A Bayesian Approach to Segmentation with Noisy Labels via Spatially Correlated Distributions
Authors: Ryu Tadokoro
Tsukasa Takagi
Shin-ichi Maeda
Venue: Accepted by TMLR ()

Abstract: In semantic segmentation, the accuracy of models heavily depends on the high-quality annotations. However, in many practical scenarios, such as medical imaging and remote sensing, obtaining true annotations is not straightforward and usually requires significant human labor. 
Relying on human labor often introduces annotation errors, including mislabeling, omissions, and inconsistency between annotators.
In the case of remote sensing, differences in procurement time can lead to misaligned ground-truth annotations. 
These label errors are not independently distributed, and instead usually appear in spatially connected regions where adjacent pixels are more likely to share the same errors.
To address these issues, we propose an approximate Bayesian estimation based on a probabilistic model that assumes training data include label errors, incorporating the tendency for these errors to occur with spatial correlations between adjacent pixels. 
However, Bayesian inference for such spatially correlated discrete variables is notoriously intractable. To overcome this fundamental challenge, we introduce a novel class of probabilistic models, which we term the \textbf{ELBO-Computable Correlated Discrete Distribution (ECCD)}. By representing the discrete dependencies through a continuous latent Gaussian field with a Kac-Murdock-Szeg\"{o} (KMS) structured covariance, our framework enables scalable and efficient variational inference for problems previously considered computationally prohibitive.
Through experiments on multiple segmentation tasks, we confirm that leveraging the spatial correlation of label errors improves robustness.
Notably, in specific tasks such as lung segmentation, the proposed method achieves performance comparable to training with clean labels under moderate noise levels.

Published in Transactions on Machine Learning Research (03/2026)
A Bayesian Approach to Segmentation with Noisy Labels via Spatially Correlated Distributions
Ryu Tadokoro
tadokororgurpnd@gaallco
Tohoku University
Tsukasa Takagi Preferved Networks, Inc.
takagi@@prfferved.pp
Shin-ichi Maeda
ichf@preferred.jp
Preferved Networks, Inc.
Reviewed on OpenReview: https: // openreview. net/ forum? id= orgfr8Kk22
Abstract
In semantic segmentation, the accuracy of models heavily depends on the high-quality anno- tations. However, in many practical scenarios, such as medical imaging and remote sensing, obtaining true annotations is not straightforward and usually requires significant human labor. Relying on human labor often introduccs annotation errors, including mislabeling, omissions, and inconsistency between annotators. In the case of remote sensing, differ- ences in procurement time can lead to misaligned ground-truth annotations. These label errors are not independently distributed, and instead usually appear in spatially connected regions where adjacent pixels are more likely to share the same errors. To address these issues, we propose an approximate Bayesian estimation based on a probabilistic model that assumes training data include label errors, incorporating the tendency for these errors to occur with spatial corrclations between adjacent pixels. However, Bayesian inference for such spatially correlated discrete variables is notoriously intractable. To overcome this fun- damental challenge, introduce novel class of probabilistic models, which term the BLBO-Computable Correlated Discrete Distribution (ECCD). By representing the discrete dependencies through a continuuus latent Gaussian field with Kac-Murdock- Szegö (KMS) structured covariance, our framework enables scalable and efficient variational infer- ence for problems previously considered computationally prohibitive. Through experiments on multiple segmentation tasks, we confirm that leveraging the spatial correlation of label er- rors improves robustness. Notably, in specific tasks such as lung segmentation, the proposed method achieves performance comparable to training with clean labels under moderate noise levels. Code is included in the supplementary materials.
1 Introduction
Semantic segmentation, which involves classifying each pixel in an image into one of several classes, a crucial task in computer vision. In supervised learning, the accuracy of segmentation models critically depends on the quality of the annotations in the training data. However, obtaining truly accurate pixel-level annotations is challenging in many practical applications. Even when expert annotators employed, errors, omissions, and subjectivity in interpretation are inevitable, leading to inconsistencies in the datascts. In particular, high inter- and intra-annolator variability is widely reported in medical imaging, where experts may have differing interpretations of the same structures. For instance, several studies Zhang et al. (2020a); Lampert et al. (2016); Rädsch al. (2023); Yang et al. (2023) highlight significant discrepancics among expert annotators; some delineate structures more generously, while others prefer more conservative annotations, Observer-dependent annotations exacerbate label noise in supervised learning. Label noise is also critical

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issue in remote sensing, where determining ground truth labels often requires field surveys over large and sometimes inaccessible regions Frénay & Verleysen (2013); Pelletier et al. (2017); Foody (2002). Due to the logistical and economic challenges of large-scale ground truth collection, researchers frequently rely on automatic labeling systems, which may introduce systematic errors. Additionally, high-quality annotated datasets remain a critical bottleneck for supervised learning, particularly in remote sensing applications where annotations are often repurposed across different types of satellite images. For example, the FattharthMap dataset Xia et al. (2023) was created by manually annotating high-resolution optical satellite images for semantic segmentation. However, these annotations are sometimes reused for synthetic aperture radar (SAR) imagery, despite differences in resolution and capture conditions Huang et al. (2021); Zhang et al. (2021); Liu et al. (2023). Additionally, changes in artificial structures or variations in land cover further contribute to label inconsistencies Fritz et al. (2009); Huang et al. (2021).
Various approaches have been proposed to mitigate the adverse effects of noisy labels. Some methods at- tempt to stop training carly to prevent the network from overfitting to noise and generating unreliable pseudo-labels Liu el. al. (2024; 2022), while others modily the loss function to be more robust against. large errors Gonzalez-Jimenez et al. (2025). Although these techniques can reduce the influence of noisy annota- tions, they do not fundamentally address the core reason why the standard supervised learning framework fails in the presence of noisy labels.
We propose a method that directly tackles the root cause of this issue. To understand this root cause, we first revisit the probabilistic assumptions underlying the standard cross-entropy loss. In supervised learning for segmentation models, training typically reduces to optimizing this loss function. This optimization implicitly follows maximum likelihood estimation (ML.E) framework under the assumption that the training data consists of independent and identically distributed samples drawn from the joint distribution of images and clean labels. However, when labels are noisy, the assumption of identically distributed data no longer holds, as the observed labels systematically deviate from clean labels, leading to performance degradation in the trained model.
To address this issue, we maintain the MLE framework but generalize the underlying probabilistic model to make it more suitable for real-world scenarios with noisy labels. Specifically, we introduce a model that explicitly accounts for the presence of noisy labels, which differ from the clean labels due to labeling errors. In practice, annolation errors tend to exhibit strong spatial correlations - mislabeling often occurs in contiguous regions. Variations in annotation criteria among experts, as well as changes in the underlying scene such as the construction or demolition of buildings or alterations in vegetation further reinforce this spatial dependency. Given this, we assume that label errors are not independent but instead exhibit spatial correlations among pixels.
However, directly modeling spatial correlations among discrete labels leads to an intractable marginal like- lihood, long-standing challenge in probabilistic modeling Koller & Friedman (2009); Sutton & McCallum (2012). Our core contribution is to resolve this challenge by proposing a novel class of tractable models, the eLBo-Computable Correlated Discrete Distribution (ECCD). This discrete distribution is repre- sented through continuous variables that follow a Gaussian distribution, where spatial correlations wetween pixels are expressed via a covariance matrix. Representing discrete variables through a Gaussian distribution successfully circumvents the intractability of summing over all possible realizations of the discrete variables. While the covariance matrix, whose number of elements scales quadratically with the number of pixels, intro- duces additional computational challenges in evaluating the ELBO, particularly in computing second-order statistics, its inverse, and its determinant. To overcome these computational intractabilities, we leverage the Kac-Murdock-Szege (KMS) matrix Fikioris (2018); Kac et al. (1953), which enables efficient computa- tions necessary for ELBO evaluation. For a detailed discussion on the iccal properties of our model, including its relation to Appendix B. To validate the effectiveness of our approach, we conduct extensive empirical evaluations on multiple segmentation tasks. Our experimental results demonstrate that our method significantly improves robustness against label noise, particularly in scenarios with moderate to high levels of spatially correlated label noise. In summary, our contributions are as follows:
Random Fields and classical MRF/CRE literature, please refer to
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1. A Novel Class of Tractable Distributions for Correlated Discrete Variables:
We introduce the ECCD, new class of probabilistic models for tractable inference high- dimensional discrete variables with stationary, exponentially decaying correlations a structure com- mon to spatial grids and temporal sequences. In this work, we apply the ECCD to learn from spatially correlated label noise in medical and remole sensing, where the model circumvents computational ional intractability by representing discrete dependencies via a continuous latent Gaussian field.
2. Efficient ELBO optimization via the KMS matrix:
The ECCD's covariance matrix introduces computational challenges that scale quadratically with the number of pixels. We overcome this by leveraging the Kac-Murdock-SAeei (KMS) matrix. To the best our knowledge, this is the first work to introduce the KMS matrix as a scalable computational tool for variational inference with correlated latent variables, enabling efficient computation of the key operations necessary for optimizing the ELBO of the ECCD.
3. Extensive empirical validation:
We perform comprehensive experiments on multiple segmentation tasks, demonstrating that our method significantly improves robustness to label noise. Our approach outperforms existing tech- niques, particularly in scenarios with moderate to high levels of spatially correlated label noise, validating the effectiveness our probabilistic formulation.
2
Related Work
Learning with noisy labels has been a significant focus in the field of machine learning, particularly clas- sification tasks Patrini et al. (2017); Yao et al. (2020); Han et al. (2018); Wei al. (2020). One research direction models the relationship between the clean label y" and the noisy label y through a transition ma- trix Patrini et al. (2017); Yao et al. (2020) that characterizes the probabilities from the clean label to the noisy label. For example, the transition probability from a clean label of k to a noisy label of j is repre- sented as ply = jly" = k). Additionally, methods have been proposed to estimate an instance-dependent transition matrix Xia et al. (2020); Berthon et al. (2021), where the transition probabilities depend on the instance also, denoted as pvy = jly" k 2). These classification methods can be adapted for semantic segmentation tasks; however, they treat label errors as being independent to each pixel, thereby ignoring the spatial correlations in label errors that may be present in annotations. This limitation is particularly critical in applications such medical imaging Zhang el al. (2020a); Shi & Wu (2021); Zhang et al. (2020b) and remote sensing Mnih & Hinton (2012); Song et al. (2022); Liu et al. (2024); Henry et al. (2021), where different types of noise may occur.
