Low-Rank Phase Retrieval Overview

• Low-rank phase retrieval is a framework for recovering structured low-rank matrices from magnitude-only data using signal or PSD constraints.
• It extends traditional rank-one phase retrieval, enabling recovery guarantees via convex methods (e.g., nuclear norm minimization) and nonconvex algorithms (e.g., alternating minimization).
• Practical insights include leveraging tailored measurement models and overparameterized factorizations to improve stability and efficiency in imaging and signal processing applications.

Searching arXiv for recent and foundational papers on low-rank phase retrieval and lifted low-rank matrix recovery. Low-rank phase retrieval refers to the recovery of a low-rank matrix, or of a collection of signals arranged as a low-rank matrix, from magnitude-only or intensity-only measurements. Across the literature, the term covers two closely related but distinct viewpoints. In one viewpoint, phase retrieval is lifted to recovery of a rank-one positive semidefinite matrix $X=xx^*$, and “low-rank phase retrieval” denotes the extension from rank one to general low-rank Hermitian or positive semidefinite matrices under rank-one quadratic measurements (Kueng et al., 2014, Kueng et al., 2016, Li et al., 2016). In the other viewpoint, one seeks a matrix $X=[x_1,\dots,x_q]$ whose columns are multiple signals or frames sharing a low-dimensional subspace, while each column is observed through its own phaseless measurements $y_{ik}=|\langle a_{ik},x_k\rangle|$ or $y_{ik}=|\langle a_{ik},x_k\rangle|^2$ (Vaswani et al., 2016, Nayer et al., 2020, Liu et al., 2018). A further extension replaces matrix structure by multilinear tensor structure, using Tucker models or structured tensor lifts when the signal ensemble is naturally multiway or when phaselessness is coupled to additional bilinear structure (Kwon et al., 2022, Liang et al., 13 Sep 2025). This suggests that low-rank phase retrieval is best understood as a family of phaseless inverse problems in which rank, subspace, positivity, or tensorial structure supplies the missing information that magnitude-only data alone cannot provide.

1. Matrix lifting and the low-rank generalization of phase retrieval

A central formulation begins from Hermitian low-rank matrix recovery under rank-one measurements. The measurements are written as

$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$

or, in operator form,

$$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$

When $X$ is rank one and positive semidefinite, $X=xx^*$, this reduces exactly to phase retrieval because

$$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$

(Kueng et al., 2014, Kueng et al., 2016, Rauhut et al., 2016).

This lifted viewpoint is the most direct bridge between ordinary phase retrieval and low-rank recovery. Standard phase retrieval becomes the special case $r=1$, while higher-rank models allow recovery of general Hermitian or PSD matrices under the same rank-one quadratic sensing architecture (Kueng et al., 2014). In this sense, the literature does not merely borrow intuition from phase retrieval; it treats phase retrieval as the rank-one endpoint of a broader low-rank matrix inverse problem.

Several works emphasize that the target may be Hermitian low rank or specifically PSD low rank. The distinction matters algorithmically. For arbitrary Hermitian matrices, the natural convex estimator is nuclear norm minimization,

$X=[x_1,\dots,x_q]$0

whereas for PSD matrices one may replace the nuclear norm by trace minimization or, in some settings, by a PSD-constrained least-squares or empirical-risk estimator (Kueng et al., 2014, Kueng et al., 2016, Li et al., 2016). Since $X=[x_1,\dots,x_q]$1 automatically, phase retrieval falls into the PSD regime.

A second major formulation treats low-rank phase retrieval directly at the matrix level without lifting a single vector. In that model, the unknown is a matrix

$X=[x_1,\dots,x_q]$2

and each column $X=[x_1,\dots,x_q]$3 is observed through its own phaseless measurements

$X=[x_1,\dots,x_q]$4

Equivalently,

$X=[x_1,\dots,x_q]$5

with a separate sensing matrix $X=[x_1,\dots,x_q]$6 for each column (Kwon et al., 2022, Liu et al., 2018, Nayer et al., 2020, Vaswani et al., 2016). This is a multiple-signal or dynamic setting rather than a single lifted rank-one matrix setting, but the rank assumption again couples all columns through a shared low-dimensional subspace.

