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+# ENHANCE MULTI-VIEW CLASSIFICATION THROUGH MULTI-SCALE ALIGNMENT AND EXPANDED BOUNDARY
+
+Yuena Lin  $^{1,\dagger}$ , Yiyuan Wang  $^{1,2,\dagger}$ , Gengyu Lyu  $^{1,*}$ , Yongjian Deng  $^{1}$ , Haichun Cai  $^{3}$ , Huibin Lin  $^{3}$ , Haobo Wang  $^{4}$ , & Zhen Yang  $^{1,*}$
+
+<sup>1</sup> College of Computer Science, Beijing University of Technology, Beijing  
+$^{2}$  Idealism Beijing Technology Co., Ltd., Beijing  
+<sup>3</sup> College of Computer and Data Science, Fuzhou University, Fuzhou  
+$^{4}$  School of Software Technology, Zhejiang University, Hangzhou
+
+yuenalin@126.com, yiyuanwang@mails.bjut.edu.cn
+
+lyugengyu@gmail.com, fjsmchc@163.com, 221010005@fzu.edu.cn
+
+wanghaobo@zju.edu.cn, {yjdeng, yangzhen}@bjut.edu.cn,
+
+# ABSTRACT
+
+Multi-view classification aims at unifying the data from multiple views to complementarily enhance the classification performance. Unfortunately, two major problems in multi-view data are damaging model performance. The first is feature heterogeneity, which makes it hard to fuse features from different views. Considering this, we introduce a multi-scale alignment module, including an instance-scale alignment module and a prototype-scale alignment module to mine the commonality from an inter-view perspective and an inter-class perspective respectively, jointly alleviating feature heterogeneity. The second is information redundancy which easily incurs ambiguous data to blur class boundaries and impair model generalization. Therefore, we propose a novel expanded boundary by extending the original class boundary with fuzzy set theory, which adaptively adjusts the boundary to fit ambiguous data. By integrating the expanded boundary into the prototype-scale alignment module, our model further tightens the produced representations and reduces boundary ambiguity. Additionally, compared with the original class boundary, the expanded boundary preserves more margins for classifying unseen data, which guarantees the model generalization. Extensive experiment results across various datasets demonstrate the superiority of the proposed model against existing state-of-the-art methods.
+
+# 1 INTRODUCTION
+
+Multi-view data are widely applied in real-world applications by combining diverse heterogeneous features to describe the same object (Wang et al. (2023a); Zhang et al. (2024b)). Unluckily, feature heterogeneity and information redundancy concealed in the multi-view data have become the obstacles in multi-view learning (MVL) (Tan et al. (2024)). The feature heterogeneity means that the multi-view data are collected from different data distributions without a unified data format, causing fusion difficulty among the view features (Wang et al. (2024a)). The information redundancy signifies the latency of unnecessary features or ambiguous data, which hinders the ability to delimit the decision boundaries appropriately and naturally engenders classification ambiguity.
+
+Existing MVL models mainly delve into alleviating the heterogeneity by finding informative common subspaces (Zhang et al. (2023a)) or ideal view weights (Hu et al. (2022)). The former explores the feature consistency and complementarity among the multi-view data (Lyu et al. (2024b); Wang et al. (2024c); Liang et al. (2024)), which neglects significant class differences for separating reliable decision boundaries. The latter presumes that the qualities or importances of views are stable for samples (Kumar & Maji (2023)), and aims at allocating appropriate weights to different views (Houfar et al. (2023); Liu et al. (2023)). However, the varied view qualities fail to match diverse samples practically (Han et al. (2022b)), which also leads to unreliable decision boundaries. For instance,
+
+![](images/60b063aba36db68944bc30bedf558937af6106113048ba4f26b1e2ca1518174f.jpg)  
+(a) Raw data
+
+![](images/a97e631d624d801beb06780ff9c699ee7cbb8bb13cfe085036e2b15ad78ca921.jpg)  
+(b) ETMC
+
+![](images/e930fe04cbadbaf26c377dfe829948df9d82335b73d69b3de386f05ff39751ec.jpg)  
+(c) PDMF  
+Figure 1: t-SNE on Scene15 dataset. The boundaries of the green class and the brown class are delineated. Compared with existing advanced models, our model shows much more distinct boundaries.
+
+![](images/acd184624b6ad435e6bd96e3ac2dcb51a9d27ea7aa191a50f8ebb9e4414701c3.jpg)  
+(d) RCML
+
+![](images/742067be496d2aab322c0d5920b49c7d359276f8de18c05ca40e0b33e85594f9.jpg)  
+(e) Ours
+
+tigers and zebras have similar stripes but different colors, which makes the model give the color view a higher weight. Nevertheless, zebras and pandas have the same colors but disparate stripes, which induces the model to weigh more for the stripe view. Such scenarios easily result in unstable qualities or significances of different views to construct unreliable decision boundaries eventually.
+
+Recent MVL works are dedicated to realizing trusted decisions by estimating the inherent uncertainties among multi-view data (Xu et al. (2023b; 2024)), and integrating multiple views at an evidence level. Actually, though they modify the logits to realize better classification decisions (Sensoy et al. (2018)), they still fail to deal with the aforementioned problems. To illustrate the issue and our motivation intuitively, we provide the t-SNE in Figure 1, which includes our model MAMC (enhance multi-view classification through multi-scale alignment and expanded boundary) and three advanced MVL models ETMC (Han et al. (2022b)), PDMF (Xu et al. (2023b)), and RCML (Xu et al. (2024)) on Scene15 dataset. The t-SNE reflects two issues of current MVL models: 1) They cannot well mine the commonality to tighten the representations; 2) They ignore to exploit the class difference for clearing the ambiguous decision boundaries. These observations imply that they are unable to reduce the feature heterogeneity and the information redundancy satisfactorily. Motivated by these observations, our model focuses on mining commonalities among view representations and learning clear decision boundaries in the representation space simultaneously to tackle the feature heterogeneity and the information redundancy. On the one hand, MAMC introduces a multi-scale alignment module to learn expressive view representations by mining the commonalities among views and class differences among prototypes; on the other hand, it expands the original class boundaries to the wider decision boundaries in the representation space, fitting ambiguous samples to clear the blurry areas between different classes adaptively. Specifically, the multi-scale alignment module includes two complementary contents: 1) An instance-scale alignment module mines the inter-view commonality by aligning the view representations of each instance. 2) A prototype-scale alignment module exploits the inter-class difference from the instances with the same labels to find appropriate decision boundaries jointly.
+
+The design of expanded boundaries is based on fuzzy set theory (Zhang et al. (2023b)), and its expansion process involves fuzzy representations and crisp representations, where the fuzzy representations are defined in the fuzzy representation space while the crisp representations are in the normal representation space. The core of boundary expansion is based on the assumption that crisp representations are a special type of fuzzy representations. It enables us to find learnable positives of the crisp representations in the fuzzy representation space, which are then employed to construct elements of the expanded boundary. Finally, the expanded decision boundary is integrated into the prototype-scale alignment process. By narrowing the expanded boundary and its corresponding class prototype while distancing other class prototypes, the inter-class difference is sufficiently exploited. Additionally, the expanded boundary reserves extra space for instances that belong to the class latently, which is beneficial for improving the model generalization. Our contributions are summarized as follows:
+
+- We propose MAMC to address the problems caused by the feature heterogeneity and the information redundancy. Through introducing the multi-scale alignment module, MAMC sufficiently mines the inter-view commonality to tighten instances and inter-class difference to delimit the decision boundaries.  
+- We propose a novel self-adaptive expanded boundary to tackle the ambiguous decision boundaries. By integrating the expanded boundary into the prototype-scale alignment module, the inter-class difference is reinforced to clear the ambiguous decision boundaries, while the representations inside the boundary are tightened.  
+- We explain the rationality of the model design theoretically, and extensive experimental results across diverse public datasets and comprehensive experimental analysis have
+
+verified that the proposed model shows significant superiority against the existing state-of-the-art models.
+
+# 2 PRELIMINARIES
+
+# 2.1 FUZZY SET THEORY
+
+Fuzzy set theory builds on classical (crisp) set theory by introducing the concept of membership degrees, which allows for fine-grained definitions of whether an element belongs to a set (Zadeh (1965)). A fuzzy set  $\mathcal{A}$  in the universe  $\mathbb{U}$  is characterized by a membership function  $f_{\mathcal{A}}(x)$  which allocates each element in  $\mathbb{U}$  with a real number in  $[0, 1]$ , where  $\mathbb{U}$  is defined as a set that contains all the elements under consideration for a particular discussion or problem, and  $f_{\mathcal{A}}(x)$  represents the membership degree of  $x$  in  $\mathcal{A}$  (Zhelezniak et al. (2019)). Fuzzy set theory provides a powerful framework for reasoning about sets with uncertainty, but the specification of membership functions depends greatly on domain knowledge. Specifically, we have the following definitions:
+
+Definition: A function  $f_{\mathcal{A}} \colon \mathbb{U} \to \mathbb{L} \subseteq \mathbb{R}$  is called a membership function.
+
+Definition: A tuple  $\mathcal{A} = (\mathbb{U}, f_{\mathcal{A}})$  including a universe  $\mathbb{U}$  and a membership function  $f_{\mathcal{A}}$  is called a fuzzy set.
+
+Following the definitions, when  $\mathbb{L} = [0,1]$ ,  $\mathcal{A}$  is a fuzzy set. When the membership function takes on only two values 0 and 1, namely  $\mathbb{L} = \{0,1\}$  according to whether  $x$  belongs to  $\mathcal{A}$ . In this case,  $\mathcal{A}$  reduces to a classical set.
+
+# 2.2 MULTI-VIEW LEARNING
+
+Joint learning on multi-view data has been proven effective on various tasks, such as clustering (Zhuge et al. (2020); Shen et al. (2024a)), and classification (Jiang et al. (2024); Lyu et al. (2024a)). Current works are roughly divided into the subspace-based (Shen et al. (2024b)) and view-weighted approaches (Zhang et al. (2024a)). The first category aims at finding a common subspace where the consensus representations are generated for all the views Wang et al. (2023b). Canonical correlation analysis (CCA) based approaches are one type of representative methods that project different views into a subspace by maximally correlating them. (Kumar & Maji (2023)) proposed D2CCA that integrates the CCA theory with the objective function to find the joint representations. (Yuan et al. (2022)) proposed a canonical  $\mathcal{F}$ -correlation framework where each feature was projected into a certain space by an arbitrary nonlinear mapping. Meanwhile, many works are dedicated to searching optimal weights for different views. (Xu et al. (2020)) proposed to make deep interactive information using view-specific information in an adaptive weighted manner, and seamlessly integrated the view-specific information with a multi-view loss fusion strategy to achieve joint decisions. (Houfar et al. (2023)) attended the distinctions between different views because of the sample importances, and proposed a dynamic learning strategy for automatically weighting views and samples. However, the subspace-based approaches mainly focus on mining feature commonality to handle feature heterogeneity but ignore the inter-class difference for delimiting the decision boundaries. And the view-weighted approaches have trouble producing generalized weights for views, which also makes it hard to find an effective boundary. In summary, these approaches fail to tackle the problem caused by information redundancy, especially when ambiguous samples are distributed around the decision boundaries. Motivated by this, we focus on mining commonalities among view representations and learning clear decision boundaries in the representation space simultaneously to alleviate the feature heterogeneity and the information redundancy.
+
+# 2.3 MULTI-VIEW CLASSIFICATION
+
+Existing multi-view classification methods can generally be divided into two categories: feature fusion-based and decision-based methods. Feature fusion-based methods focus on effectively combining features from all views (Meng et al. (2024)). MV-HFMD (Black & Souvenir (2024)) explores feature fusion by introducing a novel fusion scheme and mutual distillation to adapt neural networks for multi-view classification. Mmdynamics (Han et al. (2022a)) dynamically assesses the informativeness of both feature-level and view-level data across different samples, ensuring trustworthy integration of multiple views. However, these methods often neglect feature heterogene-
+
+ity, which undermines the effectiveness of feature fusion. For the decision-based methods, they are dedicated to making more reasonable decisions. ETMC (Han et al. (2022b)) introduces a new paradigm for multi-view learning by dynamically integrating different views at the evidence level, thereby enhancing classification reliability. RCML (Xu et al. (2024)) attends to that conflictive information in different views, and proposes a conflictive opinion aggregation strategy that exactly models the relation of multi-view common and view-specific reliabilities. MVAKNN (Fan et al. (2023)) utilizes the Dempster-Shafer theory to integrate information from each view, successfully extending adaptive KNN to a multi-view setting. IPMVSC (Hu et al. (2023)) constructs a multi-view fuzzy classification model inherited from the natural interpretability of fuzzy rule-based systems to realize interpretable classification. Many methods of this kind interact with the multiple views only in the decision phase, which cannot well integrate the information from all the views. In contrast, our model aligns view features during the representation learning phase and combines them in the decision phase, allowing for more effective and interactive information integration across all views.
+
+# 3 PROPOSED METHOD
+
+The architecture of our model is shown in Figure 2, including view-specific representation extractors, a multi-scale alignment module, and a joint classifier. To detail the model architecture conveniently, we first provide some notations.
+
+Notations. Formally, a multi-view dataset  $\mathcal{D} = \{(\{\pmb{x}_i^{(v)}\}_{v=1}^V, y_i) | 1 \leq i \leq N\}$  is defined as  $N$  i.i.d. instances with  $V$  views and corresponding labels. We use a real-value vector  $\pmb{x}_i^{(v)} \in \mathbb{R}^{D_v}$  ( $1 \leq v \leq V$ ,  $i = 1, \dots, N$ ) to denote the feature vector of  $i$ -th instance for the  $v$ -th view, where  $D_v$  is the feature dimension of the  $v$ -th view and  $K$  is the number of classes. For the labels, we use a natural number  $y_i \in \mathbb{N}$  to represent the ground truth of  $i$ -th instance and  $\hat{y}_i \in \{0, 1\}^K$  to represent the corresponding one-hot vector. To integrate information from the multiple views, our model aims at learning a desired mapping  $f: \mathcal{X} \to \mathcal{Y}$  from the multi-view dataset  $\mathcal{D}$  for better predicting the correct label of unseen instances.
+
+# 3.1 VIEW-SPECIFIC REPRESENTATION LEARNING
+
+Heterogeneous multi-view data conceal many inherent issues, such as inconsistent dimensions of view features. To address the issues, we employ multiple view-specific auto-encoders to generate exclusive view representations in the same dimension (Zhong et al. (2024); Jiang et al. (2024)). For the  $v$ -th view data  $\pmb{X}^{(v)}$ , we apply a view-specific encoder  $E^{(v)}$  to produce the view representation matrix  $\pmb{H}^{(v)}$  and a decoder  $D^{(v)}$  to restore the raw view feature, where the encoder and the decoder are constructed by multi-layer perceptions (MLPs). The reconstruction process is given by
+
+$$
+\mathcal {L} _ {r e c} = \frac {1}{V} \sum_ {v = 1} ^ {V} \left\| \boldsymbol {X} ^ {(v)} - D ^ {(v)} \left(E ^ {(v)} \left(\boldsymbol {X} ^ {(v)}\right)\right) \right\| _ {F} ^ {2}. \tag {1}
+$$
+
+# 3.2INSTANCE-SCALE ALIGNMENT
+
+The key to reducing the feature heterogeneity of multi-view data is to mine the inter-view commonality. Contrastive learning exploits commonality among data by pulling together the positives while pushing away the negatives (Wang et al. (2024b)). The instance-scale alignment module combines contrastive learning and regards all the view representations of each instance as positives while others as negatives. Concretely, the view representations  $\{h_i^{(v)}\}_{v=1}^V$  abstracted from the view-specific encoders are recognized as positives and others are negatives. Hence, the instance-scale alignment loss is
+
+$$
+\mathcal {L} _ {i n s} = - \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {u \neq v} \sum_ {i = 1} ^ {N} \log \frac {e ^ {- D i s \left(\boldsymbol {h} _ {i} ^ {(v)} , \boldsymbol {h} _ {i} ^ {(u)}\right) \cdot \tau_ {i n s}}}{e ^ {- D i s \left(\boldsymbol {h} _ {i} ^ {(v)} , \boldsymbol {h} _ {i} ^ {(u)}\right) \cdot \tau_ {i n s}} + \sum_ {r = u, v} \sum_ {j \neq i} e ^ {- D i s \left(\boldsymbol {h} _ {i} ^ {(v)} , \boldsymbol {h} _ {j} ^ {(r)}\right) \cdot \tau_ {i n s}}}, \tag {2}
+$$
+
+where  $\tau_{ins}$  is a temperature coefficient, and  $Dis(\cdot ,\cdot)$  is a distance function. By aligning the positive view representations, the critical common semantics to recognize similar samples are exploited,
+
+![](images/202df0df5e3ac96284c2c066ca2cbe9c3a4ee98daf199f1b83db135f61a3f7ed.jpg)  
+Figure 2: Three main components of the proposed model: (1) View-specific representation extractors. Multiple view-specific auto-encoders abstract critical view representations  $\{H^{(v)}\}_{v = 1}^{V}$  from multi-view data  $\{\pmb {X}^{(v)}\}_{v = 1}^{V}$ . (2) Multi-scale alignment module. The instance-scale alignment module aligns view representations  $\{h_i^{(v)}\}_{v = 1}^V$  of each instance  $x_{i}$ , where the icons with various shapes correspond to different instances and those with different colors correspond to representations from different views. The prototype-scale alignment module integrates the adaptive expanded boundaries, and then aligns the view representations with the same labels. (3) Joint classifier. All the view representations are sent into the joint classifier to obtain fused predictions.
+
+while the feature heterogeneity is mitigated. By differing the negatives, the discrepancies between different samples gradually emerge, thereby making the learned representations more suitable for delimiting the decision boundaries.
+
+# 3.3 PROTOTYPE-SCALE ALIGNMENT
+
+In contrast to the instance-scale alignment module, the goal of the prototype-scale alignment module is to exploit the inter-class difference for delimiting clear decision boundaries. The core of achieving this goal is to construct the expanded boundary.
+
+# 3.3.1 EXPANDED BOUNDARY
+
+Since the information redundancy in multi-view data easily blurs the decision boundaries, general prototype learning ignoring the inter-class difference has limited ability to work effectively. Instead, we propose an adaptive expanded boundary to assist the prototype-scale alignment module, expecting to clear the ambiguous boundaries.
+
+To sufficiently gather the class information for exploiting the inter-class difference, we collect all the view representations with the same labels to construct the class prototypes and use them to centralize the view representations. For the training data of the  $k$ -th class, the centralizing process is defined as:
+
+$$
+\boldsymbol {c} _ {k} = \frac {1}{V \left| Y _ {k} \right|} \sum_ {v = 1} ^ {V} \sum_ {y _ {i} = k} \boldsymbol {h} _ {i} ^ {(v)}, \tag {3}
+$$
+
+$$
+\boldsymbol {z} _ {i} ^ {(v)} = \boldsymbol {h} _ {i} ^ {(v)} - \boldsymbol {c} _ {k}, v = 1, \dots , V, \tag {4}
+$$
+
+where  $\pmb{z}_i^{(v)}$  is the centralized representation vector of the  $i$ -th instance in the  $v$ -th view,  $Y_k$  is the set including the training data whose labels are  $k$ , and  $|\cdot|$  is used for counting the cardinality of a set.  $\pmb{c}_k$  is the prototype vector of the  $k$ -th class, which filters the high-frequency noise by averaging.
+
+A key step of boundary expansion is to construct the fuzzy representation space with membership functions, and the Gaussian membership function is the most universal one (Li et al. (2023)). The appropriate number of membership functions imparts the expanded boundary with more adaptability. To this end, we adopt  $L$  Gaussian membership functions. For the  $l$ -th membership function, the corresponding membership degree is defined by
+
+$$
+\boldsymbol {r} _ {i, j, l} ^ {(v)} = \exp \left\{- \frac {\left(\boldsymbol {z} _ {i , j} ^ {(v)} - \boldsymbol {m} _ {j , l}\right) ^ {2}}{2 \boldsymbol {\delta} _ {j , l} ^ {2}} \right\}, l = 1, 2, \dots , L, j = 1, 2, \dots , d, \tag {5}
+$$
+
+where  $d$  is the dimension of the view representation,  $\boldsymbol{z}_{i,j}^{(v)}$  is the  $j$ -th element of  $i$ -th instance in the  $v$ -th centralized view representation vector,  $\boldsymbol{m}_{j,l}$  and  $\delta_{j,l}$  are the trainable mean and standard deviation of the  $l$ -th Gaussian membership function respectively. These membership degrees are fused to obtain the final one for producing fuzzy representations, and the process is defined as
+
+$$
+\tilde {\boldsymbol {r}} _ {i, j} ^ {(v)} = \operatorname {C o m b} \left\{\boldsymbol {r} _ {i, j, l} ^ {(v)} \right\}, l = 1, 2, \dots , L, \tag {6}
+$$
+
+$$
+\widetilde {\boldsymbol {Z}} ^ {(v)} = \widetilde {\boldsymbol {R}} ^ {(v)} \odot \boldsymbol {Z} ^ {(v)}, \tag {7}
+$$
+
+where  $\tilde{\pmb{r}}_{i,j}^{(v)}$  is the element of the membership degree matrix  $\widetilde{\pmb{R}}^{(v)}$ , and  $\odot$  is Hadamard product to transform the crisp view representation  $\mathbf{Z}^{(v)}$  into the fuzzy representation  $\widetilde{\mathbf{Z}}^{(v)}$ . Comb is the combination operator to fuse membership degrees and how to determine it will be explained later.
