"descending"}) PhaseLift is a convex programming approach to phase retrieval that recasts magnitude-only quadratic measurements as linear observations of a rank-one positive semidefinite matrix. Given measurements of the form $b_i = |\langle a_i,x\rangle|^2$, the method introduces the lifted variable $X = x x^*$, so that $b_i = \mathrm{Tr}(a_i a_i^* X)$. The nonconvex rank-one feasibility problem is then relaxed to a trace-minimization semidefinite program over the PSD cone, and the signal is recovered up to a global phase from a rank-one factorization or, in the noisy case, from the leading eigenvector of the optimizer (Candes et al., 2011).

1. Measurement model and lifted formulation

In its standard form, PhaseLift addresses recovery of an unknown vector $x \in \mathbb{C}^n$ or $\mathbb{R}^n$ from phaseless measurements

$b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$

The lifting step defines

$X = x x^* \succeq 0,$

which converts each quadratic measurement into a linear functional,

$b_i = \mathrm{Tr}(a_i a_i^* X).$

Writing $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$, one obtains the linear observation model $b = \mathcal{A}(X)$ with the additional structural constraints $X = x x^*$0 and $X = x x^*$1 (Candes et al., 2011).

PhaseLift replaces the nonconvex rank constraint by trace minimization on the PSD cone: $X = x x^*$2 In the presence of additive noise, a standard noise-aware variant is

$X = x x^*$3

and a common alternative is an $X = x x^*$4-fitting formulation that minimizes the residual $X = x x^*$5 subject to $X = x x^*$6 (Candes et al., 2011). When the optimizer is not exactly rank one, the leading eigenvector of $X = x x^*$7, scaled by the top eigenvalue and phased appropriately, is used as the estimate of $X = x x^*$8 (Huang et al., 12 Sep 2025).

The method is intrinsically invariant to a global phase factor: if $X = x x^*$9 is feasible, then $b_i = \mathrm{Tr}(a_i a_i^* X)$0 produces the same magnitude data. This ambiguity is the canonical identifiability limit in phase retrieval and is built into both the theoretical guarantees and the reconstruction procedures (Candes et al., 2012).

2. Foundational recovery theory

The original exact-recovery theorem for PhaseLift established that if the sensing vectors are sampled i.i.d. uniformly on the unit sphere, then there exist absolute constants $b_i = \mathrm{Tr}(a_i a_i^* X)$1 such that

$b_i = \mathrm{Tr}(a_i a_i^* X)$2

implies, with probability at least $b_i = \mathrm{Tr}(a_i a_i^* X)$3, that the unique minimizer is $b_i = \mathrm{Tr}(a_i a_i^* X)$4 (Candes et al., 2011). The same work also proved stability under additive noise: if $b_i = \mathrm{Tr}(a_i a_i^* X)$5 with $b_i = \mathrm{Tr}(a_i a_i^* X)$6, then the solution $b_i = \mathrm{Tr}(a_i a_i^* X)$7 to the noise-aware SDP obeys

$b_i = \mathrm{Tr}(a_i a_i^* X)$8

and the induced vector estimate satisfies

$b_i = \mathrm{Tr}(a_i a_i^* X)$9

for some phase $x \in \mathbb{C}^n$0 (Candes et al., 2011).

A subsequent sharp improvement removed the logarithmic oversampling factor for Gaussian and spherical designs. For sensing vectors drawn i.i.d. either from the complex normal law $x \in \mathbb{C}^n$1 or uniformly from the sphere of radius $x \in \mathbb{C}^n$2, there exist absolute constants $x \in \mathbb{C}^n$3 such that

$x \in \mathbb{C}^n$4

implies, with probability at least $x \in \mathbb{C}^n$5, that the PhaseLift SDP is exact for every $x \in \mathbb{C}^n$6 simultaneously (Candes et al., 2012). In the noisy setting, the same paper analyzed the $x \in \mathbb{C}^n$7-fitting program

$x \in \mathbb{C}^n$8

and proved

$x \in \mathbb{C}^n$9

together with a corresponding leading-eigenvector error bound. These bounds were described there as information-theoretically optimal up to constants (Candes et al., 2012).

