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+# LEARNING SHAPE-INDEPENDENT TRANSFORMATION VIA SPHERICAL REPRESENTATIONS FOR CATEGORY-LEVEL OBJECT POSE ESTIMATION
+
+Huan Ren $^{1,2}$  Wenfei Yang $^{1,2,3}$  Xiang Liu $^{4}$  Shifeng Zhang $^{5}$  Tianzhu Zhang $^{1,2*}$
+
+<sup>1</sup>University of Science and Technology of China
+
+$^{2}$ National Key Laboratory of Deep Space Exploration, Deep Space Exploration Laboratory
+
+$^{3}$ Jianghuai Advance Technology Center  $^{4}$ Dongguan University of Technology  $^{5}$ Sangfor Technologies rh.hr_666@mail.ustc.edu.cn {yangwf, tzzhang}@ustc.edu.cn
+
+# ABSTRACT
+
+Category-level object pose estimation aims to determine the pose and size of novel objects in specific categories. Existing correspondence-based approaches typically adopt point-based representations to establish the correspondences between primitive observed points and normalized object coordinates. However, due to the inherent shape-dependence of canonical coordinates, these methods suffer from semantic incoherence across diverse object shapes. To resolve this issue, we innovatively leverage the sphere as a shared proxy shape of objects to learn shape-independent transformation via spherical representations. Based on this insight, we introduce a novel architecture called SpherePose, which yields precise correspondence prediction through three core designs. Firstly, We endow the pointwise feature extraction with SO(3)-invariance, which facilitates robust mapping between camera coordinate space and object coordinate space regardless of rotation transformation. Secondly, the spherical attention mechanism is designed to propagate and integrate features among spherical anchors from a comprehensive perspective, thus mitigating the interference of noise and incomplete point cloud. Lastly, a hyperbolic correspondence loss function is designed to distinguish subtle distinctions, which can promote the precision of correspondence prediction. Experimental results on CAMERA25, REAL275 and HouseCat6D benchmarks demonstrate the superior performance of our method, verifying the effectiveness of spherical representations and architectural innovations.
+
+# 1 INTRODUCTION
+
+Object pose estimation, which involves predicting the 3D rotation  $\mathbf{R} \in \mathrm{SO}(3)$  and 3D translation  $t \in \mathbb{R}^3$  of observed objects, has received considerable attention from the research community due to its crucial applications in augmented reality (Marchand et al., 2015; Su et al., 2019), robotic manipulation (Liu et al., 2023a; Wen et al., 2022), and hand-object interaction (Lin et al., 2023c), etc. While many prior instance-level object pose estimation methods (Peng et al., 2019; Wang et al., 2019a; 2021a) have achieved promising performance, their dependence on CAD models restricts the generalization ability. To mitigate this problem, category-level object pose estimation has been introduced in Wang et al. (2019b), which aims to reason about the 6D pose  $\{\mathbf{R}, t\} \in \mathrm{SE}(3)$  and 3D size  $s \in \mathbb{R}^3$  of unseen objects in specific categories, without the need for their CAD models.
+
+Existing category-level methods can be mainly divided into two groups, i.e., direct regression-based methods (Chen et al., 2021; Di et al., 2022; Lin et al., 2023b) and correspondence-based methods (Wang et al., 2019b; Tian et al., 2020; Liu et al., 2023b; Lin et al., 2024). The former approaches seek to directly regress object pose in an end-to-end manner. Although conceptually simple, they struggle with the pose-sensitive feature learning due to the non-linearity of the entire pose search space SE(3) (Lin et al., 2022a; 2023b). In contrast, the latter methods focus on establishing the correspondence between camera coordinate space and the Normalized Object Coordinate Space
+
+![](images/a81a359ac97d80946d64b2e049e617ae60a963bbd6bac3164ceb77183b6873c4.jpg)  
+Figure 1: We propose a novel approach for category-level object pose estimation based on spherical representations. (a) The classical pipeline of correspondence-based methods, where representations denote the organization form of observation data, e.g., point-based representations in  $\mathbb{R}^3$ , which is straightforward and commonly adopted in previous methods. (b) The overview of comparison between our method and previous methods. Note that each XYZ position of representations and NOCS coordinates is visualized as an RGB tuple. Our method employs spherical representations to learn shape-independent transformation, yielding smaller NOCS angle errors compared to DPDN (Lin et al., 2022b), which adopts point-based representations and suffers from the shape-dependence.
+
+(NOCS) (Wang et al., 2019b), and then solve the pose through the Umeyama algorithm (Umeyama, 1991) or deep estimators (Lin et al., 2022b), as illustrated in Figure 1(a). We follow this paradigm, where the correspondences in the linear coordinate search space  $\mathbb{R}^3$  are easier to learn.
+
+Nevertheless, existing correspondence-based methods typically adopt point-based representations, where discrete points back-projected from observed depth maps serve as the representations of observation data. Subsequently, the NOCS coordinates of these points are derived by geometric normalization and alignment of object shapes, which are inherently shape-dependent. As a result, observed points with similar semantics across diverse object shapes are mapped to distinct NOCS coordinates, also known as semantic incoherence in Wan et al. (2023). For example, points on the camera lens are mapped to different coordinates in NOCS if the camera length is different. Since the correspondence prediction need to take into account both pose information and large shape variation, it is hard to learn and generalize, which results in large NOCS angle errors as demonstrated in Figure 1(b)(Top). To resolve the above issue, we propose to innovatively leverage the sphere as a shared proxy shape of objects to learn shape-independent transformation via spherical representations. Specifically, we first uniformly divide the sphere with the Hierarchical Equal Area iso-Latitude Pixelation (HEALPix) grids (Gorski et al., 2005), each of which is represented by its center anchor. Then observed points are projected onto the spherical grids, with point-wise features assigned to the corresponding spherical anchors, which serves as the new representations of observation. Since these spherical anchors are coherent across various object shapes, the spherical NOCS coordinates are shape-independent, and thus more precise correspondence prediction can be achieved by focusing only on the pose information, as shown in Figure 1(b)(Bottom).
+
+Based on the sphere representations, we introduce a novel architecture for category-level object pose estimation, termed SpherePose. Given a centralized and scale-normalized observation<sup>1</sup>, point-wise features are first extracted and then assigned to spherical anchors with HEALPix spherical projection, yielding the spherical representations. The shape-independent transformation is subsequently
+
+learned between the spherical anchors and the spherical NOCS coordinates for rotation estimation. In order to boost the precision of correspondence prediction, we further present three core designs. The first design is the SO(3)-invariant point-wise feature extraction. It is intuitive that no matter how an object is rotated, the feature of a specific point on that object should remain invariant as it is mapped to a static coordinate in NOCS. Driven by this insight, for deep image features, we employ pretrained DINOv2 (Oquab et al., 2024) features with patch-wise consensus semantics, which are robust to rotations (Chen et al., 2024). As for deep point cloud features, we propose the ColorPointNet++ network by making minimal adjustments to PointNet++ (Qi et al., 2017), which derives input features from RGB values rather than absolute XYZ coordinates, inherently ensuring SO(3)-invariance by focusing on the color information and relative proximity of point cloud. The second design is the spherical feature interaction. Due to self-occlusion, depth cameras fail to perceive the occluded regions on the backside of objects, resulting in incomplete point cloud. Consequently, after projecting point-wise features onto the sphere, there exist numerous spherical anchors with no assigned features. To this end, we adopt the attention mechanism (Vaswani, 2017) with learnable position embedding to facilitate the interaction and propagation of features among anchors, where initially unassigned spherical features can be reasoned out. Moreover, by holistically integrating the relationship among spherical anchors, we acquire comprehensive spherical features which are robust to noise. The third design is a hyperbolic correspondence loss function. For the supervision of correspondences, we derive the gradients of several typical loss functions and realize that they exhibit minor gradients around zero, which is prone to yield ambiguous prediction. To achieve more precise prediction, we leverage the computation of correspondence distance in hyperbolic space (Lin et al., 2023a), which yields higher gradients near zero and can discern more subtle distinctions.
+
+To sum up, the main contributions of this paper are as follows:
+
+- We present an innovative approach that utilizes the sphere as a proxy shape of objects to learn shape-independent transformation via HEALPix spherical representations, which addresses the semantic inconsistency of shape-dependent canonical coordinates that plagues existing correspondence-based methods with point-based representations.  
+- We introduce an architecture for category-level object pose estimation that achieves precise correspondence prediction through three core designs: SO(3)-invariant point-wise feature extraction, spherical feature interaction, and a hyperbolic correspondence loss function.  
+- Extensive experiments on existing benchmarks demonstrate the superior performance of our method over state-of-the-art approaches, verifying the effectiveness of spherical representations and architectural innovations.
+
+# 2 RELATED WORK
+
+The task of category-level object pose estimation is first introduced in Wang et al. (2019b) to predict the 6D pose and size of unseen objects in a given category, and existing methods can be categorized into two groups, i.e., direct regression-based methods and correspondence-based methods.
+
+Direct Regression-based Methods. This category of approaches intends to directly regress the object pose in an end-to-end manner. FS-Net (Chen et al., 2021) proposes to decouple the rotation into two perpendicular vectors that simplifies the prediction, and utilizes a 3D graph convolution (3D-GC) autoencoder for feature extraction. GPV-Pose (Di et al., 2022) enhances the learning of pose-sensitive features with geometric consistency, and HS-Pose (Zheng et al., 2023) further extends 3D-GC to extract hybrid scope latent features that combine both global and local geometric information. VI-Net (Lin et al., 2023b) presents a novel rotation estimation network that decouples the rotation into a viewpoint rotation and an in-plane rotation, simplifying the task by leveraging spherical representations. Based on the decoupled rotation, SecondPose (Chen et al., 2024) extracts SE(3)-consistent semantic and geometric features to further enhance the pose estimation. However, these methods struggle with the pose-sensitive feature learning due to the non-linearity of SE(3).
+
+Correspondence-based Methods. This group of approaches aims to establish the correspondence between camera coordinate space and object coordinate space, and then acquire the pose through pose fitting algorithm, such as Umayama (Umayama, 1991). As a cornerstone, the Normalized Object Coordinate Space (NOCS) is introduced in Wang et al. (2019b), which serves as a shared canonical representation to align object instances within a given category. Later, SPD (Tian et al., 2020) proposes to handle the intra-class shape variation by reconstructing the 3D object model from
+
+a pre-learned categorical shape prior, and then match the observed point cloud to the reconstructed 3D model. Inspired by it, CR-Net (Wang et al., 2021b) and SGPA (Chen & Dou, 2021) are presented to improve the reconstruction quality. DPDN (Lin et al., 2022b) learns the shape prior deformation in the feature space, and replaces the traditional Umeyama algorithm with a deep estimator to directly regress the pose and size. Query6DoF (Wang et al., 2023) and IST-Net (Liu et al., 2023b) eliminate the dependence on explicit shape priors with implicit counterparts and implicit space transformation, respectively. More recently, AG-Pose (Lin et al., 2024) proposes to detect a set of sparse keypoints to represent the geometric structures of objects and establish robust keypoint-level correspondences for pose estimation. Although these methods have demonstrated strong performance, they all adopt point-based representations, where the NOCS coordinates are derived by geometric normalization and alignment of object shapes. When handling diverse object shapes, the NOCS coordinates become semantically incoherent, which are hard to learn and generalize. To tackle this issue, Semantically-aware Object Coordinate Space (SOCS) is presented in Wan et al. (2023), where semantically meaningful SOCS coordinates are built by keypoint-guided non-rigid object alignment to the categorical average shape. However, due to the non-rigid alignment process, there is a mismatch between the geometric structure of the observed point cloud and the SOCS coordinates. This discrepancy plagues the pose fitting, as the transformation may not be optimal for all parts of the object. In this work, we instead resort to spherical representations, where the sphere is leveraged as a shared proxy shape of objects to learn shape-independent transformation between spherical anchors and spherical NOCS coordinates. Thus precise correspondence prediction can be achieved without the need to account for large shape variation, thereby enhancing subsequent pose estimation.
+
+# 3 METHOD
+
+Target. Given an RGB-D image, we first acquire the instance segmentation masks with an offline Mask R-CNN (He et al., 2017), which then yield the cropped RGB image  $\pmb{I} \in \mathbb{R}^{H \times W \times 3}$  and segmented depth image for each instance. Partially observed point cloud  $P^O \in \mathbb{R}^{N \times 3}$  is derived by back-projecting and downsampling the segmented depth image, where  $N$  denotes the number of points. With  $I$  and  $P^O$  as inputs, our method predicts the rotation  $R \in \mathrm{SO}(3)$ , translation  $t \in \mathbb{R}^3$ , and size  $s \in \mathbb{R}^3$  of the observed instance.
+
+Overview. As illustrated in Figure 2, the proposed method mainly consists of two components. First, in the spherical representation projection phase (Section 3.1), point-wise features are extracted and then assigned to spherical anchors with HEALPix spherical projection, yielding the spherical representations. Subsequently, in the spherical rotation estimation phase (Section 3.2), spherical anchors perform feature interaction with each other through the attention mechanism, and then they are mapped to the spherical NOCS coordinates for rotation estimation. In Section 3.3, we elaborate on the training and testing workflow, including additional translation and size prediction for complete category-level object pose estimation. Finally, in Section 3.4, we emphasize the distinctions between our method and some relevant works on spherical representations and the proxy shape.
+
+# 3.1 SPHERICAL REPRESENTATION PROJECTION
+
+SO(3)-invariant Point-wise Feature Extraction. Taking image  $\pmb{I}$  and normalized points  $P^2$  as inputs, we start with point-wise feature extraction. Note that for point-wise image features, we first employ a 2D feature extractor to  $\pmb{I}$  and then leverage  $\pmb{P}$  to perform point-wise selection on the extracted features. To facilitate the transformation between camera coordinate space and object coordinate space, we recognize that the point-wise features  $\pmb{F}^{P} \in \mathbb{R}^{N \times C}$  should be SO(3)-invariant:
+
+$$
+\psi_ {\boldsymbol {G}} \left(\boldsymbol {F} ^ {P}\right) = \boldsymbol {F} ^ {P}, \quad \forall \boldsymbol {G} \in \mathrm {S O} (3), \tag {1}
+$$
+
+where  $\psi_G$  indicates the transformation of features when rotating points  $P$  by rotation  $G$ . Intuitively, the feature of a particular point on an object should remain invariant regardless of the object's rotation, given that the point is mapped to a fixed coordinate in NOCS. To this end, we synthesize the features of observation with SO(3)-invariance from a comprehensive perspective. Specifically, for image features, we employ RGB values with low-level texture information and the pretrained DINOv2 (Oquab et al., 2024) features with high-level semantic information. Owing to the pretraining on large-scale datasets, DINOv2 possesses robust generalization capabilities and can provide
+
+![](images/8e9c6fbc7ca41261da978f5f24c86011cfb34fb99ae29cff083d9e0a9684ca0a.jpg)  
+Figure 2: Overview of the proposed SpherePose. Given the observation  $\mathbf{I}$  and  $P$ , we first extract SO(3)-invariant point-wise features from four distinct perspectives and assign them to the spherical anchors  $\mathbf{A}$  with HEALPix spherical projection, yielding initial spherical features  $\mathbf{F}^{\mathbf{A}}$ . Then, the Transformer encoder module is employed for spherical feature interaction and integrates comprehensive spherical features  $\mathbf{F}$ . Finally, we predict the corresponding spherical NOCS coordinates  $\mathbf{O}$  via a NOCS predictor, which is applied to the estimation of rotation  $\mathbf{R}$ .
+
+semantically consistent patch-wise features that are robust to rotations (Chen et al., 2024). As for point cloud features, we adopt radius<sup>3</sup> values with low-level spatial information, along with features with high-level geometric information from the proposed ColorPointNet++, which is a variant of PointNet++ (Qi et al., 2017). The input features of the original PointNet++ are derived from absolute XYZ coordinates, which are inherently SO(3)-equivariant. To validate the importance of SO(3)-invariance for correspondence prediction, we make minimal adjustments to adapt the architecture for extracting SO(3)-invariant features, by replacing raw XYZ coordinates with RGB values. Given the SO(3)-invariance of color attributes and kNN operation in grouping and interpolating, our ColorPointNet++ yields SO(3)-invariant point-wise features, as detailed in Appendix A.1. The final point-wise features  $F^P \in \mathbb{R}^{N \times C}$  are the concatenation of all four point-wise features that capture distinct information from the observation, followed by feature dimension reduction to  $C$ .
+
+HEALPix Spherical Projection. Once point-wise features are extracted, previous methods adopt point-based representations to directly predict the NOCS coordinates corresponding to the observed points, which causes semantic incoherence across diverse object shapes, as introduced in Section 1. Instead, we propose to employ the sphere as a shared proxy shape of objects to learn shape-independent transformation via spherical representations. Specifically, we first uniformly divide the sphere with HEALPix grids (Gorski et al., 2005) as illustrated in Figure 4(b), each of which is equal in area and represented by its center anchor  $A_{m} \in \mathbb{R}^{3}, m \in \{1, \dots, M\}$ , where  $M$  denotes the number of grids. We then project the normalized points  $P$  onto the spherical grids, and assign point-wise features  $F^{P}$  to the corresponding spherical anchors, which serve as the new representations of observation with spherical features  $F^{A} \in \mathbb{R}^{M \times C}$ . In detail, consider a grid indexed by  $m$ , we follow VI-Net (Lin et al., 2023b) to search within this grid region for the point with the largest radius value, denoted as  $P_{n}$ , and assign its feature  $F_{n}^{P}$  to the anchor  $A_{m}$ . If there is no point projected onto the grid, we set  $F_{m}^{A} = 0$ . Since these spherical anchors are coherent across diverse objects, the corresponding spherical NOCS coordinates are shape-independent.
+
+# 3.2 SPHERICAL ROTATION ESTIMATION
+
+Spherical Feature Interaction. Due to self-occlusion, depth cameras fail to perceive the occluded regions on the backside of objects, causing incomplete point cloud. This phenomenon deteriorates when projected onto spherical grids, leaving almost half of the spherical anchors without assigned features. To this end, we employ a Transformer encoder with attention mechanism (Vaswani, 2017) to facilitate feature interaction and propagation among spherical anchors  $\mathbf{A} \in \mathbb{R}^{M \times 3}$  in a global perspective, yielding comprehensive spherical features  $\mathbf{F} \in \mathbb{R}^{M \times C}$  from the initial spherical features  $\mathbf{F}^A$  with a sequence of self-attention layers and MLP blocks, formulated as follows:
+
+$$
+\begin{array}{l} \boldsymbol {F} ^ {l + 1} = \operatorname {M L P} \left(\hat {\boldsymbol {F}} ^ {l + 1}\right) + \hat {\boldsymbol {F}} ^ {l + 1}, \\ \hat {\boldsymbol {F}} ^ {l + 1} = \operatorname {A t t e n t i o n} \left(\boldsymbol {F} ^ {l}\right) + \boldsymbol {F} ^ {l}, \end{array} \quad l \in \{0, \dots , L - 1 \}, \tag {2}
+$$
+
+where  $L$  is the number of layers and  $\pmb{F} = \pmb{F}^{L}$ ,  $\pmb{F}^{0} = \pmb{F}^{A}$ . Due to the disorder of spherical anchors, the learnable position embeddings  $\pmb{E}^{pos} \in \mathbb{R}^{M \times C}$  are assigned to corresponding anchors and contribute to the computation of attention for feature propagation. In formula, the self-attention in  $l$ -th layer is defined as:
+
+$$
+\operatorname {A t t e n t i o n} \left(\boldsymbol {F} ^ {l}\right) = \operatorname {S o f t m a x} \left(\frac {\left(\boldsymbol {Q} ^ {l} \boldsymbol {W} ^ {Q}\right) \left(\boldsymbol {K} ^ {l} \boldsymbol {W} ^ {K}\right) ^ {\top}}{\sqrt {C}}\right) \left(\boldsymbol {V} ^ {l} \boldsymbol {W} ^ {V}\right), \tag {3}
+$$
+
+$$
+\boldsymbol {Q} ^ {l} = \boldsymbol {F} ^ {l} + \boldsymbol {E} ^ {\text {p o s}}, \quad \boldsymbol {K} ^ {l} = \boldsymbol {F} ^ {l} + \boldsymbol {E} ^ {\text {p o s}}, \quad \boldsymbol {V} ^ {l} = \boldsymbol {F} ^ {l}, \tag {4}
+$$
+
+where  $W^{Q / K / V} \in \mathbb{R}^{C \times C}$  denotes learnable projection matrix for query, key and value, respectively, with independent parameters across layers. Through global feature interaction and propagation, those spherical features that are initially unassigned can be reasoned out. Moreover, by attending to the relationship among spherical anchors from a holistic perspective, we integrate pose-coherent spherical features which can alleviate the interference of noise.
+
+Correspondence-based Rotation Estimation. Given the spherical features  $\pmb{F}$ , we follow previous methods (Lin et al., 2022b; 2024) to first predict the spherical NOCS coordinates  $\pmb{O} \in \mathbb{R}^{M \times 3}$  corresponding to spherical anchors  $\pmb{A}$  with an MLP-based NOCS predictor. Since the spherical anchors and NOCS coordinates are on the unit sphere, we  $L^2$  normalize the output of MLP:
+
+$$
+\boldsymbol {O} _ {m} = \frac {\operatorname {M L P} \left(\boldsymbol {F} _ {m}\right)}{\| \operatorname {M L P} \left(\boldsymbol {F} _ {m}\right) \| _ {2}}, \quad m \in \{1, \dots , M \}. \tag {5}
+$$
+
+After acquiring the correspondences between spherical anchors and spherical NOCS coordinates, there are several approaches to acquire the rotation  $\mathbf{R} \in \mathrm{SO}(3)$ , such as the Umayama algorithm (Umayama, 1991) or deep estimators (Lin et al., 2022b). We adopt the Umayama algorithm with RANSAC (Fischler & Bolles, 1981) for outlier removal as it does not require extra training.
+
+# 3.3 CATEGORY-LEVEL OBJECT POSE ESTIMATION
+
+Training. For the estimation of translation  $t \in \mathbb{R}^3$  and size  $s \in \mathbb{R}^3$ , we follow VI-Net (Lin et al., 2023b) to employ a simple and lightweight PointNet++ (Qi et al., 2017). It takes the observed point cloud  $P^O$  and the point-wise selected RGB attributes from image  $I$  as input, and directly regresses  $t$  and  $s$  with  $L^1$  loss as follows:
+
+$$
+\mathcal {L} _ {\boldsymbol {t} \boldsymbol {s}} = \| \boldsymbol {t} - \boldsymbol {t} ^ {g t} \| _ {1} + \| \boldsymbol {s} - \boldsymbol {s} ^ {g t} \| _ {1}. \tag {6}
+$$
+
+As for the estimation of rotation  $\mathbf{R} \in \mathrm{SO}(3)$ , we adopt the correspondence-based paradigm to supervise the spherical NOCS coordinates  $\mathbf{O}$ . Specifically, we generate ground truth spherical NOCS coordinates  $\mathbf{O}^{gt}$  by mapping spherical anchors  $\mathbf{A}$  into NOCS using the ground truth rotation  $\mathbf{R}^{gt}$ :
+
+$$
+\boldsymbol {O} _ {m} ^ {g t} = \left(\boldsymbol {R} ^ {g t}\right) ^ {\top} \boldsymbol {A} _ {m}, \quad m \in \{1, \dots , M \}, \tag {7}
+$$
+
+and the prediction errors  $e$  can then be measured by the distance between them, defined as:
+
+$$
+\boldsymbol {e} _ {m} = \left\{ \begin{array}{l l} \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1}, & \text {a s} L ^ {1} \text {d i s t a n c e} \\ \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {2}, & \text {a s} L ^ {2} \text {d i s t a n c e} \end{array} , \quad m \in \{1, \dots , M \}. \right. \tag {8}
+$$
+
+Previous methods employ either  $e$  directly as the loss function or the smooth  $L^1$  loss (Wang et al., 2019b), which uses a squared term  $e^2$  if  $e$  fall below a certain threshold and the term  $e$  otherwise. However, we examine the gradients of these loss functions in Figure 3 and discover that they exhibit minor gradients around zero, which is prone to generate ambiguous prediction. To promote the precision of correspondence prediction, we leverage the computation of distance in hyperbolic space (Lin et al., 2023a), which yields higher gradients near zero and can discern more subtle distinctions. In detail, the  $\operatorname{arccosh}(1 + x)$  function is employed to the errors  $e$  with  $L^2$  distance, and the final hyperbolic correspondence loss function for  $O$  is formulated as follows:
+
+$$
+\mathcal {L} _ {\text {c o r r}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \operatorname {a r c o s h} \left(1 + \boldsymbol {e} _ {m}\right) = \frac {1}{M} \sum_ {m = 1} ^ {M} \operatorname {a r c o s h} \left(1 + \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {2}\right). \tag {9}
+$$
+
+Inference. Given the cropped RGB image  $\pmb{I}$  and point cloud  $P^O$ , we first predict the translation  $t$  and size  $s$  via PointNet++, and then normalize<sup>4</sup> the point cloud as  $P = (P^O - t) / \|s\|_2$ , which is fed into our SpherePose for the estimation of rotation  $R$ .
+
+![](images/528e4d07a705b169ed6cb67c0366c0254c3906fc6b44071b5c31b84d25bd04f6.jpg)  
+Figure 3: Illustration of several correspondence loss functions and their gradients.
+
+![](images/cf63c94fccb9a2932f4ff54ea64fb9b9f9ecc51b66ff3ffe0119fdc2a85d395e.jpg)  
+Figure 4: Overview of (a) equirectangular grids and (b) HEALPix grids.
+
+![](images/ad6e35db64d294b146c8c06f5754e40ade7d3ab965f26ef7c8acbba5c3c019db.jpg)
+
+# 3.4 DISCUSSION
+
+Spherical Representations. Although several previous methods (Lin et al., 2021; 2023b; Chen et al., 2024) have also adopted spherical representations for category-level object pose estimation, there are two fundamental distinctions. Firstly, they employ the equirectangular grids (Driscoll & Healy, 1994), which are evenly divided along latitude and longitude, as shown in Figure 4(a). However, this partition of sphere is non-uniform in area, with a higher sampling density near the poles, which is suboptimal for representing the point cloud data. This characteristic is particularly detrimental for objects with diverse poses, as non-uniform sampling of distinct semantic components results in a biased distribution of sampled data. Instead, we resort to the HEALPix grids (Gorski et al., 2005), which consistently yield the uniformly sampled spherical representations across various object poses, as illustrated in Figure 4(b). This uniform spherical partition is a more appropriate choice for the capture of point cloud structures, thereby enhancing the performance of pose estimation. Secondly, they intend to directly regress the pose in an end-to-end manner, which struggle with the pose-sensitive feature learning due to the non-linearity of  $\mathrm{SE}(3) / \mathrm{SO}(3)$ . DualPoseNet (Lin et al., 2021) employs spherical convolution in the spectral domain to learn  $\mathrm{SO}(3)$ -equivariant features. However, the Fourier transforms require tensors in the Fourier domain of  $\mathrm{SO}(3)$  that scale with the cube of the bandwidth, limiting the resolution. VI-Net (Lin et al., 2023b) utilizes spherical convolution in the spatial domain to learn viewpoint-equivariant features. Although it simplifies the regression process by decoupling the rotation into viewpoint rotation and in-plane rotation, the spatial convolution has a limited receptive field. Different from them, we follow with the correspondence-based paradigm to establish the correspondences between spherical anchors and spherical NOCS coordinates, which is easier to learn in the linear space  $\mathbb{R}^3$ . As for the spherical feature extraction, we adopt the attention mechanism to facilitate the feature interaction and propagation among spherical anchors from a holistic perspective, yielding the comprehensive spherical features.
+
+Proxy Shape. SAR-Net (Lin et al., 2022a) aims to deform the category-level proxy template point cloud to align with the observed point cloud for implicitly representing its rotational state. The object rotation is then solved from the categorical and deformed template point clouds by Umeyama algorithm. Although it learns shape-independent transformation by leveraging the category-shared template shape, the representations of the proxy shape are not well exploited to facilitate correspondence prediction. Specifically, the rotation information for guidance during the deformation process is derived from a single global feature vector of the observed point cloud, which leads to difficulties with rotation-sensitive feature learning. In contrast, we resort to the sphere as a shared proxy shape to learn shape-independent transformation via spherical representations. By projecting the observed point cloud onto the spherical grids, we preserve the intact information from the observation. The comprehensive spherical features are further extracted through the interaction and integration of features among spherical anchors from a holistic perspective, which can mitigate the interference of noise and promote the correspondence prediction. The proxy shape is also incorporated in MFOS (Lee et al., 2024) for image-aligned pose representation, which transforms the pose into a 2D image by rendering the 3D coordinates of a proxy shape (e.g., a cuboid or an ellipsoid) positioned according to the pose. Since MFOS operates with only 2D images as input, the proxy shape provides a mechanism for achieving 3D geometric perception, and the object pose is recovered by solving a PnP problem from the dense 2D-3D mapping. We differ by directly learning the 3D-3D transformation of the proxy sphere between the camera and object coordinate spaces.
+
+Table 1: Performance comparison with state-of-the-art methods on REAL275 dataset.  
+
+<table><tr><td></td><td>Method</td><td>Representation</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td rowspan="6">Direct Regression</td><td>GPV-Pose (Di et al., 2022)</td><td>Point-based</td><td>-</td><td>64.4</td><td>32.0</td><td>42.9</td><td>-</td><td>73.3</td></tr><tr><td>HS-Pose (Zheng et al., 2023)</td><td>Point-based</td><td>82.1</td><td>74.7</td><td>46.5</td><td>55.2</td><td>68.6</td><td>82.7</td></tr><tr><td>GenPose (Zhang et al., 2023)</td><td>Point-based</td><td>-</td><td>-</td><td>52.1</td><td>60.9</td><td>72.4</td><td>84.0</td></tr><tr><td>DualPoseNet (Lin et al., 2021)</td><td>Spherical</td><td>79.8</td><td>62.2</td><td>29.3</td><td>35.9</td><td>50.0</td><td>66.8</td></tr><tr><td>VI-Net (Lin et al., 2023b)</td><td>Spherical</td><td>-</td><td>-</td><td>50.0</td><td>57.6</td><td>70.8</td><td>82.1</td></tr><tr><td>SecondPose (Chen et al., 2024)</td><td>Spherical</td><td>-</td><td>-</td><td>56.2</td><td>63.6</td><td>74.7</td><td>86.0</td></tr><tr><td rowspan="8">Correspondence</td><td>NOCS (Wang et al., 2019b)</td><td>Point-based</td><td>78.0</td><td>30.1</td><td>7.2</td><td>10.0</td><td>13.8</td><td>25.2</td></tr><tr><td>SPD (Tian et al., 2020)</td><td>Point-based</td><td>77.3</td><td>53.2</td><td>19.3</td><td>21.4</td><td>43.2</td><td>54.1</td></tr><tr><td>SGPA (Chen &amp; Dou, 2021)</td><td>Point-based</td><td>80.1</td><td>61.9</td><td>35.9</td><td>39.6</td><td>61.3</td><td>70.7</td></tr><tr><td>DPDN (Lin et al., 2022b)</td><td>Point-based</td><td>83.4</td><td>76.0</td><td>46.0</td><td>50.7</td><td>70.4</td><td>78.4</td></tr><tr><td>IST-Net (Liu et al., 2023b)</td><td>Point-based</td><td>82.5</td><td>76.6</td><td>47.5</td><td>53.4</td><td>72.1</td><td>80.5</td></tr><tr><td>Query6DoF (Wang et al., 2023)</td><td>Point-based</td><td>82.5</td><td>76.1</td><td>49.0</td><td>58.9</td><td>68.7</td><td>83.0</td></tr><tr><td>AG-Pose (Lin et al., 2024)</td><td>Point-based</td><td>83.7</td><td>79.5</td><td>54.7</td><td>61.7</td><td>74.7</td><td>83.1</td></tr><tr><td>SpherePose</td><td>Spherical</td><td>84.0</td><td>79.0</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr></table>
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets. We evaluate our methods on three benchmarks including CAMERA25, REAL275 (Wang et al., 2019b) and HouseCat6D (Jung et al., 2024). CAMERA25 is a synthetic dataset that contains 275K training images and 25K testing images from 6 object categories, which are generated by rendering foreground objects with real-world backgrounds using the mixed-reality technique. REAL275 is a real-world dataset that consists of 4.3K training images of 7 scenes and 2.75K testing images of 6 scenes, which shares the same categories with CAMERA25. HouseCat6D is an emerging real-world dataset that comprises 20K training frames of 34 scenes, 3K testing frames of 5 scenes and 1.4K validation frames of 2 scenes from 10 household categories. This collection encompasses a diverse range of photometrically challenging objects, including glass and cutlery, which are captured in a comprehensive manner across various viewpoints and occlusions.
+
+Evaluation Metrics. Following previous works (Wang et al., 2019b; Chen et al., 2024), we report the mean Average Precision (mAP) of  $n^{\circ}m$  cm for 6D pose estimation, which denotes the percentage of prediction with rotation error less than  $n^{\circ}$  and translation error less than  $m$  cm. We also report the mAP of Intersection over Union  $(\mathrm{IoU}_x)$  for 3D bounding boxes with thresholds of  $x\%$ .
+
+Implementation Details. For a fair comparison, we employ the same instance segmentation masks as DPDN (Lin et al., 2022b) from Mask R-CNN (He et al., 2017). The number of sampled points is  $N = 2,048$ . For DINOv2 features, images are first cropped and resized to  $224 \times 224$  and then fed into the frozen DINOv2, followed by bilinear interpolation upsampling to the original resolution for point-wise selection. After extracting four point-wise features, we concatenate them at the feature dimension, followed by a linear layer activated by GeLU (Hendrycks & Gimpel, 2016) that reduces the dimension to  $C = 128$ . The resolution of HEALPix grids is expressed by the parameter  $N_{side}$ , and we set it to 8, which results in  $M = 768$  grids. The number of the Transformer encoder layers is set to  $L = 6$  by default. All experiments are conducted on a single RTX3090Ti GPU with a batch size of 64 for 200K iterations. More details can be found in Appendix A.2.
+
+# 4.2 COMPARISON WITH STATE-OF-THE-ART METHODS
+
+Results on REAL275 and CAMERA25 Datasets. Table 1 shows the comparison of our method with existing direct regression-based methods and correspondence-based methods on REAL275 dataset. It should be noted that our approach represents a pioneering application of spherical representations within the correspondence-based paradigm. From the results we can see that SpherePose outperforms the prior state-of-the-art methods on the precision of 6D pose estimation. Specifically, when compared with direct regression-based methods, SpherePose surpasses SecondPose (Chen et al., 2024) by  $2.0\%$  on  $5^{\circ}2\mathrm{cm}$  and  $2.2\%$  on  $10^{\circ}5\mathrm{cm}$ , which also employs spherical representations and DINOv2 (Oquab et al., 2024) backbone. And when compared with correspondence-based methods, SpherePose outperforms AG-Pose (Lin et al., 2024) by  $3.5\%$  on  $5^{\circ}2\mathrm{cm}$  and  $5.1\%$  on  $10^{\circ}5\mathrm{cm}$ , which encounters the intra-class semantic incoherence arising from point-based representations. As
+
+Table 2: Performance comparison with state-of-the-art methods on HouseCat6D dataset.  
+
+<table><tr><td></td><td>Method</td><td>Representation</td><td>IoU25</td><td>IoU50</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td rowspan="4">Direct Regression</td><td>FS-Net (Chen et al., 2021)</td><td>Point-based</td><td>74.9</td><td>48.0</td><td>3.3</td><td>4.2</td><td>17.1</td><td>21.6</td></tr><tr><td>GPV-Pose (Di et al., 2022)</td><td>Point-based</td><td>74.9</td><td>50.7</td><td>3.5</td><td>4.6</td><td>17.8</td><td>22.7</td></tr><tr><td>VI-Net (Lin et al., 2023b)</td><td>Spherical</td><td>80.7</td><td>56.4</td><td>8.4</td><td>10.3</td><td>20.5</td><td>29.1</td></tr><tr><td>SecondPose (Chen et al., 2024)</td><td>Spherical</td><td>83.7</td><td>66.1</td><td>11.0</td><td>13.4</td><td>25.3</td><td>35.7</td></tr><tr><td rowspan="2">Correspondence</td><td>AG-Pose (Lin et al., 2024)</td><td>Point-based</td><td>81.8</td><td>62.5</td><td>11.5</td><td>12.0</td><td>32.7</td><td>35.8</td></tr><tr><td>SpherePose</td><td>Spherical</td><td>88.8</td><td>72.2</td><td>19.3</td><td>25.9</td><td>40.9</td><td>55.3</td></tr></table>
+
+for the precision of 3D bounding boxes, we achieve comparable performance. Similar results on CAMERA25 dataset can be found in Table 8, please refer to Appendix A.3 for the details.
+
+Results on HouseCat6D Dataset. Table 2 provides the quantitative results of existing methods on HouseCat6D dataset. As can be seen from the table, SpherePose achieves the best performance against state-of-the-art approaches by a large margin under all metrics. Specifically, SpherePose surpasses SecondPose (Chen et al., 2024) by  $6.1\%$  on  $\mathrm{IoU}_{50}$  and  $8.3\%$  on  $5^{\circ}2\mathrm{cm}$ , and outperforms AG-Pose (Lin et al., 2024) by  $10.3\%$  on  $\mathrm{IoU}_{50}$  and  $7.8\%$  on  $5^{\circ}2\mathrm{cm}$ . When compared under the  $10^{\circ}5\mathrm{cm}$  metric, our method surpasses them by almost  $20\%$  (19.6% and  $19.5\%$ , respectively). It is notable that this dataset includes objects with photometric complexities and extensive viewpoint distributions, which is extremely challenging. The significant improvements on this benchmark further demonstrate the effectiveness of the proposed method.
+
+Results of Correspondence Errors. To validate the effectiveness of the proposed spherical representations compared to point-based representations in correspondence prediction, we calculate the mean NOCS errors on the REAL275 dataset. As shown in Table 3, the NOCS errors of Sphere-Pos are lower than DPDN (Lin et al., 2022b) and AG-Pose (Lin et al., 2024) on both the angle and
+
+Table 3: Comparison of correspondence errors.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Mean NOCS Error</td></tr><tr><td>Angle (°)</td><td>Distance (cm)</td></tr><tr><td>DPDN (Lin et al., 2022b)</td><td>16.62</td><td>10.50</td></tr><tr><td>AG-Pose (Lin et al., 2024)</td><td>10.36</td><td>5.88</td></tr><tr><td>SpherePose</td><td>4.52</td><td>3.89</td></tr></table>
+
+Euclidean distance, which indicates that the proposed method achieves more accurate correspondence prediction by learning shape-independent transformation on the proxy sphere.
+
+# 4.3 ABLATION STUDIES
+
+To shed more light on the superiority of our method, we conduct comprehensive ablation studies on REAL275 dataset, as detailed below.
+
+# Efficacy of SO(3)-invariant Point-wise Feature Extraction.
+
+Table 4 illustrates the impact of different deep point cloud backbones on the performance of object pose estimation. The original PointNet++ (Qi et al., 2017)
+
+Table 4: Ablation studies on the deep point cloud backbone.  
+
+<table><tr><td>Point Cloud Backbone</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td><td>Parameters</td></tr><tr><td>None</td><td>55.9</td><td>64.6</td><td>74.3</td><td>86.7</td><td>0</td></tr><tr><td>PointNet++</td><td>56.3</td><td>64.7</td><td>73.2</td><td>84.4</td><td>1,391,104</td></tr><tr><td>ColorPointNet++</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td><td>1,389,664</td></tr></table>
+
+inherently lacks SO(3)-invariance due to the injection of SO(3)-equivariant absolute XYZ coordinates. To validate the importance of SO(3)-invariant features for correspondence-based methods, we introduce minimal adaptations to PointNet++ to derive our ColorPointNet++, which enables the extraction of inherently SO(3)-invariant point cloud features while maintaining the architecture and the number of parameters almost unchanged. As shown in the table, with comparable number of parameters, our ColorPointNet++ outperforms the original PointNet++ by  $1.9\%$  on  $5^{\circ}2\mathrm{cm}$  and  $3.8\%$  on  $10^{\circ}5\mathrm{cm}$ . Notably, PointNet++ performs even worse than no point cloud backbone at all, showing a  $2.3\%$  drop on  $10^{\circ}5\mathrm{cm}$ . These results demonstrate the effectiveness of SO(3)-invariant point-wise features for correspondence-based object pose estimation methods.
+
+Efficacy of Spherical Feature Interaction. Table 5 illustrates the impact of varying numbers  $L$  of the Transformer encoder layers. When the Transformer encoder is not employed, there is a significant drop in performance to  $20.7\%$  on  $5^{\circ}2\mathrm{cm}$ , while when two layers of encoder are applied, the performance on  $5^{\circ}2\mathrm{cm}$  goes up to  $53.6\%$ . This result demonstrates the effectiveness of spherical
+
+Table 5: Impact of the number of encoder layers.  
+
+<table><tr><td>Encoder Layers</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>0</td><td>20.7</td><td>23.9</td><td>56.2</td><td>65.8</td></tr><tr><td>2</td><td>53.6</td><td>61.3</td><td>74.0</td><td>85.8</td></tr><tr><td>4</td><td>55.7</td><td>64.5</td><td>75.2</td><td>86.5</td></tr><tr><td>6</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>8</td><td>57.9</td><td>66.7</td><td>74.9</td><td>86.2</td></tr></table>
+
+Table 6: Impact of distinct loss functions.  
+
+<table><tr><td>Loss Function</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>L1</td><td>56.0</td><td>64.6</td><td>74.4</td><td>86.1</td></tr><tr><td>Smooth L1</td><td>54.6</td><td>63.2</td><td>72.6</td><td>83.3</td></tr><tr><td>Hyperbolic L1</td><td>58.0</td><td>66.5</td><td>73.5</td><td>85.0</td></tr><tr><td>L2</td><td>54.2</td><td>62.6</td><td>72.2</td><td>83.2</td></tr><tr><td>Hyperbolic L2</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr></table>
+
+feature interaction, which yields comprehensive spherical features from a holistic perspective. As the number of encoder layers increases, the precision of pose estimation is also improved, reaching a saturation at six layers. Further increasing the number of encoder layers does not lead to enhanced performance, but rather increases the inference overhead. Therefore, we set  $L = 6$  in our method.
+
+Efficacy of the Hyperbolic Correspondence Loss Function. Table 6 presents the impact of distinct loss functions on correspondence. As for the error with  $L^1$  distance in Equation 8, we conduct evaluations using  $L^1$  loss, smooth  $L^1$  loss and hyperbolic  $L^1$  loss, respectively. Since the smooth  $L^1$  loss has minor gradients near zero, it exhibits the worst performance of  $54.6\%$  on  $5^\circ 2\mathrm{cm}$ . Conversely, the hyperbolic  $L^1$  loss augments the gradients around zero to distinguish subtle distinctions, which improves the  $5^\circ 2\mathrm{cm}$  metric from  $56.0\%$  of the  $L^1$  loss to  $58.0\%$ . With regard to the error with  $L^2$  distance in Equation 8, we also derive the hyperbolic  $L^2$  loss, which results in an improvement on  $5^\circ 2\mathrm{cm}$  from  $54.2\%$  of the  $L^2$  loss to  $58.2\%$ . We assume that the  $L^2$  distance in Euclidean space is a better measure of the correspondence errors, so we adopt the hyperbolic  $L^2$  loss for correspondence prediction. For detailed formulas of these loss functions, please refer to Appendix A.2.
+
+Choice of Spherical Grids. Table 7 provides the performance comparison of adopting different spherical grids. To ensure a fair comparison, we use  $M = 768$  grids for both HEALPix (Gorski et al., 2005) and Super-Fibonacci Spirals (Alexa, 2022) grids. For Equirectangular (Driscoll
+
+& Healy, 1994) grids, the latitude and longitude resolution is set to  $28 \times 28$ , resulting in 784 grids, which is close to 768 in number. Since the HEALPix grids are area-uniform for the partition of sphere and can better capture the structure of point cloud, it achieves  $2.5\%$  and  $4.6\%$  higher performance on the  $5^{\circ}2\mathrm{cm}$  and  $10^{\circ}5\mathrm{cm}$  metrics, respectively, compared to the Equirectangular grids. In addition, we experiment with other nearly uniform spherical partitions, such as Super-Fibonacci Spirals, which also outperforms Equirectangular grids with a  $3.6\%$  gain on  $10^{\circ}5\mathrm{cm}$ , though still slightly below the performance of HEALPix.
