Rank-Constrained Nonlinear Least-Squares Model

• Rank-constrained nonlinear least-squares models enforce low-rank structure via explicit constraints, factorization, or projection methods.
• They incorporate diverse formulations—ranging from matrix sensing to fixed-rank manifolds—to handle nonconvex and bilinear problems.
• Advanced optimization techniques and convergence theories support robust performance even in large-scale and statistically challenging settings.

A rank-constrained nonlinear least-squares model is an optimization model in which a least-squares objective is minimized under an explicit or implicit low-rank condition. A common abstract form is

$$\min_x \; \frac{1}{2}\|F(x)-y\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(G(x))\le r,$$

with nonlinearity arising either from the forward map $F$, from a bilinear or multilinear parameterization such as $UV^\top$, or from the nonconvex feasible set induced by the rank condition itself (Liu, 2023). In the literature, this class spans hard rank-constrained matrix sensing, fixed-rank manifold optimization, low-rank semidefinite programming, supervised low-rank representation learning, projected nonlinear least squares, and factorized matrix models such as NMF (Ding et al., 2021, Luo et al., 2020, Li et al., 2016).

1. Canonical formulations

The most direct formulation is a matrix least-squares problem with a hard rank bound. In rank-constrained matrix sensing, the model is

$$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$

where $\mathcal{A}$ is a linear sensing operator (Liu, 2023). Matrix completion is the special case in which $\mathcal{A}$ extracts observed entries, yielding

$$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$

(Liu, 2023).

A lifted semidefinite version appears in supervised distance preserving projection. After setting $U=PP^\top\in\mathcal{S}^d_+$, the problem becomes

$$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$

which is the rank-constrained least-squares semidefinite program (RCLSSDP) (Ding et al., 2021).

A fixed-rank manifold formulation is used in low-rank matrix trace regression and related estimation problems: $$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$ Here the least-squares residual is linear in $F$0, but the constraint set is the nonlinear manifold

$F$1

(Luo et al., 2020).

In factorized models, the rank constraint is encoded structurally. Standard Euclidean NMF solves

$F$2

with $F$3 determined by the factor width $F$4 (Chu et al., 2020). A related trace regression estimator uses $F$5 to enforce low rank in

$F$6

turning an otherwise linear problem in $F$7 into a nonlinear least-squares problem in $F$8 (Law et al., 2021).

Another distinct construction constrains the data-space representation rather than the coefficient matrix. Projected nonlinear least squares for exponential fitting replaces

$F$9

by

$UV^\top$0

where $UV^\top$1 is a low-dimensional subspace with dimension $UV^\top$2 (Hokanson, 2015). This is not a hard matrix-rank constraint, but it is a dimension-reduced low-rank representation of the residual information.

Setting	Representative objective	Rank mechanism
Matrix sensing / completion	$UV^\top$3	Hard constraint $UV^\top$4
SDP lifting	$UV^\top$5	Hard PSD rank bound $UV^\top$6
Fixed-rank manifold	$UV^\top$7	Exact-rank manifold $UV^\top$8
Factorized models	$UV^\top$9	Implicit $$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$0
Projected NLS	$$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$1	Low-dimensional residual representation

These formulations differ in parameterization and geometry, but they share the same organizing principle: least-squares fidelity is coupled to a structural restriction that suppresses degrees of freedom by low rank.

2. Sources of nonlinearity and geometric structure

A recurring misconception is that nonlinear least squares requires a nonlinear forward operator in the original variable. Several formulations show that this is too narrow. In matrix sensing, $$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$2 is linear, yet the overall problem is nonconvex because $$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$3 defines a nonconvex feasible set (Liu, 2023). In fixed-rank matrix regression, the residual map $$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$4 is linear, but optimization over $$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$5 is a nonlinear least-squares problem on a manifold (Luo et al., 2020).

Bilinear parameterizations produce a second source of nonlinearity. In constrained low-rank representation learning, the supervised term

$$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$6

is quadratic in $$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$7 when $$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$8 is fixed and quadratic in $$\min_{X\in\mathbb{R}^{m\times n}} \;\|\mathcal{A}(X)-b\|_2^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r,$$9 when $\mathcal{A}$0 is fixed, but jointly it is bilinear in $\mathcal{A}$1, so the full problem is nonconvex (Li et al., 2016). The same phenomenon appears in $\mathcal{A}$2-factorized low-rank regression and NMF (Chu et al., 2020, Law et al., 2021).

