Harnessing Feature Resonance under Arbitrary Target Alignment for Out-of-Distribution Node Detection
Authors: Shenzhi Yang
Junbo Zhao
Sharon Li
Shouqing Yang
Dingyu Yang
Xiaofang Zhang
Haobo Wang
Venue: NeurIPS 2025 poster ()
Keywords: Out-of-distribution detection
Node representation learning
Graph neural networks
Primary area: general_machine_learning

Abstract: Out-of-distribution (OOD) node detection in graphs is a critical yet challenging task. Most existing approaches rely heavily on fine-grained labeled data to obtain a pre-trained supervised classifier, inherently assuming the existence of a well-defined pretext classification task. However, when such a task is ill-defined or absent, their applicability becomes severely limited. To overcome this limitation, there is an urgent need to propose a more scalable OOD detection method that is independent of both pretext tasks and label supervision. We harness a new phenomenon called **Feature Resonance**, focusing on the feature space rather than the label space. We observe that, ideally, during the optimization of known ID samples, unknown ID samples undergo more significant representation changes than OOD samples, even when the model is trained to align arbitrary targets. The rationale behind it is that even without gold labels, the local manifold may still exhibit smooth resonance.  Based on this, we further develop a novel graph OOD framework, dubbed **R**esonance-based **S**eparation and **L**earning (**RSL**), which comprises two core modules:  (i)-a more practical micro-level proxy of feature resonance that measures the movement of feature vectors in one training step. (ii)-integrate with a synthetic OOD node strategy to train an effective OOD classifier. Theoretically, we derive an error bound showing the superior separability of OOD nodes during the resonance period. Extensive experiments on a total of thirteen real-world graph datasets empirically demonstrate that RSL achieves state-of-the-art performance.

Harnessing Feature Resonance under Arbitrary Target Alignment for Out-of-Distribution Node Detection
Shenzhi Yang
Junbo Zhao' Sharon Li Shouqing Yangl:25
Dingyu Yang
Xiaofang Zhang* Haobo Wang! 1.2.5
"hejiang University
2 Hangzhou High-Tech Zone (Binjiang) Institute of Blockchain and Data Security
Department of Computer Sciences, University of Aisconsin-Madison School of Computer Science and Technology. Soochow University
5 Innovation and Management Center, School of Software Technology(Ninggbo) Zhejiang University Corresponding to: wanghaobo@zju. edu. edu. cn
Abstract
Out-of-distribution (OOD) node detection in graphs is critical yet challenging task. Most existing approaches rely heavily on fine-grained labeled data to obtain pre- trained supervised classifier, inherently assuming the existence of a well-defined pretext classification task. However, when such a task is ill-defined or absent, their applicability becomes severely limited. To overcome this limitation, there an urgent need to propose more scalable OOD detection method that is independent of both pretext tasks and label supervision. We harness a new phenomenon called Feature Resonance, focusing on the feature space rather than the label space. We observe that, ideally, during the optimization of known ID samples, unknown ID samples undergo more significant representation changes than OOD samples, even when the model trained to align arbitrary targets. The rationale behind it is that even without gold labels, the local manifold may still exhibit smooth resonance. Based on this, we further develop a novel graph OOD framework, dubbed Resonance-based Separation and Learning (RSL), which comprises two core modules: (i)-a more practical micro-level proxy of feature resonance that measures the movement of feature vectors in one training step. (ii)-integrate with synthetic OOD node strategy to train an effective OOD classifier. Theoretically, derive error bound showing the superior separability of OOD nodes during the resonance period. Extensive experiments on a total of thirteen real-world graph datasets empirically demonstrate that RSL achieves oate--of-the-art performance. The code is available via https://ggttu...om////i how.hiitthucco//Shhhhhiiii h0ggg//h.....////eee//mmm000011 Yang2000/RSI.
Introduction
Graph-based machine learning models like Graph Neural Networks (GNNs) (Kipf and Welling 2016a
Xu al. 2018 Aluu-al-Haija al., 2019 Zhou et al. 2024] have become increasingly prevalent in applications such as social network analysis Fan et al. 2019), knowledge graphs [Back et al.) 2020], and biological networks [De Cao and Kipf] 2018). Despite the success of GNNs, detecting oul-of-distribution (OOD) nodes remains under-explored challenge. These OOD nodes differ significantly from the in-distribution (ID) nodes used during training, and their presence can severely undermine the performance and robustness of graph models. As deploying GNNs in real-world environments becomes more common, the ability to identify and handle OOD nodes is crucial for ensuring the reliability of using these models.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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To address this, most existing methods [Hendrycks and Gimpel 2016
Liang et al. 2017 Hendrycks
at. 2018 Liu et al. 2020 Wu et al. 2023 employ classifiers pretraiedd on preceding classification task to develop OOD metrics based on (i)-classifier outputs, such as Maximum Softmax Probability
(MSP) [Hendrycks and Cimpel, 2016] and Energy [Liu et al. 2020 Wu et al. 2023]; (ii)-supervised
representations, such as KNN [Sun et al. 2022) and NNGuide [Park et al. 2023]. These methods heavily rely on two key assumptions: (i) the availability of multi-class labels, and (ii) a well- defined pretext multi-class classification task. However, in practice, there exists wide range of OOD detection scenarios that fall outside these constraints. In many cases, the pretext task is not classification-foo for example, OOD detection in generative modeling [Nalisnick et al., 2018 Ren et al. 2019). regression [Lakshminarayanan et al., 2017). or reinforcement learning Nasvyytis et al. 2024 In some scenarios, there may not even be a defined pretext task all, such in one-class OOD detection [Ruff et al.) 2018). These non-classification settings lack accessible multi-class labels, making it difficult to directly apply existing methods. Therefore, there is an urgent need for label-agnostic and unsupervised approaches that can operate effectively in such contexts. To date, only few papers [Gong and Sun 2024 (Sehwag al.) [021] [Liu et al. [203]] study this practical setup, and there is still a large room for improvement, especially in the graph field at the node level.
In this paper, revisit the graph OOD detection task at the node level from new perspective and turn attention to the intrinsic similarities within the data. An intuitive idea is that the ID samples may still share some commonalities in the represen- tation space. We hypothesize that when optimizing the representation of known ID nodes, the representation of unknown ID nodes and unknown OOD nodes will change with differ- ent trajectories. Based on the hypoth- esis and using toy dataset (Figure [a)),, we design an experiment where the features of labeled ID samples are
000 hasttry
M Minuar
(a) Toy Dataset
(b) Gradient Descent Trajectory
aligned an arbitrarily fixed repre- sentation vector. Interestingly, we ob-
Intial
Training State
Labeled ID Unlabeled IDI Unlabeted OOD Node Representation
serve a distinct behavior during this optimization process: the representa- tions of unlabeled wild ID samples ex- perienced more pronounced changes than wild OOD samples, as shown in Figure b). This phenomenon closely resembles the concept of forced vibra- tion, where resonance occurs when an external force aligns with the natural frequency of an oscillator, amplifying its oscillation to a maximum. Analogously, we refer to this phenomenon as Feature Resonance: during the optimization of known ID samples, the representation of wnknown ID samples undergoes more significant changes compared to OOD samples. This phenomenon reveals the intrinsic relation- ship between ID samples, highlighting their shared underlying distribution. Evidently, this feature resonance phenomenon can be leveraged for OOD detection: weaker representation changes during known ID optimization indicate a higher likelihood of being OOD.
The Artitrary Fieed Vector
Algoment Training of Labaled Noda Represer tadions Targel
(c) Schematic of Feature Resonance
Figure 1: (a) We conduct a preliminary study on the changes in ID and OOD node representations during training using a toy dataset. (b) Projections of the representations of ID and v.(,6)
OOD nodes onto gradients: Projv [V
E.(0;)). ). (c) Schematic of Feature Resonance.
In real-world scenarios, due to the intrinsic complex pattern in data, we find that the feature resonance phenomenon still occurs but slightly differs from the ideal conditions. To illustrate this, we further propose a micro-level proxy for measuring feature resonance- by computing the movement of the representation vector in one training step. Our findings reveal that in more complex scenarios, the feature resonance phenomenon typically arises during the middle stages of the training process, whereas during other phases, it may be overwhelmed by noise or obscured by overfitting. In such cases, evaluating the entire trajectory often fails to yield satisfactory results. Fortunately, efficient OOD detection can still be achieved by calculating the micro-level feature resonance measure.

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By utilizing a simple binary ID/OOD validation se we empirically show the feature resonance period can precisely identified, and we identify more minor representation differences as OOD samples. Notably, our new micro-level feature resonance measure is still label- label-independen' by fitting a randomly fixed target, making it highly compelling in category-free and task-agnostic scenarios. Theoretical and experimental proof that micro-level feature resonance can filter a set of reliable OOD nodes with low error. Furthermore, we combine the micro-level feature resonance with the current Langevin-based synthetic OOD nodes generating strategy to train OOD classifier for more effective OOD node detection performance, which we call the whole framework as RSL; for example, the FPR95 metric is reduced by average of 15.20% compared to the current sattee-ff-heaaae art methods.
2 Method
2.1 Revealing the Feature Resonance Phenomenon
Previous studies
[Hendrycks and Gimpel 2016
Amazon
Liu et al., 2020 Wu et al.) [2023] mostly train classifier on ID nodes with maltircategory la- bels and develop selection criteria based on out- put probabilities, e.g., entropy. However, these methods become inapplicable in category-free and task-agnostic scenarios.
100
Feature Resonance Period
AUROC
FPR95
-00
:30
To address this problem, we turn our attention the intrinsic similarities within the data. An intuitive idea is that although the output space may no longer be reliable, the ID samples may still share some commonalities in the represen- tation space. We hypothesize that when opti- mizing the representation of known ID) nodes, the representation of unknown nodes and un- known OOD nodes will change with different trajectories. Motivated by this, and under the assumption some specific training process, we define a feature trajectory measure f(xx) of a sample x,:
40
50
100
150
200
250
300
Epoch
Figure 2: The performance of using resonance- based score T to detect OOD nodes varies with training progress. The higher the AUROC, the better, and the lower the FPR95, the better.
((x,)
Σ ho (x:)
he, (x,)
(1)
where ha, denote the model that performs representation transformation on a sample x;, with 0, representing its parameters at the t-th epoch.
In our preliminary experiments, we first calculate the metric under supervised conditions and observe sigmificant difference between the feature trajectories of ID samples and those of OOD samples. Specifically, we perform multi-category training on known ID nodes on two datasets with true N- category labels, Squirrel and WikiCS Imagine that during multi-category training, representations of known ID nodes within the same category align while unknown ID nodes drift toward the corresponding category centers. However, the trajectory trends and lengths of unknown II) nodes differ significantly from those of OOD nodes, with the former showing more distinct trends and longer trajectories; see Figure (c) for visual illustration. In other words, the well-defined in-distribution manifold is always shaped by samples, whose representation trajectories tend to exhibit similar behavior, which refer to as feature resonance. Conversely, OOD samples belong distinct manifold structures, making their representations less likely to converge coherently. Evidently, this feature resonance phenomenon
be leveraged for OOD detection.
Despite the promise, the abovementioned feature resonance phenomenon occurs under multi-category training. But how can we induce this phenomenon in a label-agnostic scenario without multi-category labels? ? Interestingly, we find that even when random labels are assigned to known IDD nodes for multi-category training, the trajectories of unknown ID nodes are still more significant than
'The use of validation set consistent with previous works KKazz-Samulls et al. 2022 Gong Sun
2024 Du et al. 2024a]b) and does not contain multi-category labels.
N is the number of categories, and the results above with different target vectors shown in Table

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feature resonance at the those of unknown OOD nodes. More surprisingly. on a ideal toy dataset, even when all known ID node representations are aligned toward one single random fixed target vector, the trajectories of unknown ID nodes are still longer than those of unknown OOD nodes, as shown in Figure a Green points represent unknown ID samples, blue points represent unknown OOD samples, and red points represent known ID) samples aligned to target vector. As shown in Figure modifying the representation of known ID samples results in longer representation change trajectories for unknown ID samples compared to unknown OOD samples. The experiments above indicate that the feature resonance phenomenon is label-independent and results from the intrinsic relationships between ID node representations. Therefore, this is highly suitable for category- ffree and task task-agnostic OOD detection scenarios without multi-category labels.
Since the trajectory represents a global change, we call it macroscopic feature resonance, as follows:
Definition 1. Feature Resonance (macroscopic): For any optimization objective El (Xionns's -) applied to the representations X Xnown of known ID sumples derived from any model lag()), we have II Iiggia >|| F(&:) Ippiid.
F(x.))
2.2 Utilizing the Micro-level Feature Resonance Phenomenon with An Arbitrary Target
As mentioned above, leverage the feature resonance phenomenon to detect OOD nodes. In our realistic implementations, we align the features of known ID nodes to an arbitrary target vector using mean squared error as follows:
E(ha, ( known), = IE(I| 1 e -eee-XXooow (X W l12)
where heg (Xnown) = X Known WT represent the last linear layer of the model for representation transformation and denotes an arbitrary randomly generated target vector.
But, in contrast our toy dataset, the real-world datasets typically exhibit much more complex feature attributes. As a result, the feature resonance of trajectory at the macro level is not as ideal pronounced as observed in experiments on the toy dataset. Therefore, to explore the reasons behind this issue, we delve deeper into the changes in finer-grained node representations across epochs to study the feature resonance phenomenon. Specificallyy, we study the differences in Ahe, (X.) = her4. (xx) - he, (x;) between ID samples and OOD samples. Obviously, the existence of || Ahe, (x:) lleea >|| Ahe, (Rt) lleaad necessary condition for satisfying ||
((x:) llewn x>
F(x.) Il so define || Ahg, (X;) [lagga>| || Ahg, (X:) Ilpsaa
microscopic level:
Definition 2. Feature Resonance (microscopic): For any optimization objective X(X -) applied to the known ID nodes' representations Xenown from any model (-), during the optimization process, there exists I. such that || Aho, (Xx:) Illwin Aho, (x;) Ilawne. We define the resonance-
out
based filtering score T; =|] Aho, (Rx) || The resonance- scores 7 of OOD nodes should be smaller than those of ID nodes at l.
By observing for ID samples and OOD samples, find that feature resonance does not persist throughout the entire training process but rather occurs at specific stages of training. In our ex- periments the common benchmarks, we find that during the early stages of training, the model is searching for the optimal optimization path, leading to chaotic representation changes and thus making feature resonance insignificant. However, in the middle stages of training, once the model identifies an optimization path that aligns with the patterns of the ID samples, it optimizes along the path most relevant to the features of the ID samples, and feature resonance becomes most prominent. As the model continues to optimize and enters the overfitting stage, the feature resonance phenomenon begins to dissipate. Figure 2 shows the experimental results on the Amazon dataset, and others are provided in Figure 3 of the Appendix. Through the above experiments and analyses, we find that using Fxx) to identify OOD nodes is affected by eттor accumulation and is, therefore, not reliable approach. However, there exists specific period during training when micro-level feature resonance occurs. By utilizing a validation set [Kazz-Samuels et al., 2022, Gong and Sun 2024 Du et al. we can easily identify the period during which feature resonance occurs.
2024a b).
Formally, our new feature resonance-basse based OOD nodes detector is defined as follows:
(x,) = ( S 7), s.t., T" = max AUROC(TÍ Vouss

