Polynomial Dice Loss for Medical Image Segmentation

Hiroaki Aizawa

REVIEW 2 major objections 2 minor 18 references

Polynomial Dice Loss for Medical Image Segmentation

T0 review · 2 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Polynomial Dice Loss uses Taylor expansion to adjust higher-order terms in the overlap measure for medical segmentation training.

desk verdict A Taylor-derived polynomial Dice loss that adds tunable higher-order terms and reports competitive segmentation results, but supplies almost no evidence that the approximation holds up during training. read the letter →

arxiv 2606.23373 v1 pith:VMNODFI3 submitted 2026-06-22 cs.CV

Hiroaki Aizawa This is my paper · ORCID

classification cs.CV

keywords medicalimagesegmentationDicelosspolynomialTaylorexpansionfunctiondesignoverlapmeasuresemantic

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

The paper proposes Polynomial Dice Loss as an extension of standard Dice Loss. It applies Taylor expansion to express the loss as a polynomial, which lets the contribution of higher-order terms be tuned during optimization. This gives a direct way to modify the shape of the loss surface while still measuring region overlap. The approach is tested on medical image segmentation tasks and compared to losses based on Dice and Tversky coefficients, showing competitive results.

What carries the argument

Polynomial representation of Dice Loss obtained by Taylor expansion around an operating point, which controls the weight of higher-order terms.

What would settle it

Training segmentation networks with the polynomial loss on a standard medical dataset and finding that final overlap scores are substantially lower than those from unmodified Dice Loss would show the approximation does not preserve useful training behavior.

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Extended reading notes

Core claim

By leveraging the geometric characteristics of Dice Loss and formulating the loss function as a polynomial representation via Taylor expansion, the method enables the adjustment of the contribution of higher-order components to the loss function, providing a simple way to control the loss shape and achieving competitive performance across multiple segmentation settings.

Load-bearing premise

The Taylor expansion of the Dice coefficient around the chosen operating point remains a faithful and stable surrogate for the original overlap measure when used as a training objective.

Editorial extensions

If this is right

The polynomial form supplies a direct knob for changing how much higher-order terms influence the gradient during training.
The same loss achieves performance on par with standard Dice Loss and Tversky-based losses across several segmentation benchmarks.
Data imbalance and small-lesion problems remain addressable because the base overlap measure is retained.
The formulation is presented as a lightweight modification that does not require new network architectures or training protocols.

Reading between the lines

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

The same Taylor-expansion trick could be applied to other overlap or similarity coefficients to produce tunable polynomial variants.
Choosing different expansion points might shift the loss surface in ways that favor small-object recall or boundary precision.
Because the polynomial is explicit, its derivatives are also explicit, which could simplify analysis of training dynamics compared with the original non-polynomial Dice Loss.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper's core move is to take the Dice coefficient, expand it in Taylor series around some operating point, and then treat the resulting polynomial as a trainable loss whose coefficients can be adjusted to emphasize or damp higher-order terms. That specific construction does not appear in the cited Dice variants, so the formulation itself is new.

It does a clean job of showing how the polynomial lets you dial the loss shape and then running the usual medical segmentation benchmarks against plain Dice and Tversky losses. The results are described as competitive, which is the right bar for an incremental tweak.

The soft spots are exactly where the stress-test note flags them. Nothing in the abstract (or the supplied text) gives the expansion point, the order chosen, a remainder bound, or any plot of how far the polynomial drifts from true Dice as the network output moves away from the expansion point. Early training is precisely when predictions are farthest from any reasonable operating point, so the claim that the surrogate remains faithful rests on unshown evidence. If the full paper has those checks, they need to be front and center; if not, the central justification is missing.

This is for readers who already work on loss design for class-imbalanced segmentation and want another knob to turn. It is coherent on its own terms and engages the literature honestly, so it clears the bar for a serious referee even though the current write-up leaves the approximation question open. I would send it to review and ask the authors to supply the missing stability analysis.

Referee Report

2 major / 2 minor

Summary. The paper proposes Polynomial Dice Loss for medical image segmentation by applying a Taylor expansion to the standard Dice coefficient around a chosen operating point, truncating or re-weighting higher-order terms to create a tunable polynomial loss that controls the contribution of those terms. It evaluates the approach against losses derived from Dice and Tversky coefficients, claiming competitive performance across multiple segmentation settings while providing a simple mechanism to adjust loss shape for challenges like data imbalance and small lesions.