A third source is geometric. On the fixed-rank manifold $\mathcal{A}$3, the tangent space has dimension $\mathcal{A}$4, and RISRO interprets the least-squares problem as a Riemannian Gauss–Newton problem on that manifold (Luo et al., 2020). In projected exponential fitting, the approximation quality of the projected model is governed by the principal angles between the projection subspace and the Jacobian range, making the Jacobian subspace itself the key geometric object (Hokanson, 2015).

There is also a curvature-side interpretation of rank. In large-scale machine-learning NLLS with minibatches, the Gauss–Newton term $\mathcal{A}$5 has rank at most the batch size $\mathcal{A}$6, and when $\mathcal{A}$7 the dominant curvature information lies in an $\mathcal{A}$8-dimensional subspace (Brust, 2021). In a related geometric vein, higher-order RNC-LM studies regimes where the Jacobian is effectively low rank, extremely ill-conditioned, or nearly rank deficient; there the relevant rank constraint is not on the parameter matrix but on the local metric $\mathcal{A}$9 and the model manifold’s sloppy directions (Liu et al., 8 Jul 2026).

Taken together, these cases suggest that “rank-constrained” in nonlinear least squares should be read broadly: rank may constrain the decision variable, the factorization, the residual representation, or the effective curvature model.

3. Mechanisms for enforcing low rank

The most literal mechanism is a hard algebraic constraint. Examples include $\mathcal{A}$0 in matrix sensing (Liu, 2023), $\mathcal{A}$1 in least-squares SDP (Ding et al., 2021), and $\mathcal{A}$2 on the fixed-rank manifold (Luo et al., 2020). Hard constraints preserve the intended model class exactly but make the feasible set nonconvex.

A second mechanism is factorization. Writing $\mathcal{A}$3 enforces $\mathcal{A}$4 automatically, and is the standard route in NMF, trace regression, and many low-rank learning problems (Chu et al., 2020, Law et al., 2021). This eliminates explicit rank constraints at the expense of introducing bilinear nonconvexity and gauge nonuniqueness.

A third mechanism is convex or DC relaxation. In CLRR, the intractable rank objective is replaced by the nuclear norm $\mathcal{A}$5, producing a constrained nuclear norm minimization problem. The paper explicitly notes that there is no hard constraint $\mathcal{A}$6; the nuclear norm is a soft surrogate and rank is controlled through the objective (Li et al., 2016). In RCLSSDP, the rank constraint is encoded as

$\mathcal{A}$7

for $\mathcal{A}$8, since $\mathcal{A}$9 in the PSD case. This yields the exact-penalty DC-regularized model

$$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$0

where the Ky-Fan $$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$1-norm $$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$2 preserves the top $$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$3 singular values (Ding et al., 2021).

A fourth mechanism is hard projection onto a low-rank set. In nonconvex ADMM for RCMS, the auxiliary variable $$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$4 is updated by Euclidean projection onto

$$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$5

implemented by truncated SVD: $$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$6 (Liu, 2023). This is a hard rank-enforcement step, in contrast to the soft shrinkage used in nuclear-norm ADMM.

A fifth mechanism is projection onto a low-dimensional subspace. Projected exponential fitting restricts residual matching to $$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$7, where $$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$8 has dimension $$\min_X \;\|\Omega\odot X - M\|_F^2 \quad \text{s.t.}\quad \operatorname{rank}(X)\le r$$9 and is chosen so that the subspace angles with the Jacobian remain small (Hokanson, 2015). This is not a matrix-rank constraint in the strict algebraic sense, but it is a principled low-rank reduction of the information seen by the optimizer.

These mechanisms are not equivalent. Nuclear norm regularization is a surrogate, not a hard rank bound (Li et al., 2016). Hard rank projection preserves exact rank feasibility but yields nonconvex dynamics (Liu, 2023). Factorization preserves low rank structurally but changes the optimization geometry (Chu et al., 2020). The choice among them is therefore both statistical and algorithmic.

4. Optimization methods

The algorithmic landscape is heterogeneous because the least-squares term is smooth but the rank structure may be hard, soft, geometric, or combinatorial.

For supervised low-rank representation learning, CLRR introduces an auxiliary variable $U=PP^\top\in\mathcal{S}^d_+$0 and solves the constrained nuclear-norm model by inexact augmented Lagrange multiplier. The block updates are explicit: $U=PP^\top\in\mathcal{S}^d_+$1 is obtained from a linear system, $U=PP^\top\in\mathcal{S}^d_+$2 from singular value thresholding, $U=PP^\top\in\mathcal{S}^d_+$3 from a quadratic least-squares system, and $U=PP^\top\in\mathcal{S}^d_+$4 from the proximal mapping of the $U=PP^\top\in\mathcal{S}^d_+$5 norm (Li et al., 2016). The low-rank part is enforced in the $U=PP^\top\in\mathcal{S}^d_+$6-update, while the least-squares backbone appears in the $U=PP^\top\in\mathcal{S}^d_+$7- and $U=PP^\top\in\mathcal{S}^d_+$8-updates.