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Table 1: The statisties of the real-world OOD node detection datasets. x denotes no available
multi-category labels. Notably, we do not use any true labels for all datasets.
Dataset
Squirrel
WikiCS
Cora
Citeseer
Pubmed
Chameleon
YelpChi
Amazon
Reddit
# Nodes
5,201
11,701
2,708
3,327
19.717
2,277
45,954
11,944
10,984
Features
2,089
300
1,433
3,703
500
2,325
32
25
Avg. Degree
41.7
36.9
7.8
5.5
9.0
31.7
175.2
800.2
15.3
OOD node (%)
20.0
29.5
66.7
45.7
20.8
40.2
14.5
9.5
3.3
# Calegory
5
10
3
3
2
3
x
X
where 9, = 1 indicates the OOD nodes while g- 0 indicates otherwise, and y is typically chosen to guarantee a high percentage, such as 95%, of ID data that correctly classified. Here, t is determined by the validation set Vral-
To summarize our method: we calculate a resonance- based filtering score during the transformation of known ID sample representations. By leveraging validation set, we identify the period during training when micro-level resonance most significant. Within this period, test set nodes with smaller values are more likely to be OOD nodes.
2.3 Extension with Synthetic OOD Node Strategy
Although the resonance- based filtering score effectively separates OOD nodes, recent studies [Gong and Sun 2024) suggest that training an OOD classifier with synthetic OOD nodes can improve OOD node detection. Therefore, we propose a novel framework that employs feature resonance scores to generate more realistic synthetic OOD nodes.
Specifically, define the candidate OOI) node set as Veand (ūp E Vwild : 7. I T), where T= (T) is the n-th smallest r of wild nodes, selecting nodes with the smallest n T values. The features of these nodes form Xcand- Then, we compute trainable metric based on the weighted mapping of node v's representations across K GNN layers: Ex = Wx (Ex K B. where 3x E IR is learnable parameter, and WK E R1xd transforms the node representations to the energy scalar. Then, we employ stochastic gradient Langevin dynamics (SGLD) (Welling and Teh 2011 to generate synthetic OOD nodes Vayn (01," "ha,"",,))))) with random initial features
Xayn
XI.... ,,,} as follows:
x((+)) -X(x)²
-
V 10 Ea (6))) + +6) +(1 A)Ex x (x (0)))
where / is the step size and À is trade-off hyperparameter. is the Gaussian noise sampled from multivariate Gaussian distribution N N(0 C). Unlike EnergyDef [Gong and Sun 2024], we utilize the candidate OOD nodes Veand examples to generate synthetic OOD nodes that better align with the actual OOD nodes. After obtaining the synthetic OOD nodes, we define the training set Virain Vknown U Veand U Vayn with features Xtrain and labels Yrain- The initially known ID nodes Vkuown are assigned label of 1. In contrast, the candidate OOD nodes Veaud and the generated synthetic OOD nodes Veyn assigned a label of O. We use binary cross cross-entropy loss for training:
leh - -y- (Fo (v))) + (1 y., log((1 o (FF()))))
where (-)) is the sigmod function. Similarly, we identify the OOD nodes as follows: d'. (Ea (v) < 7/). where  1 indicates the OOD nodes while g' 0 indicates otherwise, and y is chosen to guarantee high percentage, c.g., 95%, of ID data that is correctly classified.
2.4 Theoretical Analysis
Our main theorem quantifies the separability of the outliers in the wild by using the resonance-based filter score r. We provide detailed theoretical proof the Appendix C
out be the error rate of OOD data being regarded as ID at (-th epoch, i.e., ERR out = (( E Let ERR
your Ti 2 T//l walll where your denotes the set of outliers from the wild data Vwild- Then ERR has the following generalization bound:
Theorem (Informal). Under mild conditions, if ((xx,)) is B-smooth w.rl ws, Pwitd has (r, E)- discrepancy w.E. I and there is n (0,1) s.t. A (1 (1-m)22² 2)² 88, Rú 0, then where
5

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22(d/min(n2A. (y Ri,))),, (2)/nn22)), with the probability at least 0.9, for 0 < T < 0.9. is the upper bound of score T²).
d 1ax + O(V T
max d. 4 [3
ERR out T-T//V2//220 1))²
+ OVV
(6)
T²(1 77 mm
where D = 0.98n) 831 Ri and Rij, is the optimal ID risk, i.e., Rin = min w Ex f(x, e). d is the dimension of the space W, denotes the t-th epoch, and x is the OOD class-pric probability in the wild.
Practical implications of Therorem 1 The above theorem states that under mild assumptions, the error E Rouu is upper bounded. If the following two regulatory conditions hold: 1) the sizes of the labeled ID n and wild data are sufficiently large; 2) the optimal ID risk R is small, then the upper bound is mainly depended on T and t. We further study the main error of T and which we defined as t(,,t).
Theorem 2. (Informal). if 2 (1 - cyy for a small error c 2 0, then the main error 8(T, t)
satisfies that
max -0,11-//T) S 1 1-T/(V2/(2t0) - 1))2
8(T,t)= 1- 1-T/(V2/22ta - 1))2
83, Rú, 1.011 then there exists 7 E (0, 1) ensuring that A > 0 and A? > π hold, which implies that the main error 8(T, t) = 0.
2) When learning rate is small sufficiently. and if 2 2.011 /
Practical implications of Therorem 2 Theorem states that when the learning rate a is sufficiently small, the primary error 8(T, L) can approach zero if the difference between the two data distributions Fwild and Pin is greater than a certain small value. Meanwhile, Theorem 2 also shows that the primary error 8(T, t) is inversely proportional to the learning rate and the number of epochs (t). As the increases, the primary error ö(T, t) also increases, while smaller learning rate a leads to a minor primary error 6(T, t). However, during training, there exists at which the error reaches its minimum.
Experiment
In this section, we present the main experimental results, while in Appendix F.2 investigate feature resonance across datasets. Appendix F.3 compares scoring methods, |h4 evaluates time efficiency. Appendix F.5 tests RSL with different GNN encoders, and exxamiine graph-level OOD
detection. Appendices F.8
and R.
provide score distribution and node representation visualizations.
3.1 Experimental Setup
Datasets. We conduct extensive experiments evaluate RSL total of nine real-world OOD
node detection datasets: six multi-category datasets, Squirel [Rozemberczki et al. 2021]. WikiCS
Mernyei and Cangea| (2020), Cora, Citeseer, Pubmed (Kipf and Welling (0166a). and Chameleon
[Rozemberczki et al. 2021] and three binary classification fraud detection datasets: YelpChi [Rayana
and Akoglu, 2015), Amazon [McAuley and Leskoved 2013), and Reddit [Kumar al. 2019 The
statistics of these datasets are summarized in Table 1 Additionally, we validate our method on four graph-level OOD detection datasets, including ENZYMES, PROTEINS [Morris al. 2020
ClinTox, and LIPO
[Wu et al. 2018). We provide detailed dataset description in the Appendix E.3
Baselines. We assess the performance of RSL against a total of twenty-one baseline methods spanning five categories: 1) Traditional outlier detection methods, including local outlier factor (Breunig et al. 2000) like LOF-KNN and MLPAE, 2) Graph-based outlier detection models. including GCN autoencoder [Kipf and Welling 2016b). GAAN [Chen et al. 2020]. DOMINANT
[Ding et al. 2019), ANOMALOUS [Peng et al. 2018), and SL-GAD [Zheng et al., 2021] 3) Transformation-base
outlier detection approaches, such as GOAD (Bergman and Hoshen 2020) and Neulral AD [Quu et al. 2021). 4) Entropy- -hased detection techniques. including MSP. ODIN,
OE, Energy, GKDE [Zhao et al. 2020 OODGAT [Song and Wang 2022 GNNSafe (Wu et al.

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Table 2: Unsupervised OOD detection on real-world datasets. "OOM" indicates out-of-memory. "TLE" means time limit exceeded, and "." denotes inapplicability. Detectors with & use only node attributes, while share RSL's GNN backbone GCN Entropy-based methods with & use true multi-category labels, and
rely on K-means pseudo labels. Top results: 1st, 2nd.
Dataset
Squirrel
WikiC
VelpChi AUPPR
Amazon
Roddit
Method
AUROX
LOF KNN
29.57
95.21
44.06
37.48
96.25
92.57
14.26
95.10
57.88
6.95
MLPAE
43.15
24.81
97.98
70.99
63.74
77.76
51.90
24.53
92.42
74.54
51.59
57.93
52.10
5.80
94. 04.43
57.93
92.97
19.22
97.06
12.33
51.75
93.75
GAAN
38.01
22.57
98.99
58.15
16.60
93.37
14.29
19.30
96.91
53.26
6.63
98.05
52.21
5.96
91.06
DOMINANT
41.78
24.73
95.53
42.55
35.43
97.22
52.77
24.90
92.86
78.08
35.96
76.05
55.89
6.03
96.48
ANOMALOUS
51.04
29.09
96.39
67.99
54.51
92.74
00M
00M
0OM
65.12
25.15
85.34
55.18
6.40
94.10
SL-GAD
48.29
27.62
97.19
$1.87
44.
95.
56.11
26.49
93:27
82.63
56.27
51.63
6.02
94.27
GOAD*
62.32
37.51
92.28
50.65
37.2 22
99. 75
58.03
26.51
89.84
72.92
45.53
66. 36
52,89
5,36
94.
NeuTral AD
52.51
30.04
97.16
53.58
43.49
94,30
55.81
25.14
94.23
70.01
24.36
92.19
55,70
6.45
94.59
GKDEP
56.15
70.47
82.71
GODGAT
58.84
35.13
93.31
74,13
62.47
84.48
56.38
95.17
73.35
66.47
76.24
NodeSale
57,82
33.57
93.64
74.81
67.93
74.85
GRASP
61.38
36,95
90.77
78.46
71.52
71.08
OODGAT
57.78
34.66
92.61
52.76
44.71
90.02
55.97
23.07
97.93
82.54
54,94
52.10
54.62
6.05
93.85
GNNSafe
49.52
26.63
97.60
64.15
50. 85
92.63
55.26
26.68
91.40
68.51
25.39
84.31
49.63
5.36
98
NodeSafe *+
65.77
52.02
28.01
89.95
50.74
6.03
94.26
GRASP**
52.63
28.12
94.87
66.94
53.33
89.
58.05
28.67
88.11
70.31
27,81
81.29
51.82
6.91
92.04
SSD
TLE
TLE
TLE
64.29
58.45
87.12
55.39
27.88
91.63
72.49
41.82
84.27
59.74
6.21
91.15
EnergyDey
64.15
37.40
91.77
70.22
60.10
83.17
62.04
29.71
90.62
86.57
74.50
32.43
63.32
8.34
89.34
RSL, wio classifier
61.52
38.96
90.18
79.15
78.
70.38
65.42
37.08
83.53
87.43
83.31
19.56
52.37
6.97
91.
RSL w/o Van
60.16
$4,89
93.59
81.21
79.93
52.19
65.15
38.93
81.84
87.81
81.10
25 18
61.36
8.48
89.43
RSL
64.12
39.58
89.90
84.01
81.14
49.23
66.11
39.73
80.45
90.03
83,91
19,60
64.83
10.18
85,49
2023]. NodeSafe (Yang et al. methods, including EnergyDef [Gong and Sun 2024 and SSD [Sehwag et al. 202
2025], and GRASP Ma et al. 2024]. 5) Category-free detection
Additionally, we also compare our method with graph- approaches, including graph kernel combined with a detector Vishwanathan et al. 2010 Shervashidze et al.) 201 Neumann et al.
2016 Breunig et 2000, Manevitz and Yousef 2001 Liu et al. 20088. 2) graph contrastive
learning with a detector You et al. | Liu et |0088 /Sewwag el al. 202 Zhou et al. 2021
Liu al. 2023 and 3) end-to-end methods [Zhao and Akoglu 2023 Ma et al.
2022
Details of baselines and implementation are in Appendix and
respectively.
Following prior research on OOD node detection, we evaluate the detection performance using three widely recognized, threshold-indeepedeer metries; AUROC (t), AUPR (f) and FPR95()). We provide detailed metric description in the Appendix E.2
Metrics.
3.2 Main Results
Tables 2 and 3 present the main experi- mental results of various methods across nine public datasets. Specifically, when multi-class labels are unavailable, RSL sig- nificantly outperforms existing methods. For methods that require multi-class la- bels, we follow EnergyDef [Gong and Sun) 2024) by assigning pseudo-labels using K- means. On the YelpChi, Amazon, and Red- dit datasets, RSL achieves average improve- ments of 3.01% %. 7.09%, and 8.95% over the SOTA methods in terms of AUROC, AUPR, and FPR95, respectively.
Table 3: Performance comparison across methods on Cora, Citeseer, Pubmed, and Chameleon.
Dalaset
Cora
Citeseer
Pubmed
Chamelese
Method
FPIC
AURUX
MSP
70.86
$4.55
67.51
82.35 39
67.37
68.80
85.70
$7.95 36
Energy
67.54
$5.47
86.53
72.38
93.85
54.09
85.06
59.20
KNN
90.20
70.94
85.10
72.91
69.79
64.14
93.38
$7.90
ODIN
34.55
M2.42
65.80
Mabalanntis
69.68
35.45
54.62
56,85
53.19
GKDE
96.27
79.04
GPN
OODGAT
58.25
GNNSSTI
54.71
87.52
60.15
53.70
100000
ModeSale
50.32
89.11
86.
55.11
98.76
52.19
GRASP
29.70
93.50
W9.75
66.38
76.93
RSL.
25.76
94.14
33.67
90.44
59.10
75.04
When multi-class label information is available, RSL shows even more significant performance gains on heterophilic graphs. On the Squirrel, WikiCS, and Chameleon datasets, RSL achieves an average improvement of 14.93% in FPR95 over SOTA methods. This is because RSL does not rely on the homophily assumption of the graph, and thus performs well on heterophilic graphs. On homophilic graphs with multi-class labels-namely Cora, Citeseer, and Pubmed- achieves performance comparable to SOTA methods. Notably, RSL does not leverage multi-class labels for training in any
7

