Geometric Logit Decoupling for Energy-Based Graph Out-of-distribution Detection

We sincerely thank you for the detailed, constructive, and thought-provoking feedback. We appreciate that you recognized our work's clear geometric motivation, simplicity, and strong empirical results. We address the raised concerns and questions below.

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Q1: Resilience to norm-targeted perturbations.

A1: Thank you for your comment. This is an excellent question that gets to the heart of GeoEnergy's robustness. Our method is inherently resilient to such perturbations precisely because it decouples the norm from the primary classification signal (direction).

• Mechanism: In our formulation $\text{logit} = s \cdot \|\mathbf{h}\| \cdot \cos(\theta)$, the directional alignment $\cos(\theta)$ acts as a "gate." If a perturbation artificially inflates an OOD sample's norm, but its direction remains incorrect ($\cos(\theta)$ is small or negative), the resulting logit will still be low. This prevents the norm-based noise from being misinterpreted as high confidence, which is a key failure mode in standard energy-based methods.
• Empirical Evidence: Our experiments on Cora-Feature OOD scenario (Table 1) implicitly test this. In this scenario, OOD node features are generated by interpolating between ID features, which can be seen as adding noise and perturbing both the norm and direction of embeddings. GeoEnergy consistently outperforms baselines in this setting (e.g., boosting GNNSafe's AUROC from 93.44% to 94.36%), demonstrating its resilience.

In essence, GeoEnergy "defangs" the norm: it can no longer unilaterally dictate the energy score and is only effective when coupled with a correct semantic direction.

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Q2 & L2 & W3: Reliance on empirical tuning of the scaling factor $s$.

A2: Thank you for raising this critical point about real-world applicability and the limitations of relying on a validation set, especially in cold-start or distribution shift scenarios. To address this, we provide a two-fold response: an analysis of the robustness of a fixed $s$, and a concrete proposal for an adaptive $s$ framework.

• Robustness of a Fixed $s$: Our sensitivity analysis (Figure 7, Appendix B1) shows that GeoEnergy's performance is stable better than baseline across a reasonably wide range of $s$. This suggests that even if a validation set is unreliable or unavailable, a well-chosen default $s$ (e.g., $s=15$) can still provide substantial improvements over baselines, as the fundamental benefit of decoupling is a structural advantage that persists.

• A Concrete Proposal for an Adaptive $s$ Framework: To fully address the challenge, we propose a data-dependent, adaptive $s$-predictor network, which we are actively exploring for future work. This framework is particularly suited for cold-start and distribution shift scenarios.

  • Framework Idea: The framework involves pre-training a lightweight regression model (the $s$-predictor) on a meta-dataset of diverse graphs. This predictor learns to map a graph's geometric statistics (e.g., intra-class angular variance, inter-class margin, embedding norm distribution) to its optimal $s$ value.
  • Addressing Cold-Start: Once pre-trained, this $s$-predictor can be deployed on a new graph. It can predict a reasonable $s$ by analyzing the new graph's embedding geometry without requiring any labeled validation data from this new graph.
  • Addressing Distribution Shifts: The input geometric statistics are sensitive to distribution shifts. The $s$-predictor can be applied to batches of test data to dynamically adjust $s$ on-the-fly, making the system robust to shifts where a static $s$ would fail.

We will add a detailed discussion of this proposal to the limitations and future work sections of our revised paper.

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L1& W1: Lack of formal theoretical analysis for OOD detection.

A3: Thanks for your valuable feedback. Our original propositions focused on the geometric properties beneficial for classification, but did not directly formalize the link to improved OOD detection. To address this, we have developed a new theoretical analysis that provides a more direct justification for why GeoEnergy enhances the separation between ID and OOD energy scores.

Proposition 3 (ID-OOD Energy Separation Guarantee).

