Inexact Manifold Proximal Linear (IManPL)

• IManPL is an inexact proximal-linear methodology for composite optimization on Riemannian manifolds that solves local convex subproblems approximately to ensure global convergence.
• It employs adaptive inexactness criteria and Armijo-style retraction strategies to balance computational efficiency with rigorous convergence guarantees.
• IManPL variants handle both feasible and infeasible settings effectively, achieving significant speed-ups in applications like sparse PCA and spectral clustering.

Searching arXiv for the cited papers and closely related variants. arxiv_search query: (Huang et al., 2021) OR (Zheng et al., 26 Aug 2025) OR (He et al., 16 Aug 2025) OR (Xie et al., 24 May 2026) max_results: 10 arxiv_search({"query":"(Huang et al., 2021) OR (Zheng et al., 26 Aug 2025) OR (He et al., 16 Aug 2025) OR (Xie et al., 24 May 2026)","max_results":10}) Inexact Manifold Proximal Linear Algorithm (IManPL) denotes an inexact proximal-linear methodology for nonsmooth composite optimization over Riemannian manifolds or embedded submanifolds. The name appears explicitly in "A New Inexact Manifold Proximal Linear Algorithm with Adaptive Stopping Criteria" (Zheng et al., 26 Aug 2025), and closely related papers develop the same basic paradigm under adjacent names: an inexact Riemannian proximal gradient method (Huang et al., 2021), an inexact variable metric proximal linearization method (He et al., 16 Aug 2025), and a feasibility-safeguarded inexact proximal linearized method (Xie et al., 24 May 2026). Taken together, these works suggest a research line centered on solving a strongly convex local model only approximately, while retaining global stationarity guarantees, $O(\varepsilon^{-2})$-type outer iteration complexity in several settings, and Kurdyka–Łojasiewicz-based full-sequence convergence under additional assumptions.

1. Terminological scope and historical placement

The earliest formulation in the supplied corpus is "An Inexact Riemannian Proximal Gradient Method" (Huang et al., 2021), which studies minimization of the sum of a differentiable function and a nonsmooth function on a Riemannian manifold. That paper emphasizes that earlier Riemannian proximal-gradient analyses relied on solving the Riemannian proximal mapping exactly, whereas exact solution can be too expensive or impracticable. Its contribution is an inexact variant with practical accuracy conditions for the proximal mapping and convergence results based on the Riemannian Kurdyka–Łojasiewicz property.

The label IManPL is introduced directly in (Zheng et al., 26 Aug 2025) for nonsmooth and nonconvex composite optimization on Riemannian manifolds. There, each outer iteration solves a convex subproblem on a tangent space inexactly under one of two adaptive stopping criteria, followed by an Armijo-style retraction step. The paper positions IManPL against exact manifold proximal linear (ManPL) and manifold proximal gradient (ManPG) methods.

A parallel development appears in (He et al., 16 Aug 2025), where the method is an inexact variable metric proximal linearization scheme for minimizing $f(x)+g(F(x))$ over an embedded submanifold. The variable metric matrix $Q_k$ generalizes isotropic proximal regularization and yields a strongly convex tangent-space subproblem with a practical inexactness criterion.

A different but related extension is "An Infeasible Method with Feasibility Safeguard for Nonsmooth Composite Optimization Over Manifolds" (Xie et al., 24 May 2026). That method, called FSIPL in the paper, allows infeasible iterates in a bounded neighborhood of the manifold and adds a correction step plus merit-function-based nonmonotone backtracking. A plausible implication is that IManPL is best viewed not as a single fixed algorithm, but as a family of inexact manifold proximal-linear schemes sharing a common local-model philosophy.

2. Problem classes and geometric setting

Across these works, the target problem is a composite optimization problem constrained to a manifold. In (Huang et al., 2021), the formulation is

$\min_{x\in \mathcal M} F(x):=f(x)+g(x),$

where $\mathcal M$ is a finite-dimensional Riemannian manifold, $f:\mathcal M\to\mathbb R$ is continuously differentiable, and $g:\mathcal M\to(-\infty,+\infty]$ is proper and lower-semicontinuous, possibly nonsmooth.

In (Zheng et al., 26 Aug 2025), the problem is

$\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$

where $\mathcal M\subset\mathbb R^n$ is a compact Riemannian submanifold with the metric induced by the ambient Euclidean inner product, $f$ and $f(x)+g(F(x))$0 are $f(x)+g(F(x))$1-smooth with Lipschitz continuous Jacobians, and $f(x)+g(F(x))$2 is convex, Lipschitz continuous, and has a tractable Euclidean proximal operator.