For rank-constrained least-squares SDP, the main solver is an inexact proximal DC algorithm with sieving strategy (s-iPDCA). Each DC step linearizes the concave Ky-Fan term, adds a proximal quadratic, and solves a strongly convex SDP subproblem. Those subproblems are handled by a two-block accelerated block coordinate descent method on the dual, alternating a linear solve in $U=PP^\top\in\mathcal{S}^d_+$9 and a PSD-cone projection in $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$0 (Ding et al., 2021). The sieving mechanism separates serious steps from null steps according to the inexactness level.

For hard rank-constrained matrix sensing, a nonconvex two-block ADMM splits the problem into a smooth least-squares $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$1-block and a rank-constrained $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$2-block: $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$3 The $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$4-update is truncated SVD, and the $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$5-update solves the linear operator equation

$$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$6

(Liu, 2023).

RISRO takes a different route. At iteration $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$7, it builds deterministic, data-dependent sketches of the measurement matrices onto subspaces determined by the current singular spaces $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$8, solves a reduced least-squares problem in tangent coordinates $$\min_{U\in\mathcal{S}^d_+} \frac{1}{n}\|\mathcal{A}(U)-b\|^2 \quad \text{s.t.}\quad \operatorname{rank}(U)\le r,$$9, and then retracts back to a rank-$$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$0 matrix (Luo et al., 2020). The paper shows that this is exactly a Riemannian Gauss–Newton step on the fixed-rank manifold with orthographic retraction.

In projected exponential fitting, the central optimization loop is variable projection plus Levenberg–Marquardt on the nonlinear frequency variables, while projected model and Jacobian terms $$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$1 and $$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$2 are computed in closed form through geometric and generalized geometric sums (Hokanson, 2015). In low-rank signal estimation for Hankel-rank time series, a modified weighted Gauss–Newton method uses direct variable projection onto $$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$3, the GLRR-defined low-rank signal space, and attains per-iteration cost

$$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$4

for AR($$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$5) noise (Zvonarev et al., 2021).

At larger machine-learning scale, the computational bottleneck often shifts from explicit rank constraints to low-rank curvature. NLLS1 and NLLSL replace the Gauss–Newton system by

$$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$6

with stochastic rank-1 or rank-$$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$7 Jacobian estimates, exploiting the fact that $$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$8 in a minibatch (Brust, 2021). RNC-LM extends geometric LM updates by constructing higher-order Riemann-normal-coordinate corrections along a damped geodesic curve, targeting curved valleys and nearly rank-deficient Jacobians rather than explicit low-rank matrices (Liu et al., 8 Jul 2026).

These methods share a common pattern: least-squares structure is preserved in the smooth block, while rank structure is handled by SVT, hard SVD truncation, factorization, manifold retraction, variable projection, or low-rank linear algebra.

5. Theory: convergence, approximation, and inference

Several strands of theory recur across the literature: stationarity of nonconvex iterates, local high-order convergence, subspace approximation quality, and statistical prediction or testing guarantees.

For DC-regularized least-squares SDP, s-iPDCA converges globally to stationary points of the DC problem under a Restricted Isometry Property and a KL-based analysis. The serious/null-step mechanism is part of the proof, and the entire sequence converges to a stationary point rather than merely a subsequence (Ding et al., 2021). For hard rank-constrained ADMM in matrix sensing, the whole sequence $$\min_{X \in \mathbb{R}^{p_1\times p_2}} \frac{1}{2}\|y-\mathcal{A}(X)\|_2^2 \quad \text{subject to}\quad \operatorname{rank}(X)=r.$$9 converges to a KKT point under KL assumptions, and in the noiseless uniquely solvable case the limit equals the global optimum if and only if $F$00 (Liu, 2023).

On the fixed-rank manifold, RISRO admits local quadratic-linear convergence to a stationary point under RIP and a local neighborhood condition, and pure quadratic convergence in noiseless matrix sensing (Luo et al., 2020). In the statistical trace-regression setting, the same paper derives a quadratic phase followed by a minimax-order statistical floor $F$01, reached after a number of iterations that is double logarithmic in signal-to-noise and dimension (Luo et al., 2020).

Projected exponential fitting gives a complementary approximation theory. The difference between the full and projected Gauss–Newton steps is controlled by the largest principal angle $F$02 between the projection space and the Jacobian range, and the linearized D-efficiency of the projected estimator is exactly

$F$03

This quantifies when a low-dimensional projected least-squares problem preserves essentially the same information as the full one (Hokanson, 2015).