---

Table 4: Performance of RSL achievable label proportion R in the WikiCS dataset.
R= 0.0
==
R 0.2
R 0..4
= 0.6
R 1.0
AUROC(T)
84.01
86.41
87.46
87.89
88.35
88.57
89.07
AUPR(T)
81.14
83.35
84.80
85.24
85,42
86.16
86.85
FPR
49.23
44.95
42.69
41.25
39.50
39.68
38.78
Table 5: Performance of different models on the Citeseer under varying homophily ratio R values.
Model
R 0.0
H 0.1
R
0.2
R 0.3
R 0.4
R 0.5
F5S
AUROC
1
GNNSafe
60.15
84.85
52.92
86.56
47.68
87.40
45.64
89.30
42.22
90.42
37.19
91.78
NodeSafe
55.71
86.16
49,47
87.45
42.23
90.47
38.62
92.31
34.79
93.68
30.57
94.93
GRASP
35.23
89.75
30.63
91.26
27.35
94.00
16.01
96.36
13.69
97.33
10.08
98.25
RSL
33.67
90.44
28.91
91.75
16.62
94.45
9.82
96.09
7.71
97.30
4.53
98.23
of the above experiments. This highlights its label-agnostic and task-independent nature, contributing to its broader applicability.
How effective is resonance- filter score r? The experimental results in the row labeled "RSL experimenttt w/o classifier" of Table show that using the raw resonance-based score to filter OOD nodes is already more effective than the SOTA method on most datasets. On the FPR95 metric, the score т achieves an average reduction of 9.70% compared to current SOTA methods in Table 2
How effective are the synthetic OOD nodes combined with the feature resonance score? The experimental results in the row labeled "RSL w/o Vsyn" of Table 2 show that after removing the synthetic OOD nodes, the performance of the trained OOD classifirr declined to varying degrees. This indicates that synthetic OOD nodes enhance the generalization ability of the OOD classifier, allowing to detect more OOD nodes more accurately. It is worth noting that our synthetic OOD nodes, generated by leveraging real OOD nodes selected using т, better align with real-world OOD scenarios and, therefore, outperform EnergyDef.
Can label information bring gains to RSL? Although RSL can perform well in scenarios without multi-class labels, we want to investigate whether multi-class la- bels can bring similar benefits to RSL as they do for other methods. On the WikiCS dataset, we first pre-train the representa- tions of the training set's nodes using su- pervised contrastive learning loss [Kholla jet al. 2020) with training set labels at dif- ferent proportions R, and then apply RSL. When R = 1.0, it indicates that RSL, like other methods that strictly require labels, uses all the training set labels. The results in Table show that as the available label proportion increases, the ID node representations of the training set are better initialized, and RSL performs better. We believe this is because when the ID node representations are well-initialized, feature resonance is more easily induced and is more pronounced.
Table 6: The effectiveness of the resonance-based filter score T in filtering OOD nodes with different align- ment targets for known ID node representations. True multi-label means aligning ID node representations with multiple target vectors based on true multi-clase labels. Multiple random vectors means aligning ID node representations with random target vectors. A ran- dom vector means aligning ID node representations with a single target vector.
Methael
Target
Squirel
Energy!af
91.77
60.10
KXD
MSL. clsstter
label
ST95
RSL. who clasifier
Muliple nusdam vactom
61.44
$2.30
90.62
73.64
24.15
69.25
RSL wio chsitter
A candom vector 61.52
39.95
90.18
79.15
76.06
70.35
Can graph homophily bring gains to RSL. Most existing OOD node detection methods benefit from graph homophily, so we aim to explore whether RSL can also gain from it. We conduct experiments the Citeseer dataset under varying levels of homophily, by removing proportion R of heterophilous edges and adding the same proportion Rt of homophilous edges. The results in Table 5 show that as graph homophily improves, the performance of RSL also improves. We believe this is because enhanced graph homophily leads to more consistent representations among ID nodes and more pronounced differences between ID and OOD node representations, thereby making feature resonance easier to induce and more strongly expressed.
How does feature resonance occur due to different target vectors? We explore micro-level
feature resonance using different target vectors through experiments on Squirrel and WikiCS datasets with true N -ccategory labels. Based on neural collapse theory (Papyan et al. 2020 Zhou et al.)
8

---

Table 7: The effectiveness of different OOD candidate node selection strategies.
Dataeet
Squirel
WIEICS
YcipChi AUPR
Amazon
Redalft
Mechod
AUROC
FPR
RSI. w/ Conini Similartty
16.1
91.46
53.61
16.36
70.35
59.76
35.03
85389
R3.35
74.85
27.63
54.17
7:25
102.31
RSL Euclidean Disunce
39.30
90.45
73,65
74.28
63.26
52.53
24.20
93.53
53.08
18.29
93.64
62.19
6.35
90,90
RSL Milalanohis
TLE
83.18
5843
54.07
92.40
30.66
79.96
8,42
RSL EnergyDef
29.29
63.21
90.42
57.33
91.90
54.52
8.55
Rosoinss-Feee
39.58
83.91
19.60
2022], we set N target vectors that form a simplex equiangular tight frame maximizing separation. As shown in Table the "True multi-label" row demonstrates the effectiveness of this approach. Interestingly, even when random labels are assigned (the "Multiple random vectors" row) or when all ID representations align with a fixed vector (the AA random vector" row), unknown ID nodes still show larger r than unknown OOD nodes, as seen in Table These results suggest that feature resonance is label-independent, stemming from intrinsic relationships between ID node representations.
How do other OOD node selection methods perform? We aim evaluate the performance of RSL when integrated with methods other than the resonance- based score for selecting reliable OOD nodes. To ensure fairness, we used the same parameters and selected the same number of OOD nodes. From metric learning perspective, we computed the cosine similarity, Euclidean distance, and Mahalanobis distance between unknown nodes and the prototypes of known ID nodes, with smaller values indicating higher likelihood of being OOD nodes. We also applied EnergyDef for OOD node selection. The results, presented in Table 7 show that, under the same conditions, the OOD nodes selected using are more reliable than those selected by the other methods.
4
Related Work
General 0OD Detection Methods. OOD detection methods fall into three main categories: entropy-
based, density-based, and representation- basedd approaches. Entropy-based methods (e.g.. MSP
[Hendrycks and Gimpel, 2016), Energy [Liu et al. 2020), and others [Liang et al. 2017 Bendale
and Boult 2016 Hendrycks et al. 2018
Geifman and El- Clnia 2019 Malinin and Gales 2018
Jeong and Kim 2020 Chen et al.
2021 Wei et al.] 2021]] Ming et al. 2022b ) compute scores
from class distributions but rely heavily
labeled data, making them less suitable for label-free
scenarios. Density- based methods [Lee et al.) 2018 Zisselman and Tamar [020)] estimate sample
likelihoods but struggle with high- digeninnnnional,, complex data [Ren et al. 20199 Serrà et al. 2019].
Representation-bans based methods like KNN [Sun et al. 2022 and NNGuide PPar et al. [023] operate in embedding space but still require pre-trained ID classifiers. In contrast, SSD [Sehwag al. 2021
avoids label dependence by using self superviedd learning on unlabeled ID data.
General 00D Node Detection Methods. Entropy-based methods, such as MSP Hendryckkk and Gimpell (016), ODIN Liang et al. [2017), OE Hendrycks et al. [2018), Energy & Energy Fine
Liu et al. [2220], OODGAT [Song and Wang 2022], GNNSafe [Wu et al. 2023). NodeSafe (Yang) ct al. 2025). and GRASP Ma et al. [024]. J. as well as recent approaches like GOLD [Wang et al. 2025. EDBD [Um et al. 2025], and DeGEM [Chen et al. 2025 all rely the outputs of a pre- trained classifier, making them unsuitable for unsupervised settings. Graph anomaly detection methods, like DOMINANT (Ding et al., 2019] and SL-GAD [Zheng et al. [021]. detect general anomalies through reconstruction errors, but they struggle to distinguish between OOD nodes and general anomalies.
Unsupervised OOD Node Detection Methods. Unsupervised OOD node detection in graphs aims to identify OOD nodes without relying on multi-category labels and pretext classification tasks, posing unique challenges for traditional methods. Recent works [Li et al. 2022] Bazhenov et al., 2022 Liu et al. 2023 Ding and Shi 2023] explore graph-level OOD detection but not be directly applicd to nove- OOD detection due to the complexity of node dependencies. EnergyDef [Gong and
Sun [224] employs a synthetic OOD node strategy for unsupervised OOD node detection, and follow up by significantly improving OOD node detection performance in the unsupervised setting.
"The definition of the simplex equiangular tight frame is introduced in Appendix 6

---

Conclusion
In this paper, introduce the concept of Feature Resonance for unsupervised OOD node detection, demonstrating that unknown ID samples undergo more substamtial representation changes compared to OOD samples during the optimization of known ID samples, even in the absence of multi-class labels. To effectively capture this phenomenon, we propose a label-independent, micro-level proxy that measures feature vector movements a single training step. Building on this, we present the RSL framework, which integrates the micro-level feature resonance with synthetic OOD node generation via SGLD, enhancing OOD detection performance and offering an efficient and practical solution for unsupervised OOD node detection.
6 Acknowledgement
This paper is mainly supported by the NSFC under Grants (No. 62402424). Haobo Wang is also supported by the Fundamental Research Funds for the Central Universities (No. 22022025-00085) and Zhejiang Provincial Universities (No. 226-2025-00065).
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Yuning You, Tianlong Chen, Yongduo Sui, Ting Chen, Zhangyang Wang, and Yang Shen. Graph contrastive learning with augmentations. Advances neural information processing systems, 33: 5812-5823, 2020.
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A Notations, Definitions, Assumptions and Important Constants
A.1 Notations
Table Table of Notations and Descriptions
Notation
Description
Spaces
X, W
the imput space and the label space.
the hypothesis spaces.
Distributions
P wilds P Pin: P Fout
data distribution for wild data, labeled ID data and OOD data. the joint data distribution for ID data..
Pxy
Data and Models
w,
weight, input.
the average gradients on labeled ID data, uncertainty score. randomly generated unit vector.
y
target unit vector for ID node representations.
Vx
predicted vector for input x. predictor on labeled in-distribution inliers and outliers in the wild dataset. labeled ID data and unlabeled wild data. size of X size of Xwild the filtering threshold
ho.
X wild: X Yout
X Xwild
m
T
Xx
wild data whose uncertainty score higher than threshold T
Distances
F1
the radius of the hypothesis spaces W
1
l norm
Loss, Risk and Predictor
E(...)
ID loss function
Rx (ha)
the empirical risk w.r.t. predictor he, over data the risk w.r.t. predictor hg, over distribution IP Pxy. the error rate of regarding OOD as ID.
Ra xx (ho.)
ERRout
A.2 Definitions
Definition 3. (B -smooth). We say a loss function E(ho, (x), y) (defined over x Y) is 3 smooth, if forany X and y E Y
Vihoo (x).v) Ve(ho, 2),,))112 S 3l|w-Blww' -
Definition 4. (Gradient- basedd Distribution Discrepancy). Given distributions I and Q defined over X the Gradient-basse Distribution Discrepancy w.r.t. predictor fwr and loss t is
d (I. Q) =
RR - VRqhho Roo
where hi is a classifier which returns the closest one-hot vector of lw: Per (bo,, ha) Ex-pl(ha, hwo) and Ro(he, ha) = Ex-oel(he, ho)
Definition 5. (), -discrepancy). We say wild distribution IP arild has (y, {) -discrepancy w.r. an ID joint distribution Pin if > minwew Ry (ho) and for any parameter w W satisfying that Ro. fe.xxYlo (ho.) I y should meet the following condition
d (Pin, > E.
where Rexy (ha) (x ((ho
16

---

A.3 Assumptions
Assumption 1.
The parameter space W C B(wo, r1) C R2 (12 ball radius ri around Wo); (x), y) 2 0 and (ho, (x), v) B1 -smooth;
suP(x,))exxx [
bi;
sup(x.))xXxX ((heo (x). B1.
Assumption 2. f(f(x). [x) & minger {(f(x), v) where 8x returns the closest vector of the
predictor f's output on
A.4 Constants in Theory
Table 9: Constants in theory.
Constants
Description
B17 biri + B1
the upper bound loss (x), y).
M' 2((rri +b,
the upper bound of gradient-basee filtering score (Du et al. 2024a]
M = (VM'/2- (2t(
the upper bound of our resonance- based filtering score 7 at the f-th epoch a constant for simplified representation
M -
the dimensions of parameter spaces W
R
the optimal ID risk, i.e. R minwewExxe, (x(x,))
8(T, t)
the main error in Eq. 7 the discrepancy between Pin and Pwild the ratio of OOD) distribution in Pwild learning rate
B Main Theorems
Theorem 3. If AAssumptions and hold, Pwid has (7, () -diccrepancy w.r.t. Pay and there exists 7 E (0,1) s.l. A = (1 --) - 83, R > 0, then for
M Mr1 + 1)d
M2d
M + M(11 + 1)dy
Ω
(7 R ))
22(
n
r??22
with the probability least 9/10 for any T M, (here M, is the upper bound of filtering score T, at t-th epoch, i.e.. Tę < M,
ERR( < max (0, 1 A//3] + +Oly R2n +
max max (d, /m2)
(8)
T2 7)m
where A = 0.98 Ri, and R2 is the optimal ID risk, iee Ri = mimmwwwww Excr Li (x, e). is the dimension of the space W, denotes the t-th epoch, and a is the 00D class-prior probability in the wild.
M - 31r 3 + bT + B1, M - M3
(9)
Theorem 4. if A 2 (1 c)π for small error 2 O, then the main error 8(T, t) satisfies that
max /// < T/(V2/ 1))²
(TT = 1- 1/ (2ta 1))²
(10)
2) When learning rate a small sufficiently, and if 2 2.011 / 83, R + 1.011 then there exists 17 E (0, 1) ensuring that A > 0 and A > hold, which implies that the main error 8(T,t) O.
17

---

C Proofs of Main Theorems
C.1 Proof of Theorem
Step 1. With the probability at least 1 5 0
St w 28 Rín
Mri(Bir1 + bb)dd C
Mri( + b1jd
+ 431 cv
- (1-7)m v mlooo6/66/2
22og(6/6)
2 log(6/6)
+ 3M
+ M
- (1-r)m m log(6/0)/2
This can be proven by Lemma 7 in [Du et al. 2024a and following inequality
Sxn Ti 2 Ex ~ Xiiia wild
Ve(haxm (R:), haxm (Ri)) E6 (x)x" Vi(haxm (x,), 21)
Step 2.It easy to check that
X wild
X
Xout wild
Ex Numaa
Bx X am To +
Ex
X Ts. wild
Xwild
X wild
Step 3.Let
Mri Mri (Bim bijd
Mr(31r + b2)d
c(,m) 13110
+C1 (1 - π 7)m - / log/0//////// 6/0)/2
2 llg(66/)
2 2log(6/5)
3MV
M,
V
(1 π m mmlog(6/8)/2
Under the condition in Theorem in
Du al.
2024a
with the probability at least
38 > 0
100
g8
X wild
X wild m
B
X Yout
Xout wild
100
m
8,,
E(n, m)]
gg2
m
83 R
e[n,
out X
100
wild
log 6/8
98
8,3 R c(n,
22 + / (6g(6/8)
100
In this proof, we set
7og 6/6
98n
A(n, m) = _
83, R c(n,m))
T² 2m + क (6g(6//))
100
Note that A7 = 0.987722 -88, R then
1 - c(n, 4 m)) A(n, m)
where E(m) /log6/0/(1² 22m + log(6/5))
18

---

Step 4. Under the conditions in Theorem 5 in Du et al. 2024a and Proposition with the probability
least 100 56
{x
E
nut wild T S T}
1 min (1, (n, m))
(11)
Xout
wild
T/( 2ta-1
We prove this step: let Z be the uniform random variable with Swyt as its support and Z(i)
Ta/( 2ta-1 2 )², then by the Markov inequality, we have
{x
X X 74 TTH P(Z(T) < T 21a
V2
(nn, m) - -T/( 2tre-1
)²)
(12)
X wild
(
2
)2
21a
Step 5. If & (2/(1-e/MM))))) e/M'), then with the probability at least
97
- -500 0
100
{x1
E
out wild
T STH
c
2ta
1/
2 (n, m)
(13)
X
wild
/2
T
21a
where e((nm)) = + ((((m)
Step 6. we set ō = 3/100, then it is easy to see that
<(m) 1 00(
π² mm
<(n,m)>
S 0( + V (1 T)m
d m), & O V h
31 MM M
-1- VI
((1-x)m)
Step 7. By results in Steps and We complete this proof
C.2 Proof of Theorem 2
The first result is trivial. Hence,we omit it. We mainly focus on the second result in this theorem In this proof, then set
? 83 R - 0.990/((90....
831 R +
883 R2
Note that it is easy to check that
E M 2.011 R 1.111
V83,R 18011/831 + t.
Therefore,
5 2
83, R
+ 0.99m
/ 883 R3 +
V0.8
which implies that A > क Note that
(1 11-1)e n)
V0.98 (0..98883,R8 +
88, R3 + / R8 0.99m)
/83, R
which implies that A 0 We have completed this proof
19

