Discretely-constrained deep network for weakly supervised segmentation

Christian Desrosiers; Hoel Kervadec; Ismail Ben Ayed; Jizong Peng; Jose Dolz; Marco Pedersoli

REVIEW 4 major objections 5 minor 35 references

Discretely-constrained deep network for weakly supervised segmentation

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A CNN can be trained with discrete size and boundary-length constraints via ADMM, reaching near-fully-supervised Dice scores on weakly annotated medical images.

desk verdict A solid ADMM-based method for injecting discrete size and boundary priors into CNN training, with correct discrete solvers and consistent gains over penalty baselines, but the advertised 'near full supervision' result depends on giving the network the exact ground-truth foreground size. read the letter →

arxiv 1908.05770 v1 pith:PWX7QNLL submitted 2019-08-15 cs.CV

Jizong Peng , Hoel Kervadec , Jose Dolz , Ismail Ben Ayed , Marco Pedersoli , Christian Desrosiers This is my paper · ORCID

classification cs.CV

keywords weaklysupervisedsegmentationdiscreteconstraintsADMMsizeboundarylengthregularizationmedicalimageconvolutionalneuralnetworksgraphcuts

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that weakly supervised segmentation networks should be trained by enforcing constraints on hard binary labels, not on soft probability maps, and shows a concrete way to do it: split the training objective with the alternating direction method of multipliers (ADMM) into continuous network updates and discrete proposal updates. On cardiac and prostate MRI, with only a few foreground and background seeds per volume, the method reports Dice scores well above a continuous squared-penalty baseline and close to fully supervised training (left ventricle 0.901 versus 0.927 at the tightest size bound). The paper also shows that combining a discrete size constraint with boundary-length (weighted-Potts) regularization is what makes the gain robust, since size alone under-segments complex structures and CRF regularization alone has a shrinking bias. If the method works as reported, it gives practitioners a way to inject anatomical priors into deep segmentation without pixel-level annotations.

What carries the argument

The driving object is an ADMM-augmented Lagrangian with two auxiliary binary proposal vectors per image, $\hat{y}_i$ and $\tilde{y}_i$, tied to the network output by equality constraints $s_i(\theta)=\hat{y}_i$ and $s_i(\theta)=\tilde{y}_i$. The regularized proposal $\hat{y}_i$ is updated by minimizing a weighted Potts (boundary-length) energy with graph cuts, which is globally optimal for submodular pairwise terms; the size proposal $\tilde{y}_i$ is updated by solving a cardinality-constrained selection problem—pick between $S_{\min}$ and $S_{\max}$ pixels with largest utility—whose optimum is a simple ranking. The network parameters are updated by SGD, interpreted as a proximal ADMM update, and the multipliers are updated in the standard ADMM way. This machinery is what lets constraints act on hard labels rather than soft probabilities, avoiding gradient saturation and allowing large, optimal discrete steps.

What would settle it

Reset the size bounds to values derived from an independent anatomical prior rather than from the ground-truth volume and record the augmented Lagrangian after every epoch; if the Dice advantage over the penalty baseline disappears when the true size is unknown, or if the Lagrangian fails to decrease across epochs, the central claim is undermined.

Watch

Extended reading notes

Core claim

The central claim is that discrete size constraints and boundary-length regularization can be imposed on a CNN during training through an ADMM variable-splitting scheme, and that doing so beats the standard continuous penalty approach. Two binary proposal vectors are introduced for each image, one coupled to a pairwise boundary-length regularizer and one coupled to a constraint on the number of foreground pixels, with equality constraints to the network's soft predictions enforced through scaled Lagrange multipliers. Each proposal update is solved exactly: the regularized one as a submodular graph-cut and max-flow problem, the size-constrained one by sorting pixel utilities and selecting the top S_min up to S_max pixels. The network parameters are updated with ordinary mini-batch SGD, which the paper treats as a proximal ADMM step. The result, on left and right ventricle and prostate MRI with sparse seed annotations, is consistent gains over the penalty baseline in Dice, faster and more stable constraint satisfaction, and near-fully-supervised performance when the size bound is tight.