+
+Table 7: Impact of the choice of spherical grids.  
+
+<table><tr><td>Spherical Grids</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>HEALPix</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>Equirectangular</td><td>55.7</td><td>64.7</td><td>72.1</td><td>83.6</td></tr><tr><td>Super-Fibonacci Spirals</td><td>56.5</td><td>65.0</td><td>75.8</td><td>87.2</td></tr></table>
+
+# 5 CONCLUSION
+
+In this work, we investigate the prevalent issue of intra-class semantic inconsistency of the canonical coordinates within existing correspondence-based approaches for category-level object pose estimation. We attribute this issue to shape-dependent point-based representations and propose to leverage the sphere as a shared proxy shape of objects to learn shape-independent transformation via spherical representations. In light of this insight, we introduce SpherePose, a novel architecture that achieves precise correspondence prediction through three core designs: SO(3)-invariant point-wise feature extraction for robust mapping between camera and object coordinate space, spherical feature interaction for holistic relationship integration, and a hyperbolic correspondence loss function for diagnosis of subtle correspondence errors. Experimental evaluations on existing benchmarks demonstrate the superiority of our method over state-of-the-art approaches.
+
+Limitations and Future Works. Although the employment of spherical representations equipped with spherical attention mechanism yields superior performance, the computational complexity of the attention mechanism, which scales quadratically with the number of tokens, limits the resolution of spherical grids. When the sphere is evenly partitioned into  $M = 768$  grids, each grid corresponds to an area of approximately 53.71 square degrees. For future work, we may explore linear attention mechanism (Wang et al., 2020) to facilitate spherical feature interaction. This could potentially enhance the resolution of spherical partitioning and improve the quality of spherical representations.
+
+# ACKNOWLEDGEMENT
+
+This work was partially supported by the National Nature Science Foundation of China (Grant 62306294, 62071122), Basic Strengthening Program Laboratory Fund (No.NKLDSE2023A009), Dreams Foundation of Jianghuai Advance Technology Center (NO.2023M743385), Youth Innovation Promotion Association of CAS.
+
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+Sida Peng, Yuan Liu, Qixing Huang, Xiaowei Zhou, and Hujun Bao. Pvnet: Pixel-wise voting network for 6dof pose estimation. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 4561-4570, 2019. 1  
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+
+# A APPENDIX
+
+# A.1 CLARIFICATIONS ON SO(3)-INVARIANCE
+
+In this section, we first provide a detailed explanation of SO(3)-invariance and equivariance. Subsequently, we elaborate on the SO(3)-invariance of DINOv2 and ColorPointNet++, respectively.
+
+SO(3)-Invariance/Equivariance. (1) Rotation estimation network should be SO(3)-equivariant. The rotation estimation task can be defined as  $\pmb{R} = \mathrm{Net}(\pmb{P})$ , where Net denotes the rotation estimation network. The SO(3)-equivariance between the input observed point cloud  $\pmb{P} \in \mathbb{R}^{N \times 3}$  and the output estimated rotation  $\pmb{R} \in \mathrm{SO}(3)$  is then formulated as:
+
+$$
+\operatorname {N e t} \left(\psi_ {\boldsymbol {G}} (\boldsymbol {P})\right) = \psi_ {\boldsymbol {G}} \left(\operatorname {N e t} (\boldsymbol {P})\right). \tag {10}
+$$
+
+(2) Correspondence predictor should ensure SO(3)-invariance. Correspondence-based methods can be expressed as  $\mathbf{R} = \operatorname{Net}(\mathbf{P}) = \Phi (\mathbf{P},\operatorname {Corr}(\mathbf{P}))$ , where  $\Phi$  is the rotation solver (e.g., the Umayama (Umayama, 1991) algorithm) and Corr is the correspondence (NOCS) predictor. Then, SO(3)-equivariance between the network input and output is formulated as:
+
+$$
+\Phi \left(\psi_ {\boldsymbol {G}} (\boldsymbol {P}), \operatorname {C o r r} \left(\psi_ {\boldsymbol {G}} (\boldsymbol {P})\right)\right) = \psi_ {\boldsymbol {G}} \left(\Phi (\boldsymbol {P}, \operatorname {C o r r} (\boldsymbol {P}))\right). \tag {11}
+$$
+
+Given fixed reference coordinates  $\mathrm{Corr}(\pmb {P})\in \mathbb{R}^{N\times 3}$ , the rotation solver  $\Phi$  is SO(3)-equivariant with respect to the other input  $\pmb{P}$ , which is written as:
+
+$$
+\Phi \left(\psi_ {G} (\boldsymbol {P}), \operatorname {C o r r} (\boldsymbol {P})\right) = \psi_ {G} \left(\Phi (\boldsymbol {P}, \operatorname {C o r r} (\boldsymbol {P}))\right). \tag {12}
+$$
+
+Consequently, the correspondence predictor Corr needs to be designed to ensure SO(3)-invariance:
+
+$$
+\operatorname {C o r r} \left(\psi_ {\boldsymbol {G}} (\boldsymbol {P})\right) = \operatorname {C o r r} (\boldsymbol {P}). \tag {13}
+$$
+
+It means that no matter how an object is rotated, a specific point on that object should be mapped to a static coordinate in NOCS.
+
+(3) Feature extractor in correspondence-based methods should be SO(3)-invariant. Since the correspondence predictor is typically defined as  $\mathrm{Corr}(\pmb {P}) = \mathrm{MLP}(f(\pmb {P}))$ , the point-wise feature extractor  $f(P)$  in correspondence-based approaches also needs to be SO(3)-invariant, formulated as:
+
+$$
+f \left(\psi_ {\mathbf {G}} (\mathbf {P})\right) = f (\mathbf {P}), \tag {14}
+$$
+
+which is consistent with Equation 1.
+
+SO(3)-invariance of DINOv2. To illustrate the SO(3)-invariance of DINOv2 (Oquab et al., 2024), Figure 5 presents a PCA visualization of DINOv2 features from the same instance under different rotations. The visualization shows that the DINOv2 features of the camera lens remain relatively consistent across various rotations, indicating their robustness to rotation variations. However, since DINOv2 does not strictly guarantee such semantic consistency, we conclude the features as approximately SO(3)-invariant, in line with the terminology stated in Second-Pose (Chen et al., 2024).
+
+![](images/e6755de8851afef5faed14279d9c43bd95028e0bddf3e6fc9b9564b35638b81b.jpg)  
+Figure 5: PCA visualization of DINOv2 features.
+
+SO(3)-invariance of ColorPointNet++. To further illustrate the SO(3)-invariance of the proposed ColorPointNet++ network, we show the pipeline comparison between ColorPointNet++ and PointNet++ (Qi et al., 2017) in Figure 6. Given the input points  $P \in \mathbb{R}^{N \times d}$ , where  $d$  denotes the coordinate dimension and is equal to 3 in Euclidean space, and associated point attributes  $F^{attr} \in \mathbb{R}^{N \times C_0}$  (e.g., RGB colors with  $C_0 = 3$ ), PointNet++ first concatenates them to yield the input features  $F^{in} \in \mathbb{R}^{N \times (d + C_0)}$ , and then aggregates the point-wise features  $F^{out} \in \mathbb{R}^{N \times C}$  by a hierarchical grouping and propagation strategy. It should be noted that in each set abstraction level and feature propagation level, PointNet++ concatenates the absolute XYZ coordinates onto the output features. The injection of SO(3)-equivariant XYZ coordinates on features results in the output features of PointNet++ lacking SO(3)-invariance. In contrast, the proposed ColorPointNet++ is structured to
+
+![](images/52105b38f4c038327a27a3decae7a609d9773e77f90cc32de8b1c9c6c88ff7af.jpg)  
+(a) PointNet++
+
+![](images/d2f0f0a2c304861d27d04c3526e9464482f9eccb12e7db1905d0a7f450b9f7ff.jpg)  
+(b) ColorPointNet++  
+Figure 6: Overview of comparison between the PointNet++ and our ColorPointNet++ network.
+
+achieve SO(3)-invariant feature extraction by avoiding the injection of absolute XYZ coordinates on the features. Specifically, XYZ coordinates are not used for input features and are not concatenated in each level, where the input features are derived exclusively from RGB values. Given that the k Nearest Neighbor (kNN) operation employed in the grouping and interpolation processes is SO(3)-invariant, the output features of our ColorPointNet++ network therefore exhibit SO(3)-invariance.
+
+# A.2 MORE IMPLEMENTATION DETAILS
+
+Training Details. For deep image feature extraction, we use the DINOv2 (Oquab et al., 2024) model with a ViT-S/14 (Dosovitskiy et al., 2020) architecture and feature dimension of 384. And for deep point cloud feature extraction, we implement the ColorPointNet++ network incorporating 4 set abstraction levels with multi-scale grouping, and the output feature dimension is 256. In terms of HEALPix grids (Gorski et al., 2005), the resolution is expressed by the parameter  $N_{side}$  and the number of grids is equal to  $N_{pix} = 12N_{side}^2$ , where we set  $N_{side} = 8$  and yield  $M = N_{pix} = 768$ . For data augmentation, we follow previous works (Lin et al., 2023b; 2024) to use perturbations with random translation  $\Delta t \sim U(-0.02, 0.02)$ , random scale  $\Delta s \sim U(0.8, 1.2)$ , and random rotational degree sampled from  $U(0, 20)$  for each axis. Regarding network optimization, we train SpherePose using the Adam (Kingma & Ba, 2015) optimizer for 200K iterations, with an initial learning rate of 0.001 and a cosine annealing schedule.
+
+Correspondence Loss Functions. The specific formulas corresponding to the loss functions in Table 6 within the ablation study section (Section 4.3) are provided below:
+
+$$
+\mathcal {L} _ {L ^ {1}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \| O _ {m} - O _ {m} ^ {g t} \| _ {1}, \tag {15}
+$$
+
+$$
+\mathcal {L} _ {\text {S m o o t h} L ^ {1}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \left\{ \begin{array}{l l} 5 \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1} ^ {2}, & \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1} \leq 0. 1 \\ \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1} - 0. 0 5, & \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1} > 0. 1 \end{array} , \right. \tag {16}
+$$
+
+$$
+\mathcal {L} _ {\text {H y p e r b o l i c} L ^ {1}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \operatorname {a r c o s h} \left(1 + \| O _ {m} - O _ {m} ^ {g t} \| _ {1}\right), \tag {17}
+$$
+
+$$
+\mathcal {L} _ {L ^ {2}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \| O _ {m} - O _ {m} ^ {g t} \| _ {2}, \tag {18}
+$$
+
+$$
+\mathcal {L} _ {\text {H y p e r b o l i c} L ^ {2}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \operatorname {a r c o s h} \left(1 + \| O _ {m} - O _ {m} ^ {g t} \| _ {2}\right), \tag {19}
+$$
+
+Table 8: Performance comparison with state-of-the-art methods on CAMERA25 dataset.  
+
+<table><tr><td></td><td>Method</td><td>Representation</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td rowspan="4">Direct Regression</td><td>GPV-Pose (Di et al., 2022)</td><td>Point-based</td><td>93.4</td><td>88.3</td><td>72.1</td><td>79.1</td><td>-</td><td>89.0</td></tr><tr><td>HS-Pose (Zheng et al., 2023)</td><td>Point-based</td><td>93.3</td><td>89.4</td><td>73.3</td><td>80.5</td><td>80.4</td><td>89.4</td></tr><tr><td>DualPoseNet (Lin et al., 2021)</td><td>Spherical</td><td>92.4</td><td>86.4</td><td>64.7</td><td>70.7</td><td>77.2</td><td>84.7</td></tr><tr><td>VI-Net (Lin et al., 2023b)</td><td>Spherical</td><td>-</td><td>-</td><td>74.1</td><td>81.4</td><td>79.3</td><td>87.3</td></tr><tr><td rowspan="7">Correspondence</td><td>NOCS (Wang et al., 2019b)</td><td>Point-based</td><td>83.9</td><td>69.5</td><td>32.3</td><td>40.9</td><td>48.2</td><td>64.4</td></tr><tr><td>SPD (Tian et al., 2020)</td><td>Point-based</td><td>93.2</td><td>83.1</td><td>54.3</td><td>59.0</td><td>73.3</td><td>81.5</td></tr><tr><td>SGPA (Chen &amp; Dou, 2021)</td><td>Point-based</td><td>93.2</td><td>88.1</td><td>70.7</td><td>74.5</td><td>82.7</td><td>88.4</td></tr><tr><td>IST-Net (Liu et al., 2023b)</td><td>Point-based</td><td>93.7</td><td>90.8</td><td>71.3</td><td>79.9</td><td>79.4</td><td>89.9</td></tr><tr><td>Query6DoF (Wang et al., 2023)</td><td>Point-based</td><td>91.9</td><td>88.1</td><td>78.0</td><td>83.1</td><td>83.9</td><td>90.0</td></tr><tr><td>AG-Pose (Lin et al., 2024)</td><td>Point-based</td><td>93.8</td><td>91.3</td><td>77.8</td><td>82.8</td><td>85.5</td><td>91.6</td></tr><tr><td>SpherePose</td><td>Spherical</td><td>94.8</td><td>92.4</td><td>78.3</td><td>84.3</td><td>84.8</td><td>92.3</td></tr></table>
+
+Table 9: IoU metric comparison with state-of-the-art methods on HouseCat6D dataset.  
+
+<table><tr><td></td><td colspan="2">Method</td><td>Representation</td><td>IoU25 / IoU50</td><td>Bottle</td><td>Box</td><td>Can</td><td>Cup</td><td>Remote</td><td>Teapot</td><td>Cutlery</td><td>Glass</td><td>Tube</td><td>Shoe</td></tr><tr><td rowspan="4">Direct Regression</td><td>FS-Net(Chen et al., 2021)</td><td>Point-based</td><td>74.9 / 48.0</td><td>65.3 / 45.0</td><td>31.7 / 1.2</td><td>98.3 / 73.8</td><td>96.4 / 68.1</td><td>65.6 / 46.8</td><td>69.9 / 59.8</td><td>71.0 / 51.6</td><td>99.4 / 32.4</td><td>79.7 / 46.0</td><td>71.4 / 55.4</td><td></td></tr><tr><td>GPV-Pose(Di et al., 2022)</td><td>Point-based</td><td>74.9 / 50.7</td><td>66.8 / 45.6</td><td>31.4 / 1.1</td><td>98.6 / 75.2</td><td>96.7 / 69.0</td><td>65.7 / 46.9</td><td>75.4 / 61.6</td><td>70.9 / 52.0</td><td>99.6 / 62.7</td><td>76.9 / 42.4</td><td>67.4 / 50.2</td><td></td></tr><tr><td>VI-Net(Lin et al., 2023b)</td><td>Spherical</td><td>80.7 / 56.4</td><td>90.6 / 79.6</td><td>44.8 / 12.7</td><td>99.0 / 67.0</td><td>96.7 / 72.1</td><td>54.9 / 17.1</td><td>52.6 / 47.3</td><td>89.2 / 76.4</td><td>99.1 / 93.7</td><td>94.9 / 36.0</td><td>85.2 / 62.4</td><td></td></tr><tr><td>SecondPose(Chen et al., 2024)</td><td>Spherical</td><td>83.7 / 66.1</td><td>94.5 / 79.8</td><td>54.5 / 23.7</td><td>98.5 / 93.2</td><td>99.8 / 82.9</td><td>53.6 / 35.4</td><td>81.0 / 71.0</td><td>93.5 / 74.4</td><td>99.3 / 92.5</td><td>75.6 / 35.6</td><td>86.9 / 73.0</td><td></td></tr><tr><td rowspan="3">Correspondence</td><td>NOCs(Wang et al., 2019b)</td><td>Point-based</td><td>50.0 / 21.2</td><td>41.9 / 5.0</td><td>43.3 / 6.5</td><td>81.9 / 62.4</td><td>68.8 / 2.0</td><td>81.8 / 59.8</td><td>24.3 / 0.1</td><td>14.7 / 6.0</td><td>95.4 / 49.6</td><td>21.0 / 4.6</td><td>26.4 / 16.5</td><td></td></tr><tr><td>AG-Pose(Lin et al., 2024)</td><td>Point-based</td><td>81.8 / 62.5</td><td>82.3 / 62.8</td><td>57.2 / 7.7</td><td>97.1 / 83.6</td><td>97.9 / 79.6</td><td>87.0 / 66.2</td><td>63.4 / 60.9</td><td>77.2 / 62.0</td><td>100.0 / 99.4</td><td>83.4 / 53.4</td><td>72.0 / 50.0</td><td></td></tr><tr><td>SpherePose</td><td>Spherical</td><td>88.8 / 72.2</td><td>98.6 / 87.4</td><td>58.9 / 5.6</td><td>97.4 / 87.2</td><td>100.0 / 97.7</td><td>79.0 / 63.5</td><td>94.8 / 87.3</td><td>85.5 / 74.5</td><td>99.6 / 98.0</td><td>74.5 / 30.0</td><td>99.8 / 90.8</td><td></td></tr></table>
+
+# A.3 ADDITIONAL EXPERIMENTAL RESULTS
+
+Results on CAMERA25 Dataset. In Table 8, we compare our method with the existing ones for category-level object pose estimation on CAMERA25 (Wang et al., 2019b) dataset. From the results we can see that SpherePose achieves the best performance. In detail, SpherePose outperforms the state-of-the-art correspondence-based method AG-Pose (Lin et al., 2024) by  $1.0\%$  on  $\mathrm{IoU}_{50}$ ,  $1.1\%$  on  $\mathrm{IoU}_{75}$ ,  $0.5\%$  on  $5^{\circ}2\mathrm{cm}$  and  $1.5\%$  on  $5^{\circ}5\mathrm{cm}$ , respectively.
+
+Results of IoU Metric on HouseCat6D Dataset. Table 9 presents the quantitative comparison of IoU metrics between our method and existing methods on HouseCat6D (Jung et al., 2024) dataset. Once again, our SpherePose attains the state-of-the-art performance, surpassing the second-best method SecondPose (Chen et al., 2024) by  $5.1\%$  on  $\mathrm{IoU}_{25}$  and  $6.1\%$  on  $\mathrm{IoU}_{50}$ .
+
+Ablation Studies on Point-wise Features. Table 10 presents the results of detailed ablation studies on point-wise features. For low-level image features, RGB values provide the low-level texture information, bringing a performance gain of  $1.4\%$  on  $5^{\circ}2\mathrm{cm}$ . For high-level image features, the frozen pretrained DINOv2 (Oquab et al., 2024) achieves a  $4.4\%$  higher performance on  $5^{\circ}2\mathrm{cm}$  compared to
+
+Table 10: Ablation studies on point-wise features.  
+
+<table><tr><td colspan="2">Point-wise Features</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td rowspan="2">Low-level Image</td><td>RGB</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>None</td><td>56.8</td><td>65.9</td><td>75.9</td><td>87.7</td></tr><tr><td rowspan="3">High-level Image</td><td>DINOv2</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>None</td><td>48.7</td><td>57.1</td><td>69.2</td><td>80.3</td></tr><tr><td>ResNet18</td><td>53.8</td><td>62.5</td><td>72.8</td><td>83.8</td></tr><tr><td rowspan="2">Low-level Point Cloud</td><td>Radius</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>None</td><td>52.3</td><td>58.9</td><td>72.3</td><td>83.6</td></tr><tr><td rowspan="4">High-level Point Cloud</td><td>ColorPointNet++</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>None</td><td>55.9</td><td>64.6</td><td>74.3</td><td>86.7</td></tr><tr><td>PointNet++</td><td>56.3</td><td>64.7</td><td>73.2</td><td>84.4</td></tr><tr><td>HP-PPF</td><td>55.1</td><td>64.2</td><td>74.2</td><td>85.9</td></tr></table>
+
+the end-to-end trained ResNet18 (He et al., 2016). Due to the large-scale pretraining, DINOv2 produces semantically consistent patch-wise features that are robust to rotations, which facilitates correspondence prediction. For low-level point cloud features, removing the radius information results in a  $5.9\%$  drop on  $5^{\circ}2\mathrm{cm}$ , which highlights the importance of spatial structure awareness. Since the proposed ColorPointNet++ eliminates the injection of raw XYZ coordinates, its features lack implicit encoding of the underlying spatial structure, making explicit SO(3)-invariant radius information essential. For high-level point cloud features, ColorPointNet++ outperforms the original PointNet++ (Qi et al., 2017) by  $3.8\%$  on  $10^{\circ}5\mathrm{cm}$ , emphasizing the significance of SO(3)-invariant features. Additionally, We have tried HP-PPF (Chen et al., 2024), which is also SO(3)-invariant, but the performance is suboptimal compared to not using it. This might stem from the fact that HP-PPF samples 300 points from 2048 observed points to estimate point-wise normal vectors, making it prone to noise due to the sparse sampling. Moreover, as a parameter-free approach, HP-PPF lacks the deep feature extraction capabilities compared to the learnable architecture of ColorPointNet++.
+
+Table 11: Per-category results of our Sphere- Pose on REAL275 dataset.  
+
+<table><tr><td>Category</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>57.6</td><td>51.3</td><td>71.7</td><td>76.8</td><td>81.4</td><td>88.7</td></tr><tr><td>bowl</td><td>100.0</td><td>100.0</td><td>87.7</td><td>96.9</td><td>90.8</td><td>100.0</td></tr><tr><td>camera</td><td>90.9</td><td>77.5</td><td>5.4</td><td>5.7</td><td>43.9</td><td>50.1</td></tr><tr><td>can</td><td>71.4</td><td>70.6</td><td>76.5</td><td>81.6</td><td>91.9</td><td>98.4</td></tr><tr><td>laptop</td><td>84.8</td><td>75.5</td><td>59.5</td><td>92.7</td><td>60.0</td><td>97.0</td></tr><tr><td>mug</td><td>99.6</td><td>99.3</td><td>48.5</td><td>50.3</td><td>89.2</td><td>94.9</td></tr><tr><td>average</td><td>84.0</td><td>79.0</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr></table>
+
+Table 13: Per-category results of our SpherePose on CAMERA25 dataset.  
+
+<table><tr><td>Category</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>93.8</td><td>90.6</td><td>78.0</td><td>93.3</td><td>79.5</td><td>96.4</td></tr><tr><td>bowl</td><td>97.0</td><td>96.7</td><td>95.1</td><td>95.6</td><td>98.6</td><td>99.2</td></tr><tr><td>camera</td><td>94.5</td><td>91.6</td><td>69.2</td><td>72.8</td><td>78.9</td><td>86.9</td></tr><tr><td>can</td><td>92.2</td><td>91.7</td><td>96.7</td><td>97.9</td><td>97.2</td><td>98.6</td></tr><tr><td>laptop</td><td>97.9</td><td>91.4</td><td>73.1</td><td>87.4</td><td>75.6</td><td>92.1</td></tr><tr><td>mug</td><td>93.6</td><td>92.3</td><td>57.6</td><td>58.9</td><td>78.7</td><td>80.4</td></tr><tr><td>average</td><td>94.8</td><td>92.4</td><td>78.3</td><td>84.3</td><td>84.8</td><td>92.3</td></tr></table>
+
+Table 12: Per-category results of AG-Pose (Lin et al., 2024) on REAL275 dataset.  
+
+<table><tr><td>Category</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>57.7</td><td>50.3</td><td>62.0</td><td>64.9</td><td>83.4</td><td>88.0</td></tr><tr><td>bowl</td><td>100.0</td><td>100.0</td><td>88.7</td><td>94.3</td><td>94.1</td><td>99.7</td></tr><tr><td>camera</td><td>90.8</td><td>82.9</td><td>1.2</td><td>1.3</td><td>24.8</td><td>27.3</td></tr><tr><td>can</td><td>71.3</td><td>71.2</td><td>83.4</td><td>85.3</td><td>96.3</td><td>98.6</td></tr><tr><td>laptop</td><td>83.3</td><td>74.1</td><td>59.6</td><td>91.1</td><td>61.2</td><td>95.6</td></tr><tr><td>mug</td><td>99.4</td><td>98.5</td><td>32.9</td><td>33.4</td><td>88.3</td><td>89.3</td></tr><tr><td>average</td><td>83.7</td><td>79.5</td><td>54.7</td><td>61.7</td><td>74.7</td><td>83.1</td></tr></table>
+
+Table 14: Per-category results of AG-Pose (Lin et al., 2024) on CAMERA25 dataset.  
+
+<table><tr><td>Category</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>93.7</td><td>91.4</td><td>80.9</td><td>96.4</td><td>82.3</td><td>99.0</td></tr><tr><td>bowl</td><td>96.9</td><td>96.7</td><td>98.7</td><td>99.0</td><td>99.7</td><td>99.8</td></tr><tr><td>camera</td><td>89.2</td><td>84.3</td><td>57.0</td><td>60.9</td><td>73.6</td><td>81.1</td></tr><tr><td>can</td><td>92.1</td><td>92.0</td><td>99.7</td><td>99.8</td><td>99.7</td><td>99.9</td></tr><tr><td>laptop</td><td>97.5</td><td>90.8</td><td>76.1</td><td>85.9</td><td>80.6</td><td>92.4</td></tr><tr><td>mug</td><td>93.6</td><td>92.7</td><td>54.5</td><td>54.6</td><td>77.1</td><td>77.3</td></tr><tr><td>average</td><td>93.8</td><td>91.3</td><td>77.8</td><td>82.8</td><td>85.5</td><td>91.6</td></tr></table>
+
+Per-category Results. The REAL275 and CAMERA25 datasets share the same categories, including bottle, bowl, camera, can, laptop and mug. In order to conduct an in-depth analysis of the relative merits of our method, we show the per-category and average results of our Sphere-Pose and AG-Pose (Lin et al., 2024) on REAL275 dataset in Table 11 and Table 12, and on CAMERA dataset in Table 13 and Table 14, respectively. It can be observed that SpherePose outperforms AG-Pose mainly in the camera and mug categories, which have large intra-class shape variations. This result demonstrates the effective
+
+ness of our approach to learn shape-independent transformation via spherical representations, which can resolve the issue of intra-class semantic incoherence arising from point-based representations. We also provide the per-category results on HouseCat6D dataset in Table 15, which contains 10 household categories: bottle, box, can, cup, remote, teapot, cutlery, glass, tube and shoe. Note that we deal with all categories by a single model as in Lin et al. (2022b; 2023b; 2024).
+
+Convergence Speed of the Hyperbolic Correspondence Loss Function. Figure 7 visualizes the training convergence speed and accuracy comparison between our hyperbolic correspondence loss function and the traditional loss function. As discussed in Section 3.3, the hyperbolic  $L^2$  loss function is designed to improve the gradient of the traditional  $L^2$  loss function around zero, facilitating more fine-grained correspondence prediction. As shown in the figure, the hyperbolic  $L^2$  loss not only accelerates convergence during training but also achieves higher accuracy, demonstrating its effectiveness in improving correspondence prediction precision.
+
+Table 15: Per-category results of our SpherePose on HouseCat6D dataset.  
+
+<table><tr><td>Category</td><td>IoU25</td><td>IoU50</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>98.6</td><td>87.4</td><td>51.0</td><td>55.7</td><td>67.9</td><td>79.1</td></tr><tr><td>box</td><td>58.9</td><td>5.6</td><td>0.0</td><td>0.0</td><td>0.2</td><td>0.2</td></tr><tr><td>can</td><td>97.4</td><td>87.2</td><td>33.9</td><td>41.4</td><td>63.0</td><td>79.2</td></tr><tr><td>cup</td><td>100.0</td><td>97.7</td><td>2.8</td><td>3.0</td><td>60.6</td><td>68.4</td></tr><tr><td>remote</td><td>79.0</td><td>63.5</td><td>15.7</td><td>16.3</td><td>46.9</td><td>50.1</td></tr><tr><td>teapot</td><td>94.8</td><td>87.3</td><td>25.2</td><td>44.4</td><td>53.6</td><td>86.8</td></tr><tr><td>cutlery</td><td>85.5</td><td>74.5</td><td>1.4</td><td>1.6</td><td>8.2</td><td>13.0</td></tr><tr><td>glass</td><td>99.6</td><td>98.0</td><td>54.8</td><td>66.3</td><td>78.5</td><td>96.4</td></tr><tr><td>tube</td><td>74.5</td><td>30.0</td><td>1.0</td><td>1.2</td><td>5.3</td><td>6.7</td></tr><tr><td>shoe</td><td>99.8</td><td>90.8</td><td>7.3</td><td>29.3</td><td>24.5</td><td>73.2</td></tr><tr><td>average</td><td>88.8</td><td>72.2</td><td>19.3</td><td>25.9</td><td>40.9</td><td>55.3</td></tr></table>
+
+![](images/b453457419103fc015e6cf689034e92279f894368b866845bbd463ea0d5768ed.jpg)  
+Figure 7: Training comparison between the hyperbolic  $L^2$  and traditional  $L^2$  loss functions.
+
+Inference Speed and Overhead. Table 16 shows the comparison of the pose estimation performance, total number of parameters, and inference speed on REAL275 dataset. As demonstrated by the results, the increase in the number  $L$  of encoder layers leads to enhanced performance, reaching saturation at six layers. A further increase in the number of layers does not result in further perfor
+
+Table 16: Comparison of different methods in terms of performance on REAL275, inference speed and overhead. All the experiments for inference speed are conducted on a single RTX3090Ti GPU.  
+
+<table><tr><td>Method</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td><td>Parameters (M)</td><td>Speed (FPS)</td></tr><tr><td>VI-Net (Lin et al., 2023b)</td><td>50.0</td><td>57.6</td><td>70.8</td><td>82.1</td><td>28.9</td><td>29.7</td></tr><tr><td>SecondPose (Chen et al., 2024)</td><td>56.2</td><td>63.6</td><td>74.7</td><td>86.0</td><td>60.2</td><td>14.3</td></tr><tr><td>DPDN (Lin et al., 2022b)</td><td>46.0</td><td>50.7</td><td>70.4</td><td>78.4</td><td>24.6</td><td>27.4</td></tr><tr><td>AG-Pose (Lin et al., 2024)</td><td>54.7</td><td>61.7</td><td>74.7</td><td>83.1</td><td>28.9</td><td>27.5</td></tr><tr><td>SpherePose with L = 2</td><td>53.6</td><td>61.3</td><td>74.0</td><td>85.8</td><td>25.7</td><td>26.4</td></tr><tr><td>SpherePose with L = 4</td><td>55.7</td><td>64.5</td><td>75.2</td><td>86.5</td><td>26.1</td><td>25.9</td></tr><tr><td>SpherePose with L = 6</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td><td>26.5</td><td>25.3</td></tr><tr><td>SpherePose with L = 8</td><td>57.9</td><td>66.7</td><td>74.9</td><td>86.2</td><td>26.9</td><td>24.5</td></tr></table>
+
+mance gains, but rather increases the number of parameters and slows down the inference. Therefore, we set  $L = 6$  in our method. Moreover, our approach demonstrates superior performance over existing methods, retaining a comparable number of parameters and inference speed. Specifically, in comparison to the leading correspondence-based method AG-Pose (Lin et al., 2024), our SpherePose achieves a  $3.5\%$  improvement on  $5^{\circ}2\mathrm{cm}$ , with 2.4 million fewer parameters and only a slight decrease in FPS. When compared to the direct regression-based methods, SpherePose outperforms the advanced SecondPose (Chen et al., 2024) by  $2\%$  on  $5^{\circ}2\mathrm{cm}$ , with substantially fewer parameters (26.5 million and 60.2 million, respectively) and higher FPS (25.3 and 14.3, respectively).
+
+# A.4 CONCURRENT RELATED WORKS
+
+In this section, we discuss the distinctions between our approach and two concurrent related works.
+
+Correspondence-based work. Mariotti et al. (2024) propose a semantic correspondence estimation approach, which deal with the challenges of object symmetries and repeated parts by incorporating a weak geometric spherical prior to supplement leading self-supervised features with 3D perception capabilities. They project image features onto a spherical map and use viewpoint information to guide correspondence prediction. While both this work and ours utilize spherical representations, the primary focus and application are different. Mariotti et al. (2024) aim to improve semantic correspondence estimation, which involves finding local regions that correspond to the same semantic entities across images. In contrast, our SpherePose focuses on correspondence-based category-level object pose estimation, which involves establishing correspondences between observed points and normalized object coordinates, and then fitting the object pose. Mariotti et al. (2024) use spherical representations to learn 3D perception-driven semantic features, while we leverage spherical representations to learn shape-independent transformation, which is beneficial for handling large intra-class shape variations.
+
+Regression-based work. Lee & Cho (2024) propose a 3D rotation regression approach that directly predicts Wigner-D coefficients in the frequency domain, aligning with the operations of spherical CNNs. They use an SO(3)-equivariant pose harmonics predictor to ensure consistent pose estimation under arbitrary rotations, overcoming limitations of spatial parameterizations such as discontinuities and singularities. While both this work and ours aim to improve 3D rotation estimation, the operating domains and the way to estimate rotation are different. Lee & Cho (2024) focus on predicting Wigner-D coefficients in the frequency domain to capture 3D rotations, which is useful for ensuring consistent pose estimation under different rotations. Instead, our method operates in the spatial domain, leveraging spherical representations to learn shape-independent transformation rather than directly predicting rotations through the network.
+
+# A.5 VISUALIZATION
+
+Qualitative Comparison. In Figure 8, we provide the qualitative comparison of existing correspondence-based methods, i.e., DPDN (Lin et al., 2022b), AG-Pose (Lin et al., 2024) and our SpherePose on REAL275 dataset. The visualization results indicate that previous methods adopting point-based representations do not perform well when dealing with novel objects with large shape variations, such as the length variation in camera lenses and the curvature distinctions in mug han
+
+![](images/a61df54455cf56804d559f641f356b3b59f917522079d21337b0f447f63ec549.jpg)  
+Figure 8: Qualitative comparison of DPDN (Lin et al., 2022b), AG-Pose (Lin et al., 2024) and our SpherePose on REAL275 dataset. Red/Green indicates the predicted/GT results.
+
+dles. By learning shape-independent transformation on the sphere, the proposed method allows for enhanced pose estimation.
+
+More Visualization. We provide more visualization results of pose estimation from our SpherePose, including successful cases in Figure 9 and failed cases in Figure 10. We visualize all six previously unseen scenes within the REAL275 testing dataset, with three images for each scene. It can be observed that failures predominantly occur due to the omission of objects in the detection process and significant viewpoint variations that result in incomplete capture of some objects.
+
+![](images/bbcfaf67b23c749d63969fe2c667a7a8dcc3f2d0e3fd3ae5676747566e5490be.jpg)  
+Figure 9: Visualization of successful cases from our SpherePose on the 6 testing scenes of REAL275 dataset. Red/Green indicates the predicted/GT results.
+
+![](images/d6445e1d9fbc4422d6322c87c631a3a9b5dbec774ca68a01e4dbdada8ef82a2d.jpg)  
+Scene 1
+
+![](images/7a2feadb44f5b4a2c5289a7e654505bbd2028b22d47430b4a1519e699ac93e70.jpg)
+
+![](images/ecc815909fea91e2e3aec06ed7046c56abc9f871a07d098b9123a488f8b23293.jpg)
+
+![](images/c635bb4afde35f2393d619fb34f876d9af8f151f3fc753c038a93b8c5c18cc08.jpg)  
+Scene 2
+
+![](images/bb4058314e8c2af9fce88a9de77125020505bed5b3b7024e96dc8ab7488ecb0d.jpg)
+
+![](images/02b943e6f44a3ddba024f994560a862e1e8d86e660874cf14c44875613ba3bf3.jpg)
+
+![](images/5e3ace0a0eb310c3fdb024532fb807ece7dc7ccd5153e6f1accf88e4f09b2dbd.jpg)  
+Scene 3
+
+![](images/68154da035ef213f1a33513e3e7f79377c4aab23c5d36969df3f4f3658facfbf.jpg)
+
+![](images/c811f13dbc688aa96ca837abae0879314ce583d1f175cb3492d328713e169613.jpg)
+
+![](images/782780fa7b21d41838946b8a4e331640a91122ff0b36e357d7487d7148a50fed.jpg)  
+Scene 4
+
+![](images/216ab41a25a39d2c82f0a245b4372e5e9c76b2c686c48a0d63db8f17c4093abc.jpg)
+
+![](images/96b16ae9662f47bd87f5e7bf345b2581ad0076cd4848d9955c6959709bdaa9ac.jpg)
+
+![](images/475649f323c1c35877582e06bf8d2d98012e3493f3ded7595b9cf7c9dcf10e6b.jpg)  
+Scene 5
+
+![](images/1cf5686879ed8e93bfbeea21415dbccac18db69c4aa7005a429c7943bd4e47c0.jpg)
+
+![](images/585b0f39561db8e71206ccae36284d4c900f602eb5656c2b7887feed2eb6e573.jpg)
+
+![](images/e241a82a861211a1ddbc76f62c0bf191ba0a6a2f52a97792663d055e6cdde607.jpg)  
+Scene 6
+
+![](images/1a496c7b95573f5aea58f020426daba628338de0683987635db80e15f1a7f3eb.jpg)  
+Figure 10: Visualization of failed cases from our SpherePose on the 6 testing scenes of REAL275 dataset. Red/Green indicates the predicted/GT results.
+
+![](images/7944ffcd8eda56e1d476775852239428f6997ead8bd4d6ddd4dccbcb0e290b8d.jpg)

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+# LEARNING SPATIAL-SEMANTIC FEATURES FOR ROBUST VIDEO OBJECT SEGMENTATION
+
+Xin Li $^{1,2,3\dagger}$ , Deshui Miao $^{2\dagger}$ , Zhenyu He $^{2,1*}$ , Yaowei Wang $^{2,1*}$ , Huchuan Lu $^{4}$ , and Ming-Hsuan Yang $^{5,6}$
+
+$^{1}$ Pengcheng Laboratory  $^{2}$ Harbin Institute of Technology, Shenzhen  $^{3}$  Pazhou Lab (Huangpu)
+
+$^{4}$ Dalian University of Technology  $^{5}$ University of California at Merced  $^{6}$ Yonsei University
+
+# ABSTRACT
+
+Tracking and segmenting multiple similar objects with distinct or complex parts in long-term videos is particularly challenging due to the ambiguity in identifying target components and the confusion caused by occlusion, background clutter, and changes in appearance or environment over time. In this paper, we propose a robust video object segmentation framework that learns spatial-semantic features and discriminative object queries to address the above issues. Specifically, we construct a spatial-semantic block comprising a semantic embedding component and a spatial dependency modeling part for associating global semantic features and local spatial features, providing a comprehensive target representation. In addition, we develop a masked cross-attention module to generate object queries that focus on the most discriminative parts of target objects during query propagation, alleviating noise accumulation to ensure effective long-term query propagation. Extensive experimental results show that the proposed method achieves state-of-the-art performance on benchmark data sets, including the DAVIS2017 test (87.8%), YoutubeVOS 2019 (88.1%), MOSE val (74.0%), and LVOS test (73.0%), and demonstrate the effectiveness and generalization capacity of our model. The source code and trained models are released at https://github.com/yahooo-m/S3.
+
+# 1 INTRODUCTION
+
+Video Object Segmentation (VOS) aims to track and segment target objects specified in the initial frame with mask annotations in a video sequence (Xu et al., 2018; Pont-Tuset et al., 2017; Voigtlaender et al., 2019; Yang et al., 2021). This practice holds significant promise in various applications, particularly as the prevalence of video content increases in domains such as autonomous driving, augmented reality, and interactive video editing (Park et al., 2022; Bao et al., 2018). The main challenges faced by VOS include drastic target appearance change, occlusion, and identity confusion caused by similar objects and background clutter, which becomes more difficult in long-term videos.
+
+Existing VOS methods (Oh et al., 2019; Cheng et al., 2021a) typically perform video object segmentation by comparing the test frame with past frames. Specifically, they first use an association model to generate the correlated features of the test sample and target templates. Then, the target masks are predicted based on the correlated features. The pixel-wise correlated features (Wu et al., 2023) contribute to accurate mask predictions. To account for targets that appear differently over time, some methods (Yang et al., 2021; Cheng & Schwing, 2022) utilize a memory module to store these changing appearances. In addition, several recent approaches (Wang et al., 2023; Cheng et al., 2023a) introduce object queries to help distinguish different target objects to alleviate identity confusion.
+
+Although significant advances have been made in video object segmentation, existing methods do not perform well in challenging scenarios. First, when dealing with target objects with multiple complex or separate parts caused by occlusion, background clutter, and shape complexity, existing methods often generate incomplete prediction masks. This is because the pixel-wise correlation adopted by existing methods mainly focuses on detailed spatial information at the pixel level and neglects the semantic information at the instance level. Second, although object query improves the ID association
+
+![](images/3e8b95e356df50f6da371f96c9731fafdfe190c6a820dd8e3ff290a992e4d639.jpg)  
+Figure 1: Performance comparison in terms of the  $\mathcal{I}\&\mathcal{F}$  score on various datasets. The proposed method achieves favorable performance against state-of-the-art methods, including AOT_L (Yang et al., 2021), XMem (Cheng & Schwing, 2022), and Cutie (Cheng et al., 2023a) on all the datasets. All methods are trained using the YoutubeVOS and DAVIS datasets.
+
+accuracy, it does not work well in sequences with dramatic target appearance changes. Existing methods (Wang et al., 2023; Cheng et al., 2023a) update target queries based on the entire predicted sample, which can introduce noise and errors and further lead to tracking failures when targets are missed or their identities are swapped.
+
+To address the aforementioned issues, we propose a VOS algorithm, called S3, to learn spatial-semantic features and generate discriminative object queries for robust VOS. Specifically, we design a spatial semantic block with a semantic embedding module and a spatial dependencies modeling part. This block efficiently utilizes the semantic information and local details from pre-trained ViTs for VOS, avoiding the need to train all ViT backbone parameters. In addition, we propose a discriminative query that emphasizes representative target parts, enabling more reliable target representation and query updates. This approach mitigates noise accumulation during query propagation and enhances robustness in long-term videos. We evaluate the proposed method extensively on five benchmarks, including DAVIS 2017 (Pont-Tuset et al., 2017), Youtube VOS2018 (Xu et al., 2018), Youtube VOS2019, LVOS (Hong et al., 2023), and MOSE (Ding et al., 2023). Extensive experimental results show that our approach achieves state-of-the-art performance on all these datasets including the DAVIS2017 test  $(89.1\%)$ , YoutubeVOS 2019  $(88.5\%)$ , MOSE  $(75.1\%)$ , and LVOS test  $(73.0\%)$  datasets. In addition, the results achieved by only using the DAVIS and Youtube datasets for training show that our method generalizes well to different scenarios, as shown in Figure 1.
+
+In this work, we make the following contributions:
+
+- We propose a spatial-semantic network block to incorporate semantic information with spatial information for Video Object Segmentation. The spatial-semantic block integrates global semantic information from the CLS token of a pre-trained ViT backbone into the base features of the input sample and then models spatial dependencies using a spatial dependency modeling module, enhancing robust VOS performance when handling target objects with complex or separate parts.
+
+- We develop a discriminative query mechanism to capture the most representative region of the target for better target representation learning and updating. This significantly improves VOS datasets, especially in long-term VOS datasets.  
+- We demonstrate that the proposed method achieves state-of-the-art performance on five diverse datasets and evaluate the contribution of each proposed component with comprehensive ablation studies.
+
+# 2 RELATED WORK
+
+Video Object Segmentation. Video Object Segmentation (VOS) aims to separate objects from the background and identify each target object across frames based on the target mask annotations given in the initial frame. Early VOS methods (Caelles et al., 2017; Xiao et al., 2018; Voigtlaender & Leibe, 2017; Hu et al., 2017; Luiten et al., 2018) use test-time learning to adapt pre-trained segmentation models to separate the specified targets online. The test-time learning of a deep model is time-consuming and significantly reduces the run speed. To avoid online learning, propagation-based methods (Duke et al., 2021; Cheng et al., 2018; Tsai et al., 2016) use offline shift attention to model temporal correlations between adjacent frames, propagating target masks from the previous frame to the current one. Despite promising performance, these methods suffer from error accumulation caused by inaccurate predictions. Matching-based methods (Chen et al., 2018; Hu et al., 2018; Bhat et al., 2020; Yang et al., 2020) enable fast and efficient inference by comparing the target template with the test image and predicting target masks based on feature matching. For example, OML (Chen et al., 2018) and VideoMatch (Hu et al., 2018) perform pixel matching between adjacent frames for mask prediction. Several methods (Oh et al., 2019; Yang et al., 2021; Yang & Yang, 2022; Cheng & Schwing, 2022) improve the matching-based VOS framework by introducing target memory to enrich target templates and developing advanced matching strategies to generate more comprehensive correlated features. For example, SimVOS (Wu et al., 2023) and JointFormer (Mao et al., 2021) use ViT blocks (Dosovitskiy et al., 2021) to model features and patch correspondence for object modeling jointly. For better target association across frames, some recent methods (Wang et al., 2023; Cheng et al., 2023a) introduce object queries to enrich target representation. Despite good performance on simple target objects, existing methods lack modeling of object-level semantic information, so they do not work well on dealing with objects with complex structures and long-term deformation. Unlike existing methods, the proposed approach performs pixel- and object-level feature learning through a novel object-aware backbone and a query-level correlation mechanism, enabling robust VOS performance, particularly in complex and long-term video scenarios.
