Proximal Alternating Reweighted Minimization (PARM)

• PARM is an optimization framework that alternates block updates with proximal regularization and iteration-dependent reweighting to handle nonconvex, nonsmooth penalties.
• It integrates alternating minimization, proximal regularization, and surrogate majorization to transform challenging subproblems into weighted convex forms.
• PARM has been effectively applied in imaging tasks such as multiplicative-noise removal and deblurring, demonstrating measurable improvements in PSNR and SSIM.

Proximal Alternating Reweighted Minimization (PARM) denotes a class of optimization procedures that combine blockwise alternation, proximal regularization, and iteration-dependent reweighting of a difficult penalty or surrogate. In the literature considered here, the acronym appears explicitly in the multiplicative-noise removal method of Chen, Tang, and Zhao, where PARM is the solver for a nonconvex, nonsmooth nonlocal low-rank model built from patch-group matrices and a generalized rank surrogate (Liu et al., 2020). Closely related methods appear under other names, notably variable-metric composite PALM and proximal iterative re-weighted alternating minimization, which share the same structural ingredients while differing in model class, geometry, and convergence analysis (Yashtini, 2022, Adam et al., 2023).

1. Terminological scope and classification

Among the papers considered here, the explicit acronym PARM is attached to a nonlocal low-rank despeckling algorithm for multiplicative noise removal (Liu et al., 2020). Other papers are algorithmically close but use different names, and several proximal alternating methods are adjacent rather than direct matches because they omit explicit reweighting.

Paper	Core mechanism	Relation to PARM
"Multiplicative Noise Removal: Nonlocal Low-Rank Model and Its Proximal Alternating Reweighted Minimization Algorithm" (Liu et al., 2020)	Alternating updates for image and patch-group matrices with weighted nuclear-norm subproblems	Canonical explicit PARM
"Variable Metric Composite Proximal Alternating Linearized Minimization for Nonconvex Nonsmooth Optimization" (Yashtini, 2022)	PALM-type alternation with concave-composite MM weights $(\phi'\circ\psi)(y^k)$	Very close conceptual relative
"Accelerated Proximal Iterative re-Weighted $\ell_1$ Alternating Minimization for Image Deblurring" (Adam et al., 2023)	IRL1-based alternating minimization with proximal shrinkage and acceleration	Direct PARM-type method, but acronym not used
"The Proximal Alternating Minimization Algorithm for two-block separable convex optimization problems with linear constraints" (Bitterlich et al., 2018)	Variable-metric proximal alternating scheme under linear constraints	Foundational neighboring method, not reweighted
"Practical Matrix Completion and Corruption Recovery using Proximal Alternating Robust Subspace Minimization" (Wang et al., 2013)	Proximal alternation with rank and $\ell_0$ constraints plus safeguard majorization	Proximal alternating, but not fundamentally reweighted
"A Learned Proximal Alternating Minimization Algorithm and Its Induced Network for a Class of Two-block Nonconvex and Nonsmooth Optimization" (Chen et al., 2024)	Smoothing, residual PALM updates, and BCD safeguard	Learned proximal alternating, not reweighted

This pattern suggests that the term is used narrowly when the alternating-proximal scheme includes an explicit weight update derived from a concave surrogate or iterative reweighted approximation, and more broadly only by analogy when a method uses alternating proximal surrogates without such weights.

2. Structural principles

A generic PARM method usually means four things: variables are split into blocks, the algorithm alternates over those blocks, each difficult subproblem is proximalized or proximal-linearized, and a nonconvex penalty is replaced at each iteration by a weighted surrogate computed from the current iterate (Adam et al., 2023). In that sense, PARM occupies the overlap of alternating minimization, proximal minimization, and optimization-transfer or majorization-minimization.

A foundational theoretical lens is the equivalence of alternating minimization (AM), proximal minimization algorithms (PMA), and majorization-minimization (MM). In PMA, one minimizes

$f(x)+d(x,x^{k-1}),$

where $d(x,z)\ge 0$ and $d(x,x)=0$. In MM, one minimizes a majorizer $g(x\mid z)$ satisfying

$g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$

which is equivalent to PMA through

$d(x,z)=g(x\mid z)-f(x).$

For AM on a joint objective $\Phi(x,y)$, eliminating the auxiliary block induces

$\ell_1$0

Accordingly, a PARM iteration can be read simultaneously as blockwise minimization, proximal regularization, and surrogate minimization (Byrne et al., 2015).

The reweighted component typically enters through tangent majorization of a concave penalty. In CPALM, the nonsmooth block has the form

$\ell_1$1

with $\ell_1$2 concave, strictly increasing, and differentiable, and $\ell_1$3 convex. Concavity gives

$\ell_1$4

so the $\ell_1$5-subproblem becomes a proximal-linearized minimization with the weight

$\ell_1$6

multiplying the convex penalty $\ell_1$7. This is the clearest abstract reweighting mechanism in the literature surveyed here (Yashtini, 2022).

