Learning Causal Transition Matrix for Instance-dependent Label Noise

Jiahui Li; Jun Zhou; Kun Kuang; Long Chen; Tai-Wei Chang; Ximing Li

REVIEW 4 major objections 4 minor 54 references

Learning Causal Transition Matrix for Instance-dependent Label Noise

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Instance-dependent label noise becomes identifiable when each input is split into a noise-resistant component and a noise-sensitive component and the latter is recovered.

desk verdict New causal graph and training framework for instance-dependent label noise, but the central identifiability proof is invalid because conditioning on the collider X2 opens a backdoor path. read the letter →

arxiv 2412.13516 v4 pith:JHGTMQ5P submitted 2024-12-18 cs.LG

Jiahui Li , Tai-Wei Chang , Kun Kuang , Ximing Li , Long Chen , Jun Zhou This is my paper · ORCID

classification cs.LG

keywords causaltransitionmatrixinstance-dependentlabelnoiselabel-noiselearninggraphidentifiabilitydo-calculusnoise-resistantcomponentpolicygradient

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Noisy labels are usually handled by assuming the probability of a wrong label does not depend on the instance, an assumption that fails in practice. This paper replaces that assumption with a causal story: an unobservable cause $Z$ can corrupt the image, the annotator, or both, so each input $X$ is split into a noise-resistant component $X_1$ that drives the clean label and a noise-sensitive component $X_2$ that drives the noise. The central claim is that the ``causal transition matrix'' $P(\hat{Y}\mid do(Y),X)$, the noisy-label distribution after intervening on the clean label, is identifiable once $X_2$ is recovered, and it approximates the true instance-dependent transition matrix. A training framework realizes the graph with co-teaching, a decorrelation penalty, and a policy-gradient model for $X_2$. The paper itself notes that its main limitation is that the clean label is only partially observable through confidence sampling, which can strain the method at very high noise rates.

What carries the argument

The central object is the causal transition matrix $T_{cau}=P(\hat{Y}\mid do(Y),X)$: the distribution of the noisy label after an intervention forces the clean label, conditioning on the instance. The paper's graph gives it a two-part structure; $do(Y)$ removes the incoming edges of $Y$, so $X_1$, the noise-resistant parent of $Y$, drops out of the transition, while conditioning on $X_2$, the noise-sensitive component, converts the intervention distribution into the observational one $P(\hat{Y}\mid Y,X_2)$. The proof machinery is d-separation in the manipulated graphs $G_{\bar{Y}}$ and $G_Y$, using the backdoor criterion. The training machinery is a separation model that extracts $X_1$, a policy model $\pi$ that outputs $X_2$, a transition model $f_{tran}$ with Gumbel-Softmax merging, co-teaching to make $Y$ partially observable, and a policy-gradient reward that pushes $X_2$ to reduce prediction error on noisy labels.

What would settle it

Build a synthetic dataset with known clean component $X_1$ and known noise component $X_2$, generate $Y$ from $X_1$ and $\hat{Y}$ from $(Y,X_2)$ with a known transition, then train the framework without revealing $X_2$. If the estimated $P(\hat{Y}\mid do(Y),X)$ deviates systematically from the known $P(\hat{Y}\mid Y,X_2)$, or if the learned $X_2$ fails the conditional independence $\hat{Y}\perp X_1\mid Y,X_2$ on held-out data, the central identifiability claim would be falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the transition matrix does not have to be assumed instance-independent if the data generation is modeled causally. In the proposed graph, an unobservable variable $Z$ affects one or both of the input $X$ and the noisy label $\hat{Y}$; $X$ is therefore separated into $X_1$, which is independent of $Z$ and determines $Y$, and $X_2$, which interacts with $Z$ and $Y$ to produce $\hat{Y}$. Theorem 1 states that $P(\hat{Y}\mid do(Y),X)=P(\hat{Y}\mid do(Y),X_2)=P(\hat{Y}\mid Y,X_2)$, so the instance-dependent causal transition matrix is identifiable whenever the noise predictive factor $X_2$ is recovered. Theorem 2 states that $P(Y\mid do(X_1))=P(Y\mid X_1)$, so a classifier built on $X_1$ alone is unbiased. These two results together say that learning with instance-dependent label noise can be reduced to recovering the two components of $X$, and the paper proposes an end-to-end framework that does this by treating $X_2$ as the action of a policy optimized by reward.