Let $E(\mathbf{h}) = -\log(\sum\_{j=1}^{C} \exp(s \cdot \|\mathbf{h}\| \cdot \cos(\theta\_j)))$ be the energy function. Assume that for in-distribution samples $\mathbf{h} \sim \mathcal{P}\_{\text{ID}}$, their directions are well-aligned with their true class prototypes such that $\mathbb{E}[\cos(\theta\_y)] \approx 1$ for the correct class $y$. Further assume that for out-of-distribution samples $\mathbf{h}' \sim \mathcal{P}\_{\text{OOD}}$, their directions are unaligned, such that $\mathbb{E}[\cos(\theta'\_j)] \approx 0$ for all classes $j$. Then, the expected energy of ID samples is significantly lower than that of OOD samples, with the gap widening for higher-confidence ID samples:

$$\mathbb{E}\_{\mathbf{h} \sim \mathcal{P}\_{\text{ID}}}[E(\mathbf{h})] \approx -s \cdot \mathbb{E}[\|\mathbf{h}\|] \ll -\log(C) \approx \mathbb{E}\_{\mathbf{h}' \sim \mathcal{P}\_{\text{OOD}}}[E(\mathbf{h}')]$$

Proof Sketch:

The proof proceeds by analyzing the expected energy for ID and OOD samples separately.

1. For an ID sample $\mathbf{h}$: Given the assumption of good directional alignment, the logit for the correct class, $f_y = s \cdot \|\mathbf{h}\| \cdot \cos(\theta_y)$, will be approximately $s \cdot \|\mathbf{h}\|$. This term will dominate the sum inside the logarithm of the energy function $E(\mathbf{h})$.

   $E(\mathbf{h}) = -\log\left(\exp(s \cdot \|\mathbf{h}\|) + \sum_{j \neq y} \exp(f_j)\right) \approx -\log(\exp(s \cdot \|\mathbf{h}\|)) = -s \cdot \|\mathbf{h}\|$

   Taking the expectation, we find that the expected ID energy is determined by the expected norm: $\mathbb{E}[E(\mathbf{h})] \approx -s \cdot \mathbb{E}[\|\mathbf{h}\|]$. This shows that ID samples with higher confidence (larger norm) are driven to even lower energy regions.

2. For an OOD sample $\mathbf{h}'$: Under the assumption of random directional alignment, $\cos(\theta'_j) \approx 0$ for all classes $j$. Consequently, all logits become approximately zero: $f'_j \approx s \cdot \|\mathbf{h}'\| \cdot 0 = 0$.

   $E(\mathbf{h}') \approx -\log\left(\sum_{j=1}^{C} \exp(0)\right) = -\log(C)$

   The expected energy for OOD samples converges to a constant value, $-\log(C)$, which is significantly higher than the large negative energy of high-confidence ID samples and is notably independent of the OOD sample's norm $\|\mathbf{h}'\|$.

Conclusion: This analysis provides a formal argument that GeoEnergy's decoupling mechanism inherently creates a structural separation in the energy landscape. It pushes high-confidence ID samples towards very low energy values proportional to their norm, while keeping OOD samples constrained to a much higher, constant energy region. This theoretically justifies the significant improvements in ID-OOD separation observed in our experiments.

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W2: Generalization to large-scale and heterophilous graphs.

A4: Thanks for raising this important point. To explicitly address this concern, we have conducted new experiments on both large-scale and heterophilous graphs, and provide a clarification on why our method's principles extend beyond homophilous settings.

1. Effectiveness on Heterophilous Graphs: We agree the "Angular Clustering" intuition is less pronounced on heterophilous graphs. However, GeoEnergy's core mechanism—decoupling norm and direction—is not dependent on homophily. It purifies the directional signal GNNs use for classification, which is beneficial in any setting. Our new experiments on three heterophilous datasets (Texas, Wisconsin, Cornell) confirm this, showing significant and consistent improvements.

Table 1: OOD Detection Performance (AUROC ↑) on Heterophilous Datasets.

These results strongly indicate that GeoEnergy is not limited to homophilous graphs and provides robust performance gains in more challenging heterophilous environments.

2. Scalability to Large-Scale Graphs: We respectfully clarify that our original submission already included experiments on two large-scale benchmarks: ogbn-arxiv (169K nodes, 1.1M edges) and Twitch (167K nodes, 6.7M edges). As shown in Table 3 of our main paper, GeoEnergy provides substantial performance boosts on these large graphs.

3. Efficiency: We measured the inference time overhead of GeoEnergy. The experiments were conducted on a single NVIDIA A100 GPU.

Table 2: Inference Time Analysis (in milliseconds per epoch).

The results show the overhead is negligible (+ less than 2% on smaller graphs and + less than 0.5% on large-scale graphs). This is because the only modification is applied to the final classification layer, which is computationally insignificant compared to the GNN's message-passing operations. This confirms that our method is highly efficient and scalable in practice. In conclusion, through both existing and new experiments, we have demonstrated that GeoEnergy is both effective on heterophilous and scalable to large graphs, addressing the reviewers' concerns about its generalizability.