In (He et al., 16 Aug 2025), the composite structure is

$f(x)+g(F(x))$3

with $f(x)+g(F(x))$4 a $f(x)+g(F(x))$5-smooth embedded submanifold, $f(x)+g(F(x))$6 and $f(x)+g(F(x))$7 of class $f(x)+g(F(x))$8 on an open set containing $f(x)+g(F(x))$9, and $Q_k$0 closed and convex with an easy proximal mapping.

In (Xie et al., 24 May 2026), the manifold is represented extrinsically by nonlinear equality constraints,

$Q_k$1

with $Q_k$2 having full row rank in a neighborhood of $Q_k$3, and the objective is

$Q_k$4

where $Q_k$5 is continuously differentiable with $Q_k$6-Lipschitz gradient and $Q_k$7 is proper, closed, convex, and possibly nondifferentiable.

The shared geometric ingredients are the tangent space, the Riemannian gradient or Euclidean projection of the gradient onto the tangent space, and a retraction. In (Zheng et al., 26 Aug 2025), a retraction $Q_k$8 satisfies $Q_k$9 and $\min_{x\in \mathcal M} F(x):=f(x)+g(x),$0. In (He et al., 16 Aug 2025), the retraction estimates

$\min_{x\in \mathcal M} F(x):=f(x)+g(x),$1

hold for small $\min_{x\in \mathcal M} F(x):=f(x)+g(x),$2 on compact subsets. These formulations place IManPL within manifold optimization, but its distinguishing feature is the retention of the nonsmooth term in proximal form rather than full smoothing or full penalization.

3. Core proximal-linear construction

The algorithmic core is a strongly convex local model built on a tangent space or on a linearized feasible subspace. In (Huang et al., 2021), at iterate $\min_{x\in \mathcal M} F(x):=f(x)+g(x),$3 and with $\min_{x\in \mathcal M} F(x):=f(x)+g(x),$4, the tangent-space proximal-linear model is

$\min_{x\in \mathcal M} F(x):=f(x)+g(x),$5

and the exact local step would be

$\min_{x\in \mathcal M} F(x):=f(x)+g(x),$6

The next iterate is then $\min_{x\in \mathcal M} F(x):=f(x)+g(x),$7 for an inexact $\min_{x\in \mathcal M} F(x):=f(x)+g(x),$8.

In (Zheng et al., 26 Aug 2025), the Riemannian prox-linear model at $\min_{x\in \mathcal M} F(x):=f(x)+g(x),$9 and step size $\mathcal M$0 is defined on $\mathcal M$1 by

$\mathcal M$2

Under the stated Lipschitz and weak-convexity bounds, $\mathcal M$3 is $\mathcal M$4-strongly convex for $\mathcal M$5. The exact minimizer is denoted $\mathcal M$6, but the algorithm only requires an inexact point $\mathcal M$7.

In (He et al., 16 Aug 2025), the local model incorporates a variable metric:

$\mathcal M$8

with $\mathcal M$9 for some $f:\mathcal M\to\mathbb R$0. A concrete effective choice mentioned in the paper is

$f:\mathcal M\to\mathbb R$1

This preserves strong convexity while using first-order information from the composite map.

In (Xie et al., 24 May 2026), the subproblem is posed in the ambient space with linearized constraints:

$f:\mathcal M\to\mathbb R$2

Because $f:\mathcal M\to\mathbb R$3 has full row rank near $f:\mathcal M\to\mathbb R$4, this is strongly convex in $f:\mathcal M\to\mathbb R$5.

These formulations differ in geometry and notation, but their common structure is stable: linearize the smooth part, preserve the nonsmooth part in proximal or composite form, regularize quadratically, and solve only approximately.

4. Inexactness rules and stopping criteria

The defining feature of IManPL is that the local subproblem is not solved exactly. The simplest inexactness pattern in (Huang et al., 2021) imposes two conditions:

$f:\mathcal M\to\mathbb R$6

and

$f:\mathcal M\to\mathbb R$7

where $f:\mathcal M\to\mathbb R$8 is a user-chosen tolerance and $f:\mathcal M\to\mathbb R$9 is a continuous error-control function. The paper lists several common choices:

$g:\mathcal M\to(-\infty,+\infty]$0

These choices are linked to different convergence conclusions, including global stationarity, uniqueness of the limit, and local KL-based rates.

