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ICLR/2025/Hydra-SGG_ Hybrid Relation Assignment for One-stage Scene Graph Generation/full.md

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+![](images/7b2b54023912115b2a86baf085ae2bcd794dd130e66e99f483479bc915491f1b.jpg)
+
+# HYDRA-SGG: HYBRID RELATION ASSIGNMENT FOR ONE-STAGE SCENE GRAPH GENERATION
+
+Minghan Chen† Guikun Chen‡ Wenguan Wang‡ Yi Yang‡*
+
+†ReLER Lab, AAll, University of Technology Sydney
+
+$\ddagger$ ReLER Lab, CCAI, Zhejiang University
+
+# ABSTRACT
+
+DETR introduces a simplified one-stage framework for scene graph generation (SGG) but faces challenges of sparse supervision and false negative samples. The former occurs because each image typically contains fewer than 10 relation annotations, while DETR-based SGG models employ over 100 relation queries. Each ground truth relation is assigned to only one query during training. The latter arises when one ground truth relation may have multiple queries with similar matching scores, leading to suboptimally matched queries being treated as negative samples. To address these, we propose Hydra-SGG, a one-stage SGG method featuring a Hybrid Relation Assignment. This approach combines a One-to-One Relation Assignment with an IoU-based One-to-Many Relation Assignment, increasing positive training samples and mitigating sparse supervision. In addition, we empirically demonstrate that removing self-attention between relation queries leads to duplicate predictions, which actually benefits the proposed One-to-Many Relation Assignment. With this insight, we introduce Hydra Branch, an auxiliary decoder without self-attention layers, to further enhance One-to-Many Relation Assignment by promoting different queries to make the same relation prediction. Hydra-SGG achieves state-of-the-art performance on multiple datasets, including VG150 (16.0 mR@50), Open Images V6 (50.1 weighted score), and GQA (12.7 mR@50). Our code and pre-trained models will be released on Hydra-SGG.
+
+# 1 INTRODUCTION
+
+A scene graph is a data structure that describes the entities (objects) in a scene and the relations between these objects [3]. Scene graph generation (SGG) has attracted significant research attention [20, 50, 3, 43, 71] due to its ability to enhance machines' semantic comprehension of visual content. It has been widely adopted in various downstream applications, such as robotic vision and interaction [47, 60, 80, 48], image synthesis and manipulation [79, 10, 22], visual question answering [21, 66, 41], and video understanding [78, 68, 76, 81].
+
+Mainstream SGG methods [85, 64, 50, 91, 61, 29, 35] work in a two-stage fashion. First, an off-the-shelf object detector extracts all entities within an image. Then, the extracted entities are permuted, yielding  $N(N - 1)$  entity pairs for  $N$  detected entities. These entity pairs are
+
+![](images/189c4eb63a84cccd057495ef6483f47b7f6e512a962437c5e6e1fa1bd558a954.jpg)  
+Figure 1: Comparison with other SGG methods in mR@50 and training epochs on VG150 [75].
+
+used to predict the relationships between the corresponding entities. However, the two-stage methods face a critical limitation: they predict relations for all entity pairs, even though many pairs do
+
+![](images/12f210965cd0e5816ba50303828ad5c37a298b580cbc7ad6026e99b7c469807a.jpg)  
+Figure 2: (a) Previous DETR-based SGG methods such as RelTR [7] and SGTR [40] match each GT relation with only one query. (b) Our Hybrid Relation Assignment utilizes both One-to-One and One-to-Many assignments, generating more positive samples and thus accelerating training.
+
+![](images/f59ac9b5713ec03bef191bda4566d39b4bbc792ceb30a06492eedf51ac710665.jpg)
+
+not participate in any relations. This incurs heavy computational overhead and time-consuming inference, especially in complex scenes with numerous entities and intricate interactions.
+
+Recently, the simplicity of DETR-based SGG methods [7, 40] has led to an ongoing paradigm shift apart from the two-stage framework. Specifically, a sequence of visual tokens, mapped from the input image, interacts with a predefined set of relation queries in a Transformer decoder for simultaneous object detection and relation prediction. Then, One-to-One set matching (i.e., Hungarian matching [28]) is used to assign ground truth labels to the predictions (Fig. 2a). This set-prediction framework allows DETR-based SGG models to eliminate hand-designed components, such as Non-Maximum Suppression (NMS), which are commonly used in traditional two-stage methods.
+
+Unfortunately, one significant drawback of DETR-based SGG models is their slow convergence. For instance, one-stage RelTR [7] requires 150 training epochs to converge, while a recent two-stage method, PE-Net [91], only needs 32 epochs. This drawback can be attributed to the sparse relation supervision induced by Hungarian matching, which assigns each ground truth to only one relation query. The sparsity of relation annotations per image further exacerbates this issue. For instance, VG150 [75] train averages only 5.5 ground truth relation triplets per image [43]. This means that each image provides a mere 5.5 positive relation queries to optimize the loss functions. In the case of RelTR [7], which has 200 predefined relation queries, only  $2.75\%$  of these queries are positive samples per optimization step. Consequently, relation queries require more optimization steps to learn due to the limited number of positive samples for training. Furthermore, approximately  $50\%$  [26] of the plausible but suboptimally matched queries (i.e., queries with correct subject-object classification and IoU  $>0.6$  for both boxes) are simply treated as no-relation due to the One-to-One constraint of Hungarian matching. This constraint discards valuable supervisory signals by treating suboptimal yet informative queries as negative samples. These queries, while not the best matches for ground truth labels, may capture informative relational cues that could contribute to model learning. Simply assigning these queries as negative samples may introduce false negatives, which in turn leads to label noise and performance degradation [45].
+
+To accelerate the training of DETR-based SGG models, we introduce Hydra-SGG $^1$ , an efficient framework that addresses the slow convergence problem in one-stage DETR-based SGG models. The cores of Hydra-SGG are Hybrid Relation Assignment and Hydra Branch. Specifically, Hybrid Relation Assignment synergizes One-to-One and One-to-Many Relation Assignment strategies, providing over  $50\%$  more positive queries per training step than previous arts such as RelTR [7]. To further enhance this assignment strategy, we introduce an auxiliary branch called Hydra Branch. This branch is specifically designed to encourage different queries to predict duplicated relations by removing self-attention in the decoder, creating a synergistic effect with our Hybrid Relation Assignment. The branch shares all other parameters with the original decoder, and this intentional design choice leads to improved supervision signals during training ( $\S 3.3$ ).
+
+Hydra-SGG makes three main contributions to the field of SGG: First, we propose an efficient framework that effectively addresses the slow convergence problem in one-stage DETR-based SGG models through a hybrid query-label assignment and synergistic architectural design. Second, our Hybrid Relation Assignment strategy significantly increases relation supervision signals by min
+
+ing false negative samples to increase positive samples during training, providing over  $50\%$  more positive samples per training step. Third, the proposed Hydra Branch complements our assignment strategy by encouraging duplicate relation predictions during training, while maintaining inference efficiency as it shares parameters with the original decoder and is only used during training. Hydra-SGG achieves state-of-the-art performance with remarkable training efficiency, converging  $10\times$  faster than existing one-stage SGG counterparts [7, 40, 18].
+
+Extensive experiments on three challenging SGG benchmarks, VG150 [75], Open Images V6 [30], and GQA [16], demonstrate the effectiveness of our Hydra-SGG. It achieves 16.0 mR@50 on VG150 test in only 12 epochs (Fig. 1), surpassing the previous state-of-the-art one-stage method SGTR [40] by +4.0 and two-stage method PE-Net [91] by +3.6.
+
+# 2 RELATED WORK
+
+Two-stage SGG. Lu et al. [50] first proposed SGG task and designed a two-stage pipeline. Later, many SGG models have been built on it, using various techniques such as recurrent neural networks or its variants [85, 64, 75, 70, 58], visual translation embedding [88, 17], graph neural networks [83, 62, 74, 83, 77], external or internal knowledge integration [6, 12, 84, 1, 42, 82, 35, 4, 72], attention mechanisms [23, 53, 9], and Transformer-based models [51, 7, 40, 61, 8, 29, 59].
+
+Despite the promising performance, two-stage SGG methods rely heavily on manually designed modules, such as NMS, anchor generation, and entity pairing generation modules [85, 64, 9]. In addition, their designs typically involve separate stages for detection, pairing, and relation classification, resulting in a complicated pipeline that cannot be trained in a fully end-to-end manner. Furthermore, these methods predict dense entity pairs in inference, leading to high time complexity and computational burden. In contrast, this paper proposes Hydra-SGG, a one-stage SGG method that offers significant advantages over two-stage methods. Hydra-SGG enables end-to-end training and surpasses the performance of two-stage methods.
+
+One-stage SGG. One-stage methods have gained increasing attention for their simplicity and end-to-end training ability. Early works in this line adopt CNN-based one-stage detectors [46] or query-based sparse R-CNN [67] for direct relationship prediction. Recently, the DETR [2] framework significantly advances SGG and Human-Object Interaction [54, 92, 7, 40, 63, 95, 44, 25, 86, 26, 18, 38, 37, 73]. This framework enables end-to-end training by associating ground truth labels with output queries. Existing DETR-based SGG methods focus on different aspects: SGTR [40] and DSGG [13] explore query designs for relation feature extraction, SpeaQ [26] investigates relation-specific query grouping strategies to improve the specialization and discrimination of queries, while other works focus on architectural innovations [7] and lightweight frameworks [18].
+
+However, DETR-based SGG models, while eliminating NMS, face a critical challenge of slow convergence due to sparse relation supervision, requiring significantly more training epochs than two-stage counterparts (e.g., 150 for RelTR [7] vs. 32 for PE-Net [91]). Our proposed Hydra-SGG specifically addresses this limitation through a query-label assignment and architectural design while maintaining the advantages of DETR-based approaches. Hydra-SGG achieves remarkably fast convergence with only 12 epochs, surpassing both one-stage counterparts and two-stage methods to achieve state-of-the-art performance across multiple SGG datasets (16.0 mR@50 on VG150 [75], 50.1 weighted score on Open Images V6 [30], and 12.7 mR@50 on GQA [16]).
+
+DETR for Object Detection. The introduction of DETR by Carion et al. [2] marks a significant shift away from traditional CNN-based object detection models [57, 55, 56, 45, 14], adopting an end-to-end trainable approach with a novel application of the Transformer architecture and bipartite matching. While DETR streamlines the detection process, it requires 500 epochs to converge [93]. This spurs a wave of research that focuses on improving low training efficacy, including enhancing training signal [5, 15, 19, 94], the adoption of anchor boxes [49, 52], and the leverage of efficient attention mechanism [93, 11]. For instance, Co-DETR [94] introduces extra groups of object queries and DN-DETR [31] utilizes noisy queries to increase positive samples.
+
+In this paper, we present Hydra-SGG, a framework that addresses the sparse relation supervision challenge inherent in DETR-based SGG models. Our method introduces Hybrid Relation Assignment, which significantly increases the number of positive samples per training iteration, effectively mitigating the sparse relation supervision issue and accelerating the model's convergence.
+
+# 3 HYDRA-SGG
+
+§3.1 introduces our baseline model, which employs One-to-One Relation Assignment. Next, §3.2 describes Vanilla Hydra-SGG, which enhances the baseline by introducing a Hybrid Relation Assignment strategy. Finally, §3.3 details the complete Hydra-SGG model, which includes Hydra Branch (HydraBranch), an auxiliary decoder that promotes One-to-Many Relation Assignment. §3.4 performs a statistical analysis to validate our design of Hydra Branch, demonstrating its effectiveness in enhancing the One-to-Many assignment strategy.
+
+# 3.1 ONE-TO-ONE RELATION ASSIGNMENT BASELINE
+
+Problem Formulation. Given an input image, SGG models aim to generate a scene graph in the form of relation triplet:  $\langle e_{\mathrm{sub}},\rho ,e_{\mathrm{obj}}\rangle$  , where each entity  $e_{\mathrm{sub}},e_{\mathrm{obj}}\in \mathcal{E}$  is represented by a category label  $c\in \mathcal{C}$  (e.g., cat, people, car) and a bounding box  $b$  , and  $\rho \in \mathcal{P}$  is a specific relation type (e.g., on, have, ride). In this paper, we distinguish between the terms "relation" and "relation triplet". A relation refers to the interaction or relationship between entities, while a relation triplet represents the complete structure containing the subject  $e_{\mathrm{sub}}$  , object  $e_{\mathrm{obj}}$  , and their relation  $\rho$
+
+Baseline Architecture. Our baseline model is composed of a Backbone model, an Encoder, and a RelDecoder (Relation Decoder). The relation queries interact with image tokens extracted from the input image by Backbone and Encoder. The updated queries are processed by box regression, entity classification, and relation classification heads to generate the final predictions.
+
+- Backbone: Backbone maps an input image into a feature map that has a  $H \times W \times C$  dimension, where  $H$  and  $W$  denote the spatial size of the feature map, while  $C$  is the channel dimension (e.g., 2048 or 1024).  
+- Encoder: Encoder captures comprehensive spatial and contextual information across the image. Before feeding the feature map into Encoder, a  $1 \times 1$  conv layer is employed to reduce the dimension of the feature map. The enhanced image tokens are denoted as  $\pmb{F} \in \mathbb{R}^{HW \times 256}$ .  
+- RelDecoder: In RelDecoder, we introduce relation queries  $Q_{\mathrm{rel}} \in \mathbb{R}^{N \times 512}$  (■■■), which are formed by concatenating subject queries  $Q_{\mathrm{sub}} \in \mathbb{R}^{N \times 256}$  and object queries  $Q_{\mathrm{obj}} \in \mathbb{R}^{N \times 256}$  along the channel dimension (i.e., 256). Here  $N$  denotes the number of queries. RelDecoder( $Q_{\mathrm{sub}}, Q_{\mathrm{obj}}, F$ ) works as follows:
+
+$$
+\tilde {\boldsymbol {Q}} _ {\text {s u b}}, \tilde {\boldsymbol {Q}} _ {\text {o b j}} = \operatorname {S A} \left(\left[ \boldsymbol {Q} _ {\text {s u b}}, \boldsymbol {Q} _ {\text {o b j}} \right]\right) \in \mathbb {R} ^ {2 N \times 2 5 6},
+$$
+
+$$
+\bar {\boldsymbol {Q}} _ {\mathrm {s u b}} = \operatorname {C A} (\boldsymbol {F}, \tilde {\boldsymbol {Q}} _ {\mathrm {s u b}}) \in \mathbb {R} ^ {N \times 2 5 6}, \tag {1}
+$$
+
+$$
+\bar {\boldsymbol {Q}} _ {\mathrm {o b j}} = \mathbb {C} \mathbb {A} (\boldsymbol {F}, \tilde {\boldsymbol {Q}} _ {\mathrm {o b j}}) \in \mathbb {R} ^ {N \times 2 5 6}.
+$$
+
+Specifically, the subject queries  $Q_{\mathrm{sub}}$  and object queries  $Q_{\mathrm{obj}}$  first interact in a self-attention layer (SA) to obtain updated queries  $\tilde{Q}_{\mathrm{sub}}$  and  $\tilde{Q}_{\mathrm{obj}}$ . Subsequently, these updated queries interact with image tokens in cross-attention layers (CA). The output subject and object queries are fed into independent box regression heads and classification heads, producing box predictions  $\bar{b}_{\mathrm{sub}}, \bar{b}_{\mathrm{obj}} \in \mathbb{R}^{N \times 4}$  and entity class predictions  $\bar{p}_{\mathrm{sub}}, \bar{p}_{\mathrm{obj}} \in \mathbb{R}^{N \times |\mathcal{C}|}$ , respectively. The relation predictions  $\bar{p}_{\mathrm{rel}} \in \mathbb{R}^{N \times |\mathcal{P}|}$  are obtained by a relation prediction head. For example, with 300 relation queries (i.e.,  $N = 300$ ) and 50 relation classes (i.e.,  $|\mathcal{P}| = 50$ ), the output dimension of  $\bar{p}_{\mathrm{rel}}$  would be  $300 \times 50$ . Finally, the outputs of RelDecoder are combined to generate the predicted relation triplets  $\bar{R} = \{\langle (\bar{p}_{\mathrm{sub}}, \bar{b}_{\mathrm{sub}}), \bar{p}_{\mathrm{rel}}, (\bar{p}_{\mathrm{obj}}, \bar{b}_{\mathrm{obj}}) \rangle_n\}_{n=1}^N$ . Note that the  $i$ -th subject query is concatenated only with the  $i$ -th object query, and there is no permutation process.
+
+- One-to-One Relation Assignment Loss. In One-to-One Relation Assignment Loss  $\mathcal{L}_{\mathrm{o2o}}$  [7, 40], Hungarian matching (HM) is employed to find the optimal correspondence between the predicted relation triplets  $\bar{R}$  and the ground truth triplets  $R$ . The loss function can be formulated as:
+
+$$
+\mathcal {L} _ {\mathrm {o 2 o}} = \mathcal {L} _ {\mathrm {H M}} (\bar {\boldsymbol {R}}, \boldsymbol {R}). \tag {2}
+$$
+
+The classification losses (including entity and relation) and box regression losses are computed between matched predictions and ground truth labels, which is identical with the previous models [7, 40]. The Hungarian matching hyperparameters are the same as RelTR [7]. A detailed
+
+![](images/867b931e3fc773b5e6a527e3dcb97c735d83ab7e6b89c9d32783233db1b78404.jpg)  
+Figure 3: Overall pipeline of Hydra-SGG: For simplicity, FFN inside the Transformer layer are omitted. Hydra-SGG incorporates two Transformer decoders: HydraBranch and RelDecoder. HydraBranch shares its parameters with RelDecoder but removes self-attention layers. Hydra-SGG combines One-to-One and One-to-Many assignments in a synergy, generating more supervision signals.
+
+implementation of the loss function, training strategy, and model architecture is provided in Appendix. The evaluation of our baseline model is given in §4.3.
+
+# 3.2 VANILLA HYBRID RELATION ASSIGNMENT
+
+VG150 [75] contains an average of only 5.5 ground truth relation triplets per image in train. Our baseline adopts a One-to-One Relation Assignment and only assigns approximately 5 out of 300 relation queries to match the ground truth relation triplets for each image (i.e.,  $N = 300$ ). This severe sparse supervision reduces the training efficacy, as only about  $2\%$  of queries in each training step match with ground truth labels for learning, while the remaining  $98\%$  are treated as negative samples. Consequently, the model requires a significantly higher number of training steps to learn effectively, leading to slow convergence.
+
+To enrich the relation supervision signals, we propose a Hybrid Relation Assignment. Specifically, we first devise a One-to-Many Relation Assignment  $(\circ 2\mathfrak{m})$  and then embed it into the baseline to cooperate with One-to-One Relation Assignment. Given a ground truth triplet  $\pmb{r} = \langle (c_{\mathrm{sub}}, b_{\mathrm{sub}}), c_{\mathrm{rel}}, (c_{\mathrm{obj}}, b_{\mathrm{obj}}) \rangle \in \pmb{R}$  and a predicted triplet  $\bar{\pmb{r}} = \langle (\bar{p}_{\mathrm{sub}}, \bar{b}_{\mathrm{sub}}), \bar{p}_{\mathrm{rel}}, (\bar{p}_{\mathrm{obj}}, \bar{b}_{\mathrm{obj}}) \rangle \in \bar{\pmb{R}}$ , the One-to-Many Assignment score  $S_{\circ 2\mathfrak{m}}$  is calculated by:
+
+$$
+\mathcal {S} _ {\circ 2 \mathrm {m}} (\boldsymbol {r}, \bar {\boldsymbol {r}}) = \bar {p} _ {\text {s u b}} ^ {[ c _ {\text {s u b}} ]} + \bar {p} _ {\text {o b j}} ^ {[ c _ {\text {o b j}} ]} + \operatorname {I o U} \left(b _ {\text {s u b}}, \bar {b} _ {\text {s u b}}\right) + \operatorname {I o U} \left(b _ {\text {o b j}}, \bar {b} _ {\text {o b j}}\right). \tag {3}
+$$
+
+$S_{\circ 2\mathrm{m}}$  combines the predicted class probabilities and IoU scores for the subject and object bounding boxes. The notations  $\bar{p}_{\mathrm{sub}}^{[c_{\mathrm{sub}}]}$  and  $\bar{p}_{\mathrm{obj}}^{[c_{\mathrm{obj}}]}$  represent the probabilities of the ground truth classes for the subject and object, respectively. For example, if the ground truth subject class  $c_{\mathrm{sub}}$  is 3, we would extract the probability corresponding to class 3 from  $\bar{p}_{\mathrm{sub}}$ .
+
+Given  $M$  ground truth relation triplets and  $N$  predicted relation triplets, the final One-to-Many Relation Assignment score matrix, with dimensions  $M \times N$ , is computed by applying Eq. 3 to each pair of ground truth and predicted relation triplets. We keep the results with scores greater than a threshold  $T$  and select the top 6 queries for each ground truth (see details in §4.3).
+
+The proposed Vanilla Hydra-SGG applies both One-to-One and One-to-Many Relation Assignment strategies to the same set of predicted relation triplets  $\bar{R}$ . These triplets are obtained from the updated queries  $Q_{\mathrm{rel}}$  after passing through the model. By combining these two assignment strategies, we can formulate the vanilla version loss function as:
+
+$$
+\mathcal {L} _ {\text {v a n i l l a}} = \mathcal {L} _ {\mathrm {o} 2 \mathrm {o}} (\bar {\boldsymbol {R}}, \boldsymbol {R}) + \mathcal {L} _ {\mathrm {o} 2 \mathrm {m}} (\bar {\boldsymbol {R}}, \boldsymbol {R}). \tag {4}
+$$
+
+Our Vanilla Hydra-SGG provides richer relation supervision signals compared to the baseline by harmonizing the supervision of two assignment strategies. Specifically, it increases the number of positive samples by  $65.5\%$  in VG150 [75] train and  $58.7\%$  in val (Fig. 4a, b). Vanilla Hydra-SGG significantly improves training efficacy and performance ( $\S 4.3$ ).
+
+![](images/477be71c9352271e270b16c2c9d1fb1da5a42d9f974e974d23a56c7dbf7063ff.jpg)  
+Figure 4: (a) The average number of positive samples of VG150 train and val for One-to-One and Hybrid Relation Assignment. (b) The percentage increase in positive samples achieved by Hybrid Relation Assignment compared to the One-to-One baseline. (c) ADS on VG150 val. (d)-(e) The visualizations show that for the same group of queries that previously predicted different relations, removing the self-attention layers causes them to make identical predictions. The Q ID column represents the ID of each relation query.
+
+![](images/e530145fd55cd71cbde6b0d3c04b7d7b52071d5a5d1a835b058ee603a1f8da60.jpg)
+
+![](images/78b3e3838610aa70cd2a70fe9b007ac91fd40d461d5e434712e98560660e0f14.jpg)
+
+![](images/5b6115ee963c6a1b711af4c640ba7117fb09da5fbf1830de4940e0c22d2a7486.jpg)
+
+![](images/a956989e5d967e98db8f0fe6ef927d8affcb17d38ce02be0af536edc217f5e87.jpg)
+
+# 3.3 COMPLETE HYDRA-SGG
+
+Each RelDecoder layer contains self- and cross-attention layers, and a point-wise feed-forward network (FFN). The self-attention layer enables inter-query interactions [69], while cross-attention and FFN do not explicitly support query interactions. Previous studies indicate that self-attention helps inhibit duplicate predictions [2, 7]. In this work, we further find that the self-attention layer in RelDecoder is crucial to reduce duplicated relation predictions.
+
+We introduce a Diversity Score (DS) to quantify the impact of the self-attention layer on the diversity of relation predictions. Specifically, DS is defined as the number of distinct relation categories predicted by the model for a single image. For example, if the model predicts relations for 5 queries as (sit, sit, on, on, has), DS would be 3, as there are 3 distinct relation categories (sit, on, has). We calculate DS using the trained Vanilla Hydra-SGG model, with and without self-attention in the RelDecoder. Let  $\mathrm{DS}_{\mathrm{on}}^{(k)}$  and  $\mathrm{DS}_{\mathrm{off}}^{(k)}$  represent the DS for the  $k$ -th image with self-attention enabled and disabled, respectively. The average DS (ADS) across the dataset is then computed as:
+
+$$
+\mathrm {A D S} _ {\text {o n}} = \frac {1}{K} \sum_ {k = 1} ^ {K} \mathrm {D S} _ {\text {o n}} ^ {(k)}, \quad \mathrm {A D S} _ {\text {o f f}} = \frac {1}{K} \sum_ {k = 1} ^ {K} \mathrm {D S} _ {\text {o f f}} ^ {(k)}, \tag {5}
+$$
+
+where  $K$  is the total number of images. A higher  $\mathrm{ADS}_{\mathrm{on}}$  compared to  $\mathrm{ADS}_{\mathrm{off}}$  would indicate that self-attention promotes diverse relation predictions. As shown in Fig. 4c,  $\mathrm{ADS}_{\mathrm{off}}$  and  $\mathrm{ADS}_{\mathrm{on}}$  of VG150 [75] val are 4.6 and 6.6, respectively. Fig. 4d, e illustrate how the self-attention layer enables diverse relation predictions. Without self-attention, queries converge to the same relation. These examples demonstrate self-attention's role in reducing duplicate relations.
+
+The above findings reveal critical interactions between self-attention and One-to-Many Relation Assignment: i) Conflict with Self-Attention: Applying One-to-Many Relation Assignment strategy to self-attention-updated relation queries in RelDecoder of Vanilla Hydra-SGG causes a mismatch in optimization objectives, as self-attention promotes diversity while One-to-Many Relation Assignment assigns one ground truth to multiple queries. ii) Benefit without Self-Attention: Conversely, removing self-attention leads to more duplicated relation predictions, potentially synergizing with our One-to-Many Relation Assignment.
+
+To address this potential conflict and fully harness the benefits of both the self-attention layers and the One-to-Many Relation Assignment strategy, we propose HydraBranch (Hydra Branch, Fig. 3), an auxiliary decoder that shares parameters with RelDecoder but removes the self-attention layers. This multi-branch architecture decouples the learning objectives: the self-attention layers in the main RelDecoder can focus on promoting diversity in relation predictions, while HydraBranch facilitates One-to-Many Relation Assignment. This separation enables each branch to optimize its specific function without compromising the other.
+
+HydraBranch operates with the main RelDecoder in parallel, processing the same input but without the influence of self-attention layers. Specifically, the initial One-to-Many relation queries  $Q_{\mathrm{rel}}^{\mathrm{o2m}}$  (i.e.,  $Q_{\mathrm{rel}}^{\mathrm{o2m}} = Q_{\mathrm{rel}}$ ) are set to  $Q_{\mathrm{rel}}$  (i.e.,  $Q_{\mathrm{rel}}^{\mathrm{o2m}} = Q_{\mathrm{rel}}$ ) before being sent into HydraBranch. The
+
+process of HydraBranch  $(Q_{\mathrm{sub}}^{02\mathrm{m}}, Q_{\mathrm{obj}}^{02\mathrm{m}}, F)$  is as follows:
+
+$$
+\bar {Q} _ {\text {s u b}} ^ {\mathrm {o} 2 \mathrm {m}} = \mathrm {C A} \left(\boldsymbol {F}, Q _ {\text {s u b}} ^ {\mathrm {o} 2 \mathrm {m}}\right) \in \mathbb {R} ^ {N \times 2 5 6}, \quad \bar {Q} _ {\text {o b j}} ^ {\mathrm {o} 2 \mathrm {m}} = \mathrm {C A} \left(\boldsymbol {F}, Q _ {\text {o b j}} ^ {\mathrm {o} 2 \mathrm {m}}\right) \in \mathbb {R} ^ {N \times 2 5 6}. \tag {6}
+$$
+
+The One-to-Many predicted relation triplets  $\bar{R}^{\mathrm{o2m}} = \{\langle (\bar{p}_{\mathrm{sub}}^{\mathrm{o2m}},\bar{b}_{\mathrm{sub}}^{\mathrm{o2m}}),\bar{p}_{\mathrm{rel}}^{\mathrm{o2m}},(\bar{p}_{\mathrm{obj}}^{\mathrm{o2m}},\bar{b}_{\mathrm{obj}}^{\mathrm{o2m}})\rangle_{n}\}_{n = 1}^{N}$  are derived in the same manner as  $\bar{R}$  and share prediction heads with  $\bar{R}$ . Compared with Eq. 1, subject and object queries do not interact in a self-attention layer before they are sent into subsequent cross-attention layers. We apply One-to-Many Relation Assignment described in Eq. 3 to  $\bar{R}^{\mathrm{o2m}}$  and  $\pmb{R}$ . Hybrid Relation Assignment Loss is then given by:
+
+$$
+\mathcal {L} _ {\text {H y d r a}} = \mathcal {L} _ {\circ 2 \circ} (\bar {\boldsymbol {R}}, \boldsymbol {R}) + \mathcal {L} _ {\circ 2 \mathrm {m}} (\bar {\boldsymbol {R}} ^ {\circ 2 \mathrm {m}}, \boldsymbol {R}). \tag {7}
+$$
+
+By incorporating HydraBranch, the complete version of Hydra-SGG achieves 10.6 and 16.0 on mR@20 and mR@50, respectively, in just 12 training epochs, further boosting the performance compared to the vanilla version (§4.3). Note that HydraBranch is used only in training and discarded in inference, thus bringing no extra parameters or delay.