Load-bearing premise

The load-bearing premise is that each observed instance really splits into a noise-resistant part and a noise-sensitive part with the exact independencies drawn in the paper's causal graph, and that the noise-sensitive part can actually be recovered from the noisy data; if either condition fails, the causal transition matrix is not identifiable.

Editorial extensions

If this is right

Instance-dependent label noise can be handled without the instance-independent assumption $P(\hat{Y}\mid Y,X)=P(\hat{Y}\mid Y)$, as long as the noise-sensitive component of the input can be learned.
A clean classifier can be trained from the noise-resistant component alone, because $P(Y\mid do(X_1))=P(Y\mid X_1)$ removes the confounding effect of the latent noise variable.
The causal graph unifies three previously separate noise scenarios, since $Z$ may affect the instance, the annotation, or both, and all three are captured by the same transition model.
The same component-splitting idea transfers to semi-supervised denoising pipelines, where the paper's adaptation improves or matches a strong baseline on the CIFAR-N datasets.

Reading between the lines

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

The identifiability guarantee is a reduction: it says the causal transition matrix is identified once $X_2$ is recovered, but the paper gives no separate proof that the fitted policy network recovers the true $X_2$; we would read Theorem 1 as conditional on that recovery.
The authors' own ablation without the policy model remains competitive in most settings, which suggests to us that much of the accuracy gain may come from learning a noise-resistant $X_1$ rather than from the estimated causal transition matrix; an experiment with known $X_2$ would separate the two effects.
If the reduction view is right, instance-dependent label-noise learning becomes a missing-component problem, and the same causal split could be tried for other weakly supervised settings such as partial labels, crowd-sourced annotations, or noisy medical labels, though the paper does not test those.
A skeptical check would be to verify the key conditional independence $\hat{Y}\perp X_1\mid Y,X_2$ on the learned components; in the paper's own graph this independence is what makes Theorem 1 go through.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper's main idea is to split the instance X into a noise-resistant part X1 and a noise-sensitive part X2, and define a causal transition matrix P(Ŷ|do(Y),X) that is claimed to be identifiable once X2 is recovered. That graph and definition are genuinely new relative to CausalNL and NPC, and the end-to-end training framework avoids VAEs, which is a nice practical simplification. The synthetic experiments are broad, report standard deviations, and the ablation 'w.o/pg' suggests the separation model alone helps.

The problem is the proof of Theorem 1. In G_Ȳ the paper asserts Ŷ ⊥ X1 | Y, X2, but in the very graph they propose, X2 is a collider on the path X1 ← X → X2 ← Z → Ŷ. Conditioning on X2 opens that path, so the independence is false. The second independence, Ŷ ⊥ Y | X2 in G_Y, fails the same way via Y ← X1 ← X → X2 ← Z → Ŷ. Therefore the equalities P(Ŷ|do(Y),X)=P(Ŷ|do(Y),X2) and P(Ŷ|do(Y),X2)=P(Ŷ|Y,X2) are not justified. The theorem is unproven, and with it the paper's central claim of identifiable instance-dependent transition matrices.

A secondary issue: Theorem 1 is conditional on recovering X2, but the paper gives no identifiability result for X2; X2 is learned by a policy-gradient objective that minimizes the same cross-entropy, so the guarantee is at best conditional on the fitted network. The real-world results are reported as best of last 10 epochs without error bars, and code is not released, so the empirical evidence is weaker than the synthetic tables suggest.

The framework may still work as a heuristic—the ablation numbers are surprisingly strong—but the theory section does not support what the abstract promises. This paper deserves a serious referee because the problem is important and the causal graph is a real attempt to relax instance independence, but the referee should be told to check Theorem 1 carefully. I would not expect acceptance as is; a corrected proof, or an honest reframing that drops the identifiability claim, would be needed.