The literature is careful to distinguish these viewpoints. One line studies low-rank matrix recovery under quadratic rank-one measurements, with phase retrieval as the rank-one PSD special case (Kueng et al., 2014, Kueng et al., 2016, Rauhut et al., 2016, Li et al., 2016, Huang et al., 9 Jul 2026). Another line studies multiple signals or frames that jointly form a low-rank matrix and are measured column by column (Vaswani et al., 2016, Nayer et al., 2020, Liu et al., 2018). Both are properly described as low-rank phase retrieval, but they differ in their unknown object, sensing architecture, and proof methods.

2. Measurement models and structural assumptions

The dominant measurement model in the lifted literature uses rank-one sensing matrices $X=[x_1,\dots,x_q]$7. For Gaussian measurements, the vectors $X=[x_1,\dots,x_q]$8 are independent standard Gaussian vectors in $X=[x_1,\dots,x_q]$9 or $y_{ik}=|\langle a_{ik},x_k\rangle|$0, leading to sample complexities of order $y_{ik}=|\langle a_{ik},x_k\rangle|$1 for low-rank recovery and $y_{ik}=|\langle a_{ik},x_k\rangle|$2 in the rank-one phase retrieval case (Kueng et al., 2014). Structured alternatives include weighted complex projective $y_{ik}=|\langle a_{ik},x_k\rangle|$3-designs, random tight frames, Clifford-group orbits, stabilizer states, and heavy-tailed ensembles with only finite $y_{ik}=|\langle a_{ik},x_k\rangle|$4 moments (Kueng et al., 2014, Rauhut et al., 2016, Kueng et al., 2016, Huang et al., 9 Jul 2026).

The measurement structure is often decisive. Clifford-orbit measurements form a highly structured family with $y_{ik}=|\langle a_{ik},x_k\rangle|$5, where the vectors are sampled uniformly and independently from an orbit of the complex Clifford group,

$y_{ik}=|\langle a_{ik},x_k\rangle|$6

or equivalently from the orbit of projectors

$y_{ik}=|\langle a_{ik},x_k\rangle|$7

A particularly important orbit is the orbit of stabilizer states, which includes and generalizes the discrete Fourier basis by replacing linear phases with quadratic ones (Kueng et al., 2016). The paper on Clifford orbits argues that stabilizer states are “an ideal model for structured measurements for phase retrieval” because they are highly structured yet still permit recovery guarantees close to Gaussian ones (Kueng et al., 2016).

In the multi-signal LRPR setting, the measurement model is column-local rather than global over the whole matrix: $y_{ik}=|\langle a_{ik},x_k\rangle|$8 The sensing vectors are typically taken i.i.d. Gaussian and independent across both $y_{ik}=|\langle a_{ik},x_k\rangle|$9 and $y_{ik}=|\langle a_{ik},x_k\rangle|^2$0 (Vaswani et al., 2016, Nayer et al., 2020). This independence across columns is not a minor technical convenience. The literature states explicitly that the gain from low rank critically relies on different measurement vectors for different columns; if the same sensing vectors are reused for all columns, the benefit largely disappears (Vaswani et al., 2016, Nayer et al., 2020).

Low-rank phase retrieval also appears in robust and noise-corrupted settings. A prominent PSD model considers

$y_{ik}=|\langle a_{ik},x_k\rangle|^2$1

where $y_{ik}=|\langle a_{ik},x_k\rangle|^2$2 is rank $y_{ik}=|\langle a_{ik},x_k\rangle|^2$3, $y_{ik}=|\langle a_{ik},x_k\rangle|^2$4 is a sparse outlier vector, and $y_{ik}=|\langle a_{ik},x_k\rangle|^2$5 is bounded dense noise (Li et al., 2016). Another recent line replaces Gaussian or sub-Gaussian assumptions by heavy-tailed sampling vectors with independent mean-zero variance-one entries and only finite $y_{ik}=|\langle a_{ik},x_k\rangle|^2$6 moments, together with the nondegeneracy conditions

$y_{ik}=|\langle a_{ik},x_k\rangle|^2$7

(Huang et al., 9 Jul 2026). This substantially weakens the distributional assumptions under which uniform lifted recovery can be proved.