+
+# 3.4 STATISTICAL ANALYSIS OF HYDRA BRANCH
+
+To quantify how Hydra Branch enhances the One-to-Many assignment strategy, we analyze the Euclidean distances between query embedding pairs. Using 5,000 images from the VG150 val set, we compute pairwise distances for all 300 queries per image (totaling  $\binom{300}{2} = 44,850$  pairs per image) in two conditions: with and without self-attention layers. The removal of self-attention reduces the mean query distance from 7.55 to 7.23 (5% decrease). A paired t-test confirms statistical significance ( $t = 54.502$ ,  $p < 0.001$ ), with a Cohen's  $d$  effect size of 0.771 - approaching the threshold for a large effect ( $d = 0.8$ ). These results demonstrate that Hydra Branch promotes query similarity through self-attention removal, thereby facilitating One-to-Many assignment.
+
+# 4 EXPERIMENT
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets. We conduct experiments on three datasets:
+
+- Visual Genome (VG150) [27, 75] contains 150 entity and 50 relation categories. It is split into 57,723 training, 5,000 validation, and 26,446 testing images.  
+- Open Images V6 [30] features 288 entity and 30 relation categories, including 126,368 training, 1,813 validation, and 5,322 testing images with relation annotations.  
+- GQA [16] encompasses 200 entity and 100 relation types, with a split of 52,623 training, 5,000 validation, and 8,209 testing images annotated for SGG tasks.
+
+Evaluation Metrics. We focus on the scene graph detection (SGDet) setting on VG150 [27], GQA [16] and Open Images V6 [30] datasets. For VG150 and GQA, we report  $\mathrm{Recall}@\mathbf{k}$  ( $\mathbf{R}@\mathbf{k}$ ), mean  $\mathrm{Recall}@\mathbf{k}$  [6, 64] ( $\mathrm{mR}@\mathbf{k}$ ), and F-Recall [87] performance.  $\mathrm{mR}@\mathbf{K}$  calculates  $\mathbf{R}@\mathbf{K}$  for each predicate individually, then averages these values. It is important to note that recall metrics are more influenced by dataset bias [3], whereas mean recall provides a more holistic evaluation of the model's performance. F-Recall is the harmonic average of Recall and mean Recall. For Open Images V6, we follow evaluation protocols [30]: Recall@50, the weighted mean Average Precision (wmAP) for relationship detection (wmAPrel), and phrase detection (wmAPphr). The overall score, denoted as  $\mathrm{score}_{\mathrm{wtd}}$ , is calculated as a weighted average of these metrics:  $0.2 \times \mathrm{R} @ 50 + 0.4 \times \mathrm{wmAP}_{\mathrm{rel}} + 0.4 \times \mathrm{wmAP}_{\mathrm{phr}}$ .
+
+Competitors. Hydra-SGG is compared with methods from two categories: (1) Two-stage SGG methods, including MOTIFS [85], VCTree-TDE [65], BGNN [39], PE-Net [91], IS-GGT [29], VETO [61], UniVRD [90], DRM [32], CFA [34], and SHA [9]; (2) One-stage methods including SGTR [40], SSR-CNN [67], ISG [24], RelTR [7], DSGG [13], SpeaQ [26], and EGTR [18].
+
+Table 1: SGDet evaluation on VG150 [27] test (§4.2).  ${}^{ + }$  : detector pre-trained on VG150. FPS (Frames Per Second) indicates inference speed. F-Recall of Hydra-SGG is calculated based on the best results.  
+
+<table><tr><td>Method</td><td>Backbone</td><td># Epoch</td><td>FPS</td><td># Param</td><td>R@20/50/100</td><td>mR@20/50/100</td><td>F@20/50/100</td></tr><tr><td colspan="8">Two-stage methods</td></tr><tr><td>MOTIFS [85] [CVPR2018]</td><td>ResNeXt101-FPN</td><td>-</td><td>-</td><td>369.9M</td><td>25.1 / 32.1 / 36.9</td><td>4.1 / 5.5 / 6.8</td><td>7.1 / 9.2 / 11.7</td></tr><tr><td>VCTree-TDE [65] [CVPR2020]</td><td>ResNeXt101-FPN</td><td>-</td><td>-</td><td>361.3M</td><td>14.3 / 19.6 / -</td><td>6.3 / 9.3 / 11.1</td><td>8.8 / 12.4 / -</td></tr><tr><td>BGNN [39] [CVPR2021]</td><td>ResNeXt101-FPN</td><td>-</td><td>-</td><td>341.9M</td><td>23.3 / 31.0 / 35.8</td><td>7.5 / 10.7 / 12.7</td><td>11.3 / 15.5 / 19.0</td></tr><tr><td>PE-Net [91] [CVPR2023]</td><td>ResNeXt101-FPN</td><td>32+</td><td>-</td><td>-</td><td>- / 30.7 / 35.2</td><td>- / 12.4 / 14.5</td><td>- / 17.7 / 21.2</td></tr><tr><td>IS-GGT [29] [CVPR2023]</td><td>ResNet101</td><td>70</td><td>-</td><td>-</td><td>- / - / -</td><td>- / 9.1 / 11.3</td><td>- / - / -</td></tr><tr><td>VETO [61] [ICCV2023]</td><td>ResNeXt101-FPN</td><td>33+</td><td>-</td><td>-</td><td>- / 27.5 / 31.5</td><td>- / 8.1 / 9.5</td><td>- / 12.5 / 14.6</td></tr><tr><td>UniVRD [90] [ICCV2023]</td><td>CLIP ViT-B</td><td>-</td><td>-</td><td>-</td><td>- / - / -</td><td>- / 9.6 / 12.1</td><td>- / - / -</td></tr><tr><td>DRM [32] [CVPR2024]</td><td>ResNeXt101-FPN</td><td>-</td><td>-</td><td>-</td><td>- / 34.0 / 38.9</td><td>- / 9.0 / 11.2</td><td>- / 14.2 / 17.4</td></tr><tr><td colspan="8">One-stage methods</td></tr><tr><td>SGTR [40] [CVPR2022]</td><td>ResNet101</td><td>123+</td><td>-</td><td>117.1M</td><td>- / 25.1 / 26.6</td><td>- / 12.0 / 14.6</td><td>- / 16.2 / 18.9</td></tr><tr><td>SSR-CNN [67] [CVPR2022]</td><td>ResNet101</td><td>-</td><td>-</td><td>274.3M</td><td>25.8 / 32.7 / 36.9</td><td>6.1 / 8.4 / 10.0</td><td>9.9 / 13.4 / 15.7</td></tr><tr><td>ISG [24] [NeurIPS2022]</td><td>ResNet101</td><td>52</td><td>-</td><td>93.5M</td><td>- / 29.5 / 32.1</td><td>- / 7.4 / 8.4</td><td>- / 11.8 / 13.3</td></tr><tr><td>RelTR [7] [TPAM12023]</td><td>ResNet50</td><td>150</td><td>6.5</td><td>63.7M</td><td>21.2 / 27.5 / 30.7</td><td>6.8 / 10.8 / 12.3</td><td>10.3 / 15.5 / 17.6</td></tr><tr><td>DSGG [13] [CVPR2024]</td><td>-</td><td>60</td><td>-</td><td>-</td><td>- / 32.9 / 38.5</td><td>- / 13.0 / 17.3</td><td>- / 18.6 / 23.9</td></tr><tr><td>SpeaQ [26] [CVPR2024]</td><td>ResNet101</td><td>52</td><td>-</td><td>-</td><td>- / 32.9 / 36.0</td><td>- / 11.8 / 14.1</td><td>- / 17.4 / 20.3</td></tr><tr><td>EGTR [18] [CVPR2024]</td><td>ResNet50</td><td>275+</td><td>7.7</td><td>42.5M</td><td>23.5 / 30.2 / 34.3</td><td>5.5 / 7.9 / 10.1</td><td>8.9 / 12.5 / 15.6</td></tr><tr><td colspan="8">Ours</td></tr><tr><td>Hydra-SGG [ICLR2025]</td><td>ResNet50</td><td>12</td><td>5.3</td><td>67.6M</td><td>21.9 / 28.6 / 33.4 ±0.1 / ±0.2 / ±0.3</td><td>10.3 / 15.9 / 19.4 ±0.2 / ±0.2 / ±0.2</td><td>14.0 / 20.5 / 24.7</td></tr></table>
+
+Table 2: Evaluation on Open Images V6 [30] test (§4.2).+: detector pre-trained on Open Images V6.  
+
+<table><tr><td>Method</td><td>Backbone</td><td># Epoch</td><td># Param</td><td>R@50</td><td>wmAPrel</td><td>wmAPphr</td><td>scorewd</td></tr><tr><td colspan="8">Two-stage methods</td></tr><tr><td>Motifs [85] [CVPR2018]</td><td>ResNeXt101-FPN</td><td>-</td><td>369.9M</td><td>71.6</td><td>29.9</td><td>31.6</td><td>38.9</td></tr><tr><td>BGNN [39] [CVPR2021]</td><td>ResNeXt101-FPN</td><td>-</td><td>341.9M</td><td>75.0</td><td>35.5</td><td>34.2</td><td>42.1</td></tr><tr><td>PE-Net [91] [CVPR2023]</td><td>ResNeXt101-FPN</td><td>-</td><td>-</td><td>76.5</td><td>35.4</td><td>34.9</td><td>44.9</td></tr><tr><td colspan="8">One-stage methods</td></tr><tr><td>SGTR [40] [CVPR2022]</td><td>ResNet101</td><td>123+</td><td>117.1M</td><td>59.9</td><td>37.0</td><td>38.7</td><td>42.3</td></tr><tr><td>RelTR [7] [TPAMI2023]</td><td>ResNet50</td><td>150</td><td>63.7M</td><td>71.7</td><td>34.2</td><td>37.5</td><td>43.0</td></tr><tr><td>EGTR [18] [CVPR2024]</td><td>ResNet50</td><td>275+</td><td>42.5M</td><td>75.0</td><td>42.0</td><td>41.9</td><td>48.6</td></tr><tr><td colspan="8">Ours</td></tr><tr><td>Hydra-SGG [ICLR2025]</td><td>ResNet50</td><td>7</td><td>67.6M</td><td>76.0±0.2</td><td>42.8±0.2</td><td>44.1±0.2</td><td>50.0±0.2</td></tr></table>
+
+# 4.2 QUANTITATIVE COMPARISON RESULT
+
+VG150 [75] test. Table 1 reports the comparison results on VG150 test. Hydra-SGG demonstrates outstanding performance on challenging SGDet, achieving mR@20 and mR@50 scores of 10.6 and 16.0, respectively, setting a new state-of-the-art. Hydra-SGG achieves this performance with significantly shorter training time, requiring only 12 epochs. This training time is substantially shorter compared to SpeaQ [26], EGTR [18], and DSGG [13], with reductions of 40, 263, and 48 epochs, respectively. Despite the shorter training, Hydra-SGG outperforms these methods by +4.2, +8.1, and +3.0 on the mR@50. Furthermore, it even outperforms the state-of-the-art two-stage model, PE-Net [91], by a margin of 3.6 on the mR@50 metric. We speculate that the significant improvement in mR can be attributed to our One-to-Many Assignment strategy, which ensures balanced supervision across both rare and common relations by allocating a fixed number of six queries per relation category (§ 3.2). For instance, in an image containing ten "on" relations and two "sit" relations, while One-to-One Assignment would allocate ten queries to "on" and only two to "sit", our approach assigns six queries to each, resulting in a  $60\%$  increase for common relations and a substantial  $300\%$  increase for rare relations.
+
+Open Images V6 [30] test. As shown in Table 2, Hydra-SGG achieves SOTA in only 7 epochs while SGTR [40], RelTR [7] and EGTR [18] require 119, 150, and 275 epochs. Although Open Images V6 contains more than 120,000 images, its scenes are not as complex as those in VG150 [75]. Training Hydra-SGG for 7 epochs is sufficient to achieve good performance.
+
+GQA [16] test. Our experimental results demonstrate that Hydra-SGG achieves SOTA mean Recall performance on GQA. As shown in Table 3, Hydra-SGG achieves 12.7 and 15.9 for mR@50 and mR@100 respectively, surpassing previous best results. Our model employs the relatively
+
+Table 3: Evaluation on GQA [16] test (§4.2).  
+
+<table><tr><td>Method</td><td>Backbone</td><td>R@50/100</td><td>mR@50/100</td></tr><tr><td>SHA [9][CVPR2022]</td><td>ResNeXt101-FPN</td><td>25.5 / 29.1</td><td>6.6 / 7.8</td></tr><tr><td>VETO [61][ICCV2023]</td><td>ResNeXt101-FPN</td><td>26.1 / 29.0</td><td>7.0 / 8.1</td></tr><tr><td>CFA [34][ICCV2023]</td><td>ResNeXt101-FPN</td><td>-</td><td>10.8 / 12.6</td></tr><tr><td colspan="4">Ours</td></tr><tr><td>Hydra-SGG [ICLR2025]</td><td>ResNet50</td><td>23.1 / 26.8 ±0.3 / ±0.3</td><td>12.5 / 15.6 ±0.2 / ±0.3</td></tr></table>
+
+lightweight ResNet50 as its backbone, in contrast to the heavier ResNeXt101 used by other methods.
+
+Table 4: A set of ablative experiments about on VG150 [27] test (§4.3). The adopted hyperparameters are marked in red.  
+
+<table><tr><td>Method</td><td>mR@20</td><td>mR@50</td><td># Epoch</td></tr><tr><td>Baseline</td><td>8.7</td><td>12.9</td><td>50</td></tr><tr><td>Vanilla Hydra-SGG</td><td>9.9 (+1.2)</td><td>14.9 (+2.0)</td><td>12</td></tr><tr><td>Hydra-SGG</td><td>10.6 (+1.9)</td><td>16.0 (+3.1)</td><td>12</td></tr></table>
+
+(a) Key Components  
+
+<table><tr><td>Method</td><td>Total training time</td><td>mR@20</td><td>mR@50</td></tr><tr><td>RelTR [7]</td><td>50.0h (150 epochs)</td><td>6.8</td><td>10.8</td></tr><tr><td>Baseline</td><td>16.7h (50 epochs)</td><td>8.7</td><td>12.9</td></tr><tr><td>Hydra-SGG</td><td>12.0h (12 epochs)</td><td>10.6</td><td>16.0</td></tr></table>
+
+(b) Training Time  
+
+<table><tr><td>T</td><td>mR@20</td><td>mR@50</td><td>mR@100</td></tr><tr><td>0.3</td><td>10.4</td><td>14.9</td><td>18.9</td></tr><tr><td>0.4</td><td>10.6</td><td>16.0</td><td>19.7</td></tr><tr><td>0.5</td><td>9.4</td><td>15.5</td><td>19.6</td></tr><tr><td>0.6</td><td>10.3</td><td>14.9</td><td>19.1</td></tr></table>
+
+(c) Threshold  $T$  
+
+<table><tr><td># Epoch</td><td>mR@20</td><td>mR@50</td><td>mR@100</td></tr><tr><td>10</td><td>10.0</td><td>15.4</td><td>18.6</td></tr><tr><td>12</td><td>10.6</td><td>16.0</td><td>19.7</td></tr><tr><td>21</td><td>11.6</td><td>16.1</td><td>19.7</td></tr><tr><td>24</td><td>10.4</td><td>15.2</td><td>19.9</td></tr></table>
+
+(d) Training Epoch  
+
+<table><tr><td># Query</td><td>mR@20</td><td>mR@50</td><td>mR@100</td></tr><tr><td>100</td><td>8.8</td><td>14.1</td><td>17.8</td></tr><tr><td>200</td><td>9.9</td><td>15.3</td><td>18.4</td></tr><tr><td>300</td><td>10.6</td><td>16.0</td><td>19.7</td></tr><tr><td>400</td><td>10.6</td><td>15.5</td><td>18.9</td></tr></table>
+
+(e) Number of Relation Queries  
+
+<table><tr><td>Loss Ratio (1-to-1 : 1-to-M)</td><td>mR@50</td><td>mR@100</td></tr><tr><td>1 : 0.5</td><td>14.6</td><td>18.9</td></tr><tr><td>1 : 0.8</td><td>15.6</td><td>19.4</td></tr><tr><td>1 : 1</td><td>16.0</td><td>19.7</td></tr><tr><td>1 : 1.5</td><td>15.4</td><td>18.7</td></tr><tr><td>1 : 2</td><td>15.6</td><td>19.5</td></tr><tr><td>1 : 3</td><td>15.4</td><td>19.3</td></tr></table>
+
+(f) Loss Weights Between One-to-One (1-to-1) and One-to-Many (1-to-M) Losses.  
+
+<table><tr><td>Method</td><td>mR@50</td><td>mR@100</td></tr><tr><td>TDE [65][CVPR2020]</td><td>9.2</td><td>11.1</td></tr><tr><td>NICE [33][CVPR2022]</td><td>10.4</td><td>12.7</td></tr><tr><td>IETrans [87][ECCV2022]</td><td>12.5</td><td>15.0</td></tr><tr><td>CFA [34][ICCV2023]</td><td>12.3</td><td>14.6</td></tr><tr><td>VETO [61][ICCV2023]</td><td>10.6</td><td>13.8</td></tr><tr><td>NICEST [36][TPAMI2024]</td><td>10.4</td><td>12.4</td></tr><tr><td>Hydra-SGG [ICLR2025]</td><td>16.0</td><td>19.7</td></tr></table>
+
+(g) Comparisons with Unbiasing Methods.
+
+Comparisons with Unbiasing Methods. As shown in Table 4g, Hydra-SGG achieves very competitive performance on VG150 compared to these specialized debiasing methods, including TDE [65], NICE [33], IETrans [87], CFA [34], VETO [61], and NICEST [36].
+
+# 4.3 DIAGNOSTIC EXPERIMENT
+
+Key Components. We first investigate the effectiveness of our core Hybrid Relation Assignment ( $\S 3.2$ ) and Hydra Branch ( $\S 3.3$ ) in Table 4a. The first row gives the score of our baseline model ( $\S 3.1$ ). The second row corresponds to the result of Vanilla Hydra-SGG, which directly applies Hybrid Relation Assignment into RelDecoder. The third row lists the results of the complete Hydra-SGG model. Our results show that taking the proposed Hybrid Relation Assignment in the baseline model improves the mean recall and achieves  $9.9 / 14.9 \, \text{mR} @ 20 / \text{mR} @ 50$ . Furthermore, the integration of Hydra Branch with Hybrid Relation Assignment yields a synergistic effect, boosting performance substantially. Specifically, Hydra Branch increases  $\text{mR} @ 20$  from 9.9 to 10.6 and  $\text{mR} @ 50$  from 14.9 to 16.0, highlighting the complementary nature of these two strategies.
+
+Training Efficacy. In Table 4b, we compare the training time costs on VG150 [75]. All models are trained on eight NVIDIA RTX 4090 GPUs with a ResNet50 backbone. Our model achieves 16.0 mR@50 in 12 hours, whereas RelTR takes 50 hours to achieve 6.8 mR@50. This demonstrates the superior training efficacy of Hydra-SGG.
+
+Threshold  $T$ . We next study the influence of the threshold  $T$  in One-to-Many Relation Assignment in Table 4c. The threshold  $T$  controls the quality and quantity of positive relation queries. A higher value of  $T$  keeps only the high-quality positive relation queries but at the cost of reducing their number. Conversely, lowering  $T$  yields more positive samples, but their quality may decrease. This trade-off between quality and quantity directly affects the final performance of the model. Experimental results show optimal performance at  $T = 0.4$ , yielding mR@20 of 10.6 and mR@50 of 16.0. As we increase  $T$  to 0.5 and 0.6, the performance gradually decreases. This suggests that while maintaining high-quality positive samples is important, having a sufficient number of them is also crucial for the model to learn effectively.
+
+Training Epochs. We further investigate the influence of training epochs on the performance. As presented in Table 4d, Hydra-SGG achieves state-of-the-art mean Recalls in just 12 epochs, demonstrating its fast convergence. We observe further improvements when increasing the epochs to 21, i.e.,  $10.6\mathrm{mR}@20$  and  $16.0\mathrm{mR}@50$  respectively. However, the performance decreases at epoch 24, with  $\mathrm{mR}@20$  and  $\mathrm{mR}@50$  dropping to 10.4 and 15.2, respectively. This reduction in performance can be attributed to overfitting.
+
+![](images/d37b1b2fae3f20507bb2dfac8ab153c113c7337acdec3e045cf570d125d39751.jpg)
+
+![](images/bf4f50c38c5294ccb0084ac0e03ae5eeb6e542d6a71fb45a224b73c356373f04.jpg)
+
+![](images/d0bcf1a4327a1045a7e5d32dcd8cd0c15d21e80315f5e4dd658f6c32142232fb.jpg)
+
+![](images/fa8846305218cbb1d0a4598820f91bdc5e007ec3b84ee4357a1cf48cc9eda51f.jpg)
+
+![](images/dad06863b7754151beb23af4403021d7cc78f9f53c0a8abae783292e536dab09.jpg)  
+(e) Usplash Image 1
+
+![](images/1b41d9a0701105b47de56f8967c8e1667fb76262b812a10060a2362e70625f69.jpg)  
+Figure 5: Qualitative results §4.4. (a)-(d) compare Hydra-SGG and RelTR [7] on a VG150 [27] val image. We use the same color for each entity category, and the color of a predicate matches that of its subject. Differences are highlighted with red dashed rectangles  $\square$ . (e)-(g) show scene graphs generated by Hydra-SGG from images sourced from Unsplash, a platform for freely-usable images. These images are real-world, "in the wild" scenarios, demonstrating our model's capability to handle diverse and unseen visual content.
+
+![](images/7b378f53ebfdc5eb8de3194126750652b4d120b7151d156a32b54e52c69b9851.jpg)  
+(f) Usplash Image 2
+
+![](images/5a90a4cda3702facb05b8bdce50875dc5aebea48213307a048394f12853eeb45.jpg)
+
+![](images/55ab63d36b3552897e1f02e17572af0bc9135d2cb19bf8db79bf5d6f704b9ff6.jpg)  
+(g) Usplash Image 3
+
+![](images/974da9f6d4a0280dd17f626a9d56d9bae93c19c2b66c7a65a4bfd4af7b1461ad.jpg)
+
+Number of Relation Queries. Lastly, we studied the effect of the number of relation queries on model performance, as shown in Table 4e. The performance with 200 queries and 400 queries is inferior to that with 300 queries. The model achieved  $\mathrm{mR}@20/50$  of 9.9/15.3 with 200 queries, and 10.6/15.5 with 400 queries. In contrast, with 300 queries, the model reached its highest performance, with  $10.6\mathrm{mR}@20$  and  $16.0\mathrm{mR}@50$ . We hypothesize that using 200 queries may be insufficient to capture the diversity of relationships within the data. On the other hand, using 400 queries could introduce excessive noise and false positives, thereby reducing the overall precision of the model.
+
+Impact of Loss Weights. As shown in Table 4f, we conducted an ablation study to investigate the effect of different loss weights between One-to-One and One-to-Many losses. With a 1:1 loss ratio, the model achieves the best results (mR@50: 16.0, mR@100: 19.7). When the One-to-Many loss weight is reduced (e.g., 1:0.5), the model fails to fully utilize the additional positive samples available. Conversely, increasing this weight excessively (e.g., 1:3) leads to an overemphasis on One-to-Many triplets, which typically have lower quality compared to One-to-One triplets. This quality difference arises from the fundamental nature of the assignments: One-to-One assignment matches each relation label with its optimal query exclusively, while One-to-Many assignment matches a relation label with multiple queries meeting certain criteria.
+
+# 4.4 QUALITATIVE COMPARISON RESULT
+
+In Fig. 5a-d, we visualize the generated scene graph results on the image of VG150 [27] val. Both Hydra-SGG and RelTR [7] detect plausible relations, but these relations were not annotated, so these reasonable predictions become false negatives. Our Hydra-SGG detects more fine-grained relations such as rock-on-mountain and logo-on-shirt, while RelTR fails to detect such relations. In Fig. 5e-g, we visualize generated scene graph results on unseen Unsplash images.
+
+# 5 CONCLUSION
+
+This paper introduces Hydra-SGG, a one-stage DETR-based scene graph generation (SGG) model that addresses the slow convergence issue in existing DETR-based SGG models. The key contributions of our work are as follows: i) We propose a Hybrid Relation Assignment, which combines One-to-One and One-to-Many Relation Assignment strategies to increase relation supervision signals. ii) We introduce a Hydra Branch, an auxiliary decoder that encourages relation queries to predict duplicate relations, further enhancing the proposed One-to-Many Relation Assignment. These innovations work in synergy to significantly accelerate the learning process. Consequently, Hydra-SGG achieves state-of-the-art results on VG150 [75], GQA [16], and Open Images V6 [30] in just 12, 12, and 7 training epochs, demonstrating remarkable performance and efficiency.
+
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+
+# APPENDIX
+
+For a better understanding of the main paper, we provide additional details in this supplementary material, which is organized as follows:
+
+-  $\S A$  details the implementation details.  
+-  $\S B$  provides the pseudo code of Hydra-SGG.  
+-  $\S C$  shows failure cases of Hydra-SGG in VG150 [75].  
+- §D discusses our limitations, societal impact, and directions of future work.
+
+# A IMPLEMENTATION
+
+Inspired by RelTR [7], we also set relation queries to be composed of subject and object queries. We adopt the same training techniques and attention mechanisms as previous works [31, 89, 7] in Hydra-SGG. In particular, we use anchor boxes as embeddings to accelerate training and employ deformable attention [93] respectively. We input noised box-label pairs to generate both entity and relation denoising queries. To prevent information leakage, we employ attention masks. Given the absence of an attention map in the deformable decoder, we simplify the model by omitting the convolutional mask head in RelTR [7]. In particular, we concatenate the output subject and object queries and input them into an MLP. This simplification not only reduces computational complexity but also maintains performance.
+
+Both One-to-One and One-to-Many losses are composed of box regression loss, entity classification loss, and relation classification loss as previous works [2, 40, 7, 18, 63]. Specifically, we use Focal loss [45] for both relation and detection, as in previous works [49, 31, 89]. In the post-processing stage, the predictions are first ranked by the relation probability. Then we use the relation index to find corresponding subject and object predictions to compose the prediction triplets. Finally, we follow the same process as RelTR [7] that removes predictions where the subject and object are the same, as such data does not exist in VG150 [75], but we do not use this process in Open Images V6 [30].
+
+For training, we adopt the same data augmentation techniques as RelTR [7] but discard random cropping since some triplets could be incomplete. The default training epochs are 12 for VG150 [75] and 7 for Open Images V6 [30]. For VG150, the learning rate is scaled by 0.1 at epoch 11, while for Open Images V6, it is scaled at epoch 6. Open Images V6 [30] contains more than 120,000 training images, but most scenes in the dataset are simpler than those in VG150 [75], therefore training for 7 epochs is enough to achieve state-of-the-art performance. Extended training on Open Images V6 may improve the performance, but considering the dataset size, the marginal benefits are limited.
+
+Due to the inherent bias in SGG datasets, numerous unbiased algorithms have been developed [65, 34, 33, 61, 91]. While adopting these unbiased techniques typically leads to significant performance improvements, we opt for a fair comparison by evaluating Hydra-SGG against methods that do not employ such techniques.
+
+# B PSEUDO CODE
+
+The pseudo-code of Hydra-SGG is given in Algorithm S1. Our code and pre-trained models will be made publicly available.
+
+# C FAILURE CASES
+
+In this section, we present failure cases of scene graphs generated by Hydra-SGG on VG150 [75] val. In the first row (Fig. S1a-c), Hydra-SGG successfully recognizes the airplane in the image, but the predicted label is 'plane'. Although 'plane' is a plausible label, the ground truth label is 'airplane', which leads to all predicted relationships being incorrect since they don't match the ground truth. However, from a semantic perspective, the model correctly identifies these relationships and even recognizes more fine-grained relationships that are not annotated in the ground truth. For instance, the ground truth only marks the right engine of the airplane, but the model correctly identifies
+
+Algorithm 1 Hydra-SGG: PyTorch-like Pseudo-code  
+Algorithm S1: Hydra-SGG core implementation.  
+```python
+RelDecoder shares parameters with Hydra Branch   
+# memory: image tokens (L x D)   
+# Query.rel: relation queries of RelDecoder (N x D)   
+# Query rel_o2m: relation queries of HydraBranch (N x D)   
+# Loss_o2o: apply box regression, entity, and relation classification losses between matched queries and GTs by Hungarian matching.   
+# Loss_o2m: apply box regression, entity, and relation classification losses between matched queries and GTs by One-to-Many Relation Assignment.   