Referee Report

4 major / 4 minor

Summary. The paper proposes a causal-graph view of instance-dependent label noise in which the observed instance X is decomposed into a noise-resistant component X1 and a noise-sensitive component X2, with an unobserved latent Z affecting X2 and the noisy label Y-hat. On this basis it defines a 'causal transition matrix' P(Y-hat | do(Y), X), claims in Theorem 1 that this object is identifiable once X2 is recovered, and designs an end-to-end training framework combining co-teaching, a transition model, a decorrelation regularizer, and a policy-gradient objective for X2. Experiments on FashionMNIST, SVHN, CIFAR-10/100, Food101, and Clothing1M report competitive accuracy. The main theoretical claim, however, is not supported: the d-separation steps in the proof of Theorem 1 fail under the paper's own graph, and the recovery of X2 is not established as an identification result.

Significance. If the identifiability claim were valid, the paper would offer a meaningful relaxation of the instance-independent transition-matrix assumption, and the proposed framework is a plausible engineering contribution: it is end-to-end trainable, avoids generative models, and the experiments are extensive, including real-world noise and an ablation. The paper also deserves credit for explicitly modeling three noise-generation scenarios in one graph and for discussing limitations. Unfortunately, the central theoretical guarantee is unsound: the proof of Theorem 1 misapplies d-separation, and the 'identifiability' is conditional on a learned representation whose recovery is not guaranteed. The empirical results, while often strong, do not compensate for the absence of a valid identifiability theorem, so the paper's headline contribution is not established.

major comments (4)

[Causal Viewpoint for Denoising, Theorem 1 proof] The proof asserts two conditional independences that are false in the graph of Figure 2(a). First, it claims Ŷ ⊥ X1 | Y, X2 in G_Ȳ. But the path X1 ← X → X2 ← Z → Ŷ contains the collider X2, and conditioning on X2 opens this path, so X1 and Ŷ are d-connected given (Y, X2); conditioning on Y does not block the path. Second, it claims Ŷ ⊥ Y | X2 in G_Y, and the same collider path Y ← X1 ← X → X2 ← Z → Ŷ connects Y to Ŷ once X2 is conditioned on. Therefore the equalities P(Ŷ|do(Y),X) = P(Ŷ|do(Y),X2) and P(Ŷ|do(Y),X2) = P(Ŷ|Y,X2) are not justified by the graph, and Theorem 1 is unproven.
[Theorem 1 statement and Section 'Training Framework for Denoising'] The theorem is conditional on 'recovering' the noise predictive factor X2, but no identifiability result for X2 from the observed distribution (X, Ŷ) is provided. In the framework, X2 is the output of a policy network g2 trained by the policy-gradient loss of Eq. (7), whose reward is computed from the same cross-entropy LCE used by the transition model. Any representation that minimizes that cross-entropy could serve as X2, and the paper gives no argument that the optimized g2(X) coincides with the true noise-sensitive component of the proposed causal graph. The claimed identification is therefore a property of the fitted network, not of the data-generating process.
[Eq. (3), 'Training Framework for Denoising'] The decorrelation loss is written as Reg(X1) = −Σ_i \tilde{y}_{x1} log softmax(1_k). Since 1_k is the all-one vector, softmax(1_k) is the uniform distribution and log softmax(1_k) is a constant vector that does not depend on the model parameters. As written, the loss does not push the transition model's output toward a uniform distribution over class labels; it merely penalizes the sum of the output logits (or is constant if \tilde{y}_{x1} is already normalized). If the intended objective is a cross-entropy between the model's prediction and the uniform distribution, the first argument should be softmax(\tilde{y}_{x1}), not the raw output. As published, Eq. (3) does not implement the decorrelation between X1 and Ŷ that the causal-graph assumptions require.
[Causal Viewpoint for Denoising, definition of T_cau] The paper motivates T_cau = P(Ŷ|do(Y),X) as an approximation of the true transition matrix P(Ŷ|Y,X), but no statement or proof connects the interventional distribution to the observational transition matrix used in loss correction. Intervening on Y removes incoming edges of Y, whereas the transition matrix is a conditional distribution given Y and X; equality or a bounded approximation error between T_cau and T(X) is never shown. Even if Theorem 1 were correct, it would identify a do-probability, and the paper would still need an argument that training with T_cau yields consistent clean-label inference.

minor comments (4)

