LoMix: Learnable Weighted Multi-Scale Logits Mixing for Medical Image Segmentation
Authors: Md Mostafijur Rahman
Radu Marculescu
Venue: NeurIPS 2025 poster ()
Keywords: LoMix Supervision
Medical Image Segmentation
U-shaped Network
Combinatorial Multi-scale Fusion
Logits Mixing
Primary area: machine_learning_for_sciences
TL;DR: Learnable Weighted Multi‑Scale Logits Mixing

Abstract: U-shaped networks output logits at multiple spatial scales, each capturing a different blend of coarse context and fine detail. Yet, training still treats these logits in isolation—either supervising only the final, highest-resolution logits or applying deep supervision with identical loss weights at every scale—without exploring mixed-scale combinations. Consequently, the decoder output misses the complementary cues that arise only when coarse and fine predictions are fused. To address this issue, we introduce LoMix (Logits Mixing), a Neural Architecture
Search (NAS)-inspired, differentiable plug-and-play module that generates new mixed-scale outputs and learns how exactly each of them should guide the training process. More precisely, LoMix mixes the multi-scale decoder logits with four lightweight fusion operators: addition, multiplication, concatenation, and attention-based weighted fusion, yielding a rich set of synthetic “mutant” maps. Every original or mutant map is given a softplus loss weight that is co-optimized with network parameters, mimicking a one-step architecture search that automatically discovers the most useful scales, mixtures, and operators. Plugging LoMix into recent U-shaped architectures (i.e., PVT-V2-B2 backbone with EMCAD decoder) on Synapse 8-organ dataset improves DICE by +4.2% over single-output supervision, +2.2% over deep supervision, and +1.5% over equally weighted additive fusion, all with zero inference overhead. When training data are scarce (e.g., one or two labeled scans, 5% of the trainset), the advantage grows to +9.23%, underscoring LoMix’s data efficiency. Across four benchmarks and diverse U-shaped networks, LoMiX improves DICE by up to +13.5% over single-output supervision, confirming that learnable weighted mixed-scale fusion generalizes broadly while remaining data efficient, fully interpretable, and overhead-free at inference. Our implementation is available at https://github.com/SLDGroup/LoMix.

LoMix: Learnable Weighted Multi-Scale Logits Mixing for Medical Image Segmentation
Md Mostafijur Rahman Radu Marculeseu
Department of Electrical and Computer Engineering
The University of Texas Austin Austin, TX 78703
(mosta f jur. rahman, radum @uteeea. edu
Abstract
U-shaped networks output logits at multiple spatial scales, each capturing a differ- ent blend of coarse context and fine detail. Yet, training still treats these logits isolation- eitter supervising only the final, highest-resolution logits or applying deep supervision with identical loss weights at every scale- witthot exploring mixed-scale combinations. Consequently, the decoder output misses the com- plementary cues that arise only when coarse and fine predictions are fused. To address this issue, we introduce LoMix (Logits Mixing), Neural Architecture Search (NAS)-inspired, differentiable plug-and-play module that generates new mixed-scale outputs and learns how exactly each of them should guide the training process. More precisely, LoMix mixes the multi-scale decoder logits with four lightweight fusion operators: addition, multiplication, concatenation, and attention- based weighted fusion, yielding a rich set of synthetic "mutant" maps. Every original mutant map is given softplus loss weight that is co-optimized with network parameters, mimicking a one-step architecture search that automatically discovers the most useful scales, mixtures, and operators. Plugging LoMix into recent U-shaped architectures (i.e., PVT-V2-B3 backbone with EMCAD decoder) Synapse 8-organ dataset improves DICE by +4.2% over single-output super- vision, +2.2% over deep supervision, and + +1.5% over equally weighted additive fusion, all with zero inference overhead. When training data are scarce (c.g., one or two labeled scans, 5% of the trainset), the advantage grows to +9.23%, under- scoring LoMix's 's data efficiency. Across four benchmarks and diverse U-shaped networks, LoMiX improves DICE by up to + +13.5% over single-output supervision, confirming that learnable weighted mixed-scale fusion generalizes broadly while remaining data efficient, fully interpretable, and overhead free at inference. Our implementation is available /attps://gil
com/ SLDGroup/LoMi.x
Introduction
Precise delineation of organs, tumours, and lesions underpins radiotherapy planning, volumetric assessment, and computer-aided diagnosis. State-of-the-art systems almost invariably adopt U-shaped encoder-decoder architectures such as UNet [25). UNet++ (36). Attn-UNet [19], TransUNet [3], and SwinUNet (2] These models generate logit maps multiple decoder resolutions: coarse maps offer global anatomical context, whereas fine maps sharpen boundaries and reveal small pathologies.
Surprisingly, the prevailing training protocols ignore most of this multi-scale richness: they either back- pook-prop loss only from the final logits (single-output supervision) or apply deep supervision (i.e., auxiliary losses intermediate network outputs) with identical loss weight for every scale 36 8 20 23 This uniform treatment presumes that each resolution is equally informative for every anatomy, an assumption contradicted by clinical practice, where minute, high-contrast structures
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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(e.g. pancreas, gall- bladder) lean heavily on scne-scale details, whereas large homogeneous organs (e.g. liver) profit chiefly from coarse context. The resulting mismatch leaves sple-siecciic clues under-exploited, and the deficiency grows when labels are scarce.
Deep supervision is also isolationist in nature as it overlooks the synergy that emerge when coarse- and fine-grain logits are combined. Early "mutation" methods attempt to bridge this gap by summing up the decoder logits with equal weights, with MERIT [21] being such a prominent example. However, such static fusion fixes the operator, enforces equal contributions from all scales, and demands manual retuning whenever imaging protocols, organ sizes, or data volumes change. General loss balancinng schemes developed for multi-task learning, such as uncertainty weighting [4] or GradNorm [5]. do not capture the structured correlations inside a single-task, multi-scale decoder, and thus leave much of this information content untapped.
To address these limitations, introduce LoMix (Logits Mixing) convert passive deep supervision into an active, learnable ensemble of mixed-scale predictions. During training every pair of decoder stages is fused by four lightweight, differentiable operators: pixel-wise addition, multiplication. concatenation followed by 1 X 1 convolution, and attention-based weighted fusion, resulting in a rich family of "mutant" logits that explicitly blend coarse context with fine detail. Each original or mutant map is modulated by a soft-plus weight optimized jointly with the network, so the model performs a Neural Architecture Search (NAS) style selection of the most informative scales and fusion modes within the main optimization loop: that is, no extra optimizer. no validation-see -et grid search is needed. The added parameters are negligible and used only during training, leaving the FLOPs, latency, and memory footprint at test-time unchanged, while providing data-driven fusion that adapts to organ size, image contrast, and label scareity.
When integrated into a recent U-shaped network, PVT-V2-B2 backbone [34] with EMCAD decoder [24]. LoMix improves the mean DICE score on Synapse 8-organ segmentation by +4.2% over single-output supervision, +2.2% % over uniform deep supervision, and + +1.5 % over equal-weight additive fusion; of note, the gains are even larger on the harder Synapse 13-organ segmentation task. Consistent improvements on ACDC cardiac MRI and BUSI breast-tumour ultrasound across both CNN and transformer backbones confirm robustmess of LoMix. Even with only 5% of training scans available, LoMix still delivers +9.23% % DICE improvement, underscoring the data efficiency.
In summary, LoMix (i) is the first framework that jointly optimizes which decoder scales mix and how to mix them for single-task; (ii) substitutes manual loss weighting with an automatic, interpretable, NAS-inspired weighting mechanism; (iii) introduces zero inference overhead; and (iv) consistently improves performance across datasets, backbones, and annotation budgets. By allowing networks to learn their own multi-scale fusion strategy, LoMix offers principled and practical advancement that puts forth a strong baseline for data-efficient medical image segmentation.
The remainder of this paper is structured as follows: Section 2 discusses related literature, Section details the LoMix framework, Section critical ablation studies, and Section 6 concludes our findings and suggests future research directions.
S exxplain experimental evaluations, Section presents several
2 Related Work
Medical Segmentation Architectures: U-shaped encoder- networks with skip connections (e.g., U-Net) are the de facto architectures in medical image segmentation (25): they can capture fine details via multi-scale feature maps, but are limited by the locality of convolutional operations. For example, Chen et al. note that standard U-Net struggles to model long-range dependencies, motivating hybrid designs [3]. To address this, recent work has proposed transformer-base backbones for segmentation. TransUNet [3] combines a CNN encoder with a Vision Transformer to learn the global context, while the decoder recovers the spatial details. Similarly, Swin-Une! Unet 2 uses a hierarchical Swin Transformer in both encoder and decoder, demonstrating that purs-transformer U-shaped models outperform purely convolutional ones on multi-organ tasks.
Other variants adopt the Pyramid Vision Transformer (PVT) 34 as the encoder. For instance, CASCADE (20 and G-CASCADE (22) use PVT encoders with novel attention- or brapd decoders to progressively refine multi-scale features. Polyp-PVT 7 and SSFormer 33 also leverage PVT bocksones for polyp segmentation, incorporating hand-crafted fusion modules (e.g., cascaded fusion, camouflage, and locality decoders) to combine features across scales.

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More recently. EMCAD [2 introduces an efficient multi-scale convolutional attention decoder that uses depth-wise convolutions and gated attention fuse multi- resoluttoo features. While these transformer-based models generate rich multi-scale outputs, they typically use fixed fusion or skip-connection schemes and do not learn explicit weights for combining feature maps.
Deep supervision has been widely adopted to improve training of segmentation models by attaching auxiliary loss functions intermediate layers. For example, UNet++ [36] employs nested skip pathways: intermediate decoder outputs are each supervised by ground truth to encourage multi-scale consistency. In practice, however, these supervision losses are typically combined with fixed rules (e.g., simple averaging, summation, etc.).
Deep Supervision and Static Multi-Scale Fusion:
Likewise, multi-scale fusion in many models is static. Common approaches concatenate feature maps from different depths without learning the fusion weights. For example, Polyp-PVTT's 77 design includes fixed modules merge encoder features across levels, but these components have pre-defined roles and uniform weighting. Such static fusion schemes (even when effective) do not adaptively learn which scales channels to emphasize, leaving the relative contributions of multi-scale features unchanged during training.
-aase Training and Its Limitations: Some recent methods aim to exploit multi-scale logits through loss-level ensembling. MERIT [21] is such a notable example: it aggregates original multi-stage logits through additive MUTATION mixing strategy. This loss aggregation ensembles logits from different scales for final training. However, the MUTATION mixes uniformly all decoder outputs (implicitly giving equal weight of each scale) using only element-wise addition and does not include learnable parameters for fusion. Similarly, EMCAD's 24 multi-scale decoding produces several parallel predictions, but these are also combined using only additive fusion when computing loss. Hence, while mutation-based training can improve robustness via implicit ensembling, it lacks a trainable mechanism to re-weight fuse multi-scale outputs using multiple operations dynamically.
Adaptive Weighting and NAS-Inspired Fusion: Learning to balance multi-scale signals has also been studied from the perspective of multi-task learning. Notably, uncertainty weiihtted loss functions [14] and GradNorm [5] automatically tune loss weights across tasks. While these methods adaptively adjust the loss terms, they are designed for distinct tasks and do not directly apply multi-scale outputs of a single task. They also do not explicitly handle feature-level fusion. Some NAS methods have explored learnable combinations of feature maps (e.g. NAS-FPN in detection (9), but none learns operator-level fusion for multi-scale segmentation outputs. In contrast, to the best of our knowledge, LoMix is the first approach to enable fully learnable multi-scal fusion in segmentation: it synthesizes decoder outputs combinatorially (in the style of NAS cell search [8] and learns weights to aggregate losses. This allows LoMix to adaptively emphasize relevant scales and operations during training, going beyond fixed strategies used in prior work.
3 Method
Next, we first review the background on U-shaped architectures and multi-scale outputs, then formalize the problem, and describe each component (see Figure
3.1 Background: U-shaped Networks and Multi-scale Outputs
employ an encoder-decoder design with skip connections. The encoder path repeatedly downsamples the input to capture hierarchical features (see Figure aa)). while the decoder path upsamples these features to reconstruct the segmentation map (see Figure [b)). Skip connections between corresponding encoder and decoder layers ensure that fine spatial details lost during downsampling are recovered at the output.
U-shaped networks 25
24 10
To further leverage multi-scale context, many networks produce outputs at multiple decoder stages. For example, deeply supervised networks generate intermediate segmentation maps (P1, Ps, Ps. in Figure TD at different scales, thus providing richer gradients during training and allowing each decoder stage to capture structures at its respective scale.
In medical imaging, this multi-scale strategy is especially beneficial, since anatomical structures can vary greatly in size. UNet++ [36] is one instantiation of this idea, introducing dense nested skip
3

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Poria = P3 P..
Skip Connections
Ground Truth
[c) Combinatorial Mutation Module
(e) Loss
Encoder
Decoder
P
Aggregation
Stage
Stage 1
Addition
Pz
Stage 2
Stage
Concatenation
(4) NAS-inspired
Stage
Stage
Multiplication
Mutated Logits, Preut
Weight Learning
Input Image
Attention
Softolus
Stage
Stage
Original Multi- scale Logits, Porto
Weighted Fusion
Weights
Concatenation
Figure 1: The proposed LoMix supervision strategy during training. (a-b) An input image is processed by a U-shaped network, producing original multi-scale logits Porig from different stage of decoders. (c) The Combinatorial Muiation Module synthesizes additional predictions by applying four fusion operators every non-trivial original prediction subset: Addition, Concatenation, Multiplication, and a learnable Attention- Attention-Weighted Fusion (awf). (d) NAS-inspired weight-learning produces differentiable sefiplus-transformee weights w for every original logit Porig or mutated logit in P Pmutt optimized jointly with network parameters via pack-propagation. (e) All original and mutated logits are supervised by loss objective weighted by [w}.
connections and side-output layers that fuse information across scales. Such multi-scale outputs encourage consistency across resolutions and improve medical image segmentation accuracy.
3.2 Problem Definition: Logit Mixing (LoMix)
Let X denote an input image of size H x W and let Y E (1.....CC C) HxW be the ground truth with C classes. A U-shaped network with L decoder stages produces L logit maps (X)) where each Ze E Cx has spatial resolution H₃ x Wy. Upsampling each Ze to the full resolution gives logit maps (x( = of (X)) E [0, [0, HxW with a softmax (for multi-class segmentation) sigmoid (for binary segmentation), appropriate. We denote the set of these original logits by Porig. LoMix also synthesizes a collection Pimut "mutant" maps through fusion operators (addition, multiplication, concatenation, and attention-weighted fusion) in the Combinatorial Mutation Module (see Section 3.3 and Figure [c]), forming the combined set P Porig U Pmut-
For each map P. P, we introduce scalar (x E IR and convert it to positive loss weight W. softplus(a,, = In(1 + ea") > 0 by a NAS-inspired weight learning (see Sec- tion 3.4 and Figure (d). Given a segmentation loss (P,Y) (e.g. Cross-entropy + DICE), we train both the network parameters @ and the loss-weight parameters {a.} by minimizing mine.(a., Ep.EP W. feeg (X), Y) (see Section 3.5 and Figure [)). Because only the fi- nal decoder output PL is used the test time, LoMix adds zero inference overhead while adaptively learning which scales and fused combinations are most informative for robust segmentation.
3.3 Combinatorial Mutation Module (CMM)
In our LoMix framework, U-shaped network produces L logit maps Pı..... PL RC each at progressively finer scale (all upsampled to a common H x W spatial size). Let P.(p) E RC denote the logits of C' classes at pixel p E {1,..., H) x {1,..., W) from the i-th decoder output. The Combinatorial Mutation Module (CMM) creates additional fused logits by combining subsets of these logits under four operators. Specifically, for every non-trivial subset S (1,..., L} with
|S| 2 2, we define fused logit maps plop) (p) E RCC (for op E {add, mult, cat, aw f } as follows:
Addition (add): We combine each subset S of the original logit maps by element-wise addition to produce fused map p(add) (p) as Eq.
p pladd) (p) = Σ P(Pp)
(1)
SES
Intuitively, Addition fusion aggregates confidence from each subset of logit maps. Addition fusion will highlight regions where either decoder is confident (acting like an OR operation).