3. Canonical explicit PARM: nonlocal low-rank multiplicative-noise removal

The most explicit PARM formulation in the present corpus arises in multiplicative-noise removal for $\ell_1$8-look intensity images. The observation model is

$\ell_1$9

where $\ell_0$0 is the unknown clean intensity image, $\ell_0$1 is the observed noisy image, and each $\ell_0$2 follows a Gamma law with mean $\ell_0$3 and variance $\ell_0$4. The method exploits nonlocal self-similarity: for each reference patch, similar patches are stacked into a matrix

$\ell_0$5

whose columns are expected to be highly correlated and therefore approximately low-rank.

After introducing auxiliary patch-group variables $\ell_0$6, the optimization model is

$\ell_0$7

with the generalized singular-value penalty

$\ell_0$8

The implementation uses the smooth concave surrogate

$\ell_0$9

so the regularizer is nonconvex and rank-like but remains differentiable on $f(x)+d(x,x^{k-1}),$0.

The PARM algorithm alternates between $f(x)+d(x,x^{k-1}),$1-updates and an $f(x)+d(x,x^{k-1}),$2-update. The reweighting comes from linearizing the concave function $f(x)+d(x,x^{k-1}),$3 at the current singular values: $f(x)+d(x,x^{k-1}),$4 This converts the $f(x)+d(x,x^{k-1}),$5-subproblem into a proximal problem for a weighted nuclear norm,

$f(x)+d(x,x^{k-1}),$6

and weighted singular value thresholding yields the closed-form update. For the logarithmic surrogate,

$f(x)+d(x,x^{k-1}),$7

The image update aggregates the denoised patch groups through a weighted proximal map. With

$f(x)+d(x,x^{k-1}),$8

the $f(x)+d(x,x^{k-1}),$9-step is

$d(x,z)\ge 0$0

The theoretical algorithm keeps the extraction operators $d(x,z)\ge 0$1 fixed, which is essential for the convergence proof. A stronger practical variant recomputes the patch extraction by block matching at each iteration and drops the proximal dependence on $d(x,z)\ge 0$2; this improves performance empirically but is not covered by the formal proof.

The theoretical analysis assumes that $d(x,z)\ge 0$3 is inf-bounded, proper, and lower semicontinuous; $d(x,z)\ge 0$4 is monotonically increasing, concave, continuously differentiable with Lipschitz derivative; the full objective is coercive; and the proximal parameters remain bounded away from zero and infinity. Under these assumptions, the algorithm satisfies sufficient descent,

$d(x,z)\ge 0$5

a relative error bound

$d(x,z)\ge 0$6

and, if $d(x,z)\ge 0$7 is a Kurdyka-\L ojasiewicz (KL) function, finite length and convergence of the whole sequence to a critical point. Here “global convergence” means convergence from arbitrary initialization to a critical point, not to a global minimizer. The practical dynamic-block-matching version is outside that proof. Empirically, both theoretical and practical variants outperform SAR-BM3D and several classical baselines in PSNR and SSIM; the reported PSNR gain over SAR-BM3D is roughly $d(x,z)\ge 0$8 dB for $d(x,z)\ge 0$9, $d(x,x)=0$0 dB for $d(x,x)=0$1, and $d(x,x)=0$2 dB for $d(x,x)=0$3 on standard images (Liu et al., 2020).

4. PARM-type formulations beyond patch-group low rank

A direct image-deblurring instance of the same paradigm is the quadratic-penalty split

$d(x,x)=0$4

in which $d(x,x)=0$5 approximates $d(x,x)=0$6. The nonconvex term $d(x,x)=0$7 is replaced by an iterative reweighted $d(x,x)=0$8 surrogate

$d(x,x)=0$9

yielding a weighted soft-thresholding step

$g(x\mid z)$0

The $g(x\mid z)$1-block remains a convex quadratic solve,

$g(x\mid z)$2

implemented efficiently by FFT. The accelerated variant APIRL1-AM adds the FISTA/Nesterov schedule

$g(x\mid z)$3

In experiments on $g(x\mid z)$4 images with Gaussian blur of size $g(x\mid z)$5 and $g(x\mid z)$6, APIRL1-AM preserves essentially the same PSNR and SSIM as PIRL1-AM while reducing iteration count and CPU time; for example, on BSNR 30 Peppers the two methods report PSNR $g(x\mid z)$7 and SSIM $g(x\mid z)$8, but APIRL1-AM uses $g(x\mid z)$9 iterations and $g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$0 s versus $g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$1 iterations and $g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$2 s (Adam et al., 2023).

A more abstract PARM-type formulation is CPALM for

$g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$3

where $g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$4 is smooth, $g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$5 is proper lower semicontinuous, $g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$6 is convex, and $g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$7 is concave, strictly increasing, and differentiable. The algorithm alternates between

$g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$8

and

$g(x\mid z)\ge f(x), \qquad g(x\mid x)=f(x),$9

The explicit weights

$d(x,z)=g(x\mid z)-f(x).$0

arise from the tangent majorant of $d(x,z)=g(x\mid z)-f(x).$1, so the $d(x,z)=g(x\mid z)-f(x).$2-update is a reweighted convex subproblem in exactly the MM sense. The multiblock extension defines

$d(x,z)=g(x\mid z)-f(x).$3

which is even closer to the usual PARM vocabulary. The paper demonstrates this scheme on parallel magnetic resonance image reconstruction, including a log-sum regularization model and an $d(x,z)=g(x\mid z)-f(x).$4 model with explicit weights such as

$d(x,z)=g(x\mid z)-f(x).$5

depending on the chosen penalty (Yashtini, 2022).