---

D Necessary Propositions
D.I Boundedness
Proposition 1. If Assumption holds,
sup sup wEW (xx
|V[(h, (x), - Biri + +b, = 12
sup sup (x.))xXXX
(x). - Bir bbr + B1 = M,
Proof. One can prove this by Mean Value Theorem of Integrals easily. Proposition 2. If Assumption holds, for any w E W,
IVE (x), v))2 S 23, (ho, (x),
Proof. The details of the self-bounding property can be found in Appendix of Lei Ying Proposition 3. If A'ssumpptin holds, for any labeled data X and distribution I..
| Rx (ha))l S 23, Rx (ha.), Vw E W, V Rp (ho )ll - 23, Ro(ho)); Vw W.W.
(14)
(15)
Proof. Jensen's inequality implies that Rs (he.) and Rp(fw) are 31 -smooth. Then Proposition implies the results.
Proposition 4. If Assumption holds, for any W,
Ahe, (x) 21ا5 1 = M
Proof. is trivial that
x (x), ||S|| (x),,, ||S Biri + = M212
Then
x (x(h,,(x),1) |||| 2(xW -v) |1 2 Σ
Aog (x) 112 2 tAho (x)- 2t || Ahg (x) || -1
is straightforward to verify that:
M2 +
Aee, (x)ll2
< av M'/2 Mr.
2t
Here, a is the learning rate. From the inequality above, establish relationship between / /2/ o, and as follows:
V2
M' 2
2
2ta
E Experiment Details
We supplement experiment details for reproducibility. Our implementation is based on Ubuntu 20.04, Cuda 12.1, Pytorch 2.1.2, and Pytorch Geometric 2.6. I. All the experiments run with NVIDIA 3090 with 24GB memory.
E.1 Hyperparameter
As shown in Table 10
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---

Table 10: Hyper- erper-parameters for training.
Squirrei
WikiCs
CelpChi
Reddit
Pubmed
Chameleon
Learning rule (a)
0.005
0.01
0.005
0.005
0.01
0,005
0.005
0.01
0.005
by layers
1
ga(-) layers
2
Hidden states
16
16
16
16
16
16
16
16
16
Dropout
2
1
2
10
10
5
2
0.5
0.5
0.5
0.5
0.5
E.2 Metric
Following prior research on OOD node detection, we evaluate the detection performance using three widely recognized,
metrics: AUROC (t), AUPR (t) and FPR95(.). (1) AUROC measures the area under the receiver operating characteristic curve, capturing the trade-off between the true positive rate and the false positive rate across different threshold values. (2) AUPR calculates the area under the precision-recal curve, representing the balance between the precision rate and recall rate for OOD nodes across varying thresholds. (3) FPR95 is defined as the probability that OOD sample misclassified as an ID node when the true positive rate set at 95%.
E.3 Dataset Description
To thoroughly evaluate the effectiveness of RSL, we perform experiments real-world node-level and graph-level OOD detection datasets:
Node-level Datasets:
Squirrel [Rozemberezki al. 2021] : Wikipedia network where nodes correspond to
English Wikipedia articles, and edges represent mutual hyperlinks. Nodes are categorized into five classes following Geom-GCN (Pei al. 2020 annotations, with the network exhibiting
a high level of heterophily.
WikiCS [Mernyei and Cangea 2020]
This dataset consists of nodes representing articles in the Computer Science domain. Edges are based on hyperlinks, and nodes are classified into 10 categories, each corresponding to a unique sub-field of Computer Science.
YelpChi [Rayana and Akoglu 20155: Derived from Yelp, this dataset includes hotel and restaurant reviews. Legitimate reviews are labeled as ID nodes, while spam reviews are considered OOD nodes.
Amazon McAuley and Leskovec 2013]:
Contains reviews from the Musical Instrument category Amazon.com. ID nodes represent benign users, while OOD nodes correspond to fraudulent users.
Reddit Kumar et al. 2019: A dataset comprising user posts collected from various subreddits over a month. Normal users are treated as ID nodes, while banned users are labeled as OOD nodes.
Chameleon Rozemberczki al. [021 is Wikipedia network with 5 classes, where nodes represent web pages and edges represent hyperlinks between them. Node features represent several informative nouns in the Wikipedia pages, and the task is to predict the average daily traffic of the web page Fey and Lenssen [2019].
Cora [Kipf and Welling 2016a] is citation graph with 2,708 nodes, 5,429 edges, 1,433 features, and classes, widely used for node classification and link prediction. Under the Label Leave-out setting, classes treated ID and 4 as OOD.
Citeseer Kipf and Welling 2016a contains 3,327 nodes, 4,732 edges, 3,703 features, and classes. We apply the same OOD generation strategies as above, designating 3 classes as ID and 3 as OOD under the Label Leave-out setting.
PubMed [Kipt and Welling 2016a], biomedical citation graph, includes 19,717 nodes, 44,338 edges, 500 features, and 3 classes. We follow the same OOD generation and semi- supervised training procedure, using classes as ID and as OOD under the Label Leave-out setting.
For the Squirrel, WikiCS, YelpChi, Amazon, and Reddit datasets, we follow the same data preprocessing steps EnergyDef [Gong and Sun 2024 Both Squirrel and WikiCS datasets
21

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loaded using the DGL [Wang et al. 2019 package. For Squirrel, class {1] is selected as the OOD class, while {0, 2. 3, 4] are designated as ID classes. the case WikiCS, {4, 5 are chosen as OOD classes, with the remaining eight classes treated as ID. The YelpChi and Amazon datasets are processed based on the methodology described [Dou et al. 2020]. and the Reddit dataset prepared using the PyGod (Liu et al., [022] package. For the Cora and Chameleon datasets, we follow the data processing procedure used in GRASP [Ma et al. 2024).
Graph-level Datasets:
ENZYMES Morris al. [2020] is graph dataset constructed based the structural properties of protein molecules. It contains a total of 600 graphs, each representing one protein sample, across six different classes. The dataset includes 19,580 nodes and 174,564 edges, with each node having a feature vector dimension 3.
PROTEINS Morris et al. [0200] is a dataset of proteins that are classified as enzymes or non-enzymes. The dataset includes 1,113 graphs.
Clin (Wu et al.) 2018] compares drugs approved by the FDA and drugs that have failed clinical trials for toxicity reasons. The dataset includes two classification tasks for 1491 drug compounds with known chemical structures: (1) clinical trial toxicity (or absence of toxicity) and (2) FDA approval status.
LIPO (Wu et al., 2018) is a dataset included in MoleculeNet [Wu et al. 2018). It measures
the experimental results of octanol/water distribution coefficient(ogD pH 7.4).
We follow the data processing procedure used in GOOD-D [Liu et al., [023] that 90% of ID samples are used for training, and 10% of ID samples and the same number of OOD samples are integrated together for testing.
E.4 Baseline Description
Node-level Baselines:
LOF-KNN [Breunig et al. 2000] calculates the OOD scores of node attributes by assessing the deviation in local density relative to the k-nearest node attributes.
MLPAA uses an MLP-based autoencoder, where the reconstruction error of node attributes is used as the OOD score. It is trained by minimizing the reconstruction error on ID training nodes.
GCNNEE (Kipf and Welling, 2016b) swaps the MLP backbone for a GCN in the autoencoder. The OOD score is determined in the same way as MLPAE, following the same training process.
GAAN [Chen et al. 2020 is a generative adversarial network for attributes that evaluates sample reconstruction error and the confidence of recognizing real samples to predict OOD nodes.
2019 combines a structure reconstruction decoder and an attribute reconstruction decoder. The total reconstruction error for each node consists of the errors from both decoders.
DOMINANT [Ding et al.
ANOMALOUS [Peng et al. 2018 is an anomaly detection method that utilizes CUR decom- position and residual analysis for identifying OOD nodes.
-LL-GAD (Zheng al. 2021 derives OOD scores for nodes by considering two aspects: reconstruction error and contrastive scores.
GOOAD [Bergman and Hoshen, 2020) enhances training data by transforming it into inde- pendent spaces and trains classifier to align the augmented data with the corresponding transformations. OOD scores are then calculated based on the distances between OOD inputs and the centers of the transformation spaces. For graph-structured data, we use the same GNN backbone as Energy EnergyDef-h.
NeuTral AD QQu et al. 2021 uses learnable transformations to embed data into semantic space. The OOD score is determined by a contrastive loss applied the transformed data. MSP Hendrycks and Gimpel (016): Uses the maximum softmax probability as the OOD score. The method is simple but has limited performance on models with high confidence. ODDI Liang et al. [2017]: Improves OOD detection by temperature scaling and input perturbation, but is sensitive to hyperparameters.
Mahalanobis [ee al. [2018]: Calculates the feature distance between sample and data based on Mahalanobis distance, suitable for scenarios assuming Gaussian distribution.
22

---

-E Hendrycks et al. (2018): Optimizes using additional OOD data during training, relying on the availability of OOD data.
-Enegyy & Energy Fine'Tune Liu et al.
2020]: Uses an energy function instead of softmax
probabilities for OOD scoring, and can improve detection performance by fine-tuning with OOD data.
GKDE [hao 2020 predicts Dirichlet distributions for nodes and derives uncertainty as OOD scores by aggregating information from multiple sources.
CGPN[Stadlen al. [022]]: Based Bayesian posterior inference, performs OOD detection through uncertainty estimation. It is suitable for graph data but sensitive to hyperparameters.
OODGAT [Song and Wang 2022] is entropy- -based OOD detector that assumes node cate- gory labels are available. uses a Graph Attention Network as the backbone and determines OOD nodes based category distribution outcomes.
GNNNSff [Wu et al. 2023] calculates OOD scores by applying the Lxg function the output logits of GNN classifier, which is trained with multi-categun labels. The rationale for the OOD score is the similarity between the Softmax function and the Boltzmann distribution.
NodeSafe (Yang et al. 2025 reduces the occurrence of extreme energy values by enforcing consistency in the logit norms, thereby decreasing the variance within both the ID and OOD energy distributions, which enhances the performance of OOD node detection.
2024] enhances OOD node detection performance by amplifying the graph's homophily through rewiring, thereby improving the effect of score propagation.
GRASP (Ma et al.
-SSD (Sehwag et al., [021] is an outlier detector that leverages self-supervised representation
learning and Mahalanobis distance-bars -ased detection on unlabeled ID data. We use twice dropout to generate positive pairs for contrastive learning like SimCSE [Gao et al. 2021].
EnergyDef Gong and Sun (2024) uses Langevin dynamics to generate synthetic OOD nodes for training the OOD node classifier.
Graph-level Baselines:
Graph kernel detector:
This category of methods involves two main steps: first, graph kernel techniques employed to transform graphs into rester-based features Vishwanathan et al. 2010]; next, out-of-distribution (OOD) detection algorithms are applied to these feature vectors. Specifically, adopt the Weiseeiler-Lehman (WL) kernel (Shervashizze et al.) [201]
and the propagation kernel (PK) Neumann et al.) (2016) for representation, combined with
anomaly detectors such local outlier factor (L.OF) Breunig et al. 2000), one-class SVM
(OCSVM) Manevitz and Yousef [000]), and isolation forest (iF) Liu ct al. 2008).
GCL detector: These approaches leverage recent advances in Graph Contrastive Learning
(GCL) to derive graph-level embeddings, which are then assessed by OOD detection methods.
We employ two representative GCL techniques-IIforraaje (Sun al.. 20200 and GraphCI. generate node or graph representations. For detecting OOD instances, we [2008) and a Mahalanobis distance-based (MD) detector, which has demonstrated strong performance in identifying OOD data Schwag et al. [2021] Zhou al. [2021). GOOD-D Dio et al. [2023] can capture the latent ID patterns and accurately detect OOD graphs based on the semantic inconsistency in different granularities by performing hierarchical contrastive learning on the augmented graphs
You et al. [2020]
consider both the isolation forest (iF) Liu et al.
Liu
End-to-end: We also evaluate our model against end-to-end graph anomaly detection base- lines. One such approach is OCGIN Zhao and Akoglu 2202)) which utilizes a GIN encoder trained with a support vector data description (SVDD) loss. Another is GLocalKD Ma et al. which detects anomalous samples through knowledge distillation framework.
2022
E.5 Implementation Details
We adopt the same dataset settings as EnergyDef [Gong and Sun 2024], and we use GCN (Kipf and Welling 2016a| as the encoder. It is worth noting that, under this dataset setup, the features of unknown nodes are accessible. Therefore, using the features of unknown nodes during the training phase to filter reliable OOD nodes is a legitimate stralegy. Specifically, for the Squirrel and WikiCS datasets, we randomly select one and two classes as OOD classes, respectively. In the case of fraud detection datasets, we categorize a large number of legitimate entities as ID nodes and fraudsters as
23

---

Faddr
Searda
WpCK
(a) Reddit
(b) Squirrel
(c) YelpChi
(d) WikiCS
Figure 3: The performance of using resonance-based score to detect OOD nodes varies with training progress. The higher the AUROC, the better, and the lower the FPR95, the better.
OOD nodes. We allocate 40% of the ID class nodes for training, with the remaining nodes split into 1:2 ratio for validation and testing, ensuring stratified random sampling based on ID/OOD labels.
We report the average value of five runs for each dataset. The hyper-parameter are shown in Table TO The anomaly detection baselines are trained entirely based on graph structures and node attributes without requiring ID annotations. We adapt these models to the specifications of our OOD node detection tasks by minimizing the corresponding loss items solely on the ID nodes, where applicable.
More Experiments
F1
The Feature Resonance Phenomenon Induced by Different Target Vectors
We explore the phenomenon of feature resonance using different target vectors. Experiments conducted on two datasets with real N N-category labels, Squirrel and WikiCS (N represents the number of categories). First, based the neural collapse theory (Papyan et al. 2020 Zhou et al., 2022], we preset N target vectors, each representing a category. These N target vectors form an equiangular tight frame, maximizing the separation between them. The definition of the simplex equiangular tight frame is introduced as follows:
Definition Simplex ETF. [Xiao et al. 2024 A simplex equiangular tight frame (ETF) refers to collection of K equal- and -qquiaaallyeuuinuullll P-dimensional embedding vectors = ie,,... [ek] E Px K which satisfies:
K U(K TARe---KA 1x11)
E = DーVK-LU(K
(16)
K
where Ig the identity matrix, 1 an all-ones vector, and E PPPK((P K) allows a rotation.
All vectors in simplex ETF E have an equal l2 and the same pair-wise maximal equiangular angle K I-
T K
1
eky Ck2 K-1 ŌRI
Vkg, A2 [1, K]
(17)
K
where 66 = when h 22 and O otherwise.
We use MSE loss pull the representations of known ID nodes toward their corresponding target vectors based on their labels, as follows:
E(hg, (Xanown). e) = E(| Exnown - (Xinown WT) (12)
(18)
where Eknown denotes the target vector matrix corresponding to the known ID nodes.
The trajectory trends and lengths of unknown ID nodes differ significantly from those of OOD nodes, with the former showing more distinct trends and longer trajectories. We refer this the feature resonance phenomenon and leverage it to filter OOD nodes. As shown in Table under the "True multi-label" row, the experimental results demonstrate that this method is effective and performs well. Interestingly, even with random labels for known ID nodes or aligning all known ID representations to fixed target vector, unknown ID nodes consistently exhibit longer trajectories than unknown OOD nodes, shown in Table 6