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Multiplication (malt): We take the element-wise weaerii-iseeeedaaeaaiaaaaa (Hadamard) product P5 subset S of the original logit maps as in Eq. 2
p(mult)
(p) of each
(mmmtt) (p)
II P((p)
(2)
Multiplication fusion provides high confidence only where all logit maps agree (analogous to AND operation). This fusion thus focuses the intersection of the decoders' predictions, which can enhance precision by reinforcing common correct predictions and canceling out disagreements (if either logit map is uncertain, the product lowers confidence).
Concatenation (cat): We concatenate each subset S of the original logit maps chamnel-wise and apply a x 1 convolution to fuse them to produce fused map Ps pleat) (p) as in Eq.
(catt) (p) Ws [P((P]
where [P:(D) igs IR SIC is the channel-wise concatenation and Ws E RCX([IS is convolution weight matrix (with output dimension equal to one logit map). This operation allows the network to learn an optimal pixel-wise linear combination of each subset of logits. The 1 x 1 convolution combining the two inputs for each output class, potentially learning to trust one decoder more in certain regions and the other decoder elsewhere, based data.
be viewed as automatically weighting and
Attention- Weighted Fusion (awf): We introduce attention gating adaptively mix each subset S of original logits. We first compute attention scores and normalize using Eq.
exp(ár,s(p)) ás( (p) W [P:()) 0a,s(p) Ejes exp(ā,,s(D)
ãs
Then, we take the attention attention-weighhte sum P! (wf)) (p) of each subset of logits as in Eq.
p(awf) (p) Σ ar,s(p) P.(p)
(5)
iES
Attention- -eerhhed Fusion can learn to favor the logit that is likely to be correct at each pixel of the image (for instance, one pixel might be better at fine details, another at coarse structure, so the attention gate can interpolate accordingly). It generalizes the addition fusion by allowing spatially varying weighting instead of a fixed linear mix at each pixel.
These four fusion operations help produce segmentation predictions from each subset of original logits without adding significant computation (each is a simple pixel-wise operation or a 1 x conv). They are complementary: addition and multiplication are fixed arithmetic mixes (one expansive, other selective), while concatenation and attention are learnable mixes (one globally learned weight, another dynamically learned per-pixel weight). By supervising all of them, we expose the network to a wide variety of joint-decoder behaviors. The decoder stages are incentivized to cooperate because an eтror from one decoder stage can be corrected by another in fused output, thus leading to an overall more accurate ensemble of logits. In our LoMix framework, we apply all four fusion operations to every subset of original logits. Adding all non-empty subsets of the L decoder predictions introduces only 24 1 I, extra logit maps (e.g., 11 for T, 4, 26 for T, = 5); the total number of fused (mutant) logits (22 1 1-L.) Combined with the original logits, the overall
count is upper bounded by L + 4(2 - 1 - L)). which remains tractable for typical L < 5 since
U-shaped networks rarely exceed five stages in 225 20 24 12
3.4 NAS-Inspired Weight Learning
To enable the network to learn the relative importance of each logit map, we associate a trainable scalar weight with every original decoder output P. and every fused (mutated) output p(op). Concretely, let og E IR be the raw parameter for output P; and as (op) E R the parameter for fused output p((p) We map these parameters through the Softplus function to obtain strictly positive weights in Eq. 6
ma softplus(a)) = In(1 + e
w (op)
softplus(a (op)
(6)
5

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(op) (op) > 0 for all i, op. These weights w,; "'s serve as learnable By construction, Wi > 0 and Wig scaling factors on the loss contributions of each corresponding logit map. All parameters a and ag learned jointly with the network weights via backpropagation the overall training objective.
These design choices allow the model to automatically learn how much emphasis to place on each original and fused logit during training, in a manner reminiscent of architecture weighting in NAS but applicd to losses. The learned weights are shown in Appendix A.6 of Supplementary Material.
3.5 Loss Aggregation
Each output logits map is trained with standard segmentation loss (e.g., Cross-Entrope and DICE losses). For the ground-truth mask Y and the i-th original output P., we define per-output loss as in Eq. Similarly, per-output loss for each fused output psop) is defined in Eq. 8
L. 3B.cn(P: Y) + 7LDiCe(PP
_ 3L
((0) Y) 7 Y)
(8) Here, LCE is the Cross-entropp loss weighted by 3 and LDTCE is the DICE loss weighted by (here, 7 and 8,y 0). The total training loss is then formed by weighting each output's loss by the corresponding learned softplus weight and summing as in Eq.
Ltotal Σ W.C, + Σ (op) g
clP)
(9)
Weighting the loss terms (rather than directly combining logits) provides several benefits:
First, it preserves distinct supervision for each output: each P. and p((p) is individually trained and can receive gradients weighted by its own weights w. If a certain mutated logit (oo)) proves to be unhelpful or noisy, then the model can drive its w((p) toward zero, effectively ignoring its loss contribution. Conversely, an output is beneficial, its weight can be increased to emphasize it. This dynamic loss weighting is analogous to multi-task learning schemes where uncertainty or task relevance modulates loss terms.
Second, it avoids the pitfalls of weighting logits directly: mixing logits into a single predic- tion would blur their individual contributions and could hinder training of underperforming branches. Instead, our weighted loss formulation (Eq. ( allows the network to automatically focus on useful logit outputs while minimizing the impact of less informative ones, thus leading to more effective training of the ensemble of original and mutated logits.
Experimental Evaluation
We evaluate LoMix several medical image segmentation datasets. Datasets, additional results and analyses including qualitative visualization are provided the Supplementary Material.
4.1 Implementation details
Our methods are implemented and evaluated using Pytorch 1.11.., operating single NVIDIA RTX A6000 GPU equipped with 48GB of RAM. We use the PVT-EMCAD-BL as a default model 24 in our experiments with multi-scale kernels [1 x 1, 3, 5 x 5] and four stages unless otherwise mentioned. We consider all four operators (Addition, Multiplication. Concatenation, Attention- Weighted Fusion) with NAS-inspired learnable softplus weights in LoMix supervision. Only the last-stage prediction from the decoder is used as the final segmentation output. Model optimization is achieved with A
[77 optimizer with learning rate and weight decay set I 4.
4.2 Comparison with SOTA Methods
Synapse 8-organ Segmentation: Table reports Synapse 8-organ results averaged over at least three runs, comparing single-output supervision (LL), Deep Supervision (DS) (36), MUTATION 21 and

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Table Synapse 8-organ segmentation with Last Layer (LL), Deep Supervision (DS) 36) MUTA TION [21], and our LoMiX. DICE scores (%) are reported for Gallbladder (GB), Left kidney (KL), Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). f denotes the higher the better and denotes the lower the better. Results are averaged over at least three runs. Two-sided Wilcoxon signed-rank tests [35] indicate that LoMiX significantly outperforms L.I. and DS at a = 0.05. Best results are shown in bold.
Average
organ DICE (%0)
Methods
DICE (%)
HD95
mloU (%)]
Aorta
GB
KL
KR
Liver
PC
SP
SM
UNet 25 +LL
70.1
44.7
59.4
84.0
56.7
72.4
62.6
87.0
48.7
81.5
67.9
77.8
26.9
68.3
85.4
68.0
81.4
76.2
91.4
56.9
87.6
75.9
MUTATION
81.5
26.4
71.8
89.4
70.5
85.4
80.4
94.1
66.3
88.3
77.5
L.MiX (Ours)
83.6
24.3
74.6
90.4
75.3
86.3
82.5
94.3
67.9
91.8
80.2
AttUNet [9] +LL
71.7
34,5
61.4
82.6
61.9
76.1
70.4
87.5
46.7
80.7
67.7
IDS
77.9
29.9
68.1
85,4
67.5
81.4
77.4
91.0
57.2
87.1
76.3
MUTATION
82.6
19.9
73.1
88.
73.8
86.3
80.5
94.1
69.2
90.4
78.7
LoMiX (Ours)
83.0
19.4
74.1
90.0
75.5
84.4
81.4
94.5
67.2
91.3
79.7
UNat 3 I.I.
77.6
26.9
67.3
86.6
60.4
80.5
78.5
94.3
58.5
87.1
75.0
DS
82.7
17.3
73.5
86.6
68.5
87.7
84.6
94.4
65.3
90.8
83.5
MUTATION
63.7
86,9
95.5
93,1
82.7
LoMiX (Ours)
83.6
16.6
74.6
88,9
70.3
89,4
85.2
94.8
67.8
89.4
83.0
UNeXi (3I) LL
70,5
29.2
60.1
81.9
30.6
80.8
75.8
92.3
48.4
84.1
70.3
72.6
30.7
61.3
80.2
60.8
76.1
70.0
91.8
83.3
70.6
MUTATION
75.6
28.1
64..4
81.8
61.6
81.1
75.1
92.6
533.1
85.7
73.9
LoMIX (Ours)
76.8
22.7
66.2
83.8
60.9
81.1
78.3
92.8
56.2
86.6
74.5
B2
80.8
71.8
85.6
66.6
84.1
81.0
92.9
90.0
79.5
DS
81.1
20,2
70.9
83.0
70.6
82.2
80,4
94.1
64.4
90.1
83.7
MUTATION
83.0
17.8
74.3
86.9
67.6
87.1
82.1
94.4
68.7
91.8
85.6
LoMiX (Ours)
84.3
16.4
75.4
86.5
72.8
87.4
84.6
95.6
70.1
92.6
84.8
PVT-EMCAD-B2 +LL
80.9
22.9
71.2
87.1
68.0
84.9
81.1
94.6
63.1
89.8
78.9
DS
82.9
19.7
73.8
87.4
67.8
87.7
83.7
95.2
65.6
91.5
84.2
MUTATION
83.6
15.7
74.7
88.
68.9
88.1
84.1
95.3
68.5
92.2
83.9
LoMiX (Ours)
85.1
14.9
76.4
88.8
73.5
89.1
84.7
95.8
69.7
92.5
86.5
LoMiX across six representative backbones spanning convolutional and transformer families (UNet AttUNet [19), TransUNet [3) UNeXt (3I), PVT-CASCADE-B2 (20) and PVT-EMCAD-B2
25
LoMiX consistently achieves the highest average DICE and mIoU and the lowest HD95 for all backbones, improving DICE by +3-5% on average over deep supervision (DS) and by up to +13.5% over last-layer (LL) supervision, without any inference time overhead. The gains are especially pronounced on small challenging organs such as gallbladder and pancreas, indicating that learnable mixed-scale fusion recovers complementary fine detail and coarse context that prior supervision schemes fail to exploit. These results demonstrate that LoMiX a unified, lightweight, and soug-and--lat training module that generalizes across diverse U-shaped decoders and transformer backbones while remaining fully compatible with standard inference pipelines.
24
ACDC Cardiac Organ Segmentation: Table reports the cardiac organ segmentation on ACDC dataset, averaging over at least three runs. We can that across broad set of CNN and trans- former models, our
models consistently deliver superior performance. Notably, PVT-EMCAD-BL + LoMix attains new peak average DICE of 92.51%. surpassing its baseline PVT-EMCAD-BE model (92.12%) and all prior methods. It achieves the highest scores on every structure: RV 91. %, Myo 99.96% and LV 96. 15%. Even the lightweight PVT-EMCAD-BL + LoMix improves over PVT-EMCAD-BO (91.34% + 91.69%), matching or exceeding more complex cascaded and MERIT-based models. These gains confirm that LoMix's learnable, multi-scale fusion yields more accurate and robust cardiac contours- particuulpat evident in the challenging myocardium region-while maintaining identical inference complexity.
4.3 Evaluation with Limited Data
Limited Data Setup: We create subsets using 5% (1 scan), 10% (2 scans), 20% (4 scans), and 40% (7 scans) of the Synapse training set to evaluate performance under constrained supervision.
Figure 2 and Table illustrate how LoMix consistently outperforms conventional single-head ("Last- Layer") supervision as the amount of labeled data shrinks. In Table 3 and Figure A with 40% of
7

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Table Results of cardiac organ segmentation on ACDC dataset. DICE scores (%) are reported for Right ventricle (RV). Myocardium (Myo), and Left ventricle (LV). 1 (L) denotes the higher (lower) the better. Results are averaged over at least three runs. Best results are shown in bold.
Methods
Avg. DICE (%) f
RV
Myo
LV
UNet 25
87.55
87.10
80.63
94.92
Atm_ UNe
86.75
87.58
79.20
93.47
VIT+CUP
81.45
81.46
70.71
92.18
UNet
89.71
86.67
87.27
95.18
SwinUNet 2
88.07
85.77
84.42
94.03
MT-UNet 32
90.43
86.64
89.04
95.62
MISSHormer
12
90.86
89.55
88.04
94.99
PVT-CASCADE
20
91.46
89.97
88.9
95.50
TransCASCADE 20
91.63
90.25
89.14
95.50
Rolling UNet S
16
87.59
85.02
83.59
94.17
CMUNcXt 30
85.19
81.30
82.54
91.74
UNeXt [31]
84.68
81.06
81.22
91.76
Cascaded MERIT 21
91.85
90.23
89.53
95,80
PVT-GCASCADE 22
91.95
90.31
89.63
95,91
EGE-UNei 27
80.68
76.6
75.21
90.23
PVT-EMCAD-EE -BO
24
91.34
89.37
88.99
95.65
PVT-EMCAD-B3 BB2
92.12
90.65
89.68
96.02
PVT-EMCAD BB + LoMix (Ours)
91.69 +0.51
90.33
88.99
95.75
PVT-EMCAD-BES + LoMix (Ours)
92.51 +:0.47
91.41
89.96
96.15
Table 3: Performance of LoMix on Synapse 8-organ segmentation under limited data. Gallbladder (GB), Left kidney (KI., Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). 1 higher better, 1 lower better. Each row is averaged over five runs. The best entries for each limited data setting are shown in bold.
Data Fraction
Scheme
Average
Per-organ DICE (S)
DICE
HD95)
mloUp
Aorta
GB
KL
KR
Liver
PC
SM
40% (7 scans)
Last Layer
71.43
35.71
61.27
60.05
74.44
71.75
92.72
53.88
78.45
56.21
LoMix (Ours)
76.49
27.14
66.67
87.27
66.27
82.81
74.60
95.05
59.29
82.61
64.05
20% (4 scans)
Last Layer LoMix (Ours)
64.40
36.50
54.74
80.92
35.71
74.17
69.74
90.13
39.05
74.04
$1.39
69.21
29.86
$9.60
84.11
41.59
77.86
72.24
94.47
47.98
80.83
54.60
10% (2 scans)
Last Layer
55.32
46.18
45.69
70.37
27.73
66.82
60.38
83.64
31.79
69.92
31.93
LoMix (Ours)
64,53
38,58
$4.55
75.81
37.29
74.01
70.26
92.33
45.33
82.50
38.73
5% (1 scan)
Last Layer LoMix (Ours)
37.22
72.21
28.83
49.1 14
23.54
41.24
41.17
73.77
10.13
46.42
12.36
46.45
56.94
37.36
57.58
26.92
50.30
49.48
80.40
16.44
65.72
24.75
the Synapse training set, LoMix improves the mean DICE score +5.1%; at 20% data the gain is +4.8%,
and when supervision drops to only 10% and 5% the improvement margins surge to > +9%. The radar plot on the right (Figure 2 B)) shows that these improvements are not confined to single organ: LoMix raises DICE scores for every organ, with the largest boosts on small, -o-emmttsementt classes such gallbladder (GB), pancreas (PC), and stomach (SM), confirming that dynamically weighting complementary scales especially beneficial where context/details trade-offs are hardest. Hence, LoMix delivers uniform, per-organ gains and turns the U-shaped PVT-EMCADD B2 architecture into a far more data-efficient and across-organ robust learner without any added inference cost.
Ablation Study
This section describes three critical ablation studies. More ablation results and analyses are provided in the Supplementary Material.
5.1 Fusion Operation Ablation
Figure 3 shows that DICE scores increase systematically as we expand the pool of fusion operators. In Figure A), starting from single operator (AWF, leftmost bar), adding Multiply or Add provides modest gain,, while injecting the Concat operation provides the sharpest jump in mean DICE.
8