Load-bearing premise

The central claim rests on treating a fixed number of mini-batch SGD steps as a proximal ADMM update that decreases the augmented Lagrangian, and on computing the tightest reported size bounds from the exact ground-truth foreground volume, so the headline "weak" supervision includes the true target size.

Editorial extensions

If this is right

If the central claim holds, weakly supervised cardiac and prostate segmentation can be trained from a few seeds per volume to within roughly 3 Dice points of full supervision on the left ventricle when the target size is known.
The discrete ADMM formulation satisfies size constraints more stably than the continuous penalty method, with predicted-to-real size ratios staying inside the imposed bounds during training.
Combining size constraints with boundary-length regularization is necessary for complex, narrow structures like the right ventricle, where size alone is insufficient and CRF regularization alone over-shrinks.
The same variable-splitting recipe can be applied to other discrete priors without changing the network training loop, at small extra cost: one max-flow and one sort per image per epoch.
The best validation accuracy occurs early in training, around epoch 50, before convergence, which suggests early stopping is beneficial for this scheme.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An untested but natural extension is to replace the two proposal types with other discrete priors the paper cites, such as star-shape, compactness, or histogram constraints, provided each can still be solved to optimality in its proposal update.
The headline epsilon=0 result uses bounds computed from the ground-truth foreground volume, so transferring the method to a genuinely unseen organ would require deriving bounds from population statistics; the reported 10-40% relaxation results suggest the method degrades gracefully as bounds loosen.
Because the paper identifies oscillation risk when the ADMM penalty is large, adapting the penalty parameter per epoch, as standard ADMM allows, could stabilize training and is directly testable with the reported setup.
The observed early-best validation peak hints that the discrete proposals act as a strong curriculum signal; a testable variant would anneal the proposal influence over training to retain the early boost while avoiding late-training instability.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Know this paper for the ADMM splitting idea: SGD updates the network, while two discrete subproblems handle size constraints and boundary-length regularization. The discrete solvers are genuine contributions - graph cuts for the Potts term, and a simple ranking method for the size-constrained knapsack. The math in Eqs. (10)-(13) checks out, and the experiments show consistent improvements over the continuous penalty baseline on all three tasks. That part is solid.

The 'near full supervision' headline needs a qualifier, and it is a real one. The best numbers (LV 0.901 vs 0.927, RV 0.730 vs 0.870, prostate 0.807 vs 0.873) all come at ε=0, where the size bound [Smin,Smax] is exactly the ground-truth foreground size of each volume. That is not a weak annotation; it is the target size. At ε=10% the same CRF+size configuration drops to 0.884/0.719/0.795, and at ε=40% to 0.879/0.691/0.771. The paper is transparent about this in Table 2 and in the text, but the abstract's 'performance near full supervision' does not carry the caveat. The method still beats the penalty baseline at looser bounds, so the core approach is not undermined, but the strongest claim is conditional on exact size knowledge.

Two secondary concerns. First, the ADMM convergence discussion is thin: the paper calls the SGD parameter update a proximal ADMM step citing [30,31], but does not verify the required conditions (sufficient decrease, Lipschitz smoothness) for this non-convex setting. Empirically the training looks stable, so this is a theory gap rather than a demonstrated failure. Second, the hyperparameters (µ, λ, σ) are tuned per task on the validation sets, and the reported numbers are best-case. That is standard practice, but it weakens the comparative claims slightly. No code or splits are released, which hurts reproducibility but is not a fatal flaw for a 2019 preprint.

The citation pattern is fair. They build on Pathak et al. and Kervadec et al., and they cite the discrete-continuous ADMM line of work. The 'first' claim is a bit assertive given references [27,28], but they do distinguish their weakly-supervised training setting.

