Fourier Clouds: Fast Bias Correction for Imbalanced Semi-Supervised Learning

Regarding Weaknesses

1. On Batch-Averaged Amplitude Spectrum Potentially Inheriting Majority-Class Bias (W1 & Q1)

This is a deep technical question that gets to the core of our method. We argue that having the reference image's amplitude spectrum inherit the statistical properties of the training data is not a flaw, but a crucial and deliberate design choice that is key to its effectiveness.

1. Unifying the Source of Bias and its Measurement: A classifier's bias is learned from the biased statistics of the training distribution itself. To accurately measure this learned bias, the "probe" (our reference image) must share the same statistical properties as the data that created the bias. Using a "balanced" reference to measure a bias learned from an "imbalanced" distribution would create a mismatch, leading to an inaccurate and misleading bias estimate.

2. Why a "Balanced Average" is Sub-optimal: If we were to use a balanced average (e.g., by re-sampling classes), we would create a reference image whose statistics do not match the distribution the model was actually trained on. Probing the model with a statistical distribution it is "unfamiliar" with would weaken the accuracy of the bias correction.

Our experiments in Appendix D.2, which cover severe distribution mismatch scenarios, provide strong empirical support for this design. The method remains robust even when the batch statistics differ greatly from the labeled set, proving that our adaptive estimation of bias is effective.

2. On the Handling of High-Frequency Components and its Technical Meaning (W2 & Q2)

Thank you for asking for clarification on this key technical detail from lines 271-273 of our paper, where we state that we do not discard high-frequency components.

1. Technical Meaning: In image signals, high-frequency components correspond to fine textures, sharp edges, and intricate details. These details are often critical for accurate classification in tasks ranging from natural images to medical imaging.
2. Impact on Our Method: If we were to discard high-frequency components (e.g., via a low-pass filter), our reference image would become blurry and lose the rich textures present in real images. This would make it impossible to measure the classifier's bias in response to these critical high-frequency features. By retaining the complete amplitude spectrum, our reference image is statistically realistic across all frequencies. This ensures our bias estimation is comprehensive, accounting for the model's response to both coarse structures (low frequencies) and fine details (high frequencies), leading to a more accurate overall correction.

3. On Why the Improvement is More Significant on ReMixMatch (W3 & Q3)

This is a very keen observation. Our analysis indicates this phenomenon is primarily due to the internal mechanics of ReMixMatch and its particular vulnerability in long-tailed settings.

• ReMixMatch's "Distribution Alignment": A core feature of ReMixMatch is that it forces the distribution of its pseudo-labels to match the (imbalanced) distribution of the labeled data.
• Negative Effect under Distribution Mismatch: As shown by prior work and our results (Table 9), this mechanism has a significant negative impact when the labeled and unlabeled class distributions are severely mismatched. It warps the pseudo-labels towards an incorrect prior, making ReMixMatch a relatively "weaker" baseline in these challenging scenarios.
• FARAD's "Rescue" Effect: FixMatch lacks this distribution alignment module and is therefore more stable. FARAD's direct and robust debiasing mechanism effectively "rescues" the struggling ReMixMatch, leading to a more dramatic relative performance gain. While the absolute improvement on the more stable FixMatch is also large, the relative increase is smaller.

Regarding Questions

4. On Pseudo-Labeling Design Choices and Thresholding Standards

Our design choices, detailed in Appendix B (lines 1048-1051), were made to ensure a fair comparison with our primary baseline, CDMAD, and to adhere to the principles of long-tailed learning.

Specifically, we set the confidence threshold τ=0 (using all pseudo-labels) because, under severe class imbalance, any fixed high-confidence threshold would inevitably filter out nearly all samples from tail classes. This would defeat the purpose of using unlabeled data to boost minority class performance.

The core philosophy of our method is not to filter uncertain samples via thresholding, but to improve the quality of all logits (especially for minority classes) through bias correction, making them more reliable targets for consistency regularization.

5. On the Missing ReMixMatch+FARAD Results in Table 2 (W3 & Q5)

We sincerely apologize for omitting the ReMixMatch+FARAD results in Table 2 (CIFAR-100-LT). This was an oversight during the preparation of our results.

We will add this missing data to the final version of the paper to ensure the comparison is complete. For your reference, the full results on the ReMixMatch baseline are provided below.