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Average DICE (%)
(B) Orgen-wise DICE (%)
50
Last Layer LoMix
75
Layer
14,51
KI
20% Last Layer
40% LoMix
20% LoVix
(5]
60
10%L LUMix
55
555 Last Layer
575 LoMix
+9.23
40
PC
35
10%
Figure 2: Limited data evaluation on Synapse 8-organ segmentation with PVT-EMCAD-B2. LoMix uses NAS-inspired softplus weighting. t denotes higher is better. Results are averaged over five runs. Detailed results are provided Table
(AI Average DICE 186
ISI Organ (%)
85.0
$4.5
24.0
Ad5
(KAL) ANT1
Concac
Concat)
(Add) Put Canoul)
Concat
ONG.
IMSNA Cocat
(Put Danad, AWTI
(CoocatAW
(2hmm))
Oper aliom
Figure 3: Comparison of different fusion operation combinations using NAS-inspired Softplus weights and PVT-EMCAD-B2 model for Synapse 8-organ segmentation. f indicates higher is better.
Each additional operator gives the softplus search greater freedom to discover complementary scale interactions, and the trend is strictly monotonic: the full LoMix variant that activates all four operators tops the chart at 85.07% DICE, outperforming the best three-operator setting by 0.46%. The radar plot (Figure B))) shows that these improvements are broader, LoMix encloses smaller polygons for every organ, with the largest margins on hard, low-contrast structures (GB, PC, SM) while still improving saturated classes such as liver and aorta toward their performance ceiling. In short, this ablation confirms that diverse operator choice, coupled with NAS-inspired weight learning, is critical: each extra operation opens a new pathway for the model to align coarse and fine cues, and the resultant mixture consistently translates into superior, organ-robust segmentation.
5.2 Effect of NAS-Inspired Weight Learning
Figure contrasts fixed vs. learnable NAS MAS-inspired softplus loss weighting across six supervision types. In Figure [ deep supervision, single-operator fusions (addition, multiplication, concatena- tion, autentiio--wiihheed fusion (AWF)), and LoMix benefit from learning weights online instead of keeping them equal. The absolute gain ranges from +0.21% % DICE (Add) to pronounced +0.7% for Multiply, improving LoMix to 85.07% mean DICE score without changing architecture or inference cost. The radar plot (Figure 4 B)) shows that the learned variant never hurts any organ and delivers the largest jumps on the most scale-sensitive structures: gallbladder (GB) and left kidney (KL), while improving already strong classes (liver, spleen (SP)) toward the performance ceiling. Together, the results confirm that the learnable softplus weighting a universal add-on: it tightens every supervision strategy but realizes its full potential when paired with LoMix's 's rich operator pool.
5.3 Effect of LoMix on Backbone Architectures
Figure demonstrates that LoMix is architecture-agootic::: whether the backbone transformer- based (PVT-v2 (34) or purely convolutional (ResNet [II),, replacing conventional supervision with

---

85.5
(A) Average DICE (5)
[B] Organ-wise DICE (%)
Deep Sup. (Rwed)
Food
+0.45
Deep (Leamed)
$5.0
Leamned
Additinn (Tized)
Addition (Leemne)
84.5
Multipiiation (ixed)
Muttiliiation (Lce med:
$4.0
Concatenation (Fwed)
83.5
Concatenation
(fised)
$3.0
Liver
75 to it sc X0
AMF [laurred
LoMix (Fued)
Lokix (Leamned)
82.0
81.5
61.0
Deep Supp
AddRion
Multification
Concalonation
LoMkx
Figure 4: Effect of NAS-inspired Softplus weight learning on Synapse 8-organ segmentation with PYT-EMCAD-B2. Fixed = equal loss weights, Learned = NAS--nssisee softplus weights.
80
+4:13
(A) Average DICE
Layer
[3] Organ-wise DXCE 195) 7
PVT.V2 90
PVT.V2 00000
Docs Sup.
Last Layer
Sup
1V1 V2, 90- LoMo (Dura)
+8.00
00urs)
PVT V2 51 Last Layer
PVT V1 01
Dewa Sup.
82
PVT V2. 51
LoMX (Durs)
PVT 12 62
Last
80
PVL.V Deep Sup.
PVT 52- LoMk (Dun))
Becletl8 Last
Recletta Sup.
50 TD
Resletta LcMX
ResNer3t Last
ResNet31 Sup.
RssNet34 LoMi (Ou
72
5M
05
Figure 5: Comparison of different supervision schemes on Synapse 8-organ segmentation across five backbones. LoMix uses NAS-inspired softplus weighting. Sup.: Supervision.
our LoMix consistently improves performance. In Figure A), mean DICE score increases with supervision strength: Last Layer < Deep Sup. < LoMix for every network. The absolute gain delivered by LoMix over single-head supervision generalizable: +7.41-11 11.8% for ResNet variants and +3.88 6.71% on PVT-v2 variants which shows that our LoMix supervision unlocks benefits that standard decoders leave untapped. Figure B) confirms that the improvements are broader: LoMix dominates other supervisions on all eight organs, with the largest margin gains again on scale-sensitive organs such as gallbladder (GB) and pancreas (PC). Crucially, these gains come at zero inference cost, thus underscoring LoMix's 's practicality as a universal booster for existing segmentation backbones.
6 Conclusion
This work introduced LoMix, a plug-and-play, NAS-inspired module that unlocks the untapped value of multi-scale decoder logits by (i) generating a rich family of mixed-scale predictions through four differentiable fusion operators and (ii) learning, via softplus gating, how strongly each real or fused map should guide training. Extensive experiments on four medical-image tasks demonstrate that LoMix consistently outperforms both single-output supervision and classical deep supervision baselines while incurring no extra inference cost. Because LoMix exposes its learned weights as explicit scalars, the fusion it learns is interpretable, transferable across backbones, and easy to fine-tune, offering practitioners principled yet practical path to high quality segmentation without architectural redesign. Our current implementation of LoMix is limited to 2D medical segmentation tasks. Future work will extend the approach to dense prediction tasks beyond segmentation.
10

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Acknowledgments and Disclosure of Funding
This work is supported in part by the NSF grant CCF-2531882, and in part by the iMAGINE Consortium (htps://imagine.utxaal
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20

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A Appendix Supplementary Material
A.1 Training Algorithm
Algorithm I outlines the training procedure for our U-shaped segmentation network with the LoMix module. We train end-to-end with Adam [17], updating both the usual network parameters and the loss weight parameters a. Pseudocode is given in Algorithm [below.
During training, the network learns to minimize the total loss by improving all logits. As training pro- gresses, the softplus(a) weights will adjust for example, if the attention-fused output consistently awt yields lower error than others, its corresponding di may increase, giving it more influence.
In the end, we have multiple trained decoder branches and fusion modules. For inference, can either use output of the single best-performing decoder stage or combine the decoder outputs via one of the fusion strategies (e.g., attention fusion simple average) produce the final segmentation.
Thanks to our LoMix supervised training, all these outputs are optimized be accurate and comple- mentary. The result is segmentation model that embodies an ensemble of experts, trained jointly in a principled manner to maximize the segmentation quality.
A.2 Advantages of LoMix
Our LoMix-enhanen segmentation has several notable advantages:
Interpretability:
The learned weights softplus(a:) and softplus(ag" provide insights into the importance each decoder output and fusion type. For instance, if the addition-fusion outputs receive a high weight, it means that the simple aggregated prediction is consistently effective; particular logits's weight drops close to zero, then the system concludes that that particular logit is not helpful. Examining these weights can thus reveal which model finds most useful, thus offering a peek into
combinations of predictions ensemble strategy learned by the network.
Adaptability: Because the loss weights are learned, the framework adapts to different datasets and scenarios. The model can allocate more weight to certain predictions if the data benefit from that fusion. For example, on data where one decoder stage consistently outperforms the others, the learning process can put more emphasis to that decoder stage (and possibly its fused outputs) optimize performance. Conversely, if all decoders are needed (say each captures different class scale), then the weights can remain distributed.
Algorithm Training with Logits Mixing (LoMix)
Input: Labeled image dataset D ((T,, Y,) number of decoder stages L; network parameters (encoder, decoders); fusion-weight scalars { +- 0 for every original or mutant logit P. Output: Optimized model è and fusion-weight scalars {a..]
1: Initialize optimizer O for (e, (a.
2: while training not converged do
3: Sample mini-batch ((I, 10) CD
4: {Pi)/ + fe(())
> upsampled stage logits
5: Porig < (PI....,Pi), Puut 0
6:
for all non-empty subsets S C (1,..., L} with |S| 2 do pladd) p(mult)) plcat) p(awf) via Eqs. 15
7:
Generate fused logits PS
Ufpfadi pladd) p p(muutt) plear) ,S
p(aur)]
8:
Pinut +- P
9:
Liotal +0
10:
for all P. E Porig U do
11:
Wn + softplus(au)
NAS-inspired, unconstrained
Lrotal &- Crotat + We, Loogl Pu, Y6)
13: Update (O, (aw))) (.
14:
return θ {aw}

---

This adaptability alleviates the need for manual hyperparameter tuning of multi-output losses for each new application.
Training Stability and Regularization: Supervising multiple predictions (original and fused) acts as implicit regularizer and stabilizer. Indeed, it is less likely that the network will overfit or get stuck in a poor local minimum because each decoder stage guided by its own loss and by the fused losses that tie all decoder stages together. If one decoder begins to make mistakes, the others (and their combinations) still provide correct feedback, preventing the entire model from drifting off. Additionally, the learned weighting further stabilizes training by reducing the impact any particularly noisy loss term: if fused output is extremely erroneous the start, its weight can adjust downward, preventing it from exploding the gradient. Overall, we observed that LoMix yields faster convergence and more robust training than single-output or manually supp-super counterpart, thanks to these effects.
A.3 Why Softplus in NAS-inspired Weight Learning?
We choose Softplus (see Eq. 6 rather than alternatives like Softmax explicit normalization for several reasons:
No Sum Constraint. Softmax would constrain the weights to sum to one, coupling them and forcing distribution over outputs. This would prevent the model from independently suppressing noisy output (as reducing one weight necessitates increasing others). In contrast, Softplus outputs are independent and unbounded, so each weight can shrink toward zero (effectively ignoring that logit) or grow arbitrarily without affecting the sum of other weights.
Strict Positivity. Softplus guarantees a smoothly, unlike ReLU which could produce exact zeros or sigmoid which would bound weights in (0, 1). Positive weights ensure each loss term contributes non-negatively.
Smooth Gradients. Softplus is smooth and has non-vanishing derivatives for all inputs. which stabilizes learning of the weight parameters. A hard normalization or clipping could yield zero gradients for some ranges.
A.4 Datasets
We evaluate the LoMix's efficacy across seven datasets covering six segmentation tasks. Our two multi-class segmentation datasets are Synapse Multi-organs' and ACDC cardiac organs 2 The Synapse multi-organ dataset is used for abdominal organ segmentation and includes 30 abdominal CT scans with 3.779 axial slices of 512 x 512 pixels. Following the TransUNet [3, 18 scans (2,212 slices) are used for training and 12 for validation/testing. We segment eight organs: aorta, gallbladder, left kidney, right kidney, liver, pancreas, spleen, and stomach. For cardiac organ segmentation, the ACDC dataset contains 100 cardiac MRI scans segmented into three sub-organs: right ventricle (RV), myocardium (MYO), and left ventricle (LV), We follow the TransUNet protocol using 70 cases (1,930 slices) for training, 10 for validation, and 20 for testing. Our binary breast cancer segmentation dataset, BUSI (1 contains 647 images: 437 benign and 210 malignant. Our skin lesion segmentation dataset. Our three polyp segmentation datasets
Kvasir [13 (1,000 images), (28) (196 images). Furthermore, we use ISIC2018 66 (2,594 images) for skin lesion segmentation. We use 80% of the data for training, 10% for validation, and 10% for testing in BUSI, Kvasir, CVC-ColonDB, LaBioo and ISIC2018 datasets.
ClinicDB [3] (612 images), CVC-ColonDB (29 (379 images), and LITIS-arrbPolypple
A.5 Dataset Specific Implementation Details
For multi-class segmentation in Synapse Multi-organs and ACDC datasets, we use an input size of 224 x 224, and optimize the combined Cross-entropy (3=0.3) + DICE (y=0.7) loss. We train models for 300 and 400 epochs with a batch size of and 12 for Synapse and ACDC datasets, respectively.

---

Operation-wise Weights per Epoch
Operation-wise Weights per
Sum Vaights (Lag-Scaka)
Operationeen Sumt Weegits
H00
Usrkot
104
Lorcol
10'7
20
100
150
200
250 200
30
102
130
200
250
305
Fpoch
Fpoch
(a) Linear scale
(b) Log1o scale
Figure S.1: Operation wise sum of softplus loss weights during training. Each curve aggregates all logits produced by the same fusion family (Original, Add, Mul, WF. Concat) in the PVT-EMCAD-BB + LoMix run. (a) softplus weight values in linear scale, (b) the same values in log-scale.
(a) Linear scale
(b) Log1o scale
Figure S.2: Evolution of softplus weights over training epochs in Synapse 8-organ segmentation. (a) softplus values in linear-scale, (b) the same data with logarithmic color normalization, revealing the relative ordering very small weights and confirming that few predictions dominate the loss while many others are softly suppressed.
The image dimensions image dimensions are set to 352 x 352 pixels for the polyp datasets (Kvasir, CVC-ColonDB, ETIS- LaribPolypDB), respectively. We utilize multi-scale training approach, with scales of (0.75. 1.0. 1.25] and no augmentation. We use a hybrid weighted Intersection over Union (IoU) loss (I:1) and train models for 200 epochs with batches of 16 for BUSI, ISIC2018, Kvasir, CVC-ColonDB, and ETIS- LaiisLaribPolyDDB We employ random rotation and flipping as data augmentation methods in all of our experiments except the BUSI dataset. We save the best model based on validation DICE score in all datasets and report DICE score testsets except Synapse Multi-organ dataset. Only the last stage prediction chosen as final segmentation output for Synapse Multi-organ and ACDC datasets, while the predictions from all four stages are summed together to produce the final segmentation map in the BUSI, ISIC 2018, Kvasir, CVC-ColonDB,, and ETIS-Larib LTIS-LaribPolypDB datasets.
256 256 pixels for the BUSI and ISIC2018 datasets, while the
(BCE) with a weighted

---

Original Softplus Weights per Epoch
Original Softolus Weights Epoch
0.7
(1)
o3g
-0.5
ong
(3)
0.6
Softalus Weight (og10)
Softllus Weight (1og1.0)
-1.0
0.5
0.4
0.3
-2.
-.0
-35
50
100
150
200
250
300
50
100
150
200
250
300
Epoch
Eparh
(a) Linear scale
(b) Logio scale
Figure S.3: Evolution of softplus loss weights for the four original logits. Both panels track the same data over 300 training epochs: (a) the left panel plot the softplus values in linear scale; (b) the right plot rescales the y-axis logarithmically to expose tiny values.
Add Synthetic Softplus Weights per Epoch
Add Synthetic Softplus Weights per Epoch
0.0
(0.110.11
(61_(0.1.2
add. (0.1)
add_(0.1.2)
add, (0,2)
add_[0.1133;
add_ (0.2)
add_(0.1.3)
0.6
add
add_(0.2.3)
add.
Sattlls Weight (log10)
ad1.11.2)
=6d.(1.2.3)
Somplus Weightt (leg10)
-1.0
(1.2)
add_(1.2.3)
0.3
add (1.3)
ad1_[0,1.223)
add, (1.3)
add__06.1.2.3
(d.](2.3)
-1.5
[dd]
0.4
0.3
-2.5
0.2
0.3
0.0
-1.0
300
Epoch
Epoch
(a) Linear scale
(b) Logio scale
Figure S.4: Per-subset softplus weights for the addition operation during training. Curves correspond every additive mixture of decoder logits. Both the (a) linear- and (b) log-scaled plots reveal how the NAS-inspired optimization reallocates loss weight across epochs.
A.6 Weight Dynamics and Interpretability
Figure S.I shows that LoMix reallocates supervision away from the four original logits toward the far larger pool of fused logits. By epoch 50, the original logits hold < 10% of the total weight, while the learnable fusions, especially aitention-weighted fusion (AWF) and multiplication (Mult) remain dominant for the next 250 epochs. Addition and concatenation also preserve non-trivial weight, illustrating that every operator family contributes useful gradients. The log-scale panel highlights near-perfect exponential decay for all groups, with parallel slopes indicating that LoMix balances them proportionally rather than suppressing any single operator outright. Together, these curves confirm that the NAS-inspired optimization discovers a nuanced, multi-operator loss distribution in which learnable fusions drive most of the training signal, yet fixed arithmetic fusions and even the original heads are still retained to provide complementary guidance.
Figure S.2 visualizes the softplus loss weights that LoMix learns for all 48 supervised logits (4 originals is 44 fusions) over 300 epochs. In the snear-sccale map (Figure S.2p), the vast majority of weights collapse to closer to 0 after 50 epochs, while small cluster mainly the finest (last layer) original logit and few mutants created by AWE, Add, Mul, and Concat- Concat-retain higher weights. The log-scale view (Figure S.2b) makes the hierarchy clearer: the finest grain and attention-weighter weightedd fusion (awf) maps dominates over other original or synthetic maps. Thus, the differentiable search automatically reduces 48 supervision maps down to a compact, high-impact subset while discarding redundant logits.