---

Slddig Window
Overall Traectoor
F9R95
Amazon
YelpChi
Reddit
WiKCS
Souirel
Figure 4: Performance of detecting OOD nodes with different metrics. T represents the resonance- based score, the "Overall Trajectory" represents the total cumulative length of the training trajectory F(X,)= = E. and the "Sliding Window" refers to the cumulative T within window of width 10: Fio({::) 11-10 Ti.
1900
15.3
123
2:g Mirdo Mie
slcig Windo MicT
(a) WikiCS
(b) YelpChi
Figure The impact of different sliding window widths on the performance of detecting OOD nodes. When the width is 1, it corresponds to the resonance-brae based score
DED
000
Yi edon Nor
EnergyDef
(b) Score (Ours)
(c) RSL (Ours)
Figure 6: The score distribution of ID nodes and OOD nodes on Amazon.
(a) Pre-training ergyDef)
(En- (b) Pre-training (Ours)
Post-training ergyDef)
(En- (d) Post-training (Ours)
Figure 7: T-SNE visualization node embeddings on the dataset WikiCS. (a) Synthetic nodes (red) generated by EnergyDef fail to accurately represent the actual features of OOD nodes (blue), whereas ours can, as shown in (b). (c) Representations of ID (green) and OOD (blue) nodes trained with synthetic nodes generated by EnergyDef are poorly separated, whereas ours can, shown in (d).
The experiments above indicate that the feature resonance phenomenon is label-independent and results from the intrinsic relationships between ID node representations. Therefore, this is highly suitable for frre OOD detection scenarios without multi-category labels.
25

---

Table 11: Time cost (s).
Method
Squirrel
WikiCS
YelpChi
Amazon
Reddit
EnergyDef
10.94
27.11
76.51
33.81
26.44
RSL w/o classifier
5.25
4.03
5,41
5.75
3.71
RSL
11.54
17.53
74.83
36.33
38.23
Table 12: Comparison on WikiCS and Amazon datasets using different GNN encoders.
GNN Encoder
Method
WikiCS
Amazon
AUROC
AUPR
FTR@95
GCN
DeergyDet
70.22
60.10
83.17
86.57
14.30
32.43
GCN
RSL
84.01
81.14
49,23
90.03
83,91
19.60
GAT
Ene
74.22
64.15
79.80
88.20
78.40
27.88
GAT
88.01
86.37
41.02
88.28
82.90
20.48
GIN
EnergyDet
72.18
62.35
80.56
85.98
75.10
31.47
GIN
RSL
83.74
82.58
43,50
91.58
84.39
19.74
Table 13: Performance comparrssnn across different methods on graph-level OOD detection (AUROC).
Model
ID: ENZYMES OOD: PROTEIN
ID: Clin'Tox OOD: Lipo
PK-LOF
50.47=22.77
50.00=2.17
PK-OCSVM
50.46::2.77
50.06:::2.19
PK-iF
51.671:2.69
50,81=1.10
WL-LOF
52.66++2..47
51.29++3.40
WL-OCSVM
51.77::2.21
50.77:3..9
WL-iF
51.17+2.01
50.41+2.17
InfoGraph-iF
60.00++1.88
48.51
InfoGraph-MD
55.25
12.15.25.72
GraphCL-iF
61.33=2.27
47,84=0.92
GraphCL-MD
$2.87=16.11
51.58:13..64
OCGIN
$7.65=22.96
49.13::4.13
GLocclID
57.18+22.03
55.71+3.81
GOOD-Daim
61,8912.51
66. 13
GOOD-D
61.84+1.94
19.1813.61
RSL (ours)
1.89
72.03 2287
F.2 Variation of Microscopic Feature Resonance During Training
also observe the variation of the microscopic feature resonance phenomenon during the training process other datasets, as shown in Figure 3 We find that the changes Reddit, YelpChi, and WikiCS are generally consistent with Amazon, with the most significant feature resonance occurring in the middle of the training process. However, for Squirrel, the feature resonance phenomenon reaches its most pronounced level early in the training. We believe this is due to the relatively rich features in Squirrel, which allow the model to quickly identify the optimal optimization path for ID samples in the early stage of training.
Effectiveness of Different Scoring Strategies Based on Feature Resonance
We evaluate the effectiveness of three score design strategies based on feature resonance: the resonance-based score v, the global trajectory norm, and sliding window accumulation (width 10). As shown in Figure r outperforms the other two scores on most datasets. The sliding window approach performs better than the global trajectory norm. The experimental results in Figure 5 show that the sliding window width increases, the detection performance for OOD nodes gradually decreases, which indicates that finer- grained information improves OOD node detection, so we select the primary score for filtering OOD nodes in our method.
F4 Time Efficiency
compare the time consumption of our method, RSL, with the current SOTA method, EnergyDef. The experimental results are shown in Table 1T The experiments show that the overall time efficiency of RSL. comparable to that of EnergyDef, with similar time consumption across different datasets. However, it is worth noting that when we use the resonance-based score 7 alone for OOD node
26

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Table 14: Results CIFAR-10. Comparison with competitive OOD detection methods. indicates larger values are better and vice versa.
SVHN
LSUN
ISUN
Texture
Average
Method
FPRL
AUROCT
FPR.
AUROCT
FPR.
AUROCT
FPR4
AUROCT
FPRL
AUROCT
Without Contrastive Learning
MSP
59.66
91.25
45.21
93.80
54.57
92.12
66,45
88.50
$6.47
91.42
ODIN
53.78
91.30
10.93
97.93
28.44
95.51
55.59
89.47
37.19
93.55
Energy
54.41
91.22
10.19
98.05
27.52
95.59
55.23
89.37
36.83
93.56
GODIN
18.72
96.10
11.52
97.12
30.02
94.02
33.58
92.20
23.46
94.86
Mahalanobis
9.24
97.80
67.73
73.61
6.02
98.63
23.21
92.91
26.55
90.74
KNN
27.97
95.48
18,50
96.84
24.68
95.52
26.74
94.96
24.47
95.70
7 (ours)
23.50
94.85
11.48
97.80
20.93
95.67
29.22
95.28
21.28
95.90
Contrastive Learning
CSI
37.38
94.69
5.88
98.86
10.36
98.01
28.85
94.87
20.62
96.61
SSD+
1.51
99.68
6.09
33.60
95.16
12.98
97.70
13.55
97.76
KNN+
2.42
99.52
1.78
99.48
20.06
96.74
8.09
98.56
8.09
98.56
FR (ours)
3.27
99.34
0.44
99.84
9.24
98.23
14.57
97.28
6.88
98.67
Table 15: Evaluation on hard OOD detection tasks. Model is trained on CIFAR-10 with SupCon
loss.
Method
LSUN-FIX
ImageNet-FIX
ImageNot-R
Average
AUROC1
AUPRT
FPR.
AUROC1
AUPRT
FPR.
AUROC1
AUPRT
FPR.
AUROCT
AUPRT
FPR.
SSD+
95.52
96.47
29.88
94.85
95.77
32:29
93.40
94.93
45.88
94.59
95.72
36.02
KNN=
96.51
97.20
21.54
95.71
96.37
25.93
95.08
95.95
30.20
95.77
96.51
25.89
(ours)
97.10
95.13
97.74
96.29
detection, its efficiency improves significantly over EnergyDef, with an average reduction of 79.81 % in time consumption. This indicates that not only demonstrates significant effectiveness in detecting OOD nodes but also offers high efficiency.
F.5
RSL Performance with Different GNN Encoders
We conduct experiments on WikiCS and Amazon datasets using GCN (Kipf and Welling 201 GAT Velickovic et al.) 2018), and GIN [Xu et al., 2019] as the encoders, respectively. The results in Table 12 show that our RSL consistently outperforms the hate-of-theean method EnergyDef across all settings.
F.6 Graph-Level OOD Detection
Since RSL can be easily extended to independent samples beyond nodes-such as entire graphs aim to evaluate its performance on graph-level OOD detection tasks. The results Table 3 show that RSL maintains strong performance in this setting. On the ENZYMES, PROTHINS, Clin and Lipo datasets, RSL outperforms the previous strong baseline, GOOD-D. This highlights the superior scalability of RSL, demonstrating that it is not limited to node-level OOD detection.
F.7 The Generality of Feature Resonance
While our method is developed in the graph context, its core idea stems from representation dy- namics rather than graph-specific structural properties. Nevertheless, graph structure strutture-particularry homophily-car influence feature evolution and thus affect the resonance patterns. As shown in Table of the main paper, higher graph homophily correlates with more pronounced node feature resonance and improved OOD node detection performance. This is because greater homophily generally results in higher-quality node representations. Therefore, the feature resonance phenomenon itself is not solely dependent on the graph structure. To demonstrate its generality, we apply our method to standard image datasets following the setup Sun et al. 20022 strictly-using image representations extracted from 18 models trained on CIFAR-16 10 with either cross-entropy loss ("without contrastive learning") or supervised contrastive learning ("with contrastive learning"). We induce
resonance by aligning known ID features to target vector consisting of the mean of all known ID
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samples and measure step-wise changes in unknown samples. As shown in Table 14 our method remains effective on images. Models with stronger initialization (via contrastive learning) exhibit more pronounced resonance, consistent with Table 4 of the main paper. Further, Table T5 shows strong performance on a more challenging image OOD benchmark. We also evaluate our method on graph-level OOD detection (Table (3), where representations are independent like in images, and observe similarly strong results- supportng the universality the feature resonance phenomenon.
F.8 Score Distribution Visualization
We visualize the score distributions of ID and OOD nodes on the Amazon dataset obtained using different methods, as shown in Figure When using the resonance- based score (Figure (b)), the majority of unknown ID nodes show more significant representation changes compared to unknown OOD nodes. This separation of OOD nodes already exceeds EnergyDef (Figure (a)). After training with synthetic OOD nodes (Figure (c)), the separation between the energy scores of ID and OOD nodes still improves compared to EnergyDef, which demonstrates the effectiveness of RSL.
F.9 Node Representation Visualization
EnergyDef generates auxiliary synthetic OOD nodes via SGLD to train OOD classifier for category-free OOD node detection. However, we find that the synthetic OOD nodes from EnergyDef do not accurately capture the features of actual OOD nodes. As shown in Figure aa), most synthetic OOD nodes are separated from actual OOD nodes and even overlap with ID nodes, limiting the classifier's ' performance. The severe overlap between ID and OOD node representations after training by EnergyDef (Figure ()) further highlights this issue. In contrast, we use feature resonance to identify reliable OOD nodes and synthesize new ones based on these. As seen Figure bb), our synthetic OOD nodes align more closely with the actual OOD nodes. Training with these nodes results in better separation between ID and OOD node representations, shown in Figure d).
G Discussion
G.1 A Straightforward Explanation of Feature Resonance
To verify the phenomenon of Feature Resonance, we calculate the change Ahe, (X.) in the represen- tation he, (x,) of an unlabeled node from the t-th (t 0) epoch to the (t + 1)-th epoch, defined as follows:
Ahg, (Ri) ho.41 (x:) ho, (32,)
X, Vo, ( Xknown)
(19)
2alE( (xx, known
(X known W!) e)))
Term 1
Term 2
where a is the learning rate. Term I in the Equation 19 illlstratees that when the features of x; are consistent with the overall features of the labeled ID nodes X X the representation of x; undergoes a more significant change. Meanwhile, since term 2 in the Equation 19 and x; are independent, the choice of the target vector can be arbitrary. It is highly suitable for category-free OOD detection scenarios, requiring no multi-category labels as ground truth.
G.2 Why Feature Resonance Tends to Occur in the Middle Stages of Training
Empirically, observe that feature resonance peaks in the middle of training. Although it is challenging to fully explain why feature resonance is most prominent in the middle stages of training. we aim provide some theoretical insights. This aligns with the Information Bottleneck (IB) theory Tishby and Zaslavsky (2015), Saxe et al. [2019] and recent feature learning studies Allen-Zhu and Li 2022 Cao et al. 2022], which suggest that models initially memorize broad information, then gradually compress irrelevant parts while preserving task-relevan features reatures-reflecting an emerging inductive bias. This compression phase in the middle training corresponds point where irrelevant variation is reduced, allowing feature resonance to become most salient.
28

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According to the IB principle, the representation T is optimized by:
min P(T)))
1 ((x;))-((1;;))
(20)
where T(X; T) measures how much input information is retained, and (() indicates task rele- vance.
The training dynamics can be interpreted as follows:
1. Early training: TT; 1, [T; Y) is low = large information redundancy, unstable represen- tations, and little or resonance.
2. Middle training: [(X;T) T) 4. [[T; Y) 1 irrelevant information is compressed, task-relevant features are amplified, resulting in strong feature resonance.
3. Late training: possible overfitting, TT 1 again, but no further gain in I[;; Y): = represen- tations become more complex, and feature resonance diminishes.
This dynamic explains why feature resonance tends to emerge most clearly during the middle stages of training.
G.3 Differences from Gradient- Methods
It is important to note that our method RSL differs significantly from previous gradient-based methods:
1) Originating from the Commonality of Representations. Our method is based on the conjecture that there are inherent commonalities between the representations of the ID sample, which are independent of gradients.
2) No Pre-trained Multi-category Classifier Required. Gradient-based methods like GradNorm [Huang et al. 2021] compute the KL divergence between an unknown sample's softmax output from multi-category classifier and uniform distribution, using the gradient norm to distinguish OOD samples. OOD samples, with uniform softmax outputs, yield more minor gradient norms, whereas sharper outputs for ID samples produce more significant norms. Similarly, SAL (Du al., 2024a uses pseudo-label from multi-category classifier for unknown samples. continuing training to compute gradients, and identifies OOD samples via the gradient's principal component projection. These methods require pre-trained multi-category classifier, making them unsuitable for category-free scenarios without labels, whereas our RSL method avoids this limitation.
3) No Need to Compute Gradients for Unknown Samples. As shown in Equation 19 we only need the representations of unknown samples to compute our resonance-based score. This significantly enhances the flexibility of our method, as we can detect OOD samples during any optimization of known ID representations without the need to wait until after the optimization complete.
G.4 Limitations and Future Directions
While our method demonstrates strong performance in OOD detection within the graph domain, its applicability and effectiveness other domains, such as computer vision (CV), natural language processing (NLP), and multimodal data, remain largely unexplored. These domains often come with distinct data structures, noise characteristics, and task-specific challenges, which may affect the dynamics of feature resonance and the general behavior of our approach. Future work could investigate how the core principles of RSL, such as feature resonance and unsupervised separation of ID and OOD distributions, translate to domains where data representations are less structured or more abstract than in graphs.
Although our use of the validation set strictly follows the setting of the latest baseline Energy [Gong and Sun| 2024]. where the validation and test sets are constructed by randomly splitting the unknown ID and OOD nodes at 1:2 ratio, the roles of the validation set differ slightly in our approach. In Energy Def [Gong and Sun 2024), the validation set is used solely for selecting the best checkpoint. In our method, however, it serves two purposes: during Stage 1, it is used to determine the optimal threshold when identifying high-confidence OOD nodes through feature resonance, and during Stage 2, it is used to select the best checkpoint for training the OOD classifier. Nevertheless,
29