Significance. If the polynomial remains a faithful surrogate for the true Dice overlap throughout training, the method supplies an explicit, low-parameter way to reshape the loss landscape without introducing new hyperparameters beyond the expansion point and truncation order. This could be useful for tasks where standard Dice under- or over-emphasizes certain error regimes, and the competitive empirical results (if reproducible) would constitute a modest but practical contribution to the family of overlap-based losses.

major comments (2)

[Formulation (Taylor expansion of Dice coefficient)] The central claim that the Taylor-derived polynomial remains a stable training objective rests on the unstated assumption that the remainder term stays small even when network outputs are far from the expansion point (as occurs at random initialization). No radius-of-convergence argument, explicit remainder bound, or plot of approximation error versus exact Dice during training is supplied; without this, it is unclear whether the polynomial is a controlled surrogate or an uncontrolled distortion of the original loss landscape.
[Method description and experimental setup] The choice of expansion point and polynomial order is described as enabling control of higher-order components, yet no ablation or validation procedure for selecting these values is reported. If the operating point is fixed once and for all, the resulting loss is effectively a different fixed surrogate rather than a generally tunable family; this choice is load-bearing for the claim of a 'simple way to control the loss shape.'

minor comments (2)

[Abstract and §3] The abstract and introduction should explicitly state the numerical value chosen for the expansion point and the truncation order used in the reported experiments.
[Experimental results section] Figure captions and table footnotes should clarify whether the reported Dice scores are computed with the exact Dice coefficient or with the polynomial surrogate at inference time.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. We address each major comment below and will revise the paper to incorporate the suggested improvements where they strengthen the work.

read point-by-point responses

Referee: [Formulation (Taylor expansion of Dice coefficient)] The central claim that the Taylor-derived polynomial remains a stable training objective rests on the unstated assumption that the remainder term stays small even when network outputs are far from the expansion point (as occurs at random initialization). No radius-of-convergence argument, explicit remainder bound, or plot of approximation error versus exact Dice during training is supplied; without this, it is unclear whether the polynomial is a controlled surrogate or an uncontrolled distortion of the original loss landscape.

Authors: We acknowledge that the manuscript does not include a formal radius-of-convergence analysis or remainder bound. The work prioritizes the derivation of the polynomial form and its empirical utility for segmentation. In revision we will add a plot comparing the polynomial approximation error to the exact Dice coefficient over the course of training, together with a brief discussion of the observed practical stability. This empirical evidence will clarify the behavior away from the expansion point without claiming a general theoretical guarantee. revision: yes
Referee: [Method description and experimental setup] The choice of expansion point and polynomial order is described as enabling control of higher-order components, yet no ablation or validation procedure for selecting these values is reported. If the operating point is fixed once and for all, the resulting loss is effectively a different fixed surrogate rather than a generally tunable family; this choice is load-bearing for the claim of a 'simple way to control the loss shape.'

Authors: The expansion point and truncation order were determined via preliminary experiments to emphasize higher-order terms for the target challenges of class imbalance and small lesions. To substantiate the tunability claim, the revised manuscript will include an ablation study varying these parameters and reporting their effect on Dice scores across the evaluated datasets. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation uses external Taylor expansion

full rationale

The paper formulates Polynomial Dice Loss by applying the standard Taylor series expansion to the Dice coefficient around a chosen operating point and then truncating or re-weighting terms. This step relies on the external mathematical definition of Taylor expansion rather than any self-referential definition, fitted parameter, or self-citation chain. No equation in the provided text reduces the final loss to a quantity that was already an input by construction, and no load-bearing uniqueness theorem or ansatz is imported from prior author work. The central claim therefore remains independent of its own outputs.

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

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The Taylor expansion itself is treated as a standard mathematical tool rather than an invented modeling choice.

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Cite this review

Pith. "Pith review of Polynomial Dice Loss for Medical Image Segmentation." pith.science (2026). https://pith.science/paper/VMNODFI3

@misc{pith2026260623373,
  author       = {Pith},
  title        = {Pith review of: Polynomial Dice Loss for Medical Image Segmentation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VMNODFI3}},
  note         = {Machine review of arXiv:2606.23373}
}

read the original abstract

Medical image segmentation is a fundamental task for medical image processing and computer-assisted intervention, yet data imbalance and small lesion detection pose significant challenges. Dice Loss, which measures the overlap between predicted and ground truth regions, is widely used to mitigate these issues. To further emphasize its properties, we propose Polynomial Dice Loss, a polynomial extension of Dice Loss. Specifically, by leveraging the geometric characteristics of Dice Loss and formulating the loss function as a polynomial representation via Taylor expansion, we enable the adjustment of the contribution of higher-order components to the loss function. In our experiments, we evaluate the proposed method against loss functions derived from conventional Dice and Tversky coefficients. Experimental results and further analysis show that the polynomial formulation provides a simple way to control the loss shape and achieves competitive performance across multiple segmentation settings.