5. Convergence theory and analytical patterns

The available analyses indicate three recurring components in rigorous PARM and PARM-type theory: sufficient descent, a relative-error or subgradient estimate, and a KL-type argument or related Lyapunov telescoping mechanism. In the explicit despeckling PARM, the descent estimate

$d(x,z)=g(x\mid z)-f(x).$6

and the subgradient control

$d(x,z)=g(x\mid z)-f(x).$7

lead, under the KL property, to finite length and convergence of the whole sequence to a critical point (Liu et al., 2020). In CPALM, the corresponding ingredients are

$d(x,z)=g(x\mid z)-f(x).$8

and

$d(x,z)=g(x\mid z)-f(x).$9

from which bounded sequences globally converge to a critical point, again under a KL assumption (Yashtini, 2022).

At a more foundational level, the AM/PMA/MM equivalence yields objective-value convergence criteria that are independent of any specific reweighting formula. If the induced proximal distance satisfies a SUMMA- or SUMMA2-type inequality such as

$\Phi(x,y)$0

then the decreasing objective sequence converges to the infimum of the objective. This lens is particularly useful when a PARM method is formulated as alternating minimization over primal variables and implicit surrogate weights, since the same step can be reinterpreted as proximal minimization with an induced distance term (Byrne et al., 2015).

Not all PARM-type methods in the surveyed literature are equally well supported theoretically. PIRL1-AM and APIRL1-AM are algorithmically direct examples of proximal alternating reweighted minimization, but the paper does not provide a full convergence theorem for the nonconvex alternating scheme: it does not prove monotonicity of the penalized objective, boundedness of iterates, or convergence to critical points, and the statement that APIRL1-AM “exhibits the optimal $\Phi(x,y)$1 rate of convergence” is inherited from the acceleration strategy for convex proximal-gradient subproblems rather than established for the full alternating nonconvex model (Adam et al., 2023).

6. Neighboring families, boundaries, and extensions

Several proximal alternating methods are closely related to PARM yet should not be identified with it. Proximal AMA treats convex two-block problems with linear constraints,

$\Phi(x,y)$2

and adds variable-metric proximal terms $\Phi(x,y)$3 and $\Phi(x,y)$4 so that difficult subproblems can reduce to standard proximal operators. A key implementability device is the choice

$\Phi(x,y)$5

which turns the $\Phi(x,y)$6-subproblem into an explicit proximal step. The method alternates, linearizes smooth terms, and uses variable metrics, but it does not update reweighting coefficients of the kind used in IRL1 or concave-penalty MM (Bitterlich et al., 2018).

PARSuMi is another near neighbor. Its model for practical matrix completion and corruption recovery imposes the rank and corruption constraints directly,

$\Phi(x,y)$7

and alternates between a rank-constrained $\Phi(x,y)$8-update and an $\Phi(x,y)$9-constrained $\ell_1$00-update with proximal terms and a majorization safeguard. The method converges to a stationary point of the nonconvex, nonsmooth model, but the core algorithm is not reweighted; only a peripheral heuristic uses iterative reweighted Huber minimization (Wang et al., 2013).

Recent extensions preserve the proximal alternating backbone while changing the source of adaptivity. LPAM smooths a two-block nonsmooth nonconvex objective, performs residual-style PALM updates, and falls back to BCD safeguards when acceptance conditions fail; the analysis proves that a subsequence has accumulation points, each of which is Clarke stationary, but there is no formal reweighting mechanism (Chen et al., 2024). Two-step inertial Bregman PALM replaces Euclidean proximal terms by Bregman distances and adds two-step inertial extrapolation, yielding whole-sequence convergence under a KL assumption for nonconvex nonsmooth problems, yet again without explicit reweighting (Guo et al., 2023). Majorized PAM with subspace correction achieves closed-form columnwise updates for low-rank composite factorization models and proves full convergence of iterate and column-subspace sequences under KL conditions and an additional structural assumption, but its varying diagonal quantities stem from surrogate geometry rather than an explicit reweighting rule (Tao et al., 2024). Adaptive alternating minimization derived from dual adaptive proximal gradient similarly changes only the proximal parameter $\ell_1$01, not objective weights, although it extends classical AMA beyond standard globally strongly convex settings (Latafat et al., 2023).

Taken together, these developments delimit PARM rather precisely. The common backbone is proximal alternating minimization with surrogate-based simplification; the distinguishing feature of PARM, in the strict sense supported by the explicit despeckling paper and the closest related formulations, is the presence of iteration-dependent weights generated from a concave surrogate, iterative reweighted $\ell_1$02, or weighted singular-value approximation. Where that ingredient is absent, the method is better classified as proximal alternating minimization, PALM, AMA, or majorized PAM rather than PARM proper.