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we acknowledge that relying on a validation set introduces certain limitations, and in future work, we aim develop a feature resonance-buuse based induction method that can operate without the need for validation set.
Moreover, general-purpose algorithm originally designed for unsupervised scenarios, RSL inherently does not rely on label information. In situations where label data is available, especially in high-resource settings, currently leverages such information only indirectly through node or sample features, However, this represents an opportunity rather than limitation. A more deliberate integration of label signals could significantly enhance the learning process, especially for improving the discriminability between ID and OOD instances.
One promising direction is augment RSL with lightweight supervision semi supervisedd tech- niques. For example, incorporating label propagation methods could help better spread the influence of known IID categories across the feature space, strengthening the boundary between in-distribution and nut-o--distribution regions. Other techniques, such as consistency regularization, pseudo-labeling, or contrastive learning guided by label information, may also be explored bridge the gap between unsupervised robustness and label-aware precision.
G.5 Broader Impacts
On the positive side, our method can enhance the robustness and reliability of graph-based systems in various applications, such as fraud detection, cyhersecurity, and scientific discovery, by identifying anomalous out-r--distribution nodes without relying on labels. This has the potential improve safety and trust in real-world systems.
On the other hand, we recognize that misuse of OOD detection-such as for unjustified surveillance or exclusion of minority data- raise ethical concerns. To mitigate such risks, we emphasize the importance of transparent usage, fairness-aware evaluation, and domain-specific safeguards. While our method is label-free and unsupervised, it is critical to apply it with caution, particularly in sensitive or high-stakes domains.
30

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H
Algorithm Pseudo-code
Algorithm 1 Resonance-based Separate and Learn (RSL) Framework for Category-Free Free OOD
Detection
1: Input: Known ID nodes Vknown, Wild nodes Vwild, Target vector with random initial, Validation
set Vval
2: Output: OOD classifier Eg
3: Phase 1: Feature Resonance Phenomenon
4: Initialize model he with random parameters 0
5: for = 1 to T (training epochs) do
6: Optimize ho, (-) to align Vknwwn with target 6:
(ho., (nnww)), e) B(I1 BBI1 e (X Known WT) (12)
7: Calculate the representation change of 0; E Wwild Ahg, (S;) = ha (Xc: - ha, (X::
8: Compute resonance-based score =I| Ahe, (X:) ||2
9:
end for
Identify the period of feature resonance using the validation selecting / where best separates
ID and OOD nodes.
Phase 2: Candidate OOD Node Selection
12: Define candidate OOD set:
Veand fōr E Vwild : S TT
Phase 3: Synthetic OOD Node Generation
14: for each êg E Vayn (synthetic OOD nodes) do ((+1)
15:
Generate xj with random initial using:
x((+1) x(x))
V Eo(o,)) +) +(1 A)Exx (x - xx)) E~ 2
2
16: end for
17: Phase 4: OOD Classifier Training
18: Define training set Virain Vknown U Veand Veyn
19: Assign labels Ytrain for ID nodes (1) and OOD nodes (0)
20: Train Eg using binary cross-entropy loss:
lets = EuwVyeaia (yolog(((Bo(e)) + yo)log(1 ((E())))))
Return: Trained OOD classifier Ee
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38