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We hope our responses and new results have fully addressed your concerns. We remain available for any further questions or clarifications.

We sincerely thank you for the positive feedback and constructive comments. We appreciate that the reviewer recognized our work as a geometrically principled, plug-and-play, and empirically validated approach. We address each concern below in a point-by-point manner.

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W1&L1: The paper's assumption of homophily might limit its performance on heterophilic graphs.

A1: We thank the reviewer for raising this important point. We agree that evaluating performance on heterophilous graphs is crucial for assessing generalizability.

• Clarification on Mechanism: While our "Angular Clustering" observation served as an initial motivation, GeoEnergy's core mechanism—decoupling norm and direction—is not strictly dependent on high homophily. Its effectiveness relies on a more fundamental premise: that GNNs primarily use the direction of embeddings for classification. Decoupling purifies this signal from norm-based structural noise, which is beneficial regardless of the graph's homophily level. In heterophilous settings, GNNs often learn more complex, non-linear decision boundaries. Our geometric decoupling effectively simplifies the classification task for the final layer by projecting features onto a hypersphere, forcing it to focus solely on learning these complex angular separations without being distracted by noisy norm variations. This can lead to a more robust and better-generalized classifier, which is reflected in our new experimental results.

• New Empirical Evidence: To demonstrate this, we have conducted new experiments on three widely-used heterophilous datasets: Texas, Wisconsin, and Cornell. The results, showing consistent and significant improvements, strongly indicate that GeoEnergy provides robust performance gains in these challenging environments. We have added these results to the appendix of our revised paper.

Table 1: OOD Detection Performance (AUROC ↑) on Heterophilous Datasets.

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Q2&W2: The main experimental results lack diversity, which might not fully evaluate generalizability.

A2: We thank the reviewer for this important feedback on ensuring a comprehensive evaluation. We acknowledge that the diversity of our experiments could be highlighted more clearly. To address this, we summarize the broad scope of our evaluation and have included new comparisons with recent State-of-the-Art (SOTA) methods to further demonstrate the generalizability of GeoEnergy.

(1) Comprehensive Evaluation Scope (Existing & New): Our evaluation framework was designed to be diverse and rigorous, spanning multiple axes:

• Graph Scale and Type: Our experiments cover not only classic homophilous benchmarks (Cora, etc.) but also large-scale graphs (ogbn-arxiv, Twitch) and, with our new experiments, challenging heterophilous graphs (Texas, Wisconsin, Cornell, shown in Table 1).
• Distribution Shift Scenarios: We test against a wide array of OOD scenarios, including structural perturbations, feature-level shifts (interpolation), and semantic shifts (label leave-out), simulating a variety of real-world challenges.
• Backbone Models: We demonstrate that GeoEnergy consistently enhances multiple distinct backbones, including the propagation-based GNNSafe and the sample-based NodeSafe.

(2) New Benchmarks against State-of-the-Art (SOTA) Methods: To further strengthen the evaluation and address the concern about diversity of comparison, we have benchmarked GeoEnergy against two highly relevant and recent SOTA methods, GRASP[1] and GOLD[2].

Table 3: Comparison with SOTA methods on Cora-Structure (AUROC ↑).

The results show that GeoEnergy outperforms these strong, recent baselines while offering significant advantages in simplicity and efficiency as a plug-and-play module. This comparison with the latest SOTA methods provides strong evidence of our method's advanced performance and generalizability.

We believe this comprehensive evaluation, now further strengthened with new heterophily results and SOTA comparisons, robustly validates the generalizability and effectiveness of our proposed method. We have updated our manuscript to reflect these additions.

[1] Revisiting Score Propagation in Graph Out-of-Distribution Detection. NeurIPS'24

[2] GOLD: Graph Out-of-Distribution Detection via Implicit Adversarial Latent Generation. ICLR'25

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Q3&W3: The efficiency of the proposed method should be analyzed.

A3: This is an excellent point. To provide a concrete analysis of GeoEnergy's efficiency, we have measured the additional inference time introduced by our method on top of the GNNSafe backbone. Experiments were conducted on a single NVIDIA A100 GPU.

Table 4: Inference Time Analysis (in milliseconds per epoch).