+Hydra SGG forward   
+def forward(sample, target): # extract and enhance features from the input memory  $=$  Encoder(Backbone(sample))  $\#$  (L x D) # init relation queries for RelDecoder and Hydra Branch Query rel  $=$  Query rel_o2m  $=$  self.initQueries() Query rel_bar  $=$  RelDecoder(Query_rel, memory) # (N x D) Query rel_o2m_bar  $=$  HydraBranch(Query rel_o2m, memory) # (N x D) loss_1  $=$  Loss_o2o(Query_rel_bar, targets) loss_2  $=$  Loss_o2m(Query rel_o2m_bar, targets) loss_hydra  $=$  loss_1 + loss_2 loss_hydra.backup()   
+# RelDecoder   
+def RelDecoder(Query_rel, memory): Query rel  $=$  Self_attn(Query_rel) # (N x D) Query rel  $=$  Cross_attn(Query_rel, memory) # (N x D) Query rel  $=$  FFN(Query_rel) return Query rel   
+# Hydra Branch   
+def HydraBranch(Query rel_o2m, memory): Query rel_o2m  $=$  Cross_attn(Query rel_o2m, memory) # (N x D) Query rel_o2m  $=$  FFN(Query rel_o2m) return Query rel_o2m
+```
+
+![](images/2553a967e1436ac4f4d49517b95bca7b5ddd95a64979993677b5cc0f4f4793f3.jpg)  
+(a) Image 1
+
+![](images/fd7ee97bfaffa4f4fe2100aff21af74e57b74dc3792b4a1758e84e378d53e163.jpg)  
+(b) Ground truth 1
+
+![](images/02d9e4e1355cd86a3bea3e97c3749d682afe2654d828a6987fdf99adab836e55.jpg)  
+(c) Hydra-SGG 1
+
+![](images/d6f11387af1289c00a6069a8e6e47bc94b5fbffd53b3264d9d2188f491e6d0b0.jpg)  
+(d) Image 2
+
+![](images/8d882e726bce174ea3fe8e34ad3635c854da187f16ed807d080b940ca02351d8.jpg)  
+(e) Ground truth 2
+
+![](images/f998f7b840967d919897b1ad89e3627393d2c2cfe2a8e86e0d407354252cbede.jpg)  
+(f) Hydra-SGG 2  
+Figure S1: Failure cases of Hydra-SGG on VG150 [75] val.
+
+both engines. In addition, the model correctly identifies the logo and windows on the airplane, which are reasonable and correct predictions but are considered incorrect because they are not annotated in the ground truth. In the second row (Fig. S1d-f), the ground truth annotation uses a single box to label two women, which is an unreasonable annotation. The model correctly predicts two women
+
+and identifies that both women are wearing shirts and shorts, which are annotations absent in the ground truth but are semantically accurate.
+
+# D DISCUSSION
+
+Limitation Analysis. A significant limitation of the current Hydra-SGG is its inability to predict object and relation categories beyond a predefined closed set. The algorithm can only make predictions for object and relation classes that have been explicitly defined and included in the training dataset. This constraint means that the model lacks the capability to recognize or infer novel object types or relationship categories that were not present during the training phase. Such a limitation restricts the model's applicability in real-world scenarios, where encountering previously unseen objects or relations is common. In addition, since the current One-to-Many Relation Assignment rules are manually designed, they may lack flexibility and potentially introduce some noise to the training process.
+
+Societal Impact. Hydra-SGG has the potential to significantly enhance autonomous driving systems, thereby improving road safety and efficiency. For instance, when integrated into self-driving vehicles, Hydra-SGG can provide a deeper understanding of complex traffic scenarios. Beyond merely identifying vehicles, pedestrians, and road signs, it can comprehend the spatial relationships and potential interactions between these elements. This advanced scene understanding could enable autonomous vehicles to more accurately predict the behavior of other road users, leading to safer and more efficient navigation in diverse traffic conditions.
+
+Future Work. A promising direction for future research is to extend the current model towards open vocabulary SGG. This advancement would allow Hydra-SGG to recognize and predict objects and relationships beyond the predefined closed set of categories used in training.
+
+As mentioned in the limitations, since the current One-to-Many Relation Assignment rules are manually designed, they may introduce noise during training that affects learning. Therefore, another direction for future work could focus on designing better one-to-many strategies to further improve performance. For instance, we could develop a feature composition module to generate new relation triplets, or we can incorporate a relationship refine module to transfer coarse-grained relations into fine-grained relations.

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ICLR/2025/HyperDAS_ Towards Automating Mechanistic Interpretability with Hypernetworks/full.md

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+# HYPERDAS: TOWARDS AUTOMATING MECHANISTIC INTERPRETABILITY WITH HYPERNETWORKS
+
+Jiuding Sun $^{\text{♥}}$  Jing Huang $^{\text{♣}}$  Sidharth Baskaran $^{\text{♣}}$  Karel D'Oosterlinck $^{\text{♣}}$
+
+Christopher Potts Michael Sklar\* Atticus Geiger\*
+
+$^{\diamond}\mathrm{Pr}(\mathbf{Ai})^{2}\mathbf{R}$  Group Stanford University Confirm Labs Ghent University
+
+# ABSTRACT
+
+Mechanistic interpretability has made great strides in identifying neural network features (e.g., directions in hidden activation space) that mediate concepts (e.g., the birth year of a person) and enable predictable manipulation. Distributed alignment search (DAS) leverages supervision from counterfactual data to learn concept features within hidden states, but DAS assumes we can afford to conduct a brute force search over potential feature locations. To address this, we present HyperDAS, a transformer-based hypernetwork architecture that (1) automatically locates the token-positions of the residual stream that a concept is realized in and (2) constructs features of those residual stream vectors for the concept. In experiments with Llama3-8B, HyperDAS achieves state-of-the-art performance on the RAVEL benchmark for disentangling concepts in hidden states. In addition, we review the design decisions we made to mitigate the concern that HyperDAS (like all powerful interpretability methods) might inject new information into the target model rather than faithfully interpreting it.
+
+# 1 INTRODUCTION
+
+Mechanistic interpretability methods promise to demystify the internal workings of black-box language models (LMs), thereby helping us to more accurately control these models and predict how they will behave. Automating such efforts is critical for interpreting our largest and most performant models, and strides toward this goal have been made for circuit discovery (Conmy et al., 2023; Rajaram et al., 2024) and neuron/feature labeling (Bills et al., 2023; Huang et al., 2023; Schwettmann et al., 2023; Shaham et al., 2024). In the present paper, we complement these efforts by taking the first steps toward automating interpretability for identifying features of hidden representations (e.g., directions in activation space) that mediate concepts (Mueller et al., 2024; Geiger et al., 2024a).
+
+Interventions on model-internal states are the building blocks of mechanistic interpretability (Saphra & Wiegreffe, 2024). To establish that features of a hidden representation are mediators of a concept, a large number of interchange intervention (Vig et al., 2020; Geiger et al., 2020) experiments are performed on the LM. Interchange interventions change features to values they would take on if a counterfactual input were processed. For example, if the concept is  $C =$  the birth year of a person, we can fix the features  $F$  of an LM processing the input Albert Einstein was born in to the value they take for Marie Curie was a chemist. If the output changes from 1879 to 1934, we have a piece of evidence that  $F$  mediates  $C$ . The field has developed a variety of methods for learning such interventions, but all of them require a brute-force search through potential hidden representations.
+
+To address this significant bottlebeck, we propose HyperDAS, a method to automate this search process via a hypernetwork. In the HyperDAS architecture, a transformer-based hypernetwork localizes a concept within the residual stream of a fixed layer in a target LM by:
+
+1. Encoding a language description (e.g., the birth year of a person) of a concept using a transformer that can attend to the target LM processing a base prompt (e.g., Albert Einstein was born in) and a counterfactual prompt (e.g., Marie Curie was a chemist).
+
+![](images/92d7956ca05fdd4a3ce902e93c7bc915a0e349227d81ecfdaa07686327651ea7.jpg)  
+Figure 1: The HyperDAS framework, used here to find the features that mediate the concept of "country". (1) Concept Encoding A natural language description that specifies which concept to localize, "The country of a city", is encoded by a transformer hypernetwork with two additional cross-attention blocks attending to the hidden states of the target LM prompted with the base text "Vienna is in" and the counterfactual text "I love Paris". (2) Selecting Token-Positions With the encoding from step 1 as a query, HyperDAS uses selects the tokens "nna" and "Paris" as the location of the concept "country" for the base and counterfactual, respectively. (3) Identifying a Subspace With the representation from step 1 as the encoding, HyperDAS constructs a matrix whose orthogonal columns are the features for "country". (4) Interchange Intervention With the token-positions from step 2 and subspace from step 3, HyperDAS performs a intervention by patching the subspace of the hidden vector for the token "nna" to the value it takes on in the hidden vector for the token "Paris", leading the model to predict "France" from the base prompt "Vienna is in".
+
+![](images/e913e8b3d7c8f945eb567b034a45051bf36ea424f171d7e80a8708a3816b82b0.jpg)
+
+2. Pairing tokens in the base and counterfactual prompts (e.g., align "Cur" with "Ein") with an attention mechanism using the encoding from (1) as a query and token-pairs as keys/values.  
+3. Selecting features of the residual stream via a fixed orthogonal matrix that undergoes a Householder transformation (Householder, 1958) using the encoding from (1).  
+4. Patching the selected residual stream features of aligned tokens from the base prompt to the values they take on in the residual stream of aligned tokens from the counterfactual prompt.
+
+We benchmark HyperDAS on the RAVEL interpretability benchmark (Huang et al., 2024), in which concepts related to a type of entity are disentangled. For example, we might seek to separate features for the birth year and field of study of a Nobel laureate. HyperDAS achieves state-of-the-art performance on RAVEL with a single model. Greater gains are achieved when a separate HyperDAS is trained for each entity type (e.g., Nobel laureates).
+
+Finally, we address the issue of whether HyperDAS is faithful to the target model. As we use more complex machine learning tools for interpretability, there is an increasing concern that we are not uncovering latent causal structure, but instead injecting new information to steer or edit a model (Meng et al., 2022; Ghandeharioun et al., 2024). If we allow our supervised interpretability models too much power, we run the risk of false-positive signals. Thus, we conclude with a discussion of how our decisions about architecture, training, and evaluation were made to mitigate these concerns.
+
+# 2 BACKGROUND
+
+Automating Interpretability Workflows The growing size and complexity of language models demands scalable techniques for interpretability. Major directions include circuit analysis (Conmy
+
+et al., 2023; Marks et al., 2024; Rajaram et al., 2024; Ferrando & Voita, 2024), unsupervised feature learning (Huben et al., 2024; Braun et al., 2024), and feature labeling with natural language descriptions (Mu & Andreas, 2021; Hernandez et al., 2022; Bills et al., 2023; Huang et al., 2023; Shaham et al., 2024). In this work, we take steps towards automating the process of identifying features that are causal mediators of concepts using supervision from counterfactual data.
+
+Identifying Features that Mediate Concepts Interchange interventions identify neural representations that are causal mediators of high-level concepts (Vig et al., 2020; Geiger et al., 2020; Finlayson et al., 2021; Stolfo et al., 2023). Geiger et al. (2024b) and Wu et al. (2024) further extend interchange interventions to localizing concepts in hidden vector subspaces. However, these methods require an exhaustive search over all layers and tokens to measure causal effects at each position. In practice, the lack of an effective search method leads to heuristics in token selection. For example, in knowledge editing and model inspection, a widely held assumption is that the entity information is localized to the last entity token (Meng et al., 2022; 2023; Hernandez et al., 2024; Geva et al., 2023; Ghandeharioun et al., 2024), but this does not hold for all entities (Meng et al., 2022). Our proposed method directly addresses this problem by using an end-to-end optimization to automatically select the intervention site across all tokens, conditioned on the concept to localize.
+
+The RAVEL Benchmark The RAVEL benchmark evaluates how well an interpretability method can localize and disentangle entity attributes through causal interventions. An example consists of a base prompt that queries a specific attribute of an entity (e.g., Albert Einstein studied the field), a counterfactual prompt containing a different entity of the same type (e.g., Poland declared 2011 the Year of Marie Curie), an attribute targeted for intervention (e.g., field of study or birth year), and a counterfactual label for the base prompt. The label would be physics if the targeted attribute is birth year (the intervention should not affect the field of study attribute), and it would be chemistry if the targeted attribute is field of study.
+
+Distributed Interchange Interventions RAVEL supports evaluations with distributed interchange interventions on features of a hidden representation  $\mathbf{H}$  that encode an attribute in the original model  $\mathcal{M}$ . In our experiments, features are orthogonal directions that form the columns of a low-rank matrix  $\mathbf{R}$ . Given a base prompt  $\bar{\mathbf{x}}$  and a counterfactual prompt  $\hat{\mathbf{x}}$ , we perform an intervention that fixes the linear subspace spanned by  $\mathbf{R}$ :
+
+$$
+\mathbf {H} \leftarrow \bar {\mathbf {h}} + \mathbf {R} ^ {\top} \left(\mathbf {R} (\hat {\mathbf {h}}) - \mathbf {R} (\bar {\mathbf {h}})\right) \tag {1}
+$$
+
+where  $\bar{\mathbf{h}}$  and  $\hat{\mathbf{h}}$  are the values that variable  $\mathbf{H}$  has when the model  $\mathcal{M}$  is run on  $\bar{\mathbf{x}}$  and  $\hat{\mathbf{x}}$ , respectively.
+
+RAVEL Metrics The metric from the RAVEL dataset has two components. The Cause score is the proportion of interchange interventions that successfully change the attribute that was targeted, and the Iso score is the proportion of interchange interventions that successfully do not change an attribute that was not targeted. The Disentangle score is the average of these two.
+
+Distributed Alignment Search The RAVEL evaluations use distributed alignment search (DAS; Geiger et al. 2024b) as a baseline for learning a feature of a hidden representation that mediate an attribute. A rotation matrix is optimized on a RAVEL example with base input  $\bar{\mathbf{x}}$ , counterfactual input  $\hat{\mathbf{x}}$ , and counterfactual label  $y$  using the following loss:
+
+$$
+\mathcal {L} _ {\mathrm {D A S}} = \operatorname {C E} \left(\mathcal {M} _ {\mathbf {H} \leftarrow \bar {\mathbf {h}} + \mathbf {R} ^ {\top} \left(\mathbf {R} (\hat {\mathbf {h}}) - \mathbf {R} (\bar {\mathbf {h}})\right)} (\bar {\mathbf {x}}), y\right) \tag {2}
+$$
+
+where  $\mathcal{M}_{\gamma}(\bar{\mathbf{x}})$  is the output of the model  $\mathcal{M}$  run on input  $\bar{\mathbf{x}}$  with an intervention  $\gamma$ . Only the parameters  $\mathbf{R}$  are updated while the parameters of the target model  $\mathcal{M}$  are frozen.
+
+# 3 HYPERDAS
+
+To localize a concept in the layer  $l$  of a target model  $\mathcal{M}$ , a HyperDAS architecture consists of a hypernetwork  $\mathcal{H}$  that takes in a text specification  $\mathbf{x}$  of the target concept and dynamically selects token positions in the base text  $\bar{\mathbf{x}}$  and counterfactual text  $\hat{\mathbf{x}}$  and identifies a linear subspace  $\mathbf{R}$  that mediates the target concept. Selecting tokens is a discrete operation, so we "soften" the selection during training and force discrete decisions during evaluation. Our specific model is as follows.
+
+# 3.1 REPRESENTING THE TARGET CONCEPT AS A VECTOR
+
+Token Embedding A token sequence  $\mathbf{x}$  of length  $E$  that specifies the concept to localize, e.g., the country a city is in, is encoded with the embeddings of the target model  $\mathcal{M}$  to form  $\mathbf{e}_0 = \mathrm{Emb}(\mathbf{x})\in$ $\mathbb{R}^{E\times d}$ , the zeroth layer of the residual stream for the hypernetwork  $\mathcal{H}$ .
+
+Cross-attention Decoder Layers After embedding the target concept, we run a transformer with  $N$  decoder layers. Besides the standard multi-headed self-attention (MHA) and feed-forward layers (MLP), each decoder block has two additional cross-attention modules to incorporate information from the target model  $\mathcal{M}$  processing the base and counterfactual prompts  $\bar{\mathbf{x}}$  and  $\hat{\mathbf{x}}$ .
+
+Let  $\bar{\mathbf{h}}\in \mathbb{R}^{B\times L\times d}$  and  $\hat{\mathbf{h}}\in \mathbb{R}^{C\times L\times d}$  be the stacks of base and counterfactual hidden states from the base and the counterfactual input, where  $L$  is the number of layers in  $\mathcal{M}$ ,  $d$  is the hidden dimension, and  $B$  and  $C$  are the sequence length of the base and counterfactual examples, respectively. Two multi-headed cross-attention modules MHA and MHA allow  $\mathcal{H}$  to attend to  $\bar{\mathbf{h}}$  and  $\hat{\mathbf{h}}$ . Each layer of the hypernetwork  $\mathcal{H}$  can attend to every layer of the target model.
+
+For the  $p$ -th decoder layer of the hypernetwork  $\mathcal{H}$ , the three attention mechanisms are as follows:
+
+$$
+\mathbf {e} _ {p} ^ {\prime} = \operatorname {M H A} \left(\mathbf {Q} \left(\mathbf {e} _ {p}\right), \mathbf {K} \left(\mathbf {e} _ {p}\right), \mathbf {V} \left(\mathbf {e} _ {p}\right)\right) \tag {3}
+$$
+
+$$
+\mathbf {e} _ {p} ^ {\prime \prime} = \mathrm {M H A} \left(\bar {\mathbf {Q}} \left(\mathbf {e} _ {p} ^ {\prime}\right), \bar {\mathbf {K}} (\bar {\mathbf {h}}), \bar {\mathbf {V}} (\bar {\mathbf {h}})\right) \tag {4}
+$$
+
+$$
+\mathbf {e} _ {p + 1} = \mathsf {M} \hat {\mathsf {H}} \mathsf {A} \left(\hat {\mathbf {Q}} \left(\mathbf {e} _ {p} ^ {\prime \prime}\right), \hat {\mathbf {K}} (\hat {\mathbf {h}}), \hat {\mathbf {V}} (\hat {\mathbf {h}})\right) \tag {5}
+$$
+
+After the final transformer block is applied, the residual stream vector at the last token position  $\mathbf{e}_E^{(N)}\in \mathbb{R}^d$  encodes information about the concept targeted for intervention and the target model's base and counterfactual runs. This representation is used to select token-positions in the base and source texts and identify a linear subspace for intervention.
+
+# 3.2 DYNAMICALLY SELECTINGTOKEN-POSITIONS THAT CONTAIN THE TARGET CONCEPT
+
+The next step is selecting the tokens in the base and counterfactual prompts that contain the concept encoded as  $\mathbf{e}_E^{(N)}$ . We construct an "intervention score" matrix  $G \in \mathbb{R}^{B \times (C + 1)}$  with elements ranging from 0 to 1, where  $B$  and  $C$  are the number of tokens in the base and counterfactual prompts, respectively. Each element  $G_{(b,c)}$  denotes the degree to which the  $b$ -th base token is aligned with  $c$ -th counterfactual token for intervention. The additional column  $G_{(b,c)}$  corresponds to the score for retaining the  $b$ -th base token without any intervention.
+
+In Figure 1, only the base-token "nna" (the last token of the entity "Vienna") receives an intervention score of 1 when paired with the counterfactual=token "Paris". All other base tokens are not selected for intervention. The concept of "country" is localized to the last token of the city entities.
+
+To compute the element  $G_{(b,c)}$ , we use  $\bar{\mathbf{h}}_b^{(l)}$  and  $\hat{\mathbf{h}}_c^{(l)}$ , the  $l$ -th layer residual stream representation of the target model at  $b$ -th base-token and  $c$ -th counterfactual-token. We first linearly combine the two:
+
+$$
+\mathbf {g} _ {(b, c)} = F \left(\left[ \bar {\mathbf {h}} _ {b} ^ {(l)}; \hat {\mathbf {h}} _ {c} ^ {(l)} \right]\right) \tag {6}
+$$
+
+where  $F(.): \mathbb{R}^{2d} \to \mathbb{R}^d$  is a linear projection that condenses the concatenated representation into the original dimension  $d$ . For the extra column that indicates no intervention, the representation is simply the original base token representation:
+
+$$
+\mathbf {g} _ {(b, C + 1)} = \bar {\mathbf {h}} _ {b} ^ {(l)} \tag {7}
+$$
+
+Then, we weight each merged representation  $\tilde{\mathbf{h}}$  using the concept encoding  $\mathbf{e}_E^{(N)}\in \mathbb{R}^d$  and  $M$  "query" and "key" matrices  $Q^{(i)}\in \mathbb{R}^{d\times M\times \frac{d}{M}}$  and  $K^{(i)}\in \mathbb{R}^{d\times M\times \frac{d}{M}}$  ..
+
+$$
+G _ {(b, c)} ^ {\prime} = \frac {\sum_ {i = 1} ^ {M} \left(\mathbf {e} _ {E} ^ {(N)}\right) Q ^ {(i)} \cdot (\tilde {\mathbf {h}}) K ^ {(i)}}{M \sqrt {d}} \tag {8}
+$$
+
+Finally, we apply a column-wise softmax  $G = \text{ColumnSoftmax}(G')$ .
+
+Using the matrix  $G$ , we can construct the representation that we will use to intervene on each token in the base prompt. For the  $b$ -th base-token hidden states  $\bar{\mathbf{h}}_b^l$ , the intervention representation is:
+
+$$
+\tilde {\mathbf {h}} _ {b} ^ {(l)} = G _ {(b, C + 1)} \bar {\mathbf {h}} _ {b} ^ {(l)} + \sum_ {c = 1} ^ {C} G _ {(b, c)} \hat {\mathbf {h}} _ {c} ^ {(l)} \tag {9}
+$$
+
+This is how the counterfactual representation is constructed for a weighted interchange intervention (Wu et al., 2024). The counterfactual hidden states remain identical to the base hidden states (i.e., no intervention on token  $b$ ) when  $G_{(b,C + 1)} = 1$ . Conversely, if  $G_{(b,c)} = 1$  for a specific position  $c$ , the  $b$ -th base token is entirely replaced by the hidden vector for the  $c$ -th counterfactual token.
+
+# 3.3 DYNAMICALLY IDENTIFYING A LINEAR SUBSPACE THAT CONTAINS THE CONCEPT
+
+In addition to selecting token-positions, HyperDAS also dynamically identifies a linear subspace that contains the target concept, encoded as a low-rank matrix with orthogonal columns. First, we apply a multi-layer perceptron to  $\mathbf{e}_E^{(N)} \in \mathbb{R}^d$  in order to produce a new vector
+
+$$
+\mathbf {v} = \operatorname {M L P} \left(\mathbf {e} _ {E} ^ {(N)}\right) \in \mathbb {R} ^ {d}.
+$$
+
+In DAS, there is a fixed low-rank matrix with orthogonal columns  $\mathbf{R}^{\prime}$  representing a fixed subspace targeted for intervention. We use a linear algebra operation known as the Householder transformation to change  $\mathbf{R}^{\prime}$  conditional on  $\mathbf{v}$  into a new matrix  $\mathbf{R}$  that still has orthogonal columns. Given a non-zero vector  $\mathbf{v} \in \mathbb{R}^{d}$ , the Householder transformation  $\mathbf{H}$  is defined as:
+
+$$
+\mathbf {H} = \mathbf {I} - 2 \frac {\mathbf {v v} ^ {\top}}{\mathbf {v} ^ {\top} \mathbf {v}}, \tag {10}
+$$
+
+where  $\mathbf{I}$  is the identity matrix. The matrix  $\mathbf{H}$  is orthogonal and  $\mathbf{R}^{\prime}$  has orthogonal columns, which means  $\mathbf{R}^{\prime}\mathbf{H}$  has orthogonal columns. Utilizing this property, we can dynamically select the subspace based on the intervention representation  $\mathbf{e}_E^{(N)}$  by computing  $\mathbf{R} = \mathbf{R}^{\prime}\mathbf{H}$ .
+
+# 3.4 INTERVENING ON THE SUBSPACE AT THE SELECTEDTOKEN-POSITIONS
+
+After selecting token-positions and identifying a subspace, we finally intervene. For each base-token  $b$  in the  $l$ -th layer of the target model  $\mathcal{M}$ , we perform a weighted interchange intervention with the counterfactual hidden states  $\tilde{\mathbf{h}}^{(l)}$  and low-rank orthogonal matrix  $\mathbf{R}$ :
+
+$$
+\mathbf {H} _ {b} ^ {(l)} \leftarrow \bar {\mathbf {h}} _ {b} ^ {(l)} + \mathbf {R} ^ {\top} \left(\mathbf {R} \left(\tilde {\mathbf {h}} _ {b} ^ {(l)}\right) - \mathbf {R} \left(\bar {\mathbf {h}} _ {b} ^ {(l)}\right)\right). \tag {11}
+$$
+
+# 3.5 TRAINING
+
+We train HyperDAS on the RAVEL with two losses. The first simply measures success on RAVEL. The second incentivizes the model to select unique token-pairings so that performance is maintained when the token alignment matrix  $G$  is snapped to binary values during test-time evaluation.
+
+RAVEL Loss A RAVEL example consists of a base input  $\bar{\mathbf{x}}$ , counterfactual input  $\hat{\mathbf{x}}$ , target concept input  $\mathbf{x}$ , and a counterfactual label  $\mathbf{y}$ . When the target concept matches the attribute queried in the base input, the label  $\mathbf{y}$  is the attribute of  $\hat{\mathbf{x}}$ . Otherwise,  $\mathbf{y}$  is the attribute of  $\bar{\mathbf{x}}$ . The loss is:
+
+$$
+\mathcal {L} _ {\mathrm {R A V E L}} = \operatorname {C E} \left(\mathcal {M} _ {\mathbf {H} _ {b} ^ {(l)} \leftarrow \bar {\mathbf {h}} _ {b} ^ {(l)} + \mathbf {R} ^ {T} \left(\mathbf {R} \left(\tilde {\mathbf {h}} _ {b} ^ {(l)}\right) - \mathbf {R} \left(\bar {\mathbf {h}} _ {b} ^ {(l)}\right)\right) (\bar {\mathbf {x}}), \mathbf {y}\right) \tag {12}
+$$
+
+Sparse Attention Loss The construction of  $G$  allows for a single counterfactual token to be paired with a weighted combination of multiple base tokens, however during test-time evaluations we will enforce a 1-1 correspondence. Thus, we include a sparse attention loss that penalizes cases where one counterfactual token attends strongly to multiple base tokens in each row of matrix  $G$ :
+
+$$
+\mathcal {L} _ {\text {s p a r s e}} = \frac {1}{C} \sum_ {c = 1} ^ {C} \left\{ \begin{array}{l l} \operatorname {S u m} \left(G _ {(*, c)}\right) & \text {i f} \operatorname {S u m} \left(G _ {(*, c)}\right) > 1 \\ 0 & \text {i f} \operatorname {S u m} \left(G _ {(*, c)}\right) \leq 1 \end{array} \right. \tag {13}
+$$
+
+The final loss is  $\mathcal{L} = \mathcal{L}_{\mathrm{RAVEL}} + \lambda \mathcal{L}_{\mathrm{sparse}}$ , where  $\lambda$  is a real-valued weight scheduled during training.
+
+![](images/611dfca14bfb996b883921b02bcd633dc4335b7352409f77e8e7da651c73fbf6.jpg)  
+Figure 2: The "intervention score" matrix  $G$  for the counterfactual prompt "The city of Macheng's official language is" and base prompt "Springfield is a city in the country of", for which the target model will output "China". The attribute targeted for intervention is country, so the output should be "The United States". The raw intervention (left) is produced by the token-position selection discussed in Sec 3.2, and a column-wise and row-wise argmax is applied at inference time to enforce an 1-1 correspondence between the base and counterfactual tokens, detailed in Sec. 3.6.
+
+![](images/08c1880ec9030b0bb1ca29e5833b5032bac194ec6586bb500f9d918cb9918073.jpg)
+
+# 3.6 EVALUATION
+
+HyperDAS is end-to-end differentiable because discrete operations like aligning base and counterfactual tokens are "softened" using softmax operators. During the evaluation, we force these discrete decisions. In many cases, the matrix  $G$  contains non-zero weight for multiple base-counterfactual sentence token pairs (left panel in Figure 2). At inference time, we perform a double argmax operation on the intervention score to select the most important location for the intervention. For each base-counterfactual token pair, it is set to 1 if and only if this position gets the highest intervention score across its row and column.
+
+$$
+G _ {(b, c)} = \left\{ \begin{array}{l l} 1 & \text {i f} G _ {(b, c)} = \max  \left(G _ {(b, *)}, G _ {(*, c)}\right) \\ 0 & \text {o t h e r w i s e} \end{array} \right. \tag {14}
+$$
+
+The [SELF] row, representing no intervention on a base-token, is readjusted according to the discrete intervention weight (right panel in Figure 2).