+
+According to properties of membership functions, each value of  $\widetilde{\pmb{R}}^{(v)}$  is limited in [0, 1], which implies that the crisp representations can be viewed as one special case of fuzzy ones. Based on this, we regard crisp representations and fuzzy representations as positives. Besides, we assume that their discrepancies reflected by the coordinate differences are tolerable since the discrepancies between positives do not significantly violate the commonality semantics in general. Inspired by this, we implement the tolerable discrepancy to expand the original class boundary, and the boundary expansion process is depicted in Figure 3. Mathematically, the elements of the expanded boundary are transformed by the coordinates of the fuzzy representations, which are formalized as
+
+![](images/dddba76ba29b2f31525a07cdc38386dfc2523d41efb917f92c58b1301aff6e4e.jpg)  
+Figure 3: The boundary expansion process in 2-dimension space. One centralized view representation  $\triangleleft$  is firstly utilized to calculate the fuzzy representation  $\triangleleft$ , and then mapped as the boundary element  $\triangleleft$ .
+
+$$
+\overline {{\boldsymbol {Z}}} ^ {(v)} = \widetilde {\boldsymbol {Z}} ^ {(v)} + 2 \left(\mathbf {1} - \widetilde {\boldsymbol {R}} ^ {(v)}\right) \odot \boldsymbol {Z} ^ {(v)} = \left(\mathbf {2} - \widetilde {\boldsymbol {R}} ^ {(v)}\right) \odot \boldsymbol {Z} ^ {(v)}, \tag {8}
+$$
+
+where  $\overline{Z}^{(v)}$  is composed of boundary elements, 1 and 2 are two real value matrices whose each element is 1 and 2 respectively. Though simple, the expanded boundary inherits two desired properties from the fuzzy representations: 1) Semantics-positive. The boundary elements are supposed to be the positives of crisp representations, which benefit recognizing the ambiguous samples far away from the class centers. 2) Self-adaptive. The expanded boundaries are based on the learnable membership functions and could adjust adaptively to involve the ambiguous samples in the training process. These two properties aim to capture nearly all the samples inside each boundary for fully exploiting the inter-class difference.
+
+# 3.3.2 LOSS FUNCTION
+
+After expanding the original boundaries, we restore the coordinates of the centralized fuzzy representation matrix  $\overline{Z}$  by adding their corresponding class prototype  $C$  as the restored fuzzy representation matrix  $\overline{H}$ , since it is tough to separate the classes in a centralized representation space whose class centers are all origin points. The expansion of the original class boundaries leads to the new-born prototype of boundary elements  $\widetilde{C}$ , which causes a trade-off between  $C$  and  $\widetilde{C}$ . We mix up both to harvest the fused prototype  $\overline{C}$  by a linear combination as
+
+$$
+\overline {{\boldsymbol {C}}} = \eta \boldsymbol {C} + (1 - \eta) \widetilde {\boldsymbol {C}}, \tag {9}
+$$
+
+where each prototype vector of  $C$  is calculated by Eq. (3),  $\widetilde{C}$  is calculated by averaging the boundary elements of each class, and  $\eta$  is a balance coefficient to reconcile  $C$  and  $\widetilde{C}$ .
+
+To generalize for unseen data as much as possible, we expect to maximize the spectrum of each class, which is equal to maximizing the variance within each class. Meanwhile, we expect to achieve better separateness among different classes and compactness within each class, which intends to maximize the distances between different classes while minimizing the variance within each class. Considering these factors, the loss function is formalized as a minimax game by
+
+$$
+\mathcal {L} _ {p r o} = \min  _ {\mathcal {P} + \mathcal {V}} \max  _ {\mathcal {V}} (\mathcal {P} + \mathcal {V}), \tag {10}
+$$
+
+where  $\mathcal{P}$  is defined to distance different classes and  $\nu$  to tighten samples within each class while generalizing for unseen data. They could be defined as
+
+$$
+\left\{ \begin{array}{l} \mathcal {P} = - \frac {1}{N K V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {n = 1} ^ {N} \left\| \overline {{\boldsymbol {h}}} _ {n} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2} \\ \mathcal {V} = \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2} \end{array} . \right. \tag {11}
+$$
+
+For convenience, we let  $\mathcal{Q} = \mathcal{P} + \mathcal{V}$ , and propose an upper bound of  $\mathcal{Q}$  in our model, expecting to further optimize  $\mathcal{Q}$  by optimizing its upper bound. We define the upper bound  $\mathcal{Q}_{pro}$  as
+
+$$
+\mathcal {Q} _ {p r o} = - \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \log \frac {e ^ {- \tau_ {p r o} \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2}}}{\sum_ {j} e ^ {- \tau_ {p r o} \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2}}}, \tag {12}
+$$
+
+where  $\overline{c}_k$  is the fused prototype vector of the  $k$ -th class,  $\overline{h}_i^{(v)}$  is the restored boundary elements of the  $i$ -th instance in the  $v$ -th view, and  $\tau_{pro}$  is the temperature coefficient to control the compactness of classes. Generally, higher  $\tau_{pro}$  makes the classes more compact. We provide the theoretical proof that  $\mathcal{Q}_{pro}$  is an upper bound of  $\mathcal{Q}$  in the Appendix.
+
+In the prototype-scale alignment loss, we integrate all the view representations to serve as the class samples since partial view information easily releases misleading information or loses critical information, which may lead to ambiguous decision boundaries. In this way, all the view information is gathered to collaboratively form the inter-class difference for delimiting the decision boundaries. Additionally, according to (Oord et al. (2018)), contrastive learning evaluates the mutual information between the raw view data and the restored boundary elements, namely
+
+$$
+I \left(\boldsymbol {x} _ {i}, \left\{\overline {{\boldsymbol {h}}} _ {i} ^ {(v)} \right\} _ {v = 1} ^ {V}\right) \geq \log (\mathcal {N}) - \mathcal {Q} _ {p r o}, \tag {13}
+$$
+
+where  $I(\cdot)$  is used to calculate the mutual information,  $\mathcal{N}$  is the sample amount participated in the loss function. This inequation reveals that more negatives raise the lower bound and preserve more complete information from the raw data for better alignment performance. Typically, integrating all the view representations is a flexible way to multiply the amount of negatives.
+
+# 3.3.3 COMBINATION OPERATOR DETERMINATION
+
+The combination operator  $Comb$  in Eq. (6) is crucial to the expanded boundaries. On the one hand, the  $Comb$  operator cannot damage the self-adaptive ability of the expanded boundaries.
+
+On the other hand, the Comb operator controls the class variances to generalize for unseen data. The former is achieved by the learnable membership functions, and the latter is affected by the determination of the Comb.
+
+Theorem 1. When employing Min Pooling as the Comb operator, the overall variance of each class is maximized. The proof is provided in the Appendix.
+
+Therefore, when the ideal  $Comb$  operator is defined as the minimal of the membership functions, the spectrum of each class is maximized. It means that the expanded boundary could generalize more latent (unseen) samples belonging to corresponding classes. Finally, the prototype-scale alignment loss  $\mathcal{L}_{pro}$  in Eq. (10) is reduced as
+
+$$
+\mathcal {L} _ {p r o} = \min  - \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {i = 1} ^ {N} \log \frac {e ^ {- \tau_ {p r o} \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{c}} _ {i} \right\| _ {2} ^ {2}}}{\sum_ {j} e ^ {- \tau_ {p r o} \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{c}} _ {j} \right\| _ {2} ^ {2}}}. \tag {14}
+$$
+
+# 3.4 CLASSIFICATION PREDICTION
+
+During the classification process, we employ a joint classifier that adopts the parameter sharing strategy, and it has several merits: 1) Shared parameters receive information from all the views simultaneously, which is conducive to constructing a classifier specific for the multi-view data. 2) The multi-scale alignment module has mined the inter-view commonality and the inter-class difference to construct clear decision boundaries, reducing the demand on the complex classifier with a large number of parameters. The model prediction is produced by averaging the outputs of the joint classifier by
+
+$$
+\hat {\boldsymbol {p}} _ {i} = \frac {1}{V} \sum_ {v = 1} ^ {V} \boldsymbol {p} _ {i} ^ {(v)} = \frac {1}{V} \sum_ {v = 1} ^ {V} g \left(\boldsymbol {h} _ {i} ^ {(v)}\right), \tag {15}
+$$
+
+where  $g(\cdot)$  is a logistic classifier,  $\pmb{p}_i^{(v)}$  is the output of  $g(\cdot)$  in the  $v$ -th view, and  $\hat{\pmb{p}}_i$  is the final prediction. During training, we use the cross-entropy to calculate the classification loss as
+
+$$
+\mathcal {L} _ {c l s} = - \frac {1}{N} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {K} \hat {\mathbf {y}} _ {i j} \log \hat {\boldsymbol {p}} _ {i j}. \tag {16}
+$$
+
+where  $\hat{p}_i$  is the final prediction from Eq. (15), and  $\hat{p}_{ij}$  is its  $j$ -th element.  $\hat{y}_i$  refers to the one-hot vector label of the  $i$ -th instance, and  $\hat{y}_{ij}$  is the corresponding element in this vector. The overall loss  $\mathcal{L}_{all}$  comprises four parts, namely reconstruction loss, classification loss, instance-scale alignment loss, and prototype-scale alignment loss, which is given by
+
+$$
+\mathcal {L} _ {a l l} = \mathcal {L} _ {r e c} + \mathcal {L} _ {c l s} + \alpha \mathcal {L} _ {i n s} + \beta \mathcal {L} _ {p r o}, \tag {17}
+$$
+
+where  $\alpha$  and  $\beta$  are penalty coefficients.  $\mathcal{L}_{all}$  focuses on mining the commonality among different views and clearing ambiguity areas around the decision boundaries, which successfully alleviates the damage of the feature heterogeneity and the information redundancy.
+
+# 4 EXPERIMENTS
+
+In our experiment, we evaluate our model on eight public multi-view datasets including HandWritten, Scence15, PIE, CCV, Animal, 100Leaves, Hdigit, and YoutubeFace. To show the effectiveness of the proposed model, seven state-of-the-art methods are adopted to compare with, including mm-dynamics (Han et al. (2022a)), ETMC (Han et al. (2022b)), UMDL (Xu et al. (2023a)), PDMF (Xu et al. (2023b)), IPMVSC (Hu et al. (2023)), MV-HFMD (Black & Souvenir (2024)), and RCML (Xu et al. (2024)). For each dataset, we split  $80\%$  instances for training and the remainder for testing. To obtain reliable results, we implement the same dataset split ten times for all the models. More details of the datasets are provided in the Appendix.
+
+# 4.1 EXPERIMENTAL RESULTS
+
+Table 1 presents the classification results between our model and other baseline models, and we adopt Accuracy (Acc), Purity, Recall, and Macro-F1 as the metrics. From the statistical comparisons, some important observations are revealed:
+
+Table 1: Classification results (\%) with baseline models. The best results are highlighted in bold. "OOM" indicates that the models raise the out-of-memory failure.  
+
+<table><tr><td>Datasets</td><td>Metrics</td><td>mmdynamics</td><td>ETMC</td><td>UMDL</td><td>PDMF</td><td>IPMVSC</td><td>MV-HFMD</td><td>RCML</td><td>Ours</td></tr><tr><td rowspan="4">HandWritten</td><td>Acc</td><td>98.5±0.5</td><td>96.8±0.7</td><td>97.7±0.5</td><td>98.2±0.7</td><td>98.7±0.4</td><td>98.2±0.6</td><td>97.1±0.9</td><td>98.9±0.4</td></tr><tr><td>Precision</td><td>98.5±0.5</td><td>96.8±0.7</td><td>97.8±0.5</td><td>98.3±0.7</td><td>98.7±0.4</td><td>98.2±0.5</td><td>97.1±0.9</td><td>98.9±0.4</td></tr><tr><td>Recall</td><td>98.5±0.5</td><td>96.9±0.7</td><td>97.7±0.5</td><td>98.3±0.7</td><td>98.7±0.4</td><td>98.2±0.6</td><td>97.2±0.9</td><td>98.9±0.4</td></tr><tr><td>Macro F1</td><td>98.5±0.5</td><td>96.8±0.7</td><td>97.7±0.5</td><td>98.2±0.7</td><td>98.7±0.4</td><td>98.1±0.5</td><td>97.1±0.9</td><td>98.9±0.4</td></tr><tr><td rowspan="4">Scene15</td><td>Acc</td><td>62.0±2.2</td><td>66.5±1.8</td><td>63.3±0.4</td><td>67.8±1.1</td><td>71.8±1.4</td><td>80.7±1.2</td><td>70.0±1.0</td><td>81.5±0.7</td></tr><tr><td>Precision</td><td>57.3±2.8</td><td>66.6±2.5</td><td>61.2±0.7</td><td>65.8±2.8</td><td>73.8±0.7</td><td>80.6±1.6</td><td>70.6±1.3</td><td>81.0±0.6</td></tr><tr><td>Recall</td><td>60.1±2.2</td><td>65.3±1.6</td><td>62.7±0.5</td><td>65.9±1.1</td><td>70.5±0.8</td><td>80.3±1.3</td><td>68.9±1.3</td><td>80.8±0.7</td></tr><tr><td>Macro F1</td><td>57.6±2.5</td><td>62.8±1.8</td><td>61.1±0.7</td><td>63.0±1.4</td><td>69.1±0.9</td><td>80.0±1.4</td><td>67.1±1.1</td><td>80.5±0.7</td></tr><tr><td rowspan="4">PIE</td><td>Acc</td><td>72.5±3.7</td><td>90.5±2.3</td><td>71.1±3.3</td><td>88.7±3.0</td><td>91.9±2.0</td><td>86.8±2.4</td><td>91.8±2.9</td><td>93.1±1.7</td></tr><tr><td>Precision</td><td>70.6±4.0</td><td>88.0±3.8</td><td>74.6±3.9</td><td>87.6±3.8</td><td>81.7±2.1</td><td>87.7±2.7</td><td>91.1±3.4</td><td>92.8±2.1</td></tr><tr><td>Recall</td><td>70.6±3.7</td><td>89.6±3.6</td><td>71.1±3.3</td><td>87.3±4.9</td><td>82.1±1.8</td><td>88.9±2.4</td><td>91.0±3.5</td><td>93.6±1.8</td></tr><tr><td>Macro F1</td><td>67.4±3.8</td><td>87.4±3.8</td><td>69.3±3.5</td><td>85.7±4.5</td><td>80.5±2.5</td><td>83.8±6.1</td><td>89.8±3.9</td><td>92.1±2.0</td></tr><tr><td rowspan="4">CCV</td><td>Acc</td><td>29.6±1.1</td><td>42.6±1.4</td><td>36.2±1.5</td><td>50.5±1.4</td><td>42.6±1.8</td><td>53.4±0.9</td><td>42.4±1.7</td><td>54.0±1.1</td></tr><tr><td>Precision</td><td>22.8±2.8</td><td>41.4±2.6</td><td>35.5±1.3</td><td>48.5±1.3</td><td>48.3±2.0</td><td>51.1±1.1</td><td>42.3±2.2</td><td>53.0±1.3</td></tr><tr><td>Recall</td><td>23.2±1.0</td><td>37.1±1.4</td><td>33.7±1.4</td><td>45.4±1.3</td><td>35.5±1.3</td><td>50.5±0.8</td><td>37.0±1.4</td><td>50.6±1.0</td></tr><tr><td>Macro F1</td><td>20.1±1.1</td><td>36.0±1.6</td><td>33.1±1.4</td><td>45.1±1.5</td><td>34.3±1.5</td><td>50.4±0.9</td><td>35.9±1.5</td><td>51.0±0.9</td></tr><tr><td rowspan="4">Animal</td><td>Acc</td><td>56.7±1.4</td><td>56.6±0.8</td><td>34.0±1.1</td><td>57.0±0.7</td><td>39.5±1.9</td><td>59.6±0.7</td><td>56.8±1.2</td><td>60.2±1.1</td></tr><tr><td>Precision</td><td>52.4±2.2</td><td>53.3±3.2</td><td>33.7±0.9</td><td>49.5±0.6</td><td>37.3±3.9</td><td>56.5±0.6</td><td>54.5±2.5</td><td>56.7±1.3</td></tr><tr><td>Recall</td><td>49.9±1.3</td><td>49.7±1.0</td><td>30.4±0.8</td><td>47.2±0.7</td><td>33.3±2.5</td><td>53.7±0.4</td><td>49.5±1.2</td><td>53.8±0.8</td></tr><tr><td>Macro F1</td><td>50.4±1.4</td><td>49.7±1.1</td><td>30.4±0.7</td><td>46.7±0.6</td><td>33.3±2.7</td><td>54.0±0.5</td><td>49.6±1.3</td><td>54.1±0.8</td></tr><tr><td rowspan="4">100Leaves</td><td>Acc</td><td>93.5±1.5</td><td>90.8±2.1</td><td>98.4±0.8</td><td>97.7±0.7</td><td>66.1±4.1</td><td>98.3±0.2</td><td>88.6±1.5</td><td>98.5±1.1</td></tr><tr><td>Precision</td><td>93.6±1.2</td><td>90.5±2.5</td><td>98.7±0.8</td><td>97.7±0.9</td><td>70.2±3.4</td><td>98.4±0.0</td><td>88.7±2.0</td><td>98.5±1.0</td></tr><tr><td>Recall</td><td>93.9±1.1</td><td>91.3±2.4</td><td>98.4±0.9</td><td>98.0±1.0</td><td>69.0±2.0</td><td>98.9±0.2</td><td>89.4±1.4</td><td>99.0±0.7</td></tr><tr><td>Macro F1</td><td>92.7±1.3</td><td>89.3±2.7</td><td>98.3±0.9</td><td>97.5±1.0</td><td>64.0±2.7</td><td>98.4±0.1</td><td>87.0±1.7</td><td>98.5±1.0</td></tr><tr><td rowspan="4">Hdigits</td><td>Acc</td><td>99.6±0.1</td><td>90.8±2.1</td><td>98.0±0.2</td><td>99.4±0.2</td><td>97.8±0.3</td><td>84.1±2.0</td><td>98.3±0.3</td><td>99.8±0.1</td></tr><tr><td>Precision</td><td>99.6±0.1</td><td>98.4±0.2</td><td>98.0±0.2</td><td>99.4±0.2</td><td>97.8±0.3</td><td>88.9±1.7</td><td>98.3±0.3</td><td>99.8±0.1</td></tr><tr><td>Recall</td><td>99.6±0.1</td><td>98.4±0.2</td><td>98.0±0.2</td><td>99.4±0.2</td><td>97.9±0.3</td><td>87.5±2.0</td><td>98.3±0.3</td><td>99.8±0.1</td></tr><tr><td>Macro F1</td><td>99.6±0.1</td><td>98.4±0.2</td><td>98.0±0.2</td><td>99.4±0.2</td><td>97.8±0.3</td><td>86.5±2.3</td><td>98.3±0.3</td><td>99.8±0.1</td></tr><tr><td rowspan="4">YoutubeFace</td><td>Acc</td><td>56.3±0.3</td><td>71.9±2.1</td><td>OOM</td><td>85.6±0.3</td><td>28.2±0.3</td><td>83.4±0.8</td><td>52.6±1.0</td><td>87.1±0.2</td></tr><tr><td>Precision</td><td>74.8±0.6</td><td>83.2±0.8</td><td>OOM</td><td>89.2±0.4</td><td>29.9±4.0</td><td>85.6±1.0</td><td>83.1±1.0</td><td>89.5±0.4</td></tr><tr><td>Recall</td><td>46.0±0.3</td><td>67.7±3.1</td><td>OOM</td><td>84.3±0.4</td><td>5.4±0.3</td><td>82.8±0.8</td><td>38.7±1.3</td><td>86.2±0.3</td></tr><tr><td>Macro F1</td><td>53.5±0.5</td><td>73.0±2.8</td><td>OOM</td><td>86.6±0.3</td><td>5.2±0.4</td><td>83.9±0.8</td><td>46.6±1.4</td><td>87.7±0.1</td></tr></table>
+
+1) The proposed model shows extraordinary superiority among the comparing methods across all the datasets, especially on the Scene15 dataset. Compared with existing methods, our model is ideal to align all the view information while learning satisfactory decision boundaries, which are critical for the classification task.  