The proof architecture that became standard in the PhaseLift literature combines geometric analysis of the tangent space at $\mathbb{R}^n$0, injectivity or $\mathbb{R}^n$1-isometry properties of the measurement operator on that cone, and construction of dual certificates in the range of $\mathbb{R}^n$2 (Candes et al., 2011). In the Gaussian $\mathbb{R}^n$3 regime, one formulation of the certificate conditions is

$\mathbb{R}^n$4

with $\mathbb{R}^n$5 the tangent space at $\mathbb{R}^n$6; a net argument then yields universality over all signals (Candes et al., 2012).

3. Structured measurements, coded diffraction, and derandomization

A central development was the extension of PhaseLift from unstructured Gaussian measurements to coded diffraction patterns (CDP), a model tailored to Fourier magnitude measurements with random masks. In this setting,

$\mathbb{R}^n$7

where $\mathbb{R}^n$8 is a Fourier vector and $\mathbb{R}^n$9 is a diagonal mask (Huang et al., 12 Sep 2025). Under admissible random masks, exact recovery was proved when the number of patterns satisfies

$b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$0

equivalently $b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$1, with probability at least $b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$2 (Candes et al., 2013). This result showed that physically realistic masked Fourier measurements admit rigorous convex recovery guarantees.

The same coded-diffraction line has now been sharpened in the noisy regime. Under adversarial noise $b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$3 with $b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$4, and masks whose entries are i.i.d. symmetric, bounded, and satisfy $b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$5, $b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$6, Theorem 3.1 in the 2025 analysis proves that if

$b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$7

then

$b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$8

For mean-zero sub-Gaussian noise with $b_i = |\langle a_i,x\rangle|^2,\qquad i=1,\dots,m.$9, and a modified PhaseLift with the additional rank bound $X = x x^* \succeq 0,$0, the bound becomes

$X = x x^* \succeq 0,$1

with probability $X = x x^* \succeq 0,$2 when $X = x x^* \succeq 0,$3 (Huang et al., 12 Sep 2025). The same work states that these results settle the conjecture of Soltanolkotabi that CDP PhaseLift should achieve the $X = x x^* \succeq 0,$4 scaling of the Gaussian-design case, up to logarithmic factors (Huang et al., 12 Sep 2025).

PhaseLift does not fundamentally require Gaussian measurement vectors. A partial derandomization result replaced Gaussian draws by iid samples from a complex projective $X = x x^* \succeq 0,$5-design. For any exact $X = x x^* \succeq 0,$6-design with $X = x x^* \succeq 0,$7, exact recovery holds with probability at least $X = x x^* \succeq 0,$8 provided

$X = x x^* \succeq 0,$9

If $b_i = \mathrm{Tr}(a_i a_i^* X).$0, this becomes $b_i = \mathrm{Tr}(a_i a_i^* X).$1, which matches the Gaussian $b_i = \mathrm{Tr}(a_i a_i^* X).$2 scaling up to polylog-factors (Gross et al., 2013). The same work also proves a converse for arbitrary 2-designs: in prime-power dimensions, a generic 2-design can force $b_i = \mathrm{Tr}(a_i a_i^* X).$3, so $b_i = \mathrm{Tr}(a_i a_i^* X).$4 is essential for near-linear scaling (Gross et al., 2013).

There are also deterministic constructions. One explicit frame of cardinality $b_i = \mathrm{Tr}(a_i a_i^* X).$5 was shown to be 1-complete, meaning that every pure state $b_i = \mathrm{Tr}(a_i a_i^* X).$6 is the unique minimizer of the PhaseLift SDP from its associated intensity measurements (Kech, 2015). This establishes exact deterministic phase retrieval via PhaseLift with an explicit measurement design, rather than probabilistic sensing.

A recurring misconception is that Fourier oversampling alone suffices to make phase retrieval well posed. A contrary result in the structured-illumination literature shows that even heavy Fourier oversampling, up to fivefold in each dimension, can leave an ill-posed problem: both PhaseLift and classical alternating-projection methods fit the oversampled modulus perfectly yet diverge spectrally from the true solution, with relative MSE $b_i = \mathrm{Tr}(a_i a_i^* X).$7 (Candes et al., 2011). The same line of work introduced multiple structured illuminations and showed that three suitably designed diffraction patterns can uniquely determine the phase in 1D and 2D under nonvanishing Fourier and gcd conditions (Candes et al., 2011).