For constrained nonlinear least squares with convex constraints, the MM-based LM method establishes an $F$04 complexity to find an $F$05-stationary point and local quadratic convergence for zero-residual problems under a local error bound condition (Marumo et al., 2020). This result does not treat hard rank sets directly, but it is immediately relevant to low-rank models realized through factorized variables with convex side constraints.

A statistical theory is available for rank-constrained least-squares trace regression. The rank-constrained least-squares estimator $F$06 satisfies the in-sample prediction bound

$F$07

with probability at least $F$08, without any assumptions on the design matrix’s minimum restricted eigenvalue (Law et al., 2021). The same complexity argument supports a permutation test for low-rank signal detection with nontrivial power at signal strength of order $F$09, again without minimum eigenvalue assumptions (Law et al., 2021).

These results clarify a central point: direct rank constraints do not preclude meaningful theory. The nonconvexity shifts the target from global convex optimality to stationary-point analysis, local geometry, and statistically calibrated guarantees.

6. Applications, variants, and limitations

Applications are diverse because low rank can encode subspaces, latent factors, temporal structure, or effective curvature. In supervised low-rank representation learning, CLRR is evaluated on image classification, human pose estimation, and robust face recovery; the reported behavior is consistently lower classification error than unsupervised LRR, RPCA, and SRRS, lowest angle error in pose estimation against LSR, RVM, and RR, and cleaner face recovery than standard LRR (Li et al., 2016). In supervised distance preserving projection, the rank-constrained kernel extension RCKSDPP is reported as effective and competitive on ORL and YaleB face-recognition experiments (Ding et al., 2021).

Matrix sensing and completion are a second major application family. Nonconvex ADMM for RCMS specializes to a very cheap update for matrix completion,

$F$10

and, in noisy low-sampling regimes, RC-ADMM is reported to outperform NIHT at very low sampling rates while remaining much faster than NN-ADMM (Liu, 2023). R2RILS uses rank-$F$11 tangent-space least squares and is reported to recover ill-conditioned low-rank matrices near the information limit and to remain stable under additive noise (Bauch et al., 2020).

Large-scale machine learning supplies another interpretation of the topic. NLLS1 and NLLSL exploit minibatch low-rank Hessian structure and were tested on Iris classification, MovieLens 100K recommendation, and a Fashion-MNIST autoencoder; NLLS1 attained the lowest losses among the tested methods in the larger problems (Brust, 2021). RNC-LM, aimed at effectively low-rank or nearly rank-deficient Jacobians, improves convergence on curved valleys and rank-deficient problems, reduces the relative $F$12 error to the order of $F$13 on a reaction-diffusion PINN benchmark, and achieved a 34-fold speedup over standard LM on a large-scale potential-energy-surface fitting task (Liu et al., 8 Jul 2026).

Signal processing variants are equally prominent. Projected nonlinear least squares for exponential fitting is asymptotically faster than conventional nonlinear least squares and faster than HSVD for large data, while closely tracking the statistical efficiency of full NLS (Hokanson, 2015). Modified weighted Gauss–Newton for low-rank signal estimation handles AR($F$14) noise and missing data without increasing asymptotic cost (Zvonarev et al., 2021). In NMF, AR3NLS solves alternating rank-3 nonnegative least-squares subproblems in closed form and is reported to provide state-of-the-art performance in computational accuracy and CPU time (Chu et al., 2020).

The limitations are equally consistent. Direct rank minimization is combinatorial and NP-hard, which motivates nuclear-norm or DC surrogates (Li et al., 2016, Ding et al., 2021). Hard rank-constrained algorithms are globally nonconvex and generally converge only to KKT points or stationary points (Liu, 2023, Luo et al., 2020). Nuclear norm is only a surrogate and may not perfectly control rank (Li et al., 2016). SVD-based steps can dominate runtime at large $F$15 (Liu, 2023). Exact integer variants with equality, sparsity, and rank constraints are possible, but the Diophantine enumeration stage dominates runtime and scales poorly with alphabet size and dimension (Ayyar et al., 2018).

A plausible synthesis is that rank-constrained nonlinear least-squares models are best viewed not as a single optimization template but as a family of structurally related models. What unifies them is the use of least-squares fidelity to data together with low-rank structure as capacity control, denoising prior, geometric simplification, or statistical regularizer. What differentiates them is where the rank enters: directly in the variable, implicitly through factors, softly through a surrogate, or indirectly through a projected subspace or curvature model.