Reviews:
[{"review_id": "kPTiTQoWeP", "reviewer": "Reviewer_UadC", "summary": "This paper tries to address the problem of out-of-distribution node detection and proposes a feature-space-based approach. The method leverages the fact that graphs, which are commonly encountered in semi-supervised scenarios, exhibit a phenomenon during node feature training: at certain training steps, the representations of nodes belonging to the same class as those in the training set undergo larger representation shifts. The authors refer to this as the feature resonance phenomenon and use it as the basis for identifying OOD nodes. In addition, the paper incorporates weak supervision and synthetic OOD node generation to train an OOD node classifier, further enhancing the detection performance.", "questions": "See the weaknesses mentioned above.", "limitations": "No, but the authors have proposed a potential solution to be explored in future work.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. This paper introduces a novel feature resonance phenomenon from the feature space perspective. What’s especially interesting is that this phenomenon seems to happen no matter what training setup you use — it’s not tied to specific paradigms or whether labels are available. This makes it potentially useful in both supervised and unsupervised scenarios.\n\n2. The paper provides a thorough theoretical proof for the upper bound of the proposed method's error. I went through the proof and didn’t spot any major issues.\n\n3. The experiments are comprehensive, covering different setups.\n\n4. It is well-written and easy to understand.\n\nWeaknesses :\n\n1. The paper does not seem to explain why the feature resonance phenomenon tends to occur in the middle stages of training.\n\n2. Is the feature resonance phenomenon exclusive to nodes? Does it also occur in other types of data?\n\n3. Compared to the synthetic OOD node generation method in [1], what are the specific advantages of the synthetic OOD node generation method proposed in this paper?\n\n4. What are the practical application scenarios for this method?\n\n[1] Gong, Zheng, and Ying Sun. \"An energy-centric framework for category-free out-of-distribution node detection in graphs.\" Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2024.", "was_revised": true, "final_justification": "It has addressed my concerns, and I will keep my score.", "rebuttal": "Thank you very much for your comments and suggestions! We are happy you enjoyed the paper. Below are our detailed responses:\n> **W1:**\nThe paper does not seem to explain why the feature resonance phenomenon tends to occur in the middle stages of training.  \n>**A:**\nWe appreciate the reviewer's question. Empirically, we observe that feature resonance peaks in the middle of training. Although it is challenging to fully explain why feature resonance is most prominent in the middle stages of training, we aim to provide some theoretical insights. This aligns with the **Information Bottleneck (IB) theory** [1-2] and recent feature learning studies [3-4], which suggest that **models initially memorize broad information, then gradually compress irrelevant parts while preserving task-relevant features**—reflecting an emerging inductive bias. This compression phase in the middle of training corresponds to a point where irrelevant variation is reduced, allowing feature resonance to become most salient. According to IB, the representation $T$ is optimized by: \n$$\\mathop{\\text{min}}\\limits_{p(T|X)} I(X;T) -  I(T;Y)$$ \nwhere $I(X;T)$ measures how much input information is retained, $I(T;Y)$  indicates task relevance. \n>1. **Early training:** $I(X;T) \\uparrow$, $I(T;Y)$ is low $\\Rightarrow$ large information redundancy, unstable representations, and little or no resonance;\n>2. **Middle training:** $I(X;T) \\downarrow, I(T;Y) \\uparrow$ $\\Rightarrow$ irrelevant information is compressed, task-relevant features are amplified, resulting in strong feature resonance;\n>3. **Late training:** possible overfitting, $I(X;T) \\uparrow$ again, but no further gain in $I(T;Y)$ $\\Rightarrow$ representations become more complex, and feature resonance diminishes.  \n>\n>This dynamic explains why feature resonance tends to emerge most clearly during the middle stages of training.\n\n> **W2:**\nIs the feature resonance phenomenon exclusive to nodes? Does it also occur in other types of data?  \n>**A:** Feature resonance may be more pronounced in nodes from graphs with high homophily, as node features are reinforced through propagation along edges. However, since feature resonance is defined in the feature space, we believe **it naturally generalizes across different data modalities**. To validate this, we apply our method to standard image datasets **following the setup in [5] strictly**——using image representations extracted from ResNet-18 models trained on CIFAR-10 with either cross-entropy loss (\"without contrastive learning\") or supervised contrastive learning (\"with contrastive learning\"). We induce resonance by aligning known ID features to a random target and measure step-wise changes in unknown samples. As shown in **Table 1 below**, our method remains effective on images. Models with stronger initialization (via contrastive learning) exhibit more pronounced resonance, **consistent with Table 5 of the main paper**. Further, **Table 2 below** shows strong performance on a more challenging image OOD benchmark. We also evaluate our method on **graph-level OOD detection (Appendix Table 13)**, where representations are independent like in images, and observe similarly strong results—**supporting the universality of the feature resonance phenomenon**.\n\n\n**Table 1. Results on CIFAR-10.** Comparison with competitive OOD detection methods.  ↑ indicates larger values are better and vice versa.\n|Method|SVHN|SVHN|LSUN|LSUN|iSUN|iSUN|Texture|Texture|Average|Average|\n|-|-|-|-|-|-|-|-|-|-|-|\n|| FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|\n|||||**Without Contrastive Learning**|\n|MSP|59.66|91.25|45.21|93.80|54.57|92.12|66.45|88.50|56.47|91.42|\n|ODIN|53.78|91.30|10.93|97.93|28.44|95.51|55.59|89.47|37.19|93.55|\n|Energy|54.41|91.22|10.19|98.05|27.52|95.59|55.23|89.37|36.83|93.56|\n|GODIN|18.72|96.10|11.52|97.12|30.02|94.02|33.58|92.20|23.46|94.86|\n|Mahalanobis|9.24|97.80|67.73|73.61|6.02|98.63|23.21|92.91|26.55|90.74|\n|KNN|27.97|95.48|18.50|96.84|24.68|95.52|26.74|94.96|24.47|95.70|\n|FR $\\tau$ (ours)|23.50|94.85|11.48|97.80|20.93|95.67|29.22|95.28|21.28|95.90|\n|||||**With Contrastive Learning** |\n|CSI|37.38|94.69|5.88|98.86|10.36|98.01|28.85|94.87|20.62|96.61|94.38|\n|SSD+|1.51|99.68|6.09|98.48|33.60|95.16|12.98|97.70|13.55|97.76|**95.07**|\n|KNN+|2.42|99.52|1.78|99.48|20.06|96.74|8.09|98.56|8.09|98.56|**95.07**|\n|FR $\\tau$ (ours)|3.27|99.34|0.44|99.84|9.24|98.23|14.57|97.28|**6.88**|**98.67**|**95.07**|\n\n**Table 2.** Evaluation on hard OOD detection tasks. Model is trained on CIFAR-10 with SupCon loss.\n|Method|LSUN-FIX |LSUN-FIX|LSUN-FIX|ImageNet-FIX|ImageNet-FIX|ImageNet-FIX|ImageNet-R|ImageNet-R|ImageNet-R|Average|Average|Average| \n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n||AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|\n|SSD+|95.52|96.47|29.88|94.85|95.77|32.29|93.40|94.93|45.88|94.59|95.72|36.02|\n|KNN+|96.51|97.20|21.54|95.71|96.37|25.93|95.08|95.95|30.20|95.77|96.51|25.89|\n|FR $\\tau$ (ours)|96.41|97.10|21.80|95.13|95.66|26.76|97.33|97.74|15.27|**96.29**|**96.83**|**21.28**|\n\n\n> **W3:** Compared to the synthetic OOD node generation method in EnergyDef, what are the specific advantages of the synthetic OOD node generation method proposed in this paper?  \n>**A:** Great question! Although EnergyDef [6] also uses Langevin dynamics to synthesize OOD nodes, the OOD nodes generated in this way are likely to **deviate significantly from the true OOD distribution**. Our method, on the other hand, fully leverages the semi-supervised nature of graphs—we have access to nodes with known features but unknown labels. By using feature resonance, we first **identify a set of high-confidence real OOD nodes as references**, resulting in synthesized OOD nodes that **better match the true OOD distribution**. The feature visualization in **Figure 7 of the appendix** confirms this, showing that the node representations trained with our method achieve better separation between ID and OOD nodes.\n\n> **W4:**  What are the practical application scenarios for this method?  \n>**A:**\nSince feature resonance does not depend on specific proxy tasks or multi-class label supervision, we believe it can be readily generalized to a wide range of common data filtering applications, including anomaly detection and out-of-distribution detection across arbitrary numbers of categories.\n\n[1] Tishby N, Zaslavsky N. Deep learning and the information bottleneck principle[C]//2015 ieee information theory workshop (itw). Ieee, 2015: 1-5.  \n[2] Saxe A M, Bansal Y, Dapello J, et al. On the information bottleneck theory of deep learning[J]. Journal of Statistical Mechanics: Theory and Experiment, 2019, 2019(12): 124020.  \n[3] Allen-Zhu Z, Li Y. Feature purification: How adversarial training performs robust deep learning[C]//2021 IEEE 62nd annual symposium on foundations of computer science (FOCS). IEEE, 2022: 977-988.  \n[4] Cao Y, Chen Z, Belkin M, et al. Benign overfitting in two-layer convolutional neural networks[J]. Advances in neural information processing systems, 2022, 35: 25237-25250.  \n[5] Sun Y, Ming Y, Zhu X, et al. Out-of-distribution detection with deep nearest neighbors[C]//International conference on machine learning. PMLR, 2022: 20827-20840.  \n[6] Gong, Zheng, and Ying Sun. \"An energy-centric framework for category-free out-of-distribution node detection in graphs.\" Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2024."}, {"review_id": "oNqgtiHB8X", "reviewer": "Reviewer_XXzf", "summary": "The authors propose an out-of-distribution (OOD) detection method that does not rely on pretext tasks or label supervision. They identify a new phenomenon, termed Feature Resonance, which emphasizes the feature space instead of the label space. Building on this insight, the authors introduce a novel graph OOD detection framework called RSL. This framework consists of two key components: a micro-level proxy for feature resonance and a synthetic OOD node strategy used to train the OOD classifier. The effectiveness of the proposed approach is demonstrated through experiments on 13 different datasets.", "questions": "1.\tWhy does the micro-level proxy can help measure feature resonance? This is not clearly explained.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1.\tInvestigating a setting that is independent of pretext tasks and label supervision is an interesting and valuable direction.\n2.\tThe intuition behind the feature resonance phenomenon is natural and compelling.\n3.\tThe experiments demonstrate consistent improvements across various settings.\n\nWeaknesses:\n\n1.\tIs the Micro-level Feature Resonance Phenomenon commonly observed in real-world scenarios?\n2.\tExperiments on toy datasets may not accurately reflect real-world conditions. Does the phenomenon shown in Figure 1 also appear in real-world datasets? Please provide the same experiment on real datasets for comparison.\n3.\tFigure 2 only presents the OOD detection performance, but does not illustrate the feature resonance phenomenon itself.\n4.\tIn Section 2, your method assumes that samples with representations that change rapidly and are similar to known in-distribution (ID) samples are classified as ID, while those that do not change rapidly are considered OOD. Section 2.1 formalizes this phenomenon, and Section 2.2 introduces an arbitrary target to leverage it. However, the rationale behind why this approach works is not sufficiently explained; the paper focuses more on the implementation than on the underlying reasoning.\n5.\tThe technical design in Section 2.2 appears somewhat ad hoc. In Definition 2, why is it necessary to ensure the inequality and define the filtering score in this particular way? Is this the only possible way to define the score? Please clarify the motivation for this specific formulation.\n6.\tHow does Theorem 2 relate to the feature resonance phenomenon? The connection between the theoretical analysis and the design choices is unclear.", "was_revised": true, "final_justification": "The authors have done an effective rebuttal. Thank you.", "rebuttal": "Thanks very much for your insightful comments and suggestions! Our detailed responses are as follows.\n>**W1:** Is the Micro-level Feature Resonance Phenomenon commonly observed in real-world scenarios?  \n>**A:** Feature resonance may be more pronounced in nodes from graphs with high homophily, as node features are reinforced through propagation along edges. However, since feature resonance is defined in the **feature space**, we believe this phenomenon is **generally applicable across different data modalities**. To demonstrate its generality, we apply our method to standard image datasets **following the setup in [1] strictly**——using image representations extracted from ResNet-18 models trained on CIFAR-10 with either cross-entropy loss (\"without contrastive learning\") or supervised contrastive learning (\"with contrastive learning\"). We induce resonance by aligning known ID features to a random target and measure step-wise changes in unknown samples. As shown in **Table 1 below**, our method remains effective on images. Models with stronger initialization (via contrastive learning) exhibit more pronounced resonance, **consistent with Table 5 of the main paper**. Further, **Table 2** below shows strong performance on a more challenging image OOD benchmark. We also evaluate our method on **graph-level OOD detection (Appendix Table 13)**, where representations are independent like in images, and observe similarly strong results—**supporting the universality of the feature resonance phenomenon**.\n\n**Table 1. Results on CIFAR-10.** Comparison with competitive OOD detection methods.  ↑ indicates larger values are better and vice versa.\n|Method|SVHN|SVHN|LSUN|LSUN|iSUN|iSUN|Texture|Texture|Average|Average|\n|-|-|-|-|-|-|-|-|-|-|-|\n|| FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|\n|||||**Without Contrastive Learning**|\n|MSP|59.66|91.25|45.21|93.80|54.57|92.12|66.45|88.50|56.47|91.42|\n|ODIN|53.78|91.30|10.93|97.93|28.44|95.51|55.59|89.47|37.19|93.55|\n|Energy|54.41|91.22|10.19|98.05|27.52|95.59|55.23|89.37|36.83|93.56|\n|GODIN|18.72|96.10|11.52|97.12|30.02|94.02|33.58|92.20|23.46|94.86|\n|Mahalanobis|9.24|97.80|67.73|73.61|6.02|98.63|23.21|92.91|26.55|90.74|\n|KNN|27.97|95.48|18.50|96.84|24.68|95.52|26.74|94.96|24.47|95.70|\n|FR $\\tau$ (ours)|23.50|94.85|11.48|97.80|20.93|95.67|29.22|95.28|21.28|95.90|\n|||||**With Contrastive Learning** |\n|CSI|37.38|94.69|5.88|98.86|10.36|98.01|28.85|94.87|20.62|96.61|94.38|\n|SSD+|1.51|99.68|6.09|98.48|33.60|95.16|12.98|97.70|13.55|97.76|**95.07**|\n|KNN+|2.42|99.52|1.78|99.48|20.06|96.74|8.09|98.56|8.09|98.56|**95.07**|\n|FR $\\tau$ (ours)|3.27|99.34|0.44|99.84|9.24|98.23|14.57|97.28|**6.88**|**98.67**|**95.07**|\n\n**Table 2.** Evaluation on hard OOD detection tasks. The model is trained on CIFAR-10 with SupCon loss.\n|Method|LSUN-FIX |LSUN-FIX|LSUN-FIX|ImageNet-FIX|ImageNet-FIX|ImageNet-FIX|ImageNet-R|ImageNet-R|ImageNet-R|Average|Average|Average| \n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n||AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|\n|SSD+|95.52|96.47|29.88|94.85|95.77|32.29|93.40|94.93|45.88|94.59|95.72|36.02|\n|KNN+|96.51|97.20|21.54|95.71|96.37|25.93|95.08|95.95|30.20|95.77|96.51|25.89|\n|FR $\\tau$ (ours)|96.41|97.10|21.80|95.13|95.66|26.76|97.33|97.74|15.27|**96.29**|**96.83**|**21.28**|\n\n>**W2 & W3:** Does the phenomenon shown in Figure 1 also appear in real-world datasets? Figure 2 does not illustrate the feature resonance phenomenon itself.   \n>**A:**  While toy datasets offer the **clearest and most intuitive illustration** of the feature resonance phenomenon, we have conducted extensive experiments on real-world benchmarks (Amazon, Squirrel, WikiCS, YelpChi, Reddit). These show **consistent micro-level resonance patterns**, where ID and OOD nodes differ in representation dynamics across training.  \nAlthough macro-level resonance (Figure 1) may be less visible in real data due to noise, micro-level resonance is stable and general across all datasets (please refer to Figure 4 of the Appendix). Figure 2 and Figure 3 (Appendix) visualize these effects as real-data counterparts to the toy example.    \nRegarding Figure 2, the performance metrics are based on the relative representation shift of unknown ID vs. OOD samples. **Therefore, Figure 2 implicitly reflects the micro-level feature resonance phenomenon.** Since **this year's NeurIPS rebuttal does not allow PDF uploads**, we regret that we are unable to present the additional visualizations. We sincerely apologize for this limitation and **will make sure to include a Figure 1-style visualization on real datasets in the revised version.**\n\n>**W4:** ... However, the rationale behind why this approach works is not sufficiently explained.  \n>**A:** Thank you for your insightful comment. The rationale behind our approach is grounded in the observation that **ID samples tend to co-evolve in the feature space due to shared semantic structure**, whereas OOD samples lack such alignment and exhibit weaker representation dynamics, leading to measurable differences in representation shift.  \nAlthough it is challenging to fully explain why feature resonance is most prominent in the middle stages of training, we aim to provide some theoretical insights. This aligns with the **Information Bottleneck (IB) theory** [2-3], which suggest that **models initially memorize broad information, then gradually compress irrelevant parts while preserving task-relevant features**. This compression phase in the middle of training corresponds to a point where irrelevant variation is reduced, allowing feature resonance to become most salient. According to IB, the representation $T$ is optimized by: $$\\mathop{\\text{min}}\\limits_{p(T|X)} I(X;T) -  I(T;Y)$$ where $I(X;T)$ measures how much input information is retained, $I(T;Y)$ indicates task relevance. \n>1. **Early training:** $I(X;T) \\uparrow$, $I(T;Y)$ is low $\\Rightarrow$ large information redundancy, unstable representations, and little or no resonance;\n>2. **Middle training:** $I(X;T) \\downarrow, I(T;Y) \\uparrow$ $\\Rightarrow$ irrelevant information is compressed, task-relevant features are amplified, resulting in strong feature resonance;\n>3. **Late training:** possible overfitting, $I(X;T) \\uparrow$ again, but no further gain in $I(T;Y)$ $\\Rightarrow$ representations become more complex, and feature resonance diminishes.  \n>\n>This dynamic explains why feature resonance tends to emerge most clearly during the middle stages of training. We will clarify this motivation and the theoretical grounding more explicitly in the revised version.\n\n>**W5:**  In Definition 2, why is it necessary to ensure the inequality and define the filtering score in this particular way? Is this the only possible way to define the score?   \n>**A:** Thank you for raising these questions. The filtering score in Definition 2 is designed to quantify the relative feature variation of each unknown node during training by leveraging the feature resonance phenomenon. The intuition is as follows:\n According to our observations, unknown ID samples tend to exhibit stronger and more consistent feature updates during training compared to OOD samples.\n Therefore, by measuring how much a sample's representation changes across training epochs, we can estimate its likelihood of being an ID sample.\n While this formulation may not be the only possible one, **it is simple, effective, and empirically validated across multiple datasets** (see **Table 7** of the main paper for comparisons with alternative designs).\n We also chose this formulation for its interpretability—**it directly reflects the dynamic behavior of representations over time, which is central to the resonance effect.**\n\n>**W6:**  How does Theorem 2 relate to the feature resonance phenomenon?   \n>**A:**  Great question! Theorem 2 provides a theoretical justification for the timing of the feature resonance phenomenon. Specifically, it shows that **the upper bound of the filtering error related to micro-level feature changes decreases and then increases over training steps**, implying that there exists a middle stage during training where the feature representations of ID and OOD samples diverge most clearly. **This directly supports our design choice of leveraging micro-level feature resonance in a step-wise manner**, especially during this critical middle phase, for more reliable OOD detection.\n\n>**Q1:**  Why does the micro-level proxy can help measure feature resonance?   \n>**A:** Thank you for the thoughtful question. The micro-level proxy captures fine-grained, step-wise changes in representations during training. Intuitively, **feature resonance means some samples—especially unknown ID ones—react more strongly to optimization aligned with known ID patterns.** Our proxy measures this by tracking the magnitude of feature shifts over training steps. Empirically, unknown ID samples show larger representation changes than OOD samples under the same inductive bias. Thus, the micro-level proxy effectively indicates feature resonance, even without multi-class label information.\n\n[1] Sun Y, et al. \"Out-of-distribution detection with deep nearest neighbors.\"ICML'22  \n[2] Tishby N, et al. \"Deep learning and the information bottleneck principle.\" ITW'15  \n[3] Saxe A M, et al. \"On the information bottleneck theory of deep learning.\" J STAT MECH-THEORY E'19"}, {"review_id": "6EsfjLNXVE", "reviewer": "Reviewer_J96B", "summary": "Existing graph Out-of-distribution (OOD) node detection methods face a critical limitation by heavily depending on labeled data and well-defined pretext classification tasks. To overcome this, the paper introduces ​​RSL, a novel framework exploiting a newly discovered ​​Feature Resonance​​ phenomenon: during optimization, unknown in-distribution (ID) samples consistently exhibit larger representation changes than OOD samples even without real label supervision. RSL leverages this by creating a practical micro-level proxy to measure feature vector movement during a single training step and integrating it with synthetic OOD nodes; this combination enables training an effective, label-independent OOD classifier, with theoretical guarantees of superior separability during the resonance period and achieving state-of-the-art performance.", "questions": "1. The approach of \"aligning the features of known ID nodes to an arbitrary target vector\" (in Line 153) appears counterintuitive. ​​Beyond toy datasets, does the method also apply this \"align the features of known ID nodes to an arbitrary target vector\" strategy using Equation (2) to other datasets?