4. Stability, adversarial noise, and outliers

Noise robustness is a defining theme of the PhaseLift literature, but the precise scaling depends strongly on the sensing model and loss function. In the original random-sphere analysis, the Frobenius or operator-norm error scales linearly with the additive perturbation level in the constrained $b_i = \mathrm{Tr}(a_i a_i^* X).$8 formulation (Candes et al., 2011). In the Gaussian $b_i = \mathrm{Tr}(a_i a_i^* X).$9 regime, the $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$0-fitting SDP achieves the sharper bound $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$1, which improves prior $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$2-noise results by an additional $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$3 factor in the denominator (Candes et al., 2012).

For coded diffraction patterns, the recent near-optimal noisy theory makes explicit why logarithmic factors appear. The proof proceeds through robust injectivity on the tangent space $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$4, where

$$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$5

and a dual certificate constructed by a golfing scheme. If $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$6, feasibility and optimality imply $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$7 and $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$8; pairing $$\mathcal{A}(X) = (\mathrm{Tr}(a_i a_i^* X))_{i=1}^m$$9 with the dual certificate yields

$b = \mathcal{A}(X)$0

The logarithmic losses arise from union bounds over $b = \mathcal{A}(X)$1 golfing steps and concentration inequalities with tail-probability unions over $b = \mathcal{A}(X)$2 measurements (Huang et al., 12 Sep 2025).

Robustness to gross corruptions led to a distinct branch of PhaseLift variants that eliminate the explicit trace term and rely on PSD structure plus $b = \mathcal{A}(X)$3 residual minimization. In the real Gaussian model

$b = \mathcal{A}(X)$4

where $b = \mathcal{A}(X)$5 is small noise and $b = \mathcal{A}(X)$6 are arbitrary gross errors on an unknown support, the convex program

$b = \mathcal{A}(X)$7

was shown to tolerate a small, fixed fraction of gross errors with only $b = \mathcal{A}(X)$8 measurements. Specifically, there are constants $b = \mathcal{A}(X)$9 such that if $X = x x^*$00, then with probability at least $X = x x^*$01,

$X = x x^*$02

(Hand, 2015). The paper emphasizes that, in this robust formulation, the rank penalization through a trace term is unnecessary and no tuning parameter needs to be chosen (Hand, 2015).

The adversarial-outlier threshold has since been quantified sharply. For Gaussian measurements perturbed by $X = x x^*$03-bounded noise and an adversarially chosen sparse vector with support size at most $X = x x^*$04, the Robust-PhaseLift model

$X = x x^*$05

succeeds for any $X = x x^*$06 with $X = x x^*$07 measurements, even when the sparse outliers may depend on the measurement and the observation (Huang et al., 2023). The analysis is based on the robust outlier bound condition on the tangent space and on the distribution of the product of two correlated Gaussian variables. The same work constructs adaptive counterexamples showing failure when $X = x x^*$08 with high probability (Huang et al., 2023).

5. Variants, priors, and related convex formulations

Because the trace surrogate is only an indirect proxy for rank one, a number of variants refine the penalty or incorporate structure beyond low rank. One example is PhaseLiftOff, which replaces the trace-only bias by the nonconvex rank-one proxy

$X = x x^*$09

On the PSD cone this quantity is nonnegative and vanishes if and only if $X = x x^*$10. The corresponding regularized least-squares problem

$X = x x^*$11

is equivalent to rank-one least squares under the condition

$X = x x^*$12

and, for i.i.d. complex Gaussian measurements, obeys the stable bound

$X = x x^*$13

when $X = x x^*$14 (Yin et al., 2014). Computation is carried out by a convergent difference-of-convex algorithm, with each convex subproblem solved by ADMM (Yin et al., 2014).

When the unknown signal is a superposition of a few Vandermonde atoms, the PhaseLift relaxation can be coupled to atomic norm minimization. In ANM-PhaseLift, one combines the lifted PSD variable with the atomic norm $X = x x^*$15 for line spectra. The classical PhaseLift sample requirement is $X = x x^*$16 under i.i.d. Gaussian sensing, while atomic-norm recovery from linear measurements can achieve $X = x x^*$17 under the minimum separation condition

$X = x x^*$18

Simulations in the quadratic setting verify recovery from $X = x x^*$19 as small as $X = x x^*$20 samples, substantially below $X = x x^*$21, although exact RIP-type bounds were left to future work (Zhang et al., 2018).