The multi-signal literature imposes additional structure on the unknown matrix. Writing

$y_{ik}=|\langle a_{ik},x_k\rangle|^2$8

a standard assumption is right singular vector incoherence, expressed as

$y_{ik}=|\langle a_{ik},x_k\rangle|^2$9

or, in simplified form under constant condition number,

$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$0

(Nayer et al., 2020, Vaswani, 2020). This prevents a small number of columns from dominating the energy and is analogous to right incoherence in matrix completion.

3. Convex recovery methods and uniform guarantees

The foundational convex approach is nuclear norm minimization under the lifted operator $$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$1. For Gaussian rank-one measurements, if

$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$2

then with probability at least

$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$3

the constrained nuclear-norm program recovers every rank-$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$4 target stably, with error

$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$5

in the PSD case and a corresponding uniform recovery statement for all rank-$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$6 matrices under exact $$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$7-design sampling with

$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$8

(Kueng et al., 2014). The same paper emphasizes that these are uniform guarantees: one random draw of the sensing vectors succeeds simultaneously for all rank-$$y_j=\operatorname{tr}(X a_j a_j^*) = a_j^*Xa_j,\qquad j=1,\dots,m,$$9 Hermitian matrices (Kueng et al., 2014).

For PSD targets, positivity simplifies the program. One may replace nuclear norm minimization by trace minimization over the PSD cone,

$$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$0

or, in some settings, by the even simpler constrained least-squares program

$$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$1

This simplification is especially important for phase retrieval because $$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$2 automatically (Kueng et al., 2014, Kueng et al., 2016).

Clifford-orbit recovery gives a uniform, stable theorem for low-rank Hermitian matrices with measurements $$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$3 drawn from a Clifford orbit. If

$$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$4

then with probability at least

$$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$5

every Hermitian rank-$$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$6 matrix can be recovered stably via nuclear-norm minimization, with

$$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$7

For PSD targets, the same sampling rate suffices for the simpler PSD-constrained least-squares estimator, and in the phase retrieval specialization $$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$8, since $$b=\mathcal A(X)+\epsilon, \qquad (\mathcal A(Z))_j=\operatorname{tr}(Za_ja_j^*)=a_j^*Za_j.$$9, one obtains

$X$0

for stable phase retrieval from any Clifford orbit, including stabilizer measurements (Kueng et al., 2016).

Random tight-frame measurements provide another structured alternative. If the measurement vectors are the rows of a random tight frame multiplied by $X$1, and

$X$2

then with probability at least

$X$3

every minimizer of the nuclear norm program obeys

$X$4

(Rauhut et al., 2016). In the rank-one case this gives a PhaseLift-type guarantee under random tight-frame sensing.

Heavy-tailed quadratic sampling substantially enlarges the admissible sensing models. Under independent entries with finite $X$5 moments and the nondegeneracy conditions stated above, both nuclear norm minimization and PSD-constrained empirical risk minimization achieve

$X$6

up to moment-dependent constants, with uniform stable and robust recovery guarantees (Huang et al., 9 Jul 2026). This yields lifted phase retrieval guarantees with $X$7 under similarly weak moment assumptions (Huang et al., 9 Jul 2026).

A distinct robust convex formulation addresses sparse arbitrary outliers for PSD targets: $X$8 If the outlier fraction $X$9 satisfies

$X=xx^*$0

then with probability exceeding

$X=xx^*$1

the solution obeys

$X=xx^*$2

(Li et al., 2016). In the rank-one case this is a robust PhaseLift guarantee with sparse outlier corruption.

These convex results establish a broad picture. Low-rank phase retrieval in lifted form admits uniform recovery theorems under Gaussian, $X=xx^*$3-design, random tight-frame, Clifford-orbit, and heavy-tailed sensing, with positivity often allowing simplification from nuclear norm minimization to PSD-constrained least squares or empirical risk minimization (Kueng et al., 2014, Kueng et al., 2016, Rauhut et al., 2016, Li et al., 2016, Huang et al., 9 Jul 2026).