+2) For datasets that all the models receive wonderful results such as HandWritten, 100Leaves, and Hdigit datasets, the satisfactory decision boundaries are easily delimited. However, the proposed model could further improve the performance, which implies the rooted ambiguity near the decision boundaries is further alleviated.  
+3) MV-HFMD shows a very outstanding performance, which implies the significance of effective feature fusion in multi-view learning. However, it may neglect to address the feature heterogeneity that plays an important role in facilitating the feature fusion, so the model performance is somewhat limited when compared with our model.  
+4) Compared with a fuzzy multi-view learning model IPMVSC that focuses on using fuzzy rules to realize an interpretable classifier, our model attends to the feature-level issues and applies the fuzzy set theory in the representation space. Besides, from the results, we find that IPMVSC does not work well on large datasets, but our model works well on both small and large datasets.
+
+In summary, the proposed model successfully mines the commonality to tackle the feature heterogeneity, learns the adaptive boundaries to mitigate the harm from the information redundancy, and preserves margins for unseen data to enhance the model generalization.
+
+# 4.2 ABLATION STUDY
+
+To evaluate the effectiveness of the multi-scale alignment module and the rationality of the Comb operator determination, we conduct the ablation study whose results are provided in Table 2, and
+
+Table 2: Results of ablation study on classification  $(\%)$ , which are concerned with the functions of the multi-scale alignment module and the selection of Comb operator.  
+
+<table><tr><td>Lcls</td><td>Lrec</td><td>Lins</td><td>Lpro</td><td>Comb</td><td>HandWritten</td><td>Animal</td><td>PIE</td><td>CCV</td></tr><tr><td>✓</td><td>✓</td><td></td><td></td><td></td><td>96.08±0.47</td><td>55.78±0.04</td><td>84.93±0.36</td><td>42.48±0.58</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td></td><td></td><td>97.44±0.54</td><td>56.95±0.42</td><td>89.71±1.20</td><td>49.34±3.33</td></tr><tr><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td>96.76±0.68</td><td>56.95±0.65</td><td>90.81±0.82</td><td>51.01±2.39</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td></td><td>97.46±1.10</td><td>57.80±0.15</td><td>91.54±0.82</td><td>51.69±2.18</td></tr><tr><td>✓</td><td>✓</td><td></td><td>✓</td><td>Min</td><td>98.15±0.41</td><td>59.20±0.75</td><td>92.40±0.92</td><td>52.34±0.34</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>Min</td><td>98.85±0.42</td><td>60.16±1.03</td><td>93.11±1.72</td><td>53.97±1.14</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>Max</td><td>97.67±0.42</td><td>58.26±0.31</td><td>91.92±0.60</td><td>52.18±2.68</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>Mean</td><td>98.69±0.30</td><td>59.74±0.81</td><td>92.46±1.09</td><td>53.11±0.40</td></tr></table>
+
+![](images/e1ceeadcb009293238c6fdc76ae201ace50d9289d1d089f1b32ddd80fa60a5f6.jpg)  
+(a) HandWritten
+
+![](images/a01583f0c84290cf46fb5c7d14082befe4d8aca83ec605cdf85712d92235e395.jpg)  
+(b) Hdigit
+
+![](images/63488bc15fe2ad992e900055de70e2b79ccbcb3367ebdcc33a8514655e1a8e6e.jpg)  
+(c) HandWritten  
+Figure 4: Hyperparameter analysis on the HandWritten and Hdigit datasets. (a) and (b) delve into the functions of penalty coefficients  $\alpha$  and  $\beta$ , (c) and (d) delve into the functions of temperature coefficients  $\tau_{ins}$  and  $\tau_{pro}$ .
+
+![](images/3b8f480a19d6c285ee7e427760e26138b3a7a89847e214560ef943a89f3e7a68.jpg)  
+(d) Hdigit
+
+# have the following observations:
+
+1) According to the results, the instance-scale alignment and the prototype-scale alignment work effectively to improve the inter-view commonalities and inter-class differences, which are essential for addressing the problems caused by feature heterogeneity and information redundancy.  
+2) In contrast to the instance-scale alignment, the prototype-scale alignment shows greater improvements, which is attributed to exploiting the helpful inter-class differences for delimiting the clear boundaries.  
+3) Compared with different Comb operators, the Min Pooling exhibits the best performance, which demonstrates the superiority of preserving more margins for classifying unseen data.
+
+# 4.3 HYPERPARAMETER ANALYSIS
+
+We conduct the hyperparameter analysis on two penalty coefficients  $\alpha$  and  $\beta$ , and two temperature coefficients  $\tau_{ins}$  and  $\tau_{pro}$ . For  $\alpha$  and  $\beta$ , we vary their values from  $\{0.1, 0.25, 0.5, 0.75, 1\}$ , and for  $\tau_{ins}$  and  $\tau_{pro}$ , we vary their values from  $\{0.5, 0.75, 1.0, 1.5, 2.0\}$  and  $\{0.5, 0.75, 1.0, 1.2, 1.5\}$  respectively. The results on HandWritten dataset and Hdigit dataset are presented in Figure 4. Compared with the temperature coefficients, the penalty coefficients affect the results more significantly, which illustrates the effectiveness of the multi-scale alignment module. Besides, the best results are prone to larger temperature coefficients that weigh more on similar samples.
+
+# 5 CONCLUSION
+
+Feature heterogeneity and information redundancy among the multi-view data are likely to cause the difficulty of feature fusion and the ambiguity around the decision boundaries, which motivates us to propose a novel MVL model MAMC for addressing the problems. MAMC employs the view-specific auto-encoders to abstract exclusive view representations and introduces a multiscale alignment module to reduce the feature heterogeneity by mining the feature commonality. Besides, we propose a novel expanded boundary to exploit the class difference, which benefits to clear the ambiguity from the information redundancy. Extensive experiments demonstrate the superior performance and rational design of the proposed model.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the National Natural Science Foundation of China (No. 62306020), the Young Elite Scientist Sponsorship Program by BAST (BYESS2024199), the National Key Research and Development Program of China (No. 2023YFB3107100), and the Beijing Natural Science Foundation (L244009). The authors gratefully acknowledge the support of K.C.WONG Education Foundation.
+
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+Cai Xu, Jiajun Si, Ziyu Guan, Wei Zhao, Yue Wu, and Xiyue Gao. Reliable conflictive multi-view learning. In Proceedings of the AAAI Conference on Artificial Intelligence, pp. 16129-16137, 2024.  
+Jinglin Xu, Wenbin Li, Xinwang Liu, Dingwen Zhang, Ji Liu, and Junwei Han. Deep embedded complementary and interactive information for multi-view classification. In Proceedings of the AAAI Conference on Artificial Intelligence, pp. 6494-6501, 2020.  
+Yun-Hao Yuan, Jin Li, Yun Li, Jipeng Qiang, Yi Zhu, Xiaobo Shen, and Jianping Gou. Learning canonical f-correlation projection for compact multiview representation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 19260-19269, 2022.
+
+Lotfi A Zadeh. Fuzzy sets. Information and Control, 8(3):338-353, 1965.  
+Chaoyang Zhang, Zhengzheng Lou, Qinglei Zhou, and Shizhe Hu. Multi-view clustering via triplex information maximization. IEEE Transactions on Image Processing, 32:4299-4313, 2023a.  
+Chenglong Zhang, Yang Fang, Xinyan Liang, Xingyu Wu, Bingbing Jiang, et al. Efficient multi-view unsupervised feature selection with adaptive structure learning and inference. In Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, pp. 5443-5452, 2024a.  
+Chenglong Zhang, Xinyan Liang, Peng Zhou, Zhaolong Ling, Yingwei Zhang, Xingyu Wu, Weiguo Sheng, and Bingbing Jiang. Scalable multi-view unsupervised feature selection with structure learning and fusion. In Proceedings of the 32nd ACM International Conference on Multimedia, pp. 1-9, 2024b.  
+Wei Zhang, Zhaohong Deng, Te Zhang, Kup-Sze Choi, and Shitong Wang. One-step multiview fuzzy clustering with collaborative learning between common and specific hidden space information. IEEE Transactions on Neural Networks and Learning Systems, pp. 1-14, 2023b.  
+Vitalii Zhelezniak, Aleksandar Savkov, April Shen, Francesco Moramarco, Jack Flann, and Nils Y Hammerla. Don't settle for average, go for the max: Fuzzy sets and max-pooled word vectors. In International Conference on Learning Representations, pp. 1-17, 2019.  
+Qiyu Zhong, Gengyu Lyu, and Zhen Yang. Align while fusion: A generalized nonaligned multiview multilabel classification method. IEEE Transactions on Neural Networks and Learning Systems, pp. 1-10, 2024.  
+Wenzhang Zhuge, Hong Tao, Tingjin Luo, Ling-Li Zeng, Chenping Hou, and Dongyun Yi. Joint representation learning and clustering: A framework for grouping partial multiview data. IEEE Transactions on Knowledge and Data Engineering, 34(8):3826-3840, 2020.
+
+# A ADDITIONAL PROOFS
+
+# A.1 PROOF OF PROPOSITION 1
+
+Proposition 1:  $\mathcal{Q}_{pro}$  is an upper bound of  $\mathcal{Q}$ .
+
+Proof: Without losing generalization, we let  $\tau_{pro} = 1$ , and we have
+
+$$
+\begin{array}{l} \mathcal {Q} _ {p r o} = - \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \log \frac {e ^ {- \tau_ {p r o} \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2}}}{\sum_ {j} e ^ {- \tau_ {p r o} \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2}}} \\ = \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \log \frac {\sum_ {j} e ^ {- \left\| \bar {h} _ {i} ^ {(v)} - \bar {c} _ {j} \right\| _ {2} ^ {2}}}{e ^ {- \left\| \bar {h} _ {i} ^ {(v)} - \bar {c} _ {k} \right\| _ {2} ^ {2}}} \\ = \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\log \sum_ {j} e ^ {- \left\| \overline {{h}} _ {i} ^ {(v)} - \bar {c} _ {j} \right\| _ {2} ^ {2}} - \log e ^ {- \left\| \overline {{h}} _ {i} ^ {(v)} - \bar {c} _ {k} \right\| _ {2} ^ {2}}\right) \tag {18} \\ = \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\log \sum_ {j} e ^ {- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2}} + \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2}\right) \\ \approx \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\log K \mathbb {E} \left[ e ^ {- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{c}} _ {j} \right\| _ {2} ^ {2}} \right] + \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{c}} _ {k} \right\| _ {2} ^ {2}\right) \\ = \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\log \mathbb {E} \left[ e ^ {- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{c}} _ {j} \right\| _ {2} ^ {2}} \right] + \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{c}} _ {k} \right\| _ {2} ^ {2} + \log K\right). \\ \end{array}
+$$
+
+$K$  is a constant to denote the number of classes, and we ignore it to obtain a new loss  $\mathcal{Q}_{pro}^{\prime}$ , namely
+
+$$
+\mathcal {Q} _ {p r o} ^ {\prime} = \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\log \mathbb {E} \left[ e ^ {- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2}} \right] + \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2}\right). \tag {19}
+$$
+
+According to the Jensen's inequality, we could get
+
+$$
+\left. \right. \exp \left\{\mathbb {E} \left[ - \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2} \right]\right\} \leq \mathbb {E} \left[ \exp \left\{- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2} \right\}\right]. \tag {20}
+$$
+
+Considering that the logarithmic function is a monotonically increasing function, we have
+
+$$
+\mathcal {Q} _ {p r o} ^ {\prime} \geq \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\mathbb {E} \left[ - \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2} \right] + \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2}\right) \approx \mathcal {Q}. \tag {21}
+$$
+
+# A.2 PROOF OF THEOREM 1
+
+Theorem 1. When employing Min Pooling as the Comb operator, the overall variance of each class is maximized.
+
+Proof: Without losing generalization, we let  $\tau_{pro} = 1$ , and we have
+
+$$
+\begin{array}{l} \mathcal {Q} _ {p r o} = - \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \log \frac {e ^ {- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2}}}{\sum_ {j} e ^ {- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2}}} \\ = \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\log \sum_ {j} e ^ {- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| _ {2} ^ {2}} + \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2}\right) \tag {22} \\ = \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\log \sum_ {j} e ^ {- \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {j} \right\| ^ {2}}\right) + \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left(\left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \overline {{\boldsymbol {c}}} _ {k} \right\| _ {2} ^ {2}\right) \\ \approx \frac {1}{N V} \sum_ {v = 1} ^ {V} \sum_ {i = 1} ^ {N} \left(\log \sum_ {j} e ^ {- \left\| \bar {\boldsymbol {h}} _ {i} ^ {(v)} - \bar {\boldsymbol {c}} _ {j} \right\| _ {2} ^ {2}}\right) + \sum_ {k = 1} ^ {K} \operatorname {V a r} (\bar {\boldsymbol {c}} _ {k}), \\ \end{array}
+$$
+
+where  $Var(\bar{c}_k)$  stands for the variance of the  $k$ -th class cluster. Actually, it could be observed that  $\mathcal{V} = \frac{1}{NV}\sum_{v=1}^{V}\sum_{k=1}^{K}\sum_{y_i=k}\left\|\overline{\boldsymbol{h}}_i^{(v)} - \overline{\boldsymbol{c}}_k\right\|_2^2 = \sum_{k=1}^{K}Var(\bar{c}_k)$ . In practice, we weigh the crisp prototype more and set  $\eta$  to approximate 1, then we have
+
+$$
+\lim  _ {\eta \rightarrow 1} \mathcal {V} = \lim  _ {\eta \rightarrow 1} \sum_ {k = 1} ^ {K} \operatorname {V a r} \left(\bar {c} _ {k}\right) = \sum_ {k = 1} ^ {K} \operatorname {V a r} \left(c _ {k}\right). \tag {23}
+$$
+
+When applying the centralized representations for constructing the expanded boundary, the maximal overall variance within each class cluster is formalized as
+
+$$
+\begin{array}{l} \max  _ {\eta \rightarrow 1} \mathcal {V} = \max  \sum_ {k = 1} ^ {K} V a r (\boldsymbol {c} _ {k}) = \max  \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left\| \overline {{\boldsymbol {h}}} _ {i} ^ {(v)} - \boldsymbol {c} _ {k} \right\| _ {2} ^ {2} \\ = \max  \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left\| \overline {{\boldsymbol {z}}} _ {i} ^ {(v)} \right\| _ {2} ^ {2} = \max  \sum_ {v = 1} ^ {V} \sum_ {k = 1} ^ {K} \sum_ {y _ {i} = k} \left\| \left(\mathbf {2} - \tilde {\boldsymbol {r}} _ {i} ^ {(v)}\right) \odot \boldsymbol {z} _ {i} ^ {(v)} \right\| _ {2} ^ {2} \tag {24} \\ = \max  \sum_ {v = 1} ^ {V} \sum_ {i = 1} ^ {N} \left\| \left(\mathbf {2} - \tilde {\boldsymbol {r}} _ {i} ^ {(v)}\right) \odot \boldsymbol {z} _ {i} ^ {(v)} \right\| _ {2} ^ {2}, \\ \end{array}
+$$
+
+where  $K$  is the number of classes. Obviously, for each representation, when each element of  $\tilde{\pmb{r}}_i^{(v)}$  becomes smaller,  $\nu$  becomes greater, which is in line with
+
+$$
+\left\| \left(\mathbf {2} - \left\{\tilde {\boldsymbol {r}} _ {i} ^ {(v)} \right\} _ {m i n}\right) \odot \boldsymbol {z} _ {i} ^ {(v)} \right\| _ {2} ^ {2} \geq \left\| \left(\mathbf {2} - \left\{\tilde {\boldsymbol {r}} _ {i} ^ {(v)} \right\} _ {o t h e r}\right) \odot \boldsymbol {z} _ {i} ^ {(v)} \right\| _ {2} ^ {2}, \tag {25}
+$$
+
+where  $\left\{\tilde{r}_i^{(v)}\right\}_{min}$ , and  $\left\{\tilde{r}_i^{(v)}\right\}_{other}$  are the membership degree vectors, which are the outputs of Min Pooling, and other simple Comb operators such as Mean Pooling, or Max Pooling. Finally, we have
+
+$$
+\sum_ {v = 1} ^ {V} \sum_ {i = 1} ^ {N} \left\| \left(\mathbf {2} - \left\{\tilde {\boldsymbol {r}} _ {i} ^ {(v)} \right\} _ {m i n}\right) \odot \boldsymbol {z} _ {i} ^ {(v)} \right\| _ {2} ^ {2} \geq \sum_ {v = 1} ^ {V} \sum_ {i = 1} ^ {N} \left\| \left(\mathbf {2} - \left\{\tilde {\boldsymbol {r}} _ {i} ^ {(v)} \right\} _ {o t h e r}\right) \odot \boldsymbol {z} _ {i} ^ {(v)} \right\| _ {2} ^ {2}. \tag {26}
+$$
+
+This means that among the simple Comb operators, Min Pooling as the Comb could maximize the overall variance of each class.
+
+# B EXPERIMENT SETTINGS
+
+# B.1 DATASETS
+
+We provide the complete statistics of the used datasets in Table 3. HandWritten  $^{1}$  and Hdigit  $^{2}$  datasets are composed of handwritten digit images, PIE  $^{3}$  and YoutubeFace  $^{4}$  datasets collect the data on facial recognition and expression recognition, Scene15  $^{5}$  dataset contains images from 15 different scene categories, CCV  $^{6}$  dataset comprises videos under complex conditions, Animal  $^{7}$  dataset contains images of different animals, 100Leaves  $^{8}$  dataset is composed of images from 100 different leaves.
+
+Table 3: Statistics of datasets, which include the number of instances, classes, and the view dimensions.  
+
+<table><tr><td>Dataset</td><td>Instances</td><td>Classes</td><td>Views</td><td>View dimensions</td></tr><tr><td>HandWritten</td><td>2,000</td><td>10</td><td>6</td><td>240/76/216/47/64/6</td></tr><tr><td>Scene15</td><td>4,485</td><td>15</td><td>3</td><td>20/59/40</td></tr><tr><td>PIE</td><td>680</td><td>68</td><td>3</td><td>484/256/279</td></tr><tr><td>CCV</td><td>6,773</td><td>20</td><td>3</td><td>20/20/20</td></tr><tr><td>Animal</td><td>11,673</td><td>20</td><td>4</td><td>2,689/2,000/2,001/2,000</td></tr><tr><td>100Leaves</td><td>1,600</td><td>100</td><td>3</td><td>60/60/60</td></tr><tr><td>Hdigit</td><td>10,000</td><td>10</td><td>2</td><td>784/256</td></tr><tr><td>YoutubeFace</td><td>101,499</td><td>31</td><td>5</td><td>64/64/512/647/838</td></tr></table>
+
+# B.2 IMPLEMENTATION DETAILS
+
+The proposed model is implemented in Pytorch and trained with an SGD optimizer. The dimension settings of the encoder and decoder are formed by  $\{D_v, 1.4 \times 512, 1.2 \times 512, 512\}$  and  $\{512, 0.6 \times D_v, 0.8 \times D_v, D_v\}$  respectively. The number of membership functions  $L$  is set as 5. The learning rate is chosen from  $\{1e^{-3}, 3e^{-3}, 5e^{-3}, 1e^{-2}\}$ , and the coefficients  $\alpha$  and  $\beta$  are chosen from  $\{0.1, 0.25, 0.5, 0.75, 1\}$ . The temperature coefficients  $\tau_{ins}$  and  $\tau_{pro}$  are chosen from  $\{0.5, 0.75, 1.0, 1.5, 2.0\}$  and  $\{0.5, 0.75, 1.0, 1.2, 1.5\}$  respectively. All experiments are conducted on a server with 8 NVIDIA GeForce 3090 (24 GB memory each).
+
+# B.3 TRAINING ALGORITHM
+
+The proposed MAMC includes three main components of the proposed model, namely view-specific representation extractors, a multi-scale alignment module, and a joint classifier. Among the three components, the view-specific auto-encoders abstract critical view representations from multi-view data, and the multi-scale alignment module mines the inter-view commonality and inter-class difference. The joint classifier calculates the fused predictions. Algorithm 1 presents the detailed training process of the proposed MAMC.