+
+$$
+G _ {(b, C + 1)} = 1 \text {i f} \max  \left(G _ {(b, *)}\right) = 0 \tag {15}
+$$
+
+# 4 EXPERIMENTS
+
+We benchmark HyperDAS on RAVEL with Llama3-8B (Meta, 2024) as the target model. We both train a HyperDAS model on all of RAVEL at once and also train a separate HyperDAS model for each of the five entity domains in the RAVEL benchmark, i.e., cities, Nobel laureates, occupations, physical objects, and verbs. We experimented with initializing the transformer hypernetwork from pre-trained parameters, but found no advantage for this task.
+
+**Crucial Hyperparameters** We use 8 decoder blocks for the hypernetwork and 32 attention heads for computing the pairwise token position attention. The sparsity loss weight is scheduled to linearly increase from 0 to 1.5, starting at  $50\%$  of the total steps. A learning rate between  $2 \times 10^{4}$  to  $2 \times 10^{5}$  is chosen depending on the dataset. Discussion of these choices concerning the sparsity loss is in Section 4.2. For the feature subspace, we experiment with dimensions from 32 up to 2048 (out of 4096 dimensions) and use a subspace of dimension 128.
+
+Masking of the Base Prompt As the hypernetwork has access to the target attribute information from the instruction and the base attribute information from the base model states, a trivial solution
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">City</td><td colspan="2">Nobel Laureate</td><td colspan="2">Occupation</td><td colspan="2">Physical Object</td><td colspan="2">Verb</td><td rowspan="2">Average Disentangle</td></tr><tr><td>Causal</td><td>Iso</td><td>Causal</td><td>Iso</td><td>Causal</td><td>Iso</td><td>Causal</td><td>Iso</td><td>Causal</td><td>Iso</td></tr><tr><td>MDAS</td><td>55.8</td><td>77.9</td><td>56.0</td><td>93.5</td><td>50.7</td><td>88.1</td><td>85.0</td><td>97.9</td><td>74.3</td><td>79.6</td><td>76.0</td></tr><tr><td>HyperDAS</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- Asymmetric</td><td>70.8</td><td>93.9</td><td>55.4</td><td>95.1</td><td>50.4</td><td>99.1</td><td>92.7</td><td>97.2</td><td>93.0</td><td>98.9</td><td>84.7</td></tr><tr><td>- Asymmetric All Domains</td><td>58.8</td><td>90.5</td><td>47.6</td><td>92.0</td><td>75.7</td><td>82.1</td><td>92.9</td><td>94.5</td><td>86.9</td><td>95.8</td><td>80.7</td></tr><tr><td>- Symmetric</td><td>76.9</td><td>90.9</td><td>59.2</td><td>88.4</td><td>47.1</td><td>89.1</td><td>94.8</td><td>97.5</td><td>42.3</td><td>82.9</td><td>76.9</td></tr><tr><td>- Symmetric All Domains</td><td>16.8</td><td>94.7</td><td>2.0</td><td>98.8</td><td>6.1</td><td>97.3</td><td>21.6</td><td>99.3</td><td>13.6</td><td>97.8</td><td>54.8</td></tr></table>
+
+(a) Main results of HyperDAS on five domains of RAVEL with Llama3-8B. For each method, we report the results from the best layer between 10 and 15. HyperDAS achieves state-of-the-art attribute disentangling performance across the board. The MDAS baseline intervenes on fixed tokens.
+
+![](images/593bb09e3bef0e75598356be9e9541c0aca50f24559183349bf96afc81ecfca8.jpg)  
+Figure 3: RAVEL benchmark results. HyperDAS establishes a new state-of-the-art.
+
+![](images/8b7da7237edfd5e84431a52ff9adfea0d40403f3cc72d33cd551ccb96ab09ca8.jpg)
+
+![](images/6ab7c282ceaeefc5c09f87ab708cb8f79b7aee22708a2dd427b25df19eed199c.jpg)
+
+(b) The cause/iso/disentangle score of the baseline method and HyperDAS (Both symmetric and asymmetric implementation) for the entity type of "city" across the layers of Llama3-8B. For both HyperDAS and MDAS, the peak Disentangle Score happens around L15.
+
+the hypernetwork can learn is to condition the intervention location on whether the target attribute matches the base attribute. In other words, if the two attributes match, attend to a location that has causal effect on the output, otherwise attend to the extra [self] row (see Appendix A.3 for an example). This solution, however, does not find the actual concept subspace. To prevent the hypernetwork from learning this trivial solution, we apply an attention mask to the base prompt to mask out the attribute information. With the masking, the hypernetwork no longer has access to the base attribute information, and hence the localization prediction is only conditioned on the target attribute in the natural language instruction.
+
+Symmetry Intuitively, if we have localized a concept, then "get" operations that retrieve the concept and "set" operations that fix the concept should both target the same features and hidden representations. For this reason, we consider a variant of HyperDAS that enforces symmetry in the localization of base and counterfactual prompts. We can enforce symmetry between base and counterfactual inputs during token selection by randomly flipping the order of the concatenation between base and counterfactual hidden representations in Equation 6. We report full results for symmetric and asymmetric models.
+
+Multi-task DAS (MDAS) Baseline The current state-of-the-art method on RAVEL is MDAS, which uses a multi-task learning objective to satisfy multiple high-level causal criteria. MDAS requires supervised training data like HyperDAS, however, MDAS relies on manually selected token position for intervention, which in our case is the final token of the entity, e.g., "nna" in Figure 1.
+
+Results In Table 3a, we show results on RAVEL for layer 15 of Llama3-8B. In Figure 3b, we also run HyperDAS targeting every 2 layers in Llama3-8B starting from the embedding layer. The peak performance of attribute disentanglement for both MDAS and HyperDAS is around layer 15.
+
+# 4.1 LAYER-SPECIFIC INTERVENTION BEHAVIORS OF HYPERDAS
+
+HyperDAS searches for an optimal location to intervene within the target hidden state in a chosen layer. We evaluate MDAS and HyperDAS on 16 layers across the model (Figure 3b) and chose an early layer, middle layer, and deep layer for detailed study: Layer 7, Layer 15, and Layer 29.
+
+![](images/c2b9f74e7b0d9ce9bf8216970e9983714945ceef07179ba3018b3fc5420a1827.jpg)  
+Figure 4: The intervention location, in counterfactual and base sentence, picked by HyperDAS when targeting shallow (7), middle (15) and deep (29) decoder layers.
+
+Analysis presented in Figure 4 reveals that HyperDAS consistently targets entity tokens in the counterfactual input across all layers, suggesting robust detection of attribute information in the entity token's residual stream from an early stage. However, the choice of intervention location within the base input shows significant variation. For each example in the "city" entity split, we categorize the base and counterfactual token pair that gets the largest intervention weight, and classify them into the following categories: (1) BOS Token represents the beginning-of-sentence token. (2) Entity Token refers to tokens representing entities. (3) JSON Syntax includes special characters and syntactic tokens typical of JSON formatted text (e.g., opening curly brace "\{"). (4) Others comprises all tokens irrelevant to the current analysis, with "is" following the entity token being a common example in both shallow  $(36\%)$  and deep  $(29\%)$  layer bases.
+
+At very early layers, HyperDAS displays turbulent behavior, targeting random or even beginning-of-sentence tokens in the base sentence. By the middle layers, the model consistently favors the entity token for intervention, aligning with findings from Huang et al. (2024) and Geva et al. (2023). In contrast, at deeper layers, the hypernetwork learns to intervene on unintuitive positions such as syntax tokens within a JSON-formatted prompt, which were previously unknown to store attributes.
+
+# 4.2 DISCUSSION
+
+HyperDAS establishes a new state-of-the-art performance on RAVEL Our results show that HyperDAS outperforms MDAS, the previous state-of-the-art, across all entity splits at layer 15 in Llama3-8B and across all layers of Llama3-8B for the "cities" entity split.
+
+HyperDAS requires more compute than MDAS. HyperDAS is more powerful than MDAS, but also more computationally expensive. Training our HyperDAS model for one epoch on disentangling the country attribute in the city domain takes 468,923 TeraFLOPs, while training an MDAS model for one epoch on the same task takes 193,833. Thus, HyperDAS requires  $\approx 2.4\mathrm{x}$  compute.
+
+On the other hand, HyperDAS is more memory efficient for tasks like RAVEL. Our target Llama model requires 16GB of RAM. The HyperDAS model requires 52GB more in total for RAVEL, whereas MDAS requires only 4.1GB more per attribute. The memory usage of HyperDAS does not go up with additional attributes, so when trained on all of RAVEL together (23 attributes), MDAS  $(23 * 4.1 + 16 = 110.3\mathrm{GB})$  exceeds the memory usage of HyperDAS  $(52 + 16 = 68\mathrm{GB})$ .
+
+Householder vectors analysis provides a window into attribute features. To analyze the Householder vectors generated by the model, we collected vectors from each test example and categorized them according to their respective attributes. For each attribute category, a subset of 1,000 samples was randomly selected. We then computed the similarity scores between pairs of attributes by calculating the average cosine similarity across these 1,000 pairs of selected Householder vectors.
+
+![](images/5af20fc6da7dd9b2d5107551253bf675942fd6845e260dee52bdfc6b0b07b86d.jpg)  
+Figure 5: The relative position between the Householder vector (after PCA) of attributes for all the correct predictions in city domain. The clustering indicates that HyperDAS learns different subspace for each attribute.
+
+![](images/79755c3501f55007ad2d19904f7cda805717a087e8ea525a0461611044a3139b.jpg)  
+Figure 6: The cosine similarity between the Householder vectors of different attributes in the city domain, computed using 100,000 samples from each attribute. Notably, HyperDAS effectively learns a highly similar subspace for the attributes 'Longitude' and 'Latitude'.
+
+![](images/957b6bef9a2e799b16b6760ebb74084fe1d174557a347ec9fa5d0275b34ea0c9.jpg)  
+Figure 7: Intervention locations for a base/counterfactual sentences pair with Occupation entity-type selected by HyperDAS trained with different amounts of sparsity loss. This comparison illustrates the intervention locations generated by HyperDAS when trained under three different sparsity loss conditions. All three models achieved a Disentangle Score  $\approx 94.0\%$  using weighted interventions. With no sparsity loss (middle), HyperDAS tends to intervene from the last subject token in the counterfactual sentence to most tokens in the base sentence, which yields adequate performance under many-to-one constraints but not under strict one-to-one constraints. With too much sparsity loss (right), the pairwise token selection attention within HyperDAS fails, resulting in interventions that blend all hidden states. Although this approach achieves a near-perfect disentangle score with weighted intervention, the model's does not have interpretable intervention patterns and fails entirely during test time when masks are snapped to align base and source tokens one-to-one.
+
+![](images/0e4ca62e426a86125f662a38ee3dfb507d078f6f17f8f9bde234e149a00d2c21.jpg)
+
+![](images/1ba135fa41d00d5725ec428fd7b0c1bc3d014a873425caab368d1fd21b48f8f2.jpg)
+
+We analyze the geometry of the learned householder vectors, with the PCA projection shown in Figure 5. We also compute the average pairwise cosine similarity of Householder vectors sampled from within the same attribute or cross two different attributes, as shown in Figure 6. Despite an overall high cosine similarity among all Householder vectors associated with the same entity type, the Householder vectors associated with the same attribute form a tighter cluster, with a higher cosine similarity score than pairs of vectors associated with two different attributes. These per-attribute clusters might explain why the learned subspace can disentangle different attributes of the same entity, as different attributes are localized into different subspaces of the entity representation.
+
+How do we know HyperDAS uncovers actual causal structures faithful to the target model? On one hand, we should leverage the power of supervised machine learning to develop increasingly sophisticated interpretability methods. On the other hand, such methods are incentivized to "hack" evaluations without uncovering actual causal structure in the target model. We have taken several steps to maintain fidelity to the underlying model structures when training and evaluating HyperDAS, by constraining optimization flexibility to prevent inadvertently steering or editing the model with out-of-distribution interventions.
+
+![](images/284e6862065a31e9d8b5a7a4f4a93077b186e18c031180e9c43b80c835e3fd00.jpg)  
+Figure 8: The count of intervention location picked by HyperDAS at the counterfactual prompt (upper) v.s. at the base prompt (bottom) across all the attributes in the city domain on entities with three tokens. The asymmetric variant (right) of HyperDAS favors getting the attribute information from the last entity token for the majority of the counterfactual prompts ( $\geq 95\%$ ), and intervene on the second last entity token. The symmetric variant (left) favors last entity token consistently for both base and counterfactual prompt.
+
+The weighted interchange interventions used in training hacks the objective without soft constraints via loss terms. The loss term  $\mathcal{L}_{\mathrm{sparse}}$  is crucial for ensuring that HyperDAS learns a one-to-one alignment between base tokens and counterfactual tokens (Figure 2). When no sparsity loss is applied, the model aligns the final entity token (e.g., "nna" from Figure 1) to many tokens in the base sentence. These solutions fail during evaluations where token alignments are snapped to be one-to-one. Conversely, with excessive sparsity loss, the model constructs a counterfactual hidden representation that is a linear combination of many hidden states, resulting in a high flexibility optimization scheme that is closer to model steering or editing. This also fails during one-to-one evaluations. See Figure 7 for an example of these pathological settings.
+
+Often only one token is aligned between base and counterfactual inputs. The MDAS baseline performs well on the RAVEL benchmark by one token in the base and one token in the source. However, our new state-of-the-art HyperDAS model will select multiple tokens  $53\%$  of the time.
+
+Asymmetric HyperDAS targets different tokens for base and counterfactual examples. Figure 8 shows the tokens selected by the symmetric and asymmetric variants of HyperDAS. When allowed asymmetric parametrization, networks break symmetry in positional assignments; for a single input prompt, HyperDAS will select different tokens depending on whether that input is the base or counterfactual.
+
+# 5 CONCLUSION
+
+In this work, we introduced HyperDAS, a novel hypernetwork-based approach for automating causal interpretability work. HyperDAS achieves state-of-the-art performance on the RAVEL benchmark, demonstrating its effectiveness in localizing and disentangling entity attributes through causal interventions. Our method's ability to dynamically select hidden representations and learn linear features that mediate target concepts represents a significant advancement in interpretability techniques for language models. We are optimistic that HyperDAS will open new avenues for understanding and interpreting the internal workings of complex language models.
+
+Limitations HyperDAS will only be successful if the target concept is mediated by linear features. However, there is emerging evidence that non-linear mediators are a possibility (Csordás et al., 2024; Engels et al., 2024). As discussed extensively in the main text, applying supervised machine learning to interpretability has the potential to lead to false positive results. While we have taken steps to maintain fidelity to underlying model structures, future work should continue to explore the delicate balance between uncovering latent causal relationships and the risk of model steering.
+
+# ACKNOWLEDGEMENTS
+
+This research was in part supported by a grant from Open Philanthropy.
+
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+Sarah Schwettmann, Tamar Rott Shaham, Joanna Materzynska, Neil Chowdhury, Shuang Li, Jacob Andreas, David Bau, and Antonio Torralba. FIND: A function description benchmark for evaluating interpretability methods. In Alice Oh, Tristan Naumann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine (eds.), Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10 - 16, 2023, 2023. URL http://papers.nips.cc/paper_files/paper/2023/bitstream/ef0164c1112f56246224af540857348f-Abstract-Datasets_and_Benchmarks.html.  
+Tamar Rott Shaham, Sarah Schwettmann, Franklin Wang, Achyuta Rajaram, Evan Hernandez, Jacob Andreas, and Antonio Torralba. A multimodal automated interpretability agent. In *Forty-first International Conference on Machine Learning*, ICML 2024, Vienna, Austria, July 21-27, 2024. OpenReview.net, 2024. URL https://openreview.net/forum?id=mDw42ZanmE.  
+Alessandro Stolfo, Yonatan Belinkov, and Mrinmaya Sachan. A mechanistic interpretation of arithmetic reasoning in language models using causal mediation analysis. In Houda Bouamor, Juan Pino, and Kalika Bali (eds.), Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, EMNLP 2023, Singapore, December 6-10, 2023, pp. 7035-7052. Association for Computational Linguistics, 2023. doi: 10.18653/V1/2023.EMNLP-MAIN.435. URL https://doi.org/10.18653/v1/2023.emnlp-main.435.  
+Jesse Vig, Sebastian Gehrmann, Yonatan Belinkov, Sharon Qian, Daniel Nevo, Yaron Singer, and Stuart Shieber. Investigating gender bias in language models using causal mediation analysis. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 12388-12401. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper_files/paper/2020/file/92650b2e92217715fe312e6fa7b90d82-Paper.pdf.  
+Zhengxuan Wu, Atticus Geiger, Thomas Icard, Christopher Potts, and Noah Goodman. Interpretability at scale: Identifying causal mechanisms in alpaca. Advances in Neural Information Processing Systems, 36, 2024.
+
+# A APPENDIX
+
+# A.1 HYPERDAS OVER ALL DOMAINS
+
+Our results at Table 3a show that we can train HyperDAS to achieve state of the art performance on the RAVEL benchmark by training a separate model for each entity type split, which is the set up used to train the previous state of the art MDAS. To test the scalability and generalizability of HyperDAS, we train a single model across all the entity type splits and evaluate its performance.
+
+Experiment Set-up We aggregate the training split of the dataset from all 5 domains and train HyperDAS for 5 epochs. We adjust the learning rate from  $2 \times 10^{-5}$  to  $5 \times 10^{-4}$  and schedule the sparsity weight  $\lambda$  ranging from 0.75 to 1.5 starting after  $50\%$  of the total steps. This set-up allows the model to first find a stable solution across all domains with soft intervention before forcing it to converge to a single token selection.
+
+Result We report the performance of HyperDAS trained over all entity type split in Table 3a. The model performs better than MDAS but slightly worse than HyperDAS trained on individual entity type split by  $4.0\%$ . Specifically, HyperDAS-All-Domain performs worse over city and noble laureate split, better over occupation split, and on-par over physical object and verb split.
+
+# A.2 DATASET SPECIFICATION
+
+<table><tr><td>Domain/Attribute</td><td># of Cause Example</td><td># of IsolateExample</td><td># of Entity</td></tr><tr><td>City</td><td>34899/7016</td><td>49500/9930</td><td>3552/3374</td></tr><tr><td>Country</td><td>7925/1544</td><td>8250/1655</td><td>3528/2411</td></tr><tr><td>Language</td><td>6207/1252</td><td>8250/1655</td><td>3471/2221</td></tr><tr><td>Continent</td><td>8254/1658</td><td>8250/1655</td><td>3543/2567</td></tr><tr><td>Timezone</td><td>5371/1144</td><td>8250/1655</td><td>3414/1900</td></tr><tr><td>Latitude</td><td>3813/743</td><td>8250/1655</td><td>3107/1519</td></tr><tr><td>Longitude</td><td>3329/675</td><td>8250/1655</td><td>2989/1357</td></tr><tr><td>Nobel Laureate</td><td>39771/6754</td><td>44628/7600</td><td>928/928</td></tr><tr><td>Country of Birth</td><td>7218/1356</td><td>8908/1520</td><td>928/909</td></tr><tr><td>Award Year</td><td>11037/1904</td><td>8930/1520</td><td>928/926</td></tr><tr><td>Gender</td><td>854/96</td><td>8930/1520</td><td>592/149</td></tr><tr><td>Field</td><td>9518/1558</td><td>8930/1520</td><td>928/922</td></tr><tr><td>Birth Year</td><td>11144/1840</td><td>8930/1520</td><td>928/927</td></tr><tr><td>Occupation</td><td>54444/1582</td><td>29052/864</td><td>799/785</td></tr><tr><td>Work Location</td><td>24216/724</td><td>9684/288</td><td>799/708</td></tr><tr><td>Duty</td><td>12090/371</td><td>9684/288</td><td>785/522</td></tr><tr><td>Industry</td><td>18138/487</td><td>9684/288</td><td>799/600</td></tr><tr><td>Physical Object</td><td>49114/4659</td><td>35285/3636</td><td>563/563</td></tr><tr><td>Color</td><td>14707/1518</td><td>8825/909</td><td>563/563</td></tr><tr><td>Category</td><td>13540/1273</td><td>8820/909</td><td>563/562</td></tr><tr><td>Texture</td><td>14666/1265</td><td>8821/909</td><td>563/561</td></tr><tr><td>Size</td><td>6201/603</td><td>8819/909</td><td>563/528</td></tr><tr><td>Verb</td><td>70003/3806</td><td>14396/782</td><td>986/984</td></tr><tr><td>Past Tense</td><td>34043/1848</td><td>7188/391</td><td>986/975</td></tr><tr><td>Singular</td><td>35960/1958</td><td>7208/391</td><td>986/978</td></tr></table>
+
+Table 1: The details of the dataset used for the experiment, in the format of train/test splits. For every model in each setting. Methods are trained on the full dataset of that setting with 5 epochs. The prompts used by the train/test splits are completely disjoint.
+
+# A.3 DATASET PREPROCESSING
+
+HyperDAS uses an attention mechanism to gather information from the hidden states of the target model  $\mathcal{M}$  when running the base and counterfactual sentences. This makes HyperDAS overtly powerful in some situations. Consider the following input:
+
+$$
+\left\{ \begin{array}{l l} \text {B a s e} & \bar {\mathbf {x}} = \text {V i e n n a , k n o w n f o r i t s I m p e r i a l p a l a c e s , i s a c i t y i n t h e c o u n t r y o f} \\ \text {C o u n t e r f a c t u a l} & \hat {\mathbf {x}} = \text {I l o v e P a r i s} \\ \text {I n s t r u c t i o n} & \mathbf {x} = \text {L o c a l i z e t h e l a t i d u t e o f t h e c i t y} \end{array} \right. \tag {16}
+$$
+
+If the model works as intended, it will intervene on the 'Latitude' subspace, which will leave the 'Country' features intact and therefore the target model will predict Austria.
+
+However, since the model can access the hidden states  $\mathcal{M}(\bar{\mathbf{x}})$ , it knows that the queried attribute in the sentence is 'Country', which is different than the targeted attribute 'Latitude'. Through training, HyperDAS learns a shortcut to a trivial solution—performing no intervention when the target attribute is different from the one mentioned in the sentence. With this shortcut, the Isolate objective no longer works and the HyperDAS fails to learn disentangled feature subspaces for different attributes.
+
+Figure 9 shows how the HyperDAS may learn a trivial solution to the RAVEL benchmark if the relevant information (base prompt attribute) can be accessed by the model.
+
+![](images/2b2ffb270e57e564018922dffea01bd6d3581328544f3c14aab51ba20f18f696.jpg)  
+Figure 9: The trivial solution learnt by the HyperDAS on isolate examples when no mask is applied on the attribute token in the prompt. HyperDAS learns to not intervene if it sees the base prompt attribute to be different than the attribute in the instruction.
+
+Therefore, for each pair of prompts  $\bar{\mathbf{x}},\hat{\mathbf{x}}$  at training, we apply an intervention mask to all the tokens starting from the attribute mention. The hidden states from token with intervention mask is not visible to HyperDAS and therefore cannot be selected for intervention. The example becomes:
+
+$$
+\left\{ \begin{array}{l l} \text {B a s e} & \bar {\mathbf {x}} = \text {V i e n n a , k n o w n f o r i t s I m p e r i a l p a l a c e s , i s a c i t y i n t h e c o u n t r y o f} \\ \text {C o u n t e r f a c t u a l} & \hat {\mathbf {x}} = \text {I l o v e P a r i s} \\ \text {I n s t r u c t i o n} & \mathbf {x} = \text {L o c a l i z e t h e l a t i d u t e o f t h e c i t y} \end{array} \right. \tag {17}
+$$
+
+where the hidden states of the red text is masked from the HyperDAS.
+
+# A.4 LOADING HYPERDAS WITH PRE-TRAINED PARAMETERS
+
+We have also explored initializing the HyperDAS from a pretrained LM instead of initializing it from scratch. With Llama3-8b (Meta, 2024) as the target LM, we initialize the modules of HyperDAS, besides the multi-head cross-attention heads and pairwise token position scores attention heads, as the copy of the parameters from the target model. We then evaluate the performance of this variation of the model on the city dataset of RAVEL (Huang et al., 2024).
+
+![](images/cd07800e79de948b7451a5c8ebe9f2656b5c3b12218f6acc38fd459bd005ef14.jpg)  
+Figure 10: The loss curve of HyperDAS initialized from scratch or from pretrained LM while training on the city dataset of RAVEL.
+
+In Figure 10, we observe that there is no significant difference between the model initialized from scratch and the model initialized from Llama3-8b parameters. However, it remains unknown how would this difference change as the training of HyperDAS scales.
+
+# A.5 SPARSE AUTOENCODERS
+
+We experiment with different feature subspace dimension, as shown in Figure 11. We add an trained sparse autoencoder as another baseline. Following the exact same setting in (Huang et al., 2024), we train sparse autoencoder that projects the target hidden states into a higher-dimensional sparse feature space and then reconstruct the original hidden states.
+
+# A.6 ABLATION RESULTS
+
+See ablations in Table 12.
+
+# A.7 INTERVENTION PATTERNS
+
+Here we include a few demonstrations of the intervention pattern that HyperDAS generates on RAVEL, as shown in Figure 13.
+
+![](images/0e03f8ddd52b777f6b84305e5795c75cd9a31d367a8495010000f828e27108d4.jpg)  
+Figure 11: Cause (x-axis) and Iso (y-axis) scores trade-off for HyperDAS, MDAS, and SAE when using different feature size shown as the ratio  $\%$ . GOAL (1, 1) indicates the score with which the method is able to disentangle the feature subspace perfectly.  
+Figure 12: Ablation results for HyperDAS. No DAS has no rotation matrix and intervenes on entire hidden representations. No Hypernetwork replaces concept encoding via transformer with a vector lookup. No Cross Attention removes attention head submodules connecting the hypernetwork and target model.
+
+<table><tr><td>Ablation</td><td>Causal</td><td>Iso</td><td>Disentangle</td></tr><tr><td>HyperDAS</td><td>70.8</td><td>93.9</td><td>82.4</td></tr><tr><td>-No Cross Attention</td><td>68.2</td><td>83.9</td><td>76.1</td></tr><tr><td>-No DAS</td><td>0.8</td><td>97.4</td><td>49.1</td></tr><tr><td>-No Hypernetwork</td><td>15.1</td><td>46.9</td><td>31.0</td></tr></table>
+
+![](images/30f44220c9340bcf9151294e7770645e34ef825855622ab42dcf2433cd3929a4.jpg)
+
+![](images/58314309f87243bfb4d2cfe100145cc0de3a28946af4c5cb6b1530cdb8089f06.jpg)  
+(a)
+
+![](images/07ea85d1af475b413e58bf0c85abfaa485400847a091eba4bd539bb7926ce802.jpg)
+
+![](images/3bb717536c648945674ebe69680320d48cb6ad6e0ad8de67ca29a2848ba471c7.jpg)
+
+![](images/a1ca76d568e60762573c8b3f134dd4d2fb6ada95e52cee45b9a52d2f49ed4550.jpg)  
+(b)
+
+![](images/7d851e51c087477af09971eecdc77a6f7dbcc5b4cd8090c04cb7ebc513ce3bf2.jpg)
+
+![](images/9662c4fbba7311f4908ad61a577f59c3cd946a6b6fcbc26e1652f3d4e5ebef43.jpg)
+
+![](images/77829c5c5247aa3537b9df58979c90b32dc33bfc6f4d95de6248b4aa2ccc0bd5.jpg)  
+(c)
+
+![](images/56612d1b6c08c96f16f52b0cc01b0512e9bb431a97ca130837412f78e1852df1.jpg)
+
+![](images/bdbff3ac8a19deadaf97d599bd4eb30845f617317a665d539d072bf449a3cd7d.jpg)  
+Figure 13: Four types of intervention patterns.
+
+![](images/f9c5f1ccb8f504305a7685c1c1148cc330e77bfa1eca0faf13c8c59297cb86b0.jpg)  
+(d)
+
+![](images/60fb266c05e17ad690cf88976f18f07b102c2eb97bde7ac776fc0701902869ad.jpg)

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+# HYPERFACE: GENERATING SYNTHETIC FACE RECOGNITION DATASETS BY EXPLORING FACE EMBEDDING HYPERSPHHERE
+
+Hatef Otroshi Shahreza<sup>1,2</sup> and Sébastien Marcel<sup>1,3</sup>
+
+<sup>1</sup>Idiap Research Institute, Martigny, Switzerland  
+$^{2}$ École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland  
+<sup>3</sup>Université de Lausanne (UNIL), Lausanne, Switzerland
+
+{hatef.otroshi,sebastien.marcel}@idiap.ch
+
+# ABSTRACT
+
+Face recognition datasets are often collected by crawling Internet and without individuals' consents, raising ethical and privacy concerns. Generating synthetic datasets for training face recognition models has emerged as a promising alternative. However, the generation of synthetic datasets remains challenging as it entails adequate inter-class and intra-class variations. While advances in generative models have made it easier to increase intra-class variations in face datasets (such as pose, illumination, etc.), generating sufficient inter-class variation is still a difficult task. In this paper, we formulate the dataset generation as a packing problem on the embedding space (represented on a hypersphere) of a face recognition model and propose a new synthetic dataset generation approach, called HyperFace. We formalize our packing problem as an optimization problem and solve it with a gradient descent-based approach. Then, we use a conditional face generator model to synthesize face images from the optimized embeddings. We use our generated datasets to train face recognition models and evaluate the trained models on several benchmarking real datasets. Our experimental results show that models trained with HyperFace achieve state-of-the-art performance in training face recognition using synthetic datasets. Project page: https://www.idiap.ch/paper/hyperface
+
+# 1 INTRODUCTION
+
+Recent advances in the development of face recognition models are mainly driven by the deep neural networks (He et al., 2016), the angular loss functions (Deng et al., 2019; Kim et al., 2022), and the availability of large-scale training datasets (Guo et al., 2016; Cao et al., 2018; Zhu et al., 2021). The large-scale training face recognition datasets are collected by crawling the Internet and without the individual's consent, raising privacy concerns. This has created important ethical and legal challenges regarding the collecting, distribution, and use of such large-scale datasets (Nat, 2022). Considering such concerns, some popular face recognition datasets, such as MS-Celeb (Guo et al., 2016) and VGGFace2 (Cao et al., 2018), have been retracted.