4. Nonconvex formulations, alternating minimization, and landscape analysis

Nonconvex approaches avoid the dimensional cost of lifting or operate directly in a low-rank factorization. In the multi-signal LRPR setting, the principal formulation is

$X=xx^*$4

with $X=xx^*$5 or $X=xx^*$6 orthonormal and $X=xx^*$7 or $X=xx^*$8 (Vaswani et al., 2016, Nayer et al., 2020, Vaswani, 2020). The most developed algorithmic line is AltMinLowRaP, which alternates between low-dimensional phase retrieval for each $X=xx^*$9, phase estimation, and least-squares updates of the shared subspace $$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$0 (Nayer et al., 2020, Vaswani, 2020).

The improved guarantee for AltMinLowRaP states that, under right incoherence and i.i.d. Gaussian sensing,

$$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$1

and

$$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$2

suffice for geometric convergence to $$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$3-accuracy (Nayer et al., 2020). More precisely, after

$$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$4

iterations, the subspace error and matrix error satisfy

$$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$5

(Nayer et al., 2020). The paper emphasizes that this improves the sample complexity of earlier AltMinLowRaP analysis by a factor of $$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$6 for the alternating-minimization stage and a factor of $$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$7 for initialization (Nayer et al., 2020).

The earlier LRPR paper proposed LRPR1 and LRPR2 in addition to LRPR-init. LRPR1 performs projected truncated gradient descent: initialize from a spectral estimate, take one truncated-WF-like step per column, then project the resulting matrix onto rank $$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$8 (Vaswani, 2020). LRPR2 alternates among phase estimation, coefficient updates, and a least-squares update for the shared subspace under the factorization $$y_j=\operatorname{tr}(xx^* a_j a_j^*)=|\langle a_j,x\rangle|^2$$9 (Vaswani et al., 2016). Theoretical guarantees in that work focus on initialization, while extensive experiments indicate that LRPR2 has the best empirical sample complexity and LRPR1 offers a favorable speed–accuracy tradeoff (Vaswani et al., 2016).

Phase retrieval of low-rank matrices by anchored regression takes a different nonconvex-to-convex route. Given an anchor matrix $r=1$0 roughly aligned with the target $r=1$1, the paper studies the convex program

$r=1$2

subject to phaseless convex sublevel constraints. In the rank-1 phaseless blind deconvolution model, if the anchor satisfies

$r=1$3

and

$r=1$4

then the estimate is stable, with a recovery error proportional to the average noise level divided by $r=1$5 (Lee et al., 2019). For general-rank real Gaussian sensing, the theorem requires

$r=1$6

plus an anchor quality condition and an additional conditioning restriction (Lee et al., 2019).

Another nonconvex direction analyzes the quartic Burer–Monteiro objective

$r=1$7

for semidefinite low-rank sensing. In phase retrieval this reduces, for $r=1$8, to

$r=1$9

A recent landscape analysis shows that it can be helpful to overparametrize the factor rank $X=[x_1,\dots,x_q]$00, and for phase retrieval with sub-Gaussian measurements, overparametrizing by a factor at most logarithmic in the dimension allows recovery with optimal statistical sample complexity (McRae, 5 May 2025). More generally, for semidefinite matrix sensing with PSD structure and rank-1 Gaussian measurements, the same framework gives optimal $X=[x_1,\dots,x_q]$01-type sample complexity for higher-rank PSD targets (McRae, 5 May 2025). This suggests that the PSD lifted structure can make overparametrization beneficial rather than harmful.

A deterministic theory for exact non-convex phase retrieval studies Wirtinger Flow through the lifted rank-one PSD matrix $X=[x_1,\dots,x_q]$02. It identifies a sufficient condition equivalent to a restricted isometry-type property over rank-1 PSD matrices and shows that, under a uniform concentration bound on the lifted normal operator, spectral initialization and geometric convergence follow (Yonel et al., 2020). Although specialized to rank one, this work is conceptually important because it interprets nonconvex phase retrieval as recovery on the manifold of rank-1 PSD matrices and derives a weaker condition than those used in generic low-rank matrix recovery (Yonel et al., 2020).