Supplementary Data: ReMixMatch+FARAD on CIFAR-100-LT (bACC %)

These results are fully consistent with our paper's overall conclusions: FARAD provides a significant and stable performance improvement over CDMAD, regardless of the base SSL algorithm, especially on the more challenging CIFAR-100-LT dataset. Thank you for pointing out this omission.

Thank you for your message and for your support. We truly appreciate your engagement throughout this process. Please feel free to contact us if any further questions arise; we will be happy to provide a prompt response.

Regarding Weaknesses

1. On the Vague Loss Integration (W1/Q1)

We apologize for the inconsistency and lack of clarity in describing the loss integration. The reviewer is correct: our method directly corrects the logits before they are passed to the loss functions, and this correction is applied to both the supervised and unsupervised losses.

Implementation Details: Listing 3 shows the exact implementation. For the unsupervised loss, we first compute the original logits for an unlabeled sample, subtract the bias logits to get debiased logits, and then apply softmax to these debiased logits to generate higher-quality pseudo-labels for the consistency loss. For the supervised loss, we similarly use the debiased logits as the direct input to the cross-entropy function.

Correction to Algorithm 1: We acknowledge that Algorithm 1, as a high-level overview, was not precise and caused this confusion. We will thoroughly revise Algorithm 1 in the final version to be perfectly consistent with the logic in Listing 3, clearly showing how the debiased logits g* are used for both L_sup and L_unsup.

We hope this clarification and the planned correction fully resolve the reviewer's concern.

2. On the Transductive Test-Time Bias Estimation (W2/Q2)

We sincerely thank the reviewer for these crucial and insightful questions. We completely agree that a strict inductive setting is vital for fair comparison, and we appreciate the opportunity to clarify our analysis and provide a more rigorous presentation. We will address every point thoroughly.

1. Analysis of a Per-Sample Inductive Strategy and its Trade-offs

You asked if we could explore a per-sample test-time reference. We did investigate this strategy, and our findings are presented in Appendix Table 14 ("Single-Image Amp").

• Performance: This approach proved to be suboptimal. Using a single image's statistics to generate its own reference introduced too much noise for a stable bias estimation, resulting in significantly lower performance than our other inductive strategies.
• Inference Time: A per-sample execution is also the least efficient, as it cannot leverage the amortization benefits of batched FFT computations, adding a non-trivial latency to each inference call.

2. Practical Scenarios for Batch-Based Test-Time Estimation

You also asked about realistic scenarios where test-time batches are available. Such scenarios are common in many real-world applications. For instance, in medical imaging, it is standard practice to analyze a batch of image slices from a single patient's CT or MRI scan. Our batch-based strategy was designed with these practical applications in mind.

However, we fully agree this does not justify its use for standard benchmark comparisons, which must adhere to a strict inductive protocol.

3. A Rigorous Inductive Evaluation Demonstrating FARAD's Core Contribution

To fully address your core concern, we now present a comprehensive analysis of our method under a strict inductive protocol. This new ablation across different imbalance ratios not only resolves the fairness issue but also shows that our method's advantages become even more pronounced as the problem difficulty increases.

Ablation: Inductive Strategies across Different Imbalance Ratios (CIFAR-10-LT)

This analysis yields two powerful takeaways:

• The core value of our training-phase correction is amplified by difficulty. The performance gain from our training-only correction grows from a significant +3.7% in the standard setting to a massive +5.0% in the extreme setting.
• The superiority of our FARAD reference design becomes more pronounced in challenging scenarios. In a fair inductive test, the performance gap between using our proposed FARAD Reference and a simple Blank Image widens as imbalance increases (from a 0.4% advantage at γ=100 to a 0.6% advantage at γ=200).

Commitment for Manuscript Revision

Based on your invaluable feedback, we will make the following major revisions:

1. All main results will be replaced with results from the strict inductive protocol (using the Offline FARAD Reference).
2. The original transductive results will be moved to the appendix and explicitly framed as a secondary analysis.
3. This new, comprehensive ablation table will be added to the paper to transparently demonstrate our method's value under a fair evaluation protocol.

We are confident these changes fully address your concerns and more accurately highlight the robust contributions of our work. Thank you again.