Other lines of research include work by Zhang el al. Zhang et al. (2020c), which proposed a Tri-Network framework that trains three networks simultaneously, with each pair selecting reliable pixels based on their loss maps, thereby achieving robust learning even with coarse annotations. Liu et al. Liu et al. (2022) introduced mechanism to detect the moment when network begins to overfit noisy labels. Gonzalez et al. Gonzalez-Jimenez et al. (2025) designed T-Loss based on a Student-t distribution to apply logarithmic penalty on large errors, reducing undue influence from outlier pixels. Nonetheless, these methods do not take into account the spatial correlations in label errors that may occur within annotations.
Unlike the aforementioned related works, there are few studies that consider spatial correlations. Li et al. Li et al. (2021) leveraged superpixel segmentation to group pixels and smooth the network outputs within each superpixel, thereby indirectly incorporating spatial context. This method, however, relies heavily the quality of the generated superpixels. More recently, Yao et al. Yao et al. (2023) proposed approach that explicitly models spatial correlations by applying a Markov process that consists of expansion and shrinkage steps of the annotalion masks. Although this method achieves better performance, it is mainly limited to addressing noise along the boundaries. In wealdworld scenarios, segmentation labels are often corrupted by a diverse range of noise types, including omission noise, misalignment noise, boundary uncertaintics, and other systematic labeling errors Jiménez & Decacstecker (2024); Vädineanu et al. (2022); Liu et al. (2024); Muih & Hinton (2012); Jiang et al. (2021).
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Our proposed probabilistic model can incorporate spatial correlations in label errors without making sumptions about the noise types, thereby handling a wide range of noise types.
3
Problem Formulation
Example
Example 2
Logit of Label Frror (Unknown)
7
p=
O O 0
Noisy Label
y
0-255
Clean Label
(Unknown)
y *
p= 0.75
Input Image
X
(b)
Figure (a) Graphical model of our proposed method, the ECCD. The model introduces a latent Gaussian variable 7) that encodes spatial correlations of label error among adjacent pixels, thereby enabling more realistic modeling of the dependency between pixels.
(b) Conceptual illustration of spatial correlation. Examples and show how varying affects the correlation strength between the center pixel and its neighbors (top, bottom, sides, and diagonals). When adjacent pixcls are likely to contain label errors and spatial correlation is strong (p is high) label errors can inferred with high confidence, even they are rare.
3.1 Probabilistic Foundations of Standard Supervised Segmentation
In standard supervisod learning for semantic sogmentation, the training process is typically framed as the minimization the cross-entropy loss. This objective function is not arbitrary: derived from the principle of Maximum Likelihood Estimation (ML.E) under a specific, and often implicit, probabilistic model. This underlying probabilistic model makes two fundamental assumptions: (I) the provided annotations y are the truc, noise-free clean labels y", and (2) the label of each pixel yı is conditionally independent of the labels of other pixels given the input image x.
Under these assumptions, the goal is to find the model parameters that maximizes the log-likelihood of the observed data:
HW
HW
logpo(ylx) = log II (v(l)))
Σ
log (xxlx)
(1)
where HW is the total number of pixels. The segmentation model po (x.(x) outputs probability distribution over the set of possible classes y for each pixel i. Let this distribution be represented by vector of probabilities (po (p) If we represent the ground-truth label yi as one-hot vector, where lik- if pixel belongs to class k and yık 0 otherwise, the log-likelihood for single pixel can be written
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EP Mik log x). The total log- log-likelihood for the image is therefore:
HW |3
(MLE(())
Yok log po |x).
(2)
Minimizing the negative of this quantity, - -EMLE()) (f), is preciscly equivalent minimizing the standard multi- class cross-entropy loss. This derivation highlights that the validity of the cross-entropy loss is contingent the assumption that the training labels are clean and pixel-wise independent samples from the true data distribution.
3.2 A General Probabilistic Model for Learning with Noisy Labels
In many practical scenarios, particularly in medical imaging and remote sensing, the assumption that ob- served labels y are identical to the true clean labels y" does not hold. Annotations are often corrupted by noise, which can be spatially correlated. To address this, we generalize the probabilistic model by explicitly distinguishing between the observed labels and the unobserved (latent) clean labels y". This leads to the objective of maximizing the marginal log-likelihood, which integrates out the uncertainty over the unknown clean labels y":
lo log Σ Epoox.y'1ss) log (plstt)noos>ls))
Here, po xx is the probability of the clean labels y given the image x, which is modeled by the segmen-- tation network, and p(yly") represents the label noise process.
This formulation is direct generalization of the standard MLE objective described in Section 3.1. To see this, consider the noise-free case where the noise model p(yly") collapses to Kronecker delta function ö(y", y). which equals 1 if and only if yfy y, and is zero otherwise. In this scenario, the summation in Eq. (3) contains only one non-zero term, and the objective reduces precisely to maximizing log pe (y/x), which is the basis for the standard cross-entropy loss as shown in Eq. (1). However, the summation over all possible configurations of y" is computationally intractable, as the number of terms (())AW) grows exponentially with the number of pixels. One way to overcome this is to assume a pixel-independent noise process, p(yly") II, p(y v/), but this fails to capture real-world spatially correlated label noise, mentioned in Section 1.
4 The Proposed ECCD Framework
4.1 A Tractable Variational Objective via the ECCD
To overcome the intractability of the marginal likelihood while modeling spatial correlations, we resort variational inference and maximize its Evidence Lower Bound (ELBO). We introduce continuous latent variable, 17 "WW, which represents the logit of the label error probability for each pixel. This variable captures spatial correlations and indirectly imposes spatial dependencies on the labels as follows:
/ plyly". .))(m)dn / IPCMI
{ Pluilti m)) } p
p(yly') =
(4)
Figure 1a depicts the graphical model illustrating this assumption. To fully specify this generative model, which is the central component of the FCCD, we define its constituent distributions. The prior p(n)) is modeled as a Gaussian, p(n) = (nlu,, E), with its covariance matrix structured using a Kac-Murdock- Szegö (KMS) matrix Fikioris (2018); Kac el al. (1953), which we detail in Section 4.2. The proposed model can be explicitly characterized as a Logit- nigit-Gaussian Random Field. While general inference in such models is intractable, our specific restriction to the KMS structure enables efficient variational inference, which we detail in Appendix B.1. The conditional label noise model p(ys Wo m)) is defined as:
plyak 1lvic 1, 5kc ((h)) + (1 -5ke) Bke
(5)
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where ōke is the Kronecker delta, r(-) is the sigmoid function that maps the latent variable ni to label error probability, and B is a learnable class transition matrix that specifies the probability distribution over incorrect labels when an error occurs, with its elements satisfying B E [0, 10,1)i) Bee = 0 and E = L, for each true class c c y.
To derive a tractable objective for this model, we introduce a variational distribution g(y", m/y) to approx- imate the true posterior. We assume factorized form symmetric to the generative model: g(y", -ny) = ((n) II; q 1t: ma). The variational posterior for n is also a Gaussian, q(n) = (Nnlm, F), and the conditional postcrior for the clean labels is parameterized as:
Qvik = 1 1le = 1, 1.), = 6sc(1-r(m)) + (1 (n.))(n))
(6)
where A E [0, 11 x is another learnable class transition matrix modeling the inverse noise process (i.e., inferring the clean label from the noisy one), with analogous constraints (Ace = 0 and Σ Ake = 1 for each noisy class E ))). With the generative model and variational posterior fully defined, we can derive the Evidence Lower Bound (ELBO) as follows (sce Appendix A.2 for details):
log x. 0)
2 E myy) (nyy) [logpyyly" n1., 0)] LL [q(y", nly) (l |x, 0)
= Σ Boindots (m)m)) [oo [x]] Buyon) [log ((n)) + Egonn [lgg(())
Σ E 9atndalu [lo (y6, n²] + Σ E Belw)aeetlo [logp(welt nn)l
(7)
A key advantage of this formulation is that the expectations over the discrete latent labels y* in Eq. (7) are now decomposed into a sum over individual pixels. This avoids the intractable summation over all possible label map configurations that hindered the direct optimization of the marginal likelihood in (3., rendering the objective tractable. The overall goal is to optimize the model parameters O and the variational parameters of q(n) and (() [y. to maximize this lower bound. The ELBO in Eq. (7) has an important interpretation as a principled generalization of the standard cross-entropy loss. To see this, we can analyze its components.
The first term in this ELBO in Eq. (7) can be interpreted as an expected reconstruction term. By expanding the expectation over the discrete clean label v,, it becomes a cross-entropy loss with soft labels:
HW |
Bedor [log po(y [x)] = Σ M qiin = 1/y) log 1/x),
(8)
where q51k 1lys) Regno 1/9,, 7:)] is the marginal posterior probability that the clean label for pixel i is class k. This replaces the hard, fixed one-hot observed labels with soft, inferred posterior probabilities. Building on this interpretation of the first term, we can now demonstrate more formally that our entire objective is principled generalization of the standard cross-entropy loss. This is shown by examining the conditions under which the remaining teriss in the ELBO vanish.
The second and third terms together constitute the negative Kullback-Leibler (KL) divergence, -KT((()))l)(()) This term vanishes when the variational posterior identical to the prior, i.e. q(n) = p(n). Similarly, the fourth and fifth terms represent the expected log-likelihood of the observed labels under the noise model, regularized by the entropy of the label posterior. These terms also cancel each other out under the assumption of symmetric noise process, where A is symmetric and A B in Eqs. (6) and (5). See Appendix A.4 for detailed derivation of the label noise terms cancellation under the symmetric noise process assumption. This condition is met in non-trivial noisy scemarios, for instance, with a uniform noise model where an error from any class is equally likely to transition to any other incorrect class (i.e., Vk, E J, k * + Ake = Bie = 1/(D/ - 1)).