+
+With the development of generative models, generating synthetic datasets has become a promising solution to address privacy concerns in large-scale datasets (Melzi et al., 2024; Shahreza et al., 2024). In spite of several face generator models in the literature (Deng et al., 2020; Karras et al., 2019; 2020; Rombach et al., 2022; Chan et al., 2022), generating a synthetic face recognition model that can replace real face recognition datasets and be used to train a new face recognition model from scratch is a challenging task. In particular, the generated synthetic face recognition datasets require adequate inter-class and intra-class variations. While conditioning the generator models on different attributes can help increasing intra-class variations, increasing inter-class variations remains a difficult task.
+
+In this paper, we focus on the generation of synthetic face recognition datasets and formulate the dataset generation process as a packing problem on the embedding space (represented on the surface
+
+![](images/4bb61d6112b08029e927b5de5ec16e028ef013c7f4e3af2e8b7ca088c106091f.jpg)  
+Figure 1: Sample face images from the HyperFace dataset
+
+of a hypersphere) of a pretrained face recognition model. We investigate different packing strategies and show that with a simple optimization, we can find a set of reference embeddings for synthetic subjects that has a high inter-class variation. We also propose a regularization term in our optimization to keep the optimized embedding on the manifold of face embeddings. After finding optimized embeddings, we use a face generative model that can generate face images from embeddings on the hypersphere, and generate synthetic face recognition datasets. We use our generated synthetic face recognition datasets, called HyperFace, to train face recognition models. We evaluate the recognition performance of models trained using synthetic datasets, and show that our optimization and packing approach can lead to new synthetic datasets that can be used to train face recognition models. We also compare trained models with our generated dataset to models trained with previous synthetic datasets, where our generated datasets achieve competitive performance with state-of-the-art synthetic datasets in the literature. Figure 1 illustrates sample face images from our synthetic dataset.
+
+The remainder of this paper is organized as follows. In Section 2, we present our problem formulation and describe our proposed method to generate synthetic face datasets. In Section 3, we provide our experimental results and evaluate our synthetic datasets. In Section 4, we review related work in the literature. Finally, we conclude the paper in Section 5.
+
+# 2 PROBLEM FORMULATION AND PROPOSED METHOD
+
+# 2.1 PROBLEM FORMULATION
+
+Identity Hypersphere: Let us assume that we have a pretrained face recognition model  $F: \mathcal{I} \to \mathcal{X}$ , which can extract identity features (a.k.a. embedding)  $\pmb{x} \in \mathcal{X} \subset \mathbb{R}^{n\mathcal{X}}$  from each face image  $\pmb{I} \in \mathcal{I}$ . Without loss of generality, we can assume that the extracted identity features cover a unit hypersphere<sup>1</sup>, i.e.,  $||\pmb{x}||_2 = 1$ ,  $\forall \pmb{x} \in \mathcal{X}$ .
+
+Representing Synthetic Dataset on the Identity Hypersphere: We can represent a synthetic face recognition dataset  $\mathcal{D}$  on this hypersphere by finding the embeddings for each image in the dataset. For simplicity, let us assume that for subject  $i$  in the synthetic face dataset, we can have a reference face image  $I_{\mathrm{ref},i}$  and reference embedding  $\pmb{x}_{\mathrm{ref},i} = F(\pmb{I}_{\mathrm{ref},i})$ . Note that the reference face image  $I_{\mathrm{ref},i}$  may already exist in the synthetic dataset  $\mathcal{D}$ , otherwise we can assume the reference embedding  $\pmb{x}_{\mathrm{ref},i}$  as the average of embeddings of all images for subject  $i$  in the dataset  $\mathcal{D}$ . Therefore, the synthetic face recognition dataset  $\mathcal{D}$  with  $n_{\mathrm{id}}$  number of subjects can be represented as a set of reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$ .
+
+![](images/a58f109238afb56a810c5f065bceb9727657f196e118ac4471c1c10732bc5419.jpg)  
+Figure 2: Block diagram of HyperFace Dataset Generation: We start from randomly synthesized face images and extract their embeddings using a pretrained face recognition model  $F$ . The extracted embeddings are normalised and used as initial points  $\{x_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$  in our HyperFace optimization. The HyperFace optimization tries to increase the intra-class variation for synthetic identities on the manifold of the face recognition model over the hypersphere using a regularization term. The resulting points are then used by a face generator model  $G$ , which can generate synthetic face images from the embeddings.
+
+# 2.2 HYPERFACE SYNTHETIC FACE DATASET
+
+HyperFace Optimization Problem: By representing a synthetic dataset  $\mathcal{D}$  on the identity hypersphere as a set of reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i = 1}^{n_{\mathrm{id}}}$ , we can raise the question that "How should reference embeddings cover the identity hypersphere?" To answer this question, we remind that the distances between reference embeddings indicate the inter-class variation in the synthetic face recognition dataset  $\mathcal{D}$ . Therefore, since we would like to have a high inter-class variation in the generated dataset  $\mathcal{D}$ , we can say that we need to maximize the distances between reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i = 1}^{n_{\mathrm{id}}}$ . In other words, we need to solve the following optimization problem:
+
+$$
+\max  \quad \min  _ {\left\{\boldsymbol {x} _ {\text {r e f}} \right\}, i \neq j} d \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right) \quad \text {s u b j e c t t o} \quad | | \boldsymbol {x} _ {\text {r e f}, k} | | _ {2} = 1, \forall k \in \{1, \dots , n _ {\mathrm {i d}} \} \tag {1}
+$$
+
+where  $d(\cdot ,\cdot)$  is a distance function.
+
+Solving the HyperFace Optimization: The optimization problem stated in Eq. 1 is a well-known optimization problem, which is known as spherical code optimization (J. H. Conway, 1998) or the Tammes problem (Tammes, 1930), where the goal is to pack a given number of points (e.g., particles, pores, electrons, etc.) on the surface of a unit sphere such that the minimum distance between points is maximized. The optimal solutions for this problem are studied for small dimensions and the number of points (Böroczky, 1983; Hárs, 1986; Musin & Tarasov, 2012; 2015). However, for a large dimension and a high number of points, there is no closed-form solution in the literature (Tokarchuk et al., 2024). For a large dimension and a high number of points, there are different approaches for solving this optimization problem (such as geometric optimization, numerical optimization, etc.). However, for a large dimension of hypersphere (i.e.,  $n_{\mathcal{X}}$ ) and a very large number of points (i.e., the number of subjects  $n_{\mathrm{id}}$ , such as 10,000 identities in our problem), solving this optimization can be computationally expensive. To address this issue, we solve the optimization problem with an iterative approach based on gradient descent. To this end, we can randomly initialize the reference embeddings and find the optimised reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i = 1}^{n_{\mathrm{id}}}$  using the Adam optimizer (Kingma & Ba, 2015). This allows us to solve the optimization with a reasonable computation resource. For example, we can solve the optimization for  $n_{\mathcal{X}} = 512$  and  $n_{\mathrm{id}} = 10,000$  on a system equipped with a single NVIDIA 3090 GPU in 6 hours.
+
+Regularization: While we solve the optimization problem in Eq. 1 on the surface of a hypersphere, we should note that the manifold of embeddings  $\mathcal{X}$  does not necessarily cover the whole surface of the hypersphere. This means if we get out of the distribution of embeddings  $\mathcal{X}$ , we may not be able to generate face images from such embeddings. Therefore, we need to add a regularization term to our optimization problem that tends to keep the reference embeddings on the manifold
+
+Algorithm 1 HyperFace Optimization for Finding Reference Embeddings  
+1: Inputs:  $\lambda$ : learning rate,  $n_{\mathrm{itr}}$ : number of iterations,  $\{\pmb{x}_g\}_{g=1}^{n_{\mathrm{gallery}}}$ : embeddings of a gallery of face images,  
+2:  $\alpha$ : hyperparameter (contribution of regularization).  
+3: Output:  $\pmb{X}_{\mathrm{ref}} = \{\pmb{x}_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$ : optimized reference embeddings.  
+4: Procedure:  
+5: Initialize reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$   
+6: For  $n = 1,..,n_{\mathrm{itr}}$  do  
+7: Find  $\pmb{x}_{\mathrm{ref},i}, \pmb{x}_{\mathrm{ref},j} \in \pmb{X}_{\mathrm{ref}}$  which have minimum distance  $d(\pmb{x}_{\mathrm{ref},i}, \pmb{x}_{\mathrm{ref},j})$   
+8: Reg  $\leftarrow \frac{1}{n_{\mathrm{id}}} \sum_{k=1}^{n_{\mathrm{id}}} \min_{\{\pmb{x}_g\}_{\mathrm{gallery}}} d(\pmb{x}_{\mathrm{ref},k}, \pmb{x}_g)$   
+9: cost  $\leftarrow -d(\pmb{x}_{\mathrm{ref},i}, \pmb{x}_{\mathrm{ref},j})$   
+10:  $\pmb{X}_{\mathrm{ref}} \gets \pmb{X}_{\mathrm{ref}} - \text{Adam}(\nabla \text{cost}, \lambda)$   
+11:  $\pmb{X}_{\mathrm{ref}} \gets \text{normalize}(\pmb{X}_{\mathrm{ref}})$   
+12: End For  
+13: End Procedure
+
+of embeddings  $\mathcal{X}$ . To this end, we consider a set of face images  $\{I_i\}_{i=1}^{n_{\mathrm{gallery}}}$  as a gallery of images $^2$  and extract their embeddings to have set of valid embeddings  $\{\pmb{x}_i\}_{i=1}^{n_{\mathrm{gallery}}}$ . Then, we try to minimize the distance of our reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$  to the set of embeddings  $\{\pmb{x}_i\}_{i=1}^{n_{\mathrm{gallery}}}$ , which approximates the manifold of embeddings  $\mathcal{X}$ . To this end, for each reference embedding  $\{\pmb{x}_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$ , we find the closest embedding in  $\{\pmb{x}_i\}_{i=1}^{n_{\mathrm{gallery}}}$  and minimize their distance. We can write the optimization in Eq. 1 as a regularized min-max optimization as follows:
+
+$$
+\min  \quad \max  _ {\left\{\boldsymbol {x} _ {\text {r e f}} \right\}, i \neq j} - d \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right) + \alpha \underbrace {\frac {1}{n _ {\mathrm {i d}}} \sum_ {k = 1} ^ {n _ {\mathrm {i d}}} \min  _ {\left\{\boldsymbol {x} _ {g} \right\} _ {g = 1} ^ {n _ {\text {g a l l e r y}}}} d \left(\boldsymbol {x} _ {\text {r e f}, k}, \boldsymbol {x} _ {g}\right)} _ {\text {r e g u l a r i z a t i o n}}; \tag {2}
+$$
+
+$$
+\text {s u b j e c t} \quad | | \boldsymbol {x} _ {\text {r e f}, k} | | _ {2} = 1, \forall k \in \{1, \dots , n _ {\text {i d}} \},
+$$
+
+where  $\alpha$  is a hyperparameter that controls the contribution of the regularization term in the optimization. To provide more flexibility in our optimization, we consider the size of gallery  $n_{\mathrm{gallery}}$  to be greater or equal to the number of identities  $n_{\mathrm{id}}$  in the synthetic dataset (i.e.,  $n_{\mathrm{gallery}} \geq n_{\mathrm{id}}$ ).
+
+Initialization: To solve the HyperFace optimization problem in Eq. 1 using Algorithm 1, we need to initialize the reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$ . To this end, we can generate  $n_{\mathrm{id}}$  number random synthetic images  $\{I_i\}_{i=1}^{n_{\mathrm{id}}}$  using a face generator model, such as StyleGAN (Karras et al., 2020), Latent Diffusion Model (LDM) (Rombach et al., 2022). These models use a noise vector as the input and can generate synthetic face images in an unconditional setting. Then, after generating  $\{I_i\}_{i=1}^{n_{\mathrm{id}}}$  images, we can extract their embeddings using the face recognition model  $F(\cdot)$  and use the extracted embeddings as initialization values for the reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$  in Algorithm 1.
+
+Image Generation: After we find the reference embeddings  $\{\pmb{x}_{\mathrm{ref},i}\}_{i = 1}^{n_{\mathrm{id}}}$  using Algorithm 1, we can use an identity-conditioned image generator model to generate face images from reference embeddings. To this end, we use a recent face generator network (Papantoniou et al., 2024), which is based on probabilistic diffusion models. The diffusion face generator model  $G(\cdot ,\cdot)$  can generate a face image  $\pmb {I} = G(\pmb{x}_{\mathrm{ref}},\pmb {z})$  from reference embedding  $\pmb{x}_{\mathrm{ref}}$  and a random noise  $z\sim \mathcal{N}(0,\mathbb{I}^{\mathrm{DM}})$ . Therefore, by changing the random noise  $z$  and sampling different noise vectors, we can generate different samples for the reference embedding  $\pmb{x}_{\mathrm{ref}}$ . In addition, to increase intra-class variation, we add Gaussian noise  $\pmb{v}\sim \mathcal{N}(0,\mathbb{I}^{n_x})$  to the reference embedding  $\pmb{x}_{\mathrm{ref}}$ , and then normalize it to locate it on the hypersphere. In summary, we can generate different samples for each reference embedding  $\pmb{x}_{\mathrm{ref}}$  by changing  $z$  and  $\pmb{v}$  noise vectors as follows:
+
+$$
+\boldsymbol {I} = G \left(\frac {\boldsymbol {x} _ {\text {r e f}} + \beta \boldsymbol {v}}{\left| \left| \boldsymbol {x} _ {\text {r e f}} + \beta \boldsymbol {v} \right| \right| _ {2}}, \boldsymbol {z}\right), \quad \boldsymbol {v} \sim \mathcal {N} \left(0, \mathbb {I} ^ {n _ {\mathcal {X}}}\right), \boldsymbol {z} \sim \mathcal {N} \left(0, \mathbb {I} ^ {\mathrm {D M}}\right), \tag {3}
+$$
+
+Table 1: Comparison of recognition performance of face recognition models trained with different synthetic datasets and a real dataset (i.e., CASIA-WebFace). The performance reported for each dataset is in terms of accuracy and best value for each benchmark is emboldened.  
+
+<table><tr><td>Dataset name</td><td>#IDs</td><td>#Images</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>SynFace (Qiu et al., 2021)</td><td>10&#x27;000</td><td>999&#x27;994</td><td>86.57</td><td>65.10</td><td>70.08</td><td>66.79</td><td>59.13</td></tr><tr><td>SFace (Boutrós et al., 2022)</td><td>10&#x27;572</td><td>1&#x27;885&#x27;877</td><td>93.65</td><td>74.90</td><td>80.97</td><td>75.36</td><td>70.32</td></tr><tr><td>Syn-Multi-PIE (Colbois et al., 2021)</td><td>10&#x27;000</td><td>180&#x27;000</td><td>78.72</td><td>60.22</td><td>61.83</td><td>60.84</td><td>54.05</td></tr><tr><td>IDnet (Kolf et al., 2023)</td><td>10&#x27;577</td><td>1&#x27;057&#x27;200</td><td>84.48</td><td>68.12</td><td>71.42</td><td>68.93</td><td>62.63</td></tr><tr><td>ExFaceGAN (Boutrós et al., 2023b)</td><td>10&#x27;000</td><td>599&#x27;944</td><td>85.98</td><td>66.97</td><td>70.00</td><td>66.96</td><td>57.37</td></tr><tr><td>GANDiffFace (Melzi et al., 2023)</td><td>10&#x27;080</td><td>543&#x27;893</td><td>94.35</td><td>76.15</td><td>79.90</td><td>78.99</td><td>69.82</td></tr><tr><td>Langevin-Dispersion (Geissbühler et al., 2024)</td><td>10&#x27;000</td><td>650&#x27;000</td><td>94.38</td><td>65.75</td><td>86.03</td><td>65.51</td><td>77.30</td></tr><tr><td>Langevin-DisCo (Geissbühler et al., 2024)</td><td>10&#x27;000</td><td>650&#x27;000</td><td>97.07</td><td>76.73</td><td>89.05</td><td>79.56</td><td>83.38</td></tr><tr><td>DigiFace-1M (Bae et al., 2023)</td><td>109&#x27;999</td><td>1&#x27;219&#x27;995</td><td>90.68</td><td>72.55</td><td>73.75</td><td>79.43</td><td>68.43</td></tr><tr><td>IDiff-Face (Uniform) (Boutrós et al., 2023a)</td><td>10&#x27;049</td><td>502&#x27;450</td><td>98.18</td><td>80.87</td><td>90.82</td><td>82.96</td><td>85.50</td></tr><tr><td>IDiff-Face (Two-Stage) (Boutrós et al., 2023a)</td><td>10&#x27;050</td><td>502&#x27;500</td><td>98.00</td><td>77.77</td><td>88.55</td><td>82.57</td><td>82.35</td></tr><tr><td>DCFace (Kim et al., 2023)</td><td>10&#x27;000</td><td>500&#x27;000</td><td>98.35</td><td>83.12</td><td>91.70</td><td>88.43</td><td>89.50</td></tr><tr><td>HyperFace [ours]</td><td>10&#x27;000</td><td>500&#x27;000</td><td>98.50</td><td>84.23</td><td>89.40</td><td>88.83</td><td>86.53</td></tr><tr><td>CASIA-WebFace (Yi et al., 2014)</td><td>10&#x27;572</td><td>490&#x27;623</td><td>99.42</td><td>90.02</td><td>93.43</td><td>94.97</td><td>94.32</td></tr></table>
+
+where  $\beta$  is a hyperparameter that controls the variations to the reference embedding. Figure 2 depicts the block diagram of our synthetic dataset generation process. Algorithm 3 in Appendix F also present a pseudo-code of dataset generation process.
+
+# 3 EXPERIMENTS
+
+# 3.1 EXPERIMENTAL SETUP
+
+Dataset Generation: For solving the HyperFace optimization in Algorithm 1, we use an initial learning rate of  $\lambda = 0.01$  and reduce the learning rate by power 0.75 every 5,000 iterations for a total number of iterations  $n_{\mathrm{itr}} = 100,000$ . We also consider cosine distance, which is commonly used in face recognition systems for the comparison of face embeddings, as our distance function  $d(\cdot ,\cdot)$ . For the hyperparameters  $\alpha$  and  $\beta$ , we consider default values of 0.5 and 0.01, respectively, in our experiments. We also consider the size of gallery to be the same as the number of identities, and explore other cases where  $n_{\mathrm{gallery}} > n_{\mathrm{id}}$  in our ablation study. We generate 64 images, by default, per each identity in our generated datasets and explore other numbers of images in our ablation study.
+
+We use ArcFace (Deng et al., 2019) as the pretrained face recognition model  $F(\cdot)$  with the embedding dimension of  $n_{\mathcal{X}} = 512$  and use a pretrained generator model (Papantoniou et al., 2024) to generate face images from ArcFace embeddings. After generating face images, we align all face images using a pretrained MTCNN (Zhang et al., 2016) face detector model. For our regularization, we randomly generate images with StyleGAN (Karras et al., 2020) as default, and investigate other generator models in our ablation study.
+
+Evaluation: To evaluate the generated synthetic datasets, we use each generated datasets as a training dataset for training a face recognition model. To this end, we use the iResNet50 backbone and train the model with AdaFace loss function (Kim et al., 2022) using the Stochastic Gradient Descent (SGD) optimizer with the initial learning rate 0.1 and a weight decay of  $5 \times 10^{-4}$  for 30 epochs with the batch size of 256. After training the face recognition model with the synthetic dataset, we benchmark the performance of the trained face recognition models on different benchmarking datasets of real images, including Labeled Faces in the Wild (LFW) (Huang et al., 2008), Cross-age LFW (CA-LFW) (Zheng et al., 2017), CrossPose LFW (CP-LFW) (Zheng & Deng, 2018), Celebrities in Frontal-Profile in the Wild (CFP-FP) (Sengupta et al., 2016), and AgeDB-30 (Moschoglou et al., 2017) datasets. For consistency with prior works, we report recognition accuracy calculated using 10-fold cross-validation for each of benchmarking datasets. The source code of our experiments and generated datasets are publicly available<sup>3</sup>.
+
+# 3.2 ANALYSIS
+
+Comparison with Previous Synthetic Datasets: We compare the recognition performance of face recognition models trained with our synthetic dataset and previous synthetic datasets in the literature. We use the published dataset for each method and train all models with the same configuration for different datasets to prevent the effect of other hyperparameters (such as number of epochs, batch size, etc.). For a fair comparison, we consider the versions of datasets with a similar number of identities $^4$ , if there are different datasets available for each method. Table 1 compares the recognition performance of face recognition models trained with different synthetic datasets. As the results in this table show, our method achieves state-of-the-art performance in training face recognition using synthetic data. Figure 1 illustrates sample face images from our synthetic dataset. Figure 4 of appendix also presents more sample images from HyperFace dataset.
+
+Ablation Study: In our dataset generation method, there are different hyperparameters which can affect the HyperFace optimization and the generated synthetic datasets. Table 2 reports the ablation study on the number of images generated per each synthetic identity in our experiments. As the results in Table 2 show, increasing the number of images per identity
+
+improves the recognition performance of trained face recognition model, but it tends to saturate after 64 images per identity.
+
+Table 2: Ablation study on the effect of number of images  
+
+<table><tr><td>Image/ID</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>32</td><td>98.70</td><td>84.17</td><td>88.83</td><td>88.74</td><td>86.33</td></tr><tr><td>50</td><td>98.50</td><td>84.23</td><td>89.40</td><td>88.83</td><td>86.53</td></tr><tr><td>64</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr><tr><td>96</td><td>98.42</td><td>84.15</td><td>89.00</td><td>89.51</td><td>87.45</td></tr><tr><td>128</td><td>98.20</td><td>83.63</td><td>89.82</td><td>89.31</td><td>87.62</td></tr></table>
+
+Table 3 also compares the number of identities in the generated dataset, including 10k, 20k, and 50k identities. As the results in Table 3 show, increasing the number of identities improves the recognition performance of trained face recognition model on the benchmarking datasets. The re
+
+Table 3: Ablation study on the effect of number of identities  
+
+<table><tr><td>#IDs</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>10k</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr><tr><td>30k</td><td>98.82</td><td>85.23</td><td>91.12</td><td>91.74</td><td>89.42</td></tr><tr><td>50k</td><td>98.27</td><td>85.60</td><td>91.48</td><td>92.24</td><td>90.40</td></tr></table>
+
+sults in this table demonstrates that we can still increase the number of identities and scale our dataset generation without saturating the performance. The main issue for increasing the size of dataset is computation resource, which is discussed in detail in Appendix A. We can also reduce the complexity of our optimization for large number of identities, which is discussed in detail in Appendix B.
+
+Table 4 reports the recognition performance achieved for face recognition model trained with datasets with 10k identity and optimized with different numbers of gallery images. As the results in this table shows, increasing the size of gallery improves the performance of the trained model.
+
+Table 4: Ablation study on the effect of  ${n}_{\text{gallery }}$  
+
+<table><tr><td>ngallery</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>10K</td><td>98.53</td><td>84.00</td><td>88.92</td><td>89.34</td><td>85.9</td></tr><tr><td>20K</td><td>98.50</td><td>84.32</td><td>89.28</td><td>89.17</td><td>86.00</td></tr><tr><td>50K</td><td>98.72</td><td>84.23</td><td>88.72</td><td>89.19</td><td>86.85</td></tr></table>
+
+However, with 10,000 images we can still approximate the manifold of face embeddings on the hypersphere.
+
+As another ablation study, we use different source of images for the gallery set to use in our regularization and solve the HyperFace optimization. We use pretrained StyleGAN (Karras et al., 2020) as a GAN-based generator model and a pretrained latent diffusion model (Rombach et al., 2022) as a diffusion-based generator model. We use these generator models and randomly generate some synthetic face images. In addition, for our ablation study, we consider some real images from BUPT dataset (Wang et al., 2019) as a dataset of real face images.
+
+As the results in Table 5 show, optimization with images from StyleGAN and LDM lead to comparable performance for the generated face recognition dataset. However, the real images in the BUPT dataset lead to superior performance. This suggests that the synthesized images cannot completely cover the manifold of
+
+embeddings and if we use real images as our gallery it can improve the generated dataset and recognition performance of our face recognition model.
+
+Table 5: Ablation study on the type of data in gallery  
+
+<table><tr><td>Gallery</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>StyleGAN</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr><tr><td>LDM</td><td>98.65</td><td>84.35</td><td>89.17</td><td>89.09</td><td>86.35</td></tr><tr><td>BUPT</td><td>98.70</td><td>84.77</td><td>90.03</td><td>89.16</td><td>87.13</td></tr></table>
+
+We also study the effect of hyperparameters  $\alpha$  and  $\beta$  on the generated face recognition dataset. Table 6 reports the ablation study for the contribution of regularization in our optimization  $(\alpha)$ . As the results in this table show, the regularization enhances the quality of generated dataset and improves the recognition
+
+performance of face recognition model. In fact, our regularization term helps our optimization to keep the points on the manifold of face recognition over the hypersphere, and therefore improves the quality of our synthetic dataset.
+
+Similarly, Table 7 reports the ablation study for the effect of noise in data generation and augmentation (i.e., hyperparameter  $\beta$  in Eq. 3). As can be seen, the added noise increases the variation for images of each subject and increases the performance of face recognition models trained with the generated datasets. With a larger
+
+Table 6: Ablation study on the effect of  $\alpha$  
+
+<table><tr><td>α</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>0</td><td>98.40</td><td>84.15</td><td>88.87</td><td>89.31</td><td>86.48</td></tr><tr><td>0.50</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr><tr><td>0.75</td><td>98.62</td><td>84.32</td><td>89.48</td><td>89.67</td><td>86.72</td></tr><tr><td>1.0</td><td>98.55</td><td>84.72</td><td>89.10</td><td>89.76</td><td>86.63</td></tr></table>
+
+value of  $\beta$ , the generated images for each identity have more variations, which increases the performance of the face recognition model trained with our synthetic dataset.
+
+As another experiment, we consider different backbones and train face recognition models with different number of layers. As the results in Table 8 show, increasing the number of layers improves the recognition performance of trained face recognition model. While this is expected and has been observed for training us
+
+Table 8: Ablation study on the network structure  
+
+<table><tr><td>Network</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>ResNet18</td><td>98.33</td><td>81.38</td><td>88.53</td><td>86.03</td><td>85.27</td></tr><tr><td>ResNet34</td><td>98.5</td><td>83.47</td><td>88.88</td><td>88.29</td><td>86.42</td></tr><tr><td>ResNet50</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr><tr><td>ResNet101</td><td>98.73</td><td>85.43</td><td>90.05</td><td>89.54</td><td>87.52</td></tr></table>
+
+ing large-scale face recognition datasets, it sheds light on more potentials in the generated synthetic datasets.
+
+# 3.3 DISCUSSION
+
+Scaling Dataset Generation: To increase the size of the synthetic face recognition dataset, we can increase the number of images per identity and also the number of samples per identity. In our ablation study, we investigated the effect of the number of images (Table 2) and the number of identities (Table 3) on the recognition performance of the face recognition model. However, increasing the size of the dataset requires more computation. Increasing the number of images in the dataset has linear complexity in our image generation step (i.e.,  $\mathcal{O}(n_{\mathrm{images}})$ , where  $n_{\mathrm{images}}$  is the number of images in the generated dataset). However, the complexity of solving the HyperFace optimization problem with iterative optimization in Algorithm 1 has quadratic complexity (i.e.,  $\mathcal{O}(n_{\mathrm{id}}^2)$ ). Therefore, solving this optimization for a larger number of identities requires much more computation
+
+![](images/d2e509d2f5d22f2973f43b6678e5f076fe6f21e658aa6b69152740749a862b27.jpg)  
+Figure 3: Sample pairs of images with the highest similarity between face embeddings of images in synthesized dataset and training dataset of StyleGAN, which was used to generate random images for initialization and regularization in the HyperFace optimization.