+
+Query-Based VOS Frameworks. DETR (Carion et al., 2020) introduces the transformer network (Dosovitskiy et al., 2021) to object detection tasks by using queries to represent target regions. In the segmentation domain, some methods (Cheng et al., 2022; Cavagnero et al., 2024) introduce masked cross-attention modules to use queries to represent targets for segmentation tasks. Motivated by the above methods, ISVOS (Wang et al., 2023) uses additional queries generated by Mask2Former (Cheng et al., 2022) to represent instance information for target modeling, enhancing target representation in the matching-based VOS framework. To enable end-to-end training, Cutie (Cheng et al., 2023a) learns object-level features and enhances the interaction between object-level and pixel-level information, leading to improved segmentation accuracy and efficiency. However, query-based VOS methods suffer from query drift due to ineffective propagation. These methods model the entire target sample in the query, incorporating redundant information that diminishes the saliency of the target object. Instead, the proposed method generates discriminative queries for target representation and propagation, which preserves key distinguishing information of target objects and offers greater robustness to variations.
+
+Adapters for Dense Tasks. The ViT-Adapter (Chen et al., 2022) efficiently leverages pre-trained models by introducing a dense adapter for enhanced information exchange within ViT blocks, improving detection and segmentation performance. To enhance spatial hierarchical features, ViT-CoMer (Xia et al., 2024) introduces a multi-receptive field feature pyramid module to provide multi-scale spatial features. However, both methods require a complete fine-tuning step, which is inefficient, particularly in the video domain. Meanwhile, they focus on spatial information enhancement while lacking semantic feature interaction between ViT blocks and multi-scale features.
+
+To better adapt to VOS tasks, the proposed method improves adapter-based VOS frameworks in the following aspects. First, we introduce a spatial-semantic module to integrate the global semantic features from ViT to multi-scale features. This module does not need extra modules to transmit information to ViT blocks, which is more efficient. Second, the proposed spatial-semantic integration block associates semantic priors with dense local features. Third, the proposed method freezes the ViT blocks during training, eliminating the need for full fine-tuning.
+
+# 3 PROPOSED ALGORITHM
+
+Our method aims to learn comprehensive target features that contain semantic, spatial, and discriminative information for video object segmentation. This helps us cope with complex target appearance variations and ID confusion between target objects with similar appearances in long-term videos. To this end, we propose a spatial-semantic feature learning network that first embeds semantic information from a trained ViT network with multi-scale features from a trainable CNN network. Then, we model spatial dependencies upon the fused feature, enabling spatial-semantic features for VOS. In addition, to ensure that the target query retains discriminative information during long-term appearance changes, we develop a discriminative query propagation module to capture and update local representations of the target object.
+
+# 3.1 OVERALL FRAMEWORK
+
+Figure 2(a) shows the overall framework of the proposed method, consisting of two core modules: a spatial-semantic feature generation module and a pixel-and-query dual-level target association module. Given a test frame  $\mathbf{X}_t \in \mathbb{R}^{3 \times H \times W}$ , the proposed method predicts the mask of all specified objects within it based on the initial frame  $\mathbf{X}_{init} \in \mathbb{R}^{3 \times H \times W}$  and its corresponding annotation mask  $\mathbf{M}_{init} \in \mathbb{R}^{H \times W}$ . First, the feature generation module takes the test frame  $\mathbf{X}_t$  as input and generates spatial-semantic features. Then, the target association module associates the spatial-semantic features with the reference samples pixel-wise and query-wise to better represent the correlated features. Finally, a decoder predicts the mask based on these correlated features. In addition, we develop a dual-level memory module consisting of target features and target queries to better represent video targets that vary over frames. Both kinds of memory are updated every few frames using the online predicted masks, enabling adaptive target representations for effective target association.
+
+# 3.2 SPATIAL-SEMANTIC FEATURE LEARNING
+
+Figure 2(b) shows the spatial-semantic network, which consists of a semantic embedding module that incorporates semantic information from a pre-trained ViT model into multi-scale features and a spatial dependency modeling module that learns spatial dependencies based on semantic features.
+
+Semantic embedding. As the VOS task is designed for generic objects and no class labels are given, it is challenging to learn semantic representations directly from the VOS dataset during training. Fortunately, the CLS token in a trained ViT model aggregates semantic information from the entire image and provides a rich, global representation of the image content. Thus, we incorporate the semantic features generated from the CLS token with the multi-scale features generated from CNN networks to obtain detailed semantic features at different scales. To be specific, we first concatenate the CLS token  $\mathcal{F}_{cls}^{i}$  with a global token representation  $\mathcal{F}_g$ , which is generated by averagely pooling the features from a ViT block, to generate global semantic features denoted as  $(\mathcal{F}_{cls}^{i},\mathcal{F}_{g})$ . Then, we use a cross-attention operation to generate the semantic-aware features  $\mathcal{F}_{sem}^{i}$  by taking  $(\mathcal{F}_{cls}^{i},\mathcal{F}_{g})$  as the Key and Value, and the spatial-temporal features  $(\mathcal{F}_{spsem}^{i - 1})$  from the  $(i - 1)$ -th spatial-semantic block as the Query, which is formulated as:
+
+$$
+\mathcal {F} _ {s e m} ^ {i} = f _ {s o f t m a x} \left(\frac {\left(W ^ {q} \left(\mathcal {F} _ {c l s} ^ {i} , \mathcal {F} _ {g} ^ {i}\right)\right) \left(W ^ {k} \mathcal {F} _ {s p s e m} ^ {i - 1}\right) ^ {\top}}{\sqrt {d}}\right) \left(W ^ {v} \mathcal {F} _ {s p s e m} ^ {i - 1}\right), \tag {1}
+$$
+
+where  $f_{\text{softmax}}$  is the SoftMax operation, and  $W^q$ ,  $W^k$ , and  $W^v$  are the projection matrices for Query, Key, and Value.
+
+Spatial dependency modeling. We note that learning spatial dependencies between semantic features is important for the VOS model to understand the relationships between different object parts, which
+
+![](images/6d21f91883e1d886438c6aefa5ed727c46f34340577554c6ce5eeff98df26b30.jpg)  
+Figure 2: Architecture of the proposed method. (a) shows the overall framework of the proposed method comprising a feature generation part, a target association component, and a decoder-based prediction part. The feature generation part contains multiple spatial-semantic blocks, as illustrated in (b), which learn spatial-semantic features by integrating semantic priors and spatial details. (c) illustrates the discriminative query propagation process, which learns to generate discriminative queries that represent high-level object information.
+
+helps handle objects with complex structures or separate parts. Given the features of the input image patches generated by a ViT backbone model  $\mathcal{F}_{vit}^{i}$ , we adopt the Multi-scale Deformable Cross-Attention operation (Carion et al., 2020) to capture the spatial information  $\mathcal{F}_{spa}^{i}$  as:
+
+$$
+\mathcal {F} _ {s p a} ^ {i} = \sum_ {j} f _ {\alpha} \left(\mathcal {F} _ {v i t} ^ {i}, f _ {g s} \left(\mathcal {F} _ {s e m} ^ {i}, \Delta p\right)\right) \cdot f _ {g s} \left(\mathcal {F} _ {s e m} ^ {i}, \Delta p\right), \tag {2}
+$$
+
+where  $\Delta p$  indicates the deformable offsets generated based on the query features  $\mathcal{F}_{vit}^{i}, f_{gs}$  performs grid sampling on  $\mathcal{F}_{sem}^{i}$  using the offsets  $\Delta p, f_{\alpha}$  denotes the attention weights generated based on the query features  $\mathcal{F}_{vit}^{i}$  and the grid sampled features, and  $\sum_{j}$  is the summation over all sampled points. To enable an effective long-range propagation through the cascade blocks, we use the residual connection (He et al., 2016) followed by a convolution-feed-forward to generate the final spatial-semantic feature of the current block:
+
+$$
+\mathcal {F} _ {s p s e m} ^ {i} = \mathcal {F} _ {s p a} ^ {i} + f _ {C o n F F N} \left(\mathcal {F} _ {s p a} ^ {i} + \mathcal {F} _ {s p s e m} ^ {i - 1}\right). \tag {3}
+$$
+
+The spatial-semantic block leverages the strengths of both a pre-trained ViT network for providing semantic information and a multi-scale CNN network for rich spatial information, enabling comprehensive spatial-semantic feature learning for robust VOS.
+
+Figure 3(a) shows how the feature changes during the processing of the proposed model by visualizing the feature maps in different processing steps. It shows that the semantic embedding module enhances the semantic information (boundaries) of the objects in the feature maps. In contrast, after spatial dependence modeling, the feature maps pay more attention to the details of the objects. Figure 3(b) compares the features generated by different networks. Compared to ViT and ResNet models, the proposed Spatial-Semantic model exhibits better target representation capabilities with significantly higher activations in target object areas.
+
+# 3.3 DISCRIMINATIVE QUERY PROPAGATION
+
+To propagate target queries effectively across frames, we update the target queries with the most distinctive feature of the target object.
+
+![](images/c3f7761de18cfee19e57219a56d34e1af1756bf46fb8c273a8d344f9167e7cfa.jpg)  
+Figure 3: Visualized feature maps from different backbones and stages. It shows that the proposed spatial-semantic model generates effective features for target representation.
+
+Discriminative feature generation. We select the discriminative feature of a target object by comparing the target query  $\mathcal{Q} \in \mathbb{R}^{N \times C}$  with every channel activation in the correlated feature map of the target and taking the most similar one. Formally, given the feature map  $\mathcal{F} \in \mathbb{R}^{H \times W \times C}$ , the salient channel activation feature of a target is computed as:
+
+$$
+\mathcal {Q} _ {s} = \mathcal {F} [ \underset {i \in (1, H * W)} {\arg \max } (\mathcal {F Q} ^ {T} \odot \mathcal {M}) ], \tag {4}
+$$
+
+where  $\odot$  denotes the dot product,  $\mathcal{M}$  represents a binary mask of the target, and  $\mathcal{F}[i]$  denotes the  $i$ -th channel activation of the feature map  $\mathcal{F}$ . We use the strategy in Cutie (Cheng et al., 2023a) to generate the target mask  $\mathcal{M}$ , which uses the predicted segmentation results in the previous frame.
+
+Query propagation. Based on the discriminative target feature  $\mathcal{Q}_s$  generated from the target sample and the input target query  $\mathcal{Q}_{\mathrm{in}}$  at the current stage, the output query  $\mathcal{Q}_{\mathrm{out}}$  is propagated as:
+
+$$
+\mathcal {Q} _ {\text {o u t}} = \left(\alpha \odot \frac {\mathcal {A}}{\| \mathcal {A} \| _ {2}} + \mathcal {Q} _ {s}\right) \mathcal {W} _ {\text {o u t}} + \mathcal {Q} _ {\text {i n}}, \tag {5}
+$$
+
+where  $\alpha$  is a learnable parameter to scale the matrix,  $\mathcal{W}_{\mathrm{out}}$  is a learnable projection matrix, and  $\mathcal{A}$  denotes the correspondence between salient query and salient pixel features, which is computed as.
+
+$$
+\mathcal {A} = \mathcal {Q} \odot \mathcal {Q} _ {s}. \tag {6}
+$$
+
+The proposed discriminative query propagation scheme adaptively refines target queries with the most representative features, which helps deal with the challenges of dramatic appearance variations in long-term videos. We also present the similarity map between the discriminative query and the feature in the last two rows, which indicates that our proposed discriminative query can represent and distinguish different objects.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETTINGS
+
+Network. The full version of the proposed method uses a pre-trained ViT-base (Dosovitskiy et al., 2021) model as the ViT branch and constructs a simple CNN module for multi-scale feature generation.
+
+Table 1: Ablation study. All the variants are well aligned using the same settings and trained on the YoutubeVOS and DAVIS datasets. S3 denotes the proposed algorithm.  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td><td colspan="3">DAVIS 2017 test</td><td colspan="3">LVOS test</td><td colspan="3">LVOS val</td><td colspan="4">YouTube-VOS 2019 val</td><td></td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>J</td><td>F</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022) (Baseline)</td><td>53.3</td><td>62.0</td><td>57.6</td><td>81.0</td><td>77.4</td><td>84.5</td><td>50.0</td><td>45.5</td><td>54.4</td><td>49.5</td><td>45.2</td><td>53.7</td><td>85.5</td><td>84.3</td><td>88.6</td><td>80.3</td><td>88.6</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>64.0</td><td>60.0</td><td>67.9</td><td>84.2</td><td>80.6</td><td>87.7</td><td>56.2</td><td>51.8</td><td>60.5</td><td>60.1</td><td>55.9</td><td>64.2</td><td>86.1</td><td>85.8</td><td>90.5</td><td>80.0</td><td>88.0</td></tr><tr><td>+Discriminative Query</td><td>64.2</td><td>60.3</td><td>68.1</td><td>85.2</td><td>81.8</td><td>88.5</td><td>57.4</td><td>53.3</td><td>61.5</td><td>62.1</td><td>57.9</td><td>66.2</td><td>86.5</td><td>86.2</td><td>90.7</td><td>80.6</td><td>88.8</td></tr><tr><td>+ViT</td><td>64.2</td><td>60.2</td><td>68.3</td><td>85.6</td><td>82.0</td><td>89.2</td><td>58.3</td><td>53.7</td><td>62.8</td><td>58.9</td><td>54.1</td><td>63.6</td><td>86.7</td><td>86.4</td><td>90.3</td><td>81.0</td><td>88.7</td></tr><tr><td>+Spatial-semantic (S3)</td><td>68.5</td><td>64.5</td><td>72.6</td><td>86.7</td><td>82.7</td><td>90.8</td><td>66.5</td><td>62.1</td><td>70.8</td><td>67.7</td><td>63.1</td><td>72.2</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td></tr></table>
+
+The Spatial-Semantic block performs feature modeling after every block of the ViT branch. Upon the target association part, the Decoder predicts target masks based on multi-scale features. We use a soft aggregation operation for the scenario containing multiple target objects to merge the predicted masks.
+
+In all experiments, we set the number of Spatial-Semantic Blocks to  $N = 4$ .
+
+Training. We adopt similar training settings as those of Cutie (Cheng et al., 2023a). Note that, apart from ResNet pre-training, we do not pre-train other models on static images and directly trained them on video datasets (YoutubeVOS (Xu et al., 2018) and DAVIS (Pont-Tuset et al., 2017)). To enhance the performance of our model, we use the MEGA dataset constructed by Cutie, which includes the YouTubeVOS (Xu et al., 2018), DAVIS (Pont-Tuset et al., 2017), OVIS (Qi et al., 2022), MOSE (Ding et al., 2023) and BURST (Athar et al., 2023) datasets. We sample 8 frames to train the model and randomly select 3 of them to train the matching process. For each sequence, we randomly choose at most 3 targets for training. Point supervision in loss is adopted to reduce the memory requirements.
+
+For optimization, the AdamW (Kingma & Ba, 2014) optimizer is used with a learning rate of 5e-5 and a weight decay of 0.5. We train the model for 125K with a batch size of 16 on the video dataset, and 195k on the MEGA dataset. For specific parameter settings, please refer to our supplementary material. All our models are trained on a machine with 8 x NVIDIA V100 GPUs and tested using one NVIDIA V100 GPU.
+
+Inference. Our feature and query memories are updated every 5th frame during the testing phase. For longer sequences, we employ a long-term fusion strategy (Cheng & Schwing, 2022) for updating. We skip frames without targets. We use two kinds of input sizes, 480 and 600/720, for more detailed comparisons. In this section, we compare all methods with input size 480p.
+
+# 4.2 ABLATION STUDY
+
+Effect of every component. Table 1 provides a comprehensive analysis of the components of our method. The first two rows describe the current methods upon which we build improvements. Cutie improves performance by using a query transformer within the model.
+
+Baseline. We adopt XMem as the baseline model, a representative VOS framework with a well-designed memory mechanism.
+
++ Discriminative Query, which performs discriminative query generation by adaptively refining target queries with the most representative features of the query transformer. To validate the effectiveness of our proposed discriminative query propagation, we first choose ResNet50 as the backbone for feature generation (simply comparing query propagation with Cutie (Cheng et al., 2023a)). Performance gains compared to Cutie demonstrate the advantages of target modeling of discriminative queries for VOS, especially in long-term benchmarks.  
++ ViT, which directly utilizes ViTDet (Li et al., 2022b) to incorporate pre-trained model features at multiple scales. The performance indicates that directly employing ViT to generate multi-scale features is minimally beneficial for VOS tasks.  
++ Spatial-semantic, which applies the spatial-semantic blocks to integrate semantic information and multi-scale features. It is clear that the proposed spatial dependency modeling significantly boosts model performance in terms of  $\mathcal{J}\&\mathcal{F}$  on all VOS benchmarks. These improvements validate the benefits of the spatial-semantic network, which improves the semantic understanding of targets.
+
+Table 2: Effect of different settings in every component. The ablation study examines the impact of pre-trained weights and the number of queries on training results for the YoutubeVOS and DAVIS datasets. Additionally, it explores the effects of different adapters on the MEGA dataset.  
+
+<table><tr><td rowspan="2">Dataset
+Method</td><td colspan="3">MOSE-val</td><td colspan="3">DAVIS 2017 test</td><td colspan="3">LVOS test</td><td colspan="5">YouTube-VOS 2018 val</td><td colspan="4">YouTube-VOS 2019 val</td><td></td></tr><tr><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>Jn</td><td>Fn</td><td>Ju</td><td>Fu</td><td>G</td><td>Jn</td><td>Fn</td><td>Ju</td><td>Fu</td></tr><tr><td colspan="19">Effect of different pre-trained weights.</td><td></td></tr><tr><td>ResNet (He et al., 2016) (Baseline)</td><td>64.2</td><td>60.3</td><td>68.1</td><td>85.2</td><td>81.8</td><td>88.5</td><td>57.4</td><td>53.3</td><td>61.5</td><td>86.3</td><td>86.2</td><td>91.0</td><td>79.9</td><td>88.3</td><td>86.5</td><td>86.2</td><td>90.7</td><td>80.6</td><td>88.8</td></tr><tr><td>MAE (He et al., 2022)</td><td>67.0</td><td>62.6</td><td>71.3</td><td>85.3</td><td>81.3</td><td>89.3</td><td>64.4</td><td>60.2</td><td>68.7</td><td>86.4</td><td>85.6</td><td>90.2</td><td>80.6</td><td>89.0</td><td>86.4</td><td>85.4</td><td>89.7</td><td>81.2</td><td>89.3</td></tr><tr><td>DINov2 (Oquab et al., 2023)</td><td>67.5</td><td>63.5</td><td>71.6</td><td>86.1</td><td>82.7</td><td>89.6</td><td>63.2</td><td>58.7</td><td>67.6</td><td>87.3</td><td>86.6</td><td>91.7</td><td>80.9</td><td>89.9</td><td>86.7</td><td>86.2</td><td>91.0</td><td>80.6</td><td>89.1</td></tr><tr><td>ViT-small (Yang et al., 2024)</td><td>64.3</td><td>60.0</td><td>68.5</td><td>85.3</td><td>81.5</td><td>89.1</td><td>55.9</td><td>51.6</td><td>60.1</td><td>86.0</td><td>85.3</td><td>90.4</td><td>79.9</td><td>88.5</td><td>86.3</td><td>85.3</td><td>90.1</td><td>80.6</td><td>89.1</td></tr><tr><td>Depth_Anything</td><td>68.5</td><td>64.5</td><td>72.6</td><td>86.7</td><td>82.7</td><td>90.8</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.4</td><td>87.0</td><td>92.0</td><td>80.9</td><td>89.7</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td></tr><tr><td colspan="19">Effect of different query numbers.</td><td></td></tr><tr><td>32-query</td><td>68.3</td><td>64.3</td><td>72.3</td><td>85.8</td><td>81.7</td><td>89.9</td><td>64.8</td><td>60.4</td><td>69.6</td><td>86.8</td><td>86.9</td><td>92.0</td><td>79.8</td><td>88.6</td><td>86.9</td><td>86.8</td><td>91.6</td><td>80.3</td><td>88.8</td></tr><tr><td>16-query</td><td>67.7</td><td>63.6</td><td>71.8</td><td>86.6</td><td>82.7</td><td>90.2</td><td>66.4</td><td>62.0</td><td>70.9</td><td>86.9</td><td>86.7</td><td>91.8</td><td>80.1</td><td>88.9</td><td>87.0</td><td>86.6</td><td>91.6</td><td>80.6</td><td>89.3</td></tr><tr><td>8-query</td><td>68.5</td><td>64.5</td><td>72.6</td><td>86.7</td><td>82.7</td><td>90.8</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.4</td><td>87.0</td><td>92.0</td><td>80.9</td><td>89.7</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td></tr><tr><td colspan="19">Effect off different adapters.</td><td></td></tr><tr><td>Ours (Adapter (Chen et al., 2022))</td><td>72.2</td><td>67.8</td><td>76.4</td><td>87.2</td><td>84.1</td><td>90.4</td><td>71.1</td><td>66.7</td><td>75.6</td><td>87.1</td><td>86.1</td><td>91.0</td><td>81.6</td><td>89.8</td><td>87.1</td><td>85.9</td><td>90.9</td><td>81.4</td><td>90.1</td></tr><tr><td>Ours (CoMer (Xia et al., 2024))</td><td>73.2</td><td>69.1</td><td>77.3</td><td>87.0</td><td>83.2</td><td>90.8</td><td>73.4</td><td>68.8</td><td>77.9</td><td>87.3</td><td>87.1</td><td>92.2</td><td>80.7</td><td>89.4</td><td>87.4</td><td>87.0</td><td>91.7</td><td>81.3</td><td>89.6</td></tr><tr><td>Ours (Spatial-Semantic)</td><td>74.0</td><td>69.8</td><td>78.3</td><td>87.8</td><td>84.0</td><td>91.7</td><td>73.0</td><td>68.3</td><td>77.8</td><td>88.0</td><td>87.0</td><td>91.8</td><td>82.5</td><td>90.7</td><td>88.1</td><td>87.4</td><td>92.5</td><td>81.9</td><td>90.7</td></tr></table>
+
+Effect of different pre-trained weights. Table 2 also compares the performance of using different training parameters. We conduct experiments using three different pre-training parameters, and the results indicate that better pre-training parameters lead to more significant improvements in the model, especially those from Depth Anything (Yang et al., 2024) (8-query in Table 2). The Depth Anything model implicitly contains depth information about the scene, which contributes to its stronger representation capability, consistent with the findings of the Depth Anything paper.
+
+Effect of different query numbers. Table 2 shows that performance drops as the number of queries increases. This is because more queries introduce additional background noise, given the fewer targets in most VOS benchmarks, except MOSE (Ding et al., 2023).
+
+Effect of adapters. Table 2 presents a performance comparison between different adapters and our method. We replace the fusion module with various adapters and fully fine-tune them on the MEGA dataset, as indicated in related research that full fine-tuning achieves optimal performance (Chen et al., 2022; Xia et al., 2024). The experimental results demonstrate that our method is efficient in training and outperforms the other two methods in accuracy.
+
+# 4.3 COMPARISONS WITH STATE-OF-THE-ART METHODS
+
+DAVIS 2017 (Pont-Tuset et al., 2017) is a benchmark that offers densely annotated, high-quality, full-resolution videos with multiple objects of interest. Table 3 shows that our method achieves favorable performance  $(86.7\%)$  against state-of-the-art methods when trained on YoutubeVOS and DAVIS, without any pre-training. The gains are primarily due to the injection of semantic information, which helps handle sequences with object occlusion. When using the MEGA dataset as the training data, our method performs best on the DAVIS test set  $(87.8\%)$ .
+
+YouTubeVOS 2018 (Xu et al., 2018) contains 347 videos of 65 categories for training and videos for validation. For validation, there are 26 categories that the model has not seen in its training, allowing us to evaluate its generalization ability for 350 class-agnostic targets. Table 3 illustrates that our method performs better on seen and unseen categories than existing state-of-the-art models. The performance gain can be attributed to spatial-semantic feature representation and representative target correlation.
+
+YouTubeVOS 2019 is an extension of YouTube-VOS 2018, featuring a larger number of masked targets and more challenging sequences in its validation set. In Table 3, our approach exhibits competitive performance compared to the leading state-of-the-art methods. Remarkably, our model achieves the highest performance (87.5%) without relying on pre-training. This underscores the effectiveness of our method in leveraging inherent semantic-spatial information and discriminative query representation.
+
+MOSE (Ding et al., 2023) contains 2149 video clips and 5200 objects from 36 categories under more complex scenes. Compared to existing state-of-the-art methods, our model gains great improvement in this dataset. The benefits stem primarily from the discriminative query that captures more
+
+Table 3: Quantitative comparisons on the MOSE, LVOS, DAVIS 2017, YouTube-VOS 2018 & 2019 dataset. The best two results are shown in red and blue color. In the table, * denotes that the models are pre-trained using the static image datasets (Shi et al., 2016; Li et al., 2020; Zeng et al., 2019; Cheng et al., 2020; Wang et al., 2017). In the Appendix, we compare methods with external training data (e.g., large BL30K dataset for pre-training) and improved test size (600p/720p).  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td><td colspan="3">LVOS test</td><td colspan="3">DAVIS 2017 test</td><td colspan="5">YouTube-VOS 2018 val</td><td colspan="5">YouTube-VOS 2019 val</td><td rowspan="2">FPS</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td></tr><tr><td colspan="21">Trained only on the YouTube VOS, and DAVIS datasets</td></tr><tr><td>MiVOS (Cheng et al., 2021b)*</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>78.6</td><td>74.9</td><td>82.2</td><td>82.6</td><td>81.1</td><td>85.6</td><td>77.7</td><td>86.2</td><td>82.4</td><td>80.6</td><td>84.7</td><td>78.1</td><td>86.4</td><td>-</td></tr><tr><td>STCN (Cheng et al., 2021a)*</td><td>52.5</td><td>48.5</td><td>56.6</td><td>45.8</td><td>41.1</td><td>50.5</td><td>77.8</td><td>74.3</td><td>81.3</td><td>84.3</td><td>83.2</td><td>87.9</td><td>79.0</td><td>87.3</td><td>84.2</td><td>82.6</td><td>87.0</td><td>79.4</td><td>87.7</td><td>13.2</td></tr><tr><td>Swin-B-AOT-L (Yang et al., 2021)*</td><td>59.4</td><td>53.6</td><td>65.2</td><td>54.4</td><td>49.3</td><td>59.4</td><td>81.2</td><td>77.3</td><td>85.1</td><td>85.1</td><td>90.1</td><td>78.4</td><td>86.9</td><td>85.3</td><td>84.6</td><td>89.5</td><td>79.3</td><td>87.7</td><td>12.1</td><td></td></tr><tr><td>DeAOT-R50 (Yang &amp; Yang, 2022)</td><td>59.0</td><td>54.6</td><td>63.4</td><td>-</td><td>-</td><td>-</td><td>80.7</td><td>76.9</td><td>84.5</td><td>86.0</td><td>84.9</td><td>89.9</td><td>80.4</td><td>88.7</td><td>85.6</td><td>84.2</td><td>89.2</td><td>80.2</td><td>88.8</td><td>11.7</td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>79.8</td><td>76.3</td><td>83.4</td><td>84.3</td><td>83.9</td><td>88.8</td><td>77.7</td><td>86.7</td><td>84.2</td><td>83.8</td><td>88.3</td><td>78.1</td><td>86.7</td><td></td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022)*</td><td>53.3</td><td>62.0</td><td>57.6</td><td>50.0</td><td>45.5</td><td>54.4</td><td>81.0</td><td>77.4</td><td>84.5</td><td>85.7</td><td>84.6</td><td>89.3</td><td>80.2</td><td>88.7</td><td>85.5</td><td>84.3</td><td>88.6</td><td>80.3</td><td>88.6</td><td>22.6</td></tr><tr><td>ISVOS (Wang et al., 2023)*</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>82.8</td><td>79.3</td><td>86.2</td><td>86.3</td><td>85.5</td><td>90.2</td><td>80.5</td><td>88.8</td><td>86.1</td><td>85.2</td><td>89.7</td><td>80.7</td><td>88.9</td><td>5.8</td></tr><tr><td>SimVOS-B (Wu et al., 2023)</td><td>61.6</td><td>57.9</td><td>65.3</td><td>-</td><td>-</td><td>-</td><td>82.3</td><td>78.7</td><td>85.8</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>84.2</td><td>83.1</td><td>87.5</td><td>79.1</td><td>87.2</td><td>3.3</td></tr><tr><td>Cutie (Cheng et al., 2023a)*</td><td>64.0</td><td>60.0</td><td>67.9</td><td>56.2</td><td>51.8</td><td>60.5</td><td>84.2</td><td>80.6</td><td>87.7</td><td>86.1</td><td>85.5</td><td>90.0</td><td>80.6</td><td>88.3</td><td>86.1</td><td>85.8</td><td>90.5</td><td>80.0</td><td>88.0</td><td>36.4</td></tr><tr><td>JointFormer (Zhang et al., 2023)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>87.0</td><td>83.4</td><td>90.6</td><td>86.0</td><td>86.0</td><td>91.0</td><td>79.5</td><td>87.5</td><td>86.2</td><td>85.7</td><td>90.5</td><td>80.4</td><td>88.2</td><td>3.0</td></tr><tr><td>JointFormer (Zhang et al., 2023)*</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>87.6</td><td>84.2</td><td>91.1</td><td>87.0</td><td>86.2</td><td>91.0</td><td>81.4</td><td>89.3</td><td>87.0</td><td>86.1</td><td>90.6</td><td>82.0</td><td>89.5</td><td>3.0</td></tr><tr><td>S3 (Ours)</td><td>68.5</td><td>64.5</td><td>72.6</td><td>66.5</td><td>62.1</td><td>70.8</td><td>86.7</td><td>82.7</td><td>90.8</td><td>87.4</td><td>87.0</td><td>92.0</td><td>80.9</td><td>89.7</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td><td>13.1</td></tr><tr><td colspan="21">Trained on the MEGA dataset</td></tr><tr><td>DEVA (Cheng et al., 2023b)</td><td>66.5</td><td>62.3</td><td>70.8</td><td>54.0</td><td>49.0</td><td>59.0</td><td>83.2</td><td>79.6</td><td>86.8</td><td>86.2</td><td>85.4</td><td>89.9</td><td>80.5</td><td>89.1</td><td>85.8</td><td>84.8</td><td>89.2</td><td>80.3</td><td>88.8</td><td>25.3</td></tr><tr><td>Cutie (Cheng et al., 2023a)*</td><td>69.9</td><td>65.8</td><td>74.1</td><td>66.7</td><td>62.4</td><td>71.0</td><td>86.1</td><td>82.4</td><td>89.9</td><td>87.0</td><td>86.4</td><td>91.1</td><td>81.4</td><td>89.2</td><td>87.0</td><td>86.0</td><td>90.5</td><td>82.0</td><td>89.6</td><td>36.4</td></tr><tr><td>S3 (Ours)</td><td>74.0</td><td>69.8</td><td>78.3</td><td>73.0</td><td>68.3</td><td>77.8</td><td>87.8</td><td>84.0</td><td>91.7</td><td>88.0</td><td>87.0</td><td>91.8</td><td>82.5</td><td>90.7</td><td>88.1</td><td>87.4</td><td>92.5</td><td>81.9</td><td>90.7</td><td>13.1</td></tr></table>
+
+Table 4: Comparison with SAM 2 (Ravi et al., 2024). It shows that our method performs favorably against SAM 2.  
+
+<table><tr><td>Dataset</td><td colspan="3">LVOSV2-val</td><td colspan="3">LVOS-val</td><td colspan="3">DAVIS17-test</td><td colspan="3">MOSE-val</td><td colspan="3">YouTube-VOS 2019 val</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>J</td><td>F</td><td>G</td><td>J</td><td>F</td></tr><tr><td>SAM 2 (Hiera-B+)</td><td>75.8</td><td>72.0</td><td>79.6</td><td>74.9</td><td>70.2</td><td>79.6</td><td>88.3</td><td>85.0</td><td>91.5</td><td>75.8</td><td>71.8</td><td>79.9</td><td>88.4</td><td>85.2</td><td>91.6</td></tr><tr><td>SAM 2 (Hiera-L)</td><td>78.1</td><td>74.3</td><td>81.9</td><td>76.1</td><td>71.6</td><td>80.6</td><td>89.0</td><td>85.8</td><td>92.2</td><td>77.2</td><td>73.3</td><td>81.2</td><td>89.1</td><td>86.0</td><td>92.2</td></tr><tr><td rowspan="3">S3 (ViTb+)</td><td rowspan="3">(+7.9)</td><td rowspan="3">83.7</td><td rowspan="3">80.1</td><td rowspan="3">(+1.1)</td><td rowspan="3">76.0</td><td rowspan="3">71.2</td><td rowspan="3">80.8</td><td rowspan="3">(+1.5)</td><td rowspan="3">89.8</td><td rowspan="3">86.4</td><td rowspan="3">93.1</td><td rowspan="3">(+0.1)</td><td rowspan="3">75.9</td><td>71.8</td><td>79.8</td></tr><tr><td>(+0.3)</td><td>88.7</td></tr><tr><td>85.5</td><td>91.9</td></tr></table>
+
+representative target information. Additionally, the feature extraction module provides matching with more semantic and spatially informative features.
+
+LVOS (Hong et al., 2023) is a new long-term dataset to validate the robustness of VOS methods when faced with these challenges. It contains videos 20 times longer than existing VOS datasets, typically featuring a single target per video. In this benchmark, our method achieves a performance gain  $(66.5\%)$  compared to  $56.2\%$ . This improvement is attributed to the quality of memorized features and queries, which enhances the robustness of our approach in long-term video scenarios. Our method demonstrates enhanced stability and effectiveness in long-duration contexts by incorporating more discriminative queries that capture crucial target information into the object memory.
+
+Comparison with Segment Anything 2 (Ravi et al., 2024). Table 4 shows the detailed comparison results between our method and SAM2. Specifically, SAM2 used a large-scale mixed training dataset (comprising 1.  $3\%$  DAVIS, approximately  $9.4\%$  MOSE, approximately  $9.2\%$  YouTubeVOS, approximately  $15.5\%$  SA-1B, approximately  $49.5\%$  SA-V, and approximately 15.  $1\%$  Internal, approximately  $120.6\mathbf{k}$  videos). In contrast, our model is trained solely on VOS datasets (DAVIS, MOSE, YouTubeVOS, OVIS, and BURST, about  $11.4\mathbf{K}$  videos). To ensure a relatively fair comparison, we use the same model size (ViTb) and the same input image size (1024p). In all benchmarks evaluated, our method outperforms SAM2 in all metrics, especially on the challenging LVOSV2 val dataset  $(+7.9\%)$ . Even when SAM2 (Ravi et al., 2024) uses Hiera-L as a feature extractor, our model demonstrates comparable results, especially on LVOS V2 (Hong et al., 2023) val dataset ( $83.7\%$  vs  $78.1\%$ ).
+
+Visualized Results. We visualize some challenging sequences, including scenarios involving part-to-whole-changing and faint objects and those where targets closely resemble the background. In Figure 4(a), our approach shows favorable segmentation performance against the state-of-the-art methods. For example, our model accurately segments targets and provides detailed results in a video that includes gorillas, necks, and a riding man. This can be attributed to our proposed discriminative query propagation, which enables our model to capture more detailed target information by focusing on the most representative features, thereby facilitating precise segmentation in these challenging scenarios. Additionally, incorporating semantic features in our model ensures the model understands
+
+![](images/83bfa699d0b38587ec6a96214e2e1c8493f6563fbc753c6947bc092d86eb3b91.jpg)  
+(b) Qualitative comparisons with target appearance variance.  
+Figure 4: Visualized results on challenging scenarios. The proposed method performs well on these challenging sequences, while other methods suffer from semantic understanding and thus cannot segment the whole objects. (b) Our method handles the challenging sequence with fast-motion objects better than Cutie (Cheng et al., 2023a).
+
+the semantic prior of the interested targets, such as in the case of the sequence of man riding from YoutubeVOS 2019.
+
+Figure 4(b) shows the segmentation and tracking performance on long-term sequences with fast motion and dramatic appearance variations. In this sequence, two similar dogs continuously change their position and posture, making it challenging for the model to identify and track them accurately. Our VOS method performs accurately in tracking and segmenting targets, whereas Cutie fails to identify and segment the targets in this scenario. Good performance in scenarios like this is due to the semantic-aware feature and discriminative query representation learned by our model. More visualization and videos can be found in the Supplementary Material.
+
+# 4.4 LIMITATIONS
+
+Our method enhances target representation by introducing spatial-semantic features and discriminative query propagation, significantly improving numerous challenging scenarios. However, in scenes with object parts as the target (some sequences in the VOT challenge, shown in the appendix), the proposed model finds it difficult to represent a part of the target and often extends the predicted mask to the whole target. For example, when the target is the head of a person, our model may gradually shift to segmenting the entire body. This tendency arises because our method tends to interpret the target as the entire object seen during the training phase. A potential direction to improve this issue could be to let the model learn more part-aware semantic features, which also needs a new benchmark. Moreover, the improvement of the inference speed will also be addressed in our future work.
+
+# 5 CONCLUSION
+
+In this paper, we propose a spatial-semantic network to learn comprehensive representations to improve the robustness of video object segmentation in handling objects with complex structures or separate parts. In addition, we develop a discriminative query propagation module that selects representative features from target objects for query update, enabling more reliable query propagation in long-term videos. Our approach achieves favorable performance compared with state-of-the-art methods on numerous datasets. Both quantitative and qualitative results demonstrate the effectiveness of the proposed method in tracking and segmenting video objects in complex and long-term scenes.
+
+Acknowledgments This work is supported by the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2024A1515011292), the National Natural Science Foundation of China (Grant No.62476148, 62172126), and the Shenzhen Research Council (No. JCYJ20210324120202006).
+
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+Junke Wang, Dongdong Chen, Zuxuan Wu, Chong Luo, Chuanxin Tang, Xiyang Dai, Yucheng Zhao, Yujia Xie, Lu Yuan, and Yu-Gang Jiang. Look before you match: Instance understanding matters in video object segmentation. In CVPR, 2023. 1, 2, 3, 9, 15  
+Lijun Wang, Huchuan Lu, Yifan Wang, Mengyang Feng, Dong Wang, Baocai Yin, and Xiang Ruan. Learning to detect salient objects with image-level supervision. In CVPR, 2017. 5. 9  
+Qiangqiang Wu, Tianyu Yang, Wei Wu, and Antoni Chan. Scalable video object segmentation with simplified framework. In ICCV, 2023. 1, 3, 9, 14  
+Chunlong Xia, Xinliang Wang, Feng Lv, Xin Hao, and Yifeng Shi. Vit-comer: Vision transformer with convolutional multi-scale feature interaction for dense predictions. arXiv preprint arXiv:2403.07392, 2024. 3, 8  
+Huaxin Xiao, Jiashi Feng, Guosheng Lin, Yu Liu, and Maojun Zhang. Monet: Deep motion exploitation for video object segmentation. In CVPR, pp. 1140-1148, 2018. 3  
+Ning Xu, Linjie Yang, Yuchen Fan, Dingcheng Yue, Yuchen Liang, Jianchao Yang, and Thomas Huang. Youtube-vos: A large-scale video object segmentation benchmark. In ECCV, 2018. 1, 2, 7, 8  
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+
+# A APPENDIX
+
+The appendix is organized as follows:
+
+1. We provide a detailed implementation of training and inference.  
+2. We provide more experiments with different training settings and test hyperparameters.  
+3. More visualization and failure cases are shown.
+
+# A.1 IMPLEMENT DETAILS.
+
+# A.1.1 MULTI-OBJECT PROCESSING
+
+In this section, we provide additional implementation details for completeness. Our training and inference code will be released to ensure reproducibility.
+
+Following the previous work (Cheng & Schwing, 2022; Cheng et al., 2021a), we extend our methods to the multi-object setting. The predicted mask of each frame undergoes channel separation for different targets. The separated masks for each target are then concatenated with the RGB image and the aggregated mask of other targets for feature extraction. The extracted features are fused with RGB features to obtain pixel-level value features, which are then stored in memory. It is important to note that the pixel-level features stored in memory include the key (RGB feature) and the value.
+
+# A.1.2 TRAINING DETAILS
+
+As mentioned in the main paper, we train our network only on video-level datasets, which differs from other methods (Cheng et al., 2023a; Cheng & Schwing, 2022; Yang & Yang, 2022). The backbone weights are initialized from MAE pre-training, similar to the approaches used in SimVOS (Wu et al., 2023) and JointFormer (Mao et al., 2021). We perform spatial-semantic fusion every three ViT layers and five discriminative query transformer blocks for query propagation. We implement our network using PyTorch and train the model on 8 NVIDIA V100s, employing automatic mixed precision (AMP) during training.
+
+Two different training datasets are used for better improvements. The "DAVIS&YTVOS" settings combine the training sets of YouTubeVOS-2019 and DAVIS-2017, with the DAVIS-2017 dataset expanded five times to increase the data ratio. The "MEGA" set contains four different training sets, including DAVIS-2017, YouTubeVOS-2019, MOSE, and BURST datasets, with the DAVIS-2017 dataset expanded five times to increase the data ratio. To sample a training sequence, we randomly select a "seed" frame from all the frames and then randomly choose 7 other frames from the same video. We re-sample if any two consecutive frames have a temporal distance greater than the max-skip. Following the previous work (Cheng & Schwing, 2022; Cheng et al., 2023a), the max-skip is set to [5, 10, 15, 5] after  $[0\%, 10\%, 30\%, 80\%]$  of the training iterations, respectively. To avoid sampling frames without targets, we use the data statistics provided by Cutie (Cheng et al., 2023a).
+
+Table 5: Training parameters.  