In such a setting, maximizing the ELBO is equivalent to maximizing only the first term As we saw, if we assume noise-free setting, the posterior distribution q(vž yi, m))) collapses to a point mass (i.e., a Kronecker
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delta function centered at the observed label v.), making the soft labels q(y6s (r) identical to the observed one-hot labels. Consequently, our objective function reduces precisely to the standard cross-entropy loss. Therefore, our method provides principled generalization of the standard supervised learning framework, where the additional terms in the ELBO serve to regularize the posterior and enforce spatial correlation in the inferred label errors.
While the expectations over the discrete labels yf are tractable, the terms involving the continuous variable 7) pose significant computational challenge. The number of elements in n equals the number of image pixels (HW), which can be in the hundreds of thousands, and these elements are spatially correlated. For instance, the negative KI divergence term, (KL(((n)) [p())) , is analytically computable since both p(n) = (N) Σ) and ((n) = mm T) are Gaussians. Its value is given by:
-KL[(())ll((m) 1
T Σ
+ HW Tr TT (m (m--)²²3> - (m- 1))
log
-
(9)
However, evaluating this term computationally prohibitive for large images, it requires computing the determinants and inverses of the HW HW covariance matrices, in addition the trace and quadratic form terms. These challenges are efficiently addressed by leveraging the properties of the KMS-structured covariance, we describe next.
4.2 KMS-structured covariance
An n x KMS matrix R2 is type of symmetric Toeplitz matrix, where each element is defined using the parameter p E (-1,1). Its inverse, R,,1, is tridiagonal matrix. The explicit forms of Rp and Rp are given
by:
1
p
p²
p
P
0
0
1
p
n
P
+
p
P
Rp :=
p
pr
n-3
R
0
1+1 p2
pn-1
p n
2
p
0
P
and the determinant computed as Rpl (1- p1 See Appendix A.1 for mathematical deriva- tions. While the KMS matrix is well-established in fields like signal processing for modeling AR(1) pro- cesses Grenander & Szegö (1958); Brockwell & Davis (1991), its unique properties an analytic determinant - have been largely untapped as computational tool in modern machine learn- ing. Our work bridges this gap by demonstrating its utility as the key component for making the ECCD framework computationally tractable.
sparse inverse and
This matrix is useful for representing correlations among one-dimensional variables, where the correlation decays exponentially as the index distance between two variables increases regardless of their absolute posi- tions.
Suppose one-dimensional random variable x has covariance structure given by
B[(a; (2)(2; - m4)) = k-
(10)
where is the mean of r,, By utilizing the KMS matrix Rp the covariance matrix can be expressed as
Σ VRVV =-v=v
(11)
where V is a diagonal matrix whose (i, i)-th element is d,,
In the case of two-dimensional spatial forrelations, such as those in images, we utilize the Kronecker product. Suppose a two-dimensional random variable x has a covariance structure given by
E[( (Fw) Mwv)) = i
(12)
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where May is the mean of zzj. Its matrix form can be expressed as
= RR, R, = @ R R
(13)
where we assume that the wwo-dimensional pixel index (2.j) is mapped to a one-dimensional index n. V is diagonal matrix whose (nz, n)-th element corresponds to oij; and denotes the Kronecker product.
Since V is diagonal and R, is tridiagonal, the inverse covariance Σ-1 is sparse matrix. The properties of the KMS matrix provide an elegant solution to the computational challenges outlined in Section 4.1. First, the determinant of the covariance matrix can be computed analytically, bypassing the need for expensive numerical methods. Second, the sparse structure of the inverse matrix is crucial for the remaining terms. It reduces the complexity of computing the quadratic form and the trace term in the KL divergence (9) from ((HHW)²) to W(H)), which is linear in the number of pixels.
Beyond this computational speed-up, the sparse structure of the inverse also ensures that the quadratic form
(n- 1 (n 11)
(14)
involves only pairs of adjacent pixel variables. This, in turn, makes the conditional distribution p(n: my)) depend only on the neighboring pixels TN, where my denotes all variables in n except T)s and Nr represents the set of adjacent pixels of i.
In other words, the KMS-structured covariance ensures that the Gaussian distribution retains a Markov Random Field property while maintaining the tractability of both the inverse and the determinant of the covariance matrix. The effect of the correlation parameter is illustrated in Figure 1b. As shown, the model can confidently infer label errors when adjacent pixels are also likely to contain errors and the spatial correlation strong (i.c., is high). This structure corresponds the discrctization of a continuous Gaussian Random Field with an exponential kernel under the L1 distance. We provide detailed theoretical analysis of these properties in Appendix B.
4.3 Optimization
We formulate the optimization problem as follows. Let N be the number of training samples. We denote the model parameters by 0, the prior distribution paramcters by w {u, Σ}. the image-specific variational posterior parameters by {x("))
(m p( Jn=1, and the negative ELBO for the n-th sample (from (7)(7)) by L(0, w, (,,,w,w, v())). The covariance matrices E and [ are KMSSsttructured covariance matrices, each which is parameterized by the diagonal elements and one spatial correlation parameter p for KMS matrix. The spatial correlation parameter p is initialized with the same value for the prior and posterior (sce Section 5.2.3) and not optimized for the posterior either. Moreover, the label transition matrices A and B are not treated as learnable parameters:
assume a symmetric noise process in which A is symmetric and A = B in Eqs. (6) and (5). Under this symuetric noise assumption, the corresponding label noise terms in Eq. (7) cancel out and thus A and B do not explicitly appear in L(O, w. p()).
Our training procedure, outlined in Algorithm 1, employs an alternating optimization schome. For clarity, the algorithm is presented for a single sample per step, though this can be generalized to mini-batches. In each step for a given sample, the model parameters are updated K times, while the corresponding variational posterior parameters (n) are updated M times. After training, the optimized model parameters treated
0 are used for inference new images. Note that in Algorithm 1, the prior parameters as fixed, but they could also be optimized jointly with sulficiently large training dataset. Thanks to the KMS structure, gradients for the KL divergence terms (involving log-determinants and traces of inverses) are computed fully analytically, avoiding high-variance Monte Carlo estimators for these regularization terms.
5 Experiments
5.1 Dataset
Research on segmentation with noisy labels has been especially active in both medical imaging and satellite remote sensing. Tn this work, we focus two widely used benchmark datasets that represent these do-
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Mf (n) Published in Transactions on Machine Learning Research (03/2026)
Algorithm 1 Training loop for alternating optimization of model and posterior parameters.
0,w,
( (100) 1
Jt=1 f- Initialize parameters
2: repeat
3: n n Randomly sample an index from (1,...,. N
4: for k= 1,.,...,
K do
5: Compute L(0, w. =0")) and ils gradient w.r.l. 0
6: + Update the model parameters using the gradient
7: end for
for m m00,1..... -
Compute L(O, w, um and its gradient w.r.t. (n)
9:
10:
m t Update the posterior parameters using the gradient
(n)
11: end for
12: VO [n]
13:
until convergence of parameters 0
mains: the JSRT dataset for medical imaging and the WHU Building dataset for satellite imagery. Detailed descriptions of each dataset are provided below.
It is common practice noisy-label segmentation research artificially introduce label noise for evaluation because most publicly available datasets come with clean annotations Li et al. (2021); Song et al. (2022): Yao et al. (2023). Following this convention, we also employ synthetic noise in our experiments, with details of our noise generation process described in Section 5.2.1.
JSRT Datasct Shiraishi et al. (2000) is a publicly available dataset provided by the Japanese Society of Radiological Technology (JSRT), which comprises chest radiographs annotated with segmentation labels for the lung fields, heart, and clavicles. All images are of size 256 256 pixels, with 199 images designated for training and 50 for testing. Following the assumption common in noisy-label segmentation research that clean validation data is available- we evaluated our approach using the published split of 199 training images and 50 test images. Following a previous study Li et al. (2021), the Clavicle region was cropped to fixed 96 224 area for training and inference.
WHU Building Dataset Ji et al. (2018) is a publicly available dataset for building segmentation containing satellite images of Christchurch with a resolution 0.075 m covering an area of 450 km3 Manual annotations were provided for 22,000 buildings. The satellite images were downsampled to resolution of 0.3 and then divided into 512 512 tiles. Among the non-overlapping tiles, 4,736 validation, and 2,416 for testing. Similar to JSRT, adopted the published split for the training and testing data.
used for training, 1,036 for
5.2 Experimental Setup
5.2.1 Noisy Label Settings.
In the JSRT dataset, we evaluated the impact of noise cansed by boundary error, which is a critical issue in the medical imaging domain Yao et al. (2023); Zhang et al. (2020b). In accordance with previous works Xue al. (2020); Zhang et al. (2020c); Xue et al. (2020); Li al. (2021), we employed morphological transformations as synthetic noise. As illustrated in Figure 2, the noise settings consider not only boundary ambiguities (i.e., dilation or erosion) but also affine transformations. We examined the effect of noise using to represent the proportion of affected data and ( to determine its strength, setting ((a,8)} {00.3, 0.5), (0.5, 0.7). (0.7, 0.7)).
In the WHU Building dataset, we modeled three types of label noise: omission noise, where labels are missing in regions that should be labeled Song et al. (2022); Liu et al. (2024); commission error, where labels are present in regions that should not be due temporal changes Song et al. (2022); and boundary noise, which commonly occurs in segmentation tasks Song al. (2022), as illustrated in Figure We define the omission ratio (Ø), the commission ratio (C), and the boundary noise probability (A) the proportions of
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JSRT Lung
WHU Building
Inpul image
Clean label
Eroded
Dialated
Affine
Input image
Clean label
Omissiion
Comission
Fruded & Dinkated
Figure 2: Examples of Noisy Labels. medical imaging, label noise often simulated using morpho-
logical operations such as erosion, dilation, and affine transformations Li al. (2021). Conversely, building segmentation commonly involves omission noise, commission errors, and boundary noise Song et al. (2022). The pink region indicates labels present in the noisy but absent in the clean label, while the blue region indicates the opposite.
the total instances in the image that are perturbed by noise. Specifically, the noise parameters are set as (())).A) = (00.1, 0.0, (0.2, 0.05, 0.5), (0.,,0.1...7).