+
+![](images/c482e50309a12f60d18da86c36fd702339455f5981749406b361b621cc83aa06.jpg)
+
+resources. Meanwhile, most existing synthetic datasets in the literature have a comparable number of identities to our experiments. We should note that in our optimization, we considered all points in each iteration of optimization which introduces quadratic complexity to our optimization. However, we can solve the optimization with stochastic mini-batches of points on the embedding hypersphere, which can reduce the complexity in each iteration (i.e.,  $\mathcal{O}(b^2)$ , where  $b$  is the size of batch and  $b \leq n_{\mathrm{id}}$ ). We further discuss the complexity of our optimization and dataset generation in Appendix A and present further analyses for stochastic optimization, that reduces the complexity of our optimization in Appendix B.
+
+Leakage of Identity: In our dataset generation method, we used images synthesized by StyleGAN for initialization and regularization. Therefore, it is important if there is any leakage of privacy data in the images generated from StyleGAN in the final generated dataset. To this end, we extract and compare embeddings from all the generated images to embeddings of all face images in the training dataset of StyleGAN. The highest similarity score between generated images and training dataset correspond to children images (as shown in Figure 3a) which are difficult to compare visually and conclude potential leakage. Figure 3b illustrates images of highest scores excluding children. While there are some visual similarities in the images, it is difficult to conclude leakage of identity in the generated synthetic dataset. We further study the effect of identity leakage on the recognition performance of face recognition models in Appendix D.
+
+Ethical Considerations: State-of-the-art face recognition models are trained with large-scale face recognition datasets, which are crawled from the Internet, raising ethical and privacy concerns. To address the ethical and privacy concerns with web-crawled data, we can use synthetic data to train face recognition models. However, generating synthetic face recognition datasets also requires face generator models which are trained from a set of real face images. Therefore, we still rely on real face images in the generation pipeline. In our experiments, we investigated if we have leakage of identity in the generated synthetic dataset based on images used for initialization and regularization. However, there are other privacy-sensitive components used in our method. For example, we defined and solved our optimization problem on the embedding hypersphere of a pretrained face recognition model. Therefore, for generating fully privacy-friendly datasets, the leakage of information by other components needs to be investigated.
+
+We should also note that while we tried to increase the inter-class variations in our method, there might be still a potential lack of diversity in different demography groups, stemming from implicit biases of the datasets used for training in our pipeline (such as the pretrained face recognition model, the gallery of images used for regularization, etc.). It is also noteworthy that the project on which the work has been conducted has passed an Institutional Ethical Review Board (IRB).
+
+# 4 RELATED WORK
+
+With the advances in generative models, several synthetic face recognition datasets have been proposed in the literature. Bae et al. (2023) proposed DigiFace dataset where they used a computer graphic pipeline to render different identities and also generate different images for each identity by introducing different variations based on face attributes (e.g., variation in facial pose, accessories, and textures). In contrast to (Bae et al., 2023), other papers in the literature used Generative Adversarial Networks (GANs) or probabilistic Diffusion Models (PDMs) to generate synthetic face datasets. Qiu et al. (2021) proposed SynFace and utilised DiscoFaceGAN (Deng et al., 2020) to generate their dataset. They generated different synthetic identities using identity mixup by exploring the latent space of DiscoFaceGAN to increase intra-class variation and then used DiscoFaceGAN to generate different images for each identity.
+
+Boutros et al. (2022) proposed SFace by training an identity-conditioned StyleGAN (Karras et al., 2020) on the CASIA-WebFace (Yi et al., 2014) and then generating the SFace dataset using the trained model. Kolf et al. (2023) also trained an identity-conditioned StyleGAN (Karras et al., 2020) in a three-player GAN framework to integrate the identity information into the generation process and proposed the IDnet dataset. Colbois et al. (2021) proposed the Syn-Multi-PIE dataset using a pretrained StyleGAN (Karras et al., 2020). They trained a support vector machine (SVM) to find directions for different variations (such as pose, illuminations, etc.) in the intermediate latent space of a pretrained StyleGAN. Then, they used StyleGAN to generate different identities and synthesized different images for each identity by exploring the intermediate latent space of StyleGAN using linear combinations of calculated directions. Boutros et al. (2023b) proposed ExFaceGAN, where they used SVM to disentangle the identity information in the latent space of pretrained GANs, and then generated different identities with several images within the corresponding identity boundaries. Geissbühler et al. (2024) used stochastic Brownian forces to sample different identities in the intermediate latent space of pretrained StyleGAN (Karras et al., 2020) and generate different identities (named Langavien). Then they solved a similar dynamical equation in the latent space of StyleGAN to generate different images for each identity (named Langavien-Dispersion) and further explored the intermediate latent space of StyleGAN (named Langavien-DisCo).
+
+Melzi et al. (2023) proposed GANDiffFace, a hybrid dataset generation framework, where they used StyleGAN to generate face images with different identities, and then used DreamBooth (Ruiz et al., 2023) as a diffusion-based generator, to generate different samples for each identity. Boutros et al. (2023a) trained an identity-conditioned diffusion model to generate synthetic face images and proposed IDiffFace datasets. They generated different samples using an unconditional model, and then generated different samples using their conditional diffusion model (named IDiff-Face Two-Stage). Alternatively, they uniformly sampled different identities and generated different samples for each identity using their identity-conditioned diffusion model (named IDiff-Face Uniform). Kim et al. (2023) proposed DCFace, where they trained a dual condition (style and identity conditions) face generator model based on diffusion models on the CASIA-WebFace dataset. They used their trained diffusion model to generate different identities and different styles for each identity by varying identity and style conditions.
+
+# 5 CONCLUSION
+
+In this paper, we formalized the dataset generation as a packing problem on the hypersphere of a pretrained face recognition model. We focused on inter-class variation and designed our packing problem to increase the distance between synthetic identities. Then, we considered our packing problem as a regularized optimization and solved it with an iterative gradient-descent-based approach. Since the manifold of face embeddings does not cover the whole hypersphere, the regularization allows us to approximate the manifold of face embeddings and enhance the quality of generated face images. We used the generated datasets by our method (called HyperFace) to train face recognition models, and evaluated the trained models on several real benchmarking datasets. Our experiments demonstrate the effectiveness of our approach, which achieves state-of-the-art performance for training face recognition using synthetic data. We also presented an extensive ablation study to investigate the effect of each hyperparameter in our dataset generation method.
+
+# ACKNOWLEDGMENTS
+
+This research is based upon work supported by the Hasler foundation through the "Responsible Face Recognition" (SAFER) project.
+
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+
+# A COMPLEXITY AND REQUIRED COMPUTATION RESOURCE
+
+The computation required to generate the synthetic datasets in our approach has two main parts:
+
+1. HyperFace Optimization: The HyperFace optimization (Algorithm 1) considers all reference points  $\{\pmb{x}_{\mathrm{ref},i}\}_{i = 1}^{n_{\mathrm{id}}}$  in the hypersphere and maximizes their distances. Therefore, this optimization considers all pairs of points and has quadratic complexity (i.e.,  $\mathcal{O}(n_{\mathrm{id}}^2)$ ).
+
+Table 9 reports the runtime for solving the HyperFace optimization for different numbers of identities (i.e.,  $n_{\mathrm{id}}$ ) on a system equipped with a single NVIDIA 3090 GPU. Note that this optimization process cannot be parallelized.
+
+Table 9: Runtime for solving the HyperFace optimization (Algorithm 1) for different numbers of identities on a system equipped with a single NVIDIA 3090 GPU.  
+
+<table><tr><td>n_id</td><td>HyperFace Optimization Runtime</td></tr><tr><td>10k</td><td>6 hours</td></tr><tr><td>20k</td><td>11 hours</td></tr><tr><td>30k</td><td>23 hours</td></tr><tr><td>50k</td><td>84 hours</td></tr></table>
+
+We should note that instead of solving the HyperFace optimization on all pairs of points, we can solve the optimization stochastically in which in each iteration a mini-batch of points is considered and optimized. Therefore the complexity will become  $\mathcal{O}(b^2)$ , where  $b$  is size of mini-batch and  $b \leq n_{\mathrm{id}}$ . This way the complexity of our method can be independent of the number of identities and significantly reduced (especially for  $b \ll n_{\mathrm{id}}$ ). Our stochastic optimization is further studied and discussed in Section B of this Appendix.
+
+2. Image Generation: After solving the HyperFace optimization, we need to use the generator network in inference mode and generate the required number of images. Therefore, the generation of dataset has a linear complexity with respect to the number of images (i.e.,  $\mathcal{O}(n_{\mathrm{images}})$ , where  $n_{\mathrm{images}}$  is the number of images in the generated dataset). The average runtime for generating a single synthetic face image on a system equipped with a single NVIDIA 3090 GPU is 1.25 seconds. For example, generating a dataset with 500,000 images takes about 174 hours on a single NVIDIA 3090 GPU. Note that this optimization process can be parallelized, and therefore image generation can be deployed on a cluster or a farm of GPUs.
+
+# B HYPERFACE STOCHASTIC OPTIMIZATION
+
+As discussed in Appendix A, HyperFace optimization (Algorithm 1) considers all reference points  $\{\pmb{x}_{\mathrm{ref},i}\}_{i = 1}^{n_{\mathrm{id}}}$  and has a quadratic complexity  $\mathcal{O}(n_{\mathrm{id}}^2)$ . To reduce this complexity, in each iteration, we can randomly select a mini-batch of  $b$  points and only optimize the selected  $b$  reference points instead of all  $n_{\mathrm{id}}$  reference points. This way in each iteration we can compare only  $\binom{b}{2}$  pairs instead of  $\binom{n_{\mathrm{id}}}{2}$  pairs, and therefore the complexity of our optimization will become  $\mathcal{O}(b^2)$ . In the following, we first theoretically prove that the expected mini-batch gradient approximates the full gradient, and then validate it with experimental analyses.
+
+Theorem 1. Let  $X_{ref} = \{x_{ref,i}\}_{i=1}^{n_{id}}$  represent  $n_{id}$  points on a  $n_{\mathcal{X}}$ -dimensional hypersphere  $S$ . Consider an objective function:
+
+$$
+\mathcal{L}(\boldsymbol{X}_{ref}) = \frac{1}{\binom{n_{id}}{2}}\sum_{i = 1}^{n_{id}}\sum_{j = i + 1}^{n_{id}}\ell (\boldsymbol{x}_{ref,i},\boldsymbol{x}_{ref,j}),
+$$
+
+where  $\ell(\cdot, \cdot)$  denotes a pairwise function. The goal is to minimize  $\mathcal{L}(\boldsymbol{X}_{ref})$  for  $\boldsymbol{X}_{ref} = \{\boldsymbol{x}_{ref,i}\}_{i=1}^{n_{id}}$ . Suppose in each iteration, instead of computing  $\nabla \mathcal{L}(\boldsymbol{X}_{ref})$  over all  $\binom{n_{id}}{2}$  pairs, we approximate it using a random mini-batch  $B \subset \boldsymbol{X}_{ref}$  of size  $b \ll n_{id}$ . Then, the expected batch gradient approximates the full gradient:
+
+$$
+\mathbb {E} \left[ \nabla \mathcal {L} _ {B} \left(\boldsymbol {X} _ {\text {r e f}}\right) \right] = \nabla \mathcal {L} \left(\boldsymbol {X} _ {\text {r e f}}\right). \tag {4}
+$$
+
+Proof. For a batch  $B$  of size  $b$ , the batch objective is:
+
+$$
+\mathcal {L} _ {B} \left(\boldsymbol {X} _ {\text {r e f}}\right) = \frac {1}{\binom {b} {2}} \sum_ {i \in B} \sum_ {j \in B, j > i} \ell \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right). \tag {5}
+$$
+
+The full gradient of  $\mathcal{L}(\mathbf{X}_{\mathrm{ref}})$  is:
+
+$$
+\nabla \mathcal {L} \left(\boldsymbol {X} _ {\text {r e f}}\right) = \frac {1}{\binom {n _ {\mathrm {i d}}} {2}} \sum_ {i = 1} ^ {n _ {\mathrm {i d}}} \sum_ {j = i + 1} ^ {n _ {\mathrm {i d}}} \nabla \ell \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right). \tag {6}
+$$
+
+Similarly, the batch gradient is:
+
+$$
+\nabla \mathcal {L} _ {B} \left(\boldsymbol {X} _ {\text {r e f}}\right) = \frac {1}{\binom {b} {2}} \sum_ {i \in B} \sum_ {j \in B, j > i} \nabla \ell \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right). \tag {7}
+$$
+
+The expectation over all possible batches  $B$  is:
+
+$$
+\mathbb {E} \left[ \nabla \mathcal {L} _ {B} \left(\boldsymbol {X} _ {\text {r e f}}\right) \right] = \frac {1}{\binom {b} {2}} \sum_ {i = 1} ^ {n _ {\mathrm {i d}}} \sum_ {j = i + 1} ^ {n _ {\mathrm {i d}}} P [ (i, j) \in B ] \nabla \ell \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right), \tag {8}
+$$
+
+where  $P[(i,j) \in B]$  is the probability of selecting the pair  $(i,j)$  in a random batch. For uniformly sampled random batches:
+
+$$
+P [ (i, j) \in B ] = \frac {\binom {b} {2}}{\binom {n _ {\mathrm {i d}}} {2}} \tag {9}
+$$
+
+By substituting  $P[(i,j) \in B]$  into the expectation in Eq. 8, we will have:
+
+$$
+\mathbb {E} \left[ \nabla \mathcal {L} _ {B} \left(\boldsymbol {X} _ {\text {r e f}}\right) \right] = \frac {1}{\binom {n _ {\mathrm {i d}}} {2}} \sum_ {i = 1} ^ {n _ {\mathrm {i d}}} \sum_ {j = i + 1} ^ {n _ {\mathrm {i d}}} \nabla \ell \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right) = \nabla \mathcal {L} \left(\boldsymbol {X} _ {\text {r e f}}\right). \tag {10}
+$$
+
+Thus, the batch gradient is an unbiased estimator of the full gradient.
+
+![](images/0635f6f272b0d3d7bc910f558c1fae3804cffae37dba7f56616d3770122255c7.jpg)
+
+Corollary 1. A special case for Theorem 1 is when function  $\ell(\boldsymbol{x}_{\text{ref},i}, \boldsymbol{x}_{\text{ref},j})$  is defined as follows:
+
+$$
+\ell \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right) = \left\{ \begin{array}{l l} - d \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right) & (i, j) = \operatorname {a r g m a x} _ {\boldsymbol {X} _ {\text {r e f}}, i \neq j} - d \left(\boldsymbol {x} _ {\text {r e f}, i}, \boldsymbol {x} _ {\text {r e f}, j}\right) \\ 0 & \text {o t h e r w i s e .} \end{array} \right. \tag {11}
+$$
+
+Therefore, we can rewrite Algorithm 1 with a stochastic optimization as presented in Algorithm 2.
+
+# Algorithm 2 HyperFace Stochastic Optimization for Finding Reference Embeddings
+
+1: Inputs:  $\lambda$ : learning rate,  $n_{\mathrm{itr}}$ : number of iterations,  $\{\pmb{x}_g\}_{g=1}^{n_{\mathrm{gallery}}}$ : embeddings of a gallery of face images,  
+2:  $\alpha$ : hyperparameter (contribution of regularization),  $b$ : size of mini-batch.  
+3: Output:  $\mathbf{X}_{\mathrm{ref}} = \{\pmb{x}_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$ : optimized reference embeddings.  
+4: Procedure:  
+5: Initialize reference embeddings  $X_{\mathrm{ref}} = \{\pmb{x}_{\mathrm{ref},i}\}_{i = 1}^{n_{\mathrm{d}}}.$  
+6: For  $n = 1, \dots, n_{\mathrm{itr}}$  do  
+7: Sample a random mini-batch  $B\subset X_{\mathrm{ref}}$  of size  $b$  ▷ Sampling a random mini-batch  
+8: Find  $\pmb{x}_{\mathrm{ref},i}, \pmb{x}_{\mathrm{ref},j} \in B$  which have minimum distance  $d(\pmb{x}_{\mathrm{ref},i}, \pmb{x}_{\mathrm{ref},j})$  
+9: Reg  $\leftarrow \frac{1}{b}\sum_{k = 1}^{b}\min_{\{\pmb{x}_{g}\}_{\mathrm{gallery}}}d(\pmb{x}_{\mathrm{ref},k},\pmb{x}_{g})$ $\triangleright$  Calculate the regularization term  
+10:  $\mathrm{cost}\gets -d(\pmb{x}_{\mathrm{ref},i},\pmb{x}_{\mathrm{ref},j})$  
+11:  $B\gets B - \mathrm{Adam}(\nabla \mathrm{cost},\lambda)$  
+12:  $B\gets \mathrm{normalize}(B)$  To ensure that resulting embeddings B remain on the hypersphere.  
+13: Update  $B$  in  $\mathbf{X}_{\mathrm{ref}}$  
+14: End For  
+15: End Procedure
+
+To validate our theoretical analyses, we implement the HyperFace stochastic optimization (Algorithm 2) and use the optimized embeddings to generate synthetic face recognition datasets. We
+
+consider 30,000 synthetic identities and solve HyperFace stochastic optimization (Algorithm 2) for different batch sizes. In each case, after solving the stochastic optimization, we generate 50 synthetic images per identity as described in Section 2 (Image Generation). Then, we use the generated datasets to train face recognition models and evaluate the performance of trained face recognition models. Table 10 reports the performance of trained face recognition models. As the results in this table show face recognition models trained with datasets that are generated with stochastic minibatch optimization achieve comparable performance to the face recognition model trained with the dataset that is generated based on full-batch optimization. Therefore, our experimental results meet our theoretical prediction in Theorem 1.
+
+Table 10: Ablation study on the effect of number of batch size in HyperFace stochastic optimization (Algorithm 2).  
+
+<table><tr><td>Batch Size</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>1,000 (mini-batch)</td><td>98.28</td><td>85.23</td><td>91.05</td><td>91.86</td><td>89.37</td></tr><tr><td>5,000 (mini-batch)</td><td>98.62</td><td>84.98</td><td>90.73</td><td>90.41</td><td>88.97</td></tr><tr><td>30,000 (full-batch)</td><td>98.38</td><td>85.07</td><td>90.88</td><td>91.57</td><td>89.60</td></tr></table>
+
+In terms of complexity, the HyperFace stochastic optimization (Algorithm 2) requires significantly less computation resources for solving the optimization. Table 11 reports the runtime for solving the HyperFace stochastic optimization (Algorithm 2) for different batch sizes and different numbers of identities and a fixed size of gallery on a system equipped with a single NVIDIA 3090 GPU. As the results in this table show, the complexity is independent of the number of identities (i.e.,  $n_{\mathrm{id}}$ ) and depends on the size of mini-batch  $b$ . Comparing the results in Table 11 and Table 9, we can conclude that our stochastic optimization significantly reduced the complexity.
+
+Table 11: Runtime for solving the HyperFace stochastic optimization (Algorithm 2) for different numbers of identities on a system equipped with a single NVIDIA 3090 GPU.  
+
+<table><tr><td>Batch Size (b)</td><td># ID (nid)</td><td>HyperFace Stochastic Optimization Runtime</td></tr><tr><td rowspan="3">1,000</td><td>30k</td><td>0.4 hours</td></tr><tr><td>50k</td><td>0.5 hours</td></tr><tr><td>100k</td><td>0.5 hours</td></tr><tr><td rowspan="3">5,000</td><td>30k</td><td>2.2 hours</td></tr><tr><td>50k</td><td>2.2 hours</td></tr><tr><td>100k</td><td>2.2 hours</td></tr><tr><td rowspan="3">10,000</td><td>30k</td><td>8.8 hours</td></tr><tr><td>50k</td><td>8.9 hours</td></tr><tr><td>100k</td><td>8.9 hours</td></tr></table>
+
+# C SYNTHETIC DATASETS AT SCALE
+
+In Table 1 of the paper, we compared our face recognition models trained with our generated dataset and synthetic datasets in the literature. For previous datasets, we considered the available version of each dataset which has a similar number of identities (10k). In Table 3, we studied the effect of the number of identities in our dataset generation, where the results showed that we can scale our synthetic dataset and achieve a higher recognition performance. In Table 12, we compare the performance of face recognition models trained with our generated datasets and with all publicly available versions (particularly larger scale) of synthetic datasets in the literature. As the results in this table show, our generated datasets achieve competitive performance with synthetic datasets in the literature at scale. Comparing different datasets in the literature, DCFace, which outperformed previous datasets in Table 1, does not achieve the best performance on any of the benchmarks for its larger version. In contrast, Langevin-DisCo achieves a significant improvement for its larger version with 30k identities compared to its smaller version with 10k identities. However, Geissbuhler et al. (2024) reported a lower performance for their dataset with 50k identities compared to 30k identities, indicating limitations in further scaling the Langevin-DisCo dataset for more than 30k identities.
+
+Table 12: Comparison of recognition performance of face recognition models trained with the largest available versions of different synthetic datasets as well as a real dataset (i.e., CASIA-WebFace). The performance reported for each dataset is in terms of accuracy and best value for each benchmark is emboldened.  
+
+<table><tr><td>Dataset name</td><td>#IDs</td><td># Images</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>SynFace (Qiu et al., 2021)</td><td>10&#x27;000</td><td>999&#x27;994</td><td>86.57</td><td>65.10</td><td>70.08</td><td>66.79</td><td>59.13</td></tr><tr><td>SFace (Boutrós et al., 2022)</td><td>10&#x27;572</td><td>1&#x27;885&#x27;877</td><td>93.65</td><td>74.90</td><td>80.97</td><td>75.36</td><td>70.32</td></tr><tr><td>Syn-Multi-PIE (Colbois et al., 2021)</td><td>10&#x27;000</td><td>180&#x27;000</td><td>78.72</td><td>60.22</td><td>61.83</td><td>60.84</td><td>54.05</td></tr><tr><td>IDnet (Kolf et al., 2023)</td><td>10&#x27;577</td><td>1&#x27;057&#x27;200</td><td>84.48</td><td>68.12</td><td>71.42</td><td>68.93</td><td>62.63</td></tr><tr><td>ExFaceGAN (Boutrós et al., 2023b)</td><td>10&#x27;000</td><td>599&#x27;944</td><td>85.98</td><td>66.97</td><td>70.00</td><td>66.96</td><td>57.37</td></tr><tr><td>GANDiffFace (Melzi et al., 2023)</td><td>10&#x27;080</td><td>543&#x27;893</td><td>94.35</td><td>76.15</td><td>79.90</td><td>78.99</td><td>69.82</td></tr><tr><td>Langevin-Dispersion (Geissbühler et al., 2024)</td><td>10&#x27;000</td><td>650&#x27;000</td><td>94.38</td><td>65.75</td><td>86.03</td><td>65.51</td><td>77.30</td></tr><tr><td>Langevin-DisCo (Geissbühler et al., 2024)</td><td>10&#x27;000</td><td>650&#x27;000</td><td>97.07</td><td>76.73</td><td>89.05</td><td>79.56</td><td>83.38</td></tr><tr><td>Langevin-DisCo (Geissbühler et al., 2024)</td><td>30&#x27;000</td><td>1&#x27;950&#x27;000</td><td>98.97</td><td>81.52</td><td>93.95</td><td>83.77</td><td>93.32</td></tr><tr><td>DigiFace-1M (Bae et al., 2023)</td><td>109&#x27;999</td><td>1&#x27;219&#x27;995</td><td>90.68</td><td>72.55</td><td>73.75</td><td>79.43</td><td>68.43</td></tr><tr><td>IDiff-Face (Uniform) (Boutrós et al., 2023a)</td><td>10&#x27;049</td><td>502&#x27;450</td><td>98.18</td><td>80.87</td><td>90.82</td><td>82.96</td><td>85.50</td></tr><tr><td>IDiff-Face (Two-Stage) (Boutrós et al., 2023a)</td><td>10&#x27;050</td><td>502&#x27;500</td><td>98.00</td><td>77.77</td><td>88.55</td><td>82.57</td><td>82.35</td></tr><tr><td>DCFace (Kim et al., 2023)</td><td>10&#x27;000</td><td>500&#x27;000</td><td>98.35</td><td>83.12</td><td>91.70</td><td>88.43</td><td>89.50</td></tr><tr><td>DCFace (Kim et al., 2023)</td><td>60&#x27;000</td><td>1&#x27;200&#x27;000</td><td>98.90</td><td>84.97</td><td>92.80</td><td>89.04</td><td>91.52</td></tr><tr><td>HyperFace [ours]</td><td>10&#x27;000</td><td>640&#x27;000</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr><tr><td>HyperFace [ours]</td><td>50&#x27;000</td><td>3&#x27;200&#x27;000</td><td>98.27</td><td>85.60</td><td>91.48</td><td>92.24</td><td>90.40</td></tr><tr><td>CASIA-WebFace (Yi et al., 2014)</td><td>10&#x27;572</td><td>490&#x27;623</td><td>99.42</td><td>90.02</td><td>93.43</td><td>94.97</td><td>94.32</td></tr></table>
+
+Nevertheless, our method achieves improvement by scaling the number of identities (Table 3). In particular, our dataset with 50k identities and 3.2M images achieves competitive performance with large-scale synthetic datasets in the literature.
+
+# D IDENTITY LEAKAGE AND RECOGNITION PERFORMANCE
+
+In Section 3.3, we discussed identity leakage in the generated face datasets. While the leakage of identity is not evident in the generated dataset, it is important to see if identity leakage may affect the recognition performance of face recognition models. To this end, we consider the FFHQ and CASIA-WebFace datasets as two real face datasets and compare all possible pairs from our synthetic dataset with images in the real datasets. Then, for each of these real datasets, we find the top-200 pairs (synthetic-real) and exclude the corresponding synthetic image from our generated dataset. This ensures that images which may contain identity leakage are excluded from the final synthetic datasets. We use the resulting cleaned datasets to train face recognition models and compare them with the face recognition model trained on our original synthetic dataset. Table 13 reports the recognition performance of face recognition models trained with original and cleaned synthetic datasets.
+
+Table 13: Evaluation of potential identity leakage on the recognition performance.  
+
+<table><tr><td>Synthetic Dataset</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>cleaned (Ref.: CASIA-WebFace)</td><td>98.52</td><td>84.80</td><td>89.52</td><td>89.43</td><td>87.00</td></tr><tr><td>cleaned (Ref.: FFHQ)</td><td>98.77</td><td>84.53</td><td>89.35</td><td>89.36</td><td>86.67</td></tr><tr><td>original</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr></table>
+
+As the results in Table 13, removing images with similar identities does not impact the recognition performance of the trained face recognition model. However, we would like to highlight that while identity leakage may not affect recognition performance on benchmark datasets, it is an important privacy concern.
+
+# E ADDITIONAL ABLATION STUDY
+
+In Section 3, we reported ablation studies on different hyperparameters in our dataset generation. As a new experiment, we consider different optimizers for solving HyperFace optimization (full batch). We consider RMSprop, Adam, and AdamW optimizers. Table 14 compares the performance of the face recognition model trained with datasets that are generated based on different optimizers in HyperFace optimization. As the results in this table show, solving HyperFace optimization with different optimizers leads to comparable performance.
+
+Table 14: Ablation study on optimizer  
+
+<table><tr><td>Optimizer</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>RMSprop</td><td>98.47</td><td>84.73</td><td>89.18</td><td>89.27</td><td>86.83</td></tr><tr><td>AdamW</td><td>98.70</td><td>84.20</td><td>89.02</td><td>89.41</td><td>86.38</td></tr><tr><td>Adam</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr></table>
+
+As another experiment, we generate random points on the hypersphere and use random points as reference embeddings without HyperFace optimization to generate a synthetic dataset. We ensure that selected points have at least 0.3 cosine distance. Table 15 compares the performance of the face recognition model trained with the dataset based on random embeddings and HyperFace optimization. As the results in this table show, solving HyperFace optimization achieves superior performance on all benchmarks. Note that with a random selection of points on the hypersphere, there is no guarantee to be on the manifold of embeddings of the face recognition model. However, with our HyperFace optimization, we try to keep points on the face recognition manifold, which results in a dataset that leads to better performance.
+
+Table 15: Ablation study on using random points vs HyperFace optimization.  
+
+<table><tr><td>Reference Embeddings</td><td>LFW</td><td>CPLFW</td><td>CALFW</td><td>CFP</td><td>AgeDB</td></tr><tr><td>Random</td><td>98.12</td><td>83.67</td><td>86.67</td><td>88.79</td><td>84.25</td></tr><tr><td>HyperFace Optimization</td><td>98.67</td><td>84.68</td><td>89.82</td><td>89.14</td><td>87.07</td></tr></table>
+
+# F HYPERFACE DATASET GENERATION
+
+We described HyperFace dataset generation in 2. Algorithm 3 summarizes the dataset generation process in our method.
+
+# Algorithm 3 HyperFace Dataset Generation
+
+```txt
+1: Inputs:  $n_{\mathrm{id}}$ : number of synthetic identities,  $n_{\mathrm{sample}}$ : number of sample images per identity,  
+2:  $G$ : face generator model,  $\beta$ : hyperparameter (controls variations in embeddings)  
+3: Output:  $\mathcal{D}_{\mathrm{HyperFace}} = \{I\}$ : generated dataset.  