Who gets value: anyone doing weakly supervised medical segmentation, especially with size priors or boundary regularization. The method is clean, the exposition is clear, and the improvements over a strong penalty baseline are credible. I would send it to peer review. My recommendation is a revise that explicitly states the ε=0 setting is exact-size supervision in the abstract and adds an honest caveat about the unverified ADMM convergence conditions.

Referee Report

4 major / 5 minor

Summary. The paper addresses weakly supervised medical image segmentation by training a CNN under two discrete priors: a lower/upper bound on foreground size and a boundary-length (weighted Potts) regularization term. Because these terms are non-differentiable and discrete, the authors split the training objective with ADMM (Eq. 8), introducing two binary 'proposal' vectors per image that are decoupled from the continuous network output. The network parameters are updated by mini-batch SGD (Section 3.4.1); the CRF-regularized proposal is updated by solving a submodular graph-cut problem (Section 3.4.2); and the size-constrained proposal is updated by ranking pixels, since the subproblem is an equal-weight knapsack (Section 3.4.3). Experiments on three tasks (cardiac LV and RV from ACDC; prostate from PROMISE12) compare 'CRF only', 'size only', and 'CRF+size' variants against the continuous penalty baseline of Kervadec et al. [10] and a fully supervised model, reporting Dice scores, constraint-satisfaction curves, and sensitivity to the ADMM penalty parameter. The central quantitative claim is that at tight size bounds (epsilon=0) the method approaches fully supervised performance (e.g., LV 0.901 vs 0.927).

Significance. If the results hold, the paper makes a useful methodological contribution: it is, to my knowledge, the first to plug exact discrete optimizers (submodular graph cuts and cardinality-constrained ranking) into CNN training through ADMM variable splitting, and the two proposal updates are derived cleanly in Eqs. (10)-(13), each with a polynomial-time solver that is globally optimal for the subproblem. The paper also reports a complete experimental protocol (architectures, learning schedules, grid-searched hyperparameters), including an ablation of the two priors (Tables 2-3) and a constraint-satisfaction analysis (Fig. 6), which is good for reproducibility and goes beyond single-configuration reporting. The main qualification is that the 'near full supervision' headline is achieved only when the size constraint encodes the exact per-volume ground-truth foreground size (epsilon=0), which is a strong annotation; at epsilon=10-40% the gains over the baseline are smaller but still consistent for the CRF+size variant.

major comments (4)

[Section 4.1 / Table 2] The paper's strongest quantitative claim, 'performance near to full supervision,' is conditioned on epsilon=0, where the size bounds are [S_real, S_real], i.e., the exact per-volume ground-truth foreground size is supplied to the method at training time. This is a per-volume ground-truth quantity, not a weak prior; the same configuration drops to 0.884 (LV), 0.719 (RV), and 0.795 (prostate) at epsilon=10%, and to 0.879/0.691/0.771 at epsilon=40%. The abstract and conclusion should state this conditioning explicitly, and the discussion should be rebalanced around the epsilon=10-40% regime, where the contribution of the discrete formulation is still visible but more modest.
[Section 3.4.1 / Algorithm 1] The text states that the mini-batch gradient-descent update of the network parameters 'corresponds to a proximal variant of ADMM' citing references [30,31], and Section 3.5 further claims that 'the only requirement for convergence is that the update of network parameters decreases the overall loss [29].' For the non-convex network objective, the conditions of the cited proximal-ADMM analyses (Lipschitz smoothness of the smooth part, sufficient decrease of the augmented Lagrangian) are not verified, and the authors themselves note in Section 6 that strong penalty parameters cause the discrete proposals to oscillate. Please either provide empirical convergence evidence (e.g., augmented-Lagrangian or constraint-violation curves over epochs) or explicitly frame the outer loop as a heuristic whose stability is demonstrated empirically in Figures 3 and 6.
[Section 3.4.3, Eqs. (12)-(13)] There is a sign inconsistency in the size-constrained proposal update. Minimizing (1/2)||y_tilde - (s(theta) + u_tilde)||^2 over binary y_tilde, the problem stated in Eq. (12) and following from the augmented term in Eq. (8), is equivalent to maximizing sum_p (s_p(theta) + u_tilde_p - 1/2) y_tilde_p, not sum_p (s_p(theta) - u_tilde_p - 1/2) y_tilde_p as printed in Eq. (13). The printed sign also conflicts with the multiplier update u_tilde := u_tilde + (s - y_tilde) in Algorithm 1. Since the optimality of the size update is one of the two 'guarantee of solution optimality' claims of the paper, this inconsistency should be resolved (it may be a typo) rather than left as is.
[Section 5 / Table 2] The reported numbers are single runs with no variance or significance tests, which does not by itself support the adjective 'significant' used in the summary of Section 5.1; moreover, the epsilon-sweep for the CRF+size variant is non-monotonic (RV 0.734 at epsilon=20% exceeds 0.730 at epsilon=0%, and prostate 0.809 at epsilon=20% exceeds 0.807 at epsilon=0%), which is unexplained. At minimum, the paper should report repeated runs or paired significance tests for the headline comparisons and add a sentence addressing the non-monotonicity.