5.2.2 Baselines.
We compare our method against several baselines, including cross-entropy loss (CE), noisy-label learning approaches for classification tasks (CoT Han et al. (2018), TriNet Zhang et al. (2020c), JoCoR Wei et al. (2020), EM Bekker & Goldberger (2016)), noise-robust. losses for segmentation T-Loss Gonzalez-Jimenez et al. (2025), as well as techniques that account for partial spatial correlations, namely Superpixel (SP) et al. (2021) and SpatialCorrect (SC) Yao et al. (2023). For each method, we adopted the hyperparameters reported in the publicly available implementations or original papers, assuming them to be the best practices for their respective approaches. Although proposed method supports multi-label segmentation, many of the baseline methods are designed for binary segmentation. Therefore, in JSRT dataset, conducted experiments by treating each class (Tung, Heart, Clavicle) as separate binary segmentation task.
5.2.3 Implementation Details.
We use U-Net Ronneberger et al. (2015) with Elficient Tan & Le (2019) encoder, pre-trained ImageNet Deng al. (2009). The model is optimized using Adam Kingma & Ba (2014) with learning rate of 0.001 and batch sizes of 16 (JSRT) and 32 (WHU Building). The paramcters in the prior p(n) = NN)pp, E) and the initial values of the parameters in the variational posterior q(n) = (nimm, r) are set as follows. These parameters (e.g., the hyperparameters of the noise-level prior) were tuned using a 15% validation subset of the WHU Building training set. Specifically, we performed grid search over reasonable values and selected the setting that yielded the best validation performance (measured by validation ELBO). These hyperparameters
then fixed for all subsequent experiments across all datasets to evaluate the robustness of generic setting. In practice, it is also possible to tune these hyperparameters for each new dataset if validation set is available. The fixed values are: p 0.75, m = -5, = 1, = -2, and a = 1.
5.3 Results for JSRT and WHU Building dataset
5.3.1
JSRT dataset.
As shown in the first left three columns of Figure 3, report the Dice scores for each class (Lung, Heart, Clavicle) under all three noise conditions.
Under the moderate noise condition (a, 8) = (0.3, 0.5). the CE baseline, which lacks explicit noise handling, experiences a noticeable drop of 6.2%, 2.4%, and 3.8% for each organ compared to learning with clean labels. On the other hand, most noisc-robust methods exhibit only minor declines. Our proposed approach, in particular, shows slight performance degradation in Lung (only 0.3%), indicating its strong robustness to mild noise.
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Lung- a=003 0.3, A=0.
Heart a= 0.3, 8=5
Clavicle a= 0.3, #=0.5 0.5
WHU 00 000.0 0=0.
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92.2
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87.3
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0.5. / 3=00 0.7
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0.5, B 0.7
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Figure 3: Segmentation results on JSRT and WHU Building dataset. The first left three columns show the segmentation results for three classes Lung, Heart, and Clavicle in the JSRT dataset, while the right columm presents the binary segmentation results for WHU Building dataset in bar chart. The rows correspond to three different noise intensity settings tested on each dataset, with Dice scores reported for JSRT and IoU scores for WHU Building datasct. The dotted lines in the bar charts represent the performance of a model trained with standard cross-entropy loss on clean labels, serving as an approximate upper bound.
As the noise level increases to (a, = (0.5, CE undergocs a further 12.3% performance drop, and while other baselines also suffer more substantial declines, they remain relatively more stable than CE. Notably, our method continues to perform well on the Lung and Heart classes, demonstrating limited reduction from the noise-free scenario. For the Clavicle class-an an anatomically smaller and more challenging structure-bot both CE and some of the noise-robust methods experience larger drops, underscoring the sensitivity of thin or complex boundaries to label perturbations.
Under the heavier noise conditions (a, B) = (0.7,0.7) all methods face considcrable accuracy degradation. Even our approach, which remains competitive or superior for the Lung and Heart, shows more pronounced drops in Clavicle segmentation. This result highlights that extremely noisy annotations amnotations-pprtiiullle particularly for
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small or intricate regions- pose significant challenges. Nevertheless, compared to the other baselines, method tends to retain higher Dice scores and lower variance in most cases.
One notable observation in our experiments is that the Clavicle segmentation task exhibits a larger perfor- mance degradation compared to the Lung and Heart classes under severe noise. A possible explanation is related to the EM-like feedback mechanism ECCD. The method alternates between estimating the poste- rior over clean labels and updating the segmentation parameters 0. When the current segmentation model is reasonably accurate, these two steps reinforce each other: reliable posterior estimates provide improved supervision, which in turn leads to better parameter updates.
However, under severe noise especially for thin structures such as the clavicle the initial segmentation model can be highly inaccurate. In this case, the inferred posterior over label errors may become unreliable, preventing the feedback loop from functioning effectively. As a result, the model may struggle to progressively correct label noise.
As supplementary investigation, we additionally considered simple warmuu strategy, where 0 is first pretrained using standard cross-entropy before optimizing the ECCD objective. This provides a better initialization and may help stabilize the carly stage of training.
The results show that FCCD with warmup tends to yield slightly improved or comparable Dice scores for the clavicle segmentation under severe noise conditions. These observations are not conclusive, but are consistent with the hypothesis that the performance degradation may be related to optimization dynamics. Detailed results are provided in Appendix C.2.
To cxamine whether the proposed framework affects performance when labels are clean, we additionally conducted experiments on the JSRT dataset using the original clean annotations. The results show that ECCD achieves segmentation accuracy comparable to standard CE training. For example, on the lung, heart, and clavicle classes, the Dice scores of ECCD are within the confidence intervals the CE baseline. These results suggest that the proposed framework does not degrade performance when label noise is absent. Detailed results are reported Appendix C.3.
5.3.2 WHU Building dataset.
In the right column of Figure 3, we report IoU scores under three noise settings characterized by different omission, commission, and boundary probabilities (A).. For moderate noise (ó 0.1, 0.0, 0.3) and (Ø = 0.2. = 0.05, A = 0.5), CE's performance declines by 5.0% and 11.5% from the upper bound, respectively. Methods like COT, TriNet, JoCoR, and SP generally fall into the low- to mid-80% range. T- Loss and EM display a smaller gap from the clean-label upper bound, reflecting their robust design. Notably, our method reaches 87.4% for (ó = 0.1, = 0.0, À = 0.3), just 2.6% below the upper bound, and continues to outperform other basclines at the next noise level.
When the noise becomes stronger (Ø 0.3, < 0.1,A = 0.7), all methods exhibit further deterioration, although CoT, TriNet, JoCoR, and SP still maintain relatively higher loU scores. Our method remains competitive, indicating its adaptability even under substantial label corruption.
The results on the JSRT and WHU Building datasets confirm that our method effectively handles various types of label noise. In our experimental setting, this gives SC a disadvantage, as it designed specifically for boundary-related biases.
5.4 Effect of Spatial Correlation Parameter
Figure 4 (a) shows how segmentation accuracy varies with the spatial correlation parameter for JSRT Lung, JSRT Heart, JSRT Clavicle, and WHU Building datasets. In the Lung, Heart, and WHU Building datasets, performance peaks around = 0.75, suggesting that moderate correlation effectively captures spatial structure and improves segmentation. In contrast, both ignoring spatial dependencies (p 0) and assuming near-complete correlation (p 0.99) result in noticeable accuracy drops. Performance on the Clavicle class differs, remaining relatively stable from p to 0.75, likely due to its smaller, elongated
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Noisy
P (Spatial correlation value)
Region
0.1
0.5
0.75
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95
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0.8
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75
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00
poc
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(a)
(b)
Figure (a) Effect of p. Segmentation performance was evaluated across datasets based on P. which represents the degree of spatial correlation and affects segmentation performance across datasets. The results show that incorporating moderate spatial correlation enhances segmentation performance. Dice scores are reported for the Lung, Heart, and Clavicle, and IoU is used for the WHU Building dataset.
(b) Estimated label error. This figure visualizes the label error probability, computed as the sigmoid of m, the posterior mean of the logit of label error n. As shown in (a), higher accuracy estimating n leads to better segmentation performance.
shape, which reduces the benefits of spatial correlation. Figure (b) further illustrates the estimated posterior label error probability in selected noisy regions under different P values. At p 0.75, mislabeled pixels are identified more accurately, indicating that the model's label-correction mechanism is most effective under moderate correlation. In contrast, pushing p too high (e.g., 0.99) makes optimization unstable, leading to less reliable noise correction. Overall, these findings suggest that incorporating appropriate level of spatial correlation significantly enhances the label correction mechanism.
5.5 Computational Overhead
Our method maintains image-specific variational parameters, which introduces additional computational overhead in both memory and wall-clock training time. We therefore quantify this overhead and compare against standard cross-entropy (CE) training and robust baselines. In our implementation, the main addi- tional state is the posterior mean and variance of the latent Gaussian error field v,, stored as two feature maps per training image. These maps are saved on disk and loaded on-the-fly with each mini-batch. As a result, persistent GPU memory usage is almost unchanged; the primary overhead is in computation.
To quantify this overhead, we mcasured per-epoch training time and GPU memory usage on the WHU building dataset with 128 x 128 crops (197 images) an RTX A6000 (4SGB). For the per-epoch training times in Table 1, SP3 and SC measured using their official implementations. CoT, TriNet, and JoCoR were measured using the corresponding implementations included in the publicly available SP codebase.
httte///gohhbb..mm//mmmnnonss guided Nolsy Label Label_Laael Seg
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Table 1: Training time per epoch and approximate additional GPU memory usage excluding image and mask tensors (WHU dataset, 128 x 128 crops, 197 images, RTX A6000 48GB). Here 0 denotes the memory footprint of a single segmentation network, B the batch size, and H, W are the image height and width, respectively, "Correlated ELBO (Full Cov.)" denotes the same ELBO optimization without exploiting the KMS covariance structure, which requires direct numerical computation of matrix inverses and determinants. M corresponds to the number of gradient- braseent-based updates of the variational posterior in the E-step (Algorithm 1). OOM indicates GPU memory overflow.