+
+<table><tr><td>Config</td><td>DAVIS YTVOS</td><td>MEGA</td></tr><tr><td>optimizer</td><td>AdamW</td><td>AdamW</td></tr><tr><td>base learning rate</td><td>5e-5</td><td>5e-5</td></tr><tr><td>weight decay</td><td>0.05</td><td>0.05</td></tr><tr><td>droppath rate</td><td>0.15</td><td>0.15</td></tr><tr><td>batch size</td><td>16</td><td>16</td></tr><tr><td>num ref frames</td><td>3</td><td>3</td></tr><tr><td>num frames</td><td>8</td><td>8</td></tr><tr><td>max-skip</td><td>[5, 10, 15, 5]</td><td>[5, 10, 15, 5]</td></tr><tr><td>max-skip-itr</td><td>[0.1,0.3,0.8,1]</td><td>[0.1, 0.3, 0.8, 1]</td></tr><tr><td>Iterations</td><td>150,000</td><td>190,000</td></tr><tr><td>learning rate schedule</td><td>steplr</td><td>steplr</td></tr></table>
+
+Table 6: Quantitative comparisons with high-resolution input on the MOSE, LVOS, DAVIS 2017, YouTube-VOS 2018 & 2019 dataset. + means improve the input size.  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td><td colspan="3">DAVIS 2017 test</td><td colspan="6">YouTube-VOS 2018 val</td><td colspan="5">YouTube-VOS 2019 val</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td><td></td></tr><tr><td>Cutie-base (Cheng et al., 2023a)*</td><td>64.0</td><td>60.0</td><td>67.9</td><td>84.2</td><td>80.6</td><td>87.7</td><td>86.1</td><td>85.8</td><td>90.5</td><td>80.0</td><td>88.0</td><td>86.1</td><td>85.5</td><td>90.0</td><td>80.6</td><td>88.3</td><td></td></tr><tr><td>ISVOS (Wang et al., 2023)*+BL30K (Cheng et al., 2021b)</td><td>-</td><td>-</td><td>-</td><td>84.0</td><td>80.1</td><td>87.8</td><td>86.7</td><td>86.1</td><td>90.8</td><td>81.0</td><td>89.0</td><td>86.3</td><td>85.2</td><td>89.7</td><td>81.0</td><td>89.1</td><td></td></tr><tr><td>JointFormer (Zhang et al., 2023)*+BL30K (Cheng et al., 2021b)</td><td>-</td><td>-</td><td>-</td><td>88.1</td><td>84.7</td><td>91.6</td><td>87.6</td><td>86.4</td><td>91.0</td><td>82.2</td><td>90.7</td><td>87.4</td><td>86.5</td><td>90.9</td><td>82.0</td><td>90.3</td><td></td></tr><tr><td>Ours</td><td>68.5</td><td>64.5</td><td>72.6</td><td>87.1</td><td>83.1</td><td>91.1</td><td>87.4</td><td>87.0</td><td>92.0</td><td>80.9</td><td>89.7</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td><td></td></tr><tr><td>Cutie-base (Cheng et al., 2023a)*+</td><td>66.2</td><td>62.3</td><td>70.1</td><td>85.9</td><td>82.6</td><td>89.2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>86.9</td><td>86.2</td><td>90.7</td><td>81.6</td><td>89.2</td><td></td></tr><tr><td>Ours+</td><td>70.5</td><td>66.5</td><td>74.6</td><td>87.9</td><td>84.6</td><td>91.3</td><td>87.6</td><td>86.9</td><td>91.7</td><td>81.5</td><td>90.1</td><td>87.8</td><td>86.8</td><td>91.6</td><td>82.2</td><td>90.8</td><td></td></tr><tr><td>Cutie-base* (Cheng et al., 2023a) w/ MEGA</td><td>69.9</td><td>65.8</td><td>74.1</td><td>86.1</td><td>82.4</td><td>89.9</td><td>87.0</td><td>86.4</td><td>91.1</td><td>81.4</td><td>89.2</td><td>87.0</td><td>86.0</td><td>90.5</td><td>82.0</td><td>89.6</td><td></td></tr><tr><td>Ours w/MEGA</td><td>73.2</td><td>68.8</td><td>77.5</td><td>88.2</td><td>84.3</td><td>92.1</td><td>88.1</td><td>87.4</td><td>92.5</td><td>81.9</td><td>90.7</td><td>88.0</td><td>88.0</td><td>91.8</td><td>82.5</td><td>90.8</td><td></td></tr><tr><td>Cutie-base (Cheng et al., 2023a)*+ w/ MEGA</td><td>71.7</td><td>67.6</td><td>75.8</td><td>88.1</td><td>84.7</td><td>91.4</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>87.5</td><td>86.3</td><td>90.6</td><td>82.7</td><td>90.5</td><td></td></tr><tr><td>Ours+ w/MEGA</td><td>75.1</td><td>71.0</td><td>79.2</td><td>89.1</td><td>85.8</td><td>92.4</td><td>88.5</td><td>87.6</td><td>92.6</td><td>82.7</td><td>91.3</td><td>88.5</td><td>87.3</td><td>92.0</td><td>83.1</td><td>91.4</td><td></td></tr></table>
+
+We conduct data augmentation with methods of random horizontal mirroring, random affine transformations, and cut-and-paste. The training samples are also resized by crop (480 x 480). Then, random color jittering and random grayscaleing are performed in the training samples. The same augmentation is performed in the same video for stability. To ensure the images can be input into the patch embedding of ViT, we apply padding to both the images and masks so that their dimensions are multiples of 32.
+
+We initialize the model with the first frame and ground truth in the training phase. When the number of past frames is three or fewer, we use all of them as memory frames. If there are more than three past frames, we randomly sample three of them to be the memory frames. We compute the loss at all frames except the first one and perform back-propagation through time.
+
+# A.1.3 INFERENCE DETAILS
+
+During the inference phase, the model memorizes the feature every 5th frame like (Cheng & Schwing, 2022; Cheng et al., 2023a). The pixel-level memory always stores the first frame and its mask. The top-K filter is used to augment memory reading in pixel memory. We use the streaming average for object-level memory to accumulate the discriminative query representation like the operation in (Cheng et al., 2023a).
+
+# A.2 ADDITIONAL QUANTITATIVE RESULTS
+
+In this section, we provide additional ablation studies and SOTA comparisons with different settings.
+
+Table 6 shows detailed results about methods with further pre-training (e.g. JointFormer (Zhang et al., 2023) and ISVOS (Wang et al., 2023)) and improved test size (Cutie (Cheng et al., 2023a)) We note that our method achieves comparable results against SOTA methods (Cutie (Cheng et al., 2023a) and JointFormer (Zhang et al., 2023)) without any pre-training. When trained with MEGA datasets or improved test image size, our method achieves state-of-the-art results in all the benchmarks.
+
+In the main paper, we provide ablation studies for different components. To provide a broader comparison, we conduct tests using various combinations of training data and testing settings. The experimental results in Table 7 detail the impact of different components of our model on the performance, facilitating a better understanding of the effectiveness of our model.
+
+To compare with more methods on MOSE, we follow the training strategy from MOSE (Ding et al., 2023), and replace the YouTubeVOS with MOSE in the training process. Table 8 shows the results of the variants of our method. All the variants achieve great improvement in terms of  $\mathcal{J}\&\mathcal{F}$ , which further validates the effectiveness of the proposed spatial-semantic feature learning and discriminative query representation.
+
+Table 9 shows the results of our method on the more challenging VOTS2023 (Kristan et al., 2023) benchmark, which indicates that our method outperforms the winner (DMAOT) of last year on most metrics. Our method achieves the highest robustness thanks to our representation of target saliency information and the modeling of target semantic-spatial information.
+
+![](images/a2aebe0941ad5e12cea7e972355f38e956d0d12ecf9769d4cadd3d141a43d445.jpg)  
+Qualitative comparisons from various sequences.
+
+![](images/995a6821011bf51ece14e1c749abafdc4369a5d1e8607b3a8cb77be6cb7a7883.jpg)  
+Figure 5: Additional qualitative comparison against state-of-the-art methods including XMem, Swin-DeAOT, and Cutie. All the sequences are selected from YouTubeVOS 2019 and DAVIS 2017 datasets.  
+Qualitative comparisons from various sequences with appearance variance.  
+Figure 6: Additional qualitative comparison on sequence with giant appearance variance. All the sequences are picked from MOSE (Ding et al., 2023) dataset.
+
+Table 7: Model detailed ablation study.  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td><td colspan="3">DAVIS 2017 test</td><td colspan="3">LVOS test</td><td colspan="5">YouTube-VOS 2019 val</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>J</td><td>F</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td></tr><tr><td colspan="15">Trained on the YouTubeVOS, and DAVIS datasets</td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022) (Baseline)</td><td>53.3</td><td>62.0</td><td>57.6</td><td>81.0</td><td>77.4</td><td>84.5</td><td>50.0</td><td>45.5</td><td>54.4</td><td>85.5</td><td>84.3</td><td>88.6</td><td>80.3</td><td>88.6</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>64.0</td><td>60.0</td><td>67.9</td><td>84.2</td><td>80.6</td><td>87.7</td><td>56.2</td><td>51.8</td><td>60.5</td><td>86.1</td><td>85.8</td><td>90.5</td><td>80.0</td><td>88.0</td></tr><tr><td>+Discriminative Query</td><td>64.2</td><td>60.3</td><td>68.1</td><td>85.2</td><td>81.8</td><td>88.5</td><td>57.4</td><td>53.3</td><td>61.5</td><td>86.5</td><td>86.2</td><td>90.7</td><td>80.6</td><td>88.8</td></tr><tr><td>+ViT</td><td>64.2</td><td>60.2</td><td>68.3</td><td>85.6</td><td>82.0</td><td>89.2</td><td>58.3</td><td>53.7</td><td>62.8</td><td>86.7</td><td>86.4</td><td>90.3</td><td>81.0</td><td>88.7</td></tr><tr><td>+Spatial</td><td>68.2</td><td>64.0</td><td>72.4</td><td>86.2</td><td>82.4</td><td>90.1</td><td>67.4</td><td>62.9</td><td>71.9</td><td>87.3</td><td>86.4</td><td>91.3</td><td>81.5</td><td>90.3</td></tr><tr><td>+Semantic (Full)</td><td>68.5</td><td>64.5</td><td>72.6</td><td>86.7</td><td>82.7</td><td>90.8</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td></tr><tr><td colspan="15">Improved test size (600/720)</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>66.2</td><td>62.3</td><td>70.1</td><td>85.9</td><td>82.6</td><td>89.2</td><td>56.2</td><td>51.8</td><td>60.5</td><td>86.9</td><td>86.2</td><td>90.7</td><td>81.6</td><td>89.2</td></tr><tr><td>+Discriminative Query</td><td>66.4</td><td>62.4</td><td>70.1</td><td>87.9</td><td>84.6</td><td>91.2</td><td>57.4</td><td>53.3</td><td>61.5</td><td>87.1</td><td>86.3</td><td>90.6</td><td>82.0</td><td>89.5</td></tr><tr><td>+Spatial</td><td>69.9</td><td>67.5</td><td>74.1</td><td>87.0</td><td>83.7</td><td>90.2</td><td>67.4</td><td>62.9</td><td>71.9</td><td>87.5</td><td>86.8</td><td>91.6</td><td>81.6</td><td>90.2</td></tr><tr><td>+Semantic ( full )</td><td>70.5</td><td>66.5</td><td>74.6</td><td>87.8</td><td>84.6</td><td>91.3</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.7</td><td>86.8</td><td>91.6</td><td>82.2</td><td>90.8</td></tr><tr><td colspan="15">Trained on the MEGA datasets</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>69.9</td><td>65.8</td><td>74.1</td><td>86.1</td><td>82.4</td><td>89.9</td><td>66.7</td><td>62.4</td><td>71.0</td><td>87.0</td><td>86.0</td><td>90.5</td><td>82.0</td><td>90.7</td></tr><tr><td>+Discriminative query</td><td>70.6</td><td>66.5</td><td>74.6</td><td>86.6</td><td>82.7</td><td>90.5</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.5</td><td>86.0</td><td>90.6</td><td>82.8</td><td>90.6</td></tr><tr><td>+Spatial</td><td>73.5</td><td>69.1</td><td>77.7</td><td>87.6</td><td>83.8</td><td>91.5</td><td>68.8</td><td>64.4</td><td>73.1</td><td>87.9</td><td>86.9</td><td>91.8</td><td>82.3</td><td>90.4</td></tr><tr><td>+Semantic ( Full )</td><td>74.0</td><td>69.8</td><td>78.3</td><td>87.8</td><td>84.0</td><td>91.7</td><td>73.0</td><td>68.3</td><td>77.8</td><td>88.1</td><td>87.4</td><td>92.5</td><td>81.9</td><td>90.7</td></tr><tr><td colspan="15">Improved test size (600/720)</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>71.7</td><td>67.6</td><td>75.8</td><td>88.1</td><td>84.7</td><td>91.4</td><td>66.7</td><td>62.4</td><td>71.0</td><td>87.5</td><td>86.3</td><td>90.6</td><td>82.7</td><td>90.5</td></tr><tr><td>+Discriminative query</td><td>71.6</td><td>67.7</td><td>75.5</td><td>88.1</td><td>84.6</td><td>91.4</td><td>66.5</td><td>62.1</td><td>70.8</td><td>88.0</td><td>86.2</td><td>90.6</td><td>83.6</td><td>91.5</td></tr><tr><td>+Spatial</td><td>75.3</td><td>71.3</td><td>79.2</td><td>89.0</td><td>85.8</td><td>92.3</td><td>68.8</td><td>64.4</td><td>73.1</td><td>88.3</td><td>87.0</td><td>91.9</td><td>83.0</td><td>91.1</td></tr><tr><td>+Semantic(Full)</td><td>75.1</td><td>71.0</td><td>79.2</td><td>89.1</td><td>85.8</td><td>92.4</td><td>73.0</td><td>68.3</td><td>77.8</td><td>88.5</td><td>87.3</td><td>92.0</td><td>83.1</td><td>91.4</td></tr></table>
+
+Table 8: Results only trained on MOSE datasets.  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td></tr><tr><td>RDE (Li et al., 2022a)</td><td>48.8</td><td>44.6</td><td>52.9</td></tr><tr><td>STCN (Cheng et al., 2021a)</td><td>50.8</td><td>46.6</td><td>55.0</td></tr><tr><td>AOT (Yang et al., 2021)</td><td>57.2</td><td>53.1</td><td>61.3</td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022)</td><td>57.6</td><td>53.3</td><td>62.0</td></tr><tr><td>DeAOT (Yang &amp; Yang, 2022)</td><td>59.4</td><td>55.1</td><td>63.8</td></tr><tr><td>ResNet+Discriminative query</td><td>69.9</td><td>65.8</td><td>73.9</td></tr><tr><td>+Spatial</td><td>72.7</td><td>68.3</td><td>77.0</td></tr><tr><td>+Semantic</td><td>72.9</td><td>68.4</td><td>77.3</td></tr><tr><td colspan="4">Improved test size (720)</td></tr><tr><td>ResNet+Discriminative query</td><td>71.6</td><td>67.7</td><td>75.5</td></tr><tr><td>+Spatial</td><td>74.0</td><td>69.9</td><td>78.1</td></tr><tr><td>+Semantic</td><td>74.5</td><td>70.2</td><td>78.5</td></tr></table>
+
+Table 9: Results on VOTS2023 Challenge.  
+
+<table><tr><td>Dataset</td><td colspan="6">VOTS2023</td></tr><tr><td>Method</td><td>Q↑</td><td>A↑</td><td>R↑</td><td>NER↓</td><td>DRE↓</td><td>ADQ↓</td></tr><tr><td>Dynamic_DEAOT</td><td>0.59</td><td>0.69</td><td>0.84</td><td>0.07</td><td>0.1</td><td>0.57</td></tr><tr><td>M-VOSTrack</td><td>0.61</td><td>0.75</td><td>0.76</td><td>0.16</td><td>0.08</td><td>0.71</td></tr><tr><td>HQTrack</td><td>0.62</td><td>0.75</td><td>0.77</td><td>0.16</td><td>0.08</td><td>0.70</td></tr><tr><td>DMAOT</td><td>0.64</td><td>0.75</td><td>0.80</td><td>0.14</td><td>0.07</td><td>0.73</td></tr><tr><td>Ours</td><td>0.67 (+0.3)</td><td>0.74</td><td>0.86 (+0.6)</td><td>0.09 (+0.5)</td><td>0.05 (+0.02)</td><td>0.75 (+0.02)</td></tr></table>
+
+# A.3 ADDITIONAL QUALITATIVE COMPARISON
+
+We present a more qualitative comparison of most benchmarks in Figure 5 and Figure 6.
+
+In Figure 5, we categorize the scenarios into three main types: overall-to-part segmentation (man on bike and deer), small object segmentation (small monkey), blurry object segmentation (cat toy), and segmentation under varying lighting conditions (man-bike and golf). In these three cases, our method demonstrates precise segmentation compared to other methods. The semantic embedding allows our model to consider semantic information when parsing objects, spatial dependence modeling enhances the understanding of spatial positions and details of the object, and the discriminative query
+
+![](images/979b836a8248932fe294948298c79a2791f344a09bdc731b0045a91d14a80f0d.jpg)  
+(a) Visualization of the feature map output from different backbones
+
+![](images/d4443aee8890984e177cb5174346c29ff881abb26c0e07d5ea0429be2f8f6719.jpg)  
+(b) Visualization of the feature map after proposed semantic and spatial layers  
+Figure 7: Visualizations of feature maps from different architectural components.
+
+accurately represents the target. The integration of these three components enables the model to achieve excellent segmentation and tracking performance in complex and challenging scenarios.
+
+In Figure 6, we select more sequence results on MOSE, primarily divided into two categories: positional changes of similar targets and rapid movement of targets. In both cases, our method outperforms Cutie. This is attributed to our proposed spatial-semantic feature representation and discriminative query expression, which enable the model to distinguish appearance-similar targets and consistently track the target.
+
+Figure 9 compares sequences that require a semantic understanding of the target. Without the semantic embedding module, the method suffers from segmenting parts of the target. The addition of the semantic embedding module enables the model to fully understand the semantic information of the target, allowing for precise segmentation and tracking in these scenarios.
+
+Visualization of feature maps. Figure 7(a) shows that the proposed Spatial-Semantic block assigns higher activation weights to complex target objects compared to the CNN or ViT model, which indicates that the proposed model makes target representations more distinguishable than ViT and ResNet. Moreover, we show how the feature changes during the processing of the proposed model by visualizing the feature maps at different processing steps in 7(b). After passing through the semantic embedding module, the feature maps (semantic embedding) demonstrate that the module makes the contours of objects clearer, indicating richer semantic information. After passing through the spatial dependence module, the feature maps (spatial modeling) show that the details of the targets are more pronounced. The changes in the feature maps confirm that the proposed two modules significantly enhance the semantic and spatial information of the targets.
+
+We also visualize the similarity between the learned discriminative queries and the feature map using heat maps in Figure 8. The visualization shows that the learned discriminative queries distinctly express the target areas. We also mark the positions of the discriminative queries, which represent the locations of the pixels with the highest similarity.
+
+Failure cases are shown in Figure 10, in which our method tends to segment the whole instance rather than the part of it. The challenging sequences require the model to segment and track the head of the man, the flamingo, and the kangaroo. Our model interprets semantics as the entire instance during tracking and segmentation, such as a person, bird, or kangaroo, resulting in segmentation outcomes gradually favoring the whole instance.
+
+![](images/cbf909ee03de79c5d3f0d18ba6bcc59417e79e3ca221458679fc1a512b34eab1.jpg)  
+Figure 8: Visualization of the similarity map between query and features. The map indicates the discrimination of the target representation. The discriminative queries of different targets are plotted in the similarity map, which can be found in the black box.
+
+![](images/21df02a88f3efb4f11165e183bdbdb26e726d606bae8b3ecd9a1670da06c37c8.jpg)  
+Figure 9: Qualitative comparison between with and without semantic embedding block. All the sequences are picked from the YouTubeVOS 2019 dataset.
+
+![](images/452cbeb09672b207ef43906bd9e4c76abde0114a874631c8a78f12afd4a4295a.jpg)  
+Time line  
+Figure 10: Failure cases. All the sequences are picked from VOT Challenge (Kristan et al., 2023).

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+# LEARNING SPLITTING HEURISTICS IN DIVIDE-AND-CONQUER SAT SOLVERS WITH REINFORCEMENT LEARNING
+
+Shumao Zhai $^{1}$ , Ning Ge $^{1,2}$  *
+
+<sup>1</sup>Beihang University 2State Key Laboratory of Complex & Critical Software Environment (CCSE)
+
+# ABSTRACT
+
+We propose RDC-SAT, a novel approach to optimize splitting heuristics in Divide-and-Conquer SAT solvers using deep reinforcement learning. Our method dynamically extracts features from the current solving state whenever a split is required. These features, such as learned clauses, variable activity scores, and clause LBD (Literal Block Distance) values, are represented as a graph. A GNN integrated with an Actor-Critic model processes this graph to determine the optimal split variable. Unlike traditional linear state transitions characterized by Markov processes, divide-and-conquer challenges involve tree-like state transitions. To address this, we developed a reinforcement learning environment based on the Painless framework that efficiently handles these transitions. Additionally, we designed different discounted reward functions for satisfiable and unsatisfiable SAT problems, capable of handling tree-like state transitions. We trained our model using the Decentralized Proximal Policy Optimization (DPPO) algorithm on phase transition random 3-SAT problems and implemented the RDC-SAT solver, which operates in both GPU-accelerated and non-GPU modes. Evaluations show that RDC-SAT significantly improves the performance of D&C solvers on phase transition random 3-SAT datasets and generalizes well to the SAT Competition 2023 dataset, substantially outperforming traditional splitting heuristics.
+
+# 1 INTRODUCTION
+
+The propositional satisfiability problem (SAT) is one of the foundational problems in theoretical computer science. It involves determining the assignment of variables that satisfy a given Boolean formula. Due to its NP-complete nature, SAT plays a crucial role in theoretical computer science and has a wide range of applications, including software verification, hardware design verification, combinatorial optimization, and artificial intelligence, etc. Most modern complete SAT solvers primarily employ the conflict-driven clause learning (CDCL) algorithm (Marques-Silva et al., 2021), which has transformed the field by introducing clause learning mechanisms that effectively optimize the search space. CDCL solvers avoid revisiting previously explored paths and concentrate on promising areas of the search space, establishing them as the leading method for tackling complex and challenging SAT problems.
+
+In the realm of modern SAT solvers, parallelism has been embraced to fully harness the computational power of many-core machines (Hamadi & Sais, 2018). Two dominant parallelization strategies have emerged: Divide-and-Conquer (D&C) and Portfolio. The D&C strategy, often employing the guiding path technique, dynamically divides the problem into smaller, more manageable subspaces. Each subspace is independently tackled by individual sequential solvers, allowing the problem to be processed more efficiently in parallel. This method leverages problem structure to enable targeted sub-formula exploration, reducing overall solving time. The Portfolio approach adopts a competitive strategy in which multiple sequential solvers concurrently tackle the entire problem. This competition continues until one solver successfully solves the problem. Strategic diversity among solvers enhances solution coverage and success probability. Although D&C approaches seem to be the natural way to work in parallel, empirical results from annual SAT competitions indicate that Portfolio-based
+
+solvers currently lead in performance. However, D&C solvers still possess unique advantages in certain domains, such as better scalability in environments with large parallel computing power, like cloud platforms (Heisinger et al., 2020), and have been successfully employed to solve large-scale mathematical problems (Heule et al., 2016).
+
+In modern D&C SAT solvers, the problem is dynamically divided into smaller, more manageable subproblems. The splitting process entails selecting a variable and dividing the problem into subformulas by setting the variable to either True or False, which are subsequently solved using CDCL sequential solvers. This partitioning is guided by splitting heuristics, which largely determine the performance of D&C SAT solvers. An effective splitting heuristic minimizes total solving time while ensuring balanced subspaces for optimal load distribution. Splitting heuristics generally fall into two categories: look-ahead and look-back techniques. Look-ahead heuristics evaluate the potential impact of variables to minimize runtime, as demonstrated in the Cube-and-Conquer solver (Heule et al., 2011; Fleury & Heisinger, 2020), which creates smaller subspaces by choosing variables that maximize unit propagations. In contrast, look-back heuristics evaluate variables based on their historical effectiveness in the search, selecting split variables through methods like VSIDS-based variable activity (Audemard et al., 2016), the number of flips of variables (Le Frioux et al., 2019), and propagation rates (Nejati et al., 2017) of variables.
+
+Machine learning methods have already been widely applied to SAT problems (Holden et al., 2021; Guo et al., 2023; Yolcu & Poczos, 2019; Zhang et al., 2020). In recent years, GNNs and RL have shown potential in optimizing branch splitting, phase selection, clause deletion, etc., heuristics in CDCL solvers (Jaszczur et al., 2020; Kurin et al., 2020; Selsam & Bjørner, 2019; Cameron et al., 2020; Han, 2020; Wang et al., 2023; Shi et al., 2023; Liu et al., 2024). For example, Graph-Q-Sat (Kurin et al., 2019; 2020) uses GNNs and RL to optimize branch heuristics in CDCL solvers, significantly reducing the number of iterations. However, Graph-Q-Sat did not effectively reduce overall runtime due to CDCL's requirement for high-frequency variable selection calls. In contrast, splitting heuristics in D&C solvers are equally critical but invoked far less frequently, making RL and GNN-based optimization of these heuristics particularly promising. A more detailed discussion of related work is provided in Appendix A.
+
+In this paper, we present a technique that employs deep reinforcement learning to optimize the splitting euristics within parallel D&C SAT solvers, resulting in the development of the Reinforced Divide-and-Conquer SAT solver (RDC-SAT). When a split is required, the D&C solver parses information from the sequential solver, including learned clauses, variable activities, and clause LBD(Literal Block Distance), and represents this information as a graph. A GNN is then used to evaluate the importance of each variable and select the optimal split variable.
+
+To optimize this model using reinforcement learning, we developed an RL environment within the Painless (PArallel INstantiabLE Sat Solver) framework (Le Frioux et al., 2017; 2019) that efficiently manages the tree-like state transitions characteristic of D&C methods. Additionally, we designed distinct discounted reward functions for satisfiable and unsatisfiable SAT problems. We then trained a model optimized for splitting heuristics using the Distributed Proximal Policy Optimization(DPPO)(Heess et al., 2017) algorithm, resulting in the creation of RDC-SAT. Moreover, considering the computational overhead of the neural network in the SAT solver, we also implemented an approach similar to NeuroBack (Wang et al., 2023), which supports invoking the neural network only once at the initial stage to guide subsequent splits, making it suitable for GPU resource-poor computing environments. To validate our approach, we trained the model on a random dataset at the phase transition and evaluated its effectiveness and generalization on larger-scale problems. Our empirical results demonstrate that RDC-SAT outperforms baseline methods. On the random 3-SAT dataset, RDC-SAT achieved a  $25.08\%$  reduction in PAR2 score; on the SAT Competition 2023 dataset—which includes small to medium-sized SAT problems with hundreds of thousands of variables and clauses—it achieved a  $14.95\%$  reduction. Even without GPU acceleration, the insights from the initial neural network call effectively guide subsequent splits, ensuring efficient problem-solving with minimal reliance on neural network computations.
+
+# 2 PRELIMINARIES
+
+SAT problem. The Boolean Satisfiability Problem (SAT)(Biere et al., 2009) concerns the satisfiability determination of propositional logic formulas. A propositional logic formula is composed
+
+of variables and operators such as AND (conjunction,  $\land$ ), OR (disjunction,  $\lor$ ), NOT (negation,  $\neg$ ), and parentheses, typically expressed in Conjunctive Normal Form (CNF). In CNF, a formula is a conjunction of clauses, where a clause is a disjunction of literals, and a literal is either a variable (positive literal) or its negation (negative literal). For example,  $(x \lor \neg y) \land (\neg x \lor z)$  is a CNF formula, where  $x$  and  $z$  are positive literals of their respective variables, and  $\neg y$  is a negative literal. In the SAT problem, each variable can be assigned one of two values: True or False. If there is at least one assignment of variables that makes the entire formula True, then the formula is deemed satisfiable (SAT); otherwise, it is considered unsatisfiable (UNSAT). During the process of solving SAT problems, CDCL solvers generate numerous learned clauses to aid in resolving the problem efficiently. These learned clauses help reduce the search space by preventing the solver from revisiting conflicting assignments, thus facilitating quicker convergence towards a solution or proving unsatisfiability.
+
+Divide-and-Conquer SAT solvers. Modern D&C SAT solvers follow a hierarchical master-slave architecture.(Le Frioux et al., 2019) The master node manages the partitioning of the search space into smaller subspaces, which are tackled independently and in parallel by multiple sequential CDCL solvers. These subproblems are solved concurrently by worker nodes, which share learned clauses with each other to accelerate the solving process and reduce redundant computations. To ensure dynamic load balancing, D&C solvers utilize a work-stealing mechanism. When a worker node becomes idle, the master node identifies an ongoing subproblem currently being processed by another node, splits it into smaller tasks, and assigns one to the idle worker. This dynamic redistribution of tasks enhances computational efficiency. Incremental solvers further optimize this approach by enabling solvers to continue from previous states with retained learned clauses and variable activity scores, minimizing redundancy and improving overall solving speed.
+
+Graph Neural Networks. Graph Neural Networks (GNNs) (Wu et al., 2020; Zhou et al., 2020) are designed to handle data structured as graphs, allowing for the modeling of relationships and interactions within data. The GraphNets framework (Battaglia et al., 2018) utilizes an Encoder-Processor-Decoder architecture to effectively process graph-structured data. The encoder maps node and edge features into high-dimensional latent representations. The processor iteratively updates these representations through a message-passing mechanism, where each node aggregates information from its neighbors to refine its embedding. This mechanism enables the network to propagate and integrate information across the graph. Finally, the decoder reconstructs the desired outputs from the node embeddings for tasks such as node classification, graph classification, or link prediction, utilizing the embeddings either directly or through aggregation for graph-level predictions.
+
+Reinforcement Learning. Reinforcement Learning (RL) is a machine learning paradigm where an agent learns to make decisions by interacting with its environment, maximizing cumulative rewards. Central to RL is the classic Markov Decision Process (MDP), which models the decision-making problem through states  $s \in S$ , actions  $a \in A$ , state-transition probabilities  $P(s'|s,a)$ , and a reward function  $R(s,a)$ . This framework assumes the state transitions are generally linear, meaning the next state  $s'$  depends only on the current state  $s$  and action  $a$ . Advanced RL algorithms like Actor-Critic methods use a policy network (actor) to select actions and a value network (critic) to estimate the value function, improving both networks iteratively. Proximal Policy Optimization (PPO) (Schulman et al., 2017) introduces a clipped surrogate objective function to stabilize policy updates, preventing drastic changes and ensuring more reliable learning. Distributed Proximal Policy Optimization (DPPO) Heess et al. (2017) extends PPO by employing multiple agents across distributed environments, leveraging data parallelism to speed up learning while maintaining policy stability.
+
+# 3 INTERACTIVE DIVIDE-AND-CONQUER ENVIRONMENT
+
+We selected Painless (Le Frioux et al., 2017; 2019) as the foundational parallel divide-and-conquer framework. The Painless framework is a modular and high-performance parallel SAT-solving system, ideal for comparing different heuristics. It uses a master-slave architecture with a work-stealing strategy for dynamic load balancing and supports clause sharing to enhance solving efficiency. For our experiments, we used the best-performing settings identified, including MapleCOMSPS (Liang et al., 2016b;a) as the sequential solver and the "altoall" clause-sharing strategy. Building on this foundation, to construct a reinforcement learning environment suitable for optimizing splitting heuristics and handling tree-structured state transitions, we implemented features to extract SAT problem states from the SAT solver, store various information required for training using a binary
+
+tree, and retain only simple interfaces. To use Python for model training, we encapsulated the divide-and-conquer environment as a Python library using Pybind11 and created the PainlessEnv environment in Python. Specific implementation details can be found in Appendix B.
+
+# 3.1 ENVIRONMENT INTERFACE
+
+Our custom PainlessEnv environment is specifically designed for the needs of divide-and-conquer SAT problems, addressing the complexities of parallel and branching operations in SAT solving. It manages a binary tree structure to support the non-linear state transitions characteristic of these problems, unlike the linear transitions typical in traditional environments like Gym (Brockman et al., 2016). PainlessEnv primarily provides two interfaces for agent interaction: 'reset' and 'step'. The reset method initializes a new SAT problem and provides the initial state of the first subproblem that needs to be partitioned. Unlike Gym's 'step' method, which returns the next observation, reward, and done flag, our 'step' method only returns the next subproblem state and a 'done' flag. This is because, in the tree-structured state transitions, the next state is not always a direct successor of the previous state, and the reward can only be accurately determined from the tree structure. Therefore, our 'step' method does not return the reward directly.
+
+# 3.2 MANAGING STATE TRANSITIONS WITH BINARY TREES
+
+Due to the non-linear state transitions inherent in D&C problems, PainlessEnv cannot produce a continuous sequence of states and rewards for training. Instead, it utilizes a binary tree, as illustrated in Figure 1, to store all critical information during the solving process. The root node denotes the initial problem, internal nodes correspond to subproblems requiring splitting, and leaf nodes signify fully solved subproblems. When a worker requests a division or completes a solution, a new node is added to the tree with all pertinent data. After solving a SAT problem, the reinforcement learning agent retrieves training information from this binary tree. Additionally, In a D&C solver for SAT problems, once a subproblem is proven satisfiable, it indicates that the entire SAT problem is satisfiable. Typically, the D&C solver would interrupt all other active workers and return the result. However, for reinforcement learning training, the other unresolved subproblems are still meaningful. Interrupting the process leads to missing leaf nodes in the binary tree, which is suboptimal for training. To address this, even if a subproblem is confirmed as satisfiable, we continue processing until all subproblems are resolved before returning a result, ensuring a complete and informative dataset for training.
+
+# 3.3 SAT SUBPROBLEM STATE EXTRACTION
+
+Each time a split is required, the Painless environment interrupts a sequential solver and extracts basic information about the SAT problem to represent its state. MapleCOMSPS categorizes learned clauses into three levels: core, tier2, and local. Core learned clauses are considered the most crucial. They are generally derived from conflicts that involve decisions at deeper levels of the decision tree that are usually kept permanently. Tier2 learned clauses are of intermediate importance. They are often derived from conflicts at shallower depths and are kept as long as they are deemed useful. Local learned clauses are less critical and more temporary, typically generated from recent conflicts and are discarded periodically to manage memory and performance overhead. Additionally, it maintains variable activity values used in branch heuristics, the LBD values of clauses to evaluate their utility, and other statistical information such as the number of decisions and unit propagations. We extract these learned clauses at different levels, along with variable activity scores and other pertinent data, to comprehensively represent the current solving state of the SAT problem. This information is then represented as a graph, which our environment can be configured to utilize.
+
+# 4 REINFORCED DIVIDE-AND-CONQUER SAT
+
+We utilized a state representation and a GNN similar to Graph-Q-Sat (Kurin et al., 2020), with modifications to incorporate additional information that better represents the evolving state of the SAT problem. Additionally, we replaced DQN with an Actor-Critic network and trained it using DPPO.
+
+![](images/86b69ba1c06cb5939d8d46355e8b4435660aa51d14a19de110b175e54beaa9d8.jpg)  
+Figure 1: Binary tree maintained by PainlessEnv to store process information. Each node in the tree stores problem states, splitting times, solving times, split variables, satisfiability status, and other relevant information. Internal nodes represent subproblems that have been split, while leaf nodes represent solved subproblems. Dashed lines indicate satisfiable subproblems.
+
+![](images/b77bf69714d76e2b7f3da5493a686d179a86881e6472611ba0b6eb310c0ceaa0.jpg)  
+Figure 2: State representation of Boolean formula  $(x_{1}\lor x_{2})\land (\neg x_{2}\lor x_{3})$  . The first position of the node feature indicates node type: 1 for variable nodes and 0 for clause nodes, and vice versa for the second position.  $F_{x}$ $F_{c}$  ,and  $F_{g}$  represent variable activity features, clause LBD features, and global SAT problem statistics, respectively. Edge features represent the polarity of literals.
+
+# 4.1 STATE REPRESENTATION
+
+In RDC-SAT, the SAT problem is structured as a graph shown in Figure 2 where nodes represent variables and clauses, with node and edge features indicating variable polarity and type. We extract detailed information from the Maple solver, including variable data and learned clauses at various levels. Incorporating learned clauses provides a more dynamic reflection of the SAT problem's state. Variable activity levels and clause LBD values are normalized to enhance message passing efficiency. Additionally, global characteristics like restart counts, decision counts, conflict counts, and unit propagation counts are included as global features. This comprehensive representation captures the dynamic state changes in the SAT problem, enabling the GNN to process node and edge data more effectively.
+
+# 4.2 REWARD FUNCTION FOR TREE-STRUCTURED STATE TRANSITIONS
+
+In traditional reinforcement learning, rewards are typically calculated for linear state transitions, where the reward for each state is computed from the final state backward through the sequence, applying a discount factor at each step. This process involves calculating the reward for the final state first and then propagating it backward through the sequence. Correspondingly, for tree-structured state transitions, calculating the reward for each state requires first computing the discounted rewards for its child states. This necessitates using a post-order traversal to ensure that the discounted rewards of the child states are available before calculating the reward for the current state. Once the discounted rewards of the child states are obtained, they can be integrated into the discounted reward of current state using methods such as summation, averaging, taking the maximum, or the minimum of the child states' rewards. To illustrate this, consider a binary tree, where each state has two child states. Mathematically, if  $R_{L}$  and  $R_{R}$  are the discounted rewards of the left and right child states, and  $r_{s}$  is the immediate reward for the current state, the current state's reward  $R$  can be calculated as
+
+$$
+R = r _ {s} + \gamma \cdot f \left(R _ {L}, R _ {R}\right)
+$$
+
+The primary goal of this optimization is to minimize solving time. Thus, we simply use the negative solving time as the reward, making the maximization of this reward equivalent to minimizing the solving time. After solving SAT problems, we extract the splitting time and solving time from the binary tree in PainlessEnv to calculate the actual solving time for each state. For unsatisfiable problems, the total solving time is the sum of the splitting time and the longer solving time of the two subproblems. In contrast, for satisfiable problems, the total solving time is the sum of the splitting time and the shorter solving time of the satisfiable subproblems. The following formulas encapsulate
+
+the calculation of discounted reward:
+
+$$
+R _ {U} = - \left(T _ {s} + \max  \left(\Delta T _ {U c} + \gamma \cdot \left| R _ {U c} \right|\right)\right)
+$$
+
+$$
+R _ {S} = - \left(T _ {s} + \min \left(\Delta T _ {S c} + \gamma \cdot | R _ {S c} |\right)\right)
+$$
+
+Here,  $R_U$  and  $R_S$  are the discounted rewards for UNSAT and SAT problems, respectively;  $T_s$  is the splitting time, which is the time taken to analyze the problem state and decide the split variable;  $|R_{Uc}|$  and  $|R_{Sc}|$  are the absolute values of the discounted rewards for the child subproblems of UNSAT and SAT states, respectively. If a corresponding child state is a leaf node, its discounted reward is zero;  $\Delta T_{Uc}$  and  $\Delta T_{Sc}$  are the time intervals from the completion of the division to the return of the next child subproblem state in UNSAT and SAT scenarios, respectively; and  $\gamma$  is the discount factor applied to future rewards. Typically,  $\gamma$  is less than 1 to prioritize immediate rewards over future rewards. However, in our study, we set  $\gamma = 1$  because the solving process terminates within a definite time frame. This ensures that each action considers the entire solving process, and the reward directly reflects the total solving time under the current policy and action.
+
+![](images/46c348bfd39d81e86f5bc558db688b9aeed911111b0004dcf8fdfcae59bb6c4d.jpg)  
+Figure 3: Calculation of discounted reward in tree-structured state transitions.
+
+# 4.3 NEURAL NETWORK, ACTION DEFINITION, AND OBJECTIVE FUNCTION
+
+We use a Graph Neural Network (GNN) with an Encoder-Processor-Decoder architecture (Battaglia et al., 2018) to extract features, followed by an Actor-Critic network to make decisions. The Encoder transforms initial features into higher-dimensional representations. The Processor integrates local and global information through message passing. The Decoder outputs variable, clause, and global features. The RL action corresponds to selecting a variable as the split variable during the Divide-and-Conquer process. The Actor network assigns probabilities to valid variables, excluding those already chosen as split variables or eliminated during the solving process, ensuring the action space only contains currently selectable variables. The Critic network estimates the value of the current state based on clause and global features. We use the Proximal Policy Optimization (PPO)(Schulman et al., 2017) objective function. This objective combines policy loss, value loss, and entropy loss to optimize the learning process.
+
+# 4.4 TRAINING AND DEPLOYMENT
+
+Training data. We utilize randomly generated 3-SAT problems at the phase transition as our training data. The phase transition in SAT problems is characterized by a specific ratio of clauses  $(n)$  to variables  $(m)$ , given by  $n = 4.258m + 58.26m^{-\frac{2}{3}}$  (Crawford & Auton, 1996), where the likelihood of satisfiability sharply transitions from high to low as the number of clauses increases relative to the number of variables. This critical point presents the greatest computational challenge. Although these datasets may initially exhibit a uniform structure, CDCL solvers in D&C solvers can quickly generate a large number of learned clauses, making the subsequent SAT problems input to the neural network inherently rich in structural features. We selected two datasets: one with 300 variables and 1,278 clauses, and another with 350 variables and 1,491 clauses (hereafter referred to as the 300-1278 and 350-1491 datasets, respectively), each comprising 1,000 instances with a balanced ratio of satisfiable to unsatisfiable instances. These training datasets can be solved within seconds to tens of seconds, allowing for the generation of a large amount of training data. All these datasets can be solved within a reasonable timeframe, generating complete binary trees that are used for subsequent extraction of training data.
+
+Training. We implemented DPPO using a multi-agent architecture to enhance both exploration and learning efficiency. The system architecture includes five exploration agents continuously solving SAT problems and collecting training data, complemented by a single dedicated training agent. Each exploration agent runs 16 threads to solve SAT problems continuously. Upon completion, these agents parse the binary tree to extract training data, which is then transmitted to the training agent via
+
+a message queue. The training agent systematically retrieves this data from the queue, stores it in a replay buffer, and samples from this buffer to form batches for neural network training. We represent the state using a graph constructed from variables, original clauses, core learned clauses, and tier2 learned clauses, all extracted from the Sequential CDCL solver. We use the Adam optimizer to train the model. Post-training, the updated network parameters are synchronized across all exploration agents to maintain consistency and optimize policy performance. An  $\epsilon$ -greedy strategy is implemented to ensure a balance between exploration and exploitation, with the epsilon parameter decreasing progressively over time. Additionally, all rewards are normalized during training to maintain a consistent learning scale and enhance algorithmic stability. The exploration agents use only CPU resources, while the training agent uses only GPU resources. For exploration processes, we used LRB as the branch heuristics, as it performed better on the random dataset. Training took approximately 2-3 days. After completing this training, we switched the branch heuristics to VSIDS and trained a new model, which took about half a day. This training operation is hosted on a Linux server equipped with an Intel(R) Xeon(R) Platinum 8373C CPU, featuring 72 physical cores and 144 logical cores, supplemented by 256 GB of memory and a Tesla A100 GPU with 80 GB of memory. During training, we observed that even though random 3-SAT problems at the phase transition initially have just over a thousand clauses, the states used for training showed that the CDCL solver generated approximately ten to fifty thousand core and tier2 learned clauses, ensuring a rich structural representation.
+
+Deployment. After the training phase, we utilized TorchScript to export the model and integrate it with the Painless framework, thereby creating the RDC-SAT solver. This solver is a complete D&C SAT solver that no longer depends on Python, and it can dynamically generate variable probability rankings required for splitting decisions using the neural network. The solver can choose to run the neural network's forward propagation on either CPU or GPU. However, the relatively high overhead of neural networks often becomes a bottleneck in SAT solving with GNNs, especially when GPUs are unavailable and the forward propagation is done using CPUs. When multiple threads call the neural network simultaneously, resource contention may occur, slowing down the solving process. To address this and enable RDC-SAT to function effectively without GPU support, we implemented a new strategy: the neural network is used only once during the first required split to generate a ranked list of candidate split variables. For all subsequent splits, the solver directly selects split variables from this precomputed ranking. Additionally, when the problem size is very large, leading to excessive overhead from neural network calls or potential memory limitations, RDC-SAT will switch to the default splitting heuristic.
+
+# 5 EVALUATION
+
+In this chapter, we compare the performance of different splitting heuristics within the Painless framework across various datasets. We evaluate the performance of RDC-SAT against traditional splitting heuristics and analyze the cost of using GNNs.
+
+# 5.1 SETUP
+
+Solvers. We used the Painless framework to evaluate the performance of our RDC methods compared to baseline divide-and-conquer solvers. To ensure a fair comparison, we kept all configurations consistent except for the splitting heuristics. Specifically, we used FLIPS (Audemard et al., 2014b), VSIDS(Audemard et al., 2016), and PR (Nejati et al., 2017) splitting strategies from the work (Le Frioux et al., 2019), with P-CLONE-FLIPS showing the best overall performance, outperforming Treengeling and MapleAmpharos. Our RDC-SAT utilized core and tier2 learned clauses to construct the graph, employing two modes: RDC-GPU-always and RDC-CPU-once. RDC-GPU-always uses the GPU to accelerate neural network calls, selecting the split variable with GNN every time it splits. RDC-CPU-once, on the other hand, does not rely on GPU and only uses the CPU to call the GNN and generate variable sorting. For subsequent splitting, the worker uses the variable probability sorting to select split variables. Due to the poor performance of VSIDS on random datasets compared to LRB (as shown in Appendix E.2), we used Maple's LRB branch heuristics for the random dataset, and the RDC used the LRB model. In contrast, for the SAT Competition dataset, we used Maple's VSIDS branch heuristics, and the RDC used the VSIDS model.
+
+Dataset. We evaluated RDC-SAT on two datasets. (1) To investigate whether RDC-SAT has effectively learned the splitting heuristics, we randomly selected 200 SAT instances from the random
+
+dataset at the phase transition. This set includes 100 problems from the 450-1917 dataset and 100 from the 500-2129 dataset, with a 1:1 ratio of satisfiable to unsatisfiable instances. This dataset provides a suitable level of difficulty for the baseline solvers, enabling us to distinguish their performance. (2) To evaluate whether RDC-SAT, trained on random datasets, can generalize to more extensive, application-oriented datasets, we screened 305 small to medium-sized SAT problems from the 400 instances in the SAT Competition 2023. These problems, which make up the majority of the competition's instances, have a total number of variables and clauses below one million. The SAT Competition dataset includes a diverse range of difficult SAT instances, such as industrial applications and handmade problems. Since our model was trained solely on random datasets, without any knowledge of application-specific or handmade instances, this dataset serves as an ideal benchmark for evaluating the generalization capability of RDC-SAT.