minor comments (5)

[Abstract] The arXiv abstract says experiments were run 'on a benchmark cardiac segmentation dataset,' while the full-text abstract and Section 4 describe two datasets (ACDC and PROMISE12); the abstracts should be aligned.
[Section 4.1 / Figures 2-3] Figure numbering is inconsistent: a figure captioned with annotation examples is labeled Figure 2, but the text says 'Figure 3 shows examples of annotations and ground truth,' and Figure 3 is also used later for the validation Dice curves; please renumber the figures.
[Section 5.1 / Algorithm 1] The training loop returns parameters after the final epoch, but Section 5.1 reports that the highest validation accuracy is obtained around epoch 50; please specify the model-selection rule used for the numbers in Table 2 (final epoch versus best validation epoch).
[Section 4.1] There are minor typos in the dataset description: 'well-defined defined pathologies' and 'The images resolution ranges' should be corrected.
[Section 3.5] The claim of 'negligible computational overhead' for the once-per-epoch proposal updates would be more convincing with a short empirical note on wall-clock training time, given that each CRF update is a max-flow solve on 256x256 images.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central claim is an ADMM-based training method, and no predicted quantity reduces by construction to an input; the ε=0 size protocol is disclosed and is a strong annotation rather than a fitted prediction.

full rationale

I checked the derivation chain and found no step in which a claimed prediction or derived result is equivalent by construction to an input or to a self-citation. The potential concern is Section 4.1, where size bounds are defined as [Smin,Smax] = [(1−ε)×Sreal, (1+ε)×Sreal]; at ε=0 this is the exact per-volume ground-truth foreground count. This is a strong and disclosed training annotation, but it is an input to the constrained optimization, not a fitted parameter renamed as a prediction: the reported Dice still depends on localization and boundary recovery, and the same bounds are given to the penalty baseline. The ADMM splitting in Section 3.4 is a method contribution with exact discrete subproblem solves via max-flow and ranking, not a derivation that assumes its own conclusion. Section 3.4.1 explicitly admits that the network parameter update is only a proximal variant of ADMM and cites [30,31] without verifying sufficient-decrease or smoothness conditions; that is an unverified convergence assumption, a correctness risk rather than circularity. The Discussion's instability caveat for strong penalties is likewise a robustness limitation, not a circular dependency. Self-citations such as [10], [17], [25], and [28] appear as baselines or related work and are not load-bearing. The method is evaluated on external benchmarks (ACDC and PROMISE) against a fully supervised baseline and an existing penalty method, so the central empirical claims stand independently. Therefore no circular step is present, and the circularity score is 0.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The free parameters lambda, sigma, and mu are chosen by hand or by grid search on the validation set, and the reported performance is sensitive to mu (Table 3). The method relies on two structural axioms: submodularity of the Potts model for exact graph-cut optimization, and an unverified proximal-ADMM assumption for the non-convex network update. The evaluation also assumes that tight size bounds close to the true foreground size are available, which is only realistic for some applications. No new physical entities are postulated.