Method
B =
B = 16 (s/epoch)
Additional GPU memory usage (approx.)
(s/epoch)
CE
86.68
2.63
0
CoT
109.01
8.72
20
TriNet
118.47
10.50
30
JoCoR
102.13
8.55
20
SP
111.89
8.09
20 + BHW
T-Loss
92.51
2.67
0
SC
5.15
1.27
0
Pixel-wise EM
88.34
2.93
0
Correlated ELBO (Full Cov., M 1)
2303.11
O0M
0 2BWW (peak: + (HH))
ECCD (M- = 1)
164.30
5.34
0+ 2BHW (peak: + 9BHW)
ECCD (M 10)
328.89
11.77
+ 2BHW (peak: 9 + 9BHW)
Table summarizes the results. With one variational update per iteration (M = 1), our met is roughly 2x slower per epoch than standard CE training, and of the same order as other robust baselines (e.g.. Co-tcaching, JoCoR). As the number of E-step update increases, the per-epoch training time increascs. our experiment, increasing M from 1 to 10 increased the per-epoch training time from 164.30 to 328.89 (batch size B 1) and from 5.34 to 11.77 (batch size B 16). The dense covariance implementation becomes computationally infeasible even for moderate image sizes, highlighting the necessity of the KMS structure for scalable inference.
6 Conclusion
We have proposed a Bayesian framework for segmentation from noisy labels, introducing the logit of the label error probability as a continuous latent variable, n. By incorporating the KMS matrix into the covariance structure of both the prior and the variational posterior distributions, our method efficiently computes the ELBO while capturing spatial correlations among adjacent pixcls. This avoids the need to enumerate label combinations, improving computational efficiency. Our approach overcomes key limitations of conventional methods that assume pixel-wise independence or only partially model spatial correlation. Experiments on medical imaging and remote sensing datasets demonstrate superior robustness to noisy annotations when the spatial correlation parameter is set to a moderate value (e.g., p= 0.75). More broadly, ECCD provides a tractable framework for variational inference with spatially correlated discrete latent variables, which commonly arise in structured prediction problems beyond image segmentation. As such, ECCD offers a powerful tool for probabilistic modeling in domains where priors over correlated discrete variables are defined structured domains such spatial grids temporal sequences. Potential avenues for future work include extending this framework to higher-dimensional spatial problems, like 3D segmentation and volumetric data analysis. Furthermore, as the underlying KMS matrix formulation is naturally suited for modeling exponentially decaying dependencies, the ECCD holds significant. promise for domains beyond computer vision, particularly for modeling temporal correlations in time-series analysis and other sequential data.
Our method has some limitations. First, storing and optimizing posterior parameters for each training sample increases computational overhead; quantify this overhead in Section 5.5. Second, spatial correlation may
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be less beneficial when the target structures small clongated, the spatial dependencies may not contribute effectively.
Broader Impact Statement
This work introduces the ECCD, framework designed to enhance the robustness of machine learning models against spatially correlated label noise. We believe this research has the potential for significant positive societal impact, particularly in domains where high-quality data annotation a major bottleneck. In medical imaging, our method could lower the barrier to developing diagnostic support tools by enabling the use of larger, more diverse, and imperfectly labeled datasets, potentially accelerating research and improving healthcare accessibility. Similarly, in remote sensing, the ECCD can facilitate more accurate and largo-scale analysis of satellite imagery for applications such environmental monitoring, disaster response, and urban planning, by effectively handling noise inherent in large-scale data collection.
At the same time, we recognize potential negative repercussions that users of this tochnology should consider. In high-stakes applications like medical diagnosis, over-reliance on the output of any automated system, including one trained with our method, can lead to automation bias and potential misdiagnosis if not used with expert oversight. We emphasize that this technology should be used assist qualified professionals, not to replace them. Furthermore, like many remote sensing technologies, improved segmentation capabilities could be repurposed for surveillance or other applications with negative societal consequences. We advocate for the ethical use of this technology in accordance with established legal and ethical guidelines. Finally, the performance of our method relics on the assumption of stationary, exponentially decaying correlations in the label noise. Users should be aware that in scenarios where this assumption does not hold, the model's effectiveness may be reducod.
decision-support tool to
Acknowledgments
This research work was financially supported by the Ministry of Internal Affairs and Communications of Japan with scheme of "Rescarch and development of advanced technologies for user-adaptive remote sensing data platform" (JPM100316).
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Appendix
A Derivation details
In this section, we expand on several details that were only briefly mentioned in the main text. First, we provide a comprehensive explanation of the KMS matrix (see Section A.1). Next, we outline the intermediate steps in the derivation of the ELBO in our method and the derivation of the ELBO terms that account for spatial correlation (see Section A.2). We also derive the determinant and inverse of the covariance matrices E and r, which constructed using the KMS covariance (see Section A.3). In particular, Section A.3 details how exploit the sparse structure of these inverses achieve efficient computations.
A.1 KMS Matrix
The Kac-Murdock-SSan Szegö (KMS) matriz Fikioris (2018); Kac et al. (1953) is defined for positive integer E N and parameter p E (-1,1) as
P
p²
p
p
1
p
p
2
P
1
p
Rn(p):= =
(15)
3
2
P
1
p
-
1
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The (i,j) element of the KMS matrix is given by p|--a,, capturing the idea that the correlation between elements decays exponentially with the distance between indices.
Determinant of Ra(p)
To derive the determinant, start with
1
p
/²
1
P
del (p) =
P²
P
1
(16)
p n=-
p n-2
p
For = .,.... subtract p²=1 times the first row from the ith row. This operation zeroes out the first column (except the first row) and introduces factors of (1 - p²) in the diagonal the resulting submatrix.
Proceeding recursively, one obtains
det (p( (1--p?)-1
(17)
This compact expression is key for efficient computation.
Inverse of (p)
The inverse is given by
1
(p) det (p) adj(R,
(18)
where the adjugate adj( (p)) is the transpose of the cofactor matrix. Through series of elementary operations and by exploiting the Toeplitz structure, it can be shown that the cofactors also share patterned structure. In particular, one finds that
adi(Rn(p)) = (1 - p2)m- Saz
(19)
with
P 1 -P
0
0
2
p
0
S,
P
1 1+1 p
(20)
PP
p
Thus, using det (p) (1 (1-p²)m~ 1 we obtain
p²)m-1
(1 22)
Ra(p)
S
S..
(21)
(1 p2)
/
These derivations are essential in our framework for efficiently computing the covariance matrix inverse and determinant used in the Bayesian model.
A.2 Derivation of ELBO (Eq. (7))
We assume the following generative model for our segmentation task:
po ,,,, n/)) = po (y xx plyly" n) p(n)
(22)
II po (0) xx) } {II Plu,lu m)) p(n).
(23)
where the prior p(n) is designed to capture spatial correlation (via KMS covariance structure).
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We introduce the variational distribution ds
4== (n.
= ((()) II a( me Vs),
(24)
which approximates the true posterior mll,,,, Then the marginal log-likelihood of the observed labels can be lower bounded as follows:
log/o(y|x) = log
Σ Pa polyln.y' x) po 0 (x) dn
(25)
poly'n.y's po po)
log Ego
(26)
> Eg [logpolyln,y'. x)] KL 11//00001 1
(27)
We decompose the expectation term as:
Ege [lg (yll, y", x)]
(28)
Rg [log Po yy,,y",,
(29)
= Eg [log p(yln.y")
(30)
Eg
log II Ipluimee
(31)
Egg
log Plgg(uuln.. v?)
(32)
Σ Eyg pogilmyl
(33)
Σ Bygnnww ly) [logp(ui (marginalized by (maa x(.))
(34)
Σ Beie (ovetlmmaaann [logp(us p(ui,, v?)]
(35)
Σ Bycut Inand [log Mogg(usli w[]], (#i 1L valls,
(36)
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where \i denotes pixel indices other than on the image. We also decompose the negative KL divergence
term as:
KL (yb Ipo (17, y [x]
(37)
log po (0 y x
Rgg
(38)
4Φ
-By
[log po y (x) log gol
(39)
By
log py |n, x) po po( log log <ly"ly m)ggm)gg g(ll))
(40)
poly (x) = p(n))
=(())
Egg
log
po(y"Ix)
<log((n) log
gly" (,m)
log (n))
(41)
II. Po( (xc)
III, alvz lumm)))
Σ log pe (vt x) logg(n)) Σ log a( (lulll,, mi) log
(42)
Betndele bmd (v. [x]] Byt (17)( Egtn) [logp(n)]
(43)
Σ Be(nnon Int mo)))) olog ( (1.luu 7:)]
(44)
Thus, the overall ELBO becomes
logpe(ylx) 2
Σ (n(n))) a(ollann)) [log pe (v. [x]] - Eg(in) [log ((n)] + Eg(n) p(7)]
Σ olg mon) [log q("." yss n)) +
E
Egtn (lo) p(ul((" m))
(45)
A.3 Efficient ELBO Computation
In our framework, the spatially correlated prior p(n) is modeled as a Gaussian distribution with a covariance matrix that incorporates spatial structure via a Kac Murdock Szegö (KMS) matrix:
Σ= v (R, @ R2
(46)
where R2 := RH(P) E H and R, = Rw(p) Wxw are the KMS matrices for the vertical and horizontal directions, respectively, and V is diagonal matrix (c.g., V ==ol aI in a simplified setting).
Because the inverse a KMS matrix is tridiagonal, the inverse of the Kronecker product R & R very sparse (with most 9 nonzero entries per row). Hence, V aI, then
is
--'-v- (R " 1 Φ2 (R @ R² ")
(47)
Similarly, by exploiting the determinant properties of Kronecker products,
det Σ = (det V2² (det R,p) W (det R2,
II σ (-
(48)
Taking the logarithm yields
Hw
log det Σ 2
log o; + W(H 11) log(1 - -p²) H(W - log - -p²).