The adaptive stopping rules are made explicit in (Zheng et al., 26 Aug 2025). If $g:\mathcal M\to(-\infty,+\infty]$1 is the exact minimizer and

$g:\mathcal M\to(-\infty,+\infty]$2

is the subproblem residual, then two regimes are permitted. The low-accuracy condition (LACC) is

$g:\mathcal M\to(-\infty,+\infty]$3

while the high-accuracy condition (HACC) is

$g:\mathcal M\to(-\infty,+\infty]$4

The paper shows that HACC implies LACC for an appropriate $g:\mathcal M\to(-\infty,+\infty]$5. The stopping rules are called adaptive because their right-hand sides depend on the current outer iterate and current subproblem candidate.

The variable-metric paper (He et al., 16 Aug 2025) adopts a model-value-based criterion:

$g:\mathcal M\to(-\infty,+\infty]$6

where $g:\mathcal M\to(-\infty,+\infty]$7 is the exact minimizer and $g:\mathcal M\to(-\infty,+\infty]$8 is a small tolerance. Strong duality is used there to control $g:\mathcal M\to(-\infty,+\infty]$9 via a dual lower bound.

The infeasible safeguarded variant (Xie et al., 24 May 2026) uses a KKT-residual criterion. Writing

$\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$0

the algorithm requires

$\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$1

for a tolerance $\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$2 that decays to zero, for instance summably.

These criteria illustrate three distinct notions of inexactness: distance to the exact local minimizer, subproblem objective gap, and KKT residual. The literature in the supplied corpus treats all three as practical surrogates for exact local solution while preserving descent and stationarity conclusions.

5. Acceptance mechanisms, retractions, and feasibility control

After computing an inexact local step, IManPL variants use an acceptance mechanism to update the iterate. In (Huang et al., 2021), the update is the direct retraction

$\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$3

The paper also remarks that Steps 2–3 may be interpreted through an inexact Riemannian proximal mapping,

$\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$4

with the tangent-space model $\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$5 as its counterpart.

In (Zheng et al., 26 Aug 2025), the inexact tangent-space point $\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$6 is not accepted outright. Instead, an Armijo-style backtracking chooses the largest $\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$7 such that

$\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$8

and

$\min_{z\in\mathcal M} F(z):=f(z)+h(c(z)),$9

where

$\mathcal M\subset\mathbb R^n$0

The retraction is therefore integrated into a line-search globalization device rather than used only as a geometric projection.

The variable-metric formulation (He et al., 16 Aug 2025) employs an Armijo-type accept/reject test of the form

$\mathcal M\subset\mathbb R^n$1

If this fails, $\mathcal M\subset\mathbb R^n$2 is increased by a factor $\mathcal M\subset\mathbb R^n$3 and the subproblem is resolved. This produces a coupled metric-selection and acceptance loop.

The most elaborate acceptance mechanism appears in (Xie et al., 24 May 2026), where iterates may be infeasible. After forming

$\mathcal M\subset\mathbb R^n$4

the method either applies a gradient step on the squared violation function $\mathcal M\subset\mathbb R^n$5,

$\mathcal M\subset\mathbb R^n$6

or projects back to the manifold,

$\mathcal M\subset\mathbb R^n$7

Acceptance is then governed by the merit function

$\mathcal M\subset\mathbb R^n$8

together with a nonmonotone backtracking condition involving $\mathcal M\subset\mathbb R^n$9 and a summable allowance $f$0.

A central distinction therefore separates feasible and infeasible variants. The feasible algorithms maintain iterates on the manifold via retraction, whereas FSIPL permits iterates in a tubular neighborhood $f$1 and makes bounded infeasibility part of the algorithmic design.

6. Convergence theory, complexity, and empirical behavior

The convergence theory is built around sufficient decrease, vanishing model steps, and stationarity of limit points. In (Huang et al., 2021), if $f$2 is bounded below, the relevant sublevel set is compact, $f$3 is $f$4-retraction-smooth on that set, $f$5, and

$f$6

then $f$7 is nonincreasing, $f$8 has at least one accumulation point, and every accumulation point $f$9 satisfies

$f(x)+g(F(x))$00

Under the Riemannian KL property and the stronger inexactness

$f(x)+g(F(x))$01

the whole sequence converges to a single $f(x)+g(F(x))$02. If the desingularizing function is $f(x)+g(F(x))$03 with $f(x)+g(F(x))$04, then the paper states: $f(x)+g(F(x))$05 gives finite termination, $f(x)+g(F(x))$06 gives local linear convergence, and $f(x)+g(F(x))$07 gives a sublinear rate.