free parameters (3)

lambda (CRF weight) = 100 (LV/RV), 1000 (prostate)
Set by hand per dataset; controls the strength of the boundary-length regularization term in Eq. (1) and Eq. (8).
sigma (edge weight scale) = 1e-4 (LV), 1e-5 (RV), 1e-6 (prostate)
Chosen by hand to account for image contrast differences in the Potts weight w_{p,q}; affects the boundary regularization prior.
mu (ADMM penalty parameter) = 1 for CRF-only and size-only, 10 for CRF+size (best on validation)
Selected via grid search per task on the validation set. Table 3 shows Dice varies strongly with mu (e.g., size-only LV drops from 0.871 to 0.724 as mu goes from 1 to 10).

assumptions (3)

standard math The weighted Potts model is submodular, so the CRF-regularized proposal in Eq. (11) can be solved to a global optimum by graph cuts in polynomial time.
Invoked in Section 3.4.2; this is a classical result for submodular pairwise functions.
ad hoc to paper Mini-batch SGD on network parameters is a valid proximal update for ADMM in the non-convex setting, and the conditions of references [30,31] hold.
Sections 3.4.1 and 3.5; no verification of the assumptions (sufficient decrease, Lipschitz continuity) is provided, and convergence of the alternating scheme is not proven.
domain assumption Size bounds [Smin, Smax] for each image are known in advance and are derived from the true foreground size Sreal in the evaluation protocol.
Section 4.1 defines bounds as (1-epsilon) Sreal to (1+epsilon) Sreal; for epsilon=0 the exact ground-truth size is used as the constraint, which is a strong supervision signal.

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Cite this review

Pith. "Pith review of Discretely-constrained deep network for weakly supervised segmentation." pith.science (2026). https://pith.science/paper/PWX7QNLL

@misc{pith2026190805770,
  author       = {Pith},
  title        = {Pith review of: Discretely-constrained deep network for weakly supervised segmentation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWX7QNLL}},
  note         = {Machine review of arXiv:1908.05770}
}

read the original abstract

An efficient strategy for weakly-supervised segmentation is to impose constraints or regularization priors on target regions. Recent efforts have focused on incorporating such constraints in the training of convolutional neural networks (CNN), however this has so far been done within a continuous optimization framework. Yet, various segmentation constraints and regularization can be modeled and optimized more efficiently in a discrete formulation. This paper proposes a method, based on the alternating direction method of multipliers (ADMM) algorithm, to train a CNN with discrete constraints and regularization priors. This method is applied to the segmentation of medical images with weak annotations, where both size constraints and boundary length regularization are enforced. Experiments on a benchmark cardiac segmentation dataset show our method to yield a performance near to full supervision.

Figures

Figures reproduced from arXiv: 1908.05770 by the authors.

**Figure 2.** Examples of images, ground truth and annotations for the three tested datasets. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** 3D Dice on validation images of the ACDC (LV) and PROMISE prostate datasets [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Evolution of the CRF-regularized proposal ( [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

**Figure 5.** Figure 5: Visual comparison of tested methods on validation images for size bounds corre [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: Ratio between predicted and real foreground size during training, while using size [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Kervadec, J

H. Kervadec, J. Dolz, M. Tang, E. Granger, Y. Boykov, I. Ben Ayed, Constrained-CNN losses for weakly supervised segmentation, Medical im- age analysis 54 (2019) 88–99

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