3. On Incomplete/Incorrect Related Work and Citations (W3/Q3)

We sincerely apologize for the errors and omissions in our related work section and thank the reviewer for the careful proofreading. We will make the following corrections in the final version:

• W3.1 (CDMAD Citation): We apologize for this significant citation error. The correct citation for CDMAD is indeed [34], and we will correct this throughout the manuscript.
• W3.2 (FixMatch/ReMixMatch Context): The reviewer's criticism is correct. We will revise our wording to clarify that we are discussing the limitations of these methods when applied to long-tailed scenarios.
• W3.3 (Related Work Discussion): We will expand the related work section to include a more detailed discussion of the important works mentioned [32,41,43,52,56] and will better contextualize methods like ABC/RECD to highlight the uniqueness of our reference-image-based paradigm.

Regarding Questions

4. On Sensitivity to Test Batch Distribution

This is a great question. Our method mitigates sensitivity to the specific composition of a test batch through two key designs:

1. Batch Averaging: By averaging the amplitude spectrum across the entire batch, we get a smoothed, statistically stable reference that is robust to the influence of a few outlier or specific-class images.
2. Statistics vs. Semantics: The amplitude spectrum primarily encodes low-level statistical properties (e.g., textures), not high-level semantic class information. Critically, the phase randomization step is what destroys the semantics, and its effect is overwhelming and independent of the input distribution.

Our experiments in Appendix D.2 (Tables 9 and 10) provide strong empirical proof of this robustness. The method performs well even under extreme scenarios where the training and test distributions are severely mismatched. If the method were highly sensitive to the batch composition, its performance would have collapsed in these settings, which it did not.

5. On Showing Average Class Probabilities (Empirical Marginal Distribution)

This is an excellent suggestion for visually demonstrating our method's debiasing effect. We will add this analysis to the final paper. Since we cannot include new figures in this rebuttal, we present the core data in the table below. This table shows the empirical marginal distribution of predictions on a balanced CIFAR-10 test set after training on a long-tailed set ($\gamma=100$).

Table D: Predicted Empirical Marginal Distribution on Balanced CIFAR-10 Test Set

As the table shows, the FixMatch baseline's predictions are heavily skewed towards the majority classes. In contrast, FARAD's prediction distribution is significantly flatter and much closer to the ideal uniform distribution. This provides direct numerical evidence that our method successfully balances the model's predictions.

Regarding the Discussion of Limitations and Broader Impact

Thank you for pointing out the discrepancy between our checklist and the main text. The reviewer is correct that we omitted a discussion of broader impact in Section 4, and we apologize for this oversight.

We will add a dedicated "Broader Impact" paragraph to our Conclusion (Section 4) in the final version. As promised in the checklist, this section will discuss the positive societal impacts of our work (e.g., improving AI fairness on underrepresented data like rare diseases or minority groups) as well as potential risks.

Thank you for showing experiments under a strict inductive protocol. I have a few follow-up comments:

The ablation table now includes:

-FARAD (Train-Only)

-Test w/ FARAD Ref. (Offline) – this appears to apply debiasing only at inference time, without the training-phase debiasing, as described in Table 18 of the appendix.

Although showing superiority compared to a simple baseline, both of these ablation results are substantially weaker than the full FARAD results reported in the main paper (FixMatch: 88.6 ± 0.47; ReMixMatch: 88.4 ± 0.44) under CIFAR-10-LT-100.

• In light of commitment #1 for the next manuscript iteration, it would be crucial to evaluate the compatibility for the full inductive method, i.e., FARAD training + Offline Ref, across the established settings γ = 50, 100, 150, and how it compares with other relevant approaches, e.g. CDMAD.

• Also, which SSL baseline did you use for these ablations?

Regarding Weaknesses

1. On the Connection of the Title "Fourier Clouds" to the Content

We agree that the term "Fourier Clouds" could be more clearly explained in the manuscript. We chose this title as a concise and descriptive summary of our method's core.

• "Fourier" refers to the core technique we use: the Fourier transform.
• "Clouds" comes from the visual appearance of our generated random-phase images. As we state on lines 141 and 1106 of the paper, these images visually resemble "turbulent or cloud-like patterns."

In the final version of the paper, we will add a visual example (as mentioned in our response to Weakness 3) to make this "cloud-like" appearance clear. We will also add a sentence to the introduction to explicitly explain the origin of the term.

2. On the Conceptual Contribution Being an Incremental Improvement over CDMAD

We believe FARAD represents a fundamental paradigm shift, not an incremental improvement over CDMAD.