+
+# C SUPPLEMENTARY EXPERIMENTS
+
+# C.1 LEARNING CLEAR DECISION BOUNDARIES
+
+Information redundancy in the multi-view data would lead to ambiguous boundaries. To further verify that MAMC has the ability to learn clear decision boundaries, we implement our model on
+
+Algorithm 1 Training process of MAMC  
+Input: Multi-view dataset:  $\mathcal{D} = \{(\{\pmb{x}_i^{(v)}\}_{v = 1}^V,y_i)|1\leq i\leq N\}$ ; Hyperparameters  $\tau_{ins},\tau_{pro},\alpha,$  and  $\beta$ ; Training epochs  $T$   
+Output: Model parameters.  
+Initialization: Initialize the parameters of the neural network.  
+Process:  
+1: for epoch  $= 1$  to  $T$  do  
+2: Calculate the view representation matrices  $\{\pmb{H}^{(v)}\}_{v = 1}^{V}$  with view-specific auto-encoders.  
+3:  $\backslash$  Instance-scale alignment  
+4: Realize the instance-scale alignment by Eq. (2).  
+5:  $\backslash$  Prototype-scale alignment  
+6: Centralize the view representations by Eq. (3)-(4).  
+7: Calculate the boundary elements by Eq. (5)-(8).  
+8: Realize the prototype-scale alignment by Eq. (14).  
+9:  $\backslash$  Classification prediction  
+10: Calculate the classification loss by Eq. (15)-(16).  
+11: end for
+
+![](images/3025a2df613bc24fb29cc48ca7087e8d6af58874ee9bd29bb92713bfd59d54e4.jpg)  
+(a) HandWritten
+
+![](images/9307512464b224a2a484bafb92f81766c6e120b83a8a3757fe5f672017908f94.jpg)  
+(b)CCV
+
+![](images/440f193f279e5813f1a1fa9bddcd8cb84689f56192d87abcb4d0e255cfa7ff06.jpg)  
+(c) PIE
+
+![](images/745cb066262aee4ad813f19636d8e9b3c4d64dc0047059be51d3dec04f92947c.jpg)  
+(d) Animal
+
+![](images/cd623f644114216c07626c07d659ce45ed83272e4dcb134c9df0f7c379b8aefc.jpg)  
+(e) Scene15
+
+![](images/805541b13e6b2a74471a6245b45a75b2ecef357798bdfb246d949bff62fa81bd.jpg)  
+(f) Hdigit  
+Figure 5: Performance comparison on multi-view datasets with different levels of noise.
+
+manufactured ambiguous data. In our experiments, we add Gaussian noise with different levels of deviations  $\sigma$  to half of the views, and provide the comparison results in Figure 5. We discover that when the noise increases, the performance of all the models drops quickly since larger noise results in more ambiguous boundaries. Even so, the performance of our model descends more slowly than other models, especially when the noise is small. Actually, the data noise in the real multi-view data is not large, which demonstrates that MAMC exploits the desired inter-difference from all the view information to clear the ambiguous decision boundaries.
+
+![](images/125f5efdeae1aab08475288dc0d2c1f623c847c9a10375243500f5e8577f9b20.jpg)  
+(a) HandWritten
+
+![](images/757c0f4683bfa09344fea84c769356a62c7f1cb6d1fb19db99902f4f2092a16c.jpg)  
+(b) Scene15
+
+![](images/0048ee07ede6fa6d450637bdd66ac6b47bef9fdea68ed08e06c169ca5fe82e68.jpg)  
+(c) PIE
+
+![](images/f544b9c7dac9df4d98a8f44cc7d79a8e39f23853c6d573a76919ee3af72ce7ad.jpg)  
+(d) Animal
+
+![](images/b0bd8a9d41181e22763eb114f1ac00f23f14130a65a1de0d7d4635d4538e65e9.jpg)  
+(e) CCV
+
+![](images/5babd5e7535a01c865126d9f4c57f11559b1d1a8b91bfddd3e58ae65acf1c368.jpg)  
+(f) Hdigit
+
+![](images/859d865ac5ecf07ecc56ae9eef6d2948c100dcf6eb06435913ade407049e38b7.jpg)  
+(g) 100Leaves
+
+![](images/c4dd9a896bbef683bc381ee36ba81ce7065c7fc978d007158a78d9511542c798.jpg)  
+(h) YoutubeFace  
+Figure 6: The convergence analysis of MAMC across all the datasets.
+
+# C.2 LOSS ANALYSIS
+
+We conduct the convergence analysis of the proposed method across all the datasets, where experimental results of loss values are illustrated in Figure 6. It is observed that the losses decrease steadily and converge between 50 and 800 epochs across all datasets. Such a phenomenon empirically confirms the convergence of MAMC.
+
+# C.3 VISUALIZATION
+
+In this experiment, we provide the t-SNE of the comparative MVL models and raw data in Figure 7. In practice, one instance is encoded as several view representations, which are then concatenated as a final representation vector. The final representation vector is input into the t-SNE algorithm to obtain the visualization. From the results, we could find that though the raw data are ambiguous, most models disentangle different classes to some degree. However, our model shows much better results since it mines the feature commonalities to tighten samples inside each class and exploits inter-class differences for separating different classes.
+
+# C.4 COMPLEXITY COMPARISON
+
+We present the model complexity comparison in Table 4 and Table 5. Table 4 lists the complexity of all the models, and we can find that our model has the same level of complexity as some comparative models. Table 5 provides detailed computational overhead analysis including MACs, concrete running time, and Parameters. For a fair comparison, we set the same batch size and training epochs for all the models. From all the statistics, we could find that our model is not cumbersome, but is competitive with some advanced models.
+
+Table 4: Complexity comparison between the comparative models and the proposed model.  
+
+<table><tr><td>Model</td><td>Complexity</td></tr><tr><td>mmdynamics</td><td>O(N)</td></tr><tr><td>ETMC</td><td>O(N)</td></tr><tr><td>UMDL</td><td>O(N3)</td></tr><tr><td>PDMF</td><td>O(N2)</td></tr><tr><td>RCML</td><td>O(N)</td></tr><tr><td>Ours</td><td>O(N2)</td></tr></table>
+
+![](images/fbb92c8e73a708babac1fd680f94fafef541d552ea006785995eab96aa3930b7.jpg)  
+(a) Raw
+
+![](images/b5714d21ae824784679a0600d366bb3f5b6339fc031319b6a9b8dee42668f3c5.jpg)  
+(b) mmdynamics
+
+![](images/ff07e50ec5ca89bf47eb627230507df377d269f959731db6d3d56153d44c95a7.jpg)  
+(c) ETMC
+
+![](images/919972b7b70b2bf480fda583fcd42eeb34d4f79354dbc36b411094605bdb69c1.jpg)  
+(d) PDMF
+
+![](images/04a87cab69c274d61bcecd151ff47791894e352f486a216192536ced7bca5012.jpg)  
+(e) RCML
+
+![](images/3a3d311d1bcf777a798fa522e9da964b961ea3c766c4a232ad9e9ceafb3d8cf2.jpg)  
+(f) Ours  
+Figure 7: t-SNE of the comparative models and the proposed model on Scene15 dataset.
+
+Table 5: Complexity comparison among the comparative models and the proposed model.  
+
+<table><tr><td>Datasets</td><td>Metrics</td><td>mmdynamics</td><td>ETMC</td><td>UMDL</td><td>PDMF</td><td>RCML</td><td>Ours</td></tr><tr><td rowspan="3">HandWritten</td><td>Running Time</td><td>21.911s</td><td>35.739s</td><td>10057.537s</td><td>21.972s</td><td>17.064s</td><td>23.302s</td></tr><tr><td>MACs</td><td>267.871M</td><td>3.323M</td><td>1273.548M</td><td>189.989M</td><td>1.661M</td><td>1373.655M</td></tr><tr><td>Parameters</td><td>1.0543M</td><td>0.0066M</td><td>5.2389M</td><td>1.722M</td><td>0.0066M</td><td>5.353M</td></tr><tr><td rowspan="3">Scene15</td><td>Running Time</td><td>28.996s</td><td>42.596s</td><td>31938.726s</td><td>45.678s</td><td>22.922s</td><td>41.651s</td></tr><tr><td>MACs</td><td>66.604M</td><td>0.457M</td><td>132.890M</td><td>68.387M</td><td>0.457M</td><td>617.869M</td></tr><tr><td>Parameters</td><td>0.264M</td><td>0.002M</td><td>20.632M</td><td>1.098M</td><td>0.002M</td><td>3.197M</td></tr><tr><td rowspan="3">PIE</td><td>Running Time</td><td>11.525s</td><td>11.233s</td><td>2721.368s</td><td>9.214s</td><td>7.775s</td><td>17.499s</td></tr><tr><td>MACs</td><td>536.394M</td><td>17.739M</td><td>248.090M</td><td>188.088M</td><td>2.217M</td><td>996.040M</td></tr><tr><td>Parameters</td><td>2.099M</td><td>0.070M</td><td>1.429M</td><td>1.540M</td><td>0.070M</td><td>4.622M</td></tr><tr><td rowspan="3">CCV</td><td>Running Time</td><td>40.678s</td><td>61.986s</td><td>29783.373s</td><td>69.529s</td><td>37.905s</td><td>65.490s</td></tr><tr><td>MACs</td><td>56.521M</td><td>0.307M</td><td>125.338M</td><td>9.978M</td><td>0.307M</td><td>263.633M</td></tr><tr><td>Parameters</td><td>0.225M</td><td>0.001M</td><td>46.361M</td><td>1.331M</td><td>0.001M</td><td>3.323M</td></tr><tr><td rowspan="3">Animal</td><td>Running Time</td><td>650.847s</td><td>181.911s</td><td>52103.473s</td><td>221.021s</td><td>119.074s</td><td>177.946s</td></tr><tr><td>MACs</td><td>7643.710M</td><td>44.493M</td><td>1269.197M</td><td>576.355M</td><td>44.493M</td><td>9360.186M</td></tr><tr><td>Parameters</td><td>29.873M</td><td>0.174M</td><td>141.214M</td><td>10.937M</td><td>0.174M</td><td>37.353M</td></tr><tr><td rowspan="3">100Leaves</td><td>Running Time</td><td>12.697s</td><td>16.173s</td><td>7386.190s</td><td>19.825s</td><td>10.078s</td><td>44.393s</td></tr><tr><td>MACs</td><td>247.370M</td><td>4.915M</td><td>142.234M</td><td>80.740M</td><td>4.915M</td><td>672.518M</td></tr><tr><td>Parameters</td><td>0.971M</td><td>0.020M</td><td>3.113M</td><td>0.814M</td><td>0.020M</td><td>3.323M</td></tr><tr><td rowspan="3">Hdigit</td><td>Running Time</td><td>57.370s</td><td>58.236s</td><td>44589.832s</td><td>85.163s</td><td>41.918s</td><td>59.256s</td></tr><tr><td>MACs</td><td>506.520M</td><td>0.666M</td><td>211.558M</td><td>42.877M</td><td>2.662M</td><td>883.593M</td></tr><tr><td>Parameters</td><td>1.982M</td><td>0.010M</td><td>100.825M</td><td>2.614M</td><td>0.010M</td><td>4.245M</td></tr><tr><td rowspan="3">YoutubeFace</td><td>Running Time</td><td>1660.614s</td><td>598.449s</td><td>-</td><td>513.350s</td><td>523.801s</td><td>702.112s</td></tr><tr><td>MACs</td><td>4393.923M</td><td>67.456M</td><td>-</td><td>1466.827M</td><td>67.456M</td><td>7988.087M</td></tr><tr><td>Parameters</td><td>4.328M</td><td>0.066M</td><td>-</td><td>15.840M</td><td>0.066M</td><td>8.544M</td></tr></table>

2025/Enhance Multi-View Classification Through Multi-Scale Alignment and Expanded Boundary/images.zip

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2025/Enhanced Diffusion Sampling via Extrapolation with Multiple ODE Solutions/6122af0d-cbf2-435c-b71d-1b0fdc829491_origin.pdf

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2025/Enhanced Diffusion Sampling via Extrapolation with Multiple ODE Solutions/full.md

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+# ENHANCED DIFFUSION SAMPLING VIA EXTRAPOLATION WITH MULTIPLE ODE SOLUTIONS
+
+Jinyoung Choi $^{1}$ , Junoh Kang $^{1}$  & Bohyung Han $^{1,2}$   
+Computer Vision Lab.,  $^{1}$ ECE &  $^{2}$ IPAI, Seoul National University, Korea  
+{jin0.choi, junoh.kang, bhhan}@snu.ac.kr
+
+# ABSTRACT
+
+Diffusion probabilistic models (DPMs), while effective in generating high-quality samples, often suffer from high computational costs due to their iterative sampling process. To address this, we propose an enhanced ODE-based sampling method for DPMs inspired by Richardson extrapolation, which reduces numerical error and improves convergence rates. Our method, RX-DPM, leverages multiple ODE solutions at intermediate time steps to extrapolate the denoised prediction in DPMs. This significantly enhances the accuracy of estimations for the final sample while maintaining the number of function evaluations (NFEs). Unlike standard Richardson extrapolation, which assumes uniform discretization of the time grid, we develop a more general formulation tailored to arbitrary time step scheduling, guided by local truncation error derived from a baseline sampling method. The simplicity of our approach facilitates accurate estimation of numerical solutions without significant computational overhead, and allows for seamless and convenient integration into various DPMs and solvers. Additionally, RX-DPM provides explicit error estimates, effectively demonstrating the faster convergence as the leading error term's order increases. Through a series of experiments, we show that the proposed method improves the quality of generated samples without requiring additional sampling iterations.
+
+# 1 INTRODUCTION
+
+Diffusion probabilistic models (DPMs) have emerged as a powerful framework for generating high-quality samples in a wide range of applications and domains for images (Ho et al., 2020; Song et al., 2021b; Dhariwal & Nichol, 2021; Rombach et al., 2022), videos (Ho et al., 2022; Singer et al., 2022; Zhou et al., 2022; Wang et al., 2023), 3D shapes (Zeng et al., 2022), etc. While DPMs demonstrate impressive performance in data fidelity and diversity, they also have limitations, particularly their computational inefficiency due to the sequential nature of sampling. Addressing this issue is crucial for enhancing the usability of DPMs in real-world scenarios, where time constraints are critical for practical deployment.
+
+The generation process of DPMs can be formulated as a problem of finding solutions to SDEs or ODEs (Song et al., 2021b), where the truncation errors of the numerical solutions are highly correlated to the quality of the generated samples. To enhance the quality of these samples, it is essential to reduce truncation errors, which can be achieved by adopting advanced solvers or numerical techniques that improve the accuracy of numerical estimations. In this context, we aim to lower truncation errors by applying numerical extrapolation to existing sampling methods for DPMs. The key ingredient of the proposed method is Richardson extrapolation, a proven and widely used technique in the mathematical modeling of physical problems such as fluid dynamics and heat transfer, which demand high computational resources. While numerous variants and strategies have been studied (Richards, 1997; Botchev & Verwer, 2009; Zlatev et al., 2010), its application to DPMs remains unexplored. The method uses a simple linear combination of multiple numerical estimates from different resolutions of a grid to approximate the ideal solution, where the estimates are expected to converge as the resolution becomes finer, ultimately reaching the target value in the limit.
+
+We propose an extrapolation algorithm that is applied repeatedly every  $k$  denoising steps of an ODE-based sampling method to improve the accuracy of intermediate denoising steps. This is achieved by utilizing an additional ODE solution, which is estimated by a single step over an interval of  $k$  time steps. Figure 1 illustrates this concept with  $k = 2$  on time steps  $[t_i, t_{i-1}, t_{i-2}]$ , which forms a unit block of our extrapolation-based sampling. Two ODE solutions—single-step and two-step estimations at  $t_{i-2}$  from  $t_i$  can be leveraged to achieve an approximation closer to the ideal solution  $\boldsymbol{x}_{t_{i-2}}^*$ , which is unknown.
+
+The standard Richardson extrapolation assumes a uniform discretization of the time grid. However, the uniform grid of denoising time steps might be suboptimal for DPMs; a smaller time interval near the clean sample is often more beneficial (Karras et al., 2022; Song et al., 2021a) given the same number of steps. Considering such characteristics of DPMs and
+
+the advantages offered by specific time step discretizations, we propose an algorithm applicable to arbitrary schedules, with coefficients determined by the chosen configuration. We observe that our grid-aware approach yields better performance than conventional methods.
+
+![](images/3c11d451b3598c8c4c8ace2e708e016caa32696f7f0b85c2ea3b668315b3597a.jpg)  
+Figure 1: Application of the proposed extrapolation on two denoising steps  $(k = 2)$  with time steps of  $[t_i, t_{i-1}, t_{i-2}]$ .  $\hat{\boldsymbol{x}}_{t_{i-2}}^{(n)}$  denotes that  $n$  steps are used by the baseline sampler within the same interval.  $\tilde{\boldsymbol{x}}_{t_{i-2}}^{(2)}$  represents the extrapolated estimation using two ODE solutions at  $t_{i-2}$ ,  $\hat{\boldsymbol{x}}_{t_{i-2}}^{(1)}$  and  $\hat{\boldsymbol{x}}_{t_{i-2}}^{(2)}$ .
+
+Although there exist other methods applying extrapolation techniques to diffusion models, their usages of extrapolation are somewhat different from ours. For example, Zhang & Chen (2023); Zhang et al. (2023) utilize estimations from earlier steps to improve the estimation at the time step,  $t_i$ , whereas our approach adopts two denoised estimations at the same time step,  $t_i$ , to enhance accuracy at  $t_i$ . In addition, the main building block of our approach, Richardson extrapolation, is proven to enhance numerical accuracy and provides an explicit estimate of the error, which allows for a clear understanding of the convergence behavior. Furthermore, the implementation of our algorithm is simple and cost-effective because it requires no additional network evaluations and insignificant computational overhead to perform the extrapolation. We refer to the proposed sampling algorithm as RX-DPM.
+
+Our main contributions are summarized below:
+
+- We introduce an improved diffusion sampler, RX-DPM, inspired by Richardson extrapolation, which effectively increases the order of accuracy of existing ODE-based samplers without increasing NFEs.  
+- We systematically develop an algorithm for general DPM solvers with arbitrary time step scheduling starting from the derivation of a truncation error of the Euler method on a non-uniform grid.  
+- Our experiments across various well-known baselines demonstrate that RX-DPM exhibits strong generalization performance and high practicality, regardless of ODE designs, model architectures, and base samplers.
+
+# 2 RELATED WORK
+
+There exists a substantial body of research that seeks to reduce the computational burden of DPMs while maintaining their performance. One approach in this direction involves exploring alternative modeling strategies for the reverse process. For example, the networks in Salimans & Ho (2022); Song et al. (2023); Kim et al. (2024) learn alternative objectives through knowledge distillation, using the outputs obtained by iterative inferences of the pretrained (teacher) networks. On the other hand, Bao et al. (2022) model a more accurate reverse distribution by incorporating the optimal covariance, while Xiao et al. (2022); Kang et al. (2024) implicitly estimate the precise reverse distribution by utilizing GAN components.
+
+While the aforementioned methods require model training, there is also a training-free approach that interprets the generation process of a diffusion model as solving an ODE or SDE (Song et al., 2021b;
+
+Karras et al., 2022). For instance, DDIM (Song et al., 2021a) proposes to skip intermediate time steps, which is equivalent to solving an ODE using the Euler method with a large step size. To further improve sampling quality, a large volume of research (Karras et al., 2022; Dockhorn et al., 2022; Liu et al., 2022; Zhang & Chen, 2023; Lu et al., 2022; 2023) simply applies classical higher-order solvers or customizes them for diffusion models. Specifically, Karras et al. (2022) adopt the second-order Heun's method (Suli & Mayers, 2003) and Dockhorn et al. (2022) apply the second-order Taylor expansion. In addition, Liu et al. (2022) propose a pseudo-numerical solver, which combines classical high-order numerical methods and DDIM.
+
+Building on ODE-based sampling methods, various numerical techniques have been employed to further improve the sample quality. For example, IIA (Zhang et al., 2024) optimizes the coefficients of certain quantities to approximate the fine-grained integration. LA-DPM (Zhang et al., 2023) and DEIS (Zhang & Chen, 2023) adopt extrapolation techniques as our method, but with notable differences. Specifically, LA-DPM linearly extrapolates the previous and current predictions of the solution at  $t = 0$  — the original data manifold—while DEIS uses high-order polynomial extrapolation on the noise prediction function. The key distinction of our approach is that, while they adopt score (noise) predictions obtained at different time steps for each extrapolation, we utilize multiple denoised outputs obtained at the same time step.