+4: Procedure:  
+5: Solve HyperFace optimization to find reference embeddings  $X_{\mathrm{ref}} = \{x_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$  ▷ Algorithm 1 or 2  
+6: Initialize  $\mathcal{D}_{\mathrm{HyperFace}} = []$   
+7: For  $x_{\mathrm{ref}} \in \{x_{\mathrm{ref},i}\}_{i=1}^{n_{\mathrm{id}}}$  do  
+8: For  $n = 1,.., n_{\mathrm{sample}}$  do  
+9: Sample Gaussian noise  $z \sim \mathcal{N}(0, \mathbb{I}^{\mathrm{DM}})$  ▷ For diffusion model  $G$   
+10: Sample Gaussian noise  $v \sim \mathcal{N}(0, \mathbb{I}^{n_X})$  ▷ For variations in the embedding  $x_{\mathrm{ref}}$   
+11: Generate synthetic image  $I = G(\frac{x_{\mathrm{ref}} + \beta v}{||x_{\mathrm{ref}} + \beta v||_2}, z)$   
+12:  $\mathcal{D}_{\mathrm{HyperFace}} \cdot \text{append}(I)$  ▷ Store the generated image  $I$  in the dataset  $\mathcal{D}_{\mathrm{HyperFace}}$   
+13: End For  
+14: End For  
+15: End Procedure
+```
+
+# G VISUALIZATION
+
+![](images/4e80b43c7c52cd74a4d04ebed0fd0389dbdb045dc497f1ada288fd1695d3f8a1.jpg)  
+Figure 4 illustrates sample face images from the HyperFace dataset. In addition, Figure 5 and Figure 6 also show intra-class variations for two synthetic identities in the HyperFace dataset.  
+Figure 4: Sample face images of different synthetic identities from the HyperFace dataset.
+
+![](images/058440dcfcbfb25961a63e4c0613d0d5240836a3125f362ac16b478f3a203776.jpg)  
+Figure 5: Sample face images of one subject from the HyperFace dataset (intra-class variations).
+
+![](images/3d51900938c570074d5a2c9bad35353366ba84c1109def1664da2a77d9db1d05.jpg)  
+Figure 6: Sample face images of one subject from the HyperFace dataset (intra-class variations).

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+# HYPERPLR: HYPERGRAPH GENERATION THROUGH PROJECTION, LEARNING, AND RECONSTRUCTION
+
+Weihuang Wen, Tianshu Yu *
+
+School of Data Science
+
+The Chinese University of Hong Kong, Shenzhen
+
+weihuangwen1@link.cuhk.edu.cn, yutianshu@cuhk.edu.cn
+
+# ABSTRACT
+
+Hypergraphs are essential in modeling higher-order complex networks, excelling in representing group interactions within real-world contexts. This is particularly evident in collaboration networks, where they facilitate the capture of group-wise polyadic patterns, extending beyond traditional pairwise dyadic interactions. The use of hypergraph generators, or generative models, is a crucial method for promoting and validating our understanding of these structures. If such generators accurately replicate observed hypergraph patterns, it reinforces the validity of our interpretations. In this context, we introduce a novel hypergraph generative paradigm, HyperPLR, encompassing three phases: Projection, Learning, and Reconstruction. Initially, the hypergraph is projected onto a weighted graph. Subsequently, the model learns this graph's structure within a latent space, while simultaneously computing a distribution between the hyperedge and the projected graph. Finally, leveraging the learned model and distribution, HyperPLR generates new weighted graphs and samples cliques from them. These cliques are then used to reconstruct new hypergraphs by solving a specific clique cover problem. We have evaluated HyperPLR on existing real-world hypergraph datasets, which consistently demonstrate superior performance and validate the effectiveness of our approach.
+
+# 1 INTRODUCTION
+
+Hypergraphs are powerful data structures that go beyond traditional graph theory by enabling a more nuanced representation of multifaceted relationships. Unlike standard graphs, where edges connect merely pairs of nodes, hypergraphs employ hyperedges, each binding multiple nodes, thereby capturing complex, multi-way interactions with higher fidelity. This capability renders hypergraphs indispensable across a vast array of fields such as drug discovery (Kajino, 2019), electronics (Luo et al., 2024), research collaboration networks (Benson et al., 2018), and protein interaction modeling (Feng et al., 2021), where interactions often occur among more than two entities.
+
+Despite the expansive potential of hypergraphs, the generative models that craft these complex structures are still in their nascent stages. Existing models predominantly cater to simpler graph structures or are bound by pre-assumed hypergraph characteristics, which might not be universally applicable to the diverse and intricate patterns present in the actual datasets (Lee et al., 2021; Do et al., 2020; Kook et al., 2020). Recent advancements include learning-based generative models (Zuo et al., 2023; Gailhard et al., 2024) that attempt to sidestep the necessity for predefined data structures; however, these too often fail to encapsulate critical structural nuances that define hypergraph integrity.
+
+In light of these challenges, we propose a pioneering framework known as HyperPLR (Projection, Learning, and Reconstruction), which introduces a robust three-phase process for the generation of hypergraphs. Initially, HyperPLR projects the complex hypergraph onto a simplified weighted graph format, essentially maintaining key relationships. It then transitions into a learning phase, wherein it discerns the latent structures within this graph and models the inter-dynamics among the projected entities.
+
+In the final phase, we define a novel clique cover problem to reconstruct new hypergraphs from the learned weighted projection graph: Weighted Clique Edge Cover (WCEC). It is a common strategy to project hypergraphs onto graphs, but it is evident that this process results in the loss of some higher-order relationships. However, limited research has investigated the reverse direction—reconstructing hypergraphs from graph projections (Bresler et al., 2024; Wang & Kleinberg, 2024). The closest work so far is by Young et al. (2020), which proposes a maximal clique reconstruction that uses the least number of cliques to cover the projected graph, following the principle of parsimony. However, this method does not consider the number of hyperedges or the degree information of vertices. To address these issues, we use a weighted graph as the projection, design the WCEC problem, and show that solving WCEC allows efficient and more controllable hypergraph reconstruction from the projection. Additionally, we develop a fast heuristic for approximating the solution.
+
+Our extensive evaluations of various real-world datasets underscore HyperPLR's adeptness at weaving hypergraphs that faithfully reflect the underlying complexities of group interactions. Thus, HyperPLR marks a significant technological leap in hypergraph generation.
+
+The principal contributions of this paper are encapsulated in the following:
+
+- Introduction and formulation of the WCEC problem, complemented with an efficient algorithmic solution for reconstructing hypergraphs.  
+- Development of the novel HyperPLR framework, which harmoniously integrates the structure and intricate interactions within a hypergraph.  
+- Empirical validation of HyperPLR against multiple real-world datasets, showcasing enhanced efficiency and remarkable performance in hypergraph reconstruction.
+
+# 2 RELATED WORK
+
+The study of hypergraph generation has garnered increasing attention as an extension of traditional graph generation techniques, addressing higher-order relational structures present in many real-world applications. Traditional graph generation models have advanced significantly with the introduction of deep learning techniques. (Kipf & Welling, 2016; Jin et al., 2018; Bojchevski et al., 2018) generate graphs by imitating the structural properties of given graphs. This is typically achieved by learning node/edge embeddings and injecting appropriate randomness (Grover et al., 2019; Wang et al., 2019; Dai et al., 2020). However, methods relying on node/edge embeddings often focus primarily on local structures, particularly in large graphs. While graph spectral theory offers an alternative approach, some studies incorporate spectral information into deep models to capture global characteristics (Martinkus et al., 2022; Bojchevski et al., 2018; Rendsburg et al., 2020). Among these, CELL (Rendsburg et al., 2020) stands out for its efficiency and scalability, achieved by removing unnecessary operations from Bojchevski et al. (2018). Another consideration is to generate graphs with weights (edge features) (Grover et al., 2019; Niu et al., 2020; Kocayusufoglu et al., 2022), which again heavily relies on embeddings.
+
+Traditional graph generation models, while effective for molecular graphs and citation networks, are not suitable for hypergraph generation. Hypergraphs, with hyperedges connecting multiple nodes, pose challenges that standard graph models cannot address. Several methods have been proposed for hypergraph generation. HYPERPA (Do et al., 2020) uses a statistical approach, relying on three key metrics from real-world hypergraphs: node count, hyperedge size distribution, and new hyperedges per node. While HYPERPA generates hyperedges based on these metrics, its dependence on precomputed statistics and iterative generation process increases time complexity. HyperDK (Nakajima et al., 2022) generates hypergraphs that preserve local properties of nodes and hyperedges, controlled by two hyperparameters,  $d_v$  and  $d_e$ . A higher  $d_v$  value indicates that higher-order information regarding node degrees is preserved, while a higher  $d_e$  value implies the preservation of higher-order information regarding hyperedge sizes. ThERA (Kim et al., 2023) organizes nodes in a hierarchical structure across multiple levels, with nodes divided into disjoint levels and deeper levels containing more nodes. It generates hyperedges locally within each group based on a given probability. Compared to parameter-based methods, only a few deep learning models have explicitly addressed hypergraph generation to date (Zuo et al., 2023; Gailhard et al., 2024), and existing learning-based hypergraph generation approaches are limited by scalability issues.
+
+![](images/7b44303f98789885a9194bdf4d0957d591fd83732c941ac81fa9410e28e176bd.jpg)  
+Figure 1: Hypergraph's graph projection (left direction) and weighted graph projection (right direction), with line thickness referring to its weight.
+
+In summary, while deep learning techniques for graph generation have made significant strides, hypergraph generation remains an emerging area of research. Existing models, though promising, often rely on adaptations of graph generation methods and statistical approaches, with inherent limitations in scalability and evaluation. There is a growing need for more robust models specifically designed for hypergraph generation, particularly ones that can efficiently capture the complex, higher-order relationships that hypergraphs represent.
+
+# 3 PRELIMINARIES AND NOTATIONS
+
+Hypergraph. A hypergraph is a generalization of a graph where edges, called hyperedges, can connect more than two vertices. Formally, a hypergraph is defined as  $\mathcal{H} = (V,\mathcal{E})$  where  $V$  is a set of vertices and  $\mathcal{E}$  is a set of hyperedges  $(e_1,e_2,\dots ,e_m)$  with  $e_i\subseteq V$ . In a hypergraph, hyperedges can be of varying sizes, capturing higher-order relationships between sets of nodes beyond pairwise connections and we assume the hyperedges are distinct in our paper, which means  $\forall 1\leq i,j\leq m,e_i\neq e_j$ .
+
+Graph Projection. A graph projection, or clique expansion, of a hypergraph, is a transformation process that converts a hypergraph into a traditional graph. In this projection, each hyperedge of the hypergraph is mapped to a clique (a complete subgraph) in the graph, where each pair of nodes in the hyperedge is connected by an edge in the projected graph. Formally, if  $\mathcal{H} = (V,\mathcal{E})$  is a hypergraph, the projection graph  $G = \operatorname {Proj}(\mathcal{H}) = (V,\mathcal{E}')$ , where  $\mathcal{E}'$  consists all pairs  $(v_{i},v_{j})\in e_{i}$  for hyperedge  $e_i\in \mathcal{E}$ .
+
+Weighted Graph Projection. In this paper, we introduce a novel graph projection method for learning-based hypergraph generation, namely Weighted Graph Projection, which effectively preserves the frequency with which vertices co-occur within hyperedges. Formally, given a hypergraph  $\mathcal{H} = (V,\mathcal{E})$ , the weighted graph projection is to project  $\mathcal{H}$  into a weighted graph  $G_{w} = \mathrm{Proj}_{w}(\mathcal{H}) = (V,\mathcal{E}',w)$ . The  $w$  is a weight function that assigns a weight to each edge  $e'\in \mathcal{E}'$ . The common way to assign weight is  $w(v_{i},v_{j}) = \sum_{e\in \mathcal{E}}\mathbb{I}(v_{i},v_{j}\in e)$  where  $\mathbb{I}$  is an indicator function that equals 1 if both vertices  $v_{j}$  and  $v_{j}$  are part of the same hyperedge.
+
+Clique Edge Cover Problem. The Clique Edge Cover problem is concerned with covering all the edges of a graph using the minimum number of cliques (complete subgraphs). This problem is also referred to as the Minimum Clique Edge Cover (MCEC) problem. Given a graph  $G = (V, \mathcal{E}')$  a clique cover is a collection of cliques such that every edge in  $\mathcal{E}'$  belongs to at least one clique. The goal is to minimize the number of cliques in the cover. This problem is closely related to the projection of hypergraphs, as each hyperedge in a hypergraph corresponds to a clique in the projected graph, and reconstructing the original hyperedges can be seen as a clique cover problem in reverse Bresler et al. (2024).
+
+Fig. 1 illustrates the two different projection methods between hypergraph and traditional graph.
+
+![](images/d241b4f3a2a900d352bbd6c395932f63a4ba9b60068503b2e31bec9a1cd3a7fc.jpg)  
+Figure 2: An overview of HyperPLR framework. We use RED and BLUE arrowed lines to indicate flows in the training and the testing phases, respectively. Black arrowed lines are normal flows.
+
+# 4 HYPERPLR
+
+In this part, we elaborate on the framework of HyperPLR in detail. A high-level overview of the HyperPLR framework can be found in Fig. 2. As shown, the framework consists of three core blocks: "Projection", "Learning", and "Reconstruction", acting in the presented order. Given an input hypergraph  $\mathcal{H}$ , the projection block project  $\mathcal{H}$  to the corresponding graph  $G$  and weighted graph  $G_w$ . In the following learning block, we employ CELL (Rendsburg et al., 2020) to sample the new graph  $G'$ . In parallel to CELL, we adopt a GCN (Kipf & Welling, 2017) to train a weight predictor taking node features of  $G$  as input, to optimize the MSE error w.r.t. the real edge weight  $w$  in  $G_w$ . During the testing phase, the sampled graph  $G'$  is instead fed into this predictor as the graph topology, and the new weighted graph  $G'_w$  is obtained by filling predicted weights into  $G'$ . In the last block of reconstruction, GWC performs a heuristic strategy to dynamically produce the most likely overlapping cliques out of  $G'_w$ . Finally, this collection of cliques is readily converted to a new hypergraph  $\mathcal{H}'$ . We detail each block in the following sections.
+
+# 4.1 PROJECTION
+
+Projecting hypergraphs onto graphs and utilizing graph algorithms is a common technique for solving hypergraph-related problems, as it improves storage efficiency, interpretability, and allows for the use of well-established data structures and algorithms. However, hypergraph projection often results in the loss of higher-order relationships. To address this, there has been research on the limitation when reconstructing hypergraphs from their graph projections (Bresler et al., 2024; Wang & Kleinberg, 2024). Although exact reconstruction of the original hypergraph is generally infeasible, we find that with appropriate "guidance", it is possible to effectively reconstruct most of the patterns in the hypergraph. This insight motivates our investigation into the generation of such "guidance". Specifically, we demonstrate that learning to generate this guidance can enable the creation of a new hypergraph that retains most of the information from the original hypergraph's projection. In our study, we show that the weighted graph projection  $G_{w}$  serves as an effective form of guidance during the reconstruction process.
+
+On one hand, the weighted projection offers a more condensed hyperedge information into edges, reducing graph size compared to bipartite representations, which require separate hyperedge nodes. As discussed in Bojchevski et al. (2018); Rendsburg et al. (2020), bipartite graphs are difficult to analyze using spectral tools, as they do not possess steady states. On the other hand, the weighted projection provides a higher representational capacity than the unweighted projection. Notably, in the weighted projection, the weight of each edge in the graph reflects the co-occurrence frequency of its nodes within hyperedges, thereby preserving crucial information for the reconstruction of the original hypergraph.
+
+# 4.2 LEARNING
+
+Learning to generate weighted projection  $G_{w}$  that resembles the projection from the real-world hypergraph is non-trivial, as 1)  $G_{w}$  with weights are hard to model and learn, and 2) real-world hypergraph can easily scale up (to tens of thousands of vertices) with long-range dependencies like overlapping cliques or communities in large hypergraphs. We first present how this can be achieved at a high level. Considering a weighted graph with node feature  $X \in \mathbb{R}^{n \times d}$ , adjacency matrix  $A$ , and weight matrix  $W$ , we have:
+
+$$
+\begin{array}{l} \mathbb {P} (W, A | X) = \mathbb {P} (W | A, X) \mathbb {P} (A) (1a) \\ = \mathbb {P} (W | A, X) \prod_ {i, j} \mathbb {P} (A _ {i, j}) (1b) \\ = \prod_ {i, j} \mathbb {P} \left(W _ {i, j} \mid A _ {i, j}, X\right) \prod_ {i, j} \mathbb {P} \left(A _ {i, j}\right) (1c) \\ = \prod_ {i, j} \mathbb {P} _ {\alpha} \left(W _ {i, j} \mid A _ {i, j}, X _ {i}, X _ {j}\right) \mathbb {P} _ {\beta} \left(A _ {i, j}\right), (1d) \\ \end{array}
+$$
+
+where  $\mathbb{P}(\cdot)$  measures the probability. Eq. (1a) stands because  $A$  is derived solely from a spectral method, independent of  $X$ . Eq. (1b) is due to the fact that  $A$  is drawn from an edge-independent "score matrix" as in Bojchevski et al. (2018); Rendsburg et al. (2020). Eq. (1c) and (1d) are further built upon the assumption that edge weight  $W_{ij}$  is only dependent on the features of the ending nodes  $X_{i}$  and  $X_{j}$ , and the existence of this edge  $A_{i,j}$ , under two distributions parameterized by  $\alpha$  and  $\beta$ .
+
+We first introduce  $\mathbb{P}_{\beta}(\cdot)$  allowing for generating large-scale graphs with high efficiency. To this end, we employ CELL (Rendsburg et al., 2020). CELL is a graph generative model working in the spectral space by removing redundant computation from NetGAN (Bojchevski et al., 2018). In NetGAN, the following objective is optimized
+
+$$
+\min  _ {Z \in \mathbb {R} ^ {n \times n}} - \sum_ {(u, v) \in \mathcal {R}} \log \sigma_ {\text {r o w s}} (Z) _ {u, v} \quad \text {s . t .} \quad \operatorname {r a n k} (Z) \leq h, \tag {2}
+$$
+
+where  $\mathcal{R}$  is a collection of (massive) random walks and  $(u,v)$  is a transition.  $\sigma_{\mathrm{rows}}(Z)$  is a row-wise softmax function perform on the low-rank variable  $Z$ . In practice, NetGAN needs to sample millions of random walks from a graph with moderate size, which is prohibitively expensive. Besides, the extra computational burden is required as NetGAN trained a generator to produce fixed-length random walks to obtain a new graph.
+
+CELL (Rendsburg et al., 2020) argued that such a massive amount of sampling procedures and the generator in NetGAN can be avoided by only considering the spectral properties in limit. Concretely, given the adjacency matrix  $A$  of a graph, its transition matrix can be obtained via  $P = D^{-1}A$  where  $D = \mathrm{diag}(d) \in \mathbb{R}_+^{n \times n}$  is the degree matrix. Assuming the stationary state  $\pi$  of  $P$  exists and is unique, CELL considers acting an infinite number of infinite long random walks and calculates how many times a node can be visited. The time of visit for nodes can be encoded in a "score matrix"  $S$ , and the normalized score matrix in limit is
+
+$$
+\lim  _ {\substack {q \to \infty \\ T \to \infty}} \frac {S}{q T} = \operatorname {diag} (\pi) P, \tag{3}
+$$
+
+where  $q$  and  $T$  are number and length of random walks, respectively. Thus  $S$  can serve as a surrogate of random walk set  $\mathcal{R}$  in Eq. (2):
+
+$$
+\min  _ {Z \in \mathbb {R} ^ {n \times n}} - \sum_ {u, v = 1} ^ {n} S _ {u, v} \log \sigma_ {\text {r o w s}} (Z) _ {u, v}. \tag {4}
+$$
+
+Since  $\mathrm{diag}(\pi)P\propto A$  and rescaling  $S$  will not change the optima in Eq. (4), the final objective of CELL becomes
+
+$$
+\min  _ {Z \in \mathbb {R} ^ {n \times n}} - \sum_ {u, v = 1} ^ {n} A _ {u, v} \log \sigma_ {\text {r o w s}} (Z) _ {u, v} \quad \text {s . t .} \quad \operatorname {r a n k} (Z) \leq h, \tag {5}
+$$
+
+whose solution is denoted as  $Z^{*}$ . Then  $P^{*} = \sigma_{\mathrm{rows}}(Z^{*})$  can be viewed as a low-rank-regularized transition matrix. We can obtain a new score  $S^{*} = \mathrm{diag}(\pi^{*})P^{*}$ , where  $\pi^{*}$  is the stationary distribution of  $P^{*}$ . Finally, the edges of a new graph can be independently sampled according to  $S^{*}$ .
+
+Next, we discuss  $\mathbb{P}_{\alpha}(\cdot)$ . To allow generating graphs with weights, we extend CELL by providing predicted weights on the sampled edges, where the edge weight predictor is a Graph Convolutional Network (GCN) (Kipf & Welling, 2017). During the training phase, this predictor takes the vertex embeddings  $X^{(0)}$  and the adjacency matrix  $A$  as input, and outputs a set of new embedding  $X^{(l)}$ :
+
+$$
+X ^ {(l)} = \operatorname {G C N} \left(X ^ {(0)}, A\right), \tag {6}
+$$
+
+where  $l$  is the number of basic GCN layers. The loss is the MSE between the predicted weights  $X^{(l)}X^{(l)^{\top}}$  and the ground-truth weights  $W$  of the original WLIG on the observed edge set  $A$ :
+
+$$
+\sum_ {(i, j) \in A} \left(\left(X ^ {(l)} X ^ {(l) \top}\right) _ {i, j} - W _ {i, j}\right) ^ {2}. \tag {7}
+$$
+
+This weight predictor can be trained not only on a single graph projection (though we train separate embedding networks for each instance) but throughout a set of formulas, since all these formulas may share common structures on weights. In the testing stage, we feed the embedding  $X^0$  and a sampled adjacency  $A'$  from CELL into the well-trained GCN:
+
+$$
+X _ {s} ^ {(l)} = \operatorname {G C N} \left(X ^ {(0)}, A ^ {\prime}\right). \tag {8}
+$$
+
+Then weights can be derived using  $X_{s}^{(l)}X_{s}^{(l)\top}$ . We only consider the weights on the edges that appear in  $A^{\prime}$ . See the "Learning" block in Fig. 2 for an intuitive view.
+
+One remaining problem is how to obtain the initial Node Embedding  $X^{(0)}$  for each vertex. One of the most intuitive ways is to use a one-hot representation, but as the size of the hypergraph grows, the vertices number  $n$  and the required dimension increase rapidly. In practice, we train the node embedding with the Node2Vec algorithm (Grover & Leskovec, 2016) from the simple projection graph  $G$ .
+
+# 4.3 RECONSTRUCTION
+
+As each hyperedge in the hypergraph  $\mathcal{H}$  is mapped to a clique in the graph projection  $G$  or  $G_{w}$ , it is straightforward to consider the reconstruction as a clique cover problem, for example, MCEC problem (Rodrigues, 2021). Here, we give a proposition about the limitations of the MCEC:
+
+Proposition 1. One hypergraph  $\mathcal{H}$  can be reconstructed by solving the MCEC on the graph projection  $G$  if and only if each hyperedge appears as an isolated clique.
+
+Proof. A hyperedge must be a clique in the graph projection, but not vice versa. A clique may not be a hyperedge in only two cases: 1) for a hyperedge  $e$ , there is a hyperedge  $e'$  set that  $e \subset e'$ ; 2) for several hyperedges  $e_1, \dots, e_k$ , which's projections overlap together and form a bigger clique than each other. The first case would lead the solving of MCEC to ignore some hyperedges which are included by other hyperedges. The second case would lead the MCEC solving to produce the new hyperedges.
+
+Although the MCEC has its limitations, it still can be a strong baseline in the hypergraph reconstruction task (Young et al., 2020). Due to the "principle of parsimony", solving MCEC on the projection graph will ignore the frequency of node level pairwise appearance and always recover fewer hyperedges than the original hypergraph. To address these issues, inspired by MCEC, we make a step to raise a novel Weighed Clique Edge Cover (WCEC) problem and an associated efficient algorithm Greedy Weighted Cover (GWC) in a hill-climbing manner. WCEC substantially differs from MCEC by taking into account an essential but less explored factor-frequency of node level pairwise appearance.
+
+Weighted Clique Edge Cover Problem (WCEC). Though there exist several variants of clique covering problems, our problem of recovering clique covers from weighted graphs has been less studied. The most related problem in literature is Weighted Edge Clique Partition (WECP), which is
+
+proposed by Feldmann et al. (2020) and we provide a more comprehensive analysis for the connection and difference between WECP and WCEC in Appendix. A. Here, we define the WCEC problem as follows:
+
+Problem 1 (Weighted Clique Edge Cover). Given a weighted graph  $\mathcal{G}$  with weight matrix  $W$ , it asks to select  $k$  cliques from  $\mathcal{G}$ , such that the distance  $d(W, W') \geq 0$  between  $W$  and  $W'$  is minimized, where weight matrix  $W'$  is derived from a new weighted graph  $\mathcal{G}'$  by stacking  $k$  selected cliques.
+
+In our setting, we let  $d(W, W')$  be the L1-distance:
+
+$$
+d \left(W, W ^ {\prime}\right) = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \left| W _ {i j} - W _ {i j} ^ {\prime} \right|, \tag {9}
+$$
+
+where  $W_{ij}$  represents the entry of  $W$  at  $i$ th row and  $j$ th column. Comparing the proposed WCEC and MCEC, we have the following lemma:
+
+Lemma 2. Suppose a WCEC problem with weight matrix  $W$  is obtained by assigning positive weights to edges in adjacency  $A$  associated with an MCEC problem. Then a solution clique cover to this WCEC such that  $d(W, W') = 0$  must form a cover of the MCEC, but a clique cover of MCEC on  $A$  may not satisfy  $d(W, W') = 0$ .
+
+Proof. We first prove the initial part. By definition,  $d(W, W') = 0$  implies that for every  $(i,j)$ , the weight matrix  $G_w$  of the found cover satisfies  $W_{i,j}' = W_{i,j}$ . Since  $W$  is derived by assigning positive weights to the non-zero entries in  $A$ , we have  $W_{i,j}' > 0 \Rightarrow A_{i,j} = 1$  and  $W_{i,j} = 0 \Rightarrow A_{i,j} = 0$ . Therefore, the solution to the WCEC is a cover of the MCEC. For the second part, it suffices to provide an easily verifiable example, which is presented as a constructive proof in Appendix. B.
+
+Lemma 2 provides an interpretation of why the proposed framework "weighted projection +WCEC" is more powerful than "projection + MCEC" utilized by Young et al. (2020). In general, solving WCEC provides rich information for solving MCEC, but not vice versa. Nevertheless, solving either of them is NP-complete (Feldmann et al., 2020; Ullah, 2022).
+
+Greedy Weight Cover (GWC) Algorithm. Due to NP-completeness, an exact solution to WCEC can be infeasible. Therefore, we develop a specialized algorithm to obtain a local optima of WCEC via hill-climbing. Our algorithm works on a local search basis, wherein each iteration will take action with the largest gain. We term this procedure the Greedy Weighted Cover (GWC) algorithm. Concretely, GWC first enumerates or samples the cliques set from the weighted graph projection, whose sizes are below the maximal hyperedge size of the original hypergraph, using the algorithm proposed by Zhang et al. (2005). Then we create a GainTable measuring how much one can improve by decoding the corresponding clique. In each subsequent step, GWC finds a clique with the largest gain and updates the GainTable in a dynamic programming fashion, until the number of found cliques reaches a preset threshold  $m$  or distance. To prevent redundant computation, we first build an EdgeTable mapping from an edge to its related cliques (see Alg. 2), which allows GWC only updates the gain of the local cliques in each step. The overall procedure is summarized in Alg. 1.
+
+# 5 EXPERIMENT
+
+# 5.1 PROTOCALS
+
+Implementation details. Throughout all the experiments, HyperPLR is running on an Apple M1 CPU. The dimension of node embedding from Node2Vec is 50. The GCN consists of two layers, with input/output dimensions 50/128 and 128/128, respectively. We employ ADAM as the optimizer for all learning modules (i.e., Node2Vec, GCN, and CELL). For CELL, we set parameter edge_overlap_limit = 0.8 which controls the overlapping portion of the generated and the original graphs. For more details, one can refer to the supplementary material.