[Table 2] The table header reads 'Results on SVNH'; the dataset name should be 'SVHN'.
[Conclusion] The sentence 'we approa the problem from a causal viewpoint' is missing 'ch' in 'approach'.
[Analysis of the Framework] The paragraph beginning 'What our framework does?' refers to 'Theories 1 and 2'; the standard terminology is 'Theorems 1 and 2'.
[Table 1] The ablation 'Ours w.o/ pg' outperforms the full method on SYM-80 (75.8 vs 71.5) and IDN-20 (91.3 vs 91.0); the text's claim that the ablation achieves the best performance in the symmetric and asymmetric scenarios is not consistent with all entries in the table and should be qualified.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 1's identifiability guarantee is conditional on the policy-gradient 'recovery' of X2, which is the same noisy-label fit used to train the transition model, so the causal transition matrix prediction reduces to the training objective.

fitted input called prediction [Method: Causal Viewpoint for Denoising (Theorem 1); Training Framework, Policy model (Eqs. 6-7)]
"Theorem 1 The instance-dependent causal transition matrix P(Ŷ|do(Y), X) is identifiable if we recover the noise predictive factor X2. ... The transition model can be optimized using the following equation: LCE = −Σ ỹ log ŷ. (6) ... we use the policy gradient ... to obtain the noise-sensitive component X2. ... R = 1/(1−ỹ log ŷ). This reward function ensures that the reward is always greater than zero and increases as the cross-entropy decreases. ... The policy model is trained using the following loss function: Lpg = −Σ R log π(xi). (7)"

The identifiability guarantee is stated conditionally on 'recover[ing] the noise predictive factor X2.' In the framework, X2 is not obtained by an independent identification procedure; it is trained by Eq. (7), whose reward R = 1/(1−ỹ log ŷ) increases exactly as the transition model's cross-entropy to the observed noisy labels decreases. That same transition model is fit to the same noisy labels by Eq. (6), and its output is what the paper presents as the causal transition matrix P(Ŷ|do(Y),X). Thus the theorem's antecedent ('recover X2') is satisfied precisely by fitting the transition model to Ŷ, and the consequent (the causal transition matrix) is the same fitted model.

full rationale

The central circularity is in the conditional identifiability claim: Theorem 1 promises identification of the causal transition matrix if X2 is recovered, but X2 is operationalized as the representation that minimizes the same cross-entropy to the noisy labels used to fit the transition model. The guarantee is therefore conditional on the very fit it is supposed to justify, which matches the fitted-input-called-prediction pattern. Separately, the proof of Theorem 1 relies on two d-separation assertions (Ŷ ⊥⊥ X1 | Y, X2 in G_Ȳ and Ŷ ⊥⊥ Y | X2 in G_Y); in the proposed graph X2 is a collider on the path X1 ← X → X2 ← Z → Ŷ, so conditioning on X2 opens a path and the asserted independences are not supported by the graph. That is a correctness risk rather than a circularity, but it reinforces that the theorem does not provide an independent identifiability argument. The empirical benchmark results are external and could support the method's practical value, but they do not repair the circular conditional guarantee. No load-bearing self-citation chain is present; the score reflects the reduction of the theoretical claim to the training objective, not any citation practice.

Assumptions & free parameters 4 free parameters · 4 assumptions · 3 invented entities

The central claim rests on an assumed causal graph with a latent confounder Z, an assumed decomposition of X into X1 and X2, and an assumed partial observability of Y via co-teaching. The only free parameters reported are the loss weights and the merging scale β, but the architecture choices and the unspecified co-teaching schedule also influence results. None of the invented entities has independent evidence outside the paper.

free parameters (4)

α1 = 0.1
Weight for LCE in Eq. (8); chosen by hand.
α2 = 0.1 (0.01 for perturbed datasets)
Weight for policy gradient loss; text gives 0.01 for perturbed instances but Table 5 reports 0.1, an inconsistency.
α3 = 0.1 (synthetic and Food101), 0.01 (CIFAR-100N in semi-supervised)
Weight for decorrelation loss; varies across settings.
β = 0.2
Scaling of X2 in the merging function m(Y,X2)=gs(Y+β*X2), Eq. (5).