+
+# 3 PRELIMINARIES
+
+# 3.1 DIFFUSION PROBABILISTIC MODELS AS SOLVING AN ODE
+
+For  $\pmb{p}_0 = \pmb{p}_{\mathrm{data}}$  and  $\pmb{x} \in \mathbb{R}^d$ , Karras et al. (2022) defines a marginal distribution at  $t$  as
+
+$$
+\boldsymbol {p} _ {t} (\boldsymbol {x}) = s (t) ^ {- d} \boldsymbol {p} (\boldsymbol {x} / s (t); \sigma (t)), \tag {1}
+$$
+
+where  $\pmb{p}(\pmb{x};\sigma) = \pmb{p}_{\mathrm{data}}*\mathcal{N}(\pmb{0},\sigma(t)^2\pmb{I})$ , and  $s(t)$  and  $\sigma(t)$  are non-negative functions satisfying  $s(0) = 1$ ,  $\sigma(0) = 0$ , and  $\lim_{t\to\infty}\frac{\sigma(t)}{s(t)}=\infty$ . The probability flow ODE,
+
+$$
+d \boldsymbol {x} = [ \dot {s} (t) / s (t) - s (t) ^ {2} \dot {\sigma} (t) \sigma (t) \nabla_ {\boldsymbol {x}} \log p (\boldsymbol {x} / s (t); \sigma (t)) ] d t, \quad \boldsymbol {x} (T) \sim \boldsymbol {p} _ {T} (\boldsymbol {x}), \tag {2}
+$$
+
+matches the marginal distribution. By adopting the specific choices,  $s(t) = 1$  and  $\sigma(t) = t$  as in Karras et al. (2022), Equation (2) is reduced as follows:
+
+$$
+d \boldsymbol {x} = - t \nabla_ {\boldsymbol {x}} \log \boldsymbol {p} (\boldsymbol {x}; t) d t, \quad \boldsymbol {x} (T) \sim \boldsymbol {p} _ {T} (\boldsymbol {x}). \tag {3}
+$$
+
+Diffusion models now learn the score function  $\nabla_{\pmb{x}}\log \pmb{p}(\pmb{x};t)$ , which is the only unknown component in the equation. For sufficiently large  $T$ , the marginal distribution  $\pmb{p}_T(\pmb{x})$  can be approximated by  $\mathcal{N}(\pmb{x};\mathbf{0},T^2\pmb{I})$  and the generation process is equivalent to solving for  $\pmb{x}(0)$  using Equation (3) with the boundary condition,  $\pmb{x}(T)\sim \mathcal{N}(\mathbf{0},T^{2}\pmb{I})$ . Since the analytic solution of Equation (3) cannot be expressed in a closed form, numerical methods are used to solve the ODE. Given the time step scheduling,  $0 = t_0 < t_1 < \dots < t_N = T$ , the solution is given by
+
+$$
+\begin{array}{l} \boldsymbol {x} (0) = \boldsymbol {x} (T) + \int_ {T} ^ {0} - t \nabla_ {\boldsymbol {x}} \log \boldsymbol {p} (\boldsymbol {x} (t); t) d t (4) \\ = \boldsymbol {x} (T) + \sum_ {i = N} ^ {1} \int_ {t _ {i}} ^ {t _ {i - 1}} - t \nabla_ {\boldsymbol {x}} \log \boldsymbol {p} (\boldsymbol {x} (t); t) d t, (5) \\ \end{array}
+$$
+
+where each integration from  $t_i$  to  $t_{i-1}$  can be approximated by ODE solvers such as the Euler method or Heun's method.
+
+# 3.2 RICHARDSON EXTRAPOLATION
+
+Let the exact and numerical solutions at  $t = 0$  be  $V^{*}$  and  $V(h)$ , respectively, where  $h$  ( $0 < h < 1$ ) denotes the step size. If  $V^{*} = \lim_{h\to 0}V(h)$  and the order of truncation error is known, Richardson extrapolation (Richardson, 1911) identifies a faster converging sequence,  $\tilde{V} (h)$ . For instance,  $V(h)$  with a truncation error in the order of  $O(h^p)$  is expressed by
+
+$$
+V ^ {*} = V (h) + c h ^ {p} + O \left(h ^ {q}\right) \tag {6}
+$$
+
+for  $0 < p < q$  and  $c \neq 0$ . Then, for a fixed constant  $k > 1$ ,
+
+$$
+V ^ {*} = V (h / k) + \frac {c}{k ^ {p}} h ^ {p} + O \left(h ^ {q}\right). \tag {7}
+$$
+
+From Equations (6) and (7), eliminating the  $h^p$  terms, we obtain the solution
+
+$$
+\tilde {V} (h, k) = \frac {k ^ {p} V (h / k) - V (h)}{k ^ {p} - 1}, \tag {8}
+$$
+
+which has a truncation error of  $O(h^q)$ , asymptotically smaller than  $O(h^p)$ .
+
+# 4 RX-DPM
+
+Before discussing the proposed method, we first outline the algorithmic development process for the most simplified problem and then explore an extension to a general DPM solver. The application of our method, RX-DPM, to a specific solver will be referred to as RX-[SolverName].
+
+# 4.1 TRUNCATION ERROR OF EULER METHOD ON NON-UNIFORM GRID
+
+We now derive the truncation error formula for the Euler method on a non-uniform grid, based on the local truncation error, which results from a single iteration. For intuitive clarity, we consider a one-dimensional ODE of the form
+
+$$
+d x = f (x, t) d t,
+$$
+
+where  $f$  is a smooth function. Suppose that the numerical solution is obtained using the Euler method with the discretization points  $[t_i, t_{i-1}, \ldots, t_{i-k}]$  in a reverse time order, given the initial condition  $x(t_i) = x_{t_i}$ . From now on, we denote  $\hat{x}_{t_j}^{(n)}$  as the numerical solution at  $t_j$  obtained by  $n$  iterations and  $x_{t_j}^*$  as the exact solution at  $t_j$ . Given  $h = t_i - t_{i-k}$  and  $\lambda_j = \frac{1}{h} (t_{i-j+1} - t_{i-j})$  for  $j = 1, \ldots, k$ , the local truncation error formula of the one-step Euler method, derived from the Taylor expansion, is expressed as
+
+$$
+\hat {x} _ {t _ {i - 1}} ^ {(1)} = x _ {t _ {i}} - \lambda_ {1} h f \left(x _ {t _ {i}}; t _ {i}\right) = x _ {t _ {i - 1}} ^ {*} - \frac {1}{2} x _ {t _ {i}} ^ {\prime \prime} \lambda_ {1} ^ {2} h ^ {2} + O \left(h ^ {3}\right). \tag {9}
+$$
+
+Then, the truncation error of the two-step numerical solution is derived as
+
+$$
+\begin{array}{l} \hat {x} _ {t _ {i - 2}} ^ {(2)} = \hat {x} _ {t _ {i - 1}} ^ {(1)} - \lambda_ {2} h f \left(\hat {x} _ {t _ {i - 1}} ^ {(1)}\right) (10) \\ = x _ {t _ {i - 1}} ^ {*} - \frac {1}{2} x _ {t _ {i}} ^ {\prime \prime} \lambda_ {1} ^ {2} h ^ {2} + O \left(h ^ {3}\right) - \lambda_ {2} h f \left(\hat {x} _ {t _ {i - 1}} ^ {(1)}\right) (11) \\ = x _ {t _ {i - 1}} ^ {*} - \lambda_ {2} h f \left(x _ {t _ {i - 1}} ^ {*}\right) - \frac {1}{2} x _ {t _ {i}} ^ {\prime \prime} \lambda_ {1} ^ {2} h ^ {2} + O \left(h ^ {3}\right) - \lambda_ {2} h f \left(\hat {x} _ {t _ {i - 1}} ^ {(1)}\right) + \lambda_ {2} h f \left(x _ {t _ {i - 1}} ^ {*}\right) (12) \\ = x _ {t _ {i - 2}} ^ {*} - \frac {1}{2} x _ {t _ {i - 1}} ^ {* \prime \prime} \lambda_ {2} ^ {2} h ^ {2} - \frac {1}{2} x _ {t _ {i}} ^ {\prime \prime} \lambda_ {1} ^ {2} h ^ {2} + O \left(h ^ {3}\right) \quad (\because \text {E q u a t i o n (9) a n d f i s s m o o t h}) (13) \\ = x _ {t _ {i - 2}} ^ {*} - \frac {1}{2} x _ {t _ {i}} ^ {\prime \prime} \left(\lambda_ {1} ^ {2} + \lambda_ {2} ^ {2}\right) h ^ {2} + O \left(h ^ {3}\right) \quad (\because f \text {i s s m o o t h}). (14) \\ \end{array}
+$$
+
+Inductively, we can obtain the truncation error for the  $k$ -step solution as
+
+$$
+\hat {x} _ {t _ {i - k}} ^ {(k)} = x _ {t _ {i - k}} ^ {*} - \frac {1}{2} x _ {t _ {i}} ^ {\prime \prime} \sum_ {j = 1} ^ {k} \lambda_ {j} ^ {2} h ^ {2} + O \left(h ^ {3}\right), \tag {15}
+$$
+
+which approximates  $x_{t_i - k}^*$  with a truncation error of  $O(h^2)$ .
+
+# 4.2 RX-EULER
+
+We now describe RX-Euler performing extrapolation every  $k$  steps on the Euler method. Extrapolation is executed as a linear combination of two different numerical solutions  $\hat{\pmb{x}}_{t_{i - k}}^{(1)}$  and  $\hat{\pmb{x}}_{t_{i - k}}^{(k)}$  obtained by the Euler solver over a single step on the grid  $[t_i,t_{i - k}]$  and  $k$  steps on the grid  $[t_i,t_{i - 1},\dots ,t_{i - k}]$ ,
+
+respectively. To calculate coefficients for extrapolation, we use the truncation error derived in Section 4.1, which can be also applied to Equation (3) in Section 3.1, as the ideal score function is considered smooth; its derivative is Lipschitz continuous, referring to the equation in Appendix B.3 of Karras et al. (2022). From Equations (9) and (15), we derive the expressions for  $\hat{\pmb{x}}_{t_{i-k}}^{(1)}$  and  $\hat{\pmb{x}}_{t_{i-k}}^{(k)}$  for a constant  $c$  as follows:
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(1)} = \boldsymbol {x} _ {t _ {i - k}} ^ {*} + c h ^ {2} + O \left(h ^ {3}\right) \tag {16}
+$$
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(k)} = \boldsymbol {x} _ {t _ {i - k}} ^ {*} + \boldsymbol {c} \sum_ {j = 1} ^ {k} \lambda_ {j} ^ {2} h ^ {2} + O \left(h ^ {3}\right). \tag {17}
+$$
+
+Then, by solving the linear system of Equations (16) and (17), we approximate  $\boldsymbol{x}_{t_{i-k}}^{*}$  through the following extrapolation:
+
+$$
+\tilde {\boldsymbol {x}} _ {t _ {i - k}} ^ {(k)} = \frac {\hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(k)} - \sum_ {j = 1} ^ {k} \lambda_ {j} ^ {2} \hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(1)}}{1 - \sum_ {j = 1} ^ {k} \lambda_ {j} ^ {2}}, \tag {18}
+$$
+
+which involves a truncation error of  $O(h^3)$ , asymptotically smaller than  $O(h^2)$ .
+
+In the sampling process, we set the initial condition at the next denoising step,  $t_{i - k}$ , as  $\tilde{\pmb{x}}_{t_{i - 1}}^{(k)}$ , and repeatedly perform the proposed extrapolation technique every  $k$  steps. Because this approach provides provably more accurate solutions at every  $k$  steps, we can reduce error propagation and expect better quality of generated examples.
+
+The proposed method is applicable to first-order methods in general, including DDIM (Song et al., 2021a), which is arguably the most widely used DPM sampler. In this context, we bring the interpretation of DDIM as the Euler method applied to the following ODE:
+
+$$
+d \boldsymbol {y} = \epsilon_ {\theta} (\boldsymbol {x} (t), t) d \gamma , \tag {19}
+$$
+
+where  $\pmb{y}(t) = \pmb{x}(t)\sqrt{1 + \gamma(t)^2}$  and  $\gamma(t) = \sqrt{\frac{1 - \alpha_t^2}{\alpha_t^2}}$  in the variance-preserving diffusion process (Song et al., 2021b), i.e.,  $\pmb{p}_t(\pmb{x}|\pmb{x}_0) = \mathcal{N}(\alpha_t\pmb{x}_0, (1 - \alpha_t^2)\pmb{I})$ . Thus, instead of using the time grid, we compute  $\lambda_j$ 's in Equation (18) in terms of  $\gamma(t)$ 's, replacing  $t$  with the corresponding  $\gamma(t)$  in the computation, while the other procedures remain unchanged.
+
+RX-Euler (RX-DDIM) does not require additional NFEs beyond the number of time steps, as the first prediction of every  $k$ -step-interval can be stored during the computation of  $\hat{\pmb{x}}^{(k)}$  and reused to obtain  $\hat{\pmb{x}}^{(1)}$ . The only extra computation involves a linear combination of two estimates, which is negligible compared to the forward evaluations of DPMs.
+
+# 4.3 RX-DPM WITH HIGHER-ORDER SOLVERS
+
+We now present the algorithm for general ODE samplers of DPMs including high-order solvers. When the extrapolation occurs every  $k$  steps, as in Section 4.2, the error of  $\hat{x}_{t_{i-k}}^{(1)}$ , resulting from a single iteration of an arbitrary ODE method, is given by
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(1)} = \boldsymbol {x} _ {t _ {i - k}} ^ {*} + c h ^ {p} + O \left(h ^ {q}\right) \tag {20}
+$$
+
+for  $0 < p < q$  and  $\pmb{c} \neq \mathbf{0}$ . For  $\hat{\pmb{x}}_{t_{i-k}}^{(k)}$ , we extend the form of the linear error accumulation observed in Equation (17) to obtain the following equation:
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(k)} = \boldsymbol {x} _ {t _ {i - k}} ^ {*} + \boldsymbol {c} \sum_ {j = 1} ^ {k} \lambda_ {j} ^ {p} h ^ {p} + O (h ^ {q}). \tag {21}
+$$
+
+Note that Equation (21) does not hold in general; however, we consider this simplified assumption reasonable, as it is consistent with the standard assumption of Richardson extrapolation under uniform discretization (see Appendix A). Finally, by solving the linear system of Equations (20) and (21), the extrapolated solution is given by
+
+$$
+\tilde {\boldsymbol {x}} _ {t _ {i - k}} ^ {(k)} = \frac {\hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(k)} - \sum_ {j = 1} ^ {k} \lambda_ {j} ^ {p} \hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(1)}}{1 - \sum_ {j = 1} ^ {k} \lambda_ {j} ^ {p}}, \tag {22}
+$$
+
+which approximates  $\boldsymbol{x}_{t_{i - k}}^{*}$  with a truncation error of  $O(h^{q})$ , asymptotically smaller than  $O(h^{p})$ .
+
+A limitation of this approach is that estimating  $\hat{\pmb{x}}_{t_{i - k}}^{(1)}$  and  $\hat{\pmb{x}}_{t_{i - k}}^{(k)}$  through naive applications of higher-order solvers requires additional network evaluations compared to baseline sampling. However, since  $\hat{\pmb{x}}_{t_{i - k}}^{(1)}$  is accessible from the network predictions made for the computation of  $\hat{\pmb{x}}_{t_{i - k}}^{(k)}$ , applying our method to higher-order solvers does not increase the NFE provided that intermediate predictions are properly stored. We next discuss how this is achieved using specific examples of high-order ODE solvers; the generalization to other solvers is mostly straightforward.
+
+Before moving forward, we note that high-order solvers typically rely on interpolation-based techniques, such as the Runge-Kutta method (Suli & Mayers, 2003) and linear multistep method (Timothy, 2017), where the former employs evaluations at multiple intermediate points, while the latter leverages evaluations from previous steps.
+
+RX-Runge-Kutta We consider applying our method with  $k = 2$  to the second-order Runge-Kutta method. A sequence of one-step estimates are given by
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - 1}} ^ {(1)} = \boldsymbol {x} _ {t _ {i}} - \left(t _ {i} - t _ {i - 1}\right) \left(a _ {1} \mathbf {z} _ {i} + a _ {2} \mathbf {z} _ {i - \delta}\right) \text {a n d} \tag {23}
+$$
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - 2}} ^ {(2)} = \hat {\boldsymbol {x}} _ {t _ {i - 1}} ^ {(1)} - \left(t _ {i - 1} - t _ {i - 2}\right) \left(a _ {1} \mathbf {z} _ {i - 1} + a _ {2} \mathbf {z} _ {i - 1 - \delta}\right). \tag {24}
+$$
+
+where  $\mathbf{z}_j = \epsilon_\theta (\pmb {x}(t_j),t_j)$  for  $t_{j - 1} < t_{j - \delta}\leq t_j$ . Then, we can express the single combined-step estimate at  $t_{i - 2}$  as
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - 2}} ^ {(1)} = \boldsymbol {x} _ {t _ {i}} - (t _ {i} - t _ {i - 2}) \left(a _ {1} \mathbf {z} _ {i} + a _ {2} \mathbf {z} _ {i - \delta^ {\prime}}\right). \tag {25}
+$$
+
+Since  $\mathbf{z}_i$  is reusable after the calculation of  $\hat{\pmb{x}}_{t_{i - 1}}^{(1)}$ , we only need to compute  $\mathbf{z}_{i - \delta '}$ , which is approximated as  $\mathbf{z}_{i - 1}$  or  $\mathbf{z}_{i - 1 - \delta}$ , depending on the proximity of its time step. This approach allows us to efficiently extrapolate the solutions without compromising the quality of the generated samples.
+
+RX-Adams-Bashforth Suppose that, by the  $s$ -step Adams-Bashforth method, extrapolation is performed on a grid with an interval of  $h$  every  $k$  steps. For predefined  $b_{j}$ 's, we are given
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(k)} = \hat {\boldsymbol {x}} _ {t _ {i - k + 1}} + h \sum_ {j = 0} ^ {s} b _ {j} \epsilon_ {\theta} \left(\hat {\boldsymbol {x}} _ {t _ {i - k + j}}, t _ {i - k + j}\right). \tag {26}
+$$
+
+Then, we compute  $\hat{\pmb{x}}_{t_{i - k}}^{(1)}$  with a step size of  $kh$  for extrapolation as
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - k}} ^ {(1)} = \hat {\boldsymbol {x}} _ {t _ {i}} + k h \sum_ {j = 0} ^ {s} b _ {j} \epsilon_ {\theta} \left(\hat {\boldsymbol {x}} _ {t _ {i - k + j k}}, t _ {i - k + j k}\right) \tag {27}
+$$
+
+which requires no additional NFEs by storing the previous network evaluations.
+
+Algorithm 1 summarizes the procedure of the proposed method with a generic ODE solver under the assumption that  $N$  is a multitude of  $k$  for simplicity; it is simple to handle the last few steps by either adjusting  $k$  for the remaining steps or skipping the extrapolation.
+
+# 4.4 ANALYSIS ON GLOBAL TRUNCATION ERRORS
+
+We perform an analysis on global truncation errors of the Euler method and RX-Euler under the same NFEs. Assume that we solve an ODE satisfying Lipschitz condition from  $t = 1$  to  $t = 0$  with NFEs  $= N$ .
+
+Euler Since the Euler method requires a single network evaluation for each time step, the number of time steps allowed is  $N$ . Since the local truncation error of the Euler method on step size of  $h = 1 / N$  is expressed as  $ch^2 + O(h^3)$ , the global truncation error is given by
+
+$$
+\left(c h ^ {2} + O \left(h ^ {3}\right)\right) \cdot N = \frac {c}{N} + O \left(N ^ {- 2}\right) = O \left(N ^ {- 1}\right). \tag {28}
+$$
+
+Algorithm 1 Sampling of RX-DPM  
+Require:  $\epsilon_{\theta}(\cdot), N, T = t_N > \ldots > t_0 = 0$   
+1: Input:  $k, \Phi(\cdot)$  (ODE solver),  $p$   
+2:  $\boldsymbol{x}_T \sim \boldsymbol{p}_T(\boldsymbol{x})$   
+3: for  $i = 1$  to  $N$  do  
+4: # Initialization  
+5: if  $i$  mod  $k = 1$  then  
+6:  $h \gets t_{N-i+1} - t_{N-i-k+1}$   
+7:  $\hat{\boldsymbol{x}}_{t_{N-i+1}}^{(k)} \gets \boldsymbol{x}_{t_{N-i+1}}$   
+8: end if  
+9:  $\lambda_i \gets (t_{N-i+1} - t_{N-i}) / h$   
+10:  $\hat{\boldsymbol{x}}_{t_{N-i}}^{(k)} \gets \Phi(\hat{\boldsymbol{x}}_{t_{N-i+1}}^{(k)}, t_{N-i+1}, t_{N-i}; \epsilon_{\theta}(\cdot))$  # Store  $\epsilon_{\theta}(t)$ 's if necessary.  