​\n2. According to Definition 2 (in line 166-170), identifying t is of crucial importance, as the choice of t is highly sensitive. For different datasets, does the method employ distinct t values?​\n3. Section 2.1 and Section 2.2 enhance OOD detection from two distinct perspectives. How are these two components interconnected? Furthermore, does leveraging synthetic OOD nodes to fine-tune the model impact the selection of the critical parameter t defined in Section 2.1?\n4. The methodology presented in this paper appears to exhibit limited relevance to graph-specific structures. Have the authors validated their approach on standard image datasets? \n\nIf the authors can effectively address my questions, I will consider raising the rating.", "limitations": "Please see weakness and questions.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths：\n1. ​​Compared to the baseline, the method demonstrates good performance.​​\n2. The paper is well-organized, with clear writing logic making it easy to follow.​​\n3. ​​The motivation behind the method is good.​\n\nWeaknesses：\n1. Although Feature Resonance has novelty to some extent, identifying the appropriate period (in Line 183-184) is critical. While the paper uses the validation set for this, the introduction of the validation set is not specifically elaborated. Furthermore, without a validation set, the method would become inapplicable.\n2. The explanation for Equation (4) is insufficient. The paper fails to clearly justify why synthetic OOD nodes need to be generated or why they are synthesized in this specific manner.", "was_revised": true, "final_justification": "The author has addressed most of my concerns. After reviewing the other reviewers' comments and authors' response, I have decided to raise my rating to 4.", "rebuttal": "Thanks very much for your insightful comments and suggestions! Below we address the feedback and comments in detail:\n\n>**W1:** ... the introduction of the validation set is not specifically elaborated. Furthermore, without a validation set, the method would become inapplicable.  \n>**A:** **Our use of the validation set strictly follows the setup of our latest baseline EnergyDef [1]**, where the validation and test sets are constructed by randomly splitting the unknown ID and OOD nodes in a 1:2 ratio. In EnergyDef [1], the validation set is used to select the best checkpoint. In our method, the validation set serves **two roles**: during Stage 1, it is used to select the optimal $t$ when mining high-confidence OOD nodes via feature resonance; during Stage 2, it is used to select the best checkpoint for training the OOD classifier. **Importantly, this strategy of selecting the best model based on validation performance is also widely adopted in prior OOD detection works [1-6].**\n\n>**W2:** The explanation for Equation (4) is insufficient.  \n>**A:** Thank you for raising this important point. Our decision to generate synthetic OOD nodes via Equation (4) is motivated by two key considerations:  \n**First**, although we believe that **feature resonance is a fundamental phenomenon** that offers **valuable insight** into OOD detection, we also recognize the community's strong focus on achieving state-of-the-art performance (a gentle complaint with full respect). Since recent baselines (e.g., EnergyDef [1]) commonly adopt synthetic OOD generation, we include a similar component to ensure fair comparison. Importantly, this integration is not superficial—our method is **orthogonal** to existing techniques and can enhance them by using feature resonance to guide the selection of informative OOD signals, leading to more principled synthesis.    \n**Second**,in semi-supervised graph settings, many unlabeled node representations—including real OOD instances—are inherently present. We leverage feature resonance to **identify these nodes with high confidence**, then **synthesize additional OOD nodes guided by them**, as detailed in Equation (4). Compared to heuristic-based baselines like EnergyDef, our method produces samples that **better match real OOD distributions (Appendix Fig. 7)**.  \nThe effectiveness of this design is supported by Table 2, where training with our synthetic OOD nodes consistently improves performance. Table 7 further shows that **feature resonance reliably selects high-quality real OOD nodes, outperforming other strategies**.\n\n>**Q1:** ...Beyond toy datasets, does the method also apply this \"align the features of known ID nodes to an arbitrary target vector\" strategy using Equation (2) to other datasets?   \n>**A:** \nYes, **for all datasets**, we consistently align the representations of known ID samples toward an arbitrary fixed vector to induce the feature resonance phenomenon. \n\n>**Q2:** ...For different datasets, does the method employ distinct $t$ values?   \n>**A:**\nThank you for the insightful question! The optimal value of $t$ does vary across datasets; however, in practice, **we do not manually set $t$ for each individual dataset**. Instead, we select the optimal $t$ by evaluating performance on a validation set. **This approach aligns with the common practice in prior OOD detection methods [1-6]**, where validation sets are routinely used to choose the best checkpoint. That said, we also observe that the performance is relatively stable across a reasonable range of $t$ values, especially once it falls within the effective range for most datasets (as shown in Figures 2 and 3). This suggests that $t$ is not overly sensitive in practice, and our method remains robust under typical validation-based selection.\n\n>**Q3-1:** Section 2.1 and Section 2.2 enhance OOD detection from two distinct perspectives. How are these two components interconnected?   \n>**A:** Great question! In Section 2.1, we use a toy dataset to intuitively illustrate macro-level feature resonance: unknown ID samples shift more than OOD ones. However, this pattern is not consistent on real datasets (Section 2.2). To address this, we perform a finer-grained, step-wise analysis—termed micro-level feature resonance—**which consistently appears across all real datasets.**\nAlthough it is challenging to fully explain micro-level feature resonance, we aim to provide some theoretical insights. This aligns with the **Information Bottleneck (IB) theory** [7-8], which suggests that **models initially memorize broad information, then gradually compress irrelevant parts while preserving task-relevant features**.  According to IB, the representation $T$ is optimized by: $$\\mathop{\\text{min}}\\limits_{p(T|X)} I(X;T) -  I(T;Y)$$ where $I(X;T)$ measures how much input information is retained, $I(T;Y)$  indicates task relevance. \n>1. **Early training:** $I(X;T) \\uparrow$, $I(T;Y)$ is low $\\Rightarrow$ large information redundancy, unstable representations, and little or no resonance;\n>2. **Middle training:** $I(X;T) \\downarrow, I(T;Y) \\uparrow$ $\\Rightarrow$ irrelevant information is compressed, task-relevant features are amplified, resulting in strong feature resonance;\n>3. **Late training:** possible overfitting, $I(X;T) \\uparrow$ again, but no further gain in $I(T;Y)$ $\\Rightarrow$ representations become more complex, and feature resonance diminishes.  \n>\n>This dynamic shows that the compression phase in the middle of training corresponds to a point where irrelevant variation is reduced, allowing feature resonance to become most salient.\n\n>**Q3-2:** Does leveraging synthetic OOD nodes to fine-tune the model impact the selection of the critical parameter $t$ defined in Section 2.1?  \n>**A:** **The use of synthetic nodes does not affect the selection of the optimal $t$**, because $t$ is determined in the first stage to select high-confidence real OOD nodes, and these nodes guide the update of synthetic OOD representations in the second stage, the synthetic nodes do not affect the choice of $t$.\n\n>**Q4:** ...Have the authors validated their approach on standard image datasets?  \n>**A:**\nInsightful question!  While our method is originally proposed in the context of node-level OOD detection, **the core idea of feature resonance is rooted in representation dynamics**. As per your suggestion, we apply our method to standard image datasets **following the setup in [9] strictly**—using image representations extracted from ResNet-18 models trained on CIFAR-10 with either cross-entropy loss (\"without contrastive learning\") or supervised contrastive learning (\"with contrastive learning\"). We induce resonance by aligning known ID features to a random target and measure step-wise changes in unknown samples. As shown in Tables 1 and 2 below, our method remains effective on images. Models with stronger initialization (via contrastive learning) exhibit more pronounced resonance, **consistent with Table 5 of the main paper**. We also evaluate our method on **graph-level OOD detection (Appendix Table 13)**, where representations are independent like in images, and observe similarly strong results—**supporting the universality of the feature resonance phenomenon**.\n\n**Table 1. Results on CIFAR-10.** Comparison with competitive OOD detection methods.  ↑ indicates larger values are better and vice versa.\n|Method|SVHN|SVHN|LSUN|LSUN|iSUN|iSUN|Texture|Texture|Average|Average|\n|-|-|-|-|-|-|-|-|-|-|-|\n|| FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|\n|||||**Without Contrastive Learning**|\n|MSP|59.66|91.25|45.21|93.80|54.57|92.12|66.45|88.50|56.47|91.42|\n|ODIN|53.78|91.30|10.93|97.93|28.44|95.51|55.59|89.47|37.19|93.55|\n|Energy|54.41|91.22|10.19|98.05|27.52|95.59|55.23|89.37|36.83|93.56|\n|GODIN|18.72|96.10|11.52|97.12|30.02|94.02|33.58|92.20|23.46|94.86|\n|Mahalanobis|9.24|97.80|67.73|73.61|6.02|98.63|23.21|92.91|26.55|90.74|\n|KNN|27.97|95.48|18.50|96.84|24.68|95.52|26.74|94.96|24.47|95.70|\n|FR $\\tau$ (ours)|23.50|94.85|11.48|97.80|20.93|95.67|29.22|95.28|21.28|95.90|\n|||||**With Contrastive Learning** |\n|CSI|37.38|94.69|5.88|98.86|10.36|98.01|28.85|94.87|20.62|96.61|94.38|\n|SSD+|1.51|99.68|6.09|98.48|33.60|95.16|12.98|97.70|13.55|97.76|**95.07**|\n|KNN+|2.42|99.52|1.78|99.48|20.06|96.74|8.09|98.56|8.09|98.56|**95.07**|\n|FR $\\tau$ (ours)|3.27|99.34|0.44|99.84|9.24|98.23|14.57|97.28|**6.88**|**98.67**|**95.07**|\n\n**Table 2.** Evaluation on hard OOD detection tasks. Model is trained on CIFAR-10 with SupCon loss.\n|Method|LSUN-FIX |LSUN-FIX|LSUN-FIX|ImageNet-FIX|ImageNet-FIX|ImageNet-FIX|ImageNet-R|ImageNet-R|ImageNet-R|Average|Average|Average| \n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n||AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|\n|SSD+|95.52|96.47|29.88|94.85|95.77|32.29|93.40|94.93|45.88|94.59|95.72|36.02|\n|KNN+|96.51|97.20|21.54|95.71|96.37|25.93|95.08|95.95|30.20|95.77|96.51|25.89|\n|FR $\\tau$ (ours)|96.41|97.10|21.80|95.13|95.66|26.76|97.33|97.74|15.27|**96.29**|**96.83**|**21.28**|\n\n\n[1] Gong Z, et al. \"An energy-centric framework for category-free out-of-distribution node detection in graphs.\" KDD'24.  \n[2] Zheng Y, et al. \"Generative and contrastive self-supervised learning for graph anomaly detection.\" TKDE'21.  \n[3] Bergman L, et al. \"Classification-based anomaly detection for general data.\" ICLR'20.  \n[4] Qiu C, et al. \"Neural transformation learning for deep anomaly detection beyond images.\" ICML'21  \n[5] Zhao X, et al. \"Uncertainty aware semi-supervised learning on graph data.\" NIPS'20  \n[6] Song Y, et al. \"Learning on graphs with out-of-distribution nodes.\" KDD'22  \n[7] Tishby N, et al. \"Deep learning and the information bottleneck principle.\" ITW'15  \n[8] Saxe A M, et al. \"On the information bottleneck theory of deep learning.\" J STAT MECH-THEORY E'19  \n[9] Sun Y, et al. \"Out-of-distribution detection with deep nearest neighbors.\" ICML'22"}, {"review_id": "OqhCHDr8TM", "reviewer": "Reviewer_svYR", "summary": "This paper tackles the important challenge of out-of-distribution (OOD) node detection in graphs without relying on label supervision or predefined pretext tasks, which limits existing methods. The authors introduce a novel perspective based on Feature Resonance, observing that OOD nodes exhibit less representation change than unknown in-distribution nodes during training, even without labels. Building on this insight, they propose RSL, a framework that quantifies feature movement as a proxy for resonance and leverages synthetic OOD nodes to train an effective classifier. Theoretical analysis and extensive experiments on thirteen real-world datasets demonstrate the method’s strong separability and state-of-the-art performance.", "questions": "Please see the weaknesses.", "limitations": "None", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The core idea is good—the observation that ID and OOD samples follow different feature evolution trajectories, and leveraging this for OOD detection is both novel and interesting.\n2. The paper provides a degree of theoretical analysis to support its methodology.\n3. The visualizations are clear and effectively illustrate the proposed concepts.\n\nWeaknesses:\n1. The phenomenon presented in Section 2 is demonstrated under a labeled setting, raising concerns about whether it still holds in the label-free scenario that the paper aims to address.\n2. The connection between the observed phenomenon and graph-structured data is not sufficiently discussed. eg,. Is this phenomenon universal? Is it related to structural information?\n3. The experimental setup lacks clarity, particularly regarding how the detection stage t is determined and how OOD nodes are defined and separated.", "was_revised": true, "final_justification": "After reviewing the other reviewers' comments and authors' response, I have decided to maintain my positive rating.", "rebuttal": "Thanks very much for your insightful comments and suggestions! Our detailed responses are as follows.\n\n>**W1:** The phenomenon presented in Section 2 is demonstrated under a labeled setting, raising concerns about whether it still holds in the label-free scenario that the paper aims to address.  \n>**A:** \n The labels in Section 2.1 are used purely for explanatory purposes—**to verify and visualize the existence of feature resonance** by comparing ID and OOD groups. However, our actual approach in Section 2.2 does **not rely on any multi-class labels**. Instead, it leverages the alignment of all known ID representations to an arbitrary fixed target vector—meaning a vector chosen independently of any class labels—to induce and observe micro-level feature dynamics such as step-wise representation shifts. We have conducted extensive experiments on real-world benchmarks (Amazon, Squirrel, WikiCS, YelpChi, Reddit), which show **consistent micro-level resonance patterns**, where ID and OOD nodes differ in representation dynamics across training (Figure 2 and Figure 3 of the Appendix).  \n\n\n\n>**W2:** The connection between the observed phenomenon and graph-structured data is not sufficiently discussed. eg,. Is this phenomenon universal? Is it related to structural information?  \n>**A:**  Insightful question!  While our method is developed in the graph context, **its core idea stems from representation dynamics** rather than graph-specific structural properties. Nevertheless, graph structure—particularly homophily—can influence feature evolution and thus affect the resonance patterns. **As shown in Table 6 of the main paper**, higher graph homophily correlates with more pronounced node feature resonance and improved OOD node detection performance. This is because **greater homophily generally results in higher-quality node representations**. Therefore, the feature resonance phenomenon itself is not solely dependent on the graph structure.    \nTo demonstrate its generality, we apply our method to standard image datasets **following the setup in [1] strictly**—using image representations extracted from ResNet-18 models trained on CIFAR-10 with either cross-entropy loss (\"without contrastive learning\") or supervised contrastive learning (\"with contrastive learning\"). We induce resonance by aligning known ID features to a random target vector and measure step-wise changes in unknown samples. **As shown in Table 1 below**, our method remains effective on images. Models with stronger initialization (via contrastive learning) exhibit more pronounced resonance, **consistent with Table 5 of the main paper**. Further, Table 2 below shows strong performance on a more challenging image OOD benchmark. We also evaluate our method on **graph-level OOD detection (Appendix Table 13)**, where representations are independent like in images, and observe similarly strong results—**supporting the universality of the feature resonance phenomenon**.\n\n**Table 1. Results on CIFAR-10.** Comparison with competitive OOD detection methods.  ↑ indicates larger values are better and vice versa.\n|Method|SVHN|SVHN|LSUN|LSUN|iSUN|iSUN|Texture|Texture|Average|Average|\n|-|-|-|-|-|-|-|-|-|-|-|\n|| FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|FPR↓|AUROC↑|\n|||||**Without Contrastive Learning**|\n|MSP|59.66|91.25|45.21|93.80|54.57|92.12|66.45|88.50|56.47|91.42|\n|ODIN|53.78|91.30|10.93|97.93|28.44|95.51|55.59|89.47|37.19|93.55|\n|Energy|54.41|91.22|10.19|98.05|27.52|95.59|55.23|89.37|36.83|93.56|\n|GODIN|18.72|96.10|11.52|97.12|30.02|94.02|33.58|92.20|23.46|94.86|\n|Mahalanobis|9.24|97.80|67.73|73.61|6.02|98.63|23.21|92.91|26.55|90.74|\n|KNN|27.97|95.48|18.50|96.84|24.68|95.52|26.74|94.96|24.47|95.70|\n|FR $\\tau$ (ours)|23.50|94.85|11.48|97.80|20.93|95.67|29.22|95.28|21.28|95.90|\n|||||**With Contrastive Learning** |\n|CSI|37.38|94.69|5.88|98.86|10.36|98.01|28.85|94.87|20.62|96.61|94.38|\n|SSD+|1.51|99.68|6.09|98.48|33.60|95.16|12.98|97.70|13.55|97.76|**95.07**|\n|KNN+|2.42|99.52|1.78|99.48|20.06|96.74|8.09|98.56|8.09|98.56|**95.07**|\n|FR $\\tau$ (ours)|3.27|99.34|0.44|99.84|9.24|98.23|14.57|97.28|**6.88**|**98.67**|**95.07**|\n\n**Table 2.** Evaluation on hard OOD detection tasks. The model is trained on CIFAR-10 with SupCon loss.\n|Method|LSUN-FIX |LSUN-FIX|LSUN-FIX|ImageNet-FIX|ImageNet-FIX|ImageNet-FIX|ImageNet-R|ImageNet-R|ImageNet-R|Average|Average|Average| \n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n||AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|AUROC↑|AUPR↑|FPR↓|\n|SSD+|95.52|96.47|29.88|94.85|95.77|32.29|93.40|94.93|45.88|94.59|95.72|36.02|\n|KNN+|96.51|97.20|21.54|95.71|96.37|25.93|95.08|95.95|30.20|95.77|96.51|25.89|\n|FR $\\tau$ (ours)|96.41|97.10|21.80|95.13|95.66|26.76|97.33|97.74|15.27|**96.29**|**96.83**|**21.28**|\n\n\n>**W3:** The experimental setup lacks clarity, particularly regarding how the detection stage $t$ is determined and how OOD nodes are defined and separated.  \n>**A:**  Thank you for your thoughtful comments! We clarify the experimental setup as follows: Our experimental setup strictly follows the latest baseline EnergyDef [2] to ensure consistency. **Detailed configurations are provided in Appendix E.** The detection stage $t$ refers to the epoch at which micro-level feature resonance is most prominent. **We do not manually set this value**—instead, $t$ is automatically selected based on performance on a small validation set. **Our use of the validation set strictly follows the setup of the latest baseline EnergyDef [2]**, where the validation and test sets are constructed by randomly splitting the unknown ID and OOD nodes in a 1:2 ratio. In EnergyDef [2], the validation set is used to select the best checkpoint. In our method, the validation set serves **two roles**: during Stage 1, it is used to select the optimal $t$ when mining high-confidence OOD nodes via feature resonance; during Stage 2, it is used to select the best checkpoint for training the OOD classifier. **Importantly, this strategy of selecting the best model based on validation performance is also widely adopted in prior OOD detection works [2-7].**  \nAs for how OOD nodes are defined and separated, during stage one, we use the resonance-based score to rank all unlabeled nodes and select the high-confidence OOD samples. These are used in stage two for training the OOD classifier and as anchors to synthesize additional OOD representations. We will add clarifications in the revised version to make these points more explicit.\n\n\n[1] Sun Y, et al. \"Out-of-distribution detection with deep nearest neighbors.\" ICML'22   \n[2] Gong Z, et al. \"An energy-centric framework for category-free out-of-distribution node detection in graphs.\" KDD'24.  \n[3] Zheng Y, et al. \"Generative and contrastive self-supervised learning for graph anomaly detection.\" TKDE'21.  \n[4] Bergman L, et al. \"Classification-based anomaly detection for general data.\" ICLR'20.  \n[5] Qiu C, et al. \"Neural transformation learning for deep anomaly detection beyond images.\" ICML'21  \n[6] Zhao X, et al. \"Uncertainty aware semi-supervised learning on graph data.\" NIPS'20  \n[7] Song Y, et al. \"Learning on graphs with out-of-distribution nodes.\" KDD'22"}]

Author rebuttal:

Meta-review: The paper addresses OOD node detection in graphs. One major challenge is to devise a scalable OOD detection method independent of both pretext tasks and label supervision. The authors formulate a new concept called Feature Resonance, which focuses on the feature space rather than the label space. Subsequently, the authors propose a Resonance-based Separation and Learning (RSL) method. Both theoretical and empirical results demonstrate the validity of the proposed approach.

Strengths:
- The feature resonance notion is novel and a promising direction to explore.
- Solid, well-founded theoretical analysis
- Comprehensive experimental results.

Weaknesses:
- The feature resonance phenomenon on toy datasets may not reflect real-world scenarios, raising questions about its generality. 
- The validation process is not clearly explained.
- Experimental setup needs to be more detailed

Overall, there is general consensus among reviewers on the strengths of the paper. The weaknesses can be clarified or explained in the camera ready, as they have already been addressed during rebuttal as acknowledged by all reviewers.
Decision: Accept (poster)