The nonconvex literature is therefore heterogeneous. Some methods operate directly on low-rank factors in multi-signal models (Nayer et al., 2020, Vaswani et al., 2016). Others analyze quartic least-squares landscapes in the lifted PSD domain (McRae, 5 May 2025, Yonel et al., 2020). Still others use convex anchored regression over the original matrix variable to avoid higher-order lifting (Lee et al., 2019). Across these lines, initialization quality, phase estimation, and the special geometry of rank-one or PSD structure are recurring themes.

5. Bayesian, gauge-dual, and optimization-oriented perspectives

Several works approach low-rank phase retrieval from inference or optimization design rather than from sample-optimal recovery theory alone.

A Bayesian formulation for multi-signal LRPR places a Gaussian–Wishart hierarchical prior on the unknown matrix

$X=[x_1,\dots,x_q]$03

with measurement model

$X=[x_1,\dots,x_q]$04

The shared precision matrix $X=[x_1,\dots,x_q]$05 induces low-rank structure across columns, and a variational EM algorithm alternates among Gaussian posteriors for the columns, a Wishart posterior for the precision, a Gamma posterior for the noise precision, and deterministic updates of the hidden phases (Liu et al., 2018). The paper provides simulation evidence that this variational Bayesian learning approach is less sensitive to initialization and can work even with random initialization, unlike a competing alternating-minimization baseline (Liu et al., 2018).

A gauge-duality approach studies the lifted real symmetric phase retrieval problem

$X=[x_1,\dots,x_q]$06

and solves the gauge dual approximately using cheap spectral approximations. The resulting approximate dual information is used to initialize a nonconvex rank-1 least-squares refinement (Estrin et al., 2020). The contribution is not a new exact recovery theorem for low-rank phase retrieval, but rather a practical hybrid strategy that extracts the principal eigenspace of the lifted solution from the dual certificate matrix

$X=[x_1,\dots,x_q]$07

without solving the primal SDP to high accuracy (Estrin et al., 2020). In the rank-one phase retrieval case, this acts as an improved initializer for the quartic nonconvex objective.

A related optimization-oriented result studies projected SGD for convex low-rank matrix relaxations over the spectrahedron

$X=[x_1,\dots,x_q]$08

The paper does not develop a phase-retrieval-specific theorem, but it explicitly lists phase retrieval among the motivating applications. Its main result shows that if a convex lifted relaxation has a low-rank optimum satisfying an eigen-gap condition, then projected SGD with a warm start produces iterates that remain low rank with constant probability, so each projection can be computed using only a low-rank eigendecomposition (Garber, 2020). For PhaseLift-type relaxations, this suggests that large-scale convex lifted methods may be computationally viable when the target rank is small.

Another line studies PSD low-rank recovery through nonconvex regularization of the exact rank constraint rather than through nuclear norm minimization. For fixed-rank PSD matrix recovery,

$X=[x_1,\dots,x_q]$09

a nonconvex quadratic-envelope regularization preserves the global minimizers of the original constrained problem when $X=[x_1,\dots,x_q]$10, and can be optimized by forward–backward splitting or FISTA (Carlsson et al., 2019). In the Fourier phase retrieval specialization $X=[x_1,\dots,x_q]$11, the paper shows that oversampling improves the rank of the lifted operator only up to approximately factor $X=[x_1,\dots,x_q]$12 per dimension and that masked measurements are needed to stabilize the lifted inverse problem beyond that (Carlsson et al., 2019).

These optimization-oriented papers broaden the meaning of low-rank phase retrieval. The topic is not only about identifiability and sample complexity; it also concerns how lifted convex programs, dual certificates, low-rank projections, spectral initializers, and factorized nonconvex objectives can be made computationally practical in the phaseless regime (Estrin et al., 2020, Garber, 2020, Carlsson et al., 2019).