+
+All experiments were conducted on the same machine used for training, with each solver allocated 16 threads. RDC-CPU-once was also limited to using 16 CPU cores for running the neural network's forward propagation process. The timeout for each instance was set to 5000 seconds. For instances that timed out, we used the same PAR2 score as in the SAT Competition, treating the timeouts as requiring double the timeout duration to solve. To analyze the performance of different splitting heuristics, we recorded the splitting and solving time. In our small-scale tests, we found that the PR method was not well-suited for solving random problems, and thus we did not evaluate PR on the random dataset.
+
+# 5.2 EVALUATION RESULT
+
+Table 1 presents all the results. Figure 4 illustrates the comparison of wall-clock time and problems solved across different splitting heuristics on two datasets. Instances that no solver could successfully solve within the time limit were excluded from the statistics.
+
+Table 1: Comparison of Splitting Heuristics across Datasets. #SAT, #UNSAT, and #ALL indicate the number of SAT problems, UNSAT problems, and total problems solved within the time limit. Clock Time is the average wall-clock time in seconds. PAR2 is the penalized average runtime. Splits are the average number of splits per problem. Splitting Time represents the average CPU time spent by all threads on splitting SAT subproblems. Solving Time represents the average CPU time spent by all threads using the Sequential CDCL solver to solve SAT subproblems. All time values are in seconds.
+
+<table><tr><td>Dataset</td><td>Splitting Heuristics</td><td>#SAT</td><td>#UNSAT</td><td>#All</td><td>Clock Time</td><td>PAR2</td><td>Splits</td><td>Splitting Time</td><td>Solving Time</td></tr><tr><td rowspan="4">Random (141)</td><td>RDC-GPU-always</td><td>76</td><td>53</td><td>129</td><td>1064.43</td><td>1489.96</td><td>60.42</td><td>20.78</td><td>16870.24</td></tr><tr><td>RDC-CPU-once</td><td>73</td><td>53</td><td>126</td><td>1238.04</td><td>1769.95</td><td>65.84</td><td>0.624</td><td>19316.30</td></tr><tr><td>FLIPS</td><td>74</td><td>50</td><td>124</td><td>1385.80</td><td>1988.64</td><td>49.21</td><td>0.00037</td><td>22090.20</td></tr><tr><td>LRB</td><td>70</td><td>50</td><td>120</td><td>1484.81</td><td>2229.49</td><td>74.77</td><td>0.00054</td><td>23617.79</td></tr><tr><td rowspan="5">SAT COMP 2023 (140)</td><td>RDC-GPU-always</td><td>50</td><td>85</td><td>135</td><td>764.00</td><td>942.57</td><td>162.19</td><td>142.47</td><td>10678.02</td></tr><tr><td>RDC-CPU-once</td><td>50</td><td>85</td><td>135</td><td>882.10</td><td>1060.67</td><td>191.95</td><td>8.77</td><td>11737.56</td></tr><tr><td>FLIPS</td><td>50</td><td>83</td><td>133</td><td>858.25</td><td>1108.25</td><td>80.02</td><td>0.0146</td><td>12296.52</td></tr><tr><td>VSIDS</td><td>46</td><td>84</td><td>130</td><td>901.12</td><td>1258.26</td><td>108.44</td><td>0.0013</td><td>12602.74</td></tr><tr><td>PR</td><td>49</td><td>72</td><td>121</td><td>1269.57</td><td>1948.14</td><td>104.19</td><td>0.0155</td><td>19232.39</td></tr></table>
+
+![](images/656112f5b57b2db70a74dcd94807ad75901a3405c867fb7cc617e9957590915c.jpg)  
+(a) Results on Random Dataset
+
+![](images/9edca4932d524f2d6f0f8a079983c63ffbdd4f7f961cc8f6798cf00910a75495.jpg)  
+(b) Results on SAT COMP 2023 Dataset  
+Figure 4: Comparison of wall-clock time and problems solved across splitting heuristics on two datasets.
+
+Evaluation on Random 3-SAT Dataset. For the Random 3-SAT dataset, the RDC-GPU-always and RDC-CPU-once modes were evaluated against traditional heuristics FLIPS and LRB. RDC-GPU-always successfully solved 129 out of 141 problems, while RDC-CPU-once solved 126, both significantly outperforming the traditional splitting heuristics in terms of average clock time and PAR2 score. Specifically, RDC-GPU-always achieved the best performance with an average clock time of 1064.43 seconds and a PAR2 score of 1489.96. Compared to the best baseline method, FLIPS, RDC-GPU-always reduced the PAR2 score by approximately  $25.08\%$ , demonstrating the model's capability to effectively learn and apply splitting heuristics from the random dataset, thus significantly enhancing solving efficiency. Even RDC-CPU-once, which only utilizes the initial SAT problem state for its analysis, reduced the PAR2 score by approximately  $10.99\%$ , showing substantial performance improvement over traditional heuristics.
+
+Generalization to SAT Competition Dataset. The RDC-SAT model, trained solely on random 3-SAT problems at the phase transition without any additional dataset information, was further evaluated on the SAT Competition dataset to test its generalization capability. This dataset includes a diverse range of problems, many of which are application-specific, providing a comprehensive test of RDC-SAT's robustness. Both RDC-GPU-always and RDC-CPU-once demonstrated competitive performance. Specifically, both methods successfully solved 135 problems, with RDC-GPU-always achieving the best average clock time of 764.00 seconds and a PAR2 score of 942.57. Compared to the best traditional method, FLIPS, which had a PAR2 score of 1108.25, RDC-GPU-always reduced the PAR2 score by approximately  $14.95\%$ , showcasing its effectiveness across varied problem types. RDC-CPU-once, while solving the same number of problems, recorded slightly higher average clock time and PAR2 score of 1060.67, yet still achieved a PAR2 reduction of approximately  $4.29\%$ , indicating substantial performance improvement even with limited reliance on neural network calls. These results underscore the ability of RDC-SAT to generalize well beyond its training conditions.
+
+Analysis of Splitting Time and Solving Time. To analyze the solving performance of our methods, we consider two key metrics: Splitting Time, which represents the total splitting time across all threads, and Solving Time, which represents the total solving time across all threads. In SAT solvers, incorporating a GNN is beneficial only if its performance improvements outweigh the overhead. In D&C solvers, the average number of splits per problem is relatively low; for example, in the random dataset, it ranges from 49.21 to 74.77 across various splitting heuristics (see Table 1). Therefore, even if the GNN is invoked at each split, it does not introduce significant performance overhead. Figures 5a and 5b illustrate the relationship between the splitting time per split and the total number of nodes for RDC-GPU-always and RDC-CPU-once on the two datasets. From the figures, it can be observed that the Splitting Time is approximately proportional to the total number of nodes. Notably, in RDC-GPU-always, the average time per split is less than 1 second. In RDC-CPU-once, although the initial GNN call takes slightly longer, subsequent splits are faster because they utilize precomputed variable rankings. When considering total CPU time, the Splitting Time of RDC-GPU-always accounts for only  $0.12\%$  and  $1.33\%$  of the total Solving Time on the random and SAT Competition datasets, respectively. Similarly, for RDC-CPU-once, it accounts for just  $0.003\%$  and  $0.075\%$ . Despite the slightly higher Splitting Time compared to baseline methods, the RDC methods significantly reduce the overall Solving Time. Specifically, RDC-GPU-always reduces the Solving Time by approximately  $23.63\%$  and  $13.17\%$  on the random and SAT Competition datasets, respectively, compared to the best-performing baseline, FLIPS. RDC-CPU-once achieves reductions of approximately  $12.56\%$  and  $4.54\%$ , respectively. Therefore, although the Splitting Time of both RDC-SAT methods is marginally higher than that of the baseline methods, they substantially decrease the Solving Time, leading to superior overall performance compared to the baselines.
+
+# 6 DISCUSSION
+
+In this paper, we propose a method to optimize splitting heuristics in D&C SAT solvers using reinforcement learning. We built a reinforcement learning environment tailored for Divide-and-Conquer tasks and trained a model, resulting in the RDC-SAT solver. Compared to baseline methods, RDC-SAT achieves significant performance improvements in wall-clock time. To the best of our knowledge, this is the first work addressing tree-structured state transitions induced by Divide-and-Conquer tasks using reinforcement learning. However, our focus was on exploring the potential of reinforcement learning and GNNs in optimizing splitting heuristics, rather than constructing a state-of-the-art (SOTA) parallel D&C SAT solver. Therefore, we did not modify any other components of
+
+Figure 5: Analysis of splitting Times for RDC-SAT on two dataset.  
+![](images/ce31fb0862bd6961aabca00358ba7d302b2d2fff649a39690b70567b76d2a18d.jpg)  
+(a) Splitting Time on Random Dataset. RDC-GPU-always: Avg. 72803.28 nodes, 0.344s per split. RDCCPU-once: Avg. 9043.16 nodes, 0.624s per split.
+
+![](images/cfe5c8e020675260a3a635bba34fe667413948a36717a7d8338efd8eece5659f.jpg)  
+(b) Splitting Time on SAT Competition 2023 Dataset. RDC-GPU-always: Avg. 248184.79 nodes, 0.878s per split. RDC-CPU-once: Avg. 191098.54 nodes, 8.77s per split.
+
+the Painless framework or train a general model on large-scale datasets containing multiple types of instances, leaving room for further optimization to reach better performance.
+
+Integration of GNNs with SAT Solvers. The integration of GNNs with SAT solving has become a hot research area in recent years. However, for SAT solvers, which are performance-critical tools, the overhead introduced by GNNs—including computational cost and GPU memory usage—is non-negligible, making it challenging to scale to very large SAT problems. Leading SAT solvers have not adopted GNNs. For example, attempts to swap CDCL solvers' branch heuristics with GNNs show that while GNNs can improve decision-making, their significant overhead might extend overall solving time. Our work expands the application of GNNs in SAT solving to a new direction. In a Divide-and-Conquer solver, the splitting heuristic significantly impacts performance, but its invocation frequency is relatively low, allowing RDC-SAT to improve wall-clock time. To extend GNNs to larger-scale SAT problems and reduce overhead, future efforts should prioritize reducing both the size and computational cost of the neural networks. Currently, the GNN employed in RDC-SAT is relatively large, resulting in substantial GPU memory consumption. We plan to reduce the size of the neural network to decrease memory usage and improve efficiency. For large SAT problems, sampling methods can be explored to construct the graph. Techniques such as sampled message passing could further reduce the overhead of GNNs. Minimizing GNN calls is another crucial focus. For example, using the RDC-once method proposed in this work, the GNN is called only initially or at specific points to generate information that guides subsequent decisions. Currently, RDC-once only calls the neural network once initially, without using any new information generated during the solving process. In the future, we will explore a hybrid approach that integrates initial GNN-generated information with default strategies for decision-making. Additionally, training specialized models for domain-specific problems(Li et al., 2022) may yield significant performance improvements for these applications.
+
+Further Improvements for RDC-SAT. The development of D&C solvers has lagged behind portfolio-based parallel solvers in recent years. Many D&C solvers, including the Painless framework we used, have not incorporated high-performance sequential solvers like Kissat and CaDiCal or recent optimizations. RDC-SAT currently uses default configurations of the Painless framework, such as MapleCOMSPS as the sequential solver. Replacing MapleCOMSPS with better-performing solvers, optimizing other components, and exploring different configurations could lead to substantial improvements. Additionally, since our experiments aimed to evaluate RDC-SAT's performance, we trained the model only on the random dataset without training a general model. Training on larger and more diverse datasets could enhance RDC-SAT's effectiveness and generalization. Furthermore, RDC-SAT uses limited information to represent states, focusing on key features of variables and clauses. Future work could explore incorporating additional features to achieve greater performance enhancements.
+
+In future work, we plan to implement these optimizations and evaluate the performance of RDC-SAT on larger datasets to further enhance its effectiveness.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by National Key Research and Development Program of China Grant 2022YFB3305102, Natural Science Foundation of China Grant 61902011, and State Key Laboratory of Complex & Critical Software Environment (CCSE-2024ZX-16).
+
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+
+# A RELATED WORK
+
+# A.1 DIVIDE-AND-CONQUER SAT SOLVERS AND THEIR SPLITTING HEURISTICS
+
+Divide-and-Conquer (D&C) SAT solvers have gained prominence due to their ability to leverage parallel computing resources to tackle large and complex SAT problems efficiently. These solvers divide the problem into smaller subproblems, which are then solved independently. A crucial aspect of their performance is the heuristic used to select the best split variable, which significantly impacts the overall solving time and the balance of the resulting subspaces.
+
+Treengeling (Fleury & Heisinger, 2020) and Paracooba (Heisinger et al., 2020) are advanced SAT solvers that employ the Cube-and-Conquer method to harness parallelism. Treengeling utilizes a look-ahead heuristic during the Cube phase to divide the SAT problem into numerous subproblems, or cubes. This heuristic selects the split variable based on the number of unit propagations, aiming to create balanced subspaces. Treengeling generates more cubes than the available computational cores to ensure efficient utilization of all resources. Each cube is then solved independently by sequential CDCL solvers in the Conquer phase. Similarly, Paracooba extends the Cube-and-Conquer method to a distributed environment. It divides the SAT problem into many cubes using a look-ahead heuristic and distributes these cubes to different nodes across a network. Each node runs a CDCL solver to independently solve its assigned subproblems. By aggregating the computational power of multiple machines, Paracooba effectively handles significantly larger SAT problems, ensuring better load balancing and reducing overall solving time.
+
+In contrast to look-ahead techniques, some solvers use look-back methods that dynamically analyze the search performed by the solver, as well as formula statistics, to identify the best candidate at the current splitting point. For example, Ampharos (Audemard et al., 2016) picks the variable with the highest VSIDS activity, while MapleAmpharos (Nejati et al., 2017) uses a propagation-rate (PR) based splitting heuristic. This heuristic ranks variables by the ratio of the propagations they cause to the number of times they are branched on, selecting the highest-ranking variables for splitting. Another explored track is the number of flips of the variables (Audemard et al., 2014b;a). A flip occurs when a variable is propagated to the reverse of its last propagated value. Ranking the variables according to the number of their flips and choosing the highest one as a split variable helps to generate search subspaces with comparable computational time. This approach can also be used to limit the number of variables on which the look-ahead propagation is applied by preselecting a predefined percentage of variables with the highest number of flips. MaplePainlessDC (Nejati et al., 2020) uses a machine learning-based splitting heuristic to optimize divide-and-conquer parallel Boolean SAT solvers. The paper developed pairwise ranking models and minimum ranking models using a random forest classifier, comparing and selecting split variables based on structural features of the formula and dynamic probing statistics.
+
+The Painless Framework is a modular and high-performance parallel SAT solving framework that supports the implementation of various components, including multiple splitting heuristics. It allows developers to create parallel SAT solvers that can compete with state-of-the-art solvers. The framework implements several splitting heuristics, ensuring its solvers can match the performance of leading SAT solvers. Our reinforcement learning environment and the RDC-SAT solver are developed based on the Painless framework. This framework was also used for comparative experiments to evaluate the effectiveness of our approach.
+
+# A.2 OPTIMIZING SAT SOLVING WITH GNNS AND RL
+
+Graph Neural Networks (GNNs) and Reinforcement Learning (RL) have been increasingly applied to optimize SAT solving, showing promising results. Research like Graph-Q-Sat has utilized GNNs to learn branch heuristics in CDCL solvers, replacing VSIDS. Similarly, RL techniques have been used to train heuristics to enhance decision-making in SAT solvers. However, the overhead of calling GNNs for each variable decision in CDCL solvers is very high, making it difficult to apply to real-world problems. Consequently, recent research in this field has focused on reducing the overhead and frequency of GNN calls.
+
+Graph-Q-Sat(Kurin et al., 2020) is an approach that improves the heuristics of SAT solvers by employing GNNs and reinforcement learning. It uses value-based reinforcement learning to train a branching heuristic that significantly reduces the number of iterations compared to VSIDS. However,
+
+it requires calling GNNs at each decision point, which incurs high computational costs and does not reduce the actual wall-clock solving time. To mitigate this issue, other research, such as NeuroCore (Selsam & Bjørner, 2019), predicts the unsatisfiable core through periodic GNN inferences, reducing the frequency of GNN calls. While this approach enhances solver performance, it still demands substantial computational resources and relies heavily on GPUs. Neuroglue (Han, 2020) enhances SAT solvers by predicting glue variables, which are critical for efficient clause learning. This method periodically updates variable scores using GNN-based insights, significantly improving solver performance without incurring substantial computational costs. NeuroBack (Wang et al., 2023) builds upon these methods by making offline GNN predictions about variable phases before the SAT solving begins, thus eliminating the need for frequent online inferences and reducing computational costs. NeuroBack guides CDCL solvers like Kissat with a single GNN call for phase predictions, increasing performance on recent SAT competition benchmarks. SATformer (Shi et al., 2023) uses a GNN to extract clause representations only once, reducing computational costs, and applies a Transformer to predict unsatisfiability. Integrated into solvers like CaDiCaL and Kissat, SATformer improves solving efficiency over baseline methods. NeuroSelect (Liu et al., 2024) employs a GNN to guide the deletion of learned clauses, making only a single GNN call per instance. On instances with fewer than 400,000 nodes from the SAT Competition 2022, NeuroSelect improves performance over Kissat These methods primarily optimize branch heuristics and phase heuristics in CDCL solvers, but the overhead of GNNs remains a challenge.
+
+In contrast, our approach focuses on the importance of splitting heuristics in Divide-and-Conquer (D&C) solvers. The invocation frequency of these heuristics is relatively low, making it feasible to use GNNs without incurring high computational costs. Our RDC-GPU method leverages GNNs to significantly optimize D&C solver performance with minimal GNN calls. Even without GPUs, we can adopt a strategy similar to NeuroBack. By using RDC-CPU-once, we make a single GNN call to guide subsequent splitting decisions, thereby optimizing splitting heuristics while minimizing GNN overhead.
+
+# B IMPLEMENTATION DETAILS OF INTERACTIVE DIVIDE-AND-CONQUER ENVIRONMENT
+
+# B.1 DIVIDE-AND-CONQUER IN PAINLESS FRAMEWORK
+
+The Painless framework (Le Frioux et al., 2017; 2019) is a modular and high-performance parallel SAT-solving framework that is both flexible and ideal for comparing different algorithms. Within Painless, the D&C algorithm is implemented using a master-slave architecture that employs a workstealing strategy to achieve dynamic load balancing. Whenever a CPU core becomes idle, the master thread interrupts an active worker, splits its subproblem, and assigns it to two separate workers. Painless also supports clause sharing, allowing multiple workers to exchange learned clauses to speed up the solving process. This design makes the framework highly adaptable and efficient, empowering researchers to explore and compare various heuristics for parallel SAT solving.
+
+The Painless framework incorporates several D&C solvers, which can compete effectively with the leading solvers available. For the basic configuration in our experiment, we used the best-performing settings identified through experimentation. This includes utilizing MapleCOMSPS (Liang et al., 2016b) as the sequential solver, the "altoall" clause-sharing strategy, and the "clone" strategy. Specifically, MapleCOMSPS, which won the SAT Competition in 2016, serves as the sequential solver. The "altoall" clause-sharing strategy ensures that all solvers directly share clauses with an LBD below 4, which helps to improve the efficiency of the solving process. Additionally, the "clone" strategy allows a new worker, after requesting and dividing a partition, to duplicate the original solver and tackle the next problem. This helps to reuse learned clauses and internal solver information to accelerate problem-solving.
+
+Our Interactive Divide-and-Conquer Environment is primarily divided into two parts. First, we encapsulate the necessary components in the PainlessSolver within the Painless C++ environment, derived from painless-v2, The basic configuration is based on P-CLONE-FLIP (Le Frioux et al., 2019), and use Pybind11 to create a shared library. Then, we create a custom PainlessEnv in Python to facilitate interaction with the agent. PainlessSolver handles the solving process,
+
+![](images/b28ea337dacadb93bd9791abebdfdda73f76323c2ded29410c95652f4da16e68.jpg)  
+Figure 6: Interactive Divide-and-Conquer environment
+
+while PainlessEnv provides the interface for the agent to interact, allowing for dynamic state transitions and reward calculations.
+
+# B.2 PAINLESSSOLVER IN C++
+
+Algorithm 1: PainlessSolver::step  
+Input: action  
+Output: state, done  
+1 if there is a pending splitting request then  
+2 pass action as split variable to worker;  
+3 while not done do  
+4 if no globalEnding, no pending splitting requests, and no finished workers then  
+5 wait for signal;  
+6 parse finished workers and insert as leaf nodes in binary tree;  
+7 if there is a pending splitting request then  
+8 state = create graph from subproblem;  
+9 parse worker and insert as internal node in binary tree;  
+10 return state, done = false;  
+11 if globalEnding then  
+12 return root, done = true;
+
+We designed a new class, PainlessSolver, to handle all interactions and maintain statistics with the Python environment. When solving a new SAT problem, a PainlessSolver instance is created, initializing the Master thread, Worker threads, and Sharer thread, and setting the initial solving state. The Master thread manages the Worker threads, which use a CDCL solver to solve SAT problems. The Sharer thread handles the sharing of learned clauses among the Worker threads. Additionally, the PainlessSolver thread executes the logic described in the corresponding pseudocode. This class uses the solve() method to start the entire solving process. The interface built with Pybind11 enables the direct transmission of data generated by Painless to Python algorithms for processing and the reception of decision outputs from the Python environment to control the solving process. The core function step() of PainlessSolver is shown in Algorithm 1.
+
+Due to Python's Global Interpreter Lock (GIL), only one thread can execute Python bytecode at a time. In a D&C solver, multiple workers may request splits simultaneously. To manage these requests,
+
+we use a queue. When the master thread detects idle threads, it interrupts a worker, adds its instance to the pending split queue, and signals the PainlessSolver thread(line 5). The worker then blocks, waiting for the split variable. The PainlessSolver thread dequeues the worker instance, parses the problem state, adds it as an internal node to the binary tree, and sends the state to the Python environment(line 7-10). After processing, the PainlessSolver thread returns the split variable, allowing the worker to continue solving(line 2). While one worker waits, others continue solving their subproblems. Additionally, PainlessSolver parses finished workers and inserts them as leaf nodes in the tree(line 6). Once solving is complete, it returns the root node and sets done to true. Locks and signals ensure thread-safe communication and protect data structures.
+
+Finally, we used Pybind11 to package PainlessSolver into a shared library. This allows the shared library to be directly imported into Python, enabling interaction with PainlessSolver through its provided interfaces.
+
+# B.3 PAINLESSENV IN PYTHON
+
+To facilitate interaction between the agent and the environment, we defined a custom environment similar to gym. Env. OpenAI's gym library provides a widely-used framework for custom reinforcement learning environments. However, gym. Env is designed for linear state transitions, where each state follows a sequential order, and rewards are computed based on this continuous sequence. In these environments, the step function returns the "next state," which is not suitable for our case. In a divide-and-conquer reinforcement learning environment, each split creates two subproblems, forming a binary tree structure rather than a direct "forward" relationship between states. Additionally, it is not possible to immediately obtain the reward in divide-and-conquer, as this information is embedded within the tree structure. To address these challenges, we created a custom class PainlessEnv, mimicking the basic structure of the gym library but without inheriting it. This allows us to define state transitions and reward logic flexibly, accurately handling parallel and tree-structured transitions.
+
+PainlessEnv accepts a set of SAT problem paths and related configuration parameters as initial inputs, such as solver timeout, CPU core count, and levels of learned clauses. It provides reset and step methods for interaction. The reset method fully reinitializes the solving environment, creating a new PainlessSolver instance, loading a new SAT problem, and starting the solving process. The agent returns a split variable, which the step method passes to the PainlessSolver environment, then waits for the next split state while recording necessary training data. Upon solving the problem or timeout, PainlessEnv retrieves the root node of the binary tree, analyzes the solving process, and calculates the reward for each action. It performs a post-order traversal from the root node to compute the discounted rewards for all internal node states. This tree structure details the solving paths from root to leaf nodes, storing all this data in a replay buffer for subsequent machine learning model training.
+
+# C DETAILS OF REINFORCED DIVIDE-AND-CONQUER
+
+# C.1 NEURAL NETWORK
+
+Our approach utilizes a modified Graph Neural Network (GNN) structure similar to Graph-Q-Sat, replacing the DQN structure with an Actor-Critic network. As shown in Figure 7, the neural network encodes the SAT problem information as a graph. The Encoder transforms initial features into higher-dimensional representations, the Processor performs iterative message passing between nodes, edges, and global attributes, and the Decoder outputs variable, clause, and global features. The Actor network evaluates the variable features, outputs the probabilities for selecting each variable after removing invalid split variables, and the Critic network uses clause and global features to assess the current state's value. We implemented our neural network using PyTorch(Paszke et al., 2019) and PyTorch Geometric(PyG)(Fey & Lenssen, 2019).
+
+We adopted Graph Networks (Battaglia et al., 2018) as the basis of our GNN architecture. This framework is known for its versatility in handling various graph structures. The GN framework consists of three main parts: Encoder, Processor, and Decoder. The Encoder independently converts node, edge, and global features into higher-dimensional representations using MLPs. The Processor, the core component, updates these features through multiple layers of message passing, aggregating information from connected nodes and edges, and applying batch normalization to ensure scale
+
+![](images/8442dc9d5d66d81d145a746cf7c4cc3c5a26137a9995f803d1f56c345b43f0f3.jpg)  
+Figure 7: Network of RDC-SAT.
+
+invariance across nodes. The Decoder simplifies the complex graph structure into actionable outputs, retaining node and global features for the subsequent PPO usage while discarding edge features. Layer normalization (Ba et al., 2016) is applied within MLPs to expedite training convergence and reduce sensitivity to input scale variations.
+
+The Actor-Critic network leverages features derived from the GNN to perform variable selection and value estimation. The Actor network, responsible for decision-making, processes the variable features through an MLP and applies a softmax function to produce a probability distribution over the variables, excluding already chosen or invalid ones via a masking strategy. The Critic network evaluates the cumulative rewards from the current state using clause and global features. It employs attention pooling networks to aggregate clause nodes and global features, which are then fed into an MLP to generate the state value. This architecture allows the system to separate decision-making from value assessment, facilitating more precise policy adjustments and long-term reward optimization.
+
+# C.2 SETTINGS AND HYPERPARAMETERS
+
+We implemented a Distributed Proximal Policy Optimization (DPPO) framework, as shown in Figure 8, to accelerate data collection and training using one training process and multiple exploration processes. This framework includes a message queue for transferring collected data from exploration processes to the training process and a parameter server for synchronizing updated network parameters from the training process to the exploration processes. The message queue allows real-time transmission of states, actions, rewards, and subsequent states to the training process, which then stores them in a replay buffer for training. After updating the network parameters, the training process synchronizes these updates to the parameter server. Exploration processes check the parameter server for updates before solving new problems and update their parameters accordingly.
+
+Table 2 shows hyperparameters were used in our experiments:
+
+# D DATASETS
+
+# D.1 TRAINING DATA
+
+For our task involving reinforcement learning and graph neural networks, the most critical factors are problem size, difficulty, and structural richness. The ideal dataset should have a moderate problem size, which demands less memory during training; moderate difficulty, allowing for the rapid collection of substantial training data; and rich structure, enabling the GNN to learn more information. Although datasets like SAT Competition possess rich structural features, their generally high difficulty and larger size significantly slow down the training process for reinforcement learning. In contrast, random datasets offer highly controllable size and difficulty, and the learning clauses generated during the process can introduce additional structural features. Therefore, we chose to use a random dataset for our experiments.
+
+Our training dataset comes from the dataset provided by the study(Cameron et al., 2020). This study provides 11 datasets ranging from 100 to 600 variables, each containing 5000 satisfiable (SAT) and
+
+![](images/7e90953df85b5034f6eb36e87121da043a4f5a6b4058fb2f7cb51ebe01455d47.jpg)  
+Figure 8: Architecture of. Exploration agents interact with the PainlessEnv on CPU, collect training data, and send it to the message queue. The training agent on GPU samples data from the replay buffer to update the model parameters, which are then synchronized back to the exploration agents.
+
+5000 unsatisfiable (UNSAT) problems. We randomly selected 500 SAT and 500 UNSAT problems from the datasets with 300 and 350 variables as our training dataset. These problems have solving times ranging from a few seconds to several tens of seconds, and during the solving process, they generate thousands to tens of thousands of core and tier2 learned clauses. Consequently, the resulting graphs have relatively rich structural features.
+
+# D.2 EVALUATION DATA
+
+To investigate whether RDC-SAT has effectively learned the splitting heuristics, we randomly selected 200 SAT instances from the random dataset at the phase transition. This set includes 100 problems from the 450-1917 dataset and 100 from the 500-2129 dataset, with a 1:1 ratio of satisfiable to unsatisfiable instances. This dataset provides a suitable level of difficulty for the baseline solvers, enabling us to distinguish their performance.
+
+To evaluate whether RDC-SAT, trained on random datasets, can generalize to other types of more extensive, application-oriented datasets, we focused on small to medium-sized SAT problems from the SAT Competitions dataset. Some problems in the competition are extremely large, with many SAT instances having tens of millions of clauses. Applying GNNs to these problems would consume significant memory and computational resources. For such large-scale problems, our RDC-SAT solver automatically switches to the default splitting heuristics. Therefore, we primarily focus on the performance of various methods on small to medium-sized SAT problems. We first screened 400 problems from the SAT Competition 2023 (Balyo et al., 2023) dataset and selected those with a total number of variables and clauses less than one million, resulting in 305 SAT problems. These small to medium-sized problems constitute the majority of the SAT Competition.
+
+# EXPERIMENT DETAILS
+
+# E.1 BASELINES SELECTION
+
+The Painless framework provides three splitting heuristics: based on Flips, PR, and using CDCL solver branch heuristics to select split variables. We used these three methods for our evaluations. However, in a small-scale test, we found that the PR method did not perform well on the random dataset, leading us to exclude it from our random dataset tests. Additionally, we chose not to use MaplePainlessDC as a baseline. MaplePainlessDC(Nejati et al., 2020) is also a D&C solver based on the Painless framework, which uses a machine learning method to optimize the splitting heuristic. While the performance reported in the original paper exceeds that of Treengeling, it has been observed
+
+Table 2: Hyperparameters used in our experiments  
+
+<table><tr><td>Parameter</td><td>Value</td><td>Description</td></tr><tr><td colspan="3">PainlessEnv Configuration</td></tr><tr><td>timeout</td><td>1000</td><td>Maximum time (in seconds) allowed for exploration agent</td></tr><tr><td>ncpus</td><td>16</td><td>The number of threads used by each exploration agent</td></tr><tr><td>use_learnt</td><td>2</td><td>In addition to the original clause, two levels of learning clauses, core and tire2, are used to build the graph</td></tr><tr><td>node feats</td><td>4</td><td>Number of node features</td></tr><tr><td>edge feats</td><td>2</td><td>Number of edge features</td></tr><tr><td>global feats</td><td>4</td><td>Number of global features</td></tr><tr><td>use_LRB</td><td>true</td><td>Use LRB as branch heuristic</td></tr><tr><td colspan="3">Network Configuration</td></tr><tr><td>encoder_out_dims</td><td>(32, 32, 32)</td><td>Dimensions for the encoder outputs (node, edge, global)</td></tr><tr><td>core_out_dims</td><td>(64, 64, 32)</td><td>Dimensions for the core outputs (node, edge, global)</td></tr><tr><td>decoder_out_dims</td><td>(32, 1, 32)</td><td>Dimensions for the decoder outputs (node, edge, global)</td></tr><tr><td>core_steps</td><td>4</td><td>Number of processing steps in the Processor</td></tr><tr><td>n Hidden</td><td>1</td><td>Number of hidden layers in Processor&#x27;s MLP</td></tr><tr><td>hidden_size</td><td>64</td><td>Size of hidden layers in GNN&#x27;s MLP</td></tr><tr><td>e2v_aggregator</td><td>sum + BN</td><td>Aggregation method for edge to vertex. After sum aggregation, we use batch normalization(Ioffe &amp; Szegedy, 2015) to normalize the features</td></tr><tr><td>independent_block_layers</td><td>0</td><td>Number of hidden layer of Encoder and Decoder</td></tr><tr><td>n HiddenActor</td><td>2</td><td>Number of hidden layer of actor&#x27;s MLP</td></tr><tr><td>n Hidden_critic</td><td>2</td><td>Number of hidden layer of critic&#x27;s MLP</td></tr><tr><td>activation_function</td><td>LeakyRelu(Maas et al., 2013)</td><td>Activation function unsed in MLP</td></tr><tr><td colspan="3">DPPO Configuration</td></tr><tr><td>optimizer</td><td>Adam(Kingma &amp; Ba, 2014)</td><td>Optimizer used</td></tr><tr><td>lr_gnn</td><td>1e-3</td><td>Learning rate for the GNN</td></tr><tr><td>lr_ACTor</td><td>1e-3</td><td>Learning rate for the actor network</td></tr><tr><td>lr_critic</td><td>1e-3</td><td>Learning rate for the critic network</td></tr><tr><td>policy_loss_weight</td><td>1.0</td><td>Weight of the policy loss</td></tr><tr><td>value_loss_weight</td><td>0.5</td><td>Weight of the value loss</td></tr><tr><td>entropy_loss_weight</td><td>0.1</td><td>Weight of the entropy loss</td></tr><tr><td>eps Clip</td><td>0.2</td><td>Clipping parameter for PPO</td></tr><tr><td>epsilon_start</td><td>1.0</td><td>Starting value for epsilon in epsilon-greedy exploration</td></tr><tr><td>epsilon_final</td><td>0.01</td><td>Final value for epsilon in epsilon-greedy exploration</td></tr><tr><td>epsilon Decay</td><td>200</td><td>Decay rate for epsilon</td></tr><tr><td>normalize Reward</td><td>True</td><td>Whether to normalize rewards</td></tr><tr><td>gamma</td><td>1.0</td><td>Discount factor for future rewards</td></tr><tr><td>train_epochs</td><td>1</td><td>Number of training epochs</td></tr><tr><td>train_max_steps</td><td>10000</td><td>Maximum number of training steps</td></tr><tr><td>sample_max_nodes</td><td>300000</td><td>Maximum number of nodes of one batch to sample</td></tr><tr><td>num_of_process</td><td>5</td><td>Number of exploration processes</td></tr></table>
+
+from the experimental results of SAT Competition 2020(Nejati & Ganesh) that MaplePainlessDC can produce incorrect results in a small number of SAT instances. This potential unreliability in solving performance led us to exclude MaplePainlessDC as a baseline in our evaluations.
+
+# E.2 EXPLANATION FOR USING LRB OVER VSIDS IN RANDOM DATASET EXPERIMENTS
+
+The decision to use different branching heuristics for random and SAT Competition benchmarks stems from the observed poor performance of VSIDS on random datasets, as commonly noted in empirical studies. We conducted a small experiment using VSIDS as the branching heuristic on 100 instances from the 400-1704 random dataset with a time limit of 1000 seconds. The results of this experiment are presented in Table 3. This performance is significantly worse compared to using LRB as the branching heuristic (see Table 4 for a comparison). Therefore, we opted for Maple's LRB branching heuristics for both training and evaluation on the random datasets. It is also worth noting that even with VSIDS, RDC-SAT still significantly outperforms the baseline.
+
+Table 3: Performance Comparison of Splitting Heuristics on 400-1704 random dataset with VSIDS as Branching Heuristics  
+
+<table><tr><td>Splitting Heuristics</td><td>Clock time (s)</td><td>PAR2</td><td>Solved</td></tr><tr><td>RDC-GPU-always</td><td>388.23</td><td>588.18</td><td>80</td></tr><tr><td>RDC-CPU-once</td><td>405.48</td><td>703.34</td><td>77</td></tr><tr><td>FLIPS</td><td>460.70</td><td>720.72</td><td>74</td></tr><tr><td>VSIDS</td><td>502.28</td><td>852.20</td><td>65</td></tr></table>
+
+# E.3 MORE EVALUATION RESULT ON RANDOM DATASET
+
+Table 4 shows a detailed comparison of various splitting heuristics across different datasets. This table includes all instances, including those that no solver could solve within the time limit.
+
+Firstly, for relatively easy SAT problems, such as those in the 350-1491 dataset, using the default splitting heuristics like FLIPS and LRB can yield quick results. In these cases, the overhead of using a GNN-based approach might actually slow down the process. For example, FLIPS and LRB both performed better than RDC-GPU-always and RDC-CPU-once in terms of average solving time. This is because the simpler problems do not require the sophisticated analysis that GNN provides, and the additional computation introduces unnecessary delay. However, as the problem difficulty increases, the advantages of RDC-SAT become more apparent. In the 450-1917 dataset, RDC-GPU-always significantly outperformed traditional methods, with a total average solving time of 526.21 seconds compared to FLIPS' 827.07 seconds and LRB's 780.34 seconds. On the most challenging dataset, 500-2129, RDC-GPU-always not only achieved a lower average solving time but also solved more instances than the baselines. Specifically, RDC-GPU-always solved 30 instances in this dataset, whereas the best baseline method solved only 26 instances, representing a  $15.38\%$  increase in the number of solved instances.
+
+Additionally, the table reveals that the PR splitting heuristic performed very poorly on the 350-1491 dataset, with an average solving time of 61.65 seconds. This performance suggests that the PR method may not be suitable for solving random SAT problems. Due to this, we did not test the PR method on larger datasets.
+
+Table 4: Comparison of splitting Heuristics across datasets  
+
+<table><tr><td>Dataset</td><td>Splitting Heuristics</td><td>SAT Avg (s)</td><td>#SAT</td><td>UNSAT Avg (s)</td><td>#UNSAT</td><td>Total Avg (s)</td><td>PAR2</td></tr><tr><td rowspan="5">random 350-1491</td><td>RDC-GPU-always</td><td>12.34</td><td>50</td><td>19.80</td><td>50</td><td>16.03</td><td>16.03</td></tr><tr><td>RDC-CPU-once</td><td>11.16</td><td>50</td><td>17.99</td><td>50</td><td>14.58</td><td>14.58</td></tr><tr><td>FLIPS</td><td>10.76</td><td>50</td><td>17.22</td><td>50</td><td>13.99</td><td>13.99</td></tr><tr><td>LRB</td><td>11.02</td><td>50</td><td>17.96</td><td>50</td><td>14.49</td><td>14.49</td></tr><tr><td>PR</td><td>22.36</td><td>50</td><td>103.83</td><td>50</td><td>61.65</td><td>61.65</td></tr><tr><td rowspan="4">random 400-1704</td><td>RDC-GPU-always</td><td>19.14</td><td>50</td><td>63.38</td><td>50</td><td>41.27</td><td>41.27</td></tr><tr><td>RDC-CPU-once</td><td>18.29</td><td>50</td><td>64.35</td><td>50</td><td>41.32</td><td>41.32</td></tr><tr><td>FLIPS</td><td>18.16</td><td>50</td><td>80.27</td><td>50</td><td>49.21</td><td>49.21</td></tr><tr><td>LRB</td><td>17.90</td><td>50</td><td>67.84</td><td>50</td><td>42.87</td><td>42.87</td></tr><tr><td rowspan="4">random 450-1917</td><td>RDC-GPU-always</td><td>291.48</td><td>50</td><td>760.94</td><td>49</td><td>526.21</td><td>576.21</td></tr><tr><td>RDC-CPU-once</td><td>353.63</td><td>50</td><td>835.91</td><td>49</td><td>594.75</td><td>644.77</td></tr><tr><td>FLIPS</td><td>390.83</td><td>50</td><td>1263.31</td><td>48</td><td>827.07</td><td>927.07</td></tr><tr><td>LRB</td><td>341.64</td><td>50</td><td>1219.04</td><td>47</td><td>780.34</td><td>930.34</td></tr><tr><td rowspan="4">random 500-2129</td><td>RDC-GPU-always</td><td>3072.05</td><td>26</td><td>4777.20</td><td>4</td><td>3924.63</td><td>7424.63</td></tr><tr><td>RDC-CPU-once</td><td>3397.68</td><td>23</td><td>4804.05</td><td>4</td><td>4100.86</td><td>7750.86</td></tr><tr><td>FLIPS</td><td>3255.54</td><td>24</td><td>4898.27</td><td>2</td><td>4076.91</td><td>7776.91</td></tr><tr><td>LRB</td><td>3627.84</td><td>20</td><td>4898.62</td><td>3</td><td>4263.23</td><td>8113.23</td></tr></table>
+
+# E.4 DOMAIN-SPECIFIC BENCHMARK COMPARISON OF RDC-SAT AND SOTA PORTFOLIO SOLVERS
+
+SAT Competition results indicate that D&C solvers generally lag behind portfolio approaches in average performance. However, D&C solvers demonstrate unique advantages in specific scenarios.
+
+For instance, we conducted a small-scale experiment on a dataset containing 100 randomly selected instances from the miter benchmarks, which are used to verify circuit equivalence. The timeout was set to 1000 seconds, with all methods using 16 threads and VSIDS as the branch heuristic for the Sequential Solver. Using the model trained in our work. Table 5 compares our RDC-SAT with the D&C solver FLIPS (Le Frioux et al., 2019) and PRS (Chen et al., 2023) – the SAT Competition 2023 Parallel Track champion portfolio solver.
+
+Table 5: Performance Comparison on Circuit Equivalence Verification  
+
+<table><tr><td>Method</td><td>Solved</td><td>Avg. Time (s)</td><td>PAR2</td></tr><tr><td>RDC-SAT (D&amp;C)</td><td>80</td><td>313.64</td><td>513.64</td></tr><tr><td>FLIPS (D&amp;C)</td><td>76</td><td>333.66</td><td>573.66</td></tr><tr><td>PRS (Portfolio)</td><td>79</td><td>385.44</td><td>595.44</td></tr></table>
+
+This result shows that D&C solvers can outperform Portfolio solvers in specific domains. In this experiment, RDC-SAT solved the most instances and achieved the lowest PAR2, reducing PAR2 by  $13.74\%$  compared to the SOTA Portfolio solver PRS. The RDC-SAT model, trained on a random dataset, generalized effectively to the circuit equivalence dataset.
+
+# F THREATS TO VALIDITY
+
+During the training phase of our D&C SAT solver using reinforcement learning, several potential threats to validity may impact the results. Firstly, due to Python's Global Interpreter Lock (GIL) enforcing single-threaded execution for Python bytecode, we use a message queue to manage all split requests. When multiple threads simultaneously request splits, only one worker can call the neural network to obtain the split point, while other threads enter a waiting state. This restriction may lead to inefficiencies and does not fully replicate the actual D&C solver's runtime environment. Additionally, during training, multiple threads simultaneously calling the neural network, along with the training process itself, can cause GPU resource contention, leading to delays in split decisions and affecting training performance. Consequently, we opted to use CPUs for neural network calls during training to avoid GPU contention. However, this results in overestimated split times during training, as it does not accurately reflect the performance of the D&C solver in a real-world environment. Furthermore, our current setup limits the efficient use of computational resources, particularly GPUs. If more computational resources were available, such as additional GPUs for running the DPPO framework, this issue could be significantly mitigated. Enhanced computational resources would allow for more realistic training conditions, better reflecting the actual operational environment of the D&C solver, and potentially leading to more effective learning outcomes. These factors may result in the model's performance not reaching its optimal potential. With more computational resources to train the model and better simulate actual environments, RDC-SAT could achieve better results.
+
+Another potential threat to validity pertains to the experimental evaluation of our solvers. All experiments were conducted on a machine equipped with a CPU featuring 144 logical cores, an A100 GPU, and 256 GB of RAM. Due to the extensive time required for the experiments, we executed multiple solvers in parallel, allocating 16 threads to each solver. For the RDC-CPU-once configuration, the neural network calls were restricted to using a maximum of 16 threads. Although the number of threads used per solver is less than the total logical cores, the total number of threads during the SAT Competition experiments exceeded the number of physical cores, potentially leading to resource contention. This contention could affect performance, but it should impact all solvers equally, thus maintaining fairness in the comparisons. We primarily used small to medium-sized SAT problems, so we did not observe significant memory contention.