---

Mui Synthetic Softplus Weights
Epoch
Mui Synthetic Softplus Weights Egoch
0.7
mult (0.3)
mut_
mun_10.33
(0,122
0.6
mult (0.3)
(0.2_
m6.1.6.1133
-0.5
mut, (0.3)
mult (0.2.)
mitt_001135
Softalus Weight (ogg00)
mult (1.25
(1,23)
Softglus Weight (log100
mutt (1.2]
mult, (1.23)
0.5
mult, (1.5)
(ut1_0,1.223))
mult (1.33
mult (0.12.3)
-1.5
mul_(2.95
0.4
0-3
2.5
-35
50
100
150
200
250
300
50
100
150
200
250
300
Epoch
Epach
(a) Linear scale
(b) Logio scale
Figure S.5: Per-subset softplus weights for the multiplication operation during training. Curves correspond to every multiplication mixture of decoder logits, Both the (a) linear- and (b) log-scaled plots reveal how the NAS-inspired optimization reallocates loss weight across epochs.
Concat Synthetic Softplus Weights per Epoch
Concat Synthetic Softplus Weights per Epoch
concat, (0,1)
concat, (0,1.3)
concat, (0,1)
concat, (0.1.2)
0.6
concat, (0,3)
concat, (0,2.3)
-0.5
concat (0.3)
cancat (0.2.3
(0.1.3)
Softplus Weightt (lgg0))
0.5
concal [1.2)
concat [1.3)
concat (0,12233
concat (1,2.3)
Softplus Weight Flgg00
-1.0
concat (1.3)
concal (1,2)
concat (0.1.23)
concat (1.23)
[2.3)
-1.5
0.4
0.3
0.2
3.00
-3.5
50
100
130
200
250
300
50
100
150
200
250
300
Epoch
Epach
(a) Lincar scale
(b) Log1o scale
Figure S.6: Per-subset softplus weights for the concafenation operation during training. Curves correspond to every concatenation mixture of decoder logits. Both the (a) linear- and (b) log-scaled plots reveal how the NAS-inspired optimization reallocates loss weight across epochs.
Original. Figure S.3 shows how LoMix optimizes the weights to four original decoder outputs during training. Both inear-scale (Figure aand log-scale (Figure S..3)) reveal that the fine-scale heads (orig_2 2/ origs) retaining more influence than the coarser ones during all training epochs. This trend mirrors the full heatmap in Figure [S.2] the NAS-inspired optimization quickly reallocates the supervision to a handful of high-utility fused logits, allowing LoMix to concentrate the gradient signal where it is most beneficial.
Addition (Add). Figure S.4 shows that all additive subsets decay smoothly, with the full 4-logit sum (.dd0,1,2.3) holding the largest weight throughout training. This indicates that the optimizer values a coarse, presolutioonagnostii blending of logits for global consistency even late into training.
Multiplication (Mult). Figure demonsttssss that multiplicative subsets start at the same magni- tude as Add, leaving the 3- and 4-branch products dominant after 150 epochs. This pattern suggests that the optimizer relies on multiplicative interactions primarily when multiple scales jointly agree, using them selective gating rather than expansive fusion.
Concatenation (Concat). All Concat curves in Figure S.6 lie almost on top of each other, revealing that concatenation quickly learns down-weight raw channel stacks. The uniform and steady decay implies that concatenation contributes mainly in the early epochs to speed convergence, with its influence reduces as finer operators take over.
5

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WF Synthetic Softplus Weights Epoch
WF Synthetic Softplus Weights Epoch
0.7
aw1_(0.3)
wf_(0,112)
[
(0,1.2
0.6
em. (0.3)
moL_06.1.3
(0.2))
-0.5
aw|(0.2)
awt. (0.3)
sw__0..1.3
(0.2.3)
Softalus Weight (ogg00)
awl aw|_(1.3)
not (1,2,3
Softllus Weighht (log10)
awt. (1.23
ant (1.23)
0.5
awd, (1.3)
mof (0,1,,,)))3
awi
anf (0.113)))))3)
.w._(2.3)
-1.5
[2.33
0.4
0.3
-25
-35
50
100
150
200
250
300
50
100
150
200
250
300
Epoch
Epach
(a) Linear scale
(b) Logio scale
Figure S.7: Per-subset softplus weights for the attention-weighted fusion (AWF) operation during training. Curves correspond to every AWF mixture of decoder logits. Both the (a) linear- and (b) lcalsc plots reveal how the NAS- AAS-inspired optimization reallocates loss weight across epochs.
Table S1: Results of Synapse 8-organ segmentation. DICE scores (%) are reported for individual
organs, Results of UNet, AttnUNet, PolypPVT, SSFormerPVT, TNet,UNet, and SwinUNet taken from [24]. Gallbladder (GB), Left kidney (KL., Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). 1 (4) denotes the higher (lower) the better. means missing data from the source.
Results are averaged over five runs. Best results are shown in bold.
Methods
Average
Per-organ DICE (S)
DICE (%)
HD955
mloU
GB
KL.
KR
Liver
SP
SM
UNet
25
70.11
44.69
59.39
84.00
56.70
72:41
62.64
86 98
48.73
81.48
67.96
AttnUNe
71.70
34.47
61.38
82.61
61.94
76.07
70.42
87.54
46.70
80.67
67.66
UNet++
34
80.01
08
89.15
70.99
83.37
79.21
94.00
61.23
86.38
75.79
Deepl abx 3lus-
79.37
23.43
69.52
83.84
63.56
85.31
82.29
94.04
59,22
$8.61
78.12
SSFomner
67.23
82.78
63.74
80.72
93.53
87.
PolypPTT 21
78.08
25.61
67.43
82.
66.14
81.21
73.78
94.37
59.34
88.05
79.4
TransUNet g1
77.61
26.90
67,32
86.56
60.43
80.54
78.53
94,33
58.47
87.
75.00
SwinLNet 2]
81.76
65.95
82.32
93.73
53.81
MT-UNet
78.59
26.59
87.92
64.99
81.47
77.29
93.06
59.46
87.75
76.81
MISSForner
96
18.20
86
68.65
85,
82.00
94.41
91.92
PVT-CASCADE
81.06
20.23
70.88
83.01
70.59
82.23
80.37
94.08
64.43
90.10
83,69
TrasCASCADE
20
82,68
17.34
73.48
86.63
68.48
87.
84.56
94,43
65.33
90.79
83.52
Rolling-LINet-S
77.18
85.77
CMUNeXi-S [0]
75.20
28.16
64.37
83.96
61.34
77.01
78.04
91.53
$1.50
85,36
72.85
EGE- UNE--UNei [
62.28
51.22
48.83
70.43
51.35
68.28
59.32
86.70
42.25
67.06
52.84
PVT-GCASCADE
83,28
15.83
73.91
86.50
71.71
87.07
83.77
95.31
66.72
90.84
83.58
UNeXi [1]]
72.60
30.68
61.30
80.20
60.82
76.13
69.96
91.80
48.04
83.27
70.64
PVT-EMCAD--O
94.57
92.66
PVT-EMCAD--B 24
83.63
15.68
74.65
88.14
68.87
88.00 08.08
84.10
95.26
68.51
92.17
83.92
PVT-EMCADD-B0 LoMix (Ours)
+:0.9
16,80
73.44
87.41
86.67
83,77
95.41
92.70
83.02
PVT-EMCAD-B2 LoMix (Ours)
85.07 11.3
14.85
76.41
88.84
73.51
89.07
84.71
95.76
69.74
92.47
86.44
Attention- Weighted Fusion (AWF). Figure S.7 shows that AWF subsets preserve comparatively above other operators. This
higher weights for longer- the full 4-logit persistence shows the optimizer's 's strong preference for spatially adaptive weighting, thus confirming AWF's key role in exploiting complementary decoder resolutions throughout training.
A.7 Learned Loss Weights Comparison of the Best Epoch
At convergence (Figure S.8), the NAS-inspired optimizer concentrates most of the loss weight on attention- fusion (AWF) combinations, followed by smaller but still meaningful allocation to multiplicative and additive mixes, while concatenation paths receive almost negligible weight. Notably, among the original logits only the fine grain stages (2, 3) receive higher weights, confirming that the network relies primarily on multi-scale mixtures rather than any single head. This distribution echoes our ablation study: the model learns to emphasize operators that can reconcile complementary spatial cues (AWE, Mult, Add), while weighting logits that add parameters without clear synergy (Concat), thereby producing the highest DICE performance without manual tuning.

---

Softplus Weights by Combination at Last Epoch
[0)
(1)
[2)
(3)
add (0.1)
add add_(0.2)
add [0.3
add_
add
a2d]
add (0.1,)
add_
add (0,2,3)
add
add_ (0,1,0,,
m011_
[0,2]
[0,3_
mult mult_[1,2]
mult (2.3)
mult [0,1,2)
mult [0,1.3)
mult_[0,2.3]
mult (1,2,33
mult {0.1,2.3)
awf [0.1)
aw,[[0
awf (0,3)
awf (1,2)
awf (1,3)
awf (2,3)
awf [0,1,)
[0,10)
[0,23)
awf (1,2,3]
awf awf_ (0,1,2,33
[oncat_[O.1]
concat_(0.2)
concat_(0.3)
concat_
Group
concat_(1.3)
Original
concat_(2.3)
Add
concat_[0.1.2]
Mult
concat- [0.1.3)
concat co.cat_(0.2.3)
AWF
concat [1,2,3) concat_(O,1,2.))
Concat
0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 0.00035 Softplus Weight
Figure S.8: Learned softplus weights of the best LoMix epoch. Each bar corresponds the final softplus weight assigned to one logit combination, grouped by fusion operator (Original decoders, Addition, Multiplication, AWE, and Concatenation). Longer bars indicate that combination is more strongly trusted by the learned loss during training.
A.8 Results of Synapse 8-organ Segmentation
Table shows that LoMix achieves clearly superior multi-organ segmentation Synapse compared to prior CNN- and transformer- based methods. In particular, our LoMix variants attain the highest average DICE, lowest HD95, and highest mloU of any approach. For example, the best LoMix model (85.07%) outperforms PVT-EMCAD-B2 (83.63%) while also exceeding TranSCASCADE (82.68%) and MISSFormer (81.96%). The gains especially large small, difficult organs: e.g. LoMix
7

---

Table S2: Results Synapse 13-organ segmentation. DICE scores (%) are reported for individual organs. Gallbladder (GB), Left kidney (KL), Right kidney (KR), Pancreas (PC), Spleen (SP), Esophagus (Eso), Stomach (SM), Inferior Vena Cava (IVC), Portal and Splenic Veins (Veins), Left
Adrenal Glands (LAG)), and Right Adrenal Glands (RAG). Results are averaged over five runs. Best
results per method are shown in bold. LoMix achieves the best average DICE and lowest HD95.
Methods
Averuge
Per-organ (1G
DICE
HLD95.
58.06
71.33
50.37
Deep Sup
6006
67
70.40
95.68
77.29
65.41
0.00
IVFEMMAD--B 23
76.21
13.56
61.64
91.34
83.40
56,78
63.
72.49
95.35
84.86
57.42
79.37
67.74
66.35
52.58
53.50
PYTEMCAD--2
LoMix (Ours)
76.90
12.42
65.49
90.24
84.06
87.54
71.08
74,40
95.74
85.68
$7.05
90.65
68.31
69.63
51.34
54,07
Table S3: Results of breast cancer and skin lesion segmentation. We reproduce the results of SOTA methods using their publicly available implementations with our 80: 10: 10 train-val-test splits. The mean DICE scores (%) of testset over five runs are reported. #FLOPs of all the methods are reported for 256 x 256 imputs. Best results are shown in bold.
Methods
#Params
#FLOPs
BUSI
ISIC2018
UNet 25
34.53M
65.53G
74.04
86.67
UNet++
9.16M
34.65G
74.76
87.46
AttnUNel
19
34.88M
66.64G
74.48
87.05
DeepLaby34
39.76M
14.92G
76.81
88.64
PraNet
32.55M
6,93G
75.14
88.46
UACANet 15
69.16M
31,51G
76.96
88.72
SSFormer-I.
66.22M
17,28G
78.76
90.25
PolypPVT
25.1
5.30G
79.35
90.36
TransUNet 8
105.320 M2.
38.52G
78.01
89.04
SwinUNet 2
27.17M
6,20G
77.38
88.66
UNeXt [31
1.7M
0.57G
74.71
87.78
CMUNeXI 30
3.15M
7.37G
77.34
87.51
Rolling-UNett
16
1.78M
2.10G
76.38
87.35
PYT-CASCADE-B2
20
34.12M
7.62G
79.21
90.41
PVT-EMCAD BO
224
3,92M
0,84G
79.80
90.70
PVT-EMCAD-B3 241
26.76M
5.60G
80.25
90.96
PYT-EMCAD-BO LoMix (Ours)
3.92M
0.84G
80.47+11.04
90.77-0.63
PVT-EMCAD-B2- + LoMix (Ours)
26.76M
5.60G
81.32-11.21
91.18:10.72
significantly improves pancreas and gallbladder DICE versus previous models. Moreover, integrating LoMix into lightweight PVTv2 backbones yields consistent boosts: the PVT-EMCAD-BO- + LoMix (82.60%) exceeds the DICE of PYT-BO-EMCAD (1.97%) and similarly the PVT-EMCAD-B2 + LoMix (85.07%) outperforms the PVT-EMCAD-B2 baseline (83.63%). Crucially, LoMix is applied only during training (no extra inference cost). yet consistently pushes the SOTA across all key metrics. The reason behind the performance gain is that LoMix's adaptive multi-scale logit fusion produces more accurate and robust abdominal organ segmentation, particularly for small, challenging structures such as the pancreas and gallbladder.
A.9 Results of Synapse 13-organ Segmentation
Table S2 shows that integrating LoMix into the PVT-EMCAD-B2 network provides the strongest 13- organ performance reported to date on Synapse. LoMix improves the mean DICE to 76.90%: a gain of +9.8% over single-head supervision and +6.5% over uniform deep supervision, while simultancously reducing HD95 from 15.56 to 12.42. More importantly, improvements are not limited to one or two easy structures. Indeed, LoMix achieves the best DICE 10 of 13 organs, including difficult soall-volume classes such as the gallbladder (+4.3%), esophagus (+3. 1.3.1% 1%), and portal and splenic veins (+3.8%). It also revives the previously "dead" adrenal- gland predictions, pushing DICE from to 51-5144% Larger, context-driven organs such as liver and spleen see further improvement, and aorta performance remains on par with the best prior result. These gains confirm that LoMix's learnable, mixed-scale supervision improves both boundary-sensittee and context-dependent structures, thus delivering uniformly stronger and more anatomically faithful segmentation without introducing any inference-tine time overhead.
8