6. Tensor-structured extensions, blind deconvolution couplings, and open directions

Matrix low-rankness is not always the most natural structure. For video or image sequences, reshaping each frame into its spatial dimensions and time axis yields a tensor

$X=[x_1,\dots,x_q]$13

and Tucker structure may reduce the number of unknowns from roughly $X=[x_1,\dots,x_q]$14 to

$X=[x_1,\dots,x_q]$15

The Tucker-Structured Phase Retrieval method models

$X=[x_1,\dots,x_q]$16

and alternates among spectral frame initialization, HOSVD, low-dimensional phase retrieval for the temporal factors $X=[x_1,\dots,x_q]$17, phase updates, and CGLS updates for $X=[x_1,\dots,x_q]$18 (Kwon et al., 2022). The paper does not provide formal recovery guarantees, but on real video datasets it reports lower reconstruction error than matrix-based baselines in both underdetermined and over-determined regimes when the Tucker ranks are chosen appropriately (Kwon et al., 2022). A plausible implication is that multilinear low-rank structure can regularize low-rank phase retrieval more effectively than matrix rank when the signals are intrinsically multiway.

A different tensor extension arises when phase retrieval is coupled to blind deconvolution. In a partially coherent imaging model,

$X=[x_1,\dots,x_q]$19

with $X=[x_1,\dots,x_q]$20, the lifted object becomes a third-order structured rank-one tensor

$X=[x_1,\dots,x_q]$21

A 2025 paper studies a surrogate tensor sensing problem

$X=[x_1,\dots,x_q]$22

and gives a landscape analysis showing benign population geometry and local linear convergence of Riemannian gradient descent under a tensor restricted isometry property (Liang et al., 13 Sep 2025). This is not a general LRPR theorem, but it shows how phaseless quadratic structure combined with an additional bilinear nuisance parameter naturally leads from low-rank matrices to structured low-rank tensors (Liang et al., 13 Sep 2025).

The topic continues to develop in more direct amplitude-based formulations as well. A 2025 paper studies real-valued low-rank phase retrieval from amplitude measurements

$X=[x_1,\dots,x_q]$23

and proves that the rank-constrained nonlinear least-squares estimator

$X=[x_1,\dots,x_q]$24

achieves

$X=[x_1,\dots,x_q]$25

with high probability under Gaussian measurements and

$X=[x_1,\dots,x_q]$26

The same paper gives comparable guarantees for constrained and penalized nuclear-norm formulations and introduces a strong restricted isometry property for matrices tailored to amplitude measurements (Ge et al., 29 Sep 2025). This suggests that a more direct, non-lifted matrix LRPR theory can recover near-degree-of-freedom sample complexity for Gaussian linear maps, at least in the real-valued setting (Ge et al., 29 Sep 2025).

Several open problems recur across the literature. The multi-signal LRPR survey lists reduction of sample complexity from $X=[x_1,\dots,x_q]$27 toward $X=[x_1,\dots,x_q]$28, removal of sample splitting, extension to practical Fourier and masked-Fourier measurements, and treatment of dynamic low-rank evolution as major directions (Vaswani, 2020). Tensor-structured LRPR lacks recovery and convergence theorems and still relies on trial-and-error rank selection (Kwon et al., 2022). Anchored regression for general-rank LRPR still requires restrictive anchor and conditioning assumptions (Lee et al., 2019). Overparametrized semidefinite landscape results are currently strongest for PSD sensing and rank-1 sub-Gaussian phase retrieval, not arbitrary complex LRPR models (McRae, 5 May 2025). Heavy-tailed convex theory covers lifted quadratic sampling, but not the full range of structured non-Gaussian acquisition operators encountered in practice (Huang et al., 9 Jul 2026).

Taken together, the literature presents low-rank phase retrieval as a technically diverse area at the intersection of phase retrieval, low-rank matrix recovery, matrix completion, tensor methods, and nonconvex optimization. Its unifying principle is simple: phaseless data are insufficient without structure, and low rank—whether expressed as a PSD lift, a shared subspace, a nuclear norm, a Tucker tensor, or a structured factorization—provides that structure in a form amenable to both analysis and computation (Kueng et al., 2014, Vaswani et al., 2016, Nayer et al., 2020, Kwon et al., 2022, Ge et al., 29 Sep 2025).