assumptions (4)

ad hoc to paper A latent variable Z exists and influences X2 and Ŷ as in Figure 2(a).
Assumed to explain the noise process; no evidence or falsifiable prediction is provided for Z.
ad hoc to paper X can be separated into a noise-resistant component X1 and a noise-sensitive component X2 satisfying the conditional independences used in Theorems 1 and 2 (e.g., Ŷ ⊥ X1 | Y, X2, Y ⊥ X2 | X1).
This decomposition is enforced by the architecture and loss functions, not derived from data.
domain assumption Confidently sampled noisy labels are treated as clean labels Y (partial observability of Y via co-teaching).
Used in Eq. (1) and throughout the training framework; fails at high noise rates as the authors note in the Limitations.
standard math The backdoor criterion and do-calculus are validly applied to the proposed graph to derive P(Y|do(X1)) and P(Ŷ|do(Y),X).
The paper invokes Pearl (1995); but the graph may not satisfy the required d-separation conditions, so the derivation is not justified.

invented entities (3)

X1 (noise-resistant component)
purpose: Component of X that directly causes Y and is independent of Z; used to train a clean classifier.
Produced by a 1x1 conv g1(X); no external handle, its noise-resistance is imposed by the decorrelation loss.
X2 (noise-sensitive component)
purpose: Component of X that mediates noise to Ŷ; used to model the transition matrix.
Produced by a policy network g2(X) trained with policy gradient; no independent measurement.
Z (latent confounder)
purpose: Unobserved factor affecting X2 and Ŷ; justifies the do-operation.
No falsifiable prediction; the graph's utility rests on this latent variable but no evidence is given.

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Cite this review

Pith. "Pith review of Learning Causal Transition Matrix for Instance-dependent Label Noise." pith.science (2026). https://pith.science/paper/JHGTMQ5P

@misc{pith2026241213516,
  author       = {Pith},
  title        = {Pith review of: Learning Causal Transition Matrix for Instance-dependent Label Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JHGTMQ5P}},
  note         = {Machine review of arXiv:2412.13516}
}

read the original abstract

Noisy labels are both inevitable and problematic in machine learning methods, as they negatively impact models' generalization ability by causing overfitting. In the context of learning with noise, the transition matrix plays a crucial role in the design of statistically consistent algorithms. However, the transition matrix is often considered unidentifiable. One strand of methods typically addresses this problem by assuming that the transition matrix is instance-independent; that is, the probability of mislabeling a particular instance is not influenced by its characteristics or attributes. This assumption is clearly invalid in complex real-world scenarios. To better understand the transition relationship and relax this assumption, we propose to study the data generation process of noisy labels from a causal perspective. We discover that an unobservable latent variable can affect either the instance itself, the label annotation procedure, or both, which complicates the identification of the transition matrix. To address various scenarios, we have unified these observations within a new causal graph. In this graph, the input instance is divided into a noise-resistant component and a noise-sensitive component based on whether they are affected by the latent variable. These two components contribute to identifying the ``causal transition matrix'', which approximates the true transition matrix with theoretical guarantee. In line with this, we have designed a novel training framework that explicitly models this causal relationship and, as a result, achieves a more accurate model for inferring the clean label.

Figures

Figures reproduced from arXiv: 2412.13516 by the authors.

**Figure 2.** (a) The proposed causal graph for learning with noisy labels. (b) The training framework of our method. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** The design of the transition model. Modeling the observable contributing factors. Initially, we utilize a separation model to separate the noise-resistant component X1 from X, which can be represented as X1 = g1(X), where g1(·) denotes any neural network. Subsequently, we infer the clean label Y using a classifier Y = f(X1). As mentioned above, Y is unobservable in the causal graph, so it is necessary to sample con… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Model performance on the CIFAR10 Dataset. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Causal graphs for data generation process. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: (a) The ground truth transition matrix. (b) The transition matrix of our method. (c) The transition matrix of our [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: Samples and feature map of X1 in (a) CIFAR10 with 30% instance-dependent label noise (b) CIFAR10 with 30% instance-dependent label noise and perturbation of 0.6 intensity. 𝑋 Classifier Block Gradient Module Loss Function 𝑋! 𝑋" Transition Encoder Main Head 𝑌 Pseudo Head…

**Figure 8.** Figure 8: Denoising training framework in Semi-Supervised [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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