---

Table S4: Results of polyp segmentation. We reproduce the results of SOTA methods using their publicly available implementations with our 00:10:10 10 train val-test splits. The mean DICE scores (%) of testset over five runs are reported. Best results are shown in bold.
Methods
WParams
Kvasir CVC-ColonDB
ETIS- ETIS-LariPooyyDDB
UNet [25
34.53M
82.87
83,95
76.85
UNet++
36
9.16M
83.36
87.88
77.40
AttnUNet
19
34.88M
83.49
86.46
76.84
DeepLaby3+
39.76M
89.06
91.92
90.73
PraNet
32.55M
84.82
89.16
83.84
UACANet
15
69.16M
90.17
91.02
89.77
SSForme--
33
66.22M
91.47
92.11
90.16
PolypPVT
25
91.56
91.53
89.93
TransUNet
105.32M
91.08
91.63
87.79
SwinUNet
27.17M
89.59
89.27
85,10
UNeXt 31
1.47M
77.88
83.84
74.03
CMUNeXi 30
3.15M
78.41
83.25
76.12
Rolling-UNet-S
16
1.78M
75.93
82,48
73.26
PYT-CASCADE-B2
20
34.12M
92.05
91.60
91.03
PYT-EMCAD-BO
24
3.92M
91.95
91.71
91.65
PYT-EMCAD--B2
24
26.76M
92.75
92.31
92.29
PYT-EMCAD- LoMix (Ours)
3.92M
92.34.10.96
93.31 1..86
92.74.10.79
PVT-EMCAD-B2 + LoMix (Ours)
26.76M
93.45=0.87
93.98=+0.68
93.10=:0.96
Table S5: Effect of input resolution on Synapse 8-organ segmentation (1 higher is better, lower is better). Each row is averaged over five runs. The best results are shown in bold.
Average
organ DICE (%)
Resolution
DICE
HD95
mIoU
Aorta
GB
KL
KR
Liver
PC
SP
SM
224 x 224
85.07
14.85
76.41
88.84 88.84
73.51
89.07
84.71
95.76
69.74
92.47
86.47
256 x 256
85.45
12.18
77.05
89.07
72.90
89.26
84.88
95.68
72.65
92.48
86.71
x 512
87.25
14.49
79.52
91.58
78.05
88.96
85.73
96.33
76.96
92.41
87.98
A.10
Results of Breast Cancer and Skin Lesion Segmentation
Table $3 shows the evaluation of ultrasound breast-tumout (BUSI) and dermoscopic skin lesion (ISIC2018) benchmarks and again demonstrates that LoMix can improve the DICE score of efficient networks without increasing their computations. When integrated onto the PVT-EMCAD-BI network. LoMix improves DICE scores by +1 1.07% BUSI and +0.22% on ISIC 2018, surpassing heavyweight designs such as DeepLabv3+ and SSFormer-L. In all cases, the gains do not require architectural changes at the test time, confirming that LoMix's 's learnable mixed-scale supervision translates into tangible DICE score improvements, even small, noise-prone medical datasets, without compromising the compactness efficiency of the underlying model.
A.11 Results of Polyp Segmentation
Table benchmarks LoMix on three challenging colon-polyp datasets against different CNN, transformer, and lightweight hybrid methods. LoMix improves mean DICE of the PVT-EMCAD-BO network by +0.4-1.6% on Kvasir, CVC-ColonDB, and ETIS-LaribPolypDD. thus outperforming substantially larger models such DeepLabv3+ and SSFormer-L. Coupling LoMix with the PVT- EMCAD-B2 establishes a new SOTA on all three datasets, thus surpassing the best baseline of PVT-EMCAD-B2 by up +1.7% while matching its inference parameter count and runtime. Gains are achieved without modifying the network at inference, confirming that LoMix's 's learnable mixed- scale supervision improves polyp delineation accuracy and data efficiency without sacrificing the compactness of the underlying architecture.
A.12 Effect of Input Resolution on Synapse 8-organ Segmentation
Table S5 shows that LoMix capitalizes on every pixel is given. When the input image enlarged
from 224 224 the mean DICE rises from 85.07% to 85.45% while HD95 reduces by 2.7, indicating

---

Table S6: Comparison of different fusion operation combinations using NAS-inspired Softplus weights and PVT-EMCAD-BR B2 model for Synapse 8-organ segmentation. Gallbladder (GB), Left kidney (KL), Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). t (4) indicates
higher (lower) is better. LoMix achieves the highest average DICE.
Operation
DICE
HD95,
mloU
Aorta
GB
KL
KR
Liver
PC
SP
SM
AWF
82.91
19.05
73.88
88.30
68.04
87.30
83.13
95.42
65.85
91.39
83.84
(Mult, AWF)
83.45
18.84
74.44
88.00
68.41
86.50
81.96
95.95
69.94
92.43
84.38
(Add,Muli)
83.51
22.99
74.63
88.12
68.35
86.25
83.54
95.24
70.09
90.75
85.74
Mult
83,55
17.47
74.33
88.20
69.34
88.02
83.84
95.06
66.59
91.27
86.09
Add
83.84
20.15
74.84
88.04
73.78
87.77
83,37
95.35
68.47
90.18
83.79
(Ad),AWF)
83.95
19.77
75.08
88.38
71.09
86.82
82.82
95.52
69.24
92.36
85,37
(Add,Concat)
83.96
15.74
75.07
88,
72.06
87.53
84.12
95.24
69.18
90.54
84,52
Murd,Mutr,AWF)
84.01
21.62
74.92
89.44
71.40
88.78
84.24
94.98
68.56
91.56
83.11
(Add,,oncat,,WW))
84.10
17.42
75.23
87.65
71.09
88.24
84.10
95.88
68.55
91.45
85.86
(Mult, Concat)
84.23
16.25
75.49
88.46
69.21
89.35
82.82
95.31
68.81
92.97
86.89
MAdddMult,Concal
84.32
19.14
75.35
88.50
71.79
87.59
83.18
95.02
70.18
92.78
85,48
Concut
84,45
19.33
75.53
88,12
71.12
88.32
84.73
96.05
69.80
91.22
86.23
84.61
20.24
75.75
88, 83
72.60
87.82
83,58
95,90
70.14
91.56
86.45
(Concat.AWF)
84.71
20.39
76.15
88.94
71.54
87.83
85.40
96.12
69.18
92.11
86.56
LoMix (Ours)
85.07
14.85
76.41
88.84
73.51
89.07
84.71
95.76
69.74
92.47
86.47
crisper boundaries at only a modest memory cost. Doubling the image again to 512x512 unlocks further leap to 87.25 % DICE score, setting new highs on six of eight organs: gallbladder gains +4.54% pancreas +7.22% stomach +1.51 %, and even large structures such as aorta and liver surpass 91.5% and 96.3% DICE, respectively. The pattern confirms that LoMix's 's learnable multi-scale fusion continues to integrate fine detail without over-fitting, thus scaling with resolution.
A.13 Detailed Results of Fusion Operation Ablation
This section extends Section by listing the full numeric results in Table 56 is restricted to single operator-c.g, only the attention-weighter fusion (AWF) or only element- wise multiplication- mean DICE remains below 83.6%. Two-operator mixtures raise performance into the mid-84% range, and every additional operator yields a further monotonic gain because the softplus search can explore a richer set of cross-scale interactions. The best three-operator recipe (Concat Multiply AWF) reaches 84.71 % DICE, but the full LoMix variant, which activates all four operators, pushes the average to 85.07%. Organ-wise scores show that only the complete LoMix combination provides balanced gains across the entire anatomy set, confirming our claim that operator diversity-coupled with NAS-style weight learning- is essential for fully exploiting complementary coarse-to- fine cues.
When the search space
A.14 Detailed Results of NAS-Inspired Weight Learning
This section extends Section 5.2 by presenting the full numbers in Table Across all five supervision settings, replacing uniform loss weights with our NAS-inspired softplus weights provides a clear net win. For plain deep supervision the learned variant improves mean DICE from 82.90% to 83.17% and, more importantly, reduces HD995 by 4 point, indicating sharper boundaries. The benefit is modest, but consistent for single-operator fusions- +0.21 % DICE for Add, +0.70% for Multiply, +0.39% for Concat, and +0.25% for AWT-while simultancously lowering or matching HD95 in every case. Crucially, when the full operator pool is active, the learned weights unlock LoMix's headroom: DICE improves from 84.6 34.62% to a new high of 85.07 % and HD95 falls by over point, with eight-of-eight organs improving remaining steady, thus confirming that adaptive reiweiigt helps the network emphasize whichever resolutions (and fusions) are most informative for each anatomy. Because weights are pruned scalars after training, these improvements come at inference cost.
A.15 Detailed Results of Different Supervision Across Backbones
This section augments Section 5.3 with the full numbers in Table S8 bones inclu-incl three transformer variants DYT_V2_Bon networks (ResNet18/34) our LoMix shows the supervision hierarchy: Last-Layer < Deep Supervi-
Across all five back- and two purely-convolutional
10

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Table S7: Effect of NAS-inspired Softplus weight learning on Synapse 8-organ segmentation with PVT-EMCAD-BA Fixed uniform loss weights of 1, Learned = our NAS- softplus weights. Gallbladder (GB), Left kidney (KI.), Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). t (L) denotes higher (lower) is better. Results are averaged over five runs. The best setting for
each fusion is shown in bold.
Fusion
Scheme
Average
Per-organ DICE (%)
DICE
HD95,
m
Aorta
GB
KL
KR
Liver
PC
SP
SM
Doep Sup.
Fixed
82.90
19.70
73.84
87,43
67.80
87.66
83,75
95.18
65.63
91.53
84.19
Learned
83.17
15.35
74.23
87.29
70.13
87.17
83.74
95.52
67.66
90.45
83.40
Add
Fixed
83,63
15.68
74,65
88,
68.87
88.08
84,10
95.26
68.51
92.17
83.92
Learned
83.84
17.15
74.84
88.04
73.78
87.77
83.37
95.35
68.47
90.18
83.79
Multiply
Fixed
82.85
19.16
73.37
88.61
67.19
86.62
82.85
94.93
68.09
90.27
84.21
Learned
83.55
17.47
74.33
88.2
69.34
88.02
83.84
95.06
66.59
91.27
86.09
Concat
Fixed
84.14
17.77
75.15
87.11
70.51
88.25
83,70
95.71
68.27
92.70
86.86
Learned
84.45
19.33
75.53
88.12
71.12
88.32
84,73
96.05
69.80
91.22
86.23
AWF
Fixed
82,66
20.56
73.44
88.21
69.69
87.48
81.96
95. 05.05
67.36
89,20
82.34
Learned
82.91
19.05
73.88
88,30
68.04
87.30
83.13
95.42
65.85
91.39
83.84
LoMix
Fixed
84.62
18.08
75.72
88.82
72.35
87.72
84.10
95.56
70.46
91.87
86.08
Learned
85.07
14.85
76.41
88.84
73.51
89.07
84.71
95.76
69.74
92.47
86.47
Table S8: Comparison of different supervision schemes on Synapse 8-organ segmentation across five backbones. LoMix uses NAS-inspired softplus weighting. Gallbladder (GB), Left kidney (KL), Right kidney (KR). Panereas (PC), Spleen (SP), and Stomach (SM). Results are averaged over multiple runs. Best result per model is bolded.
Model
Scheme
Average
Per-organ DICE (SO)
DICE
HD95)
mloU
Aorta
GB
KL
KR
Liver
PC
SP
SM
V2__v2_B
Last Layer
75.72
19.95
66.69
85.78
66.16
84.98
80.10
94.08
30.29
88.70
75.71
Deep Sup.
81.38
21.88
71.66
86.01
65.18
87.31
83.03
94.23
65.14
90.63
79.52
LoMix (Ours)
82,43
17.32
73.10
87.42
68.10
86.84
83,18
95.35
62.96
91.84
83.76
PVT_V2_B1
Last Layer
79.77
27.30
69.85
87.15
65.95
82.84
79.83
94,26
60.54
89.77
77.
Deep Sup.
82.23
25.45 25.46
72.60
85.54
69.00
85
81.02
95.42
67.90
90.86
82.68
LoMix (Ours)
83.64
21.05
74.52
88.20
69.57
88.22
83.64
95.04
69.90
90.58
84.00
VVT_V2.B2
Last Layer
80.94
22.89
71.18
87.14
68.00
84.87
81.10
94.63
63.08
89.82
78.86
Deep Sup.
82.90
19.70
73.84
87.43
67,80
87.66
83.75
95.
65.63
91.53
84.19
LoMix (Ours)
85.07
14.85
76.41
88.84
73.51
89.07
84.71
95.76
69.74
92.47
86.47
ResNet18
Layer
70.25
27.26
60.24
80.79
59.12
81.06
78.
91.13
15.68
88.10
67.94
Deep Sup.
79.80
20.63
69.74
85.95
67.
84.65
81.06
93.72
57.72
89.72
78.49
LoMix (Ours)
82.12
20.37
72.54
87.03
70.89
83.60
80.96
94.57
65.47
92.42
82.04
ResNet34
Last Layer
75.50
33.65
65.57
86.05
63.30
83.17
79.74
93.20
37.70
86.96
73.87
Deep Sup.
81.07
18.04
71.38
85.78
68. 13
86.78
82.75
94.06
60.93
90.10
80.06
LoMix (Ours)
82.91
15.31
73.61
87.08
73.61
87.67
83.77
94.36
66.54
91.55
78.73
sion < LoMix. Switching from single-head training to LoMix improves the mean DICE by +6.71 % on the lightweight PVT S___2_BB, by +4.13% on the large PVT_V2_BB V2_ and by a striking +1 +11.87% on ResNet18, while simultancously reducing HD95 by 2.63-18.34 34. Organ-wise scores echo this trend: LoMix delivers the best DICE for most classes on every backbone, with the sharpest jumps on the most scale-sensitive structures (GB, PC, SM) yet still improving saturated organs such as aorta and liver towards the performance ceiling. These consistent architecture-agnen gains confirm its value as a plug-and and--ndd-p-d -lay supervision method for both CNN and transformer networks.
A.16 Cross-dataset evaluation on polyp segmentation
Clinical deployment demands robustness across scanners and sites. Our evaluation has already addressed overfitting risk by spanning multiple, heterogeneous public datasets and modalities (i.c., abdominal CT, cardiac MRI, breast ultrasound, dermoscopy, colonoscopy), each collected with different protocols and devices. Yet LoMiX improves on every dataset without manual tuning, because it actually behaves like regularizer. In fact, LoMiX implicitly ensembles diverse multi-scale logits only during training, thus reducing the chance of overfitting to dataset-specific biases.
11

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Table S9: tross-aatasethoospital generalization of LoMiX. Using the PVT-EMCAD-BL network, all models are trained for 200 epochs on the Kvasir polyp-segmentation training set (900 images); the epoch with the best DICE (%) on the Kvasir validation split (100 images) is saved. Then we evaluated CVC-ColonDB,, EETIS-LaribPolyppe
the generalizability on three external test sets without further tuning.
Methods
CVC-ClinieDB
CVC-ColonDB
ETIS-LaribPolypDE
Last Layer
80.59
75.31
71.64
Deep Supervision
81.87
76.16
75.84
MUTATION
81.88
76.39
75.97
LoMiX (Ours)
83.08
77.70
77.01
Table $10: MedNeXt's (26 performance on 3D Synapse 8-organ segmentation with Last Layer (LL), Deep Supervision (DS), MUTATION, and LoMiX. DICE scores (%) reported for Gallbladder (GB),
Left kidney (KL), Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM).
Methods
Avg. DICE
Avg. HD95
Aorta
GB
LK
RK
Liver
PC
SP
SM
MedNeXI-M K3 LT.
86.22
6.62
91.96
75.59
87.11
85.14
96.81
78.31
91.26
83.61
MedNeXi-M.
86.63
78.91
85,94
92.10
84.43
MedNeXi-M M.MieXiiM_KCC K3 MUTATION
86.84
6.04
91.72
79.94
90.97
86.62
96.58
76.65
90.55
81.69
MedNeXt-MM K3 LoMiX (Ours)
87.19
4.84
91.81
79.87
90.54
86.65
96.68
76.95
90.63
84.37
To make this explicit, we include cross-dataset experiment (e.g., train on one dataset/hospital, and test another) as shown in S9 LoMiX produces the highest DICE on every dataset which confirm its superior ability to generalize across hospitals and acquisition devices.
A.17 3D Feasibility and Results of Different Supervision
LoMiX is dimension-agnostic:: it operates on C-channel class logit maps at the loss level. Extending LoMiX 3D is straightforward: can be done simply replacing the 2D convolutions and bilinear upsampling with 3D convolutions and trilinear upsampling, everything else remains unchanged. The fusion/weighting logic remains identical, and the cost still scales with the (small) number of decoder stages (rarely exceeding five stages). When GPU memory is low, either in 2D 3D, standard optimizations such as gradient checkpointing or caching logits on CPU further reduce compute/memory requirements without altering the algorithm, thus keeping LoMiX practical on modest hardware (< 5 GB extra GPU memory required for backpropagation to process a 96x96x96 volume with a four-stage network and 9 output classes).
To show the feasibility of LoMiX with 3D networks, the new preliminary results of 3D MedNeXt (66 with LoMiX are reported in Table STO Our results (in bold) demonstrate that LoMiX achieves the best average DICE and HD95 scores among all supervisions.
12