As the results clearly indicate, the computational overhead introduced by GeoEnergy is negligible. This is because our modification only affects the lightweight final classification layer and does not alter the expensive message-passing operations of the GNN backbone. This confirms that our method is highly efficient and scalable in practice. We have added this analysis to the appendix of our revised paper.

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We hope that our point-by-point responses, along with the new experiments and analyses, have thoroughly addressed all of the reviewer's concerns. We are more than happy to engage in further discussion and clarify any remaining questions.

We sincerely thank you for the detailed and insightful feedback, which has helped us to significantly improve our paper. We are encouraged that the reviewer found our work to be technically solid and addressing a relevant and under-explored issue. We address each of the weaknesses and questions below.

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W1&W4: The method's reliance on the embedding norm, and whether a fully norm-independent function would be cleaner.

A1: Thanks for this insightful question, as it allows us to clarify a core contribution of GeoEnergy. We agree with your premise: in standard GNNs, the embedding norm is often entangled with non-semantic structural factors (like node degree), acting as a source of noise. Our work is motivated by precisely this problem, which we term "coupling-induced misdetection."

Our core insight is that decoupling transforms the norm's role. After normalizing classifier weights, the model must prioritize learning correct semantic directions. Subsequently, the norm is "freed" to become a purified signal of model confidence, an emergent property of the optimization. The model learns to assign larger norms to high-confidence ("easy") samples and smaller norms to low-confidence ("hard") samples. This enhances the model's expressiveness, which explains the improved ID accuracy we observe.

To directly test the reviewer's hypothesis, we conducted a new ablation study comparing our full method against a fully norm-independent variant (where the embedding norm is also normalized).

Table 1: Ablation on the embedding norm on Cora-Structure (AUROC / ID ACC).

The results unequivocally show that removing the purified norm significantly degrades performance, for both OOD detection (AUROC) and ID classification (ID ACC). This provides strong empirical evidence that preserving the norm is a critical and beneficial design choice.

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W2&W5: Generalization to large-scale and heterophilous graphs.

A2: Thanks for this critical feedback. To address this, we provide new experiments and clarifications.

(1) Effectiveness on Heterophilous Graphs: While our "Angular Clustering" motivation may be less pronounced in heterophilous settings, GeoEnergy's core decoupling mechanism is general. To prove this, we conducted new experiments on 3 heterophilous datasets, which confirm that our method provides significant and consistent improvements.

Table 1: OOD Detection Performance (AUROC ↑) on Heterophilous Datasets.

(2) Scalability to Large-Scale Graphs: We respectfully clarify that our original submission already included experiments on two large-scale benchmarks: ogbn-arxiv (169K nodes, 1.1M edges) and Twitch (167K nodes, 6.7M edges). As shown in Table 3 of our main paper, GeoEnergy provides substantial performance boosts on these large graphs.

(3) Efficiency: We measured the inference time overhead of GeoEnergy. The experiments were conducted on a single NVIDIA A100 GPU.

Table 2: Inference Time Analysis (in milliseconds per epoch).

The analysis confirms the overhead is negligible and decreases with graph scale, as our modification only affects the lightweight final layer. This demonstrates that GeoEnergy is both efficient and highly scalable.

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W3: The theoretical justification is trivial and lacks depth.

A3: Thanks for your valuable feedback. To address this, we have added a new, more formal analysis (Proposition 3) to our revised manuscript. This proposition provides a direct theoretical justification for the improved ID-OOD energy separation. We present a sketch below.

Proposition 3 (ID-OOD Energy Separation Guarantee).

Let $E(\mathbf{h}) = -\log(\sum\_{j=1}^{C} \exp(s \cdot \|\mathbf{h}\| \cdot \cos(\theta\_j)))$ be the energy function. Assume that for in-distribution samples $\mathbf{h} \sim \mathcal{P}\_{\text{ID}}$, their directions are well-aligned with their true class prototypes such that $\mathbb{E}[\cos(\theta\_y)] \approx 1$ for the correct class $y$. Further assume that for out-of-distribution samples $\mathbf{h}' \sim \mathcal{P}\_{\text{OOD}}$, their directions are unaligned, such that $\mathbb{E}[\cos(\theta'\_j)] \approx 0$ for all classes $j$. Then, the expected energy of ID samples is significantly lower than that of OOD samples, with the gap widening for higher-confidence ID samples:

$$\mathbb{E}\_{\mathbf{h} \sim \mathcal{P}\_{\text{ID}}}[E(\mathbf{h})] \approx -s \cdot \mathbb{E}[\|\mathbf{h}\|] \ll -\log(C) \approx \mathbb{E}\_{\mathbf{h}' \sim \mathcal{P}\_{\text{OOD}}}[E(\mathbf{h}')]$$

Proof Sketch:

The proof proceeds by analyzing the asymptotic behavior of the energy for ID and OOD samples.