Figures

Figures reproduced from arXiv: 2606.23373 by the authors.

**Figure 1.** Overview of the proposed Polynomial Dice Loss. Dice Loss is decomposed into a scale term and an angular alignment term, and the angular term is expanded as a polynomial by Taylor approximation. The truncation order K in DropDice and the leading coefficient ϵ1 in PolyDice-1 control the penalty assigned to misaligned predictions. loss function becomes a key lever for stable training and reliable detection of small str… view at source ↗

**Figure 2.** DropDice Loss for s = 0.1: smaller K penalizes large angular errors more strongly, while larger K approaches Dice Loss. 3 2 1 0 1 2 3 1.0 1.2 1.4 1.6 1.8 Loss Dice Loss PolyDice1Loss ( = -0.3) PolyDice1Loss ( = -0.15) PolyDice1Loss ( = 0.0) PolyDice1Loss ( = 0.15) PolyDice1Loss ( = 0.3) PolyDice1Loss ( = 0.5) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** Qualitative segmentation results on the Synapse dataset using TransUNet. Each row shows one sample, with columns corresponding to the input image, ground truth, Dice Loss, Tversky Loss, DropDice Loss, and PolyDice-1 Loss. In these examples, the proposed losses preserve some organ regions more clearly than the baselines, consistent with the observed quantitative trends on Synapse [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Effect of the truncation order K in ℓ (K) DropDice for UNet. Dice scores are shown for four datasets, illustrating that the best order is dataset dependent: larger K often approaches or improves upon Dice Loss, while smaller orders can be beneficial on some datasets. t…

**Figure 6.** Figure 6: Effect of the coefficient ϵ in ℓPolyDice-1 for UNet. We vary the single coefficient that controls the leading angular penalty and report Dice scores on each dataset, showing that validation-based tuning of ϵ affects performance and that the optimal value differs acros…

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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

In: ISBI

Abraham, N., Khan, N.M.: A novel focal tversky loss function with improved at- tention u-net for lesion segmentation. In: ISBI. pp. 683–687. IEEE (2019)

2019

[2] [2]

saliency maps from physicians

Bernal, J., Sánchez, F.J., Fernández-Esparrach, G., Gil, D., Rodríguez, C., Vilar- iño, F.: Wm-dova maps for accurate polyp highlighting in colonoscopy: Validation vs. saliency maps from physicians. Computerized medical imaging and graphics 43, 99–111 (2015)

2015

[3] [3]

Bernard, O., Lalande, A., Zotti, C., Cervenansky, F., Yang, X., Heng, P.A., Cetin, I., Lekadir, K., Camara, O., Ballester, M.A.G., et al.: Deep learning techniques for automatic mri cardiac multi-structures segmentation and diagnosis: is the problem solved? IEEE transactions on medical imaging37(11), 2514–2525 (2018)

2018

[4] [4]

Medical Image Analysis p

Chen, J., Mei, J., Li, X., Lu, Y., Yu, Q., Wei, Q., Luo, X., Xie, Y., Adeli, E., Wang, Y., et al.: Transunet: Rethinking the u-net architecture design for medical image 12 Hiroaki Aizawa segmentation through the lens of transformers. Medical Image Analysis p. 103280 (2024)

2024

[5] [5]

Feng, L., Shu, S., Lin, Z., Lv, F., Li, L., An, B.: Can cross entropy loss be robust to label noise? In: IJCAI. pp. 2206–2212 (2021)

2021

[6] [6]

arXiv preprint arXiv:2412.07219 (2024)

Fukuda, K., Aizawa, H.: Taylor outlier exposure. arXiv preprint arXiv:2412.07219 (2024)

work page arXiv 2024

[7] [7]

Jha, D., Smedsrud, P.H., Riegler, M.A., Halvorsen, P., De Lange, T., Johansen, D., Johansen, H.D.: Kvasir-seg: A segmented polyp dataset. In: MMM. pp. 451–462. Springer (2020)

2020

[8] [8]

Computers in Biology and Medicine168, 107695 (2024)

Kato, S., Hotta, K.: Adaptive t-vmf dice loss: An effective expansion of dice loss for medical image segmentation. Computers in Biology and Medicine168, 107695 (2024)

2024