Reviews:
[{"review_id": "f0WsEi73Qg", "reviewer": "Reviewer_vjqb", "summary": "Neural networks of UNet like shape are widely used in medical image segmentation. These networks include an encoder that processes the input image to a sequence of increasingly abstract representations with lower dimension, and a decoder that combines these representations. Classically when training these neural networks, a signal is generated only by comparing the final layer with the target labels but work on deep supervision has shown that improvements can be made by introducing such a signal for different layers in the network. This work introduces a new kind of supervision method that generates a set of “fusion maps” by up sampling different stages in the decoder and then fuse them together with various operators. The fusion maps are then weighted together and added to the loss. Importantly, the weights are learnable which allows for more flexible supervision compared to previous work.", "questions": "**Question 1:** On line 145: it looks like the number of channels in each decoder layer is constant. Is this a requirement? In that case how would a UNet architecture that increase the number of channels when getting closer to the bottle neck be compatible with the introduced extension?\n\n**Question 2:** Would it be computationally feasible to generalize the method to a 3D setting? Is there some kind of exponential increase in the number of required fusion maps that gets out of hand?\n\n**Question 3:** Are the W_S in Equation (3) normalized to 1? I find it surprising that figure S.8 in the supplementals show that addition fused logit maps get weights so much higher than the concat ones, yet still are shown to be of the most important in the ablation study. Is there an explanation to why this is the case?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Quality:**\nThis work is technically sound.\n\n**Clarity:**\nThis paper is for the most part clearly written and well-structured. However, I think it would be nice to list all of the experiments as tables (some results seem to only be presented in text which makes it hard to find when jumping around in the paper).\n\n**Significance:**\nComparing and evaluating medical image segmentation methods is hard due to lack of a large consensus test suite. This has lead to that very many different papers have been written on architectural improvements of the UNet with claims of improving Dice, but few actually convince the community of improvements good enough to be worth actual implementation. I think this paper unfortunately is likely to fall into this category.\n\n**Originality:**\nFrom what I can tell the leap from [18] doesn’t seem to technically be very high. However, the work is thorough and the results show improvement over the previous results in [18].", "was_revised": true, "final_justification": "My main concern is that the cost of implementing this methodology is so big that it outweighs the benefits. When dealing with image segmentation, particularly in 3D, the GPU memory tradeoff is always present - using a couple of gigabytes for supervision to gain 2% Dice is potentially a couple of gigabytes that could be used for some other technique to gain 2% Dice. However, the authors provided me with results that showed 3D feasible and patiently explained the memory requirements. Even though these requirements sound steep, I think it is compensated by the attractiveness of the method being automatic and in particularly that it does not affect inference. Furthermore, I think that further work could investigate trimming down the memory requirements. Based on the above, I will change my score to borderline accept.", "rebuttal": "We sincerely thank the reviewer for the constructive feedback. Our responses are given below.\n### **Q1: On line 145: it looks like the number of channels in each decoder layer is constant. Is this a requirement? In that case how would a UNet architecture that increase the number of channels when getting closer to the bottle neck be compatible with the introduced extension?**\n**Response:** In our problem definition (Lines 143–145), **C is the number of segmentation classes (C = #classes)**, *not* the UNet’s internal feature-channel width. LoMix does *not* require constant channels in the decoder, i.e., a UNet that widens toward the bottleneck is fully compatible. The only requirement is that each decoder stage output is projected onto a C-channel class logit map at a common resolution so these maps can be mixed. We will rephrase the description to make this clear.\n\n--- \n### **Q2: Would it be computationally feasible to generalize the method to a 3D setting? Is there some kind of exponential increase in the number of required fusion maps that gets out of hand?**\n**Response:** Yes, generalizing LoMiX to 3D is straightforward: simply replace the 2D convs and bilinear upsampling with 3D convs and trilinear upsampling, everything else remains unchanged. This is because we **fuse predictions (class logits), not wide feature tensors**: our goal is to synthesize intermediate “pseudo‑ensembled” predictions during training for richer supervision, with zero inference overhead.\n\nAdding all non‑empty subsets of the $L$ decoder predictions introduces only $2^L−L−1$ extra logit maps (e.g., $11$ for $L=4$, $26$ for $L=5$); since U‑shaped networks **rarely exceed five stages**, these maps are finite, lightweight, and used only during training, so the incremental training compute/memory cost remains feasible and inference is still zero‑overhead. When GPU memory is low, either in 2D or 3D, standard optimizations such as **gradient checkpointing** or **caching logits on CPU** further *reduce* compute requirements without altering the algorithm, thus keeping LoMiX practical on modest hardware (< 5 GB extra GPU memory required for backpropagation to process a 96×96×96 volume with a four-stage network and 9 output classes). To show the feasibility of LoMiX with 3D networks, the new preliminary results of 3D MedNeXt with LoMiX are reported in Table R1 below.\n\n**Table R1:** MedNeXt's performance on 3D Synapse 8-organ segmentation with Last Layer (LL), Deep Supervision (DS), MUTATION, and LoMiX. DICE scores (%) reported for Gallbladder (GB), Left kidney (KL), Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). Our results (in bold) demonstrate that LoMiX achieves the best average DICE and HD95 scores among all supervisions.\n\n|**Methods**|**Avg. DICE**|**Avg. HD95**|**Aorta**|**GB**|**LK**|**RK**|**Liver**|**PC**|**SP**|**SM**|\n|-|-|-|-|-|-|-|-|-|-|-|\n|MedNeXt-M_K3 + LL|86.22|6.62|91.96|75.59|87.11|85.14|96.81|78.31|91.26|83.61|\n|MedNeXt-M_K3 + DS|86.63|8.31|91.61|78.91|90.40|85.94|94.72|74.95|92.10|84.43|\n|MedNeXt-M_K3 + MUTATION|86.84|6.04|91.72|79.94|90.97|86.62|96.58|76.65| 90.55|81.69|\n|**MedNeXt-M_K3 + LoMiX (Ours)**|**87.19**|**4.84**|**91.81**|**79.87**| **90.54**|**86.65**|**96.68**|**76.95**|**90.63**|**84.37**|\n---\n\n### **Q3: Are the W_S in Eq. (3) normalized to 1? I find it surprising that figure S.8 in the supplementals show that addition fused logit maps get weights so much higher than the concat ones, yet still are shown to be of the most important in the ablation study. Is there an explanation to why this is the case?**\n**Response:** No, $W_S$ in Eq. (3) is the $1\\times1$ convolution weight matrix used inside the *concat* fusion; $W_S$ is a standard linear projection and it is **not normalized**. The bars in Fig. S8, however, depict the **Softplus loss weights *w*** from Eq. (6) for the best-epoch: independent scalar factors that rescale each loss term and are likewise unconstrained, so their magnitudes should not be interpreted as direct \"importance\" scores.\n\nThe apparent mismatch arises because *concat*’s contribution is better reflected by its **complementary signal** and **training dynamics**, not by its final scalar weight. Channel-wise concatenation injects cross‑stage information that additive/multiplicative operators may not capture, which is why ablating *concat* degrades DICE. Moreover, Fig. S8 shows *only the best epoch*; earlier curves (Supplementary Figs. S1-2, S6) reveal that *concat* fused logits receive higher weights during earlier training epochs, helping shape the representation before the optimizer redistributed emphasis. In short, smaller final $w$ does *not* imply low utility; *concat* remains important because of the distinct supervision it provides. We will clarify this aspect in our revised paper.\n\n---\n### **W1: It would be nice to list all of the experiments as tables.**\n**Response:** Thank you for the suggestion. All quantitative results that appear as plots in the main paper are already reported as tables in the Supplementary pdf file (i.e., Tables S3, S6-S8 correspond to Figs. 2–5). For the camera‑ready version, we will *(i)* add explicit cross‑references (in the main text) to these tables, and *(ii)* move key results into compact tables in the main paper as space permits. This will make it easier to locate the experimental results without having to search through the entire text.\n\n---\n### **W2. ...very many different papers have been written on architectural improvements of the UNet with claims of improving Dice, but few actually convince the community of improvements good enough to be worth actual implementation. I think this paper unfortunately is likely to fall into this category.**\n**Response:** **LoMiX is not another UNet variant**, but a **new supervision strategy at training‑time.** To clarify, we do not alter the encoder/decoder blocks, we do not add new stages or resolutions, hence **inference remains exactly the same (i.e., no additional parameters, zero latency overhead).** Our paper proposes a fundamentally new methodology, rather than a new architecture for efficient and accurate image segmentation.\n\nGetting into more details, during training, **LoMiX mixes the existing multi‑stage logit outputs to synthesize additional intermediate predictions**, thus enriching supervision *without* introducing new resolution features. **A NAS‑inspired, Softplus weighting scheme** then automatically *learns* how much each original or fused logit should influence the overall loss. This is why our contribution is about **how to supervise, not about a new architecture.**\n\nThe benefit of LoMiX is most evident in **limited‑data scenarios** (see Fig. 2 and Supplementary Table S3), where supervision is the bottleneck: indeed, by implicitly ensembling stage predictions during training, LoMiX provides richer, more diverse signals and yields consistent DICE gains across various networks and datasets. In other words, rather than contributing with yet another UNet variant, we offer a new, general, **architecture‑agnostic** supervision strategy that the community can adopt with minimal code changes and **no inference overhead**.\n\n--- \n### **W3. From what I can tell the leap from [18] doesn’t seem to technically be very high. However, the work is thorough and the results show improvement over the previous results in [18].**\n**Response:** **LoMiX** makes a significant advancement over MUTATION [18] by **turning supervision into an automatic, learnable, and architecture‑agnostic process** that enables:\n- **Richer fusion space for diverse logit synthesis:** Indeed, instead of a single, fixed combinatorial mutation, LoMiX defines a library of operators (Add, Mult, Concat, AWF, etc.) and systematically considers **all non‑empty subsets** of stage logits. This yields a broad spectrum of intermediate predictions, thus capturing complementary spatial cues that *cannot* be captured by using simply a single operator (i.e., Add) as in [18].\n- **Fully automatic supervision via NAS‑inspired Softplus weighting:** We replace the uniform weights (i.e., 1.0) in [18] with a Softplus weighting scheme that *learns* in an end‑to‑end fashion how much each original or fused logit should contribute to the loss. As such, no heuristic tuning and no per‑dataset tweaking are needed; the weighting adapts itself during training, making supervision fully automatic.\n- **Training‑only, plug‑and‑play module with zero inference overhead:** LoMiX operates purely at the logit/loss level. We do *not* modify the encoder/decoder blocks, we do *not* add stages, we do *not* keep any extra heads at inference time. The deployed network remains identical: no extra parameters, no additional latency, directly mitigating any practical adoption barriers of our approach.\n- **Targets data‑limited scenarios:** Our largest gains appear when data are limited: the scenario where the supervision quality (not the network architecture) is critical. By implicitly ensembling multi‑stage predictions during training, LoMiX delivers consistent DICE improvements across all backbones and datasets. This trend is *not* observed in [18].\n- **Rigorous analysis and robustness checks:** We go beyond headline numbers: operator and subset ablations (Fig. 3 and Supplementary Table S6), weight‑dynamics curves over training (Supplementary Figs. S1-S8), cross‑architecture evaluations (Fig. 5 and Supplementary Table S8), and data‑efficiency assessments (Fig. 2 and Supplementary Table S3), all substantiate *why* and *how* LoMiX works. This breadth of experimental evidence is *absent* in [18].\n\nTo sum up, LoMiX introduces a **new, general, learnable, and fully automatic supervision method**; this is a clear technical and practical contribution beyond the fixed, single-operation mutation strategy of [18].\n\n[18] Rahman, M.M. and Marculescu, R. Multi-scale hierarchical vision transformer with cascaded attention decoding for medical image segmentation. MIDL, 2023."}, {"review_id": "CtsJOHfsgg", "reviewer": "Reviewer_5cjt", "summary": "The authors introduce LoMix, a method for enhancing 2D semantic medical image segmentation through mixed-scale fusion of decoder features. LoMix combines multi-scale decoder logits using operations such as addition, multiplication, concatenation, and attention-based fusion within a Combinatorial Mutation Module. A Neural Architecture Search (NAS)-inspired strategy co-optimizes a softplus-weighted loss alongside model parameters to adaptively select effective fusion strategies and scales. The approach is integrated into UNet-like architectures (e.g., PVT-V2-B2 with EMCAD decoder) and evaluated on Synapse and ACDC datasets, including low-data settings and extensive ablations on fusion operations, NAS weighting, and backbone choices.", "questions": "1. While the NAS-inspired loss weighting helps mitigate supervision imbalance across fusion paths, all paths are still computed and backpropagated during training. Can the authors justify this design in light of the significant computational overhead?\n2. The authors claim LoMix improves small and challenging organ segmentation. Can the authors provide any analysis or visualizations showing which decoder scales or fusion types (e.g., concatenation, multiplication) are most responsible for these improvements?\n3. In Table 2, the performance differences between PVT-EMCAD-B with and without LoMix appear small. Have the authors conducted any statistical tests to confirm that these improvements are significant?\n4. Why are HD95 and mIoU not reported for the multi-organ segmentation results in Table 2, despite being included for Synapse in Table 1? Including them would allow more complete comparisons.", "limitations": "The authors adequately addressed the limitations and potential negative societal impact of their work", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n1. Data-driven and adaptive fusion strategy: Unlike traditional deep supervision or fixed mutation-based fusion methods, LoMix introduces a dynamic mechanism that learns how to fuse multi-scale decoder logits during training. By leveraging a combinatorial set of operations (addition, multiplication, concatenation, attention) and jointly optimizing them through a NAS-inspired weighting scheme, LoMix adapts the fusion process to the specific characteristics of the data. This makes the method flexible to the semantic granularity and scale variance present in medical images, enabling a more informed, data-driven integration of decoder features instead of relying on manually designed fusion heuristics.\n2. Robustness in limited data regimes: Medical image segmentation often suffers from scarce annotated data due to the high cost of expert labeling. LoMix demonstrates strong performance even under constrained training data conditions. This robustness stems from its ability to dynamically prioritize the most informative scales and fusion paths, effectively regularizing the learning process and mitigating overfitting. This characteristic makes LoMix particularly practical for real-world clinical settings where annotation budgets are limited.\n\nWeaknesses\n1. While the NAS-inspired loss weighting partially addresses the supervision imbalance across the combinatorially generated fused logits, all fusion paths are still computed and backpropagated regardless of their utility. This results in significant computational overhead, especially as the number of decoder stages increases. Moreover, some fusion operators, particularly multiplicative fusion, remain sensitive to low-confidence predictions early in training, potentially introducing unstable gradients before the loss weights converge to suppress them. \n2. The authors state (line 256) that LoMix helps in segmenting small and challenging abdominal organs, yet no analysis is provided to visualize which fusion types or decoder scales contribute most to those improvements. This would strengthen the claim and help explain LoMix’s behavior. Additionally, in Table 2, the performance gain between PVT-EMCAD-B with and without LoMix appears modest, and no statistical significance analysis is presented to support the observed differences. Also, unlike Table 1 (Synapse), Table 2 omits HD95 and mIoU metrics, which would be relevant for assessing organ boundary quality and overall segmentation agreement, especially since table space permits their inclusion.