1. For an ID sample $\mathbf{h}$: Given the strong directional alignment assumption ($\cos(\theta\_y) \approx 1$), the logit of the correct class $f\_y = s \cdot \|\mathbf{h}\| \cdot \cos(\theta\_y)$ dominates the logits of incorrect classes. The energy function $E(\mathbf{h}) = -\log(\sum\_{j} \exp(f\_j))$ is therefore well-approximated by its largest term: $E(\mathbf{h}) \approx -\log(\exp(f\_y)) = -f\_y \approx -s \cdot \|\mathbf{h}\|$ This demonstrates that the energy of a high-confidence ID sample is a large negative value, linearly scaled by its purified norm, which now represents model confidence.

2. For an OOD sample $\mathbf{h}'$: Under the assumption of random alignment ($\cos(\theta'\_j) \approx 0$), all logits become approximately zero, irrespective of the norm's magnitude: $f'\_j = s \cdot \|\mathbf{h}'\| \cdot \cos(\theta'\_j) \approx 0$. The energy score thus converges to a constant: $E(\mathbf{h}') \approx -\log\left(\sum\_{j=1}^{C} \exp(0)\right) = -\log(C)$ This shows the OOD energy is confined to a much higher (less negative) value, effectively independent of the potentially large and noisy norm of the OOD sample.

In conclusion, this analysis formally shows that our decoupling mechanism structurally separates the energy landscape by driving ID energy down with confidence (norm), while confining OOD energy to a high, constant floor. This provides a principled justification for the improved OOD separation we observe empirically.

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Q1: Lack of comparison with recent SOTA methods.

A4: Thanks for pointing out these important and recent SOTA works. To address this, we have conducted extensive new experiments to benchmark GeoEnergy against these methods under fair and relevant settings. More detailed comparsion will be added to our revised manuscript.

(1) Direct Comparison with SOTA Methods:

We benchmarked our GNNsafe+GeoEnergy against the SOTA methods mentioned. To ensure a fair comparison, we strictly followed their original experimental setups.

• vs. GRASP on Common Benchmarks: We compare against GRASP on both homophilous and heterophilous graphs.

Table 3: Comparison with GRASP (AUROC ↑).

The results show our method consistently outperforms GRASP, highlighting that generating a cleaner initial score is a more effective strategy.

• vs. SPREAD on Label Leave-out: We use the "label leave-out" setting, a key benchmark in SPREAD.

Table 4: Comparison with EDBD on Cora (AUROC ↑).

Even though EDBD is specialized for its proposed OOD setting, our general-purpose method achieves superior performance, demonstrating the fundamental power of geometric decoupling.

• vs. GOLD in Non-OOD Exposure Setting: We compare against GOLD in the setting without any OOD exposure during training.

Table 5: Comparison with GOLD on Cora (AUROC ↑).

Our method surpasses GOLD without relying on any complex data generation or adversarial training.

(2) GeoEnergy as a Universal Enhancement Module:

Beyond being a strong standalone method, GeoEnergy's plug-and-play nature allows it to enhance other SOTA methods. To demonstrate this, we integrated GeoEnergy into GRASP.

Table 6: GeoEnergy enhances GRASP (Cora, AUROC ↑).

The results show GeoEnergy significantly boosts GRASP's performance. This empirically proves our geometric decoupling is orthogonal and complementary to score propagation. Thus, GeoEnergy acts as a fundamental building block to enhance a wide range of graph OOD detection frameworks

(3) Methodological Differences:

• vs. GOLD: GeoEnergy is a post-hoc module, whereas GOLD is a training-time framework. Our method offers significant advantages in simplicity, efficiency, and ease of integration into any pre-trained GNN.
• vs. SPREAD: SPREAD introduces a new problem setting (spreading OODs). Our work focuses on improving the fundamental scoring function for the conventional node-level OOD task.

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We hope our responses and new results have fully addressed your concerns. We welcome any further questions during the discussion period and are happy to provide additional clarifications.