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+# LEARNING STRUCTURED UNIVERSE GRAPH WITH OUTLIER OOD DETECTION FOR PARTIAL MATCHING
+
+Zetian Jiang $^{1\dagger}$ , Jiaxin Lu $^{3\dagger}$ , Haozhao Fan $^{1}$ , Tianzhe Wang $^{1}$ , Junchi Yan $^{12*}$
+
+$^{1}$ Sch. of Computer Science & Sch. of Artificial Intelligence, Shanghai Jiao Tong University  
+$^{2}$ Shanghai Artificial Intelligence Laboratory  
+$^{3}$ Department of Computer Science, University of Texas at Austin
+
+{maple_jzt, usedtobe, yanjunchi}@sjtu.edu.cn, lujiaxin@utexas.edu
+
+# ABSTRACT
+
+Partial matching is a kind of graph matching where only part of two graphs can be aligned. This problem is particularly important in computer vision applications, where challenges like point occlusion or annotation errors often occur when labeling key points. Previous work has often conflated point occlusion and annotation errors, despite their distinct underlying causes. We propose two components to address these challenges: (1) a structured universe graph is learned to connect two input graphs  $\mathbf{X}_{ij} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$ , effectively resolving the issue of point occlusion; (2) an energy-based out-of-distribution detection is designed to remove annotation errors from the input graphs before matching. We evaluated our method on the Pascal VOC and Willow Object datasets, focusing on scenarios involving point occlusion and random outliers. The experimental results demonstrate that our approach consistently outperforms state-of-the-art methods across all tested scenarios, highlighting the accuracy and robustness of our method.
+
+# 1 INTRODUCTION
+
+Graph matching has been a fundamental task in traditional computer vision for a long time, applied in areas such as structure from motion (Vijayanarasimhan et al., 2017), object tracking (Nam & Han, 2016; Iqbal et al., 2017), optical flow (Sun et al., 2018), stereo matching (Luo et al., 2016; Chang & Chen, 2018), and pose estimation (Gasse et al., 2019). With the rise of deep learning, integrating graph matching and neural networks has led to significant improvements in accuracy and efficiency (Zanfir & Sminchisescu, 2018; Wang et al., 2019), making it more suitable for downstream tasks. Beyond these, graph matching has also shown great potential in other fields, such as bioinformatics for protein structure comparison (Luo et al., 2021) and drug design (Kriege et al., 2019), as well as in social network analysis for community detection and network alignment (Chen et al., 2020).
+
+In particular, graph matching aims to find node correspondence between graph-structured data via both node and edge affinity (Gold & Rangarajan, 1996; Cho et al., 2010). It is a well-known NP-Hard problem and is often formulated as the quadratic assignment problem (QAP):
+
+$$
+\begin{array}{l} \max  _ {\mathbf {X}} \operatorname {v e c} (\mathbf {X}) ^ {\top} \mathbf {K} \operatorname {v e c} (\mathbf {X}) \\ s. t. \quad \mathbf {X} \in \{0, 1 \} ^ {n \times n}, \mathbf {X} \mathbf {1} _ {n} = \mathbf {1} _ {n}, \mathbf {X} ^ {\top} \mathbf {1} _ {n} = \mathbf {1} _ {n}. \\ \end{array}
+$$
+
+(1)
+
+Here,  $\mathbf{X}$  represents a binary permutation matrix, where  $\mathbf{X}_{ij} = 1$  indicates that node  $i$  in the first graph corresponds to node  $j$  in the second graph.  $\mathbf{K} \in \mathbf{R}^{n^2 \times n^2}$  is a hand-crafted or learned affinity matrix whose diagonal (off-diagonal) encodes node (edge) affinities.
+
+While theoretically elegant, the QAP's assumption of perfect one-to-one node correspondence between graphs rarely holds in real-world applications. Practical scenarios often involve partial matching, where only subsets of nodes between graphs correspond, as illustrated in Figure 1. Traditional approaches attempted to address this limitation by introducing dummy nodes with zero affinity. However, this solution proves inadequate for handling real-world challenges such as occlusions and annotation errors.
+
+![](images/667aa2463760c5aa4bf8c871a8ed9e08796a35ea3052b8a4f87372404355109f.jpg)  
+Figure 1: Overview of the problem setting and UGM. Partial Matching: Due to occlusion, viewpoint differences, and annotation errors, only a subset of key points on each graph can be matched. Left: Prior works have limited capability in detecting and filtering outliers. Right: UGM employs two key components to improve partial matching performance: (1) a universe graph that serves as a bridge to aggregate matches from different graphs, and (2) an outlier filter to remove spurious matches. This design ensures that only inliers are matched, significantly enhancing partial matching performance.
+
+Several recent works have explored ways to overcome this limitation. BBGM (Rolínek et al., 2020a), for instance, bypasses the constraint that each node must find a specific match by learning a negative affinity score. However, this approach heavily relies on the neural network's capacity, and the actual performance gains are often limited. GCAN (Jiang et al., 2022) reformulates graph matching as an integer linear programming (ILP), introducing a placeholder node for each graph to absorb outliers. While this helps, learning for ILP remains a challenging task. AFAT (Wang et al., 2023) predicts which node match to discard by analyzing matched points and the affinity matrix. However, as a post-processing step, its effectiveness is limited because outliers may have already disrupted the initial matching, weakening its ability to correct errors. Another approach, URL (Nurlanov et al., 2023), learns a universe point representation, which serves as a central reference for key point alignment across all images. Although it addresses the issue of point occlusion, it still suffers from redundant matches as the erroneous annotation outliers fail to find the corresponding universe point representation.
+
+A key insight often overlooked by these literature is the distinction between point occlusion and annotation errors: occlusion-caused outliers may find correspondences in other graphs, while annotation errors are inherently meaningless. Building on this observation, we propose Universe Graph Matching (UGM), which addresses these challenges through two key innovations: (1) A latent universe graph learning approach that utilizes the structured universe graph to serve as a bridge for matching graph pairs  $\mathbf{X}_{ij} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$ . This method leverages the inherent structural information from input graphs, enhancing the accuracy and robustness of the learned universe graph for handling point occlusions. (2) An energy-based out-of-distribution detection method that effectively identifies and removes annotation errors at test time. Through training with a margin loss, we effectively differentiate between the energy scores of erroneous annotation outliers and valid in-distribution keypoints. By removing annotation errors, we significantly reduce the complexity of the partial matching problem (as we will discuss in Sec. 3.3), thereby improving overall matching accuracy.
+
+# Our primary contributions are:
+
+- We propose Latent Universe Graph Learning, which enhances universe matching robustness through combined node features and structural information.  
+- We are the first to introduce the Energy-based OOD Detection for annotation error identification, enabling pre-matching outlier filtering for improved accuracy.  
+- We evaluate the proposed UGM approach on keypoint matching tasks under occlusion and random outlier settings using both Pascal VOC and Willow datasets. Our experimental results demonstrate that UGM consistently outperforms state-of-the-art methods, highlighting the effectiveness and robustness of our method.
+
+# 2 RELATED WORKS
+
+# 2.1 CLASSICAL GRAPH MATCHING
+
+Two-graph matching has been widely studied, which involves constructing graphs for the objects to be matched and aligning the nodes and edges between the two graphs. (Gold & Rangarajan, 1996) solves graph matching via a graduated assignment (GA) algorithm, which computes the partial derivative
+
+of the objective function with Taylor expansion to turn the GM problem into a linear assignment problem. RRWM (Cho et al., 2010) constructs an association graph for matching graphs and applies the random walk algorithm with matching constraints. RRWHM (Lee et al., 2011) is the extended version of RRWM where the association graph is built with higher-order geometric information extracted from the graph. DS++, DS* (Dym et al., 2017; Bernard et al., 2018) design different ways to tightly relax GM to a convex problem and then project the results back to the original solution space. LPMP (Swoboda et al., 2017) proposes several additional Lagrangian relaxations of the graph matching problem and o leading solvers for this problem optimize the Lagrange decomposition duals with sub-gradient and dual ascent updates.
+
+In addition, numerous works have focused on matching multiple graphs, which is called Multi-Graph Matching. There are mainly two forms of methods. The first solves the problem in discrete space, which includes composition-based method CAO (Yan et al., 2015b), MGM-Floyd (Jiang et al., 2021), tree-structure-based approach MatchOpt (Yan et al., 2015a), DPMC (Wang et al., 2020b), and Majorize-Minimization framework based approach M3C (Lu et al., 2024). Another line of works (Pachauri et al., 2013; Huang & Guibas, 2013; Chen et al., 2014; Wang et al., 2018) etc. tries to relax the MGM problem into a continuous space and re-project the solution to obtain a discrete matching result.
+
+# 2.2 DEEP LEARNING OF GRAPH MATCHING
+
+Deep learning has recently been applied to graph matching on images (Zanfir & Sminchisescu, 2018), whereby CNN extracts node features from images followed by spectral matching and is learned using a regression-like node correspondence supervision. This work is improved via introducing GNN to encode structural (Wang et al., 2019; Wang et al., 2020) or geometric (Zhang & Lee, 2019) information, with a combinatorial loss based on cross-entropy loss, and Sinkhorn network (Adams & Zemel, 2011) as a differential matching solver. The work (Yu et al., 2020) extends PCA (Permutation loss and Cross-graph Affinity GM) (Wang et al., 2019) by edge embedding and Hungarian-based (Kuhn, 1955) attention mechanism to stabilize end-to-end training. BBGM (Blackbox Deep Graph Matching) (Rolinić et al., 2020b) proposes a better front-end feature extraction backbone with Spline Convolution (Fey et al., 2018), and the gradient is backpropagated by a fitting linear gradient of the discrete graph matching solver (Pogancić et al., 2019). NGM (Neural Graph Matching Nets) (Wang et al., 2021; 2020a) proposes to address the most general Lawler's QAP form, based on the novel feature extractors e.g. (Rolinić et al., 2020a) with the proposed learnable graph matching solver. DLGM (Deep Latent Graph Matching) (Yu et al., 2021), based on BBGM, predicts consistent graph topology utilizing deterministic and generative models to improve the matching quality. GCAN (Jiang et al., 2022) formulates the graph matching problem as an Integer Linear Programming problem and explores a novel node feature extraction framework with attention. AFAT (Wang et al., 2023) uses the attention model to predict which node match to discard by analyzing matched points and the affinity matrix. COMMON (Lin et al., 2023) adopts a contrastive learning approach, utilizing the infoNEC loss to supervise training similarity matrices for both edges and nodes. URL (Nurlanov et al., 2023) introduces a novel universe point representation learning method, applying universe key point matching to the graph matching field for the first time.
+
+# 2.3 OUT-OF-DISTRIBUTION DETECTION
+
+Out-of-distribution (OOD) detection is a growing field aimed at identifying data points that deviate from the distribution on which a model was trained. This is crucial for improving the robustness and safety of machine learning systems. Many OOD detection methods can be applied to pre-trained models without modifying the training process. These methods often rely on statistics from the model's output, such as maximum softmax probability, to distinguish between in-distribution (ID) and OOD data (Hendrycks & Gimpel, 2016; Liang et al., 2017). Energy-based OOD detection methods, on the other hand, assign an energy score to each input, where OOD data typically results in higher energy values (Liu et al., 2020; Bao et al., 2024). Generative models, such as VAEs and GANs, have also been used to estimate the likelihood that a given data point belongs to the training distribution (Nalisnick et al., 2018). More recently, self-supervised tasks have been employed to improve OOD detection by learning representations that are highly correlated with the training distribution, enabling the model to identify deviations (Sehwag et al., 2021). Contrastive learning has further advanced OOD detection by ensuring that OOD data points are positioned far from ID data in the learned representation space (Winkens et al., 2020).
+
+![](images/1c0c14abd129676c7ba68f9b9e04e1156b3a01436f2ce0106c6e5d9b3bfb2576.jpg)  
+Figure 2: Pipeline of Universe Graph Matching. The pipeline consists of two main components: 1) structured universe graph learning, which includes the training of node and edge affinities, and 2) outlier ood detection through energy score. The dark blue arrows indicate the training phase, while the light blue arrows represent the inference phase.
+
+# 3 METHODOLOGY
+
+In this section, we will introduce our method Universe Graph Matching (UGM) from two parts: structured universe graph learning and outlier ood detection in Section 3.1 and 3.2, respectively. The overall pipeline is shown in Figure 2, and more implementation and training details are provided in Appendix B. Meanwhile, we also discuss UGM in perspective of problem formulation and analyze its strength as well as drawbacks in Section 3.3.
+
+# 3.1 STRUCTURED UNIVERSE GRAPH LEARNING
+
+We begin by defining the structured universe graph, which serves as the foundation for our approach. The universe graph  $\mathbf{U} = \{\mathbf{N}_u\in \mathbf{R}^{n_u\times d},\mathbf{E}_u\in \mathbf{R}^{m_u\times d}\}$  is composed of both node embeddings  $\mathbf{N}_u$  and edge embeddings  $\mathbf{E}_u$ . The number of nodes  $n_u$  corresponds to the distinct types of key points present in the input graphs. We define this latent  $\mathbf{U}$  as a complete, directed graph, where every pair of nodes is connected by a directed edge, resulting in  $m_{u} = n_{u}\times (n_{u} - 1)$  edges. This ensures that any edge in the input graph will have a corresponding edge in the universe graph. Moreover, the directed edges capture the inherent asymmetry in real-world relationships between key points, such as varying geometric or contextual dependencies, offering greater flexibility in encoding complex structural information.
+
+We now introduce the pipeline to match the latent universe graph  $\mathbf{U}$  to the input graph  $\mathbf{G}_i$ . The input graph  $\mathbf{G}_i = \{\mathbf{N}_i \in \mathbf{R}^{n_i \times d}, \mathbf{E}_i \in \mathbf{R}^{m_i \times d}\}$  consists of node features  $\mathbf{N}_i$  and edge features  $\mathbf{E}_i$ . As in previous works, these features are initially extracted using a CNN-based backbone, followed by refinement through a GNN. Node affinity  $\mathbf{K}_n$  and Edge affinity  $\mathbf{K}_e$  are constructed by:
+
+$$
+\mathbf {K} _ {n} = \mathbf {N} _ {i} \mathbf {N} _ {u} ^ {\top}, \mathbf {K} _ {e} = \mathbf {E} _ {i} \mathbf {E} _ {u} ^ {\top}. \tag {2}
+$$
+
+The matching between the universe graph  $\mathbf{U}$  and the input graph  $\mathbf{G}_i$  is then obtained using a traditional graph matching solver, such as LPMP (Swoboda et al., 2017):
+
+$$
+\mathbf {X} _ {i u} = G M S o l v e r \left(\mathbf {K} _ {n}, \mathbf {K} _ {e}\right). \tag {3}
+$$
+
+In our training process, we use the ground truth universe matching  $\mathbf{X}_{iu}^{gt}$ , instead of ground truth pairwise matching  $\mathbf{X}_{ij}^{gt}$ , as the supervision signal.  $\mathbf{X}_{iu}^{gt} \in \{0,1\}^{n_i \times n_u}$  is a permutation matrix which denotes the ground truth correspondence between input graph  $\mathbf{G}_i$  and universe latent graph. In fact, the universe matching set  $\{\mathbf{X}_{iu}|1 \leq i \leq N\}$  and the pairwise matching set  $\{\mathbf{X}_{ij}|1 \leq i,j \leq N\}$  are fully equivalent and can be obtained via non-negative factorisation (Bernard et al., 2019):
+
+$$
+\mathcal {X} _ {p} = \left( \begin{array}{c c c c} \mathbf {X} _ {0 0} & \mathbf {X} _ {0 1} & \dots & \mathbf {X} _ {0 N} \\ \mathbf {X} _ {1 0} & \mathbf {X} _ {1 1} & \dots & \mathbf {X} _ {1 N} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf {X} _ {N 0} & \mathbf {X} _ {N 1} & \dots & \mathbf {X} _ {N N} \end{array} \right) = \left( \begin{array}{c} \mathbf {X} _ {0 u} \\ \mathbf {X} _ {1 u} \\ \vdots \\ \mathbf {X} _ {N u} \end{array} \right) \left(\mathbf {X} _ {0 u} ^ {\top} \quad \mathbf {X} _ {1 u} ^ {\top} \quad \dots \quad \mathbf {X} _ {N u} ^ {\top}\right) = \mathcal {X} _ {u} \mathcal {X} _ {u} ^ {T}. \tag {4}
+$$
+
+Even if some pairwise matchings are missing, we can still recover the universe matching by first applying spectral decomposition and then projecting the result back into the discrete space (Pachauri et al., 2013). It is worth noting that, in most computer vision tasks, the matching matrix is typically
+
+generated from keypoint annotations, and these annotations naturally serve as the ground truth universe matching.
+
+In practice, we cannot directly supervise  $\mathbf{X}_{iu}$  with  $\mathbf{X}_{iu}^{gt}$ , as most graph matching algorithms are non-differentiable. Even when using differentiable algorithms, such as RRWM (Cho et al., 2010), the high number of iterations would make gradient-based optimization difficult. Therefore, instead of supervising the solution, we directly act on the node and edge affinities. By treating the universe's nodes and edges indices as labels, we apply the cross-entropy loss to the node affinity  $\mathbf{K}_n$  and edge affinity  $\mathbf{K}_e$  for training:
+
+$$
+\mathcal {L} _ {\text {g r a p h}} = C E \left(\operatorname {s o f t m a x} \left(\mathbf {K} _ {n}\right), y _ {n}\right) + C E \left(\operatorname {s o f t m a x} \left(\mathbf {K} _ {e}\right), y _ {e}\right) \tag {5}
+$$
+
+where
+
+$$
+y _ {n} = \arg \max  \left(\mathbf {X} _ {i u} ^ {g t}\right), y _ {e} = y _ {n} (E d g e I D (0)) \times n _ {u} + y _ {n} (E d g e I D (1)). \tag {6}
+$$
+
+Note  $EdgeID \in \mathbb{N}^{2 \times m}$  denotes directed edges in the graph.  $EdgeID(0)$  denotes the indices of the source nodes and  $EdgeID(1)$  denotes the indices of the target nodes. Another important consideration in the matching process is the diversity of categories. The input may consist of pairs from different categories, each ideally requiring its own distinct latent universe graph. A straightforward approach would be to learn a separate graph representation for each category. However, this method would cause the latent universe graph to grow excessively as the number of categories increases. Consequently, the complexity of the graph matching solver could escalate quadratically or even worse, making it impractical for large-scale problems.
+
+To address this, we propose learning a single latent universe graph, where key points from different categories are mapped to the same universe node. For example, the left front wheel of a car and the left handlebar of a bicycle can share the same universe node embedding. In fact, a single universe node embedding is shared among all categories, allowing the graph's size to be limited by the maximum number of points across all categories. The same principle applies to edge embeddings, which further reduces space complexity.
+
+However, since the latent universe graph shares node and edge embeddings for all categories, the embeddings  $\mathbf{N}_u$  and  $\mathbf{E}_u$  inherently lack category-specific information. To reintroduce this information for affinity construction, we add a class embedding  $\mathbf{C}$  to each key point feature  $\mathbf{F}$  extracted by CNN-based backbone. This ensures that the nodes  $\mathbf{N}_i$  and edges  $\mathbf{E}_i$  of the input graph still carry the necessary class information. The class embedding is derived from the global features  $\mathbf{H}$  of the input images pair as follows:
+
+$$
+\mathbf {C} = \operatorname {E n c o d e r} \left(\mathbf {H} _ {i}, \mathbf {H} _ {j}\right), \mathbf {F} _ {i} = \operatorname {C o n c a t} \left(\mathbf {F} _ {i}, \mathbf {C}\right), \mathbf {F} _ {j} = \operatorname {C o n c a t} \left(\mathbf {F} _ {j}, \mathbf {C}\right). \tag {7}
+$$
+
+# 3.2 OUTLIER DETECTION
+
+Although the structured universe graph effectively handles point occlusion, it cannot address error annotation outliers, as these outliers do not have corresponding nodes in the universe graph. We frame the identification of these erroneous annotation outliers as an out-of-distribution (OOD) detection problem.
+
+Out-of-distribution detection typically refers to a model's ability to identify and appropriately handle data that deviates significantly from its training distribution. In the context of keypoint annotation, the correctly annotated keypoints adhere to a well-defined, coherent distribution that is consistent across all data. In contrast, erroneously annotated outliers are generated through random and unpredictable factors, causing them to deviate substantially from this established distribution. This inherent difference in data characteristics enables us to frame the outlier detection problem within an OOD detection framework, leveraging the disparity between the structured nature of valid annotations and the stochastic nature of annotation errors.
+
+Following Liu et al. (2020), we define an energy score function for each node  $v_{i}$  of input graph:
+
+$$
+\mathcal {E} \left(v _ {i}\right) = - T \cdot \ln \left(\sum_ {j} e ^ {K _ {n} (i, j) / T}\right), \tag {8}
+$$
+
+where  $K_{n}(i,j)$  denotes the node affinity between  $v_{i}$  on input graph and  $v_{j}$  on universe graph, and  $T$  is a temperature constant. When  $v_{i}$  is an inlier, it should have a corresponding point in the latent universe graph, resulting in a relatively low energy score. Conversely, when  $v_{i}$  is an outlier, all the values of  $K_{n}(i,j)$  will be relatively low, leading to a higher energy score.
+
+While the energy score can be used for a pre-trained neural network directly, the energy gap between in-distribution and out-of-distribution data may not always be optimal for differentiation. To address this, we also introduce a margin loss that fine-tunes the neural network to explicitly create a larger energy gap:
+
+$$
+\mathcal {L} _ {\text {e n e r g y}} = \mathbb {E} _ {v \sim \mathcal {D} _ {i n}} (\max  (0, \mathcal {E} (v) - m _ {i n})) + \mathbb {E} _ {v \sim \mathcal {D} _ {o u t}} (\max  (0, m _ {o u t} - \mathcal {E} (v))), \tag {9}
+$$
+
+where  $\mathcal{D}_{in}$  is the in-distribution training data,  $\mathcal{D}_{out}$  is the out-of-distribution training data,  $m_{in}$  and  $m_{out}$  are the margins for in- and out-of-distribution data, respectively. This loss assigns lower energy scores to in-distribution data and higher scores to OOD data. By contrastively shaping the energy landscape during training, the model becomes more effective at distinguishing between in- and out-of-distribution data.
+
+To apply the energy score in our network, we first compute the energy  $\mathcal{E}$  for each node in the input graph  $\mathbf{G}_i$  based on its affinity with nodes in the latent universe graph  $\mathbf{U}$ . Nodes with energy scores above a certain threshold  $\tau$ , indicating a high likelihood of being outliers, are removed before the graph matching process:
+
+$$
+\mathbf {G} ^ {\text {f i l t e r e d}} = \left\{\mathbf {V} = \left\{v _ {i} \mid \mathcal {E} \left(v _ {i}\right) \leq \tau \right\}, \mathbf {E} = \left\{e _ {i j} \mid \mathcal {E} \left(v _ {i}\right) \leq \tau , \mathcal {E} \left(v _ {j}\right) \leq \tau \right\} \right\} \tag {10}
+$$
+
+We use the newly obtained graph  $\mathbf{G}^{\mathrm{filtered}}$  to construct the filtered node affinity  $\mathbf{K}_n^{\mathrm{filtered}}$  and edge affinity  $\mathbf{K}_e^{\mathrm{filtered}}$ , and then feed it into the graph matching solver to obtain the matching result.
+
+$$
+\mathbf {X} _ {i u} ^ {\text {f i l t e r e d}} = G M S o l v e r \left(\mathbf {K} _ {n} ^ {\text {f i l t e r e d}}, \mathbf {K} _ {e} ^ {\text {f i l t e r e d}}\right). \tag {11}
+$$
+
+This preprocessing step improves the quality and accuracy of the subsequent matching between the input graph and the latent universe graph. After the matching process, the removed outlier points are reintroduced into the solution to maintain the integrity of the input data.
+
+$$
+\mathbf {X} _ {i u} \left(v _ {i}\right) = \left\{ \begin{array}{l l} \mathbf {0} ^ {\top} & \mathcal {E} \left(v _ {i}\right) > \tau , \\ \mathbf {X} _ {i u} ^ {\text {f i l t e r e d}} \left(v _ {i}\right) & \mathcal {E} \left(v _ {i}\right) \leq \tau . \end{array} \right. \tag {12}
+$$
+
+# 3.3 DISCUSSION
+
+Universe Graph Matching essentially decomposes the ambiguous partial matching problem into two well-defined subgraph matching problems. Partial matching, where only parts of the input graphs are aligned, is difficult to formalize with clear optimization objectives and constraints. Conventional approaches, such as adding dummy nodes or relaxing the condition  $\mathbf{X1}_n = \mathbf{1}_n$  to  $\mathbf{X1}_n \leq \mathbf{1}_n$ , have proven inadequate. Our key contribution lies in introducing latent universe graph, which enables us to decompose the partial matching problem between two graphs as  $\mathbf{X}_{ij} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$ . This reformulation shifts the objective from maximizing the match  $\mathbf{X}_{ij}$  between the input graphs to maximizing the match  $\mathbf{X}_{iu}, \mathbf{X}_{ju}$  between each input graph and the latent universe graph. Since the latent universe graph contains all points and edges, the input graphs become its subgraphs, turning the solution of  $\mathbf{X}_{iu}$  into a well-defined subgraph matching problem. This more precise and structured problem definition in universe graph matching improves performance over traditional pairwise graph matching on partial matching tasks.
+
+However, the latent universe graph alone cannot fully address the error annotation points, which lack corresponding matches in the universe graph and violate the assumption that input graphs are subgraphs of the universe graph. While introducing dummy nodes in to the universe graph  $\mathbf{U}$  could potentially account for these outliers, this approach faces several challenges. Estimating the appropriate number of dummy nodes is difficult, and the randomness of error annotations makes it complicate to learning effective dummy node embeddings. Our solution adopts an OOD detection approach, treating error annotation outliers as out-of-distribution points and excluding them before solving the subgraph matching problem.
+
+Despite its advantages, our approach still faces several challenges: 1) Universe latent graph is inherently limited to closed-set problems, relying on class information and training data, which restricts generalization to unseen categories. However, few-shot learning can significantly alleviate this limitation, as shown in our experiments in Appendix ??, demonstrating the method's potential to adapt to new classes. 2) The universe graph is a complete graph; however, the distribution of edges in the input graphs is uneven. This mismatch leads to suboptimal learning of many edge embeddings, with some edges either absent or appearing only rarely in the training data. 3) While the universe graph is more stable and structured than universe point representation, the universe graph cannot fully prevent errors in  $\mathbf{X}_{iu}$  from propagating to all  $\mathbf{X}_{ij}$  matches. All in all, expanding graph partial matching to a more general form, both in terms of mathematical modeling and finer design, requires a long-term exploration.
+
+# 4 EXPERIMENT
+
+# 4.1 PROTOCOL
+
+Datasets We evaluate our method on Pascal VOC (Everingham et al., 2010) and Willow Object Class (Cho et al., 2013), two widely recognized datasets.
+
+The Pascal VOC (Everingham et al., 2010) with Berkeley annotations (Bourdev & Malik, 2009) contains images with bounding boxes surrounding objects of 20 classes. We follow the standard data preparation procedure of NGM and BBGM (Wang et al., 2021; Rolínek et al., 2020b). Each object is cropped to its bounding box and scaled to  $256 \times 256$  px. We follow the unfiltered setting (Rolínek et al., 2020b), which preserves all key points for graph pairs, to evaluate our method's performance under point occlusion. Moreover, we randomly sample the coordinates on images to add outliers, which serve as error annotation key points. We also follow prior works to split the train and test dataset, where training data includes 7,020 images and test data includes 1,682 images.
+
+The Willow Object Class (Cho et al., 2013) contains images from Caltech-256 (Griffin et al., 2007) and Pascal VOC 2007 (Everingham et al.), which consists of 256 images from 5 categories: 40 cars, 40 motorbikes, 50 ducks, 66 wine bottles, and 109 faces. Each image is annotated with the same ten key points. Following standard procedure, we also rescale the image to  $256 \times 256$  px. In experiments, we randomly drop 0 - 6 key points for each image for the 'occlusion' setting. We also add 1 - 10 random outliers as we do in PascalVOC for the 'random outlier' setting. We choose 20 images from each category as our training dataset and leave others for evaluation.
+
+Compared Method We compare our methods UGM with previous work of two kinds: traditional solver with pre-defined affinity and deep graph matching learning model. For the learning-free traditional solvers, we consider four works: GAGM (Gold & Rangarajan, 1996), RRWM (Cho et al., 2010), BPF (Wang et al., 2017), and ZAC (Wang et al., 2020). We generate node and edge affinities using the pre-trained BBGM learning framework and input these affinities into each solver to obtain their respective matching results. For learning-based models, we compare our method with NGMv2 (Wang et al., 2021), BBGM (Roliniak et al., 2020b), DLGM (Yu et al., 2021), GCAN (Jiang et al., 2022), AFAT (Wang et al., 2023), and URL (Lin et al., 2023). To report the performance of prior works, we adhere to the following principles: 1) If a prior work's experimental settings align with ours, we directly report the performance metrics provided in their paper. 2) If the experimental setting is unique to our work, we attempt to replicate the prior methods under their original experimental settings and then adapt them to our setting to obtain comparable performance metrics. However, we failed to replicate the methods of DLGM and URL because they did not release publicly available code. Therefore, we did not report their performance in some of the experiments (Table 2 and Table 3).
+
+Evaluation Metric We see the matching prediction as a binary classification task for each entry. Therefore, F1 score is applied as an evaluation metric, which is introduced by BBGM (Roliniak et al., 2020b). The F1 score is calculated as follows:
+
+$$
+\mathbf {F} _ {1} = 2 \cdot \frac {\text {P r e c i s i o n} \cdot \text {R e c a l l}}{\text {P r e c i s i o n} + \text {R e c a l l}} = \frac {2 \mathrm {T P}}{2 \mathrm {T P} + \mathrm {F P} + \mathrm {F N}} \tag {13}
+$$
+
+where TP denotes true positive, FP denotes false positive, and FN denotes false negative.
+
+# 4.2 PERFORMANCE ON PASCAL VOC
+
+To thoroughly evaluate the performance of our proposed UGM method, we conducted experiments on the Pascal VOC dataset under two distinct settings: the unfiltered setting, where all key points are preserved without the introduction of additional noise, and the random outlier setting, where two random outliers are further added to each input image. All the results are shown in Table 1 and 2.
+
+Our proposed UGM method demonstrates superior performance in both setting, surpassing the state-of-the-art (SOTA) models by a significant margin. Specifically, in the unfiltered setting, UGM outperforms the best-performing SOTA model by  $2.2\%$  in terms of average F1 score. Similarly, in the random outlier setting, UGM still leads the comparison, exceeding the next best method by  $4.8\%$ . This consistent improvement across both settings highlights the robustness and generalization ability of our approach.
+
+A comparison of the two tables reveals how much performance degrades between the unfiltered and random outlier settings, allowing us to evaluate the robustness of learning-based models in the presence of outliers. Our UGM method experiences a performance drop of  $8.8\%$ , ranking second
+
+Table 1: F1 Score performance on Pascal VOC under 'unfiltered' setting. All the key points are preserved, and no additional outliers are added. Top: learning-free graph matching solver with pre-learned affinity. Middle: deep learning graph matching model. Bottom: out method UGM.  
+
+<table><tr><td>Method</td><td>aero</td><td>bike</td><td>bird</td><td>boat</td><td>bottle</td><td>bus</td><td>car</td><td>cat</td><td>chair</td><td>cow</td><td>table</td><td>dog</td><td>horse</td><td>motor</td><td>person</td><td>plant</td><td>sheep</td><td>sofa</td><td>train</td><td>tv</td><td>avg</td></tr><tr><td>GAGM</td><td>37.3</td><td>60.3</td><td>47.7</td><td>39.1</td><td>75.0</td><td>53.7</td><td>36.1</td><td>59.2</td><td>34.8</td><td>56.2</td><td>40.2</td><td>54.4</td><td>51.9</td><td>54.3</td><td>39</td><td>83.8</td><td>48.8</td><td>19</td><td>58.3</td><td>68.7</td><td>50.9</td></tr><tr><td>RRWM</td><td>39.4</td><td>63.7</td><td>52.1</td><td>39.7</td><td>76.5</td><td>57.9</td><td>28.5</td><td>64.3</td><td>37.9</td><td>60.3</td><td>43</td><td>57.9</td><td>54.2</td><td>54</td><td>43.3</td><td>85</td><td>50.4</td><td>22</td><td>66.1</td><td>69.9</td><td>53.3</td></tr><tr><td>BPF</td><td>39.2</td><td>66.5</td><td>51.2</td><td>40.0</td><td>77.0</td><td>59.0</td><td>26.7</td><td>63.8</td><td>36.9</td><td>60.6</td><td>49.7</td><td>58.1</td><td>54.3</td><td>58.2</td><td>43.6</td><td>84.9</td><td>50.7</td><td>23</td><td>66.4</td><td>70.4</td><td>54.0</td></tr><tr><td>ZAC</td><td>43.5</td><td>62.8</td><td>57.5</td><td>43.9</td><td>72.1</td><td>61.2</td><td>33.6</td><td>69</td><td>38.3</td><td>64.1</td><td>46.4</td><td>66.2</td><td>61.8</td><td>61.4</td><td>46.4</td><td>83.2</td><td>58.1</td><td>27.9</td><td>69.8</td><td>72.6</td><td>57.0</td></tr><tr><td>NGMv2</td><td>45.5</td><td>65.3</td><td>55.3</td><td>45.8</td><td>88.4</td><td>64.3</td><td>45.9</td><td>58.6</td><td>43.3</td><td>59.1</td><td>39.2</td><td>55.7</td><td>58</td><td>65.3</td><td>44.4</td><td>95.4</td><td>50.3</td><td>41.2</td><td>72.4</td><td>81.8</td><td>58.8</td></tr><tr><td>BBGM</td><td>42.7</td><td>70.9</td><td>57.5</td><td>46.6</td><td>85.8</td><td>64.1</td><td>51</td><td>63.8</td><td>42.4</td><td>63.7</td><td>47.9</td><td>61.5</td><td>63.4</td><td>69</td><td>46.1</td><td>94.2</td><td>57.4</td><td>39</td><td>78</td><td>82.7</td><td>61.4</td></tr><tr><td>DLGM</td><td>43.8</td><td>72.9</td><td>58.5</td><td>47.4</td><td>86.4</td><td>71.2</td><td>53.1</td><td>66.9</td><td>54.6</td><td>67.8</td><td>64.9</td><td>65.7</td><td>66.9</td><td>70.8</td><td>47.4</td><td>96.5</td><td>61.4</td><td>48.4</td><td>77.5</td><td>83.9</td><td>64.8</td></tr><tr><td>GCAN</td><td>45</td><td>66.7</td><td>60.6</td><td>49.7</td><td>89.7</td><td>66.3</td><td>65.2</td><td>64.9</td><td>45.5</td><td>66.9</td><td>54.4</td><td>63.1</td><td>62.5</td><td>63.5</td><td>55</td><td>96.1</td><td>63.5</td><td>49.7</td><td>80.6</td><td>83.6</td><td>64.6</td></tr><tr><td>AFAT</td><td>47.1</td><td>70.8</td><td>58.1</td><td>45.8</td><td>90.8</td><td>66.5</td><td>49.6</td><td>58.8</td><td>50.6</td><td>64.6</td><td>47.2</td><td>60.5</td><td>62.3</td><td>65.7</td><td>46.3</td><td>95.4</td><td>52.7</td><td>47.4</td><td>74.2</td><td>83.8</td><td>62.0</td></tr><tr><td>URL</td><td>62.7</td><td>75.2</td><td>73</td><td>56.7</td><td>93.7</td><td>66.2</td><td>76.7</td><td>69.2</td><td>64.9</td><td>76.6</td><td>44.6</td><td>74.4</td><td>78.8</td><td>80.9</td><td>62.5</td><td>96.9</td><td>70.3</td><td>55.4</td><td>73.6</td><td>82.1</td><td>71.7</td></tr><tr><td>UGM</td><td>57.4</td><td>76.3</td><td>71.2</td><td>55.1</td><td>91.7</td><td>72.3</td><td>80.3</td><td>72.3</td><td>70.1</td><td>74.6</td><td>61.5</td><td>75.3</td><td>71.6</td><td>76.1</td><td>63.8</td><td>97.2</td><td>73.6</td><td>60.0</td><td>93.1</td><td>84.8</td><td>73.9</td></tr></table>
+
+Table 2: F1 Score performance on Pascal VOC under 'random outlier' setting. Two random outliers are further added to each input image. Top: learning-free graph matching solver with pre-learned affinity. Middle: deep learning graph matching model. Bottom: out method UGM.  
+
+<table><tr><td>Method</td><td>aero</td><td>bike</td><td>bird</td><td>boat</td><td>bottle</td><td>bus</td><td>car</td><td>cat</td><td>chair</td><td>cow</td><td>table</td><td>dog</td><td>horse</td><td>motor</td><td>person</td><td>plant</td><td>sheep</td><td>sofa</td><td>train</td><td>tv</td><td>avg</td></tr><tr><td>GAGM</td><td>26.7</td><td>49.3</td><td>36.6</td><td>25.1</td><td>61.9</td><td>40.7</td><td>9.0</td><td>45.4</td><td>22.5</td><td>39.3</td><td>13.1</td><td>41.0</td><td>37.8</td><td>39.1</td><td>27.9</td><td>59.8</td><td>29.1</td><td>19.7</td><td>26.0</td><td>54.9</td><td>35.2</td></tr><tr><td>RRWM</td><td>28.3</td><td>54.6</td><td>40.3</td><td>27.4</td><td>70.6</td><td>46.3</td><td>9.0</td><td>49.8</td><td>23.5</td><td>42.8</td><td>15.1</td><td>44.8</td><td>38.1</td><td>39.1</td><td>30.3</td><td>66.8</td><td>29.3</td><td>18.1</td><td>27.1</td><td>64</td><td>38.3</td></tr><tr><td>BPF</td><td>25.8</td><td>55.2</td><td>38.8</td><td>26.0</td><td>70.7</td><td>45.7</td><td>9.5</td><td>49.4</td><td>22.9</td><td>40.2</td><td>14.4</td><td>44.1</td><td>36.9</td><td>40.0</td><td>29.0</td><td>68.6</td><td>28.6</td><td>18.8</td><td>28.1</td><td>63.8</td><td>37.8</td></tr><tr><td>ZAC</td><td>37.2</td><td>62.3</td><td>50.8</td><td>34.0</td><td>76.0</td><td>55.9</td><td>23.8</td><td>63.7</td><td>33.2</td><td>58.8</td><td>37.7</td><td>59.4</td><td>55.8</td><td>55.4</td><td>39.1</td><td>76.8</td><td>49.3</td><td>29.0</td><td>49.1</td><td>68.9</td><td>50.8</td></tr><tr><td>NGMv2</td><td>35.6</td><td>60.2</td><td>45.0</td><td>34.3</td><td>72.1</td><td>51.3</td><td>40.2</td><td>50.2</td><td>29.5</td><td>49.4</td><td>23.7</td><td>47.6</td><td>49.8</td><td>51.5</td><td>38.7</td><td>69.3</td><td>42.9</td><td>40.8</td><td>45.0</td><td>62.2</td><td>47.0</td></tr><tr><td>BBGM</td><td>34.4</td><td>63.9</td><td>44.1</td><td>35.0</td><td>78.5</td><td>57.6</td><td>21.5</td><td>54.6</td><td>33.7</td><td>50.3</td><td>37.4</td><td>49.8</td><td>50.8</td><td>54.7</td><td>29.6</td><td>79.8</td><td>44.1</td><td>23.5</td><td>44.5</td><td>76.1</td><td>48.2</td></tr><tr><td>GCAN</td><td>44.4</td><td>61.7</td><td>51.2</td><td>47.5</td><td>78.8</td><td>72.3</td><td>48.9</td><td>58.0</td><td>50.2</td><td>62.8</td><td>53.0</td><td>54.6</td><td>60.3</td><td>61.2</td><td>51.5</td><td>83.2</td><td>62.6</td><td>48.8</td><td>70.0</td><td>84.8</td><td>60.3</td></tr><tr><td>AFAT</td><td>41.2</td><td>66.9</td><td>51.4</td><td>36.5</td><td>82.7</td><td>61.8</td><td>37.9</td><td>56.9</td><td>34.8</td><td>55.7</td><td>28.0</td><td>54.4</td><td>55.3</td><td>60.3</td><td>38.2</td><td>86.3</td><td>48.8</td><td>32.6</td><td>57.1</td><td>77.1</td><td>53.1</td></tr><tr><td>UGM</td><td>49.9</td><td>72.2</td><td>63.5</td><td>48.8</td><td>85.3</td><td>70.1</td><td>71.8</td><td>66.5</td><td>54.6</td><td>67.4</td><td>49.5</td><td>67.4</td><td>65.2</td><td>66.2</td><td>57.6</td><td>88.0</td><td>62.2</td><td>47.2</td><td>65.6</td><td>83.4</td><td>65.1</td></tr></table>
+
+among the learning-based models, just behind GCAN. In comparison, models like NGMv2 and BBGM exhibit larger performance declines of  $11.8\%$  and  $13.2\%$ , respectively. The relatively low decrease in UGM's performance demonstrates its robustness and adaptability, particularly in noisy and complex environments, making it a reliable option for real-world graph matching tasks.
+
+We also perform an random outlier pressure test on our method UGM. Results are shown in Figure 3. Classes like sofa, train, and table see a significant performance drop (32% - 61%) as the number of random outliers increases. This decline is primarily due to the limited amount of training data, particularly for classes like 'table', which has only 28 images, and 'sofa', which has 73 images in the train dataset. The fewer images available, the lower the quality of the constructed latent universe graph, making it more difficult to distinguish between inliers and error annotation outliers. On the other hand, classes like TV, boat, horse, and bus are more robust, with relatively smaller drops in performance (9.9% - 13.9%).
+
+# 4.3 PERFORMANCE ON WILLOW OBJECT
+
+In this experiment, we evaluate the performance of various learning-based graph matching models on the Willow dataset under three settings: occlusion, random outlier, and occlusion + random outlier. These settings simulate real-world challenges, where key points may be missing (occlusion), or noisy data (outliers) may be introduced. All the results are shown in Table 3.
+
+Our proposed UGM method demonstrates superior performance across all three settings. In the occlusion setting, UGM achieves the highest average F1 score of  $91.9\%$ , outperforming the next best model, GCAN, by a significant margin of  $9.7\%$ . In the random outlier setting, UGM again leads with an average score of  $85.4\%$ , surpassing AFAT, the second-best model, by  $1.7\%$ . Finally, in the occlusion + random outlier setting, UGM shows exceptional robustness, maintaining the highest average score of  $71.1\%$ , which is  $5.9\%$  higher than the second-place model.
+
+It is worth noting that the extent of our lead varies across the three settings.
+
+- In the occlusion setting, UGM demonstrates a significant advantage. This suggests that the latent universe graph structure offers a considerable benefit compared to pairwise matching approaches,
+
+![](images/887bc5aae116965dfe94650f2c4fb04c188e70fa3d50342050fa357ad089e018.jpg)  
+Figure 3: Random outlier pressure test of UGM on Pascal VOC. All the key points are preserved, and additional outliers are added to each input image.