+11: # Extrapolation (Equation (22))  
+12: if  $i$  mod  $k = 0$  then  
+13:  $\hat{\boldsymbol{x}}_{t_{N-i}}^{(1)} \gets \Phi(\boldsymbol{x}_{t_{N-i} + h}, t_{N-i} + h, t_{N-i}; \epsilon)$  # No NFE required.  
+14:  $\tilde{\boldsymbol{x}}_{t_{N-i}}^{(k)} \gets \frac{\hat{\boldsymbol{x}}_{t_{N-i}}^{(k)} - \sum_{j=i}^{i+k-1} \lambda_j^p \hat{\boldsymbol{x}}_{t_{N-i}}^{(1)}}{1 - \sum_{j=i}^{i+k-1} \lambda_j^p}$   
+15:  $\boldsymbol{x}_{t_{N-i}} \gets \tilde{\boldsymbol{x}}_{t_{N-i}}^{(k)}$   
+16: end if  
+17: end for  
+18: return  $\boldsymbol{x}_{t_0}$
+
+RX-Euler If RX-Euler performs extrapolation every  $k$  steps, the extrapolation happens  $N / k$  times, where  $N$  is equal to the NFEs for the Euler method. The local truncation error for RX-Euler over each  $k$  steps, which has the interval of  $h = k / N$ , is expressed as  $c^{\prime}h^{3} + O(h^{4})$  and therefore the global truncation error is given by
+
+$$
+\left(c ^ {\prime} h ^ {3} + O \left(h ^ {4}\right)\right) \cdot \frac {N}{k} = \frac {k ^ {2} c ^ {\prime}}{N ^ {2}} + O \left(N ^ {- 3}\right) = O \left(N ^ {- 2}\right). \tag {29}
+$$
+
+RX-Euler exhibits a higher convergence rate of the global truncation error compared to the Euler method by one order of magnitude. Using the same approach, we can also demonstrate that the proposed method achieves faster convergence of the global truncation error for higher-order solvers.
+
+# 5 EXPERIMENT
+
+# 5.1 IMPLEMENTATION DETAILS
+
+We conduct the experiment with EDM (Karras et al., 2022), Stable Diffusion  $\mathrm{V}2^{1}$  (Rombach et al., 2022), DPM-Solver (Lu et al., 2022), and PNDM (Liu et al., 2022) using their official implementations and provided pretrained models. Throughout all experiments, we retain the default settings from the official codebases, except for additional hyperparameters related to the proposed method. For experiments with EDM, DPM-Solver, and PNDM as backbones, we generate  $50\mathrm{K}$  images and compute FID (Heusel et al., 2017) using the evaluation code provided in their implementations. To evaluate Stable Diffusion V2 results, we use the PyTorch implementation for the computation of  $\mathrm{FID}^2$  and CLIP score<sup>3</sup> with the patch size of  $32 \times 32$ .
+
+# 5.2 VALIDITY TEST
+
+We first evaluate the effectiveness of RX-Euler under the EDM backbone for  $k \in \{2, 3, 4\}$ , where smaller  $k$  values correspond to more frequent extrapolation over the same number of time steps. As
+
+shown in Figure 2, RX-Euler consistently achieves significantly better FID scores than the Euler method across all values of  $k$ . In particular, when extrapolation is applied every two steps, i.e.,  $k = 2$ , our approach achieves the best performance across a wide range of NFEs. This indicates that more frequent extrapolation leads to more accurate intermediate predictions, effectively mitigating error accumulation in the final samples. Meanwhile, the curves for  $k \geq 3$  remain closer to that of  $k = 2$  than to the Euler method, empirically validating the reduced truncation error derived in Equation (18) for general  $k$ . In other words, even sparse extrapolation still has a significant impact on the output quality. For the rest of our results, we set  $k = 2$ .
+
+We also compare the proposed method with conventional Richardson extrapolation, as formulated in Equation (8) with  $k = 2$ , which employs fixed coefficients. This baseline is labeled as Naïve ( $k = 2$ ) in Figure 2. When comparing RX-Euler ( $k = 2$ ) with Naïve ( $k = 2$ ), we find that our method produces superior results. This implies that extrapolation coefficients adapted to arbitrary time step scheduling are more effective than fixed coefficients derived from uniformly discretized time steps. In other words, the proposed grid-aware coefficients enable more effective extrapolation than deterministic ones, better capturing the varying importance of precision in DPMs over time (Karras et al., 2022).
+
+![](images/a053eafc9c837c58605b5b184256d36770c64283dae9e6e09b3a94659c3d6ae2.jpg)  
+Figure 2: Effect of extrapolation on the Euler method with different  $k$ 's.
+
+![](images/6b72f19c3b6253ba44a428007c8527016915f786a1ca53b446498ba73c045e0f.jpg)
+
+# 5.3 QUANTITATIVE COMPARISONS ON EDM BACKBONE
+
+We compare RX-Euler with other methods on four different datasets—CIFAR-10 (Krizhevsky & Hinton, 2009), FFHQ (Karras et al., 2019), AFHQv2 (Choi et al., 2020), and ImageNet (Deng et al., 2009)—using the EDM (Karras et al., 2022) backbone. Following standard practice, we evaluate the performance of class-conditional generation on CIFAR-10 and ImageNet while testing unconditional generation on the other two datasets. In this experiment, we include Heun's method, LA-DPM (Zhang et al., 2023), and IIA (Zhang et al., 2024) for comparisons with our approach, RX-Euler. Note that both Heun's method and RX-Euler are second-order numerical solvers, offering higher accuracy than the Euler method, while LA-DPM and IIA are techniques refining baseline sampling. To reproduce the results of LA-DPM, we use the Euler method, as it achieves better performance than Heun's method. For IIA (Zhang et al., 2024), we present results from the better-performing variant, selected between IIA and BIIA, as indicated in the original paper.
+
+Figure 3 presents a comparison of FID scores across the evaluated methods over a broad range of NFEs. RX-Euler consistently outperforms the other approaches, particularly at lower NFEs, demonstrating its effectiveness in fast sampling scenarios. While it occasionally falls behind LA-DPM or IIA on CIFAR-10 within specific intervals, its overall performance remains more stable and superior across the other three datasets, which pose greater challenges due to higher resolutions and increased data diversity.
+
+In the comparison between RX-Euler and Heun's method, while RX-Euler excels at lower NFEs, Heun's method performs slightly better at larger NFEs. This implies that selecting a more appropriate solver for each interval—between RX-Euler (extrapolation) and Heun's method (interpolation)—could yield better results. In this regard, one might expect that in the early steps—where predictions are closer to noise and thus less accurate—interpolation tends to be more stable than extrapolation. Based on this reasoning, we experiment with a hybrid approach of RX-Euler and Heun's method, labeled as RX+EDM in Figure 3. We find strong performance of this approach when RX-Euler is selectively applied to the middle or last (low-noise) few steps, outperforming both Heun's method and RX-Euler. This indicates that there is still room for improvement in our algorithm and provides another direction for future work.
+
+![](images/fca60c59554a95dbd3b74ac16c82852657cc5b0cd8710b352a593b42bc4cf467.jpg)  
+Figure 3: FIDs of RX-Euler, Heun's method (labeled as EDM), LA-DPM and IIA (or BIIA) by varying NFEs on the CIFAR-10, FFHQ, AFHQv2, and ImageNet datasets using the EDM backbone.
+
+![](images/570c47be29d6dbeab6643eb9aab4574dec86c3bd7b359908397c1d89b00e6a30.jpg)
+
+![](images/ab92a587b1186f8b7d9e18482e8933c00dc8f2c5274d45f045af723a7837d4c7.jpg)
+
+![](images/4a5ccea0a74a93779b2e4b366ee04eec2068396b4b9d6c1d25a5493334bed630.jpg)
+
+![](images/d152a2e617a7c31fe2dcb194b589b123a0dc9f19c7a8782571664937dbc3e559.jpg)  
+Figure 4: Qualitative results of DDIM and RX-DDIM based on Stable Diffusion V2. RX-DDIM produces sharper and more detailed backgrounds, especially evident in the left example. On the right, RX-DDIM generates more realistic giraffes, whereas DDIM struggles at NFEs = 50, failing to properly render the giraffe.
+
+Table 1: FID and CLIP scores of DDIM and RX-DDIM using Stable Diffusion V2.  
+
+<table><tr><td>NFEs</td><td colspan="2">15</td><td colspan="2">20</td><td colspan="2">30</td><td colspan="2">50</td></tr><tr><td>Method \ Metric</td><td>FID (↓)</td><td>CLIP (↑)</td><td>FID (↓)</td><td>CLIP (↑)</td><td>FID (↓)</td><td>CLIP (↑)</td><td>FID (↓)</td><td>CLIP (↑)</td></tr><tr><td>DDIM</td><td>19.15</td><td>31.727</td><td>18.43</td><td>31.716</td><td>19.00</td><td>31.750</td><td>18.65</td><td>31.711</td></tr><tr><td>RX-DDIM</td><td>17.24</td><td>31.629</td><td>17.12</td><td>31.721</td><td>17.62</td><td>31.781</td><td>17.83</td><td>31.727</td></tr></table>
+
+# 5.4 COMPARISONS ON STABLE DIFFUSION
+
+We apply RX-DDIM to Stable Diffusion V2, which provides various conditional generations. For evaluation, we generate  $10\mathrm{K}512\times 512$  images from unique text prompts in the COCO2014 (Lin et al., 2014) validation set and compute FID and CLIP scores on resized  $256\times 256$  images. As shown in Table 1, our method also performs well on large models. However, we observe that RX-DDIM yields lower CLIP scores at NFEs  $= 15$ , which we attribute to classifier-guidance scales. According to Rombach et al. (2022), optimal classifier-free guidance scales differ across models. Since the default setting is tuned for DDIM, RX-DDIM may benefit from further optimization. Figure 4 presents qualitative comparisons between DDIM and RX-DDIM, highlighting the superior image quality of RX-DDIM. Notably, RX-DDIM generates images with more vivid colors, sharper textures, and more realistic object depictions, leading to an overall more natural appearance. We provide more examples for qualitative comparisons between DDIM and RX-DDIM in Figures 13 and 14 of Appendix F.
+
+Table 2: FID scores of DPM-Solvers (Lu et al., 2022) and RX-DPMs applied to DPM-Solvers on CIFAR-10 and LSUN Bedroom datasets. All baseline results are reproduced under the same setting as RX-DPMs.  
+
+<table><tr><td rowspan="2">Method\ NFEs</td><td colspan="4">CIFAR-10 (32×32)</td><td colspan="4">LSUN Bedroom (256×256)</td></tr><tr><td>9</td><td>10</td><td>12</td><td>15</td><td>9</td><td>10</td><td>12</td><td>15</td></tr><tr><td>DPM-Solver-2</td><td>-</td><td>15.06</td><td>11.33</td><td>7.36</td><td>-</td><td>14.67</td><td>11.38</td><td>6.44</td></tr><tr><td>RX-DPM-Solver-2</td><td>-</td><td>12.94</td><td>9.80</td><td>6.53</td><td>-</td><td>12.66</td><td>10.13</td><td>5.72</td></tr><tr><td>DPM-Solver-3</td><td>12.39</td><td>-</td><td>6.76</td><td>5.00</td><td>8.79</td><td>-</td><td>5.37</td><td>4.04</td></tr><tr><td>RX-DPM-Solver-3</td><td>11.50</td><td>-</td><td>6.62</td><td>4.85</td><td>8.12</td><td>-</td><td>5.18</td><td>4.04</td></tr></table>
+
+Table 3: FID scores of two types of PNDM (Liu et al., 2022) and RX-DPMs applied to each solver on CIFAR-10, CelebA and LSUN Church datasets. Note that S-PNDM and F-PNDM require 1 and 9 additional NFEs to the number of time steps, respectively. The baseline results are copied from PNDM (Liu et al., 2022).  
+
+<table><tr><td rowspan="2">Method \ # of steps</td><td colspan="3">CIFAR-10 (32×32)</td><td colspan="3">CelebA (64×64)</td><td colspan="3">LSUN Church (256×256)</td></tr><tr><td>5</td><td>10</td><td>20</td><td>5</td><td>10</td><td>20</td><td>5</td><td>10</td><td>20</td></tr><tr><td>S-PNDM</td><td>18.3</td><td>8.64</td><td>5.77</td><td>15.2</td><td>12.2</td><td>9.45</td><td>20.5</td><td>11.8</td><td>9.20</td></tr><tr><td>RX-S-PNDM</td><td>19.69</td><td>7.64</td><td>4.72</td><td>11.56</td><td>9.22</td><td>6.89</td><td>21.15</td><td>10.96</td><td>8.96</td></tr><tr><td>F-PNDM</td><td>18.2</td><td>7.05</td><td>4.61</td><td>11.3</td><td>7.71</td><td>5.51</td><td>14.8</td><td>8.69</td><td>9.13</td></tr><tr><td>RX-F-PNDM</td><td>-</td><td>6.60</td><td>3.99</td><td>-</td><td>7.10</td><td>4.99</td><td>-</td><td>8.85</td><td>9.41</td></tr></table>
+
+# 5.5 COMPARISONS ON HIGHER-ORDER SOLVERS
+
+We further apply the proposed method to advanced ODE samplers with higher-order accuracy. Table 2 presents the effectiveness of RX-DPM when applied to DPM-Solvers (Lu et al., 2022) on CIFAR-10 and LSUN Bedroom (Yu et al., 2015). Among the variations of DPM solvers, we utilize the single-step versions of DPM-Solver-2 and DPM-Solver-3. Note that, since a single-step DPM-solver- $n$  can be considered as an  $n^{\text{th}}$ -order Runge-Kutta-like solver, we apply RX-DPM with  $p = n + 1$  in Equation (22) for the DPM-solver- $n$ . Additionally, we compare our method with another accelerated diffusion sampler, DEIS (Zhang & Chen, 2023), for class-conditioned image generation on ImageNet  $(64 \times 64)$  in Table 4 of Appendix B.1. As shown in the results, RX-DPM consistently achieves the best performance across all NFEs.
+
+As another type of advanced sampler, we consider PNDM (Liu et al., 2022). S-PNDM and F-PNDM employ linear multistep methods, specifically the 2-step and 4-step Adams-Bashforth methods, respectively, except for the initial few time steps. Accordingly, we apply RX-DPM with  $p = 3$  and  $p = 5$  in Equation (22) for S-PNDM and F-PNDM, respectively. The results on the CIFAR-10, CelebA (Liu et al., 2015), and LSUN Church (Yu et al., 2015) datasets are presented in Table 3. While RX-DPM improves performance in most cases, an exception arises with F-PNDM on the LSUN Church dataset, where RX-DPM does not provide a clear advantage. Upon analysis, we observe that the baseline performance of F-PNDM is highest when using 10 time steps and degrades as the number of steps increases (Liu et al., 2022). Since RX-DPM enhances accuracy by leveraging improvements in the baseline solver with finer time steps, its effectiveness is limited when the baseline itself deteriorates under finer discretization. A similar phenomenon with F-PNDM on LSUN datasets has also been reported in IIA (Zhang et al., 2024).
+
+# 6 CONCLUSION
+
+We introduced RX-DPM, an advanced ODE sampling method for DPMs that leverages extrapolation based on two ODE solutions derived from different discretizations of the same time interval. Our algorithm computes the optimal coefficients for arbitrary time step scheduling without additional training and incurs no extra NFEs by utilizing past predictions. This approach effectively reduces truncation errors, resulting in improved sample quality. Extensive experiments on well-established baseline models and datasets confirm that RX-DPM surpasses existing sampling methods, offering a more efficient and accurate solution for DPMs.
+
+# ACKNOWLEDGEMENTS
+
+This work was partly supported by Samsung Research, Samsung Electronics Co., Ltd., and the Institute of Information & communications Technology Planning & Evaluation (IITP) grants [RS-2022-II220959 (No.2022-0-00959), (Part 2) Few-Shot Learning of Causal Inference in Vision and Language for Decision Making; No.RS-2021-II212068, AI Innovation Hub (AI Institute, Seoul National University); No.RS-2021-II211343, Artificial Intelligence Graduate School Program (Seoul National University)] funded by the Korea government (MSIT).
+
+# ETHICS STATEMENT
+
+This paper utilizes pretrained diffusion probabilistic models to generate high-quality images while prioritizing efficiency. As a result, our approach does not inherently introduce hazardous elements. Nonetheless, we acknowledge the potential for unintended misuse, including the synthesis of inappropriate or sensitive content.
+
+# REPRODUCIBILITY STATEMENT
+
+We provide detailed implementation instructions and reproducibility guidelines in Section 5.1, along with the pseudo-code of the core algorithm in Algorithm 1. The full implementation is available at https://github.com/jin01020/rx-dpm.
+
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+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, University of Toronto, Toronto, Ontario, 2009.  
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+
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+
+# A JUSTIFICATION ON EQUATION (21)
+
+Assuming a uniform grid as in the context of conventional Richardson extrapolation in Section 3.2, and the ODE solver with  $O(h^{p + 1})$  of local truncation error formula, we have
+
+$$
+V ^ {*} = V (h) + c h ^ {p} + O \left(h ^ {p + 1}\right) \quad \text {a n d} \tag {30}
+$$
+
+$$
+V ^ {*} = V \left(\frac {h}{k}\right) + c \left(\frac {h}{k}\right) ^ {p} + O \left(h ^ {p + 1}\right) \tag {31}
+$$
+
+since we expect the  $O(h^p)$  of the global truncation error. Then, we have the following extrapolated solution with a truncation error of  $O(h^{p + 1})$  by Richardson extrapolation (Equation (8)):
+
+$$
+\tilde {V} (h, k) = \frac {k ^ {p} V (h / k) - V (h)}{k ^ {p} - 1}. \tag {32}
+$$
+
+Now, considering the case of Equations (20) and (21) with uniform discretization, we have
+
+$$
+V ^ {*} = V (h) + c ^ {\prime} h ^ {p + 1} + O \left(h ^ {p + 2}\right) \quad \text {a n d} \tag {33}
+$$
+
+$$
+V ^ {*} = V \left(\frac {h}{k}\right) + k c ^ {\prime} \left(\frac {h}{k}\right) ^ {p + 1} + O \left(h ^ {p + 2}\right), \tag {34}
+$$
+
+each correspondingly. Then, the extrapolated solution  $\tilde{V}_{\mathrm{ours}}$  obtained from solving linear system of Equations (33) and (34) becomes
+
+$$
+\tilde {V} _ {\text {o u r s}} = \frac {k ^ {p} V (h / k) - V (h)}{k ^ {p} - 1}, \tag {35}
+$$
+
+which turns out to be exactly the same as Equation (32). Thus, we believe our approach can be considered to employ assumptions shared by those used in common practices of Richardson extrapolation and also can reduce global errors which is backed by experimental results as well.
+
+# B MORE QUANTITATIVE RESULTS
+
+# B.1 COMPARISON WITH DEIS
+
+We compare DEIS variants (Zhang & Chen, 2023), DPM-Solvers, and RX-DPMs applied to DPM-Solvers on class-conditioned ImageNet  $(64\times 64)$  in Table 4. The results clearly demonstrate that our method outperforms all other approaches across all NFEs.
+
+# B.2 DPMs WITH OPTIMAL COVARIANCES
+
+Although our method is designed for ODE solvers, we also conduct experiments with SN-DPM and NPR-DPM (Bao et al., 2022) on CIFAR-10 (Krizhevsky & Hinton, 2009) and CelebA datasets (Liu et al., 2015) to evaluate its performance when applied to SDE solvers. We use the official codes and provided pretrained models as described in Section 5.1. SN-DPM and NPR-DPM are two different models that correct the imperfect mean prediction in the reverse process of existing models through optimal covariance learning.
+
+To incorporate our method into stochastic sampling, we decompose it into a deterministic sampling component and a stochastic component, and apply our method to the deterministic sampling part. Specifically, within each  $k$ -step interval, we execute RX-DPM algorithm using the deterministic sampling component and then add the stochasticity term afterward. In this way, our method uses the stochasticity of the baseline sampling method in a limited manner compared to the baseline sampling with the same NFEs. For a better understanding of our implementation, we provide the diagrams of the proposed method in Appendix C.