In (Zheng et al., 26 Aug 2025), the key theorem gives $f(x)+g(F(x))$08 outer iteration complexity when $f(x)+g(F(x))$09 is used as the stationarity criterion. The paper also states that any limit point exists by compactness and satisfies the first-order stationarity condition

$f(x)+g(F(x))$10

Its sufficient-decrease lemma yields

$f(x)+g(F(x))$11

once the line-search accepts $f(x)+g(F(x))$12.

In (He et al., 16 Aug 2025), the stated results include finite inner-loop termination, sufficient decrease

$f(x)+g(F(x))$13

vanishing steps $f(x)+g(F(x))$14, stationarity of cluster points, and $f(x)+g(F(x))$15 outer iterations together with $f(x)+g(F(x))$16 calls to a subproblem solver for obtaining an $f(x)+g(F(x))$17-stationary point. If the inner solver is a dual first-order method, the paper further deduces

$f(x)+g(F(x))$18

Under a KL property of the constructed potential function on the set of cluster points, the whole sequence converges. For KL exponent $f(x)+g(F(x))$19, the paper states locally linear decay for $f(x)+g(F(x))$20 and a sublinear rate for $f(x)+g(F(x))$21.

In (Xie et al., 24 May 2026), the main guarantees are finite termination of backtracking, subsequential convergence to KKT points, and an $f(x)+g(F(x))$22 outer iteration complexity bound under summable inexactness $f(x)+g(F(x))$23 and summable nonmonotone allowance $f(x)+g(F(x))$24. If a suitable auxiliary function satisfies a Kurdyka–Łojasiewicz property, then the full sequence converges to a single stationary point.

Empirically, the principal testbeds are sparse PCA and sparse spectral clustering. The sparse PCA model in (Huang et al., 2021) is

$f(x)+g(F(x))$25

on the Stiefel manifold, with $f(x)+g(F(x))$26 having i.i.d. $f(x)+g(F(x))$27 entries, columns centered and scaled, $f(x)+g(F(x))$28, small $f(x)+g(F(x))$29 such as $f(x)+g(F(x))$30, $f(x)+g(F(x))$31, $f(x)+g(F(x))$32, polar-decomposition retraction, and BB-step initialization for $f(x)+g(F(x))$33. The reported observation is that more accurate subproblem solves reduce outer iterations but increase per-iteration cost; IRPG-L converges in the fewest iterations but often costs the most time overall, while IRPG-G offers the lowest total time for moderate accuracy.

The 2025 IManPL paper (Zheng et al., 26 Aug 2025) reports sparse spectral clustering on $f(x)+g(F(x))$34 with synthetic datasets and nine single-cell RNA-seq sets, and sparse PCA with synthetic $f(x)+g(F(x))$35. In representative SSC timings, ManPL versus IManPL-(LACC) versus IManPL-(HACC) is reported as $f(x)+g(F(x))$36, $f(x)+g(F(x))$37, and $f(x)+g(F(x))$38 seconds on Synthetic 1, and $f(x)+g(F(x))$39, $f(x)+g(F(x))$40, and $f(x)+g(F(x))$41 seconds on Synthetic 2. On the Macosko dataset the reported times are $f(x)+g(F(x))$42, $f(x)+g(F(x))$43, and $f(x)+g(F(x))$44 seconds, and on Zeisel $f(x)+g(F(x))$45, $f(x)+g(F(x))$46, and $f(x)+g(F(x))$47 seconds. The paper states that IManPL achieves roughly an order-of-magnitude speed-up when $f(x)+g(F(x))$48 is large, with essentially identical clustering accuracy as measured by NMI. For SPCA, adaptive IManPL-ASSN(LACC) is reported to match or outperform ManPG-Ada in CPU time by $f(x)+g(F(x))$49–$f(x)+g(F(x))$50 when subproblems are harder, and APG-based subproblem solves are generally slower than ASSN.

The feasibility-safeguarded variant (Xie et al., 24 May 2026) evaluates sparse PCA and sparse spectral clustering on the Stiefel manifold and reports that it outperforms ARPG and SLPG in CPU time on sparse PCA while achieving the same objective and constraint accuracy, and that only a few projection steps are needed. On sparse spectral clustering, it is compared with RADA-PGD and MPGDA and is reported to achieve similar cluster objectives at roughly half the CPU time and fewer iterations on synthetic data.

Taken as a whole, the literature portrays IManPL as a manifold analogue of inexact prox-linear optimization in which local models are solved only to controlled accuracy. The theoretical record in the supplied papers consistently supports global convergence to stationary points, KL-based refinement to full-sequence convergence and local rates, and strong practical performance when subproblem accuracy is matched to the outer progress requirement.