+
+Table 3: F1 Score performance on Willow Object Class. Occlusion: randomly drop 0-6 key points for each input image. Random Outlier: randomly add 1-10 outliers for each input image. Occlusion + Random Outlier: both operations are applied. Only the learning model's performance is reported.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">Occlusion</td><td colspan="6">Random Outlier</td><td colspan="6">Occlusion + Random Outlier</td></tr><tr><td>car</td><td>duck</td><td>face</td><td>motor</td><td>bottle</td><td>avg</td><td>car</td><td>duck</td><td>face</td><td>motor</td><td>bottle</td><td>avg</td><td>car</td><td>duck</td><td>face</td><td>motor</td><td>bottle</td><td>avg</td></tr><tr><td>NGMv2</td><td>80.9</td><td>73.8</td><td>84.9</td><td>85.7</td><td>76.7</td><td>80.4</td><td>78.9</td><td>66.6</td><td>84.3</td><td>63.1</td><td>76</td><td>73.8</td><td>46.3</td><td>45.2</td><td>61.6</td><td>42.9</td><td>54.6</td><td>50.1</td></tr><tr><td>BBGM</td><td>77.5</td><td>78.8</td><td>92.7</td><td>83.1</td><td>82.3</td><td>82.9</td><td>65.1</td><td>60.7</td><td>85.5</td><td>71.6</td><td>65.5</td><td>69.7</td><td>36.2</td><td>35.5</td><td>57.2</td><td>35.0</td><td>52.4</td><td>43.3</td></tr><tr><td>GCAN</td><td>81.4</td><td>74.4</td><td>88.5</td><td>88.3</td><td>78.2</td><td>82.2</td><td>74.8</td><td>75.7</td><td>92.8</td><td>77.1</td><td>83.5</td><td>80.8</td><td>43.7</td><td>51.7</td><td>60.2</td><td>45.8</td><td>56.9</td><td>51.7</td></tr><tr><td>AFAT</td><td>77.1</td><td>73.7</td><td>88.0</td><td>76.6</td><td>86.4</td><td>80.4</td><td>82.2</td><td>77.7</td><td>92.7</td><td>77.2</td><td>88.6</td><td>83.7</td><td>70.8</td><td>54.7</td><td>79.1</td><td>53.8</td><td>67.9</td><td>65.2</td></tr><tr><td>UGM</td><td>87.7</td><td>86.2</td><td>99.5</td><td>90.2</td><td>96.0</td><td>91.9</td><td>88.1</td><td>72.9</td><td>96.8</td><td>82.5</td><td>86.5</td><td>85.4</td><td>64.4</td><td>62.1</td><td>89.6</td><td>65.1</td><td>74.5</td><td>71.1</td></tr></table>
+
+as Universe Graph Matching essentially decomposes the ambiguous pairwise partial matching problem into two well-defined subgraph matching problems, thereby improving its ability to handle missing key points.
+
+- In the random outlier setting, the performance gap between UGM and other models is narrower. While our OOD (out-of-distribution) detection technique offers a clear advantage, the improvement is less pronounced. This is likely due to the fact that our approach does not introduce additional training parameters, which limits the potential for further gains.  
+- Finally, in the occlusion + random outlier setting, UGM once again achieves a clear lead. This demonstrates the strong synergy between the latent universe graph learning and our energy based OOD detection technique. The two techniques work in harmony, with no conflicting or contradictory effects, allowing UGM to excel in the most challenging conditions.
+
+# 4.4 ABLATION STUDIES
+
+The ablation study results presented in Table 4 highlight the contribution of each key component—class embedding, universe graph, and outlier filter—to the overall performance. Without the outlier filter component, we matched all the points in the input graph to the latent universe graph. Without edge learning, we only learned the universe node representation and used a linear assignment solver to obtain the matching results between the input graph and the universe point representation. Without class embedding, we directly used the local key point features extracted by the backbone to learn the node embeddings.
+
+Table 4: Ablation study on Pascal VOC with 'random outlier' setting. All the key points are preserved, and two random outliers are added to each input image. F1 score is reported.  
+
+<table><tr><td>class embedding</td><td>edge learning</td><td>outlier filter</td><td>F1 score</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>65.12</td></tr><tr><td>✓</td><td>✓</td><td>x</td><td>56.88</td></tr><tr><td>✓</td><td>x</td><td>✓</td><td>63.49</td></tr><tr><td>x</td><td>✓</td><td>✓</td><td>58.71</td></tr></table>
+
+These results demonstrate that each component plays a unique role in enhancing the model's performance. Specifically, removing the outlier filter has the most significant negative impact, underscoring its importance in handling noise and maintaining robust graph matching.
+
+Furthermore, we also conduct a sensitivity test for hyperparameters on Pascal VOC under the 'random outlier' setting, as shown in Figure 4. In this sensitivity analysis, we investigate the effects of varying three hyperparameters—temperature  $T$ , threshold  $\tau$ , and the pair of values for  $m_{in}$  and  $m_{out}$ . Unless
+
+![](images/39c27db9dff28639018f7396e35835ccbd820142c0c9a8300a2ac06212d3ae44.jpg)  
+Figure 4: Sensitivity test for hyperparameters on Pascal VOC under 'random outlier' setting. Temperature  $T$  in energy function, OOD threshold  $\tau$ , and  $m_{in}, m_{out}$  in energy loss are tested.
+
+![](images/5f8d7c3858b646310c1a32346bdcdf45a8d28af9372c07193078ba1b0a529f8d.jpg)
+
+![](images/4ce9a751cee315f835ceb36ecff2643ebde62ae1ac9de63284ee68c375e387d0.jpg)
+
+![](images/f54e086d47df54d4e23df8b3811337135eac7d5266511d851894a76c6c53aa4e.jpg)
+
+![](images/50bcdd39d39e7a542e90a200f60ce60682413fe693db28eaf18937aba694ee84.jpg)  
+Figure 5: Visualization on Pascal VOC of UGM and peer methods. The green dashed lines represent correct matches, while the red dashed lines represent incorrect matches. Ground truth matching is shown at the bottom.
+
+specified otherwise, the default parameter values are set to  $T = 1.0$ ,  $m_{in} = -3$ ,  $m_{out} = -6$ , and  $\tau = (m_{in} + m_{out}) / 2$ .
+
+For the temperature experiment, we tested values ranging from 0.8 to 10. The results show that performance peaks at temp=1.0 with an average value of 65.1, and gradually decreases as the temperature increases. The threshold experiment explores a range from -1.5 to -6. The average performance is highest at  $\tau = -2.5$  with a value of 66.37, and the performance decreases consistently as the threshold becomes more negative. Although adjusting  $\tau$  can achieve a higher F1 score, we are concerned that this might lead to some overfitting in the model. Therefore, we still report the model results using the default value  $\tau = (m_{in} + m_{out}) / 2$  in other contexts. For the  $m_{\mathrm{in}}$  and  $m_{\mathrm{out}}$  experiments, we analyze both their margin (difference) and absolute offset. Performance peaks at (-3, -6), where the margin is 3, but the overall impact of varying the range (from 1 to 7) or shifting their absolute values is minimal.
+
+# 5 CONCLUSION
+
+In summary, this work presents a novel approach to addressing the challenges of partial matching in graph-based keypoint alignment tasks. We introduce Structured Universe Graph Learning to effectively resolve point occlusion by connecting input pairs via a learned latent graph. Additionally, by incorporating both node features and structural information, the robustness of the matching process is further enhanced. We also adopt Energy-based Out-of-Distribution Detection to filter out annotation errors before matching, improving the overall quality of the matching process. Through extensive evaluations of the Pascal VOC and Willow Object datasets, our method consistently outperforms state-of-the-art techniques, particularly in challenging scenarios involving both point occlusion and random outliers, demonstrating the effectiveness and robustness of our approach.
+
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+
+# APPENDIX
+
+# A NOTATION
+
+We first present all notations used in this paper for a better understanding of proposed algorithms and to facilitate the following discussion.
+
+Table 5: Main notations and description used in this paper.  
+
+<table><tr><td>Notations</td><td>Descriptions</td></tr><tr><td>N</td><td>Node embedding or node feature on the graph.</td></tr><tr><td>E</td><td>Edge embedding or node feature on the graph.</td></tr><tr><td>U</td><td>U = {Nu ∈ Rnun×d, Eu ∈ Rmu×d} denotes the latent universe graph, where nu and mu represents the node and edge number of the latent universe graph.</td></tr><tr><td>Gi</td><td>Gi = {Ni ∈ Rni×d, Ei ∈ Rmi×d} denotes the input graph, where ni and mi represents the node and edge number of the input graph.</td></tr><tr><td>EdgeID</td><td>EdgeID ∈ N2×m denotes directed edges in the graph. EdgeID(0) denote the indices of the source nodes and EdgeID(1) denote the indices of the target nodes.</td></tr><tr><td>Kn</td><td>Kn ∈ Rni×nu denotes the node affinity between the input graph and the universe graph.</td></tr><tr><td>Ke</td><td>Ke ∈ Rmi×mu denotes the edge affinity between the input graph and the universe graph.</td></tr><tr><td>Xij</td><td>Xij ∈ {0,1}ni×nu denotes the matching between the input graphs.</td></tr><tr><td>Xiu</td><td>Xi ∈ {0,1}ni×nu denotes the matching between the input graph and the universe graph.</td></tr><tr><td>C</td><td>Class embedding for image pair.</td></tr><tr><td>Hi</td><td>Global feature extracted by backbone for the input image.</td></tr><tr><td>Fi</td><td>Key point feature extracted by backbone for the input image.</td></tr><tr><td>E(vi)</td><td>Energy score for node vi.</td></tr><tr><td>τ</td><td>Energy score threshold for outlier filter.</td></tr><tr><td>Din, Dout</td><td>Din, Dout denotes in-distribution data and out-of-distribution data respectively.</td></tr></table>
+
+# B IMPLEMENTATION DETAIL
+
+We utilize the standard feature extractor pipeline with a few modifications.
+
+- Fisrt of all, we replace the VGG16 (Simonyan & Zisserman, 2014) backbone with ResNet50 (He et al., 2016). We compute the outputs of  $relu3\_5$ ,  $relu4\_1$  of the ResNet50 network pre-trained on ImageNet (Krizhevsky et al., 2012), to obtain feature  $\mathbf{F}_1$  and  $\mathbf{F}_2$ , respectively. These features are then concatenated to create the key point feature  $\mathbf{F}$ :
+
+$$
+\mathbf {F} = \text {C O N C A T} \left(\mathbf {F} _ {1}, \mathbf {F} _ {2}\right) \tag {14}
+$$
+
+Class embedding is added to  $\mathbf{F}$  as introduced in Equation 7, and we apply an MLP layer to reduce the feature dim to keep the same dimension with the output of the VGG16-based backbone.
+
+- Then we feed the obtained feature  $\mathbf{F}$  and the graph adjacency  $\mathcal{A}$  into the geometric feature refinement component. The graph adjacency  $\mathcal{A}$  is generated using Delaunay triangulation (Delaunay et al., 1934) based on keypoint locations. We apply SplineConv (Fey et al., 2018) to encode higher-order information and the geometric structure of the entire graph into node-wise features  $\mathbf{F}^n$ :
+
+$$
+\mathbf {N}, \mathbf {E} = S p l i n e C o n v (\mathbf {F}, \mathcal {A}) \tag {15}
+$$
+
+The Spline Conv operation is calculated as follows:
+
+$$
+\mathbf {F} _ {i} = \frac {1}{| \mathcal {N} (i) |} \sum_ {j \in \mathcal {N} (i)} \mathbf {F} _ {j} \cdot h _ {\Theta} \left(\mathbf {F} _ {i} - \mathbf {F} _ {j}\right) \tag {16}
+$$
+
+where  $\mathbf{F}_i$  represents the node feature of  $v_i$ ,  $\mathcal{N}(i)$  denotes the neighborhood of  $v_i$ , and  $h_\Theta$  denotes a kernel function defined over the weighted B-Spline tensor product basis.
+
+Algorithm 1 Universe Graph Matching  
+Require: Input images  $I$  input key points  $P$  , learnable parameters  $\theta$  , learning rate  $\eta$  , epoch number  $E$  , margins  $m_{in},m_{out}$  , Temperature  $T$  , OOD threshold  $\tau$    
+Ensure: Trained parameters  $\theta^{*}$  and universe latent graph  $\mathbf{U}^*$    
+1: Initialize  $\theta$  randomly   
+2: Initialize node and edge embedding of universe latent graph U randomly   
+3: for  $e = 1$  to  $E$  do   
+4: for image pair  $I_{i},I_{j}$  in train/test dataset do   
+5: # extract features   
+6: Extract key point feature F by CNN backbone via Eq. 14   
+7: Add class embedding to F via Eq. 7   
+8: Refine F with Spline Conv to obtain node N and edge feature E via Eq. 15 and Eq. 16   
+9:   
+10: # build affinity   
+11: Construct node affinity  $\mathbf{K}_n$  and edge affinity  $\mathbf{K}_e$  via Eq. 2   
+12: Calculate energy  $\mathcal{E}$  for each node via Eq. 8   
+13: Filter out random outlier with OOD threshold  $\tau$  via Eq. 10   
+14:   
+15: if training then   
+16: # loss and update   
+17: Calculate permutation loss with  $\mathbf{K}_n^{\mathrm{filtered}}$  and  $\mathbf{K}_e^{\mathrm{filtered}}$  for both  $I_{i}$  and  $I_{j}$    
+18: Calculate energy loss with  $\mathcal{E}$ $m_{in}$  ,and  $m_{out}$  via Eq. 9 for both  $I_{i}$  and  $I_{j}$    
+19: Final loss  $\mathcal{L} = \mathcal{L}_{\mathrm{permutation}} + \mathcal{L}_{\mathrm{energy}}$  , and compute gradient  $\nabla_{\theta}\mathcal{L},\nabla_{\mathbf{U}}\mathcal{L}$    
+20: Update parameters:  $\theta \leftarrow \theta -\eta \nabla_{\theta}\mathcal{L}$    
+21: Update universe embedding:  $\mathbf{U}\gets \mathbf{U} - \eta \nabla_{\mathbf{U}}\mathcal{L}$    
+22: else   
+23: # build pairwise matching   
+24: Use LPMP solver to obtain universe matching  $\mathbf{X}_{iu},\mathbf{X}_{ju}$  via Eq. 11 and Eq. 12   
+25: Build pairwise matching by  $\mathbf{X} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$    
+26: end if   
+27: end for   
+28: end for   
+29: return  $\theta^{*}$  , U\*
+
+Node and edge affinity construction, universe matching, and outlier filter process have been introduced in Section 3.1 and 3.2. We finally build pairwise matching by  $\mathbf{X}_{ij} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$ .
+
+All the models are trained on on a Linux workstation with i9-10920X CPU@3.50GHz CPU, one RTX3090, and 128GB RAM. The training and inference algorithm is shown in Algorithm 1. In default, we train our UGM with hyper parameters  $m_{in} = -6$ ,  $m_{out} = -3$ ,  $T = 1.0$ ,  $\tau = -4.5$ ,  $\eta = 1e - 3$ , and  $E = 15$ .
+
+# C BACKBONE COMPARISON
+
+To systematically evaluate the impact of network architecture on performance, we conducted comprehensive experiments comparing VGG and ResNet as backbone models across all methods. This analysis ensures fair comparison by providing results using consistent architectures while also quantifying how architectural choices influence each method's effectiveness.
+
+As demonstrated in Table 6, our method consistently outperforms existing approaches across multiple feature extractor configurations, including VGG+Spline2D, VGG+Attention, and ResNet+Spline2D. The sole exception occurs with VGG+Spline2D+3D, where URL achieves better results through its utilization of spline3D features. However, due to the unavailability of URL's implementation, we were unable to integrate this feature extractor into our UGM model. To facilitate a fair comparison, we adopted ResNet50 as the common backbone architecture.
+
+While evaluating architectural impacts, we observed that not all baseline methods exhibited performance improvements when transitioning to ResNet. To maintain scientific rigor and transparency, we have included these results in this appendix, allowing for a more proper discussion of architectural sensitivities across different approaches.
+
+Table 6: Backbone fairness comparison on Pascal VOC with unfiltered setting. The performance with the VGG backbone is reported by their paper, and the performance with the ResNet backbone is reproduced via publicly available code.  $-$  denotes that the method has not released its code.  
+
+<table><tr><td>Method</td><td>Backbone</td><td>GNN</td><td>F1 score</td><td>Backbone</td><td>GNN</td><td>F1 score</td></tr><tr><td>NGMv2</td><td>VGG</td><td>Spline2D</td><td>58.8</td><td>ResNet</td><td>Spline2D</td><td>57.46</td></tr><tr><td>BBGM</td><td>VGG</td><td>Spline2D</td><td>61.4</td><td>ResNet</td><td>Spline2D</td><td>64.45</td></tr><tr><td>DLGM</td><td>VGG</td><td>Spline2D</td><td>64.8</td><td>ResNet</td><td>Spline2D</td><td>-</td></tr><tr><td>GCAN</td><td>VGG</td><td>Attention</td><td>64.6</td><td>ResNet</td><td>Attention</td><td>64.67</td></tr><tr><td>AFAT</td><td>VGG</td><td>Attention</td><td>62</td><td>ResNet</td><td>Attention</td><td>59.55</td></tr><tr><td>URL</td><td>VGG</td><td>Spline2D</td><td>67.6</td><td>ResNet</td><td>Spline2D</td><td>-</td></tr><tr><td>URL</td><td>VGG</td><td>Spline2D+3D</td><td>71.7</td><td>ResNet</td><td>Spline2D+3D</td><td>-</td></tr><tr><td>UGM</td><td>VGG</td><td>Spline2D</td><td>70.5</td><td>ResNet</td><td>Spline2D</td><td>73.9</td></tr></table>
+
+# D EFFECT OF ENERGY MARGIN LOSS
+
+![](images/7744428b666dbf78cc0a51c248d85856f8b85e373c644a03a153604727902829.jpg)  
+Figure 6: Left: Energy histogram of ID and OOD samples without energy loss finetuning, Right: Energy histogram of ID and OOD samples with energy loss finetuning
+
+![](images/d0e11d3a7b41fd5cb87d98ff044c6cb636864f3dee88a10197861036defcb341.jpg)
+
+The effect of energy margin loss is shown in Figure 6. Left shows the energy distribution of indistribution (ID) and out-of-distribution (OOD) samples before applying the Energy Margin Loss, where the energy values for both ID and OOD samples overlap significantly, making it difficult to distinguish between them. After finetuning with the Energy Margin Loss, the energy values for ID samples shift toward lower energy levels, while OOD samples are pushed toward higher energy levels. This separation of energy distributions effectively enhances the model's ability to distinguish between ID and OOD samples, demonstrating the efficacy of the Energy Margin Loss in improving model robustness.

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+# LEARNING SUCCESSOR FEATURES WITH DISTRIBUTED HEBBIAN TEMPORAL MEMORY
+
+Evgenii Dzhivelikian
+
+MIPT, AIRI
+
+Moscow, Russia
+
+dzhivelikian.ea@phystech.edu
+
+Petr Kuderov
+
+AIRI, MIPT
+
+Moscow, Russia
+
+kuderov@airi.net
+
+Aleksandr Panov
+
+AIRI, MIPT
+
+Moscow, Russia
+
+panov@airi.net
+
+# ABSTRACT
+
+This paper presents a novel approach to address the challenge of online sequence learning for decision making under uncertainty in non-stationary, partially observable environments. The proposed algorithm, Distributed Hebbian Temporal Memory (DHTM), is based on the factor graph formalism and a multi-component neuron model. DHTM aims to capture sequential data relationships and make cumulative predictions about future observations, forming Successor Features (SFs). Inspired by neurophysiological models of the neocortex, the algorithm uses distributed representations, sparse transition matrices, and local Hebbian-like learning rules to overcome the instability and slow learning of traditional temporal memory algorithms such as RNN and HMM. Experimental results show that DHTM outperforms LSTM, RWKV and a biologically inspired HMM-like algorithm, CSCG, on non-stationary data sets. Our results suggest that DHTM is a promising approach to address the challenges of online sequence learning and planning in dynamic environments.
+
+# 1 INTRODUCTION
+
+Modelling sequential data is one of the essential tasks in artificial intelligence as it has many applications, including decision-making and learning world models (Moerland et al., 2023), natural language processing (Min et al., 2021), conversational AI (Dwivedi et al., 2023), time series analysis (Eraslan et al., 2019), and video and music generation (Ji et al., 2020). One of the classical approaches to modelling sequential data is to form a representation that stores and condenses the most relevant information about a sequence, and to find a general transformation rule of this information through the dimension of time (Lipton et al., 2015; Harshvardhan et al., 2020; Mathys et al., 2011). We refer to the class of algorithms that use this approach as temporal memory (TM) algorithms because they essentially model the cognitive ability of complex living organisms to remember the past experience and make future predictions based on that memory (Hochreiter & Schmidhuber, 1997; Friston et al., 2016; 2018; Parr & Friston, 2017).
+
+Temporal memories differ in their ability to capture data dependencies and the extent to which they can generalise their predictions to unseen sequences. There are two extremes known in psychology as cognitive maps Tolman (1948), which extract commonalities or schemas generalising experience, and episodic memory Tulving et al. (1972), which stores fragments of experience separately. They both can be viewed as temporal memories, however, the former is slower and require a lot of data to built up a useful schema, according to contemporary computational models based on HMM (George et al., 2021) and Transformers (Dedieu et al., 2024), while the latter is instant and is usually viewed as a data buffer (Zhang et al., 2021).
+
+In this paper, we explore advantages of an episodic-like memory in addressing the problem of online sequence learning for planning and decision-making under uncertainty, which can be formalized as reinforcement learning (RL) for a partially observable Markov decision process (POMDP) (Poupart, 2005; Singh et al., 2021). Inferring the hidden state in a partially observable environment is a sequence modelling problem, since it requires processing a sequence of observations to obtain enough information about the hidden state. One of the most efficient representations of hidden states for discrete POMDPs is the Successor Representation (SR), which disentangles hidden states and goals
+
+given by the reward function (Dayan, 1993; Samuel J. Gershman, 2018; Millidge & Buckley, 2022). The Successor Features framework is an extension of SR to continuous POMDP, using the same idea of value function decomposition, but instead for features of a hidden state (Barreto et al., 2017; Touati et al., 2022). Temporal memory (TM) algorithms can be used to make cumulative predictions about future states and their features, forming SR or SF. This work shows that the proposed algorithm, Distributed Hebbian Temporal Memory (DHTM), can dynamically form SFs for navigation tasks in Gridworld and AnimalAI (Fountas et al., 2020) environments.
+
+The most prominent TM algorithms, such as a Recurrent Neural Network (RNN) (Boden, 2002) or LSTM (Hochreiter & Schmidhuber, 1997), use backpropagation to capture data relationships, which is known to be unstable due to recurrent nonlinear derivatives. They also require complete data sequences to be available during training. Although the gradient vanishing problem can be partially circumvented in a way that Receptance Weighted Key Value (RWKV) (Peng et al., 2023) or Linear Recurrent Unit (LRU) (Orvieto et al., 2023) models do, the problem of online learning, i.e. learning on non-stationary data sets, is still a viable issue. Unlike graphical models such as HMM (Eddy, 2004), RNN models and their descendants also lack a probabilistic theoretical foundation, which is advantageous for modelling sequences captured from stochastic environments (Salaun et al., 2019; Zhao et al., 2020). There is little research on TM models that can be used in fully online adaptive systems interacting with partially observable stochastic environments with access to only one sequence data point at a time, a common case in reinforcement learning (Jahromi et al., 2022).
+
+We propose a Distributed Hebbian Temporal Memory (DHTM) algorithm based on factor graph formalism and a multi-compartment neuron model inspired by the HTM neuron (Hawkins & Ahmad, 2016). Factor graph is a flexible graphical representation of probabilistic distributions, which is commonly used in combination with sum-product algorithm (Loeliger, 2004). We found this representation particularly convenient in our attempt of probabilistic interpretation of the HTM framework. The resulting graphical structure of our model is similar to that of the factorial HMM (Ghahramani & Jordan, 1995). An important feature of our model is that the transition matrices for each factor are stored as different components (segments) of artificial neurons, which makes computation very efficient in the case of sparse transition matrices. Our TM forms sequence representations entirely online and uses only local Hebbian-like learning rules (Hebb, 2005; Churchland & Sejnowski, 1992; Lillicrap et al., 2020), which avoid the drawbacks of gradient methods and make the learning process much more sample efficient than gradient methods. This comes at the expense of hidden state generalization, since DHTM's hidden state uniquely encodes each observation sequence. Different trajectories leading to the same position in the partially observable Gridworld can result in different hidden states inferred by DHTM. In essence, this makes DHTM a model of episodic memory, or an adaptive trajectory buffer, with built-in machinery for making multistep future predictions. In this light, we also show that an agent forming SF with DHTM can be interpreted as a kind of episodic control (EC) (Lengyel & Dayan, 2007; Pritzel et al., 2017; Ritter et al., 2018; Emukpere et al., 2023).
+
+Though our approach is similar to EC, we are not aware of other works, demonstrating its application to partially observable environments. There are other temporal memories that, similarly to DHTM, use Hebbian rules and learn online, like Asymmetric Hopfield Network (AHN) (Chaudhry et al., 2023), Temporal Predictive Coding (tPC) (Tang et al., 2023) and Hierarchical Temporal Memory (Hawkins & Ahmad, 2016). Compared to DHTM, AHN is a first-order memory. That is, it is not able to disentangle identical observations that correspond to different underlying environmental states. Therefore, it is not suitable for partially observable environments. In contrast, tPC has a hidden layer that, potentially, makes it a high-order memory. Despite that it has been tested only in MDP environments (Tang et al., 2024) so far, this method paves a promising direction, which we aim to compare with in future work. Finally, the algorithm that inspired our work, HTM, is also a Hebbian high-order temporal memory, which, opposed to DHTM, doesn't allow for probabilistic predictions due to its all-or-nothing computations.
+
+The proposed TM is tested as an episodic memory for an RL agent architecture navigating in a Gridworld environment and a more challenging AnimalAI testbed (Crosby et al., 2020). The results demonstrate that our algorithm outperforms classical LSTM, RWKV and a biologically inspired HMM-like world model CSCG (George et al., 2021) in changing environments and is in order of magnitude more sample efficient than Dreamer V3 (Hafner et al., 2023). Another advantage of our algorithm is that it can be implemented on neuromorphic processors using only local learning rules.
+
+Our contribution to this work includes the following key elements:
+
+- A novel episodic memory model, called DHTM, which is based on a factor graph formalism and features an efficient, bio-inspired implementation.  
+- We demonstrate that the DHTM can operate fully online, using only local Hebbian-like learning rules.  
+- We show that DHTM learns at a speed comparable to model-based table episodic control while supporting distributed representations. This capability significantly broadens its applicability.
+
+# 2 BACKGROUND
+
+# 2.1 REINFORCEMENT LEARNING
+
+This paper considers decision-making in a partially observable environment (Poupart, 2005). The environment is defined by its state transition function  $P(s,a,s^{\prime}) = Pr(s^{\prime} \mid s,a)$ , determining its dynamics, where  $s^{\prime}, s \in S$  is called state space,  $a \in A$  — action space. Partial observability means that an agent has access to observations  $o \in O$ , which only partially inform about the environment's state  $s$  at each moment.
+
+In reinforcement learning (RL), the goal for the agent is set through the reward function  $R(s): S \to \mathbb{R}$ . The task of RL is to find the agent's policy  $\pi(a|s): S \times A \to [0,1]$  that maximizes the expected return  $G = \mathbb{E}[\sum_{t=0}^{T} \gamma^l R_t]$ , where  $T$  is an episode length and discount factor  $\gamma$  sets the importance of remote rewards. Value-based methods usually aim to estimate the Q-function given a policy  $\pi: Q^\pi(s_t, a_t) = \mathbb{E}[\sum_{l \geq t} \gamma^{l-t} R(s_{l+1}) | s_t, a_t, \pi]$ . For an optimal value function  $Q^*$ , an optimal policy can be defined as  $\pi(a|s) = \operatorname*{argmax}_{a} Q^*(s, a)$ .
+
+One of the drawbacks of value-based methods is that they require learning a new Q-function when the reward function changes, even if the environment remains the same. To get around this, the dynamics of the environment and the reward function can be decoupled, which leads us to the successor representation framework.
+
+# 2.2 SUCCESSOR REPRESENTATION
+
+Successor representations are such representations of hidden states from which we can linearly infer the state value given the reward function (Dayan, 1993). For a discrete state space, such representation can be derived just by rearranging sums in the equation for value function:
+
+$$
+V \left(h _ {t} = i\right) = \mathrm {E} \left[ \sum_ {l = 0} ^ {\infty} \gamma^ {l} R _ {t + l + 1} \mid h _ {t} = i \right] = \sum_ {j} \mathrm {S R} _ {i j} R _ {j}, \tag {1}
+$$
+
+where  $\gamma$  is a discount factor,  $i$ -th row in the matrix SR is the Successor Representation of a state  $i$ , and  $\mathrm{SR}_{ij} = \sum_{l=0}^{\infty} \gamma^{l} p(h_{t+l+1} = j \mid h_{t} = i)$ .  $R_{j}$  is a reward for observing the state  $j$ .
+
+An extension of SR for large or continuous state spaces that we use is Successor Features (Barreto et al., 2017), which is based on a similar idea of value function decomposition, but in a feature space instead (see details in Appendix A).
+
+It is easy to show that, similarly to the Q-function, both SR and SF can be obtained using dynamic programming. However, this limits its use to relatively stable environments, since changes in the transition function can't be taken into account immediately.
+
+The next step to improve an agent's adaptability is to use a world model to approximate the environment's transition function and use it directly to form SR according to its definition. And one of the most basic such world models is the Hidden Markov Model.
+
+# 2.3 HIDDEN MARKOV MODEL
+
+A partially observable environment can be approximated by a Hidden Markov Model (HMM) with hidden state space  $H$  and observation space  $O$ . Generally,  $H$  is not equal to environements state
+
+space  $S$  since the environemnt's true structure is usually unknown. For simplicity, we assume that actions are fully observable and that information about them is contained in hidden state. For the process of length  $T$  with state values  $h_{1:T} = (h_1,\dots ,h_T)$  and observations  $o_{1:T} = (o_1,\dots ,o_T)$ , the Markov property yields the following factorization of the generative model:
+
+$$
+p \left(o _ {1: T}, h _ {1: T}\right) = p \left(h _ {1}\right) \prod_ {t = 2} ^ {T} p \left(h _ {t} \mid h _ {t - 1}\right) \prod_ {t = 1} ^ {T} p \left(o _ {t} \mid h _ {t}\right). \tag {2}
+$$
+
+In the case of a discrete hidden state variable, a time-independent stochastic transition matrix can be learned with the Baum-Welch algorithm (Baum et al., 1970), a variant of the expectation maximization (EM) algorithm. To compute the statistics for the expectation step, it uses the forward-backward algorithm, which is a special case of the sum-product algorithm (Kschischang et al., 2001).
+
+Factor graph is a convenient graphical representation (Loeliger, 2004) of probabilistic models like HMM (see Fig.1A for an example). Formally, a factor graph is a bipartite graph, where one part consists of factors (squares), which, replacing conditional probabilities, define dependencies between random variables (circles) constituting the other part. Such representation also allows extension of forward-backward algorithm to sum-product algorithm. As shown in section 3.1, we base DHTM on ideas from the sum-product algorithm and the Factorial HMM, while adapting them for online learning.
+
+# 3 DISTRIBUTED HEBBIAN TEMPORAL MEMORY
+
+# 3.1 FACTOR GRAPH MODEL
+
+Distributed Hebbian Temporal Memory is based on the sum-product belief propagation algorithm in a factor graph (see Figure 1A). Analogous to Factorial HMM (Ghahramani & Jordan, 1997), we divide the hidden space  $H$  into subspaces  $H^k$ . There are four sets of random variables (RV) in the model:  $H_{t-1}^i$ -latent variables representing hidden states from the previous time step (context),  $H_t^k$ -latent variables for the current time step, and  $\Phi_t^k$ -feature variables. All random variables have categorical distributions. RV state values are denoted by corresponding lowercase letters:  $h_{t-1}^i$ ,  $h_t^k$ ,  $\varphi_t^k$ ,  $o_t^{lm}$ .
+
+In this work, each variable  $\Phi_t^k$  is considered independent and has a separate graphical model for increased computational efficiency. However, in practice, hidden variables of the same time step are statistically interdependent. We introduce their interdependence through a segment computation trick that goes beyond the standard sum-product algorithm (see Appendix B for details).
+
+The model also has three types of factors:  $M_{t-1}^i$ —messages from previous time steps,  $F_c^k$ —context factor (generalized transition matrix),  $F_e^k$ —emission factor. We assume that messages  $M_{t-1}^i$  contain posterior information from time step  $t-1$ , so we don't plot observable feature variables for previous time steps in Figure 1A.
+
+We will discuss only the upper block of the graph in Figure 1A, which is DHTM itself. The lower block—an encoder—is described in Appendix D. The only requirement for the encoder is that its output should be represented as states of categorical variables (features) for the current observation.
+
+The main routine of the DHTM is to estimate distributions of hidden state variables given by the equation 3, the computational flow of which is schematically shown in Figure 1B:
+
+$$
+p \left(h _ {t} ^ {k}\right) \propto \sum_ {\Omega_ {k}} \prod_ {i \in \omega_ {k}} M _ {t - 1} ^ {i} \left(h _ {t - 1} ^ {i}\right) F _ {c} ^ {k} \left(h _ {t} ^ {k}, \Omega_ {k}\right), \tag {3}
+$$
+
+where summation is over all possible hidden RV states in  $\Omega_{k} = \{h_{t - 1}^{i}:i\in \omega_{k}\}$ ,  $\omega_{k} = i_{1},\ldots ,i_{n}$  set of previous time step RV indexes included in  $F_{c}^{k}$  factor,  $(n + 1)$  —factor size.
+
+# 3.2 NEURAL IMPLEMENTATION
+
+For computational purposes, we translate the factor graph shown in Figure 1A to the Hebbian learning neural network architecture shown in Figure 1B (for a biological interpretation of the model,
+
+![](images/e97ae17619be265711aef579069328b7ed1c4cb50027fd827152b7f60621a08c.jpg)  
+Figure 1: A. Example of a factor graph for the DHTM. The input to the model is a sequence of observations  $o_{t}$ . The encoder block forms categorical features  $\Phi_t^k$ . Each feature  $\Phi$  has its own explaining hidden variable, which may depend on hidden variables of the other features and on itself from the previous time step.  $F_{c}^{k}$  and  $F_{e}^{k}$  are context and emission factors for the corresponding variables. Unary factors  $M_{t - 1}^{i}$  called messages represent accumulated information about previous time steps. B. Neural implementation. Random variables are represented by cell clusters (white circles), where each cell corresponds to a state and its spike frequency to the probability of the state  $p(h_{t}^{k})$ . The dendritic segments of the cell correspond to the context factor values  $f_{l}$  for a given combination of states (active presynaptic cells). Segments' excitations  $E_{l}$  are combined to determine the cell's spike frequency  $p(h_{t}^{k})$ . Segment synaptic weights reflect the specificity of the combination of presynaptic cells for the segment. Emission factors  $F_{e}^{k}$  are fixed and represented by minicolumns within a variable.
+
+![](images/7ba7d6259521186633103bdbe3aa2ad97dcb2b766ddb9b417513f0b2936b4cd3.jpg)
+
+see Appendix C). Each RV can be viewed as a set of neurons representing the states of the RV, i.e.  $p(h_t^k) = p(c_t^j = 1)$ , where  $j$ -is the index of a neuron corresponding to the state  $h_t^k$  and  $c_t^j \in \{0,1\}$  cell's activity. Since RVs are categorical, only one cell per variable can be active at a time and their activity is defined through sampling of RV's distribution.
+
+Since factors  $F_{c}^{k}$  represent dependencies between variables, in neural implementation we view them as collections of connections between cell groups corresponding to RVs that  $F_{c}^{k}$  takes as arguments. Each factor's entry maps a particular combination of states  $h_{t}^{k}, h_{t-1}^{i_{1}}, \ldots, h_{t-1}^{i_{n_{l}}}$  to a factor value  $f_{l}$ . We found it useful to represent each such entry as a separate computational unit, which we call segment (alluding to dendritic segments of a biological neuron). As shown in Figure 1B, segment  $l$  connects its presynaptic cells  $h_{t-1}^{i_{1}}, \ldots, h_{t-1}^{i_{n_{l}}}$  (one per each RV), constituting its receptive field, to the cell it excites  $h_{t}^{k}$ . The factor  $f_{l}$  value can be thought of as a segment's excitation efficiency, i.e. how strong its presynaptic pattern influences the cell's activation.
+
+Reformulation of belief propagation step in equation 3 in terms of segmental computations results in the following equations:
+
+$$
+p \left(h _ {t} ^ {k}\right) = \operatorname {s o f t m a x} _ {j \in \text {c e l l s} \left[ H _ {t} ^ {k} \right]} \left(\max  _ {l \in \operatorname {s e g} (j)} \left(E _ {l}\right)\right), \tag {4}
+$$
+
+$$
+E _ {l} = \log f _ {l} + w _ {l} \log \frac {1}{n _ {l}} \sum_ {u \in \operatorname {r e c} (l)} w _ {u l} m _ {u} + \sum_ {u \in \operatorname {r e c} (l)} \left(1 - w _ {u l}\right) \log m _ {u}, \tag {5}
+$$
+
+where cells  $[H_t^k ]$  --indexes of cells corresponding to states of the variable  $H_{t}^{k}$ $seg(j)$  all segments for cell  $j$ $w_{ul}$  synapse efficacy or neuron's specificity for segment, such that  $w_{ul} = p(s_l = 1|c_{t - 1}^u = 1)$ $w_{l} = \frac{1}{n_{l}}\sum_{u}w_{ul}$  average synapse efficacy for segment  $l$  , and  $n_l$  number of cells in segment's receptive field rec(l).  $m_{u}$  is presynaptic cell's activation probability  $p(c_{t - 1}^u = 1)$  , which carries messages from the previous time-step and corresponds to entry in  $M_{t - 1}^{i}$  For detailed derivation and rationale behind these equations, see Appendix B.
+
+Initially, cells have no segments. As learning progresses, new non-zero connections, grouped into segments, are created. In equation 5, we benefit from having sparse factors because their complexity depends linearly on the number of non-zero entries or segments. And that's usually the case in our
+
+model due to the one-step Monte Carlo learning and the special form of the emission factors  $F_{e}^{k}$ :
+
+$$
+F _ {e} ^ {k} \left(h _ {t} ^ {k}, \varphi_ {t} ^ {k}\right) = \mathbb {I} \left[ h _ {t} ^ {k} \in \operatorname {c o l} \left(\varphi_ {t} ^ {k}\right) \right], \tag {6}
+$$
+
+where  $\mathbb{I}$  indicator function,  $\mathrm{col}(\varphi_t^k)$  is a set of hidden states connected to the feature state  $\varphi_t^k$  forming a column. The shape of the emission factor is inspired by the presumed columnar structure of the neocortex and has been shown to induce a sparse transition matrix in HMM (George et al., 2021).
+
+The next step after computing the  $p(h_t^k)$  distribution is to incorporate information about the current feature states  $p(h_t^k \mid \varphi_t^k) \propto p(h_t^k) \mathbb{I}[h_t^k \in \mathrm{col}(\varphi_t^k)]$ . After that, the learning step is executed. The loop closing step of our TM algorithm is to assign the posterior of the current step  $p(h_t^k \mid \varphi_t^k)$  to  $M_{t-1}^i$ .
+
+# 3.3 LEARNING
+
+DHTM learns  $f_{l}$  and  $w_{ul}$  weights by Monte-Carlo Hebbian-like updates. First,  $h_{t - 1}^{i}$  and  $h_t^k$  are sampled from their posterior distributions:  $p(h_{t - 1}^{i}\mid \varphi_{t - 1}^{i})\propto M_{t - 1}^{i}$  and  $p(h_t^k\mid \varphi_t^k)$  respectively. Cells that correspond to sampled states become active. For cell  $j$  corresponding to  $h_t^k$  we check if there are segments matching the pattern of sampled states (active cells) from the previous time-step  $\{h_{t - 1}^i\}_i$ . If there are no such segments, the new segment connecting sampled cells is created. For all newly created segments and segments with receptive fields matching active cells from the previous time-step, its activity  $s_l$  is set to 1. Then  $f_{l}$  is updated according to the activity of the segment  $s_l$  and the activity of its cell  $c_t^j$ , so that  $f_{l}$  is proportional to several coincidences  $s_l = c_t^j = 1$  in the recent past (exponential moving average of coincidence):
+
+$$
+\Delta f _ {l} = \alpha \left(c _ {t} ^ {j} - f _ {l}\right) s _ {l}, \tag {7}
+$$
+
+where  $\alpha \in [0,1)$  is the segment's learning rate.
+
+That is,  $f_{l}$  is trained to be equal to the cell's average activation frequency when the segment is active. It's similar to Baum-Welch's update rule (Baum et al., 1970) for the transition matrix in HMM, which, in effect, counts transitions from one state to another, but in our case, the previous state (context) is represented by a group of RVs.
+
+Weights  $w_{ul}$  are also updated by the Hebbian rule to reflect the specificity of a presynaptic  $u$  for activating a segment  $l$ . That is, they are targeted to represent the probability  $p(s_l = 1 \mid c_{t-1}^u = 1)$  that segment  $s_l$  is active, given that cell  $u$  was active at the previous time step. In our algorithm, it is approximated as an exponential moving average of the frequency activation of segment  $s_l$ , given  $c_{t-1}^u = 1$ :
+
+$$
+\Delta w _ {u l} = \beta \cdot \mathbb {I} \left[ c _ {t - 1} ^ {u} = 1 \right] \cdot \left(\mathbb {I} \left[ s _ {l} = 1 \right] - w _ {u l}\right), \tag {8}
+$$
+
+where  $\beta \in [0,1)$  — learning rate.
+
+# 3.4 AGENT ARCHITECTURE
+
+We incorporate DHTM as a part of an RL agent. The same agent is used for other memories tested. The agent consists of a memory model and a feature reward function. The memory model generates SF by predicting cumulative future distributions of feature variables  $\Phi^k$ . We assume that the reward function can be decomposed linearly in feature space as  $R_{t} = \frac{1}{n}\sum_{k=1}^{n}r_{i}^{k}$ , where  $r_{i}^{k}$  is a reward associated with feature state  $\varphi_{t}^{k} = i$  and  $n$ -number of feature variables.  $r_{i}^{k}$  is learned during interaction with the environment and is used in combination with SF representations to estimate the action value function (see Appendix A for details).
+
+The agent training procedure is outlined in Algorithm 1. For each episode, the memory state is reset to a fixed initial message with RESET MEMORY() and the variable action is set to a fixed initial action. An observation returned by an environment (obs) is encoded as a set of categorical variables. In the OBSERVE() routine, memory learns to predict next feature states as described in section 3. An agent learns associations to map feature states and rewards in the REINFORCE() function:
+
+$$
+r _ {i} ^ {k} \leftarrow r _ {i} ^ {k} + \alpha \mathbb {I} \left[ \varphi_ {t} ^ {k} = i \right] \left(R _ {t} - r _ {i} ^ {k}\right) \tag {9}
+$$
+
+where  $\alpha$  is a learning rate,  $R_{t}$  —a reward for the current time step.
+
+Algorithm 1 General agent training procedure  
+1: for episode=1..n do  
+2: RESET MEMORY()  
+3: action  $\leftarrow$  initial_action  
+4: while (not terminal) and (steps  $<$  max_steps) do  
+5: obs, reward  $\leftarrow$  STEP()  
+6: features  $\leftarrow$  ENCODE(PREPROCESS(obs))  
+7: OBSERVE(features, action)  
+8: REINFORCE(reward, features)  
+9: action  $\leftarrow$  SAMPLE_ACTION()  
+10: ACT(action)  
+11: end while  
+12: end for
+
+We include actions in DHTM by forcing some of the hidden variables  $H_{t}^{k}$  to represent actions. That is, we assume that information about action is contained in the hidden state of the model. For example, if we have 4 actions, we set 4 states for one of the hidden variables and set its state from observing the action.
+
+In SAMPLE_ACION(), for each action and current state, TM predicts feature states up to time step  $T$  under policy  $\pi$ , and the predictions are combined to form SF (equation 10). For simplicity, we always build SF under a uniform policy, which doesn't guarantee optimality, but
+
+we found that it doesn't make a significant difference to iterative policy improvement in our experimental setups. The planning horizon  $T$  is determined dynamically from the distribution of  $\Phi_{t + l}^{k}$ . For each prediction step  $l$ , we check whether the distribution of  $\Phi_{t + l}^{k}$  is close to uniform (according to Kullback-Leibler distance) or whether any of states with positive reward has probability above a threshold; in that case, the prediction cycle is stopped. The action value is determined by combining the corresponding SF and feature state rewards as shown in equation 11. Finally, the action is sampled from the softmax distribution over the action values.