1. Problem Reframing: From "Zero-Information" to "Optimal-Information". CDMAD's paradigm asks, "How can we measure bias with a zero-information reference?" Our work asks a deeper question: "What is the optimal information content for a reference image to most accurately measure classifier bias?" Our answer—semantically blank but statistically representative—marks a shift from eliminating all information to preserving only the critical statistical information.

2. Theoretical Grounding: Addressing the Core Bias of CNNs. It is well-established that CNNs have a strong "texture bias." CDMAD's blank reference completely ignores this. FARAD is the first reference-image debiasing method to directly address this. By preserving the amplitude spectrum, which encodes texture and frequency statistics, our method precisely probes and corrects this core bias inherent to CNNs.

3. Validation Through Performance: The consistent and significant ~3% performance gain across all scenarios validates our new conceptual framework. A simple incremental change would struggle to explain such a robust and universal advantage.

Therefore, FARAD's contribution is a new, more theoretically profound, and practically effective paradigm for bias estimation, not just a simple replacement of the reference image.

3. On the Lack of Visual Examples for Random-Phase Images

This is a very pertinent criticism. We agree that a visual example is crucial for understanding our method, and we apologize for this oversight.

We will add a high-quality comparative figure to the final version of the paper. This figure will show (a) an original image, (b) the blank reference image used by CDMAD, and (c) our generated "Fourier Cloud." This will provide valuable intuition and support our core claims of "semantic-agnosticism" and "statistical-representativeness."

4. On Missing Important Literature in the Related Work Section

Thank you for pointing out these important related works. We will expand our literature review in Appendix A of the final version to discuss these four papers and clarify the connections and differences.

• vs. DASO: DASO focuses on optimizing the pseudo-labeling process. Our method, FARAD, offers a more upstream correction by debiasing the model's raw logits before any pseudo-labels are generated. FARAD addresses the root cause of biased pseudo-labels, making it complementary to methods like DASO.
• vs. ACR (Towards Realistic...): ACR is an elegant adaptive method that estimates the unlabeled data's true class distribution. In contrast, FARAD is an estimation-free approach. We probe the classifier's internal bias directly with a statistically representative reference, avoiding the need to explicitly estimate the target distribution, which may make our method more robust when distribution estimation is difficult.
• vs. Twice Class Bias Correction (TCBC): TCBC uses a complex, two-stage correction based on distribution estimation. FARAD provides a simpler, unified mechanism that corrects based on the classifier's internal learned state, rather than relying on an estimate of the external data distribution.
• vs. InPL: InPL uses energy scores to select reliable "in-distribution" samples for pseudo-labeling. FARAD's contribution is orthogonal and complementary. We focus on improving the logit quality of all samples, not on selecting which samples to use. In principle, FARAD's logit debiasing could be combined with InPL's sample selection to achieve further gains.

Regarding Questions

1. On the Method's Applicability in a Fully-Supervised Long-Tailed Setting

This is an excellent question about our method's versatility. FARAD's bias correction mechanism is general and can be seamlessly applied to fully-supervised long-tailed learning.

The application is straightforward: during standard supervised training, for each batch of labeled data, we generate a reference image, compute the bias logits, and subtract them from the original logits before calculating the standard cross-entropy loss. This actively counteracts the model's tendency to accumulate bias towards majority classes.

To provide empirical evidence, we conducted a new experiment on CIFAR-100-LT ($\gamma=100$):

Table C: Performance of FARAD in Fully-Supervised Long-Tailed Learning (Accuracy %)

The results show that FARAD not only significantly outperforms the baseline but also surpasses the strong and classic Logit Adjustment method, demonstrating that it is a powerful and general bias correction technique.

2. On the Uncertainty of Test-Time Predictions

This is a key question for practical deployment. Our default strategy of dynamically generating a reference image for each test batch does introduce prediction uncertainty, which is designed to adapt to the statistics of the current batch.

However, we recognize the importance of deterministic predictions. We have already evaluated a solution for this in Appendix D.5 and Table 18. The "Offline Fixed Reference" strategy provides a perfect solution:

• It uses a single, pre-computed, fixed reference image (based on the entire training set's statistics) for all test samples.
• As shown in Table 18, this approach ensures fully deterministic predictions with only a negligible drop in performance (e.g., 82.6% vs. 83.6% bACC), while still far outperforming all baselines.

Users can therefore choose the inference strategy that best fits their deployment needs.

3. On the Missing Baseline Results in Table 1

We apologize for the missing entries in Table 1 and thank the reviewer for pointing this out.