PhaseLift has also been generalized from vectors to unitary matrices. In the unitary setting, the unknown is $X = x x^*$22, the measurements are $X = x x^*$23, and the lifted variable is $X = x x^*$24 with additional Jamiolkowski-state constraints. With unitary 4-design measurements, the noisy PhaseLift program achieves uniform exact recovery with a near-optimal number of samples; with unitary 2-designs, it recovers almost all signals up to a constant additive error under a non-spikiness condition (Kimmel et al., 2015). This matrix-valued extension is directly motivated by quantum process tomography.

These variants show that “PhaseLift” names both a specific trace-minimization SDP and a broader lifting paradigm. A plausible implication is that many later methods should be understood as structured or geometry-aware descendants of the original convex lifting idea rather than as outright replacements.

6. Applications and empirical practice

PhaseLift has been integrated into inverse problems well beyond textbook phase retrieval. In a stochastic time-fractional diffusion inverse problem, the unknown discrete source $X = x x^*$25 is recovered from phaseless Fourier data using two masks, $X = x x^*$26 and $X = x x^*$27 with $X = x x^*$28 drawn i.i.d. from Bernoulli$X = x x^*$29. The implementation uses the TFOCS library and an accelerated proximal scheme of Auslender–Teboulle/Nesterov type, with stopping criterion

$X = x x^*$30

For the reported numerical experiments, the typical problem size is $X = x x^*$31, $X = x x^*$32, and reconstructions of $X = x x^*$33 and $X = x x^*$34, both corrupted by $X = x x^*$35 uniform noise in the variance estimate, agree closely with the true $X = x x^*$36 for $X = x x^*$37 and $X = x x^*$38 (Gong et al., 2020).

In integrated photonics, PhaseLift was used to reconstruct the transfer matrix of linear-optical networks from classical coherent-light intensity data. For each output port $X = x x^*$39, the measurement model is

$X = x x^*$40

so each row of the transfer matrix reduces to a standard phase-retrieval instance. The paper employs a PSD-constrained $X = x x^*$41-loss formulation without an explicit trace penalty, proves a stable bound

$X = x x^*$42

and reports that for random $X = x x^*$43 unitaries the average circuit fidelity between the PhaseLift reconstruction and a two-photon reference reconstruction is $X = x x^*$44 (Suess et al., 2020). The simulated phase transition appears near $X = x x^*$45 even with $X = x x^*$46 additive Gaussian noise on intensities (Suess et al., 2020).

In photoemission orbital tomography, Robust Sparse PhaseLift augments the trace term by an entrywise $X = x x^*$47 sparsity penalty and an $X = x x^*$48 data-fidelity term,

$X = x x^*$49

subject to $X = x x^*$50. The method uses known atomic positions and basis functions inside the measurement operator and is reported to identify molecular orbitals, including three-dimensional phases, from a single PMM. The paper further states that adsorption-induced molecular deformations can be perfectly discriminated with an accuracy of $X = x x^*$51 $X = x x^*$52 (Niki et al., 2023).

The broader influence of the lifting paradigm extends into quantum time-series estimation. “Quantum Phaselift” estimates the rank-one matrix $X = x x^*$53 associated with a discrete Loschmidt amplitude rather than estimating $X = x x^*$54 directly, proves that a $X = x x^*$55 bandwidth suffices for generic signals, and demonstrates recovery for signals of size exceeding 100 time points using only a few million measurement shots (Patel et al., 9 Feb 2026). Although this is a distinct framework, it preserves the central PhaseLift principle: move the inference task from nonlinear phase variables to a linear measurement model over PSD rank-one matrices.

Across these applications, a consistent pattern emerges. Whenever the forward model naturally produces magnitude-squared linear functionals, and whenever either randomness, structure, or auxiliary masks make the lifted operator sufficiently injective, PhaseLift provides a mathematically explicit route from phaseless data to stable convex recovery.