(49)
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The efficient computation of the trace terms in the ELBO is similarly achieved by noting that Σ-1 is sparse, which reduces the computational burden when evaluating expressions such as ir ir(rr-1) and tr((#= mm m) -)
A.4
Derivation of Label Noise Terms Cancellation under Symmetric Noise Process Assumption
We consider the label noise terms, which the last two terms of the ELBO in Eq. (7):
Bo(n))oo lwo) [log q() [96. mc] +
Σ E Batna)oaa [log p(u: 1y m))
(50)
Σ B Boln)alut (u.):)
[log po: n:)] Σ Balnndast bommd) [log (vv le ()))
(51)
Σ B Boln) (m))
plg W7 mk) - log ( lus, m)]
(52)
Σ Botn)) Botyy (n())) (im) ( (o m) - log ( we mm)]
(53)
The contribution of pixel for a given v and an observed (noisy) label can be written as
Fo(o, mk) = Bylu [log p(1o 1 mh) log yvv lit. 1, m:)]
(54)
It is sufficient show (o nk) = for every i, Wes o under the symmetric noise process assumption. Expanding over E y, Fi m)) can be expressed as follows:
F((o.)))
Σ
q/1 1/y 1, m/) p(yio 1lyic 1,))) log qiyi 1lyio 1, 1,)]
(55)
ccY
Σ
qiii 1/9t = 1, [logpll) 11vio = 1,) log q11i 1/yin = 1, 1,
(56)
CEV/
(c7a)
+ gVVio 1l1io = m:) [log = 1 = I, n²) - log alvic 1l1io = 1, m;]
(57)
Σ
Aco [log(r(m)))))))))))) Boc) log(r(m))Aoo)] + (1 r(n)) log(1 - r(n)) log (ogg)))(r)))
(58)
rE)
(c#0)
Boc A(1:) log r(jk) Aco
Σ
(59)
(cf0)
Boc
Σ
Aco log
(60)
Aco
(cFo)
Under the symmetric noise process assumption, in which A is symmetric and A = B, obtain
Boc
Anc
Ace
F(o.m) = Σ A(n))co log
Σ
r(n.)co log
Σ r(n:A.o log
0.
(61)
Aco
Ano
A
cE) (c/0)
cEY (000)
cE)
(02t0)
Therefore,
Σ Egtn F((o.
0,
(62)
which shows that the last two terms of ELBO in Eq. (7) cancel each other out.
B Theoretical Analysis and Discussions
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B.1 Marginal discrete distribution induced by the latent Gaussian field
Our model can be explicitly characterized as a Model with logistic link function). Specifically, models the log-odds (logits) of the label error probability as spatially correlated Gaussian process, which is then mapped to discrete outcomes via a rogistic/sottmaxx function. From modeling standpoint, the induced discrete distribution belongs to the same general con- struction as ppoitttobiit models with a latent Gaussian field (often described as Gaussian-ooplle= type con- structions in the statistical literature). While this general class of models is well-known, inference is typically computationally intractable, requiring expensive MCMC or approximations like INL.A (Integrated Nested Laplace Approximation). Our specific contribution is identifying a subclass of these models that admits efficient, closed-form variational inference. By restricting the covariance of the latent field to the KMS structure, we enable the analytic computation of the ELBO (involving determinants and inverses) in O(N) time, which is not possible for general Gaussian random fields. To the best of our knowledge, this specific KMS-based, ELBO-computable subclass has not been studied under standard name. For this reason, we refer to it as an ELBO-Computee key computational property. Regarding the dependency structure: the interactions are strictly pairwise in the latent Gaussian space (defined by the inverse covariance matrix E- However, in the induced marginal discrete distribution p(yly") (after integrating out 11), the dependencies do not generally reduce to simple pairwise interactions (like in a standard Ising or Potts model). Instead, the marginalization induces dense, higher-order dependencies among the discrete labels. We view this as an advantage, as it allows the model to capture complex, long-range error correlations (blobs) that are difficult to represent with simple pairwise discrete MRFs.
Random Field (or a Latent Gaussian
Correlated Discrete Distribution (ECCD) to highlight this
B.2 Variational family, approximation bias, and gradients
We explicitly acknowledge the inherent approximation bias resulting from our choice of the variational family. Optimizing the ELBO is equivalent to minimizing the Kullback-Leibler (KL) divergence K LLyg(v" ly, ())(n))lp(v)) (1ll:p((""""")))))) m)). between the joint variational postcrior and the true joint posterior. Our variational formulation imposes specific structural constraints: 1) the label posterior is factorized as q(y"ly v) = II, ((V lys, ne), ignoring direct dependence on æ and assuming the functional form of Eq. (5) with a fixed class transition matrix A B (shared across images) rather than optimizing A per image; and 2) the latent field posterior is restricted to a Gaussian distribution q(n) [N//m, with KMS-structured covariance, where (m, [) are optimized for each image. Since the true posterior over the latent field 77 is generally non-Gaussian (and likely multimodal) due to conditioning on discrete labels via a non-linear likelihood, the optimal variational posterior within this constrained family does not converge to the true posterior. We accept this variational gap to achieve computational tractability and efficient inference.
A Approximation Bias:
Gradients: Regarding the optimization, key advantage of our KMS-based framework is that the gradi- ents with respect to the correlation parameters are computed fully analytically for the regularization terms. Specifically, the KL divergence term in the ELBO contains log det E and Tr(--11)) Because we utilize the properties of KMS matrices to derive closed-form analytical expressions for the determinant and the sparse inverse
--1 (as detailed in Appendix A. the gradients of these terms with respect to the correlation pa- rameter are exact and do not require Monte Carlo estimators. (Note: The expected likelihood term involves an expoctation over q(n), which estimated via the reparameterization trick, but the computationally heavy covariance-related terms are handled analytically.)
B.3 Identifiability between segmentation parameters and noise correlations
We address the potential identifiability issues between the segmentation parameters 0 and the noise correla- tion parameters:
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1. Structural separation: independence vs spatial correlation. The clean labels are modeled by a standard discriminative segmentation network with conditional independence across pixels: pe(y"(z) = II:=1 are independent for 7 j. The network may produce smooth probability maps (because CNNs lend to output smooth logits), but at the distributional level this still a fully factorized model: can control the first-order moments (per-pixel marginals), not the second-order moments (covariances) of the label field. By contrast, the noise model introduces a latent Gaussian field 77 with KMMS-ttructurd covariance, and the resulting marginal distribution p(yly") exhibits non- zero spatial correlation between yi and yj (for nearby pixels) whenever p > 0. This correlation is stationary and controlled by the KMS-structured covariance; it is not something the factorized po(y" (a)) can reproduce. Any systematic clustering of errors cannot be explained away by a different choice of in a factorized model; it must be attributed to the correlated label noise component.
" pe (vī [r). Thus, conditioned on the image the random variables y and y
2. Practical behavior and remaining ambiguity. In practice, identifiability is further helped by the fact that the segmentation model po(}"(>) learns image-dependent structure (e.g., anatomy) across the datasct, whereas the KMS noise model captures image-agnostic, stationary spatial clustoring of residuals. Trying to explain correlated annotation noise purely via 9 would require the network memorize idiosyncratic "noise blobs", which generalizes poorly; the stationary noise model is much more parsimonious fit. Empirically, across all our experiments we have not observed qualitatively different parameter configurations with essentially identical ELBO but very different predictions.
B.4 Asymptotic and scaling behavior of the KMS approximation
Convergence to Continuous Field: The KMS covariance corresponds to a discretized exponential kernel, where the correlation decays exponentially with distance. By parameterizing P = exp(-Ah), this structure can be interpreted as a discrete approximation of a continuous Gaussian random field with an exponential covariance function. As the spatial resolution increases (i.c., grid spacing - 0) while scaling the correlation parameter as p p=c - E-Ah, the discrete field converges to a continuous Gaussian Random Field with a separable exponential covariance kernel defined under the L, distance: k(u, v) exp(-Alju v/h1) exp(-A|ur vil [x)eexp(xX (-X/us v2)), where E I are spatial coordinates. This implies that the inductive bias of our model is an anisotropic spatial correlation aligned with the coordinate axcs.
Regarding boundary conditions, it is important to clarify that the KMS (Toeplitz) matrix represents the exact covariance of a stationary process observed on a finite grid, without impos- ing artificial boundary conditions. Unlike spectral approximations that use Circulant matrices (which implicitly assume periodic boundaries, wrapping the image edges), the KMS structure correctly models non-periodic field on finite domain. Therefore, there are implicit boundary artifacts at the image edges that would distort structures like clavicles. The primary limitation for thin, elongated structures (like clav- ieles) comes not from boundary conditions, but from the anisotropy of the L1 metric: since the correlation decays based on Manhattan distance, the effective correlation length is direction-dependent (stronger along axes), which may not perfectly align with diagonal anatomical structures.
Boundary Conditions:
C Additional Experimental Results
C.1
Noise- wiise-type-wiee Results
C.1.1 JSRT
We conduct experiments for three types of synthetic label perturbations erosion, dilation, and affine trans- formations each with different. noise level settings ((a, 33) = (00.3, 0.5), (0.5, 0.7), (0.7.0.7)) on JSRT Lung and Clavicle datasets.
We report noise-type-wist results on the JSRT Lung and Clavicle datasets in Tables 2. Across both structures, Dice scores tend to decrease as the noise level increases, indicating that stronger perturbations cause
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Table 2: Dice scores on JSRT dataset for each noise type and noise level.