Our experimental setup strictly follows that of our main baseline, CDMAD, to ensure a fair comparison. The entries that are missing in our table are also missing in the original CDMAD paper. Our own attempts to reproduce these specific baselines did not yield stable and reliable results.

To maintain the rigor of our comparison, we decided to only report results that we could either directly cite from CDMAD or successfully reproduce under the exact same experimental conditions. We assure the reviewer that all presented data was obtained under identical settings, ensuring a fair comparison to our method. We will add a footnote to the final version to clarify this.

Regarding Weakness 1: Robustness to Extreme Domain Shift

We agree that robustness under extreme domain shift is a key measure of generalization. To validate FARAD's performance in such challenging scenarios, we conducted a dedicated experiment in Appendix D.8 of our paper. In this experiment, we created a harsh setting that combines both class distribution mismatch and domain feature shift: we used the color version of CIFAR-10-LT ($\gamma_l=100$) as the labeled dataset and a grayscale version with a different imbalance ratio ($\gamma_u=50$) as the unlabeled dataset.

As shown in Table 23 of the appendix, while all methods' performance decreased due to the domain shift, our FARAD (72.7% bACC) still significantly outperformed CDMAD (70.2% bACC) and the FixMatch baseline (68.4% bACC). This result provides strong evidence of FARAD's robustness. The underlying reason is that even when the image color space changes, the frequency energy distribution (i.e., the amplitude spectrum) corresponding to intrinsic textures and structures is largely preserved. FARAD captures these more fundamental statistics to estimate bias, making it more robust than methods like CDMAD that rely on simpler color-based cues.

Regarding Weakness 2: Lack of Training and Inference Time Comparison

We thank the reviewer for this important question regarding computational efficiency. We acknowledge this comparison was missing from the original manuscript. To address this, we have conducted a new experiment and will add the following detailed time comparison to the final version of our paper.

Table A: Training and Inference Time on CIFAR-10-LT (single NVIDIA A6000)

As shown in Table A, while a naive FFT implementation introduces overhead, our batch-wise Real-to-Complex (R2C) FFT optimization (detailed in Section 2.6 and Figure 2) is highly effective. With this optimization, FARAD adds only about 6.2% to the training time per epoch. We believe this minor cost is an excellent trade-off for the consistent ~3% bACC performance improvement across all imbalance scenarios. This result demonstrates that our method achieves significant gains while maintaining high scalability.

Regarding Weakness 3: Lack of Evaluation on Other Datasets like Medical Images

This is an excellent forward-looking suggestion. To address the concern about generalization to different domains, we have followed the reviewer's advice and conducted a new experiment on a real-world, challenging, long-tailed medical imaging dataset: NIH ChestX-ray14. This dataset is well-known for its natural and severe class imbalance, making it an ideal testbed for long-tail learning algorithms.

We constructed a semi-supervised long-tailed classification task on this dataset. The results are as follows:

Table B: Performance on Long-Tailed NIH ChestX-ray (Mean AUC)

The results show that FARAD not only outperforms the baseline but also achieves a 1.8% improvement over CDMAD. This provides strong evidence that FARAD's effectiveness is not limited to natural images. This advantage stems from the nature of medical imaging, where diagnoses often rely on textures, tissue morphology, and frequency patterns. By preserving the amplitude spectrum, FARAD generates reference images that are statistically consistent with real pathological textures, allowing it to more accurately probe and correct the model's bias towards common diseases (majority classes). We will add this experiment to our final version to fully demonstrate our method's generalization capabilities.

Regarding Weakness 4: Dependence on GPU FFT Libraries

Thank you for this practical question about deployment. We acknowledge that our efficient implementation benefits from modern GPU-accelerated libraries, which are standard for deep learning training.

However, for resource-constrained inference scenarios, we have already evaluated a solution in Appendix D.5 (Table 18). The "Offline Fixed Reference" strategy addresses this perfectly:

1. After training, we pre-generate one or more fixed reference images from the training set's overall statistics.
2. During inference, we use these static images directly, requiring no FFT calculations on the deployment device.

As shown in Table 18, this approach maintains a strong performance (e.g., 82.6% bACC vs. 83.6% for the online version), still significantly outperforming all baselines while completely removing the dependency on real-time FFT computation. This provides a practical and efficient pathway for deploying FARAD across diverse hardware environments.