+
+![](images/9c957ada53b539207baa090817424bffe68ec257b734abcfa5b888480399d656.jpg)
+
+![](images/ef26010eb6dfc5e16eb9aff2174bc705f4479580abf3a5d7b56dc4fb944082ad.jpg)
+
+Dataset. Following the standard approach outlined in Nakajima et al. (2022); Lee et al. (2021); Kim et al. (2023), we evaluate performance on five datasets across three domains: contact (contact-high-school, contact-primary-school), email (email-Eu, email-Enron), and drug (NDC-classes) (Benson et al., 2018). In the contact datasets, each node represents an individual, and each hyperedge represents a group communication event involving all participating individuals. In the email datasets, each node represents a user, and each hyperedge corresponds to an email, consisting of the sender and all recipients. In the drug dataset, each hyperedge corresponds to an NDC code for a drug, and the nodes represent the substances composing the drug. More details can be found in Appendix. C.1.
+
+Baselines. We compared the performance of our proposed model against four existing hypergraph generation methods: HyperDK (Nakajima et al., 2022), Hyperlap (Lee et al., 2021), Hyperlap+ (Lee et al., 2021), and THERA (Kim et al., 2023). HyperDK, Hyperlap, and Hyperlap+ are static generators, primarily suited for producing static hypergraphs, while THERA generates dynamic graphs, such as temporal hypergraphs. Since HyperDK is highly sensitive to parameter settings, we evaluated it under two configurations: HyperDK $_{0,0}$ , which aims to generate hypergraphs that preserve the average node degree and the average hyperedge size, and HyperDK $_{1,1}$ , which seeks to preserve the degree of each individual node and the size of each hyperedge.
+
+# 5.2 RESULTS
+
+Graph statistics. The results of the comparisons between our model and state-of-the-art methods across five datasets, using various evaluation metrics, are presented in Table 1. We further compare some of the advanced structural properties in Appendix. C.2. Each dataset was generated five times, and the average results were reported. The graph properties evaluated include hypergraph structure-level metrics (density, average hyperedge size, average node degree), graph projection-level metrics ("G coefficient" and "G modularity"), and bipartite graph representation-level metrics ("B modularity"). These properties provide a comprehensive overview of the graph's structural characteristics. The detailed definitions about the metrics can be found in Appendix. C.3.
+
+Representation ability and reconstruction ability. We investigate how much the graph structure statistics can retain after feeding a hypergraph into different consecutive phases of HyperPLR from projection  $G_{w}$ , learning  $G_{w}^{L}$ , and reconstruction  $G_{w}^{R}$ . Results are in Table 2. We also visualize this process using the email-Enron instance in Fig. 3. It is evident that the main structures of the WLIGs are well-preserved across all three phases.
+
+![](images/77a76ab065d13962b9ba790da92503740ca286b346c49252d5f7c8a350d892af.jpg)  
+Figure 3: Visualization of weighted graph projection of hypergraph email-Enron at different phases. Each node corresponds to one vertex and each edge (with thickness) indicates inter-node incidence (with frequency). (a) Weighted graph projection  $G_{w}$  derived from original hypergraph; (b) Generated Weighted graph projection  $G_{w}^{L}$  based on (a) using hyperPLR; (c) Weighted graph projection  $G_{w}^{R}$  of generated hypergraph from (b) with GWC. Zoom in for a better view.
+
+![](images/43dba963d371ee88ae8c05468c5a8b6255e1a8efb34809f34f8d46ef668286d7.jpg)
+
+![](images/b6b23ce8727d29f7a4f278ae33e4aa67b58dc173d540f21a1c2ce31aa213e193.jpg)
+
+Table 1: Results of hypergraph generation in terms of graph metrics with 5 real-world hypergraphs  
+
+<table><tr><td></td><td>density</td><td>average size</td><td>average degree</td><td>G coefficient</td><td>G modularity</td><td>B modularity</td></tr><tr><td>contact-high-school</td><td>23.908</td><td>2.327</td><td>55.633</td><td>0.504</td><td>0.582</td><td>0.754</td></tr><tr><td>HyperDK0,0</td><td>272.161</td><td>3.089</td><td>840.796</td><td>0.999</td><td>0.000</td><td>0.324</td></tr><tr><td>HyperDK1,1</td><td>142.573</td><td>2.178</td><td>310.567</td><td>0.861</td><td>0.016</td><td>0.459</td></tr><tr><td>Hyperlap</td><td>23.938</td><td>2.327</td><td>55.701</td><td>0.341</td><td>0.394</td><td>0.622</td></tr><tr><td>Hyperlap+</td><td>23.908</td><td>2.327</td><td>55.633</td><td>0.632</td><td>0.742</td><td>0.747</td></tr><tr><td>THERA</td><td>23.908</td><td>2.609</td><td>62.382</td><td>0.354</td><td>0.405</td><td>0.831</td></tr><tr><td>HyperPLR</td><td>24.055</td><td>2.703</td><td>65.011</td><td>0.471</td><td>0.556</td><td>0.649</td></tr><tr><td>contact-primary-school</td><td>52.496</td><td>2.419</td><td>126.979</td><td>0.526</td><td>0.284</td><td>0.635</td></tr><tr><td>HyperDK0,0</td><td>229.896</td><td>3.129</td><td>719.429</td><td>1.000</td><td>0.000</td><td>0.319</td></tr><tr><td>HyperDK1,1</td><td>143.343</td><td>2.285</td><td>327.516</td><td>0.924</td><td>0.010</td><td>0.437</td></tr><tr><td>Hyperlap</td><td>52.496</td><td>2.419</td><td>126.979</td><td>0.496</td><td>0.281</td><td>0.620</td></tr><tr><td>Hyperlap+</td><td>52.496</td><td>2.419</td><td>126.979</td><td>0.655</td><td>0.496</td><td>0.640</td></tr><tr><td>THERA</td><td>52.496</td><td>2.767</td><td>145.252</td><td>0.446</td><td>0.150</td><td>0.758</td></tr><tr><td>HyperPLR</td><td>52.714</td><td>2.648</td><td>139.593</td><td>0.491</td><td>0.281</td><td>0.600</td></tr><tr><td>email-Enron</td><td>10.573</td><td>3.009</td><td>31.818</td><td>0.593</td><td>0.352</td><td>0.656</td></tr><tr><td>HyperDK0,0</td><td>52.045</td><td>3.188</td><td>165.933</td><td>0.958</td><td>0.007</td><td>0.322</td></tr><tr><td>HyperDK1,1</td><td>39.596</td><td>2.885</td><td>114.252</td><td>0.796</td><td>0.040</td><td>0.355</td></tr><tr><td>Hyperlap</td><td>10.232</td><td>3.131</td><td>32.034</td><td>0.646</td><td>0.275</td><td>0.576</td></tr><tr><td>Hyperlap+</td><td>10.203</td><td>3.131</td><td>31.944</td><td>0.841</td><td>0.582</td><td>0.738</td></tr><tr><td>THERA</td><td>10.203</td><td>3.386</td><td>34.545</td><td>0.674</td><td>0.166</td><td>0.736</td></tr><tr><td>HyperPLR</td><td>10.723</td><td>3.244</td><td>34.782</td><td>0.593</td><td>0.351</td><td>0.569</td></tr><tr><td>email-Eu</td><td>25.077</td><td>3.426</td><td>85.909</td><td>0.492</td><td>0.371</td><td>0.675</td></tr><tr><td>HyperDK0,0</td><td>156.130</td><td>3.125</td><td>487.965</td><td>0.723</td><td>0.015</td><td>0.327</td></tr><tr><td>HyperDK1,1</td><td>114.040</td><td>2.887</td><td>329.223</td><td>0.788</td><td>0.035</td><td>0.353</td></tr><tr><td>Hyperlap</td><td>25.778</td><td>3.621</td><td>93.338</td><td>0.623</td><td>0.280</td><td>0.571</td></tr><tr><td>Hyperlap+</td><td>25.860</td><td>3.621</td><td>93.633</td><td>0.785</td><td>0.745</td><td>0.751</td></tr><tr><td>THERA</td><td>24.868</td><td>4.696</td><td>116.792</td><td>0.837</td><td>0.099</td><td>0.793</td></tr><tr><td>HyperPLR</td><td>29.081</td><td>3.135</td><td>91.155</td><td>0.446</td><td>0.319</td><td>0.493</td></tr><tr><td>NDC-classes</td><td>0.937</td><td>5.922</td><td>5.550</td><td>0.611</td><td>0.610</td><td>0.741</td></tr><tr><td>HyperDK0,0</td><td>36.112</td><td>3.583</td><td>129.392</td><td>0.311</td><td>0.038</td><td>0.303</td></tr><tr><td>HyperDK1,1</td><td>31.083</td><td>3.594</td><td>111.711</td><td>0.754</td><td>0.058</td><td>0.296</td></tr><tr><td>Hyperlap</td><td>1.089</td><td>6.166</td><td>6.718</td><td>0.672</td><td>0.456</td><td>0.575</td></tr><tr><td>Hyperlap+</td><td>1.098</td><td>6.166</td><td>6.767</td><td>0.750</td><td>0.793</td><td>0.836</td></tr><tr><td>THERA</td><td>0.999</td><td>6.282</td><td>6.277</td><td>0.793</td><td>0.520</td><td>0.892</td></tr><tr><td>HyperPLR</td><td>1.210</td><td>5.501</td><td>6.655</td><td>0.791</td><td>0.623</td><td>0.733</td></tr></table>
+
+# 5.3 ANALYSIS AND DISCUSSION
+
+The experimental results demonstrate several key findings regarding the performance and capabilities of the proposed HyperPLR framework:
+
+Graph Statistics Comparison. Towards graph-level metrics, HyperPLR outperformed traditional hypergraph generation models, including HyperDK, Hyperlap, and ThERA, across various datasets. Remarkably, despite not receiving any direct structural parameters from the original hypergraph,
+
+Table 2: Results of the graph metrics for the weighted graph projection  $G_{w}$  derived from original hypergraph; Generated Weighted graph projection  $G_{w}^{\bar{L}}$ ; and weighted graph projection  $G_{w}^{R}$  of generated hypergraph  
+
+<table><tr><td>Hypergraph</td><td>Gw coefficient</td><td>Gw modularity</td><td>GL coefficient</td><td>GWLmodularity</td><td>GRcoefficient</td><td>GRWmodularity</td></tr><tr><td>contact-high-school</td><td>0.504</td><td>0.582</td><td>0.388</td><td>0.546</td><td>0.472</td><td>0.556</td></tr><tr><td>contact-primary-school</td><td>0.526</td><td>0.284</td><td>0.453</td><td>0.308</td><td>0.491</td><td>0.277</td></tr><tr><td>email-Enron</td><td>0.593</td><td>0.352</td><td>0.508</td><td>0.433</td><td>0.593</td><td>0.347</td></tr><tr><td>email-Eu</td><td>0.492</td><td>0.371</td><td>0.366</td><td>0.448</td><td>0.446</td><td>0.315</td></tr><tr><td>NDC-classes</td><td>0.611</td><td>0.610</td><td>0.452</td><td>0.538</td><td>0.788</td><td>0.625</td></tr></table>
+
+HyperPLR consistently demonstrates competitive similarity across nearly all graph metrics when compared to other parameter-based methods, underscoring its superior potential in hypergraph generation. Additionally, the clustering coefficient distribution of the hypergraphs generated by HyperPLR exhibits the highest degree of similarity to real-world datasets. Furthermore, HyperPLR is parameter-insensitive, delivering stable and robust results with default parameter settings. For example, the default rank number of CELL Rendsburg et al. (2020) is 9. To investigate how much this parameter impacts the generated results, we conduct extra experiments on varying rank  $= 8, 10, 12$ . We also present three generated weighted graph projections in Appendix. D. Additionally, a brief discussion on the trade-off between the fidelity and diversity of HyperPLR is provided in Appendix. E. All these result indicate that the HyperPLR framework, despite not requiring parameter tuning specific to each dataset, can generalize effectively across different domains. This robustness and parameter insensitivity highlight HyperPLR's advantage in practical applications, where real-world datasets can vary widely in structure.
+
+Representation and Reconstruction Ability. Table 2 and Figure 3 provide insights into how HyperPLR maintains structural properties throughout the projection, learning, and reconstruction phases. The close similarity between the weighted graph projections derived from the original hypergraph  $(G_w)$  and the generated hypergraph  $(G_w^R)$  demonstrates that HyperPLR effectively preserves critical structural information. The consistency in clustering coefficients and modularity across the different phases (as seen in Table 2) suggests that the proposed weighted graph projection method captures essential high-order relationships. This is further confirmed by the visualization in Figure 3, where the main structures are visibly well-retained, implying that the framework's learning and reconstruction processes are efficient in preserving hypergraph topology.
+
+Limitations and Future Directions. Despite its strengths, HyperPLR has limitations. It requires significant computational resources, especially for larger datasets, and the GWC heuristic algorithm, while effective, doesn't guarantee optimality. Future research could explore advanced optimization techniques like reinforcement learning or MCTS for improving clique selection. Additionally, the complexity of HyperPLR is constrained by the exponential nature of the maximal clique enumeration problem, though it remains manageable for real-world hypergraphs. Future work could also focus on accelerating clique enumeration using hyperedges as prior information.
+
+# 6 CONCLUSION
+
+Deep hypergraph generation models have demonstrated their ability to capture the complex distribution of hypergraphs by leveraging the rich information embedded within them to generate more realistic structures. In this paper, we address a novel clique cover problem—WCEC—for reconstructing hypergraphs from weighted graph projections, and we propose an efficient and effective framework, HyperPLR, for generating real-world hypergraphs. The weighted graph projection shows strong representational power compared to existing inexact graph representations, while also significantly enhancing learning and generation against the exact representation. This is the first work to achieve both. Our method demonstrated superior efficiency and stability compared to previous state-of-the-art methods across various graph-based metrics. In the future, we aim to extend HyperPLR's capabilities to additional generation tasks, such as bipartite network generation or SAT instance generation, in order to capture the rich information inherent in these structures.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the National Science and Technology Major Project of China under Grant 2022ZD0116408. This work is also supported by the Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence (2023B1212010001). We would like to thank all the anonymous reviewers for their insightful comments. We also sincerely thank Chaolong Ying and Huizhi Zhu for their valuable discussions, which greatly contributed to the development of this work.
+
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+Zhishang Luo, Truong Son Hy, Puoya Tabaghi, Michael Defferrard, Elahe Rezaei, Ryan Carey, W. Rhett Davis, Rajeev Jain, and Yusu Wang. DE-HNN: an effective neural model for circuit netlist representation. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li (eds.), International Conference on Artificial Intelligence and Statistics, 2-4 May 2024, Palau de Congressos, Valencia, Spain, volume 238 of Proceedings of Machine Learning Research, pp. 4258-4266. PMLR, 2024. URL https://proceedings.mlr.press/v238/luo24a.html.  
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+
+# A WECP PROBLEM
+
+Problem 2 (Weighted Edge Clique Partition). Given a graph  $\mathcal{G}$  with weight function  $w_{e}:E(G)\to \mathbb{Z}^{+}$  and a positive integer  $k$ , it asks to select at most  $k$  cliques such that each edge appears in exactly as many cliques as its weight (if it exists, otherwise output NO).
+
+Note that although the WECP may appear similar to the WCEC problem, they have essential differences. In WECP, the objective is to partition the edges such that each edge is covered exactly as many times as indicated by its given weight. This means that if an edge has a weight greater than one, it must be covered by multiple cliques, and no edge can be covered more or fewer times than specified. This strict covering constraint makes the problem more restrictive. The solution either finds such a partition or determines that it is impossible, framing WECP as a Constraint Satisfaction Problem (CSP).
+
+In contrast, the WCEC problem aims to minimize the distance between the original weight matrix  $W$  of the graph and a derived weight matrix  $W'$ , which is formed by stacking the selected cliques. In this sense, it approximates or covers the edge weights using cliques. Unlike WECP, there is no strict requirement for how many times each edge must be covered; instead, the focus is on minimizing the difference between the original and derived weight distributions. Therefore, WCEC is essentially a combinatorial optimization problem.
+
+# B ADDITIONAL PROOF FOR THE SECOND PART OF LEMMA 2
+
+Proof. Consider a simple hypergraph  $\mathcal{H} = (V, \mathcal{E})$  where  $V = \{v_1, v_2, v_3, v_4\}$  and  $\mathcal{E} = \{(v_1, v_2, v_4), (v_2, v_3, v_4), (v_1, v_3)\}$ . The graph projection  $G$  of  $\mathcal{H}$  will be a complete graph. If  $G$  is used as the input for the minimum clique edge cover problem, the optimal solution will be a clique  $c = (v_1, v_2, v_3, v_4)$ . In the WCEC setting, let  $W$  be the weighted matrix of the weighted graph projection of hypergraph  $\mathcal{H}$ , and let  $W'$  be the weighted matrix derived from the solution of the MCEC. Then we have:
+
+$$
+W = \left[ \begin{array}{c c c c} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 2 \\ 1 & 1 & 0 & 1 \\ 1 & 2 & 1 & 0 \end{array} \right], \qquad W ^ {\prime} = \left[ \begin{array}{c c c c} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{array} \right].
+$$
+
+It is evident that  $d(W, W') = \sum_{i=1}^{n} \sum_{j=1}^{n} |W_{ij} - W_{ij}'| = 2$ , which is not the optimal solution for WCEC problem.
+
+# C EXPERIMENTAL DETAILS
+
+The code of our experiments is publicly available at https://github.com/LOGO-CUHKSZ/HyperPLR.
+
+# C.1 DATASETS
+
+Our experimental evaluation utilizes five real-world datasets from Benson et al. (2018):
+
+- contact-high-school: This dataset is constructed from interactions recorded by wearable sensors at a high school, consisting of 327 nodes, 172,035 timestamped hyperedges, and 7,818 unique hyperedges. The projected graph contains 5,818 edges and 6,151 maximum cliques.  
+- contact-primary-school: This dataset is constructed from interactions recorded by wearable sensors at a primary school, consisting of 242 nodes, 106,879 timestamped hyperedges, and 12,704 unique hyperedges. The projected graph contains 8,317 edges and 100,153 maximum cliques.  
+- email-Enron: In this dataset, nodes represent email addresses at Enron, and each hyperedge comprises the sender and all recipients of an email. The dataset contains 143 nodes,
+
+Table 3: Comparative result of advanced properties of the ground-truth hypergraphs and generation hypergraphs.  
+
+<table><tr><td>Diameter</td><td>contact-high-school</td><td>contact-primary-school</td><td>email-Enron</td><td>email-Eu</td><td>NDC-classes</td></tr><tr><td>HyperDK0,0</td><td>0.500</td><td>0.333</td><td>0.500</td><td>0.667</td><td>0.778</td></tr><tr><td>HyperDK1,1</td><td>0.250</td><td>0.333</td><td>0.250</td><td>0.333</td><td>0.533</td></tr><tr><td>Hyperlap</td><td>0.200</td><td>0.067</td><td>0.250</td><td>0.333</td><td>0.422</td></tr><tr><td>Hyperlap+</td><td>0.250</td><td>0.333</td><td>0.050</td><td>0.133</td><td>0.289</td></tr><tr><td>TheRA</td><td>0.250</td><td>0.000</td><td>0.250</td><td>0.667</td><td>0.222</td></tr><tr><td>HyperPLR</td><td>0.250</td><td>0.000</td><td>0.000</td><td>0.033</td><td>0.000</td></tr><tr><td>Triangles Num</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>HyperDK0,0</td><td>167.117</td><td>21.456</td><td>41.427</td><td>156.738</td><td>184.785</td></tr><tr><td>HyperDK1,1</td><td>84.440</td><td>15.988</td><td>10.867</td><td>41.227</td><td>79.077</td></tr><tr><td>Hyperlap</td><td>0.599</td><td>0.409</td><td>3.315</td><td>8.998</td><td>3.536</td></tr><tr><td>Hyperlap+</td><td>0.942</td><td>0.284</td><td>1.517</td><td>2.981</td><td>2.124</td></tr><tr><td>TheRA</td><td>0.468</td><td>0.198</td><td>2.451</td><td>3.725</td><td>1.638</td></tr><tr><td>HyperPLR</td><td>0.316</td><td>0.354</td><td>0.055</td><td>0.742</td><td>0.281</td></tr><tr><td>Degree Distribution</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>HyperDK0,0</td><td>23.026</td><td>23.026</td><td>23.026</td><td>23.026</td><td>23.026</td></tr><tr><td>HyperDK1,1</td><td>21.002</td><td>22.635</td><td>12.406</td><td>8.490</td><td>3.960</td></tr><tr><td>Hyperlap</td><td>5.958</td><td>8.072</td><td>9.294</td><td>6.946</td><td>4.415</td></tr><tr><td>Hyperlap+</td><td>5.336</td><td>8.812</td><td>9.023</td><td>6.448</td><td>4.474</td></tr><tr><td>TheRA</td><td>6.634</td><td>10.530</td><td>7.986</td><td>15.315</td><td>14.325</td></tr><tr><td>HyperPLR</td><td>5.065</td><td>7.657</td><td>4.639</td><td>7.424</td><td>4.020</td></tr><tr><td>Singular Value Distribution</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>HyperDK0,0</td><td>6.212</td><td>5.730</td><td>4.653</td><td>6.422</td><td>6.750</td></tr><tr><td>HyperDK1,1</td><td>5.462</td><td>5.147</td><td>4.513</td><td>6.345</td><td>7.008</td></tr><tr><td>Hyperlap</td><td>5.369</td><td>5.043</td><td>4.470</td><td>6.293</td><td>6.589</td></tr><tr><td>Hyperlap+</td><td>5.372</td><td>5.056</td><td>4.520</td><td>6.319</td><td>6.560</td></tr><tr><td>TheRA</td><td>5.377</td><td>5.054</td><td>4.465</td><td>6.339</td><td>6.620</td></tr><tr><td>HyperPLR</td><td>5.347</td><td>5.032</td><td>4.451</td><td>6.278</td><td>6.432</td></tr></table>
+
+10,883 timestamped hyperedges, and 1,512 unique hyperedges. The projected graph contains 1,800 edges and 10,883 maximum cliques.
+
+- email-Eu: This dataset includes email addresses at a European research institution, with hyperedges representing the sender and all recipients of an email with the same timestamp. The dataset consists of 998 nodes, 234,760 timestamped hyperedges, and 25,027 unique hyperedges. The projected graph contains 29,299 edges and 237,231 maximum cliques.  
+- NDC-classes: In this dataset, each hyperedge corresponds to a drug, and the nodes are the class labels assigned to the drugs. The dataset consists of 1,161 nodes, 49,724 timestamped hyperedges, and 1,088 unique hyperedges. The projected graph contains 6,222 edges and 624 maximum cliques.
+
+# C.2 ADVANCED PROPERTIES
+
+Inspired by the reviewer, we expand the evaluation metrics to some advanced properties like SHyRe in Wang & Kleinberg (2024). The result is presented in Table 3. Notably, the simplicial closure metric from SHyRe evaluates the temporal evolution process of dynamic hypergraphs, is not applicable to our work or the baseline static generation models. Therefore, we use a similar concept, triangle numbers, which is a widely used graph structural metric (Sansford et al., 2023). For density, we have already presented the results in Table 1, so it is not repeated here. Similar to the approach in SHyRe, we standardized the metrics for triangle numbers and diameter using the formula:  $\frac{|x_1 - x_2|}{x_1}$ , where  $x_1$  and  $x_2$  represent the values of the ground truth and the generated graphs, respectively. For degree distribution and singular value distribution, we compared their cross-entropy. A smaller value indicates better alignment for all the metrics. From the results, we observe that HyperPLR outperforms or is competitive with other models across most datasets and metrics.
+
+# C.3 EVALUATION METRICS
+
+We evaluate the training and generated hypergraphs using the following six metrics:
+
+- Density: While multiple definitions of hypergraph density exist in different contexts, we adopt a most straightforward definition as the ratio of the number of hyperedges to the number of nodes, given by Density = |E| / |V|.  
+- Average Hyperedge Size: The average hyperedge size is calculated as: Average Hyperedge Size =  $\frac{1}{|\mathcal{E}|}\sum_{e\in \mathcal{E}}|e|$ .  
+- Average Node Degree: The average node degree represents the average number of hyperedges in which each node participates, given by: Average Node Degree =  $\frac{1}{|V|}\sum_{v\in V}d(v)$ .  
+- Clustering Coefficient of Graph Projection: The clustering coefficient is computed for the 2-section (or graph projection) of the hypergraph, where each hyperedge induces a clique among its nodes. This coefficient measures the likelihood that two neighbors of a node are also neighbors. It is calculated as:  $C = \frac{1}{|V|} \sum_{v \in V} C(v)$ , where  $C(v)$  is the local clustering coefficient of node  $v$  in the projected graph.  
+- Modularity of Graph Projection: The modularity of the graph projection measures the quality of a partition of the graph into communities, with higher values indicating a stronger community structure. It is given by:  $Q = \frac{1}{2m}\sum_{i,j}\left[A_{ij} - \frac{k_ik_j}{2m}\right]\delta (c_i,c_j)$ , where  $A_{ij}$  represents the adjacency matrix,  $k_{i}$  and  $k_{j}$  are the node degrees,  $m$  is the total number of edges, and  $\delta (c_i,c_j)$  equals 1 if nodes  $i$  and  $j$  belong to the same community, and 0 otherwise.  
+- Modularity of Bipartite Graph Representation: The modularity of the bipartite graph representation measures the quality of a partition of the bipartite graph, considering nodes representing original vertices and hyperedges. The formula for modularity is similar to that used for the projected graph.
+
+# D IMPACT OF RANK VALUE IN CELL
+
+![](images/042e430b75aeeaec961b4fb922000c11db3ebaefa4e197e284710e86ed2b2ea2.jpg)  
+(a) rank  $= 8$
+
+![](images/2c833f1579748bde899058f55a842cad140dec11fa9114f1a46c8e2f4b3c8c57.jpg)  
+(b) rank  $= 10$  
+Figure 4: Visualization of generated weighted graph projection of Hypergraph email-Enron with different rank values. Zoom in for a better view.
+
+![](images/a22c97e7dfda903f66a3fdb2527d7f6e4aa8b652b45eb05153f8ff244f4df752.jpg)  
+(c) rank  $= 12$
+
+From fig. 4, we can conclude that, in terms of graph structure, there is almost no influence from rank value. However, rank greatly impacts the generation speed in the CELL module. High rank can accelerate the generation greatly but lose some diversity in the generated hypergraph, which is also discussed in Rendsburg et al. (2020).
+
+# E DIVERSITY OF THE GENERATION
+
+To the best of our knowledge, this is an essential yet unresolved problem in the graph (and hypergraph) generation task. Below, we make the best effort to provide a fair comparison and discussion of the diversity of hypergraphs generated by our model and HyperLap. Despite the lack of discussion on diversity in hypergraph generation directly, Velikonivtsev et al. (2024) proposed a theoretical framework for evaluating graph diversity. In summary, the paper defines the diversity of a set of graphs  $S$  as  $\text{Diversity}(S) = F(D(G, G') : G, G' \in S)$ , where  $D(G, G')$  is a distance measure between two graphs, and  $F$  is a function that computes the overall diversity from the set of pairwise
+
+distances. Building on this theory, we conducted the following experiments. First, we converted the hypergraphs generated by HyperLap, HyperLap+, and HyperPLR into bipartite graph representations for each dataset<sup>1</sup>. In this representation, one set of vertices corresponds to the hyperedges, and the other set corresponds to the vertices connected by these hyperedges. This conversion is lossless with respect to the original hypergraph structure.
+
+Given the large size of the graphs, classical graph distance measures like graph edit distance are NP-hard and computationally prohibitive. Instead, we followed the suggestions of Tsitsulin et al. (2018) and utilized two spectral-based distance measures introduced in: NetLSD-heat and NetLSD-wave. For the function  $F$ , we adopted the sum of pairwise distances between elements, as it is the most natural and widely used approach. The Table 4 presents the results of our experiments. From the results, we can see that in most cases, HyperPLR achieves graph diversity that either surpasses or is competitive to that of random generators like HyperLAP and HyperLap+. At the same time, we believe that the concept of diversity in the hypergraph generation task requires a more targeted definition and evaluation method.
+
+Table 4: NetLSD-Heat&NetLSD-Wave results across various datasets.  
+
+<table><tr><td>NetLSD-Heat</td><td>contact-high-school</td><td>contact-primary-school</td><td>email-Enron</td><td>email-Eu</td><td>NDC-classes</td></tr><tr><td>HyperLap</td><td>0.006</td><td>0.002</td><td>0.011</td><td>0.015</td><td>0.094</td></tr><tr><td>HyperLap+</td><td>0.007</td><td>0.056</td><td>0.009</td><td>0.007</td><td>0.087</td></tr><tr><td>HyperPLR</td><td>0.008</td><td>0.006</td><td>0.017</td><td>0.022</td><td>0.354</td></tr><tr><td colspan="6">NetLSD-Wave</td></tr><tr><td>HyperLap</td><td>0.070</td><td>0.033</td><td>0.158</td><td>0.262</td><td>1.216</td></tr><tr><td>HyperLap+</td><td>0.086</td><td>0.975</td><td>0.097</td><td>0.127</td><td>0.562</td></tr><tr><td>HyperPLR</td><td>0.113</td><td>0.107</td><td>0.183</td><td>0.251</td><td>1.479</td></tr></table>

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