Lung
Clavicle
Noise type
(a, 3)
CE
T-Loss
Ours
CE
T-Loss
TriNet
Ours
(0.3, 0.5)
89.0
88.4
94.9
92.2
87.9
89.2
88.7
88.0
Erosion noise
(0.5, 0.7)
77.2
74.3
84.8
79.3
76.8
73.3
82.4
71.3
(0.7,0.7)
66.5
58.1
72.5
48.9
67.8
72.4
76.6
56.3
(0.3, 0.5)
91.2
92.9
92.0
95.1
87.8
89.9
83.2
89.4
Dilation noise
(0.5, 0.7)
79.8
81.0
82.3
82.2
83.2
83.1
76.4
83.9
(0.7,0.7)
78.4
78.2
79.9
76.1
79.3
79.7
73.6
80.1
(0.3, 0.5)
94.4
95.2
96.4
97.6
89.7
92.0
87.2
88.8
Affine noise
(0.5, 0.7)
89.6
87.7
93.5
94.0
80.7
78.5
81.3
82.2
(0.7, 0.7)
78.1
75.5
86.5
85.8
75.2
71.9
71.6
66.6
Table 3: ToU scores on WHU Building dataset with different. noise types and noise levels.
Noise type
Noise level
CE
T-Loss
Ours
0.1
81.4
80.6
86.7
Omission noise
0.3
65.4
62.8
77.3
0.1
82.9
85.5
87.4
Commission noise
0.3
80.1
84.5
88.0
larger mismatch between the noisy supervision and the latent clean labels. The proposed method remains competitive under dilation and some affine perturbations, whereas its performance drops more substantially under erosion, particularly in the high-noise setting. One possible explanation is that erosion shrinks the foreground mask and may reduce the usefulness of spatially correlated cues for label correction, especially for thin structures such as the clavicle.
C.1.2 WHU Building dataset
We conduct experiments for two types of synthetic label perturbations, omission noise and commission noise, with two noise levels each (omission ratio ø E {0.1, commission ratio Ç E {0.1, 0.33) on the WHC Building dataset. Due to the large size of the original dataset, use 30% of the data in this supplementary experiment.
Table reports the loU results. As shown in Table 3, our method achieved higher loU than CE and T-Loss for both omission and commission noise at all tested noise levels.
C.2 Supplementary Analysis of Warmup Initialization
In this section, we provide supplementary analysis related to the discussion in Section 5.3.1 regarding the effect of initialization on the EM-like feedback mechanism in ECCD.
We conduct experiments on the JSRT dataset (Clavicle) under the severe noise setting (a. 3) = 0.7), We compare different initialization strategics for the sogmentation parameters A: (i) no warmup (direct ECCD optimization from random initialization), and warmup using cross-entropy pretraining for (ii) 25, (ii) 50, and (iv) 100 epochs. Each setting is repeated three times, and report the mean and standard deviation.
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-0.14
84
-0.16
80.90
$1.38
Dice Score (%)
-0.18
No warmup
ELBO
Warmup epochs
-0.20
Warmup 50 epochs
Warmup 100 epochs
-0.22
78
-0.24
-0.26
0
25
50
100
25
50
75
100
125
150
175
200
Warmup Epochs
ECCD Epoch (b)
(a)
Figure 5: (a) Final Dice scores for different warmup lengths (mean standard deviation). (b) ELBO during ECCD training for different warmup lengths.
Segmentation performance the JSRT dataset with clean annotations. Mean Dice score and standard deviation are reported.
Table 4:
Method
Lung
Heart
Clavicle
CE
98.0 + 0.0
94.9 0.0
93.1 + 0.1
T-Loss
98.1 = 0.0
95.0 +.2
93.9 + 0.1
Ours
97.9 + 0.0
94.8 0.1
92.7 + 0.0
Figure 5 (a) summarizes the final Dice scores, while Figure 5 (b) shows the evolution of the ELBO during training. We observe that longer warmup lends to result in higher ELBO values and more stable optimization. However, the differences in task performance, measured by Dice score, remain relatively small across settings.
Overall, these results are not conclusive, but consistent with the hypothesis that initialization may influence the optimization dynamics of ECCD.
C.3 Results with Clean Labels on JSRT
To evaluate whether the proposed method affects performance when labels are noise-free, we conducted additional experiments the JSRT datasct using the original clean annotations. We compare the standard cross-entropy loss (CE), the robust. loss T-Loss, and our ECCD method.
Table reports the mean Dice scores over three runs together with the standard deviation. Overall, ECCD achioves performance comparable to CE across all three anatomical structures. The results indicate that the proposed framework does not significantly degrade segmentation performance when labels are clean.
26


Reviews:
[{"review_id": "xJMgEFfpuD", "reviewer": "Reviewer_YuBG", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper addressed the problem of semantic segmentation with noisy labels. The paper makes the assumption that the label noise/errors are in spatially connected regions. The key contribution is the introduction of a latent factor/variable that models the ‘structured’ noise present in the annotation. Through variational inference, the method approximates the true label posterior for each training sample and trains the segmentation model using that label. \n\nStrengths:\n\n- The assumption that pixel annotations should be spatially correlated, holds true in general.\n- Modeling the unknown noise through a latent variable, and the intractable inference was addressed through variational inference. Although, there are existing works (especially in partial multi-label learning) which use similar techniques for true label estimation, this paper particularly uses structured KMS for efficient variational computation.\n- The spatial correlation is constrained to KMS structure, which makes the ELBO computation efficient.\n- Definite improvements over other methods for Lungs and Heart segmentation.\n\nWeaknesses:\n\n- Although the method itself is a good contribution, there are very limited experiments to clearly establish the practical effectiveness of the method. For example, the paper doesn’t perform experiments on datasets like other works do: e.g., ISIC and Shenzhen datasets, as done in Gonzalez-Jimenez et al. (2025) and Yao et al (2023) papers.\n- Even in the experiments added in the paper, the method mainly does well on Lung and Heart segmentation but not on Clavicle. In fact, for Clavicle, it performs worse than some of the other methods, even at smaller noise scales. Similarly, for WHU dataset, it only performs better than other methods at medium-level noise. There is a clear need to do significantly more experiments to evaluate if the method does well in other cases.\n- In general,  the method might do well on dense segmentation or objects with larger segmentation volume but not on smaller object segments. This is evident in results on Clavicle.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper assumes noisy label generation process and estimates the true labels through variational inference. By training the model on the estimated true labels, the method produces results which outperform other methods in at least a few settings. That shows that their assumption/claim are reasonable in some settings.", "audience_yn": "Yes", "audience_explanation": "Yes. The idea of modeling label noise, estimating the true label based on spatial correlation, and using KMS to enforce the correlation structure for efficient computation is quite interesting and would be useful for more applications.", "requested_changes": "- Please take a look at the weaknesses and address them. Mainly, I am concerned about the limited empirical evaluation and the method’s performance.\n- If you could provide results for each kind of noisy label setting e.g., eroded, dilated, affine, and then for WHU: Omission, Comission etc. it would provide clearer understanding of the performance of the method. I would imagine that the method would not do well in affine transformations and in WHU building case for extreme noise.\n- How are the parameters in the prior p(\\eta) and initial values in q(\\eta) are tuned? Explain.\n- How would the fourth and fifth terms cancel out? It is not clear if there is an assumption of symmetric noise process? Please elaborate.\n- Appendix A.2 need to be fixed and explained in detail. There are missing terms from (26) and (37) onwards.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "jZ7m6W4TXM", "reviewer": "Reviewer_855A", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper tackles semantic segmentation with noisy labels in settings such as medical imaging and remote sensing, where expert disagreement, misalignment and automatic labeling introduce structured errors. Instead of assuming label noise is pixel-wise independent, the authors explicitly model spatially correlated label errors and derive a Bayesian framework that generalizes standard cross-entropy training. \n\nSummary of contributions:\n1.They propose a new class of probabilistic models that explicitly capture spatial dependencies in correlated noisy labels for segmentation masks.\n\n2.They make the resulting optimization problem computationally efficient and tractable by exploiting a Kac–Murdock–Szegö (KMS) matrix structure.\n\n\nStrength of the paper:\n1.The paper is well written, clearly motivates the limitations of standard cross-entropy under label noise, and provides solid mathematical derivations that show how their ELBO objective reduces to standard training as a special case.\n2.The analysis of the spatial correlation parameter ρ (e.g., Figure 4) gives good intuition on how model assumptions relate to performance and helps interpret the behavior of the method.\n\nLimitations:\n1. The method requires storing and optimizing variational posterior parameters per training sample, which increases memory and computation compared to standard segmentation training or simpler noisy-label methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The main claim—that explicitly modeling spatially correlated label noise leads to improved robustness compared to methods that assume independent noise—is supported by the experiments: their approach clearly outperforms baselines in high-noise regimes and achieves comparable performance under low-noise conditions, which is consistent with their hypothesis.\n\nThe second claim—regarding computational efficiency—is supported in a more limited sense. The use of the KMS matrix structure is convincingly shown to make the required linear algebra operations within their Bayesian framework more tractable and efficient than a naïve implementation. However, when compared to standard segmentation baselines, their overall method still appears computationally heavier, so this efficiency claim would benefit from clearer empirical comparisons (e.g., time and memory usage) against other methods.", "audience_yn": "Yes", "audience_explanation": "The problem addressed—robust segmentation under structured, spatially correlated label noise—is highly relevant in practical settings, especially in medical imaging and remote sensing, where imperfect and inconsistent annotations are the norm rather than the exception. Making the training procedure explicitly aware of such noise is an important direction for reliable deployment of segmentation models. The proposed framework and findings could therefore be of clear interest to TMLR’s audience, both for practitioners facing noisy supervision in real applications and for researchers interested in developing or extending methods for learning with noisy labels.", "requested_changes": "Comments for securing my recommendation for acceptance:\n- The proposed framework is designed for settings where the type and amount of label noise are unknown, and the model is not explicitly informed about the noise level. It would be very helpful to discuss how the method behaves in the opposite regime, i.e., when training on essentially clean data. If we train the model on completely clean labels (without injecting noise), does the performance degrade compared to standard training, and if so, by how much? Even if a full experimental study is beyond the scope of this paper, providing some intuition or empirical evidence for this scenario would be valuable, as many readers may wonder whether explicitly modeling label noise can harm performance when the actual noise level is very low or zero.