+
+In Table 5, we show the results on NPR-DPM and SN-DPM along with the vanilla DDIM and LA-DPM (Zhang et al., 2023), which is another extrapolation-based sampling method. We observe that our method outperforms the compared methods in most cases, although a performance degradation is noted with RX-SN-DDIM on the CIFAR-10 dataset. This implies that our approach of solving the ODE might offset the benefits of the model's optimization. Despite this, we observe significant performance improvements in the most extreme case, NFEs = 10.
+
+Table 4: Comparisons of DEIS variants, DPM-Solvers and RX-DPMs applied to DPM-Solvers on class-conditioned ImageNet (64×64). All results of DEIS and DPM-Solvers are copied from DEIS (Zhang & Chen, 2023) except for the result of DPM-Solver-3 with NFEs = 9.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">NFEs</td></tr><tr><td>9</td><td>10</td><td>12</td><td>18</td><td>30</td></tr><tr><td>tAB-DEIS</td><td>-</td><td>6.65</td><td>3.99</td><td>3.21</td><td>2.81</td></tr><tr><td>ρAB-DEIS</td><td>-</td><td>9.28</td><td>6.46</td><td>3.74</td><td>2.87</td></tr><tr><td>DPM-Solver-2</td><td>-</td><td>7.93</td><td>5.36</td><td>3.63</td><td>3.00</td></tr><tr><td>ρMid-DEIS</td><td>-</td><td>9.12</td><td>6.78</td><td>4.00</td><td>2.99</td></tr><tr><td>RX-DPM-Solver-2</td><td>-</td><td>6.11</td><td>5.61</td><td>3.64</td><td>2.93</td></tr><tr><td>DPM-Solver-3</td><td>7.45</td><td>-</td><td>5.02</td><td>3.18</td><td>2.84</td></tr><tr><td>ρKutta-DEIS</td><td>-</td><td>-</td><td>13.12</td><td>3.63</td><td>2.82</td></tr><tr><td>RX-DPM-Solver-3</td><td>7.08</td><td></td><td>3.90</td><td>2.36</td><td>2.18</td></tr></table>
+
+Table 5: FID scores for CIFAR-10 and CelebA on DDIM, NPR-DDIM and SN-DDIM models. The values for each baseline and LA-DDIM results are copied from Zhang et al. (2023).  
+
+<table><tr><td>Dataset</td><td colspan="3">CIFAR-10 (32×32)</td><td colspan="3">CelebA (64×64)</td></tr><tr><td>Method \ NFEs</td><td>10</td><td>25</td><td>50</td><td>10</td><td>25</td><td>50</td></tr><tr><td>DDIM</td><td>21.31</td><td>10.70</td><td>7.74</td><td>20.54</td><td>13.45</td><td>9.33</td></tr><tr><td>RX-DDIM</td><td>14.78</td><td>8.42</td><td>6.30</td><td>18.31</td><td>10.54</td><td>6.88</td></tr><tr><td>NPR-DDIM</td><td>13.40</td><td>5.43</td><td>3.99</td><td>14.94</td><td>9.18</td><td>6.17</td></tr><tr><td>LA-NPR-DDIM</td><td>10.74</td><td>4.71</td><td>3.64</td><td>14.25</td><td>8.83</td><td>5.67</td></tr><tr><td>RX-NPR-DDIM</td><td>6.35</td><td>3.92</td><td>3.34</td><td>11.58</td><td>6.61</td><td>3.98</td></tr><tr><td>SN-DDIM</td><td>12.19</td><td>4.28</td><td>3.39</td><td>10.17</td><td>5.62</td><td>3.90</td></tr><tr><td>LA-SN-DDIM</td><td>8.48</td><td>3.15</td><td>2.93</td><td>8.05</td><td>4.56</td><td>2.93</td></tr><tr><td>RX-SN-DDIM</td><td>7.50</td><td>5.12</td><td>4.40</td><td>5.20</td><td>2.72</td><td>2.25</td></tr></table>
+
+# C DIAGRAMS
+
+Figure 5 compares the diagrams of an ODE solver, the proposed method with an ODE solver, and the proposed method with an SDE solver.
+
+# D COMPUTATIONAL COST
+
+We compare the computational time and GPU memory usage of the Euler method and RX-Euler using the EDM backbone in Tables 6 and 7, respectively. For measurements, we set the batch size to 128 and use 10-step sampling on an A6000 GPU. The average runtime per batch is measured for computational time. The additional operations introduced by our method, which consist of linear combinations of precomputed values, result in negligible computational overhead compared to the time required for the network forward pass. Furthermore, as the model size increases, the relative overhead diminishes (e.g., only  $0.11\%$  increase for ImageNet class-conditional sampling). Similarly, GPU memory usage increases slightly with RX-Euler, primarily due to the storage of previous predictions. However, this increase is minimal and decreases as model or data size increases, showing a similar trend to that observed with computational time.
+
+# E LIMITATIONS
+
+As our method is primarily designed for an ODE solver, to integrate it with an SDE solver (or a stochastic sampling method), we partially apply the stochasticity component of the SDE solver as demonstrated in Appendix B.2. Consequently, in some cases, the effectiveness is offset because the full effects of stochasticity are not captured. However, in scenarios where NFE is very limited, which are of greater interest to us, the combined effect of RX-DPM and stochastic sampling has been empirically shown to be highly beneficial. We leave the development of methods that can perform better in more general cases for future work. Additionally, in the extension of the RX-Euler algorithm to a higher-order solver in Section Section 4.3, there remains room for improvement since we impose assumptions about linear error propagation. We believe that relaxing these assumptions or deriving more accurate equations could further enhance the performance.
+
+![](images/55df582f69f03fb851037d7250ff32eb90754197724d51bd87f172373730ff64.jpg)  
+(a) One step of an ODE solver
+
+![](images/38748f7f913feaceb1a83fe64eb1f99fd47ded90385efff95a995cd6b8239d7c.jpg)  
+(b)  $k$  steps of RX-DPM applied to an ODE solver
+
+![](images/380a9b682f6c28717554f493e67631bf0f9e530f38954dcf2a9a33f82ede2807.jpg)  
+(c)  $k$  steps of RX-DPM applied to an SDE solver  
+Figure 5: Digarams of the baseline and the proposed sampling methods. The blue-bordered boxes in (b) and (c) indicate that the corresponding operation does not require network evaluation. The ODE solver in (c) refers to the deterministic sampling component of the SDE solver.
+
+Table 6: Comparison of per-batch computation times between the Euler method and RX-Euler with the EDM backbone. The reported values represent the average runtime across 100 measurements (in seconds).  
+
+<table><tr><td></td><td>CIFAR-10 cond. (32x32)</td><td>FFHQ (64x64)</td><td>ImageNet cond. (64x64)</td></tr><tr><td>Euler</td><td>1.737 ± 0.028</td><td>3.895 ± 0.023</td><td>6.436 ± 0.033</td></tr><tr><td>RX-Euler</td><td>1.743 ± 0.031</td><td>3.903 ± 0.025</td><td>6.443 ± 0.039</td></tr></table>
+
+Table 7: Comparison of GPU memory usage (MiB) during inference between the Euler method and RX-Euler with the EDM backbone.  
+
+<table><tr><td></td><td>CIFAR-10 cond. (32x32)</td><td>FFHQ (64x64)</td><td>ImageNet cond. (64x64)</td></tr><tr><td>Euler</td><td>2929</td><td>9433</td><td>12661</td></tr><tr><td>RX-Euler</td><td>3033</td><td>9477</td><td>12705</td></tr></table>
+
+# F QUALITATIVE RESULTS
+
+We provide qualitative results using the EDM backbone in Figures 6 to 12 and Stable Diffusion V2 in Figures 13 and 14.
+
+![](images/4c96192bbe5fd0a89016fff4732d63bee8f8f53329d1cde975355d6aa8b0a530.jpg)  
+Euler (NFEs = 10, FID: 15.88)
+
+![](images/ce4469651a5f478e344091e9b975a221084215fa60764bf59d30a6b78ea8d119.jpg)  
+EDM (NFEs = 11, FID: 14.46)
+
+![](images/95a9987257432742f38caa3dc9b03b3252352d32c28f23abbfd86d5f30e6c3fb.jpg)  
+RX-Euler (NFEs = 10, FID 4.35)  
+Figure 6: Qualitative results of CIFAR-10 of different sampling methods with EDM backbone.
+
+![](images/f2c50ce7813ab3f0beb65431cb6f239efeb9115389e0930c2509f5c28278f2c5.jpg)  
+RX+EDM (NFEs = 10, FID 4.26)
+
+![](images/4174c8a5d094adb57c45282ba3cbe6937d22ceec57bea6a98dbbcb135277cb8f.jpg)  
+Euler (NFEs  $= 9$  ,FID:24.27)
+
+![](images/a44ba6f97d5bc9d55647d04829f666d9b116d583122155c487d7b306ecbbbfbd.jpg)  
+EDM (NFEs = 9, FID: 57.12)
+
+![](images/b670118da72d04b28f5ca775945418064499a2e1a69c658c2c1a29002ed5a5cf.jpg)  
+RX-Euler (NFEs  $= 8$  FID:12.94)  
+Figure 7: Qualitative results of FFHQ of different sampling methods with EDM backbone.
+
+![](images/689cab9dd729bdb08fe7a27549564bef103334aba7c757971a3d0173deb357df.jpg)  
+RX+EDM (NFEs = 9, FID: 6.83)
+
+![](images/33fdd586e4d6c515ebb3e2fcde41bcd9a507fc28c38c8ed33d2ca47a743e4801.jpg)  
+Euler (NFEs = 11, FID: 18.30)
+
+![](images/90ca6127739e92e1edefdfd95adf6e3c3c648f0d7a8059b5e6e9bfc4896e5a01.jpg)  
+EDM (NFEs = 11, FID: 29.34)
+
+![](images/29a948b822c27ca170113aa22cfc4520ae76cbe0a97681ab7941807831f9265e.jpg)  
+RX-Euler (NFEs = 10, FID: 7.20)  
+Figure 8: Qualitative results of FFHQ of different sampling methods with EDM backbone.
+
+![](images/caeee28f3a6621b32d5d320a6b18de5a615f0f1cc13cc0a316e235bb5109c5c4.jpg)  
+RX+EDM (NFEs = 11, FID: 5.50)
+
+![](images/7abce3392a58632fa2bce60b6c202951c9a826bba108c626389e393ea877b6a1.jpg)  
+Euler (NFEs  $= 9$  ,FID:11.81)
+
+![](images/e3c417fd90fc9416b80777e6d766e1740038c7326d134b4b6c7a366af9c9126e.jpg)  
+EDM (NFEs = 9, FID: 27.92)
+
+![](images/7012d7e386f5efa96fd04cd0825074b544c01b104488187ae6c26f2884c68386.jpg)  
+RX-Euler (NFEs  $= 8$  ,FID:6.49)  
+Figure 9: Qualitative results of AFHQv2 of different sampling methods with EDM backbone.
+
+![](images/d3051fc35101e7c4805818082691e650964275f87ed10c2235a87fe979ff3fd2.jpg)  
+RX+EDM (NFEs = 9, FID: 3.88)
+
+![](images/a734e88f63720e26a78361720a3da7b6bfcc9fe55ac10b394c82db58f940b3b3.jpg)  
+Euler (NFEs  $= 11$  FID:8.94)
+
+![](images/4e5d5d6a8b69e5daf66569adf32ba0009a57d08bbb87d7873489cef692d415e6.jpg)  
+EDM (NFEs = 11, FID: 13.61)
+
+![](images/4754504601dc913d40ea056c4bbbfa2f6a9b5aede8ebfd84f6f4f18ab933cc2b.jpg)  
+RX-Euler (NFEs = 10, FID: 4.02)  
+Figure 10: Qualitative results of AFHQv2 of different sampling methods with EDM backbone.
+
+![](images/8b687836b47423b6a11797fec3825c6afcc32f929652b69ccb056d0c7028d3ab.jpg)  
+RX+EDM (NFEs = 10, FID: 3.72)
+
+![](images/148f228fe6769ff3ff6da9ced43ec2f6754085d838adf76d98bc94016390438a.jpg)  
+Euler (NFEs  $= 9$  ,FID:20.49)
+
+![](images/e0b5cf14169f97ef00ff28ad17cbc2825ea71a7a7e39ec40b71ddf54f17c5205.jpg)  
+EDM (NFEs  $= 9$  ,FID:35.34)
+
+![](images/7ea03c64f7972e19ac8e68a30ea4a09b528ced5e72e653d81950ed469bb9d369.jpg)  
+RX-Euler (NFEs = 8, FID: 12.38)  
+Figure 11: Qualitative results of ImageNet of different sampling methods with EDM backbone.
+
+![](images/0ed8df7c485cdd89fd0437bf73264ffa3aa218177e36a946eb384e3e7582c1a8.jpg)  
+RX+EDM (NFEs = 9, FID: 7.94)
+
+![](images/32e3f739c62f852ede116d74d1e2db772f17e1d901c6236026f2f0ff1edeaa41.jpg)  
+Euler (NFEs = 11, FID: 14.84)
+
+![](images/11551a65f21d41a6e18a2397c2fe7a3befad6ba4733cf034f5f91d9f95087306.jpg)  
+EDM (NFEs = 11, FID: 15.35)
+
+![](images/8f6c60c6dbbdaab40e25c26aceaa0684f06a5b90ad09222a39819e43bd77d762.jpg)  
+RX-Euler (NFEs = 10, FID: 6.95)  
+Figure 12: Qualitative results of ImageNet of different sampling methods with EDM backbone.
+
+![](images/4d97c8056a1aee47695ee333410f398e8a694b2a56d260c8e6432813673e7ba8.jpg)  
+RX+EDM (NFEs = 11, FID: 5.83)
+
+![](images/c888938817a12a1961c184d4af749986283a11b87b6edf271bd031f31c58dca0.jpg)  
+Figure 13: Qualitative results on Stable Diffusion V2 of DDIM and RX-DDIM.
+
+![](images/c2f531f24f53f49fd3da20a9a1a5cd8ce3c888c7f339a7739c9712002272cd25.jpg)  
+Figure 14: Qualitative results on Stable Diffusion V2 of DDIM and RX-DDIM.

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+# ENHANCING COGNITION AND EXPLAINABILITY OF MULTIMODAL FOUNDATION MODELS WITH SELF-SYNTHESIZED DATA
+
+Yucheng Shi $^{1}$  Quanzheng Li $^{2}$  Jin Sun $^{1}$  Xiang Li $^{2}$  Ninghao Liu $^{1}$
+
+<sup>1</sup>School of Computing, University of Georgia
+
+$^{2}$ Department of Radiology, Massachusetts General Hospital and Harvard Medical School
+
+# ABSTRACT
+
+Large Multimodal Models (LMMs), or Vision-Language Models (VLMs), have shown impressive capabilities in a wide range of visual tasks. However, they often struggle with fine-grained visual reasoning, failing to identify domain-specific objectives and provide justifiable explanations for their predictions. To address the above challenge, we propose a novel visual rejection sampling framework to improve the cognition and explainability of LMMs using self-synthesized data. Specifically, visual fine-tuning requires images, queries, and target answers. Our approach begins by synthesizing interpretable answers that include human-verifiable visual features. These features are based on expert-defined concepts, and carefully selected based on their alignment with the image content. After each round of fine-tuning, we apply a reward model-free filtering mechanism to select the highest-quality interpretable answers for the next round of tuning. This iterative process of synthetic data generation and fine-tuning progressively improves the model's ability to generate accurate and reasonable explanations. Experimental results demonstrate the effectiveness of our method in improving both the accuracy and explainability of specialized visual classification tasks.
+
+# 1 INTRODUCTION
+
+Large Multimodal Models (LMMs) have recently demonstrated remarkable performance in visual descriptive tasks, excelling in applications such as image captioning and visual question answering (Liu et al., 2024a; Alayrac et al., 2022; Li et al., 2022; Liu et al., 2024b). However, recent research (Kim & Ji, 2024; Geigle et al., 2024) reveals a significant drop in their effectiveness when applied to knowledge-intensive, domain-specific visual classification tasks. We observe that even advanced models like LLaVA-1.5 (Liu et al., 2024a) can only achieve a mere  $12.2\%$  classification accuracy in the Stanford Dogs dataset (Khosla et al., 2011). Further investigation reveals that current LMMs struggle to utilize key visual features for reasoning in classification and fail to provide justifiable explanations for their predictions, as illustrated by the examples in Figure 1.
+
+User: Identify the bird species and explain your reasoning.
+
+![](images/d3b1f041b9930f596b9ae5d4fe7717089f4e37de93f4cbe5a07e7caf42eacf41.jpg)  
+LMM: The bird species shown in this image is a seagull. The reason for this identification is that the bird is flying over the ocean, which is a typical habitat for seagulls.  
+Label: Black footed Albatross  
+Figure 1: LLaVA-1.5 struggles to utilize key visual features in images for reasoning and explaining predictions in classification tasks.
+
+![](images/75eee3e08bef3a29679e1c853a3c9310716ae59dc89046b3aa84de553ec3735f.jpg)  
+LMM: The bird species is a black crow. The reason for this identification is that the bird is perched on a tree branch, which is a common behavior for crows.  
+Label: Groove billed Ani
+
+The core issue stems from insufficient domain-specific alignment, as the model struggles to recognize key visual features and link them to the correct labels. To address this problem, we propose enhancing the LMM's domain-specific cognition and explainability through fine-tuning (Touvron et al., 2023; Gu et al., 2021). However, this approach is hindered by a lack of data, as creating high-quality, feature-level image annotations is both complex and resource-intensive (Liu et al., 2024c). While labeling images by category and identifying key features for each class independent of the image is manageable, annotating the specific visual characteristics per image requires an extensive workload. Moreover, this level of detailed annotation goes beyond the capacity of standard annotators and current LMMs (Chen et al., 2024a), making it impractical to scale.
+
+The biggest challenge now is synthesizing high-quality training data, specifically interpretable target answers. Given a dataset with images and labels, a naive approach would be to use only labels as target answers. However, training on such data may result in shortcut learning, where models pick up spurious correlations instead of truly understanding key visual features (Geirhos et al., 2020). While including general label-associated features as target answers might seem beneficial, it often results in overly generic explanations that lack the image-specific details necessary for accurate interpretation. We illustrate these shortcomings with examples in Figure 2.
+
+To tackle this challenge, we propose a framework that allows LMMs to self-synthesize interpretable answers without relying on explicit image-specific annotations. For a given image, we first leverage the LMM's captioning ability to generate descriptions, which are then used to identify visual features relevant to that specific image. Each description may only cover part of the key features, but by collecting a large set of descriptions from the LMM, we can approximate the true distribution of the image's features, reducing the incompleteness in individual descriptions. We provide a formal justification for this approach in Section 3.2. Moreover, to ensure precise identification, we apply an information bottleneck technique to select the most relevant features. Once the image-specific concepts are identified, they are rewritten into interpretable answers.
+
+For the training procedure, we also design an iterative fine-tuning approach to further improve performance over a one-shot training scheme. We begin by extracting image-level features and transforming them into interpretable answers, which, together with the corresponding images and queries, form the initial training dataset. Fine-tuning on this data results in an updated model that can generate more accurate answers. The
+
+![](images/2e151d624d5845ff64d776fd8d02b00ceb44056fa65288116330aee9bcf1ab0e.jpg)
+
+Q: What is this bird's species? Explain your reasoning.
+
+Synthesized Answer
+
+![](images/a394a337a619a689b72926ea6ad51d3a52dff226e8b023ce9c7a422df2f8e82e.jpg)
+
+only label
+
+A: This is a Black Footed Albatross.
+
+#label and general label-level features
+
+A: ... Black Footed Albatross, because ... black feet; long, narrow wings; ..., hooked bill... dark eyes
+
+![](images/f81109e8f87842df04a5fbc33090196715df72b0f9b0402805b5579c42c6b2cc.jpg)
+
+A: ... a Black Footed Albatross based on its large, hooked bill, dark feathers, small, dark eyes on either side of its head ....
+
+Figure 2: Examples of synthetic answers for query Q. Training with the first two types leads to shortcut learning or overgeneralization.
+
+updated model is then used to repeatedly generate answers, with the best one selected for the next round of fine-tuning. This self-boosting process progressively improves the LMM's ability to deliver reliable explanations.
+
+In summary, our contributions are threefold: (1) We propose a novel framework that improves LMMs' interpretable visual classification abilities without requiring extensive manual labeling, (2) We introduce an information-theoretic approach to select interpretable visual concepts for each image and a reward model-free filtering method to ensure high-quality data selection from synthetic outputs, and (3) We develop an iterative process of data synthesis and model fine-tuning to progressively enhance LMMs' cognitive abilities and explainability.