\n3. Although the authors acknowledge that LoMix is designed for 2D images and they recognize the weakness, all datasets used in experiments are intrinsically 3D. This mismatch between method and data dimensionality may limit its applicability in clinical practice unless a 3D-compatible version is developed.", "was_revised": true, "final_justification": "The authors have extensively addressed all of my comments through detailed explanations and additional experiments. I have also reviewed the concerns raised by other reviewers. While I partially agree with the two remaining weaknesses highlighted by reviewers xQqf (overfitting) and vjqb (memory requirements), I have no further concerns from my side. I therefore maintain my score as 5 (Accept).", "rebuttal": "### **Official Response to Reviewer 5cjt**  \n\nWe sincerely thank the reviewer for the constructive feedback. Below, we address each concern and outline how we plan to incorporate these suggestions in our revised manuscript.\n\n---\n### **W1.1 & Q1. While the NAS-inspired loss weighting partially addresses the supervision imbalance across the combinatorially generated fused logits, all fusion paths are still computed and backpropagated regardless of their utility. This results in significant computational overhead, especially as the number of decoder stages increases. Can the authors justify this design in light of the significant computational overhead?** \n\n**Response:** Thank you for the thoughtful critique. **Why keeping all the fusion paths?**  \n\nWe deliberately **retain and back-propagate all fusion paths** for three intertwined reasons:  \n\n- **Diversity + automatic weighting**: We deliberately enumerate all non‑empty subsets to **maximize logit diversity** as different operator/subset pairs emphasize different spatial scales and error modes. Our NAS‑inspired Softplus weights then assign **distinct, learned contributions** to each fusion path; the unhelpful fusion paths are automatically down‑weighted, while the useful ones retain influence. Pruning too early would remove good candidates before the optimizer can assess them.  \n\n- **Computational overhead is modest and training‑only**: All fusions operate on **C‑channel class logits** (C = #classes, typically small), *not* on wide feature tensors. Even with 4–5 decoder stages, the extra tensors are at most a few dozen small maps and a few 1×1 (or 1×1×1) convs. In practice, this adds only to the training FLOPs/memory, while **inference remains unchanged (i.e., zero overhead)**. When GPU memory is low **for training**, standard optimizations such as **gradient checkpointing or caching logits on CPU** further reduce compute/memory requirements without altering the algorithm, thus keeping LoMiX practical on modest hardware. \n\n- **Optional pruning is easy**: If desired, one can prune paths whose weights stay near zero after a short warm‑up or sample a top‑K set; this is orthogonal to LoMiX and straightforward to add.  \n\nTo sum up, computing and backpropagating all paths is a conscious design choice to ensure **maximal supervision diversity** while letting the NAS‑inspired weighting **learn which fused logits matter most**. The added cost is small, confined to training, and the result is consistent performance gains without any inference overhead.  \n\n### **W1.2. Moreover, some fusion operators, particularly multiplicative fusion, remain sensitive to low-confidence predictions early in training, potentially introducing unstable gradients before the loss weights converge to suppress them.** \n\n**Response:** We multiply **logits**, *not* probabilities, and initialize all Softplus weights to small values, so no path dominates early. If an operator does inject noisy gradients, its weight shrinks. Empirically, our training is stable (see **weight dynamics** in Supplementary Figs. S1-S8), and removing multiplicative paths can hurt accuracy (see **operator ablations** in Fig. 3 and Supplementary Table S6), thus indicating they provide complementary information if properly weighted.  \n\n--- \n### **W2.1 & Q2. The authors state (line 256) that LoMix helps in segmenting small and challenging abdominal organs, yet no analysis is provided to visualize which fusion types or decoder scales contribute most to those improvements. This would strengthen the claim and help explain LoMix’s behavior. Can the authors provide any analysis or visualizations showing which decoder scales or fusion types (e.g., concatenation, multiplication) are most responsible for these improvements?** \n\n**Response:** We do report already **organ‑wise DICE scores for *every* operator combination in Table S6 and the learned weight dynamics in Supplementary Figs. S1-S8**. Together, these results show that paths involving the *higher-resolution decoder stages* and *AWF/Mult operators* consistently correlate with the *largest* gains on small/low‑volume abdominal organs, the scenario highlighted in line 256-257 (Section 4.2).  \n\nTo make this linkage more clear, we will revise the camera-ready version as follows:  \n\n- Add clear cross‑references in the main text from the small‑organ claim to Table S6 and Supplementary Figs. S1-S8.  \n\n- Provide qualitative overlays (baseline vs. LoMiX) on representative small organs to visually substantiate the quantitative trend. \n\n- Add visualization of synthesized/fused predictions.  \n\nThese revisions will directly connect specific fusion paths and scales to the observed boosts on challenging organs, thus strengthening both the interpretability and credibility of our claim. \n\n--- \n### **W2.2. & Q3. Additionally, in Table 2, the performance gain between PVT-EMCAD-B2 with and without LoMix appears modest, and no statistical significance analysis is presented to support the observed differences. Have the authors conducted any statistical tests to confirm that these improvements are significant?** \n\n**Response:** Although the average DICE score gain on PVT‑EMCAD‑B2 looks small, it is **consistently and statistically reliable** (see Table R2 below). Using a two‑sided Wilcoxon signed‑rank test over 12 test scans (Holm‑corrected), LoMiX is significantly better than every training baseline: MUTATION (p = 0.0122), Deep Supervision (p = 0.0049), and Last Layer (p = 0.0015). We will report these p‑values in main Table 1 to show significance. Again, this improvement comes at **zero inference overhead**.  \n\n**Table R2:** DICE (%) on 12 test scans of BTCV 8-organ segmentation with PVT‑EMCAD‑B2. p-values (two‑sided Wilcoxon signed‑rank [3], Holm‑corrected [4]) compare each baseline to LoMiX; all are < 0.05, indicating LoMiX’s improvement is statistically significant despite the modest mean gap. \n\n| **Method** | **DICE (%)** | **p‑value (vs. LoMiX, Holm‑corrected)** | **Significant (α = 0.05)** | \n|---------|---------|--------|---------| \n| **LoMiX (Ours)** | **85.07** | - | - | \n| MUTATION | 83.63 | 0.0122 | Yes | \n| Deep Supervision | 82.90 | 0.0049 | Yes | \n| Last Layer | 80.94 | 0.0015 | Yes | \n\n--- \n\n### **W2.3. & Q4. Also, unlike Table 1 (Synapse), Table 2 omits HD95 and mIoU metrics, which would be relevant for assessing organ boundary quality and overall segmentation agreement, especially since table space permits their inclusion. Why are HD95 and mIoU not reported for the multi-organ segmentation results in Table 2, despite being included for Synapse in Table 1? Including them would allow more complete comparisons.** \n\n**Response:** We agree that HD95 and mIoU are informative for boundary quality and overall agreement. They were omitted from the initial Table 2 to stay consistent with the EMCAD paper’s [5] reporting protocol, but we will definitely include both metrics in the revised paper. \n \n---\n### **W3. Although the authors acknowledge that LoMix is designed for 2D images and they recognize the weakness, all datasets used in experiments are intrinsically 3D. This mismatch between method and data dimensionality may limit its applicability in clinical practice unless a 3D-compatible version is developed.** \n\n**Response:** LoMiX is **dimension‑agnostic**: it operates on C‑channel class **logit maps** at *the loss level*. Extending LoMix to 3D is **straightforward**: simply replace Conv2d with Conv3d and bilinear with trilinear upsampling; the fusion/weighting logic remains identical and the cost still scales with the (small) number of decoder stages (rarely exceeding five stages). When GPU memory is low, either in 2D or 3D, standard optimizations such as **gradient checkpointing or caching logits on CPU** further reduce compute/memory requirements without altering the algorithm, thus keeping LoMiX practical on modest hardware (< 5 GB extra GPU memory required for backpropagation to process a 96×96×96 volume with a four-stage network and 9 output classes). \n\nOf note, we also **reported purely 2D tasks** (ISIC2018 skin lesions; Kvasir/CVC-ColonDB/ETIS-LaribPolypDB polyps in Supplementary Tables S2 and S4). For the intrinsically 3D datasets, we followed the commonly used memory‑efficient slice‑based protocol as in [5,6], while computing metrics on full volumes. To show the feasibility of LoMiX with 3D segmentation networks, the new preliminary results of 3D MedNeXt with LoMiX are reported in Table R3 below. \n\n**Table R3:** MedNeXt's performance on 3D Synapse 8-organ segmentation with Last Layer (LL), Deep Supervision (DS), MUTATION, and LoMiX. DICE scores (%) are reported for Gallbladder (GB), Left kidney (KL), Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). Our results (in bold) demonstrate that LoMiX achieves the best average DICE and HD95 scores among all supervisions.\n\n|**Methods**|**Avg. DICE**|**Avg. HD95**|**Aorta**|**GB**|**LK**|**RK**|**Liver**|**PC**|**SP**|**SM**|\n|-|-|-|-|-|-|-|-|-|-|-|\n|MedNeXt-M_K3 + LL|86.22|6.62|91.96|75.59|87.11|85.14|96.81|78.31|91.26|83.61|\n|MedNeXt-M_K3 + DS|86.63|8.31|91.61|78.91|90.40|85.94|94.72|74.95|92.10|84.43|\n|MedNeXt-M_K3 + MUTATION|86.84|6.04|91.72|79.94|90.97|86.62|96.58|76.65| 90.55|81.69|\n|**MedNeXt-M_K3 + LoMiX (Ours)**|**87.19**|**4.84**|**91.81**|**79.87**| **90.54**|**86.65**|**96.68**|**76.95**|**90.63**|**84.37**|\n---\n\n### **References** \n[3] Wilcoxon, F. Individual comparisons by ranking methods. Biometrics bulletin, 1945.\n\n[4] Holm, S. A simple sequentially rejective multiple test procedure. Scandinavian journal of statistics, 1979.\n\n[5] Rahman, M.M., et al. Emcad: Efficient multi-scale convolutional attention decoding for medical image segmentation. CVPR, 2024. \n\n[6] Chen, J., et al. Transunet: Transformers make strong encoders for medical image segmentation. arXiv preprint arXiv:2102.04306, 2021."}, {"review_id": "yVzF83NDI8", "reviewer": "Reviewer_xQqf", "summary": "This paper proposes the LoMix (Logits Mixing) method, which significantly improves the performance of medical image segmentation through a learnable multi-scale feature fusion strategy. Its core innovation lies in the introduction of four lightweight fusion operations (addition, multiplication, concatenation, and attention-weighted fusion) combined with a NAS-style weight learning mechanism to dynamically optimize the contributions of features at different scales. Experiments show that LoMix achieves significant performance gains across multiple datasets and backbone networks, and performs particularly well in data-scarce scenarios. The method does not need to increase the inference overhead, is efficient and versatile, and provides a new technical idea for medical image segmentation.", "questions": "See Weaknesses 1 and 2", "limitations": "See Weaknesses 3 and 4", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. A learnable multi-scale Logits mixing strategy is proposed for the first time, which dynamically combines the decoder outputs of different scales through four lightweight fusion operations to generate rich \"mutant\" feature maps.\n2. A NAS-style weight learning mechanism is introduced to automatically optimize the loss weights for different fusion methods and scales, without the need for manual parameter tuning or additional validation set search.\n3. The proposed method achieves a plug-and-play design with zero inference overhead, introducing computation only in the training phase without affecting the model inference speed or memory footprint, which is suitable for practical deployment.\n4. The proposed method provides a certain degree of generality and interpretability. It is compatible with CNN and Transformer backbone networks (e.g., PVT, ResNet), and the learned weights can intuitively explain the contributions of different scales.\n5. The effectiveness of the proposed method is fully validated in multiple sets of experiments conducted on multiple datasets. The experimental results show that the proposed method outperforms SOTA methods on several medical image segmentation tasks.\n\nWeaknesses:\n\n1. The paper does not discuss the convergence of multi-scale fusion optimization. For example, are the weights of all fusion operations eventually stable? Are there redundancies?\n2. Different image modalities (e.g., ultrasound vs. CT) may require different fusion strategies, but the paper does not discuss parameter migration or modality adaptation approaches.\n3. Medical image analysis often requires 3D segmentation, but the paper has not yet verified the scalability and capability of LoMix on 3D architectures.\n4. The experiments were based only on publicly available benchmarks, and the generalization ability of LoMix on the data from different acquisition devices or hospitals has not been tested, which may conceal the risk of overfitting.", "was_revised": true, "final_justification": "The authors have addressed most of my concerns. Regarding the generalization ability of LoMix, although the results in Table R5 show overfitting, the performance of the method proposed in the paper is still superior to other methods. After comprehensively considering the authors' responses and the quality of the paper itself, I am inclined to maintain my initial score.", "rebuttal": "We sincerely thank the reviewer for the constructive feedback. Below, we address each concern and outline how we plan to incorporate these suggestions in our revised manuscript.  \n\n### **Q1: The paper does not discuss the convergence of multi-scale fusion optimization. For example, are the weights of all fusion operations eventually stable? Are there redundancies?** \n\n**Response:** Our multi‑scale fusion **does converge and self‑suppresses redundancies.** As the Softplus weights are co‑optimized with the segmentation model, absolute values drop with the overall loss, so after a short warm‑up their **relative ordering stabilizes** (see Supplementary Figs. S1–S8). Two consistent patterns emerge: \n\n- **By scale:** *Low‑resolution, high‑level logits* (early decoder stages) are driven to relatively much smaller weights, while *high‑resolution, fine‑grain logits* (later stages) retain larger weights, indicating that less informative logits are automatically down‑weighted.  \n\n- **By operator:** *AWF* and *Add* fusions receive comparatively higher weights, *Mult* gets moderate weights, and *Concat* gets the lowest weights in later epochs (see Supplementary Fig. S1–S8). Still, even these low‑weighted *Concat* fused logits matter; our ablations (Fig. 3 and Supplementary Table S6) show that they supply **complementary segmentation cues** that improve DICE despite their small scalar values. \n\n--- \n### **Q2. Different image modalities (e.g., ultrasound vs. CT) may require different fusion strategies, but the paper does not discuss parameter migration or modality adaptation approaches.** \n\n**Response:** We agree that ultrasound, CT, MRI, and endoscopy have very different characteristics. In fact, **LoMiX is designed precisely to learn such differences rather than hard‑code a single fusion recipe**. Our Softplus loss weights are **co‑optimized with the segmentation network**, so the model automatically explores the full fusion space and **down‑weights the less useful fused logits** while **relatively upweighting the informative ones** for each dataset/modality. In practice, all weights numerically shrink as the overall loss decreases, but their **relative pattern stabilizes**: *low‑resolution/early‑stage logits* and certain operators (e.g., *Concat*) receive relatively much smaller final weights, while *high‑resolution logits* and *AWF/Add* remain influential; this is **evidence of data‑driven adaptation** to various datasets/modalities (see Supplementary Figs. S1-S8). \n\nBecause this weighting is learned jointly, no separate “parameter migration” or modality-specific tuning is needed: training (or fine‑tuning) on a new modality simply yields a different weight configuration. We already demonstrate this across heterogeneous modalities, such as **CT organs** (see Table 1), **cardiac MRI** (Table 2), **ultrasound** (Table S2), **skin lesions** (Table S2), and **colon polyps** (Table S4), without changing the architecture or hand‑tuning weights. We will make this modality‑adaptation mechanism more explicit in the camera-ready version. Equally important, **LoMiX works at training‑time**, so clinical deployment is unaffected (i.e., zero extra parameters or latency overhead during inference).  \n\n--- \n### **W1. Medical image analysis often requires 3D segmentation, but the paper has not yet verified the scalability and capability of LoMix on 3D architectures.**  \n\n**Response:** Generalizing LoMiX to 3D is straightforward: simply replace the 2D convs and bilinear upsampling with 3D convs and trilinear upsampling, everything else remains unchanged. This is because we **fuse predictions (class logits), not wide feature tensors**: our goal is to synthesize intermediate \"pseudo‑ensembled\" predictions during training for richer supervision, with zero inference overhead.  \n\nAdding all non‑empty subsets of the $L$ decoder predictions introduces only $2^L−L−1$ extra logit maps (e.g., $11$ for $L=4$, $26$ for $L=5$); since U‑shaped networks **rarely exceed five stages**, these maps are finite, lightweight, and used only during training, so the incremental training compute/memory cost remains feasible and inference is still zero‑overhead. When GPU memory is low, either in 2D or 3D, standard optimizations such as **gradient checkpointing** or **caching logits on CPU** further *reduce* compute requirements without altering the algorithm, thus keeping LoMiX practical on modest hardware (< 5 GB extra GPU memory required for backpropagation to process a 96×96×96 volume with a four-stage network and 9 output classes). To show the feasibility of LoMiX with 3D segmentation networks, the priliminary results of 3D MedNeXt with LoMiX are reported in Table R4 below.