\n\nComments to strengthen the work:\n- 4.3 Optimization section:    In this section you write: “We denote the parameters of the prior distribution as \\(\\omega\\), the image-specific parameters of the variational posterior as \\(\\{\\nu^{(n)}\\}_{n=1}^N\\).”  \n    It would improve readability to clarify more explicitly what these parameters are and how they relate to the notation introduced earlier.\n\n- Section 5.2.3 (Implementation Details).\nYou write: “These parameters were tuned using 15% of the WHU Building training set and validation sets, then fixed for all experiments.”\nPlease elaborate on the tuning procedure. For example:\nWere these parameters treated as trainable variables jointly optimized with the network parameters, but only on that 15% subset?\nWhy are they not kept trainable during the full training on the entire dataset?\nIs there an issue of optimization stability or overfitting that motivates this design choice?\nClarifying this would make the implementation details easier to reproduce and interpret. \n\n- Figure 3. \nThe meaning of the dotted line is not clear, and it appears only in some subplots but not others. Please explain what the dotted line represents in the caption and, if relevant, why it is only present in some panels.\nIn addition, your method is shown with standard deviation bars while other methods are not. Please clarify in the caption (or main text) whether the baselines were also run with multiple seeds and, if so, why their variability is not shown.\n\n- Figure 4.\nIn the left panel, for the Clavicle class, it appears that \\(\\rho = 1.0\\) can yield better performance than the other \\(\\rho\\) values, which seems somewhat at odds with the pattern observed for the other datasets and with the right panel. In the text you mention that, due to the small area, the result is not very dependent on \\(\\rho\\), but I could not find any discussion of the specific behavior at \\(\\rho = 1.0\\).  \n    I suggest adding a brief explanation of this case (e.g., why \\(\\rho = 1.0\\) behaves differently here, or whether this is within expected variance) to better align the figure with the textual discussion.", "broader_impact_concerns": "No concern", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "UF0orZK8JW", "reviewer": "Reviewer_SiDx", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper addresses semantic segmentation with noisy labels. The key insight is that annotation errors in real-world data (medical imaging, remote sensing) are not independently distributed across pixels, they exhibit spatial correlations. Adjacent pixels are more likely to share the same labeling errors.This paper proposes a Bayesian framework called ELBO-Computable Correlated Discrete Distribution (ECCD) for semantic segmentation with spatially correlated label noise. The key contributions are:\n\n1. Novel probabilistic model: The authors introduce a continuous latent variable $\\boldsymbol{\\eta}$ representing the logit of label error probability, with spatial correlations encoded through a Gaussian prior with structured covariance.\n\n2. Computational tractability via KMS matrices: By structuring the covariance matrix using the Kac-Murdock-Szegö (KMS) matrix, the authors achieve:\n   - Closed-form determinant: $|\\boldsymbol{\\Sigma}| = \\prod_i \\sigma_i^2 \\cdot (1-\\rho^2)^{W(H-1)+H(W-1)}$\n   - Sparse (tridiagonal) inverse enabling $O(HW)$ complexity for ELBO computation\n\n3. Empirical validation: Experiments on JSRT (medical) and WHU Building (remote sensing) datasets demonstrate improvements over baselines, particularly at moderate noise levels.\n\nStrengths:\n- Elegant mathematical formulation connecting spatial correlation modeling to computational tractability\n- Principled generalization of cross-entropy loss (reduces to CE when $q(\\boldsymbol{\\eta})=p(\\boldsymbol{\\eta})$ and noise-free)\n- First application of KMS matrices for variational inference in this context\n\nWeaknesses:\n- The exponentially decaying isotropic correlation assumption may not match real noise patterns\n- Per-sample variational parameters $\\{\\nu^{(n)}\\}_{n=1}^N$ create memory overhead\n- Limited analysis of when the method fails (e.g., clavicle results)", "claims_and_evidence_yn": "Yes", "claims_explanation": "1. **Mathematical derivations are correct:** The ELBO derivation (Appendix A.2) properly applies variational inference principles. The factorization assumptions are clearly stated and the marginalization over discrete $\\mathbf{y}^*$ is handled correctly.\n\n2. **KMS matrix properties are well-established:** The determinant and inverse formulas (Appendix A.1) are classical results that the authors correctly apply.\n\n3. **Experimental improvements are demonstrated:** The proposed method outperforms baselines across most conditions, with particularly strong results for lung segmentation (97.7% Dice at moderate noise, only 0.3% below clean).\n\n**Concerns:**\n\n1. **Correlation structure mismatch:** The KMS structure assumes:\n$$\\text{Corr}(\\eta_{ij}, \\eta_{uv}) = \\rho^{|i-u|+|j-v|}$$\n\nThis is an **isotropic, stationary, axis-aligned** correlation. Real label noise (e.g., from annotator disagreement on organ boundaries) likely follows the **object boundary geometry**, not a grid-aligned exponential decay. The paper does not analyze how well this assumption matches actual noise patterns.\n\n2. **Missing ablation on correlation structure:** The authors vary $\\rho$ (Figure 4a) but do not compare against alternative correlation structures (e.g., RBF kernel, boundary-aware correlations). This limits understanding of whether KMS specifically or spatial correlation generally drives improvements.\n\n3. **Inconsistent clavicle results:** Performance on clavicle (thin, elongated structure) shows minimal benefit from spatial correlation. The authors acknowledge this but don't provide theoretical analysis of when their assumptions break down.\n\n4. **Synthetic noise evaluation only:** All experiments use synthetically generated noise. While common in the literature, this means the claimed match between model assumptions and \"real-world\" noise patterns is not empirically validated.\n\n5. **Statistical significance:** Error bars are shown but no formal statistical tests are reported. Some improvements appear within noise margins (e.g., WHU at highest noise level).", "audience_yn": "Yes", "audience_explanation": "1. **Practical relevance:** Noisy labels are ubiquitous in medical imaging and remote sensing. Methods that explicitly model noise structure address a genuine bottleneck.\n\n2. **Methodological novelty:** The introduction of KMS matrices for scalable variational inference is a technique that could transfer to other problems (time series, 3D data, sequential models).\n\n3. **Theoretical interest:** The ECCD framework provides a principled probabilistic treatment that connects to classical graphical model theory (Markov Random Fields) while maintaining computational tractability.\n\n4. **Reproducibility:** Code is provided, and the method uses standard architectures (U-Net + EfficientNet).", "requested_changes": "1. **Justify the KMS correlation structure for label noise:**\n   \n   The paper claims label errors exhibit spatial correlations but never validates that the specific KMS structure $\\rho^{|i-u|+|j-v|}$ is appropriate. Please:\n   - Empirically measure correlation patterns in real noisy annotations (e.g., from multi-annotator datasets like LIDC-IDRI)\n   - Or provide theoretical justification for why exponentially decaying axis-aligned correlations model annotation errors\n   - Or acknowledge this as a convenient approximation and discuss limitations\n\n2. **Clarify the variational posterior structure:**\n\nIn Eq. (6), the conditional posterior $q(y_{ik}^* = 1 \\mid y_{ic} = 1, \\eta_i)$ depends explicitly on $\\eta_i$, yet the paper also defines $q(\\boldsymbol{\\eta})$ as a Gaussian distribution. This creates a dependency structure: since $q(y_i^* \\mid y_i, \\eta_i)$ varies with $\\eta_i$, the marginal $q(y_i^* \\mid y_i)$ obtained by taking the expectation over $q(\\eta_i)$ is generally \\emph{not} a simple categorical distribution.\n\n-- How is the expectation over $\\eta_i$ computed in practice? Is it Monte Carlo sampling, numerical integration, or does a closed-form expression exist?\n\n--  Algorithm~1 mentions gradients with respect to $\\nu^{(n)}$ but does not specify what parameters are contained within $\\nu^{(n)}$. Please clarify the definition and role of $\\nu^{(n)}$ in the optimization.\n \n\n3. **Address the computational overhead rigorously:**\n   \n   The paper claims overhead is \"comparable to methods requiring multiple models\" but:\n   - How does memory scale with $N$ (training set size)?\n   - What are wall-clock training times vs. baselines?\n   - For the WHU dataset with $N=4736$ samples, each requiring posterior parameters $\\{\\mathbf{m}^{(n)}, \\boldsymbol{\\Gamma}^{(n)}\\}$ where each is $512 \\times 512 = 262,144$ dimensional, this is substantial.\n\n4. **Provide statistical significance analysis:**\n   \n   Given the variance shown in error bars (e.g., Figure 3, WHU at high noise), please report p-values or confidence intervals for claims of improvement.\n\n5. **Ablation on correlation structure:**\n   \n   Compare KMS against:\n   - Independent noise ($\\rho=0$) — already shown\n   - Alternative kernels (e.g., Matérn, RBF approximated by low-rank)\n   - Boundary-aware correlations (e.g., higher correlation along predicted edges)\n\n6. **Analysis of failure cases:**\n   \n   The clavicle results show spatial correlation provides minimal benefit. Please analyze:\n   - Is this because the structure is elongated (violating isotropy)?\n   - Or because boundary noise dominates interior noise?\n   - Provide guidance on when practitioners should expect the method to help.\n\n7. **Extension to non-stationary correlations:**\n   \n   Discuss how the framework could accommodate spatially-varying $\\rho(i,j)$, which would better model real annotation patterns where uncertainty varies (e.g., higher at boundaries).\n\n8. **Clarify the connection to MRF/CRF literature:**\n   \n   The sparse inverse $\\boldsymbol{\\Sigma}^{-1}$ implies a Gaussian MRF structure. The paper mentions this briefly but could strengthen the theoretical contribution by connecting to classical results on Gaussian MRFs and explaining how ECCD differs from standard CRF approaches for segmentation.", "broader_impact_concerns": "The authors provide a thoughtful broader impact statement addressing:\n- Automation bias in medical diagnosis\n- Potential for surveillance applications in remote sensing\n- Limitations when model assumptions don't hold\n\n**No additional concerns required.** The statement appropriately identifies risks and emphasizes the tool should assist, not replace, human experts. The acknowledgment of assumption-dependent performance is appropriate.\n\nOne minor addition could address: implications for annotation labor practices if methods that tolerate noise become standard, potentially reducing incentives for quality annotation infrastructure.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)