+
+$$
+\mathrm {S F} _ {t + T} ^ {\pi} \left(\varphi^ {k} = j \mid h _ {t}\right) = \sum_ {l = 0} ^ {T} \gamma^ {l} p _ {\pi} \left(\varphi_ {t + l + 1} ^ {k} = j \mid h _ {t}\right) \tag {10}
+$$
+
+$$
+Q ^ {\pi} \left(h _ {t}, a _ {t}\right) = \sum_ {j k} \sum_ {h _ {t + 1}} \mathrm {S F} _ {t + T} ^ {\pi} \left(\varphi^ {k} = j \mid h _ {t + 1}\right) \cdot p \left(h _ {t + 1} \mid h _ {t}, a _ {t}\right) \cdot r _ {j} ^ {k}, \tag {11}
+$$
+
+# 3.5 DHTM AND EPISODIC CONTROL
+
+Algorithm 2 Episodic memory learning  
+Input:  $\varphi_{t + 1},a_t$    
+1:  $h_{t + 1}\leftarrow \mathcal{D}_{a_t}(h_t)$    
+2:  $\varphi_{t + 1}^{*}\gets \operatorname {argmax}_{\varphi}F_{e}(h_{t + 1},\varphi)$    
+3: if  $h_{t + 1}$  is null or  $\varphi_{t + 1}^{*}$  is not  $\varphi_{t + 1}$  then   
+4:  $h_{t + 1}\gets \underset {h\notin K}{\operatorname{argmax}}F_{e}(h,\varphi_{t + 1})\# K$  is the set of keys for all actions   
+5:  $\mathcal{D}_{a_t}(h_t)\gets h_{t + 1}$    
+6: end if   
+7:  $h_t\gets h_{t + 1}$
+
+It can be shown that DHTM combined with the agent architecture described in the previous section is similar to episodic control.
+
+DTHM performs best when its hidden state space is large. Since the hidden state for unpredicted observations is formed randomly, it's important that there are no accidental state collisions when remembering transitions. That is, DHTM tries to remember new sequences without interfering with old memories. In effect, this makes DHTM a buffer of agent trajectories with a built-in recall engine. Episodic Control is based on a similar idea, and to investigate its difference from DHTM, we implemented an Episodic Control agent, a simpli
+
+fied counterpart of DHTM, reflecting its core property that presumably helps it solve RL tasks.
+
+Episodic Control (EC) is a general computational approach to storing and reusing an agent's experience, inspired by the notion of episodic memory from psychology (Tulving et al., 1972). The core of EC is a dictionary  $\mathcal{D}_a = (K_a, V_a)$  that stores key-value pairs for each discrete action  $a \in A$ . In our EC version,  $K_a$  is an array of hidden states  $h_t$  and  $V_a$  is an array of corresponding subsequent hidden states  $h_{t+1}$ . If the key state  $h_t$  already exists in the dictionary, its value is rewritten. That is,  $\mathcal{D}_a$  represents the transition matrix of a deterministic HMM for a feature variable  $\Phi_t$ . The hidden state  $h_t$  and the feature state  $\varphi_t$  have the same dependence as in DHTM (equation 6), but in EC we assume that there are infinite hidden states per feature state. To predict the next state, we just have to look up  $\mathcal{D}_a$  for the current state  $h_t$ , but if there is no entry for  $h_t$  or the prediction does not match the observed state  $\varphi_{t+1}$ , then the new state is formed. Unlike DHTM, the new state is not completely
+
+![](images/90caac5288a149f9b4d86c7f404e4557a43b3b7d51282698e8ca85d95d8e395b.jpg)  
+Figure 2: POMDP 5x5 Gridworld foraging task using different types of memory: DHTM, CSCG, LSTM, RWKV. All agents, except for DHTM, are trained in two modes. The first mode is with oracle, which tells the agent to reinitialize memory and clear the trajectory buffer during environmental change. The second mode is without oracle, so the agent continues using its current strategy to gather new trajectories and update memory. There is also an episodic control agent (e c) among baselines, which uses a simple dictionary to store trajectories and make predictions. A. Dynamic of steps required for the agent to reach the goal. In the 300th episode, the goal is blocked by a wall, so the agent needs to find a new optimal trajectory. B. Total steps taken during 1000 episodes.
+
+![](images/f4673f4cfcd5d45dcdbfbfd88e0d569b6356db976e4ac907f169f53de64b4c17.jpg)
+
+random, but to avoid collisions, the algorithm makes sure that this state hasn't been chosen before. The learning procedure of the EC agent's memory  $\mathcal{D}_a$  is summarized in the algorithm 2.
+
+Since  $\mathcal{D}_a$  can be transformed into a transition matrix, the process of forming SF can be reduced to one in Sec. 3.4. However, since the resulting transition matrix is sparse, it's more efficient to consider  $\mathcal{D}_a$  as a graph representation and to perform SF formation as a breadth-first search (see Appendix E.3 for details).
+
+# 4 EXPERIMENTS
+
+We test our model in a reinforcement learning task in the Gridworld environment and in a more challenging AnimalAI 3D environment. This section shows how different memory models affect the adaptability of an RL agent. For each setup, the results are averaged over five experiments with different seed values, and one standard deviation is shown everywhere, except for boxplots. See Appendix E for details on the baselines, their model parameters, and training regimes. See Appendix F for a more detailed description of the environments and setups<sup>1</sup>.
+
+# 4.1 GRIDWORLD
+
+The first test was conducted in partially observable Gridworld in a 5x5 alternating maze with a stationary end state and the agent's starting position. The agent can observe only the colour of the cell it is in and the colour of an obstacle it encounters, while being punished in addition to the constant step penalty. Each episode ends when the reward is collected or when the maximum number of action steps is reached. In each episode, we measure the number of action steps required for the agent to reach the goal.
+
+Each trial lasts 1000 episodes and consists of two phases. In the first phase, there are no obstacles on the shortest path to the terminal state. In the second phase, after 300 episodes, the previously optimal path is blocked by a wall, so the agent has to take a detour to get the reward. The experiment aims to show the ability of a memory to relearn the structure of the maze in the test time. If the agent is able to reach the end state in the second phase, it suggests that the model can forget, or at least suppress,
+
+![](images/541c334c1ae94e4613f5f18a5a79ec7f67e9a6a86c2762973286dbcbab023cb1.jpg)  
+Figure 3: Amount of action steps the agent performs to reach the goal in a  $10 \times 10$ -meter room in AnimalAI environment using DHTM, CSCG, LSTM, RWKV and EC dictionary. The agent and the goal have the same initial positions throughout the episodes. Each time-step, the agent gets an encoded image of a first-person view of the environment and produces one of three actions: turn left, turn right, or go forward. A. Dynamic of steps required for the agent to reach the goal. B. Total steps taken during 1000 episodes by the agent with DHTM with K-Means encoder and Categorical VAE. For the latter, DHTM's context factor size is varied (i.e., how many previous time-step hidden variables are considered for prediction).
+
+![](images/bb3b03e1a53386ec026e14c02e32f682fd10c8f58b36c2e55e3530f4576dd008.jpg)
+
+the memory of the old shortest path when it is blocked. Otherwise, the agent will continue to hit the wall and fail to reach the terminal state.
+
+In this experiment, we compare our memory model DHTM with CSCG (George et al., 2021), LSTM (Hochreiter & Schmidhuber, 1997) and RWKV (Peng et al., 2023) within the same agent architecture that uses memory to form successor features and evaluate actions (see Sec. 3.4). All memories, except DHTM, are trained in two modes. The first mode is with an oracle that tells the agent to reinitialize memory and clear the trajectory buffer just in time for the environmental change. The second mode is without an oracle, so the agent continues to use its current strategy to collect new trajectories and update memory in the second phase. There is also a simple episodic control agent (EC) among the baselines, which uses a dictionary to store trajectories and make predictions (see Section 3.5 for details).
+
+The results in Figure 2 show that CSCG effectively can't adapt to the environmental change in the online regime, but only if it is retrained from scratch on new trajectories. And, apparently, LSTM and RWKV are only able to learn a predictive model for a small horizon, since they perform significantly worse than other baselines in the second phase, even with oracle enabled. In contrast, DHTM and EC are able to adapt online even without explicit memory's trainable parameters reinitialisation. Another important observation is that the EC agent performs on par with DHTM, suggesting that its mechanism can explain most of DHTM's performance in this task. However, see Appendix H.2 for experiments that show their difference in scalability.
+
+# 4.2 ANIMALAI
+
+AnimalAI is a 3D environment capable of providing rich visual input to an agent in the form of RGB images. The goal of the experiment described here is to show that DHTM can handle distributed representations and is scalable to environments with larger state spaces.
+
+The agent's task is to reach a sphere of radius  $0.5\mathrm{m}$ , which is located in a square area of size 10 by 10 meters. The start position of the agent and the position of the ball are the same between episodes. In each episode, we measure the number of actions (steps), which are AnimalAI's standard left/right turn and forward walk, the agent takes to reach the goal, with the maximum number of actions limited to 40. Each action is repeated three times and three frames are skipped, so one agent action step corresponds to three steps in the environment.
+
+We test DHTM and EC agents using two different encoders: K-Means and Categorical VAE (Jang et al., 2017). Both encoders are pre-trained on RGB images collected with a uniform strategy during 2000 episodes in the same room used for agent testing. K-Means represents an image as a single categorical variable with 50 states, and Categorical VAE is configured for distributed representation with five variables with 50 states each (see Appendix D for details). In case of EC agent, since it doesn't support distributed representations, we map each unique VAE encoding to one feature state using a simple dictionary. Since the resulting number of states for such feature variable is large (typically more than 1000), it's infeasible to use the same mapping for CSCG. We also didn't find much difference for LSTM and RWKV using K-Means or VAE, so we only keep the results for VAE encoder.
+
+The results are shown in Figure 3A. It can be seen that after about 150 episodes, the agent with DHTM and the EC agent reach the goal stably, although the DHTM agent with distributed encoding (with a factor size of 4) converges to a slightly shorter path on average. The agent with CSCG slowly improves its strategy, but LSTM and RWKV agents don't show any learning on this time-step scale. To understand a common learning time-step scale for contemporary model-based algorithms, we also tested a small version of Dreamer V3 (18M parameters) (Hafner et al., 2023), which successfully solves the same task after about  $35 \times 10^{3}$  action steps (1000 episodes) versus  $3.5 \times 10^{3}$  steps (150 episodes) for DHTM (see Appendix H.1 for details).
+
+To show that DHTM can't simply be replaced by multiple parallel EC dictionaries or CSCGs in the case of distributed encoding, we also vary the context factor size of DHTM, that is, how many previous time-step variables are taken into account to predict the current state of a variable. The factor size of 1 corresponds to the graph of the Factorial HMM, with parallel HMMs. It can be seen from Figure 3B that in our case of categorical VAE, knowing only the previous state of the variable is not enough to reliably predict rewarding features, since the agent doesn't converge to the shortest path. We also tested the variant (factor 1*) where each feature variable is represented by three hidden states, to ensure that the result for factor size 1 is not due to DHTM's hidden state collisions. DHTM with a factor that takes into account more than one previous time-step hidden variable (excluding the action variable) performs significantly better. For diagrams depicting factor graphs used in these experiments, see Appendix E.4. For further experiments showing the importance of distributed encoding in DHTM, see Appendix H.3.
+
+# 5 CONCLUSION
+
+This paper introduces a novel episodic memory model, DHTM. It combines ideas from belief propagation, episodic control, and neurophysiological models of the neocortex, resulting in a fast probabilistic sequence memory.
+
+The experiments show that our memory model can quickly learn observation sequences in changing environments and effectively reuse them for RL tasks, on par with model-based table episodic control. We also show that, unlike CSCG and table EC, DHTM is able to handle distributed representations and account for statistical dependencies between variables of these representations. Potentially, it extends the application of our algorithm to higher state spaces, exploiting their factorizations for more efficient computations and to reduce the number of stored parameters.
+
+Our results suggest that DHTM can be viewed as an adaptive trajectory buffer for a slower generalizing algorithm, such as a Neural State Space Model (Jiang et al., 2023). Through prediction error learning, DHTM can selectively discard and replace obsolete transitions in observed trajectories without disrupting relevant experience. It can also provide an initial policy for an agent through episodic control until a slower but more accurate world model takes over. These features of a trajectory buffer are highly desirable in the case of online learning in non-stationary environments.
+
+Further improvements of DHTM can be directed towards the implementation of an automatic feature space factorization discovery to determine an optimal context factor size during learning. To avoid excessive segment growth in noisy environments, it's also possible to implement an adaptive segment activation threshold. One of the main limitations of this study is that the evaluation of the algorithms is limited to basic RL setups due to the simple agent architecture. More advanced use cases could be demonstrated using better exploration strategies and in combination with other memory models, which is also a matter of future research.
+
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+Achille Salaun, Yohan Petetin, and François Desbouvries. Comparing the modeling powers of rn and hmm. In 2019 18th iee international conference on machine learning and applications (icmla), pp. 1496-1499. IEEE, 2019.  
+Samuel J. Gershman. The Successor Representation: Its Computational Logic and Neural Substrates. The Journal of Neuroscience, 38(33):7193, August 2018. doi: 10.1523/JNEUROSCI.0151-18.2018. URL http://www.jneurosci.org/content/38/33/7193.abstract.
+
+Gautam Singh, Skand Peri, Junghyun Kim, Hyunseok Kim, and Sungjin Ahn. Structured world belief for reinforcement learning in pomdp. In Marina Meila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp. 9744-9755. PMLR, 18-24 Jul 2021. URL https://proceedings.mlr.press/v139/singh21a.html.  
+Jochen F. Staiger and Carl C. H. Petersen. Neuronal circuits in barrel cortex for whisker sensory perception. Physiological Reviews, 101(1):353-415, 2021. doi: 10.1152/physrev.00019.2019. URL https://doi.org/10.1152/physrev.00019.2019. PMID: 32816652.  
+Greg J. Stuart and Nelson Spruston. *Dendritic integration: 60 years of progress.* Nature Neuroscience, 18(12):1713-1721, Dec 2015. ISSN 1546-1726. doi: 10.1038/nn.4157. URL https://doi.org/10.1038/nn.4157.  
+A.K Subramanian. Pytorch-vae. https://github.com/AntixK/PyTorch-VAE, 2020.  
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+Ahmed Touati, Jérémy Rapin, and Yann Ollivier. Does Zero-Shot Reinforcement Learning Exist? September 2022. URL https://openreview.net/forum?id=MYEap_OcQI.  
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+Konstantinos Voudouris, Ibrahim Alhas, Wout Schellaert, Matthew Crosby, Joel Holmes, John Burden, Niharika Chaubey, Niall Donnelly, Matishalin Patel, Marta Halina, José Hernández-Orallo, and Lucy G. Cheke. Animal-ai 3: What's new & why you should care, 2023. URL https://arxiv.org/abs/2312.11414.  
+Xiaohan Zhang, Lu Liu, Guodong Long, Jing Jiang, and Shenquan Liu. Episodic memory governs choices: An RNN-based reinforcement learning model for decision-making task. Neural Networks, 134:1-10, February 2021. ISSN 0893-6080. doi: 10.1016/j.neunet.2020.11.003. URL https://www.sciencedirect.com/science/article/pii/S0893608020303889.  
+Jingyu Zhao, Feiqing Huang, Jia Lv, Yanjie Duan, Zhen Qin, Guodong Li, and Guangjian Tian. Do rnn and lstm have long memory? In International Conference on Machine Learning, pp. 11365-11375. PMLR, 2020.
+
+# A VALUE FUNCTION DECOMPOSITION
+
+In our agent model, we approximate the reward function  $\mathbf{R}(\mathbf{s})$  as a sum:
+
+$$
+R _ {t} = \frac {1}{n} \sum_ {k = 1} ^ {n} r \left(\varphi_ {t} ^ {k}\right), \tag {12}
+$$
+
+where  $r(\varphi_t^k)$  is a reward associated with state  $\varphi_t^k$ ,  $n$ -number of feature variables. Then, similarly to the Successor Representation idea (see Section 2.2), the value function can be represented as:
+
+$$
+\begin{array}{l} V \left(h _ {t}\right) = \mathrm {E} \left[ \sum_ {l = 0} ^ {\infty} \gamma^ {l} R _ {t + l + 1} \mid h _ {t} \right] = \sum_ {l = 0} ^ {\infty} \gamma^ {l} \mathrm {E} \left[ \frac {1}{n} \sum_ {k = 1} ^ {n} r \left(\varphi_ {t} ^ {k}\right) \mid h _ {t} \right] \\ = \frac {1}{n} \sum_ {k = 1} ^ {n} \sum_ {j} \sum_ {l = 0} ^ {\infty} \gamma^ {l} p (\varphi_ {t + l + 1} ^ {k} = j \mid h _ {t}) r _ {j} ^ {k} \\ = \frac {1}{n} \sum_ {k = 1} ^ {n} \sum_ {j} \mathrm {S F} _ {j} ^ {k} \left(h _ {t}\right) r _ {j} ^ {k}, \tag {13} \\ \end{array}
+$$
+
+where  $\mathrm{SF}_j^k (h_t) = \sum_{l = 0}^{\infty}\gamma^l p(\varphi_{t + l + 1}^k = j\mid h_t),h_t = (h_t^1,\dots,h_t^n)$  -hidden state vector of variables  $\{H_t^k\}_{k}$ .
+
+# B DETAILS OF NEURONAL IMPLEMENTATION
+
+As shown in Figure 1B, each RV can be viewed as a set of neurons representing the states of the RV, i.e.  $p(h_{t}^{k}) = p(c_{t}^{j} = 1)$ , where  $j$ -is the index of a neuron corresponding to the state  $h_{t}^{k}$ . Cell activity is binary  $c_{t}^{j} \in \{0,1\}$ , and the probability can be interpreted as a spike rate. Factors  $F_{c}^{k}$  and  $M_{t - 1}^{i}$  can be represented as vectors, where elements are factor values for all possible combinations of RV states included in the factor. Let's denote the elements of the vectors as  $f_{l}$  and  $m_{u}$  respectively, where  $l$  corresponds to a particular combination of state values  $h_{t}^{k}, h_{t - 1}^{i_{1}}, \ldots, h_{t - 1}^{i_{n_{l}}}$ , and  $u$  indexes all neurons representing states of RVs from previous time steps.
+
+Inspired by biological neural networks, we group the connections of a neuron into dendritic segments. A segment acts as an independent computational unit that detects a particular input pattern (a contextual state) defined by its receptive field. In our model, a segment connects the factor value  $f_{l}$  and the excitation  $E_{l}$  induced by the segment  $l$  to the cell to which it is connected. The segment is active, i.e.  $s_{l} = 1$  if all its presynaptic cells are active, otherwise  $s_{l} = 0$ . Computationally, a segment transmits its factor value  $f_{l}$  to the cell to which it is attached if the context matches the corresponding state combination.
+
+We can now rewrite equation 3 as the following:
+
+$$
+p \left(h _ {t} ^ {k}\right) \propto \sum_ {l \in \operatorname {s e g} (j)} L _ {l} f _ {l} ^ {k}, \tag {14}
+$$
+
+where  $L_{l} = \prod_{u\in \operatorname{rec}(l)}m_{u}$  is segment's likelihood as long as messages are normalized to probability distributions,  $\mathrm{seg}(j)$  — indexes of segments that are attached to cell  $j$ ,  $\operatorname{rec}(l)$  — indexes of presynaptic cells that constitute receptive field of a segment with index  $l$ .
+
+Initially, cells have no segments. As learning progresses, new non-zero connections, grouped into segments, are created. In equation 14, we benefit from having sparse factor value vectors because their complexity depends linearly on the number of non-zero components. And that's usually the case in our model due to the one-step Monte Carlo learning and the special form of the emission factors  $F_{e}^{k}$ :
+
+$$
+F _ {e} ^ {k} \left(h _ {t} ^ {k}, \varphi_ {t} ^ {k}\right) = \mathbb {I} \left[ h _ {t} ^ {k} \in \operatorname {c o l} \left(\varphi_ {t} ^ {k}\right) \right], \tag {15}
+$$
+
+where  $\mathbb{I}$  indicator function,  $\operatorname {col}(\varphi_t^k)$  is a set of hidden states connected to the feature state  $\varphi_t^k$  forming a column. The shape of the emission factor is inspired by the presumed columnar structure
+
+of the neocortex and has been shown to induce a sparse transition matrix in HMM (George et al., 2021).
+
+The segment likelihood  $L_{l}$  resulting from the sum-product algorithm is computed as if the presynaptic cells were independent. However, this isn't usually the case for sparse factors. To account for their interdependence, we substitute the following equation for segment log-likelihood:
+
+$$
+\log L _ {l} = w _ {l} \log \frac {1}{n _ {l}} \sum_ {u \in \operatorname {r e c} (l)} w _ {u l} m _ {u} + \sum_ {u \in \operatorname {r e c} (l)} \left(1 - w _ {u l}\right) \log m _ {u}, \tag {16}
+$$
+
+where  $w_{ul}$  — synapse efficacy or neuron specificity for segment, such that  $w_{ul} = p(s_l = 1|c_{t - 1}^u = 1)$ ,  $w_{l} = \frac{1}{n_{l}}\sum_{u}w_{ul}$  average synapse efficacy for segment  $l$ , and  $n_l$ -number of cells in segment's receptive field  $rec(l)$ .
+
+The idea behind the formula is to approximate between two extreme cases. The first is  $p(s_{l} = 1|c_{t - 1}^{u} = 1) \to 1$  for all  $u$ , which means that all cells in the receptive field are dependent and part of a cluster, i.e. they fire together. In this case,  $p(s_{l}) = m_{u}$  for all  $u$ , but we also reduce the prediction variance by averaging across different  $u$ . In the opposite case,  $p(s_{l} = 1|c_{t - 1}^{u} = 1) \to 0$  for all  $u$ , presynaptic cells don't cluster and the segment activation probability is just a product of the activation probability of each cell.
+
+The resulting equation for belief propagation in DHTM is the following:
+
+$$
+p \left(h _ {t} ^ {k}\right) = p \left(c _ {t} ^ {j} = 1\right) = \operatorname * {s o f t m a x} _ {j \in \text {c e l l s} \left[ H _ {t} ^ {k} \right]} \left(\max  _ {l \in \operatorname {s e g} (j)} \left(E _ {l}\right)\right), \tag {17}
+$$
+
+where  $E_{l} = \log f_{l} + \log L_{l}$ , cells  $[H_t^k ]$  —indexes of cells representing states for the variable  $H_{t}^{k}$ . Here we also approximate the logarithmic sum with the max operation, inspired by the neurophysiological model of segment aggregation by cells (Stuart & Spruston, 2015).
+
+The next step after computing the  $p(h_t^k)$  distribution parameters is to incorporate information about the current feature states  $p(h_t^k \mid \varphi_t^k) \propto p(h_t^k) \mathbb{I}[h_t^k \in \mathrm{col}(\varphi_t^k)]$ . After that, the learning step is executed. The loop closing step of our TM algorithm is to assign the posterior of the current step  $p(h_t^k \mid \varphi_t^k)$  to  $M_{t-1}^i$ .
+
+# C BIOLOGICAL INTERPRETATION
+
+Neural implementation of the DHTM is inspired by neocortical neural networks (see Fig. 4). Hidden variables  $H^{k}$  may be considered as populations of excitatory pyramidal neurons in cortical layer L2/3 of somatosensory areas, with lateral inhibition modelled as softmax function. Staiger & Petersen (2021) showed that neurons in this layer are responsible for temporal context formation.
+
+The neuronal activity at timestep  $t$  can be thought to carry messages  $M_{t-1}^{k}$ . Messages are propagated through synapses of dendritic segments, which correspond to factors  $F_{c}^{k}$ . Dendritic segments of biological neurons are known to be coincidence detectors of their synaptic input (Stuart & Spruston, 2015). That is, each segment can be thought of as a detector of a particular combination of active cells (states in our interpretation). If we associate a segment's parameter with factor value, a collection of dendritic segments represent a mapping from state space to factor values, representing context factor  $F_{c}$ .
+
+Our computational model of neuron and its dendrites is based on HTM neuron model (Hawkins & Ahmad, 2016). In this model, each segment has a receptive field and activates when a number of active cells in this receptive field reaches a threshold. The neuron is active if any of its segments are active. We use the same activation rule, but the threshold is fixed to the receptive field size. Our algorithm of segment's growth on demand is also similar to HTM's. In principle, DHTM can be considered as a probabilistic interpretation of HTM, allowing belief propagation.
+
+Similarly to HTM framework, our temporal memory can be interpreted as a model of cortical layers. Feature variables  $\Phi_t^k$  may be considered to represent cells of a granular layer (L4), as they are known to be the main hub for sensory excitation for L2/3. L2/3 cells that have common sensory input from the layer L4 are modelled as columns for particular feature states  $\operatorname{col}(\varphi_t^k)$  (Mountcastle, 1997).
+
+![](images/dff6c9ed50eb7cd707902252739bbc6ff8a2c2a7115ce3b67c07eb62d1da3246.jpg)  
+Figure 4: Biological view of the neural implementation of the DHTM. Variables  $H_{t-1}$  correspond to populations of neurons that have common sensory input and lateral inhibitory competition. Dendritic segments correspond to factor values  $f_l$ . Spike frequency of a neuron reflects state probability  $p(h_t^k)$  of a variable.
+
+# D ENCODING OBSERVATIONS
+
+Since DHTM is designed to process categorical random variables, we need a categorical encoder for RGB images to solve AnimalAI tasks.
+
+To compare our model to CSCG and EC in AnimalAI, we use the approach described in Guntupalli et al. (2023) to quantize observation space. K-Means (implementation from scikit-learn) with 50 clusters pretrained on  $4 \times 10^{4}$  64x64 RGB observations from trajectories sampled by uniform policy. To encode an image, the distance from each cluster is calculated and the index of the closest cluster is returned. In that way, AnimalAI's observation space is represented by one categorical variable.
+
+In order to test DHTM's ability to deal with distributed encoding, we also trained Categorical VAE (Jang et al., 2017) with the latent space represented by five categorical variables with 50 states each. We use a slightly modified implementation from Subramanian (2020). The only difference is that we modify the loss function by adding mean squared reward prediction error, which is linearly calculated from the latent codes to ensure the linear decomposition in feature space (equation 12). The model is trained on  $6 \times 10^{4}$  64x64 RGB images for 20 epochs with batch size 64. An image is encoded by first passing it through VAE's encoder block, then, to ensure stable encoding, instead of sampling from the latent distribution, the state with the highest logit is chosen for each variable.
+
+# E IMPLEMENTATION DETAILS
+
+As mentioned in Section 3.4, we incorporate DHTM as a part of an RL agent, which has a memory model and a feature reward function. All tested memory models share the same pipeline—conditioned on actions, they learn sequences of categorically encoded observations and are used to form SFs.
+
+In all experiments in Section 4.1, the agent uses softmax exploration with temperature equal to 0.04, except for EC agent we set softmax temperature to 0.01, since for higher temperatures its policy is unstable. For SF formation the agent uses  $\gamma = 0.8$  and  $\gamma = 0.9$  for Gridworld and AnimalAI experiments respectively. Maximum SF horizon is set to 50, but it is usually shorter since we employ early stop condition based on feature variables distribution. That is, if the prediction is close to uniform or the probability of a state with associated positive reward meets a threshold, the
+
+planning is interrupted. We use probability thresholds equal to 0.05 and 0.01 for Gridworld and AnimalAI respectively.
+
+In scaling Gridworld experiments (Appendix H.2), we increase maximum planning horizon up to 100 steps, and we also had to decrease agent's temperature to 0.02 for DHTM and to 0.002 for EC agent. Discount factor is 0.7 for DHTM and 0.9 for EC. We found that, in  $10 \times 10$  maze, DHTM's strategy becomes more stable if we increase the positive reward threshold up to 0.4. EC agent converges faster if we increase its softmax temperature, however, it converges to less optimal trajectory on average.
+
+In the noise tolerance AnimalAI experiment (Appendix H.3), for DHTM we use the same set of agent parameters, except we set the goal probability threshold to 0.5. We also set the goal detection threshold equal to the mean positive reward since we found that noise variable states mostly get reward assignment slightly above zero, in contrast to signal variables, in which only several states get positive reward assignment.
+
+# E.1 LSTM AND RWKV
+
+LSTM baseline was implemented with a single LSTMCell from PyTorch library (Paszke et al., 2019). It is supported by an additional symexp-layer to encode input before passing it to the LSTM cell and a symexp-layer to decode the LSTM cell's output from the LSTM's hidden state back to the input representation, where symexp activation function,  $\mathrm{symexp}(x) = \mathrm{sign}(x)e^{|x| - 1}$ , is a reverse of symlog function:  $\mathrm{symlog} = \mathrm{sign}(x)\log(|x| + 1)$ .
+
+The similar way we implemented RWKV baseline: a single RWKV layer supported by single-layer encoder and decoder. Current public RWKV implementation is a fast evolving framework (Bo, 2021), and for the increased performance it is tightly bound to the offline batch training common for the transformer architectures. In our case we needed a so-called sequential mode for online learning similar to LSTM. Thus, we adapted another public implementation mentioned in the official documentation (RWKV in 150 lines of code).
+
+For both Gridworld and AnimalAI tasks, LSTM and RWKV with a hidden state size of 100 are trained on a buffer of 1000 trajectories with learning rate 0.002 on 50 randomly sampled batches of size 50 every 10 episodes.
+
+We also incorporated some notion of random variables in hidden states by splitting the hidden state of the tested RNNs into groups. In all experiments the hidden state represents 10 categorical variables with 10 states. That is, RNN is forced to learn 10 categorical distributions with multi-cross-entropy loss to explain the observed sequences, which is a somewhat close to the multi-categorical hidden state representation used in Dreamer V2/V3 (Hafner et al., 2023). The idea of using symexp activation function, mentioned above, is inspired by Dreamer too, and is used to remedy the problem of learning extreme probability values. Without symexp the neural network has to represent zero probability with high negative logit values and one-probability with high positive logit values, which is hard to reach with low learning rate and may lead to instabilities. Thus, symexp function makes it faster to reach target values in log space.
+
+# E.2 CSCG
+
+CSCG baseline was implemented using code from the repository accompanying the paper (https: //github.com/vicariousinc/naturecomm_cscg). CSCG was trained on buffers of 500 action steps with 1000 EM steps until convergence. We iteratively calculated the exponential moving average of transition matrices obtained for different batches with a smoothing coefficient  $\alpha = 0.05$  as described in the paper (Dedieu et al., 2019). This smoothed transition matrix was used as an initialization for the next training batch and for inference. The first batch of observations is gathered using the uniform policy, and later batches are gathered using the current best policy.
+
+In so-called oracle mode, the transition matrix is trained from scratch every 500 steps. The buffer is discarded at the environmental change and the transition matrix is reinitialised.
+
+# E.3 EPISODIC CONTROL
+
+Here, we provide an algorithm for SF formation using EC dictionary  $\mathcal{D}_a$  (see Algorithm 3). In contrast to other memory algorithms, EC dictionary doesn't estimate state transition probabilities, but it stores connections between them in all-or-nothing fashion. Therefore, on each time step, all predicted observations are assumed equally probable. But it's not a limitation in our set of experiments, since we tested our algorithms in deterministic environments.
+
+Algorithm 3 SF formation for EC  
+Input:  $h_0,\gamma \in (0,1),T$    
+Output: SF   
+1: SF  $\leftarrow$  array of zeros   
+2: PS  $\leftarrow \{h_0\}$  # PS is the set of all states (nodes) on the current BFS depth   
+3: goal_found  $\leftarrow$  false   
+4: for  $l = 1..T$  do   
+5: PS  $\leftarrow \bigcup_{a\in A}\{\mathcal{D}_a(h)\}_{h\in \mathrm{PS}}$  # get next depth nodes assuming the policy is uniform   
+6: counts  $\leftarrow$  array of zeros   
+7: for all  $h\in \mathrm{PS}$  do   
+8:  $\varphi \gets \operatorname {argmax}F_e(h,\varphi)$    
+9: count  $\varphi =$  counts+1   
+10: goal_found  $\leftarrow r_{\varphi} > 0$  # check if the feature state is rewarding   
+11: end for   
+12: SF  $=$  SF  $^+\gamma^{l - 1}$  NORMALIZE(counts)   
+13: if goal_found is true then   
+14: break   
+15: end if   
+16: end for
+
+# E.4 DHTM
+
+In the Gridworld task and AnimalAI with K-Means encoder, the factor-graph for DHTM consists of three hidden variables  $H$  with 40 states per one feature state. Each hidden variable gets the same copied observation, and their predictions are averaged. Using several hidden variables here is essential, as it significantly increases the amount of trajectories memory is able to store by decreasing collision probability for hidden state representations of different trajectories.
+
+In AnimalAI experiments, hidden variables state spaces are of size 40 states per one feature state. For Categorical VAE encoder, we use one hidden variable per feature variable, except the case (factor  $1\star$ ) that tests that information from one feature variable is not sufficient for correct prediction, in which three hidden variables per observation variable are used and the factor of size three connecting to only hidden copies of the same feature variable. Examples of typical DHTM's graphical models used in our experiments are presented on Figure 5. It's important to note that edge colours denote separate graphical models, but the variables are shared (copied) between them. So there are no loops in these models. For simplicity, we define context factor size as the number of hidden variables from the previous time-step it has connections to, but it also connects to one hidden variable of the current step and the action variable.
+
+The most important DHTM's parameters, except the factor size, are factor learning rate  $\alpha$  and its initial value  $f_0$ . For better adaptability in changing and stochastic environments, the initial factor value should be set closer to zero, but in stationary environments, it can be safely set  $f_0 = 1$ , allowing faster convergence. In Gridworld we use  $\alpha = 0.1$  and  $f_0 = 0.05$ , and in AnimalAI we set  $\alpha = 0.001$  and  $f_0 = 1$ . We also set the limit on the number of segments  $5 \times 10^4$ , but practically never reach it in our experiments.
+
+![](images/f9a7e443692e243b7b875b3bd777815057c75c5f5a025cd013276eecc6ad5d26.jpg)  
+Figure 5: Examples of factor-graphs for two main setups. Edge colours show that edges correspond to separate graphical models with copied variables. All context factors are also connected to action variable  $A_{t-1}$ . A. Graphical model for one feature variable. In that case, feature value is copied for three hidden variables, to increase hidden state space and avoid memory collisions. B. Graphical model for Categorical VAE distributed encoding, example for factor of size 3 (i.e. takes into account tree hidden variables from the previous time-step).
+
+![](images/3f7b668fe7423b78d70fdf1c5a08472a7b3dc2c7a7b9a0114f854ba8b31e0850.jpg)
+
+# F EXPERIMENTAL SETUPS
+
+# F.1 GRIDWORLD
+
+We use a simple homebrew Gridworld implementation. Each Gridworld position maps to a colour, reward, and terminal state indicator. There are two special colours for the world border and obstacles. Each time-step, an agent gets the colour index of the current agent's position. An agent is able to see obstacles or border colour when trying to go through them, which just replace current position colour for one step without actually changing the agent's position. We also punish an agent trying to go through a wall or world border in addition to base step reward punishment to facilitate faster convergence to optimal trajectories.
+
+Samples of setups used in our experiments are depicted in Figure 6. Additional setups for scaling experiments are shown in Figure 7. Here, the red colour corresponds to obstacles (negative numbers) and the light gray colour to the reward (number 4). Letters A and G correspond to the agent's initial position and goal, correspondingly. Floor colours are resampled for each trial, but we ensure that there are no more than five floor colours, including the terminal state. In each episode, the agent and the terminal rewarding state have the same initial positions. The episode ends when the terminal state is entered or the limit of action steps is reached.
+
+# F.2 ANIMALAI
+
+AnimalAI is a testbed inspired by experiments with animals (Crosby et al., 2020). The environment consists of 3D area surrounded by a wall and many different objects that can be placed using a configuration file, including walls, food, ramps, trees, movable obstacles, and more. We tested our agent in a  $10 \times 10$ -meter room surrounded by walls (see Fig. 8). Each timestep, the agent gets an RGB image of a first-person view of the environment and is able to influence its behaviour by choosing one of three actions: turn left, turn right, or go forward. The agent and food item have the same fixed initial positions for each episode. The episode ends if the food item is collected or the maximum episode time is reached. To make an agent's actions more discretized, we repeat its choice three times and skip three frames.
+
+![](images/a6ee4366a898a2edff0a64ac5f7f9303cd04d0fc3bd11103faffda5e694b4d68.jpg)  
+Figure 6: Schematic view of POMDP Gridworld setups. The left picture shows the initial setup of the environment, with the reward delivered by the light gray square (number 4). The right picture represents the environment after the reward is blocked by obstacles (red squares or negative numbers). The letter A depicts the agent's initial position and the letter G—the goal.
+
+![](images/e23657e3bf04e049b7d8d637549272ada924b4e6a8e94d6dab5152d3c0f87f6f.jpg)
+
+![](images/a7b3ee5c1f04a0c993ec369272ca7a187a21df8e7979c4248d296800bebcd6b5.jpg)  
+Figure 7: Schematic view of POMDP Gridworld setups for scale experiments. From left to right:  $5 \times 5$ ,  $7 \times 7$ ,  $10 \times 10$ . The letter A depicts the agent's initial position and the letter G—the goal.
+
+![](images/ef233b8dab17e1a3b65ee40ce60fff01b99eb7f73c9e376839d8bb5ee23b5f8a.jpg)
+
+![](images/55585ae2a2fe373e7f2271cd1f8e4a4a654ecb31f7893d2c3dabb6a727e59059.jpg)
+
+![](images/1cf9064f71211905eaba375e7765a68927849cc64b864cd00a6ed70bca5ef7b7.jpg)  
+Figure 8: Sample of the agent's first-person view of our setup in AnimalAI environment. The green ball represents a goal, a food item to be collected.
+
+![](images/32613895544b1cb3d3993a329b63d2a363d92644035c9a2b9e9521b8f9a53445.jpg)  
+Figure 9: Dreamer V3 (18M) score in AnimalAI experiment described in this paper.
+
+# G CAPACITY
+
+In this section, we provide basic theoretical analysis of DHTM's capacity. Since the hidden state for unpredicted feature state transition is generated randomly, the main source of errors is accidental collision of hidden states. Due to special form of the emission factor (see equation 6), the collision is possible only for the same feature state. Here we analyse the capacity of DHTM for factor grpah depicted in Figure 5. Let's assume that the stream of feature states is sampled from uniformly distributed categorical random variable. Therefore, the probability to have a collision  $p_c = \frac{1}{n_\varphi n_v n_h}$  where  $n_v$  — number of hidden variables,  $n_\varphi, n_h$  are the number of states for feature and hidden variables correspondingly. Therefore, the probability of having no collisions for a  $l$  transitions is equal to  $(1 - p_c)^l$ . Let's set a threshold  $c \in (0,1)$ , which determines our confidence in that the sequence of transitions is collision free. Solving the equation  $1 - c = (1 - p_c)^l$  with respect to  $l$ , we get:
+
+$$
+l = \frac {\log (1 - c)}{\log \left(1 - \frac {1}{n _ {\varphi} n _ {v} n _ {h}}\right)}. \tag {18}
+$$
+
+Using the Taylor expansion for  $n_{\varphi}n_v n_h \to +\infty$ , we can see that  $l$  grows linearly with the total number of states  $n_{\varphi}n_v n_h$ . Therefore, DHTM's capacity scales linearly with the number of neurons. This, however, can be easily improved if we set additional prior on hidden state activation probability, which would take into account hidden state usage, but we leave it for future research.
+
+# H ADDITIONAL EXPERIMENTS
+
+# H.1 DREAMER
+
+In order to ground our experiments to state-of-the-art model based algorithms, as an additional baseline, we test the smallest version of Dreamer V3 with 18 million parameters. We use the implementation described in Voudouris et al. (2023), which is based on Hafner et al. (2023) and adapted for AnimalAI.
+
+We tested it with default parameters in the same setup as described in Section 4.2, except we don't apply frame skip. The agent successfully solves the task after  $105 \times 10^{3}$  action steps. Taking into account the frame skip used in our experiments with DHTM, it's approximately  $35 \times 10^{3}$  action steps versus  $3.5 \times 10^{3}$  for the DHTM agent to solve the task.
+
+# H.2 SCALABILITY
+
+In this section, we explore DHTM's scalability properties and compare them to the EC agent. In RL experiments, like presented in section 4, the number of segments largely depends on the RL algorithm used, its exploration strategy, and how fast it converges to a stable policy. We tested DHTM and EC agents in empty Gridworld of different sizes:  $5 \times 5$ ,  $7 \times 7$  and  $10 \times 10$ , where the agent and the goal are placed in opposite corners (see Fig. 7). As shown in Figure 10A, the DHTM-based
+
+agent requires fewer episodes to find the goal; however, the policy becomes less stable as the size of the Gridworld grows. As a consequence of faster convergence, DHTM produces fewer segments than the number of EC's dictionary entries throughout experiments. It also can be seen that for both DHTM and EC agents, the number of memory units grows non-linearly with the state space size (see Fig. 10B).
+
+![](images/b7dc866a1f980f64254c54eb6faff2024ae0141dbbbc1478a93c543e7a4282ef.jpg)  
+Figure 10: Comparison of EC and DHTM-based agents in foraging task in POMDP Gridworld of different sizes: 5x5, 7x7 and 10x10. A. Dynamic of steps required for the agent to reach the goal. B. Number of segments grown (or dictionary entries for EC agent) after 300 episodes with respect to state space size.
+
+![](images/42820ea17e67cfa028b22c7eae95b8583442e0317177958e67489d6fa9f45c0d.jpg)
+
+In principle, the number of segments reflects the number of transitions stored, which is proportional to the amount of experience provided to the memory. To show that, we tested DHTM and EC agent in 10x10 empty Gridworld using the same memory parameters as for experiments in section 4.1, but under uniform strategy. That is, agents freely explored the environment during 100 episodes and at the end the number of segments for DHTM and dictionary size for EC was recorded for each episode length. Figure 11 shows that DHTM's number of segments and EC's dictionary size grow linearly with the maximum episode length. In other words, since DHTM lacks generalisation abilities it largely serves as a trajectory buffer, in the same sense as a dictionary in EC.
+
+DHTM's computational complexity (time and memory) of the prediction step grows linearly with respect to the number of segments. Therefore, to use it for huge environments, there is need to prune unnecessary segments or remember only selected episodes. This is a common problem among episodic-like memories. The current approach to form SF also should be adjusted to be scalable, since it requires maximum planning horizon to be equal to the episode length. Among the improved approaches could be to employ n-step TD learning, hierarchical planning or SF caching, to recalculate it less frequently.
+
+# H.3 NOISE TOLERANCE
+
+In order to show the key difference between DHTM and its simplified EC version, we tested agents in the same AnimalAI setup as described in 4.2, but we add one noise variable to the VAE encoder. It's easy to see that the naive mapping from distributed VAE encoding to one categorical variable used for EC doesn't work in this case, since even identical observations will likely be encoded differently due to the noise variable. In contrast, since DHTM is able to handle distributed representations directly (as the name implies), it still can use non-noise variables independently to learn transitions.
+
+In this experiment, we vary DHTM's factor size (see Fig. 12). Since factors formed completely random, probability to include the noisy variable increases with factor size and this consequently worsen the performance of the agent. If the factor size is set to the total number of variables, DHTM becomes effectively unfactorised, similarly to its EC counterpart.
+
+![](images/f0dfbfbd6043498714d22f44c4dc462a709de084d890f71a4cd91c0efb533264.jpg)  
+Figure 11: Comparison of DHTM's number of segments with EC agent's dictionary size depending on episode length in Gridworld after uniform exploration during 100 episodes.
+
+![](images/f1243b4add336176d9c977179d4ed2e51d67a1f8b2ac4724177d37201eac4b3b.jpg)  
+Figure 12: Number of action steps the agent takes to reach the goal in a  $10 \times 10$ -meter room in the AnimalAI environment, using DHTM and EC models with a distributed Categorical VAE encoder that includes one additional noisy variable (six variables in total). DHTM's factor size is varied, determining the number of variables sampled for each factor.
+
+# I GLOSSARY
+
+Categorical Random Variable—a discrete random variable that can take on finite  $K$  possible states.
+
+Cortical Column or Minicolumn—a population of neurons in the neocortex that spans across layers and shares sensory input.
+
+Dendritic segment—a group of synapses (neuron's connections) that acts as an independent computational unit affecting the resulting neuron's activity.
+
+Factor Graph—bipartite graph representing the factorization of a probability distribution, with one part representing factor nodes and another—random variables.
+
+Factor size—number of variables connected to the factor. For simplicity, we refer only to the number of previous hidden states, used for prediction.
+
+Hidden Markov Model (HMM)—statistical model of a stochastic process where state probability depends only on previous state of the process.
+
+Multi-compartment neuron model—a model of neuron that divides neuron's connections into groups (segments) of different types (compartments), where each group may be considered as partly independent computational unit and groups of each compartment may affect the neuron's activity differently.
+
+Oracle—a mythical creature that gives up to our algorithms insightful information about the environment that is usually hidden.
+
+Successor Representations (SR)—a discounted sum of future [one-hot encoded] observations.
+
+Successor Features (SF)—a generalization of SR, a discounted sum of future latent states.
+
+Temporal Memory (TM)—in this work by this term we mean “memory for sequences”.

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