\n\n**Table R4:** MedNeXt's performance on 3D Synapse 8-organ segmentation with Last Layer (LL), Deep Supervision (DS), MUTATION, and LoMiX. DICE scores (%) are reported for Gallbladder (GB), Left kidney (KL), Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). Our results (in bold) demonstrate that LoMiX achieves the best average DICE and HD95 scores among all supervisions.\n\n|**Methods**|**Avg. DICE**|**Avg. HD95**|**Aorta**|**GB**|**LK**|**RK**|**Liver**|**PC**|**SP**|**SM**|\n|-|-|-|-|-|-|-|-|-|-|-|\n|MedNeXt-M_K3 + LL|86.22|6.62|91.96|75.59|87.11|85.14|96.81|78.31|91.26|83.61|\n|MedNeXt-M_K3 + DS|86.63|8.31|91.61|78.91|90.40|85.94|94.72|74.95|92.10|84.43|\n|MedNeXt-M_K3 + MUTATION|86.84|6.04|91.72|79.94|90.97|86.62|96.58|76.65| 90.55|81.69|\n|**MedNeXt-M_K3 + LoMiX (Ours)**|**87.19**|**4.84**|**91.81**|**79.87**| **90.54**|**86.65**|**96.68**|**76.95**|**90.63**|**84.37**|\n---    \n\n### **W2: The experiments were based only on publicly available benchmarks, and the generalization ability of LoMix on the data from different acquisition devices or hospitals has not been tested, which may conceal the risk of overfitting.** \n\n**Response:** We agree that clinical deployment demands robustness across scanners and sites. Our current evaluation already addresses overfitting risk by spanning **multiple, heterogeneous public datasets and modalities** (i.e., abdominal CT, cardiac MRI, breast ultrasound, dermoscopy, colonoscopy), each collected with different protocols and devices. Yet LoMiX improves on every dataset without manual tuning.  \n\nBecause LoMiX is **training‑only, parameter‑efficient, and does not increase model capacity** (e.g., parameters and FLOPs), it actually **behaves like a regularizer**. In fact, LoMiX implicitly ensembles diverse multi‑scale logits only during training, thus **reducing the chance of overfitting to dataset‑specific biases**.  \n\nTo make this explicit, in the revised paper, we will include a **cross‑dataset experiment** (e.g., train on one dataset/hospital, and test on another) as shown in Table R5 below.  \n\n**Table R5:** Cross‑dataset/hospital generalization of LoMiX. Using the PVT‑EMCAD‑B2 network, all models are trained for 200 epochs on the **Kvasir polyp‑segmentation training set** (900 images); the epoch with the best DICE (%) on the Kvasir validation split (100 images) is saved. Then we evaluated the generalizability on **three external test sets** (CVC‑ClinicDB, CVC‑ColonDB, ETIS‑LaribPolypDB), without further tuning. LoMiX produces the highest DICE on every dataset which confirm its superior ability to generalize across hospitals and acquisition devices. \n\n| **Methods** | **CVC‑ClinicDB** | **CVC‑ColonDB** | **ETIS‑LaribPolypDB** | \n|------------|------------|------------|-------------| \n| Last Layer | 80.59 | 75.31 | 71.64 | \n| Deep Supervision | 81.87 | 76.16 | 75.84 | \n| MUTATION | 81.88 | 76.39 | 75.97 | \n| **LoMiX (Ours)** | **83.08** | **77.70** | **77.01** | \n\n---"}, {"review_id": "2YPKfVMSHq", "reviewer": "Reviewer_h4U4", "summary": "This paper introduces LoMix (Logits Mixing), a novel training module designed to improve the performance of U-shaped networks in medical image segmentation. It assigns a learnable weight to every loss term.\n\nNumerous experiments were conducted on datasets such as Synapse Multi-organs, ACDC cardiac organs, and BUSI. They also tested their method using limited data (5%, 10%, etc.), and the experiments demonstrated promising results.\n\nMoreover, a comprehensive ablation study was performed on the fusion operation, the loss term, and the backbone. Show that every module is working.", "questions": "1. Clarify the effectiveness of different modules: how do learnable weights compare with other dynamic loss weights methods? How are your late fusion methods different from others? My opinion would improve if the authors could differentiate their work from methods that use similar individual components in experiments.\n\n2. Carefully choose your comparison method. It is very confusing to include so many irrelevant methods. In fact, the ablation study should be the main result. And the author should write the main table in this way: UNet/UNet+other fusion methods/UNet+ours; PVT-EMCAD/PVT-EMCAD+other fusion methods/PVT-EMCAD+ours. I would improve my rating if the author could improve the main results table. \n\n3. Argue more about the novelty of the module. In my opinion, applying long-standing, off-the-shelf methods to a classical problem can hardly be considered novel research for NeurIPS; It is definitely not the \"first approach to enable fully learnable multi-scale fusion in segmentation\". Please see [1] [2], they are very classical papers in computer vision and were written nearly 8 years ago. I would suggest citing the paper in the related work and arguing more about the difference between the proposed methods and these classical methods. My rating could be improved if the author could convince me of the novelty of this paper.\n\n\n[1] Liu, Shu, et al. \"Path aggregation network for instance segmentation.\" Proceedings of the IEEE conference on computer vision and pattern recognition. 2018.\n\n[2] Zhao, Qijie, et al. \"M2det: A single-shot object detector based on multi-level feature pyramid network.\" Proceedings of the AAAI conference on artificial intelligence. Vol. 33. No. 01. 2019.", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The paper is well written, and they conducted many experiments. \n2. I like the ablation part. They did thorough ablation studies: different fusion operations, fixed loss term vs learnable loss term, different backbones.\n\nWeaknesses:\n1. Confusing comparison, the author compares their method with UNet, SwinUNet, TransUNet, PVT-CASCADE, and others (10+ methods) for no specific reason. Since the proposed method is built upon PVT-EMCAD, the only truly relevant baseline is PVT-EMCAD, whose worst-case performance should be equivalent to PVT-EMCAD (i.e., when all irrelevant loss terms become 0). The inclusion of other methods is unnecessary, as these exact comparisons were already published in the original PVT-EMCAD paper. If you check the tables from this paper and PVT-EMCAD, they are the same. \n\n2. The novelty of this paper is limited. There are two modules introduced in this paper: multi-scale feature(logit) fusion and learnable loss weights. The two concepts are widely used in computer vision. The first module looks to me like a late layer fusion module (add, concatenation, multiplication, attention weighted fusion), and the second module is learnable loss weights. All these modules are off-the-shelf. It's hard to say this is the \"first approach to enable fully learnable multi-scale fusion in segmentation\" (line 121).\n\n3. Missing comparison: Since one of the major contributions is introducing learnable weights, they should compare their NAS-based loss weights with other dynamic loss methods. Is there any specific reason or technical barriers that stop doing such a comparison?", "was_revised": true, "final_justification": "I commend the author for their detailed experiment and ablation study. The author attempts to introduce a multi-scale loss term with learnable weights, referred to as LoMix, into the training process.\n\nAlthough the author claims to solve a \"single task (segmentation) with dozens of highly correlated, multi-scale logit losses synthesized using our multi-operator combinatorial fusion,\" they do not provide a specific design for this process.\n\nThe entire process consists of setting the weights as nn.Parameters. The only constraint is to ensure the weights are positive (enforced via a softplus function), which is arguably a default requirement when dealing with loss values.\n\nFor this reason, I would give a rating of 3.", "rebuttal": "We sincerely thank reviewer for the constructive feedback. Our responses are given below.\n \n### **Q1.1. How do learnable weights compare with other dynamic loss weights methods?**\n**Response:** To our knowledge, there is **no prior method that dynamically balances dozens of multi‑scale, single‑task supervision losses** the way LoMiX does. Prior work falls into two categories: \n- **Deep supervision in segmentation (fixed or hand‑scheduled weights):** Classical deeply‑supervised/multi‑scale segmentation papers use side heads but keep their weights *constant or manually decayed*, e.g., U‑Net++ [7] and DSN [8]. None of these adapt weights data‑dependently across *combinatorial multi-scale* heads.\n- **Dynamic loss weighting designed for multiple (heterogeneous) tasks:** GradNorm [9] and Uncertainty weighting [10] address *multi‑task* learning where the number of losses is small (typically 2–5). Their normalization schedules or gradient‑matching heuristics *may not scale well* to dozens of highly correlated single‑task losses (one per fusion path).\n\nLoMiX fills this gap by reformulating the single‑task, multi‑scale supervision as a **multi-operator, combinatorial, fully learnable problem at logit/loss level**. Each fusion path gets its own Softplus weight, so the optimizer can automatically suppress the unhelpful paths and emphasize the useful ones. In our ablations, fixed/uniform weights baselines underperform our NAS‑inspired weighting significantly (Fig. 4 and Supplementary Table S7). If a truly comparable single‑task, multi‑scale dynamic weighting method exists, we will gladly cite and benchmark it.\n\n--- \n### **Q1.2. How LoMiX different from other late-fusions?**. \n**Response:** This is how LoMiX is different: \n- **Where/when it operates:** We fuse only during training, thus having zero extra parameters or latency overhead at inference. Late‑fusion papers [1,2] keep their fusion blocks at inference, thus adding overhead.\n- **What is fused:** We fuse multi-scale decoder logits, *not* high‑dimensional features, using multiple operators (Add, Mult, Concat, AWF) over all non‑empty subsets, yielding a combinatorial, diverse set of supervision signals rather than a single fixed fusion path.\n- **Why we fuse:** Our goal is a stronger, more diverse supervision, i.e., an implicit ensemble that improves gradients (especially in data‑scarce regimes), not a deeper architecture.\n- **How do we optimize:** Each original/fused logit gets a NAS‑inspired, Softplus loss weight learned end‑to‑end; prior works [1,2] do *not* auto‑weight dozens of fusion paths.\n- **Empirical evidence:** Operator/subset ablations (Fig. 3 and Supplementary Table S6) and weight‑dynamics plots (Figs. S1-S8) show that different fusion types/resolutions do matter, and the learned weights self-rebalance over training.\n\nTo sum up, LoMiX is **not** a late fusion block with off‑the‑shelf operations; instead, it is a **new combinatorial, and automatically weighted supervision methodology at training-time**. This clearly differentiate it from methods that use similar components (names), but with completely different purposes.\n\n---\n### **Q2 & W1. Comparison methods and restructuring main table.** \n**Response:** The confusion stems from considering LoMiX a PVT‑EMCAD add‑on. In fact, LoMiX is a **new architecture‑agnostic supervision methodology at training‑time**, so we must show that it works across multiple networks, not just one.\n\nFig. 5 and Table S8 already show that LoMiX scales well across different CNN/Transformer encoders with the EMCAD decoder. To restructure Table 1 as per-network progression, we run *new* experiments with different supervision across representative CNN, MLP and Transformer hybrid networks. The results show that LoMiX achieves the best DICE for every network (see Table R6). We will add the full restructured Table 1 in the revised paper. This will directly show \"what does LoMiX add beyond existing supervisions?\"\n\n**Table R6:** Synapse 8-organ segmentation with Last Layer (LL), Deep Supervision (DS) [7], MUTATION [18], and LoMiX. DICE scores (%) reported for Gallbladder (GB), Left kidney (KL), Right kidney (KR), Pancreas (PC), Spleen (SP), and Stomach (SM). We report results of only three networks here for space limits.\n\n|**Methods**|**Avg. DICE**|**Avg. HD95**|**mIoU**|**Aorta**|**GB**|**LK**|**RK**|**Liver**|**PC**|**SP**|**SM**|\n|-|-|-|-|-|-|-|-|-|-|-|-|\n|UNet + LL|70.1|44.7|59.4|84.0|56.7|72.4|62.6|87.0|48.7|81.5|67.9|\n|+ DS|77.8|26.9|68.3|85.4|68.0|81.4|76.2|91.4|56.9|87.6|75.9|\n|+ MUTATION|81.5|26.4|71.8|89.4|70.5|85.4|80.4|94.1|66.3|88.3|77.5|\n|**+ LoMiX (Ours)**|**83.6**|**24.3**|**74.6**|**90.4**|**75.3**|**86.3**|**82.5**|**94.3**|**67.9**|**91.8**|**80.2**|\n|TransUNet + LL|77.6|26.9|67.3|86.6|60.4|80.5|78.5|94.3|58.5|87.1|75.0|\n|+ DS|82.7|17.3|73.5|86.6|68.5|87.7|84.6|94.4|65.3|90.8|83.5|\n|+ MUTATION|83.0|17.0|73.9|89.3|63.7|86.9|83.0|95.5|69.6|93.1|82.7|\n|**+ LoMiX (Ours)**|**83.6**|**16.6**|**74.6**|**88.9**|**70.3**|**89.4**|**85.2**|**94.8**|**67.8**|**89.4**|**83.0**|\n|PVT‑EMCAD‑B2 + LL|80.9|22.9|71.2|87.1|68.0|84.9|81.1|94.6|63.1|89.8|78.9|\n|+ DS|82.9|19.7|73.8|87.4|67.8|87.7|83.7|95.2|65.6|91.5|84.2|\n|+ MUTATION|83.6|15.7|74.7|88.1|68.9|88.1|84.1|95.3|68.5|92.2|83.9|\n|**+ LoMiX (Ours)**|**85.1**|**14.9**|**76.4**|**88.8**|**73.5**|**89.1**|**84.7**|**95.8**|**69.7**|**92.5**|**86.5**|\n---\n### **Q3 & W2. Novelty and claim of this paper...**\n**Response:** We respectfully disagree with this assessment, as we propose:\n- **A new supervision methodology**, not a new architecture. Prior works [1, 2] aggregate high-dimensional features inside the network and keep those aggregation steps at inference. In contrast, LoMiX operates at the **logit/loss level, during training only**. We synthesize intermediate resolution *prediction* maps (class logits) from existing multi-stage/resolution decoder outputs only during training, thus having **zero extra parameters and latency overhead** at inference. Our focus is to **optimize supervision** for **any U-shaped network**.\n- **Operator-diverse combinatorial logit fusion:** Prior works [1,2] fuse a *fixed* set of multi-scale wide feature tensors. In contrast, LoMiX systematically considers **all non‑empty subsets** of multi-scale decoder logits and applies a **set of operators** (Add, Mult, Concat, AWF) to generate a rich set of intermediate resolution predictions. This exhaustive, resolution- and operator-diverse **logit space** has never been explored for segmentation supervision. This makes our contribution unique and worthwhile.\n- **Fully automatic, NAS-inspired loss weighting:** Rather than hand‑tuning or using uniform loss weights, we introduce a NAS‑inspired Softplus weighting mechanism that *learns* how much each original/fused logit should contribute to the loss. This way, dozens of fusion paths are jointly optimized, thus making the combinatorial space tractable.\n- **Data-scarce medical segmentation:** Our largest gains appear where supervision quality matters most, i.e., in **limited-data settings** which is common in practical medical imaging. Prior work [1,2] targets large-scale natural-image detection with abundant labels. We show that rethinking supervision (instead of architecture) yields consistent DICE improvements across diverse medical datasets/networks with limited annotated data.\n- **Thorough analyses:** We provide operator/subset ablations (Fig. 3 and Supplementary Table S6), weight-dynamics curves (Supplementary Figs. S1-S8), per-organ results (Tables 1-2, and Supplementary Tables S1, S3, S5-S8), and cross-architecture studies (Fig. 5 and Supplementary Table S8). This demonstrates generality and interpretability rather than an on-off tweak.\n\nTo mitigate reviewer concerns, we will discuss [1,2] in the related work, and rephrase the claim in line 121: \"In contrast, to the best of our knowledge, LoMix is the *first approach* to enable fully learnable multi-scale *logit* fusion for improved supervision in segmentation.\"\n\n---\n### **W3. Compare NAS-based loss weights with other dynamic loss methods.**\n**Response:** The \"dynamic loss\" schemes (GradNorm [9], Uncertainty weighting [10], etc.) target **multi‑task settings** with a few heterogeneous losses. In contrast, our LoMiX tackles a **single task** (segmentation) with dozens of highly correlated, multi‑scale logit losses synthesized using our multi-operator combinatorial fusion. Consequently, we designed a new NAS‑inspired Softplus weighting mechanism that *(i)* assigns each path an independent positive scalar, *(ii)* requires no global normalization or manual weights, and *(iii)* scales well to many multi-scale loss signals, which is exactly what we need in our setup. To keep the evaluation focused, we compared against the **closest relevant baselines for our setting**: multi‑scale/deep supervision with **fixed or uniform weights** (e.g., deep supervision [7], MUTATION [18]). Those baselines directly isolate the effect of **making supervision learnable**, which is our core contribution.\n\nIn principle, there is no fundamental barrier to plugging in multi‑task dynamic methods; however, they are simply **not the right comparisons for a single‑task, multi-scale loss supervision problem**. If the reviewer still deems such a comparison essential, we will include it in the revised paper.\n\n---\n### **References**\n[1] Liu, S., et al. Path aggregation network for instance segmentation. CVPR, 2018.\n\n[2] Zhao, Q., et al. M2det: A single-shot object detector based on multi-level feature pyramid network. AAAI, 2019.\n\n[7] Zhou, Z., et al. Unet++: Redesigning skip connections to exploit multiscale features in image segmentation. TMI, 2019.\n\n[8] Lee, C.Y., et al. Deeply-supervised nets. AIS, 2015.\n\n[9] Chen, Z., et al., Gradnorm: Gradient normalization for adaptive loss balancing in deep multitask networks. ICML, 2018.\n\n[10] Kendall et al., Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. CVPR, 2018."}]

Author rebuttal:

Meta-review: The paper presents a novel training-time supervision method to improve U-shaped networks for medical image segmentation. It addresses the limitations of traditional supervision, which either focuses only on the final layer or uses equally weighted deep supervision on multi-scale logits. 

The most important reason for the decision to accept the paper was the authors' successful rebuttal, which directly and convincingly addressed the main weaknesses. While the paper was initially seen as "borderline," the authors provided crucial additional experiments and clarifications that demonstrated the method's feasibility and robustness. The authors’ detailed explanations about the computational requirements for a 3D setting and their provision of preliminary 3D results alleviated a major reviewer concern.
Decision: Accept (poster)