+
+# 2 PRELIMINARY
+
+Problem Statement. We aim at developing a Large Multimodal Model (LMM) for explainable visual classification. Let  $f_{\theta}$  denote the LMM model,  $X$  be the input image, and  $q$  be the query prompt. The model's answer is denoted as  $\hat{y} = f_{\theta}(X,q)$ , where  $\hat{y}$  is expected to correctly predict the label and explain its prediction by using the visual features observed in the image. To build such a model, a straightforward strategy is to fine-tune the LMM with a training dataset that contains the ground-truth answer for each input image. However, most available datasets  $\mathcal{D} = \{(X_i,c_i)\}_{i=1}^N$  only consist of images  $X_i$  and class labels  $c_i$  (Khosla et al., 2011). To solve this, we propose a data synthesis approach that transforms the raw dataset into an augmented dataset  $\mathcal{D}^* = \{(X_i,q_i,y_i)\}_{i=1}^N$  with queries and explainable answers. While generating queries  $q_i$  is straightforward, the challenge lies in synthesizing explainable answers  $y_i$ , which must include detailed visual features  $Z_i^*$  that humans can identify and use for explanations. Therefore, the key problem is developing a method to automatically annotate visual features  $Z_i^*$  for each image  $X_i$ , given their labels  $c_i$ .
+
+Visual Fine-Tuning. Visual fine-tuning adapts a pre-trained LMM to understand specific visual information by training on image-text pairs. Typically, an LMM  $f_{\theta}$  consists of a vision encoder that extracts visual embeddings from an input image  $X$ , a projector that maps these embeddings into the language embedding space, and a language model that processes the combined visual and
+
+![](images/ab846451a16e8abf75af3c9ecbd502d02a5d8bc6b4d9bf6f683cd9e82575f535.jpg)  
+Figure 3: Our framework: An iterative approach of data synthesis and model fine-tuning.
+
+textual information (Liu et al., 2024b;a; Lin et al., 2024). Formally, given a round of conversation containing an image  $X$ , a question  $q$  and an answer  $y$ , the model is trained to maximize the likelihood of generating the target answer:  $\mathcal{L}(\theta) = \sum_{i=1}^{|y|} \log p_{\theta}(y^i | X, q, y^{<i})$ . Here,  $y^i$  is the  $i$ -th token of the answer  $y$ , and  $|y|$  is the length of the answer. The fine-tuning process typically optimizes performance by freezing the pre-trained visual encoder to preserve learned visual representations while updating the projector and language model parameters to improve language understanding for visual inputs (Liu et al., 2024b; Lin et al., 2024). Recognizing the critical role of fine-tuning data quality in model performance, our research proposes synthesizing high-quality conversation data to improve performance.
+
+# 3 METHODOLOGY
+
+# 3.1 OVERVIEW
+
+Our approach leverages synthetic data for visual fine-tuning to enhance both the cognitive ability and explainability of LMMs, as illustrated in Figure 3. There are two major steps:
+
+1. Image-level Visual Concept Selection: Given an image and its label, our first task is to extract a set of image-specific concepts that explain the connection between the image and label. We propose a selection method that identifies a subset of concepts most relevant to the image content while ensuring the subset is concise. Using these concepts, we prompt the LMM to rewrite them into textual answers that explain the label. These answers are used for the first round of fine-tuning.  
+2. Reward Model-Free Rejection Sampling: After the initial fine-tuning, the model can generate synthetic answers, which can be used for next-round fine-tuning, but their quality still varies. To filter out low-quality explanations and only select the best quality one as training data, we use previously selected concepts as filtering criteria, quantifying alignment between explanations and the concepts. The synthetic answer with the best-aligned explanation is then selected and combined with the corresponding image and query, forming a new data pair for subsequent rounds of tuning.
+
+# 3.2 STEP 1: IMAGE-LEVEL VISUAL CONCEPT SELECTION
+
+In this step, our goal is to identify the visual concepts present in a given image. Let  $X$  represent the true content of the image and  $c$  its class label. Each label class  $c$  is associated with a set of expert-defined visual concepts  $Z$ , which can be obtained by consulting domain experts or using a large language model. However, not all concepts in  $Z$  will necessarily be present in the image  $X$ . Therefore, we aim to select a subset of concepts,  $Z^{*} \subseteq Z$ , that are observable in the image  $X$ .
+
+To achieve this, we propose concept selection by leveraging the Information Bottleneck (IB) principle, which seeks a compressed representation that preserves maximal information about another variable (Tishby et al., 2000). In our context, we need to find  $Z^{*}$  that maximizes its mutual information with the image content  $X$ , i.e.,  $I(X;Z^{*})$ , while minimizing the redundancy by penalizing the mutual information between the selected concepts  $Z^{*}$  and the full concept set  $Z$ , i.e.,  $I(Z^{*};Z)$ .
+
+Formally, we define the optimization problem:
+
+$$
+Z ^ {*} = \arg \max  _ {Z ^ {\prime} \subseteq Z} \left[ I \left(X; Z ^ {\prime}\right) - \beta I \left(Z ^ {\prime}; Z\right) \right], \tag {1}
+$$
+
+where  $\beta$  is a Lagrange multiplier that balances relevance and redundancy. However, directly computing  $I(X;Z^{\prime})$  is intractable due to the high dimensionality and complexity of the image space. To address this, we introduce an intermediate variable: a set of image descriptions  $D = \{d_{1},d_{2},\ldots ,d_{n}\}$  generated by prompting an LMM (i.e., the base LLaVA-1.5 (Liu et al., 2024a)), with instructions like "Please describe the image." Each description  $d_{i}$  attempts to capture some aspects of the image content  $X$ . By increasing the number of collected descriptions  $n$  with different prompts, we aim to approximate the true distribution of  $X$ .
+
+This approach is analogous to assembling pieces of a puzzle: Each description provides partial information about the image, and collectively, they form a more complete representation. Similarly, multiple potentially partial descriptions generated from different prompts can collectively approximate the true image content. Under this intuition, we formalize the approximation in Theorem 1, with proof provided in the appendix.
+
+Theorem 1. Let  $X$  be the true image content with label  $c$  and  $D = \{d_1, d_2, \ldots, d_n\}$  be independent and identically distributed (i.i.d.) samples from  $P(D|X)$ . Let  $Z$  be an expert-defined concept list about label  $c$ . Under the assumptions of conditional independence and convergence (Assumptions 1 and 2), as  $n \to \infty$ , the mutual information  $I(D;Z)$  converges to  $I(X;Z)$ :
+
+$$
+\lim  _ {n \to \infty} I (D; Z) = I (X; Z).
+$$
+
+In practice, we cannot sample an infinite number of descriptions, and LMMs may generate inconsistent or contradictory descriptions due to hallucinations or uncertainties. To mitigate this, we design high-quality prompts and encourage diverse responses to improve the reliability of the generated descriptions. By doing so, we assume that most of the descriptions will accurately reflect the image content. This assumption is further validated through experiments in Section 4.4. From another perspective, if a feature is not consistently covered by image descriptions, it means that the model is not certain about its presence, which will naturally result in lower MI scores for the associated concepts, reducing their likelihood of being selected. Using the set of descriptions  $D$ , we reformulate the IB objective in Equation 1 as:
+
+$$
+Z ^ {*} = \arg \max  _ {Z ^ {\prime} \subseteq Z} \left[ I (D; Z ^ {\prime}) - \beta I \left(Z ^ {\prime}; Z\right) \right]. \tag {2}
+$$
+
+However, computing mutual information in high-dimensional space directly remains challenging. Therefore, we employ the InfoNCE loss (Oord et al., 2018) as a lower-bound estimator of mutual information. For each concept  $z_{j} \in Z$ , we calculate an InfoNCE score  $s_{j}$ :
+
+$$
+s _ {j} = \sum_ {d _ {i} \in D} \log \frac {\exp \left(\operatorname {s i m} \left(\mathbf {e} _ {d _ {i}} , \mathbf {e} _ {z _ {j}}\right) / \tau\right)}{\exp \left(\operatorname {s i m} \left(\mathbf {e} _ {d _ {i}} , \mathbf {e} _ {z _ {j}}\right) / \tau\right) + \sum_ {d _ {k} \in \bar {D}} \exp \left(\operatorname {s i m} \left(\mathbf {e} _ {d _ {k}} , \mathbf {e} _ {z _ {j}}\right) / \tau\right)}, \tag {3}
+$$
+
+where  $\mathrm{sim}(\cdot ,\cdot)$  denotes the cosine similarity between embeddings,  $\mathbf{e}_{d_i}$  and  $\mathbf{e}_{z_j}$  are the embeddings of description  $d_{i}$  and concept  $z_{j}$ , respectively, and  $\tau$  is a temperature parameter.  $\bar{D}$  are the descriptions for other images as negative samples. We can easily obtain the above embeddings through an off-the-shelf language embedding model (e.g., BERT (Devlin, 2018)). Next, we approximate  $I(D;Z^{*})$  as the sum of InfoNCE scores for the selected concepts:  $I(D;Z^{*})\approx \sum_{z_{j}\in Z^{*}}s_{j}$ . To minimize  $I(Z^{*};Z)$ , we selectively include concepts to reduce redundancy. Given that  $Z^{*}$  is a subset of  $Z$ , we have:  $H(Z^{*}) = I(Z^{*};Z) = -\sum_{z_{i}\in Z^{*}}p(z_{i})\log p(z_{i})$ , where  $H(Z^{*})$  is the entropy of  $Z^{*}$ . A selected subset  $Z^{*}$  with a smaller size and higher probabilities for its  $z_{i}$  elements will result in lower entropy. Combining these approximations, our selection criterion for  $Z^{*}$  becomes:
+
+$$
+Z ^ {*} = \left\{z _ {j} \in Z \mid s _ {j} > \mu + \hat {\beta} \sigma \right\}, \tag {4}
+$$
+
+where  $\mu$  and  $\sigma$  are the mean and standard deviation of the InfoNCE scores  $\{s_j\}_{j=1}^{|Z|}$ , respectively. The parameter  $\hat{\beta}$  controls the trade-off between including relevant concepts and avoiding redundancy. The selected concepts  $Z^{*}$  are not only relevant to the image but also capture the most informative features unique to the class label, providing strong evidence for the classification result and serving as reasonable explanations. Once we obtain  $Z^{*}$ , we can generate an explainable answer for a classification query on the image  $X$ . Specifically, we prompt the base LMM with these concepts to produce a coherent explanation. The prompts used are detailed in the appendix. After gathering the image and query-answer pairs, we can use them to fine-tune our LMMs.
+
+# 3.3 STEP 2: REWARD MODEL-FREE REJECTION SAMPLING
+
+After the initial round of fine-tuning with explainable visual query-answer pairs, the LMMs have significantly improved their ability to generate reasonable explanations. This improvement allows us to leverage the current fine-tuned model to generate new data for subsequent training rounds. However, the quality of newly generated data can vary considerably in terms of label accuracy and explanation quality. Training on low-quality data could lead to performance degradation. To address this issue, we propose a rejection sampling technique that filters out low-quality outputs.
+
+Rejection sampling, also known as Best-of-N, is an inference-time strategy that generates multiple candidates and selects the best one for further tuning (Touvron et al., 2023; Stiannon et al., 2020). In our work, we adapt this idea for visual fine-tuning to iteratively improve LMM performance. Our proposed rejection sampling process begins by generating a series of answer candidates, using the fine-tuned model  $f_{\theta}^{T}$  from the last round  $T$ , and then identify the best answer from these candidates. For language-only conversations, this selection is typically performed by a reward model, which assigns higher rewards to answers aligning with desired criteria (Touvron et al., 2023). The answer with the highest reward is then selected and used for the next round  $(T + 1)$  of training. However, in the visual domain, finding a reliable reward model remains challenging, as noted by (Chen et al., 2024a). To address this, we propose a reward model-free data filtering method to select the highest quality interpretable answers.
+
+Specifically, we leverage the selected concept set  $Z^{*}$  from Section 3.2 as a reference to evaluate explanation quality. Our aim is to select the answer that best aligns with these relevant concepts. Formally, let  $Y = \{y_{1}, y_{2}, \ldots, y_{m}\}$  represent the set of answers generated by the fine-tuned model for a given image. These answers are obtained by prompting the model with questions like "What is the [item] in this image? Please provide your reasoning." The "item" here is set to be an coarse-level label, like bird, airplane. Our goal is to select the answer  $y^{*} \in Y$  that maximizes the mutual information with  $Z^{*}$ :
+
+$$
+y ^ {*} = \arg \max  _ {y _ {i} \in Y} I \left(y _ {i}; Z ^ {*}\right). \tag {5}
+$$
+
+We approximate  $I(y_{i};Z^{*})$  using the InfoNCE score:
+
+$$
+s _ {i} ^ {\prime} = \sum_ {z _ {j} \in Z ^ {*}} \log \frac {\exp \left(\sin \left(\mathbf {e} _ {y _ {i}} , \mathbf {e} _ {z _ {j}}\right) / \tau\right)}{\exp \left(\sin \left(\mathbf {e} _ {y _ {i}} , \mathbf {e} _ {z _ {j}}\right) / \tau\right) + \sum_ {z _ {k} \in Z , z _ {k} \notin Z ^ {*}} \exp \left(\sin \left(\mathbf {e} _ {y _ {i}} , \mathbf {e} _ {z _ {k}}\right) / \tau\right)}, \tag {6}
+$$
+
+where  $\mathbf{e}_{y_i}$  is the embedding of answer  $y_{i}$ . We select the answer with the highest score:  $y^{*} = \arg \max_{y_{i}\in Y}s_{i}^{\prime}$ . The InfoNCE score provides a quantitative measure of how well the generated answer aligns with the relevant concepts. Therefore, our method eliminates the need for a separate reward model, which is particularly beneficial given the lack of reliable reward models for multimodal data (Chen et al., 2024a). Additionally, we add another empirical constraint: the selected answer should contain the correct label  $c$ ; otherwise, it will be discarded.
+
+In summary, our framework, outlined in Algorithm 1, enhances the model's fine-grained classification capabilities by iteratively fine-tuning on diverse, high-quality, synthetic visual classification query-answer pairs. Step 1 identifies the most informative concepts in the image, while Step 2 selects the explanations that best align with these concepts. This two-step approach improves both the accuracy and interpretability of the model's predictions, enabling it to perform more effectively in complex visual classification tasks.
+
+# 3.4 THEORETICAL JUSTIFICATION
+
+Our approach is grounded in information theory, specifically the maximization of mutual information among the image  $X$ , the expert-defined concepts  $Z$  for the whole class, and the generated answer  $Y$  with explanations. Theoretically, the problem of finding the best  $Y^{*}$  is equivalent to maximizing the three-way mutual information:
+
+$$
+\max  _ {Y ^ {*}} I (X; Y; Z), \tag {7}
+$$
+
+where  $Y^{*}$  captures the maximum amount of information from both the image and the expert-defined concepts. However, directly optimizing this objective is intractable due to the high dimensionality of  $X$  and  $Y$  (Poole et al., 2019). To make the problem manageable, we decompose it into two subproblems corresponding to our two-step method.
+
+In Step 1, we select a subset of concepts  $Z^{*} \subseteq Z$  that maximizes  $I(X;Z^{*})$ , capturing the most relevant concepts for the image. This aligns with the Information Bottleneck principle, where  $Z^{*}$  serves as a distilled subset of  $Z$  that retains maximal information about  $X$ . In Step 2, we select the generated explanation  $Y^{*}$  that maximizes  $I(Y^{*};Z^{*})$ , ensuring that the generated explanation closely aligns with the selected concepts. With this decomposition, we establish a lower bound on the three-way mutual information (proof provided in appendix).
+
+Theorem 2. Let  $X, Y$ , and  $Z$  be discrete random variables. Define:
+
+$$
+Z^{*} = \operatorname *{arg  max}_{Z^{\prime}\subseteq Z}I(X;Z^{\prime}) - \beta I(Z^{\prime};Z)
+$$
+
+$$
+Y^{*} = \operatorname *{arg  max}_{\substack{Y^{\prime}\subseteq Y}}I(Y^{\prime};Z^{*})
+$$
+
+Then, the following inequality holds:
+
+$$
+I (X; Y; Z) \geq
+$$
+
+$$
+I (Y ^ {*}; Z ^ {*}) + I (X; Z ^ {*}) - I (Z ^ {*}; Z).
+$$
+
+This theoretical foundation justifies our two-step optimization process, ensuring that our method effectively captures the essential information shared among the image, concepts, and explanations.
+
+# 4 EXPERIMENTS
+
+Algorithm 1 Iterative synthesis and fine-tuning.  
+Ensure: Fine-tuned LMM  $f_{\theta}$  for accurate, interpretable explanations  
+```txt
+Require: Dataset  $\mathcal{D} = \{(X_i,c_i)\}_{i = 1}^N$  , Concept Sets  $\mathcal{Z} = \{Z_i\}_{i = 1}^{|C|}$  of every label, Pretrained LMM  $f_{\theta}^{0}$  , synthetic query set  $Q = \{q_{i}\}_{i = 1}^{N}$  , Descriptive query  $q_{d}$
+```
+
+2: for each  $(X_{i},c_{i})\in \mathcal{D}$  do  
+```txt
+1: Initialize training set  $\mathcal{D}^* \gets \emptyset$
+```
+
+```txt
+3:  $Z \gets \mathcal{Z}[c_i]$
+```
+
+```txt
+4:  $D_{i}\gets f_{\theta}^{0}(X_{i},q_{d})$
+```
+
+```txt
+5: for  $z_{j}\in Z$  do
+```
+
+```javascript
+6:  $s_{ij} \gets \operatorname{InfoNCE}(D_i, z_j)$
+```
+
+```txt
+7: end for
+```
+
+```javascript
+8:  $\mu_{i}\gets \mathrm{mean}(\{s_{ij}\}),\sigma_{i}\gets \mathrm{std}(\{s_{ij}\})$
+```
+
+```latex
+9:  $Z_{i}^{*}\gets \{z_{j}\in Z\mid s_{ij} > \mu_{i} + \beta \sigma_{i}\}$
+```
+
+```latex
+10:  $y_{i}\gets$  assemble  $\left(Z_i^*\right)$
+```
+
+```txt
+11:  $\mathcal{D}^* \gets \mathcal{D}^* \cup \{(X_i, q_i, y_i)\}$
+```
+
+```txt
+12: end for
+```
+
+```txt
+13: FT: Fine-tune  $f_{\theta}^{0}$  on  $\mathcal{D}^*$  to obtain  $f_{\theta}^{1}$
+```
+
+```txt
+14: repeat
+```
+
+```txt
+15: for each  $(X_{i, - })\in \mathcal{D}$  do
+```
+
+```latex
+16:  $Y_{i}\gets f_{\theta}^{T}(X_{i},q_{i})$
+```
+
+```txt
+17: for  $y_{ij}\in Y_i$  do
+```
+
+```latex
+18:  $s_{ij}^{\prime}\gets \mathrm{InfoNCE}(Z_i^*,y_{ij})$
+```
+
+```txt
+19: end for
+```
+
+```txt
+20:  $y_{i}^{*}\gets \arg \max_{y_{ij}}s_{ij}^{\prime}$
+```
+
+```txt
+21:  $\mathcal{D}^* \gets \mathcal{D}^* \cup \{(X_i, q_i, y_i^*)\}$
+```
+
+```txt
+22: end for
+```
+
+```txt
+23: Iterative FT: Fine-tune  $f_{\theta}^{T}$  on  $\mathcal{D}^*$
+```
+
+```txt
+24: until Max iterations  $T^{m}$
+```
+
+```txt
+25: return  $f_{\theta}^{m}$
+```
+
+We conduct experiments to address the following questions. Q1: How effectively does our method improve performance on fine-grained visual classification tasks? Q2: How does our method enhance the explainability of the fine-tuned model? Q3: How does our concept selection strategy compare to baseline methods? Q4: Does our training scheme mitigate shortcut learning? Q5: How usable is our trained model?
+
+# 4.1 EXPERIMENTS SETTINGS
+
+Implementation Details: We evaluated our approach on a variety of datasets, including fine-grained classification datasets (CUB-200 (Wah et al., 2011), Stanford Dogs (Khosla et al., 2011), FGVC-Aircraft (Maji et al., 2013)), medical datasets (HAM10000 (Tschandl et al., 2018), Chest X-Ray for Pneumonia (Kermany et al., 2018)), and the Plant Disease Dataset (PLD) (Vipoooool, 2020). By testing on datasets from diverse domains, we demonstrate the versatility of our proposed framework with self-synthesized data. Our experiments used Llava-1.5-7B (Liu et al., 2024a) as the base LMM, and for the Chest X-Ray datasets, we employed its medical version (Li et al., 2024b). Additionally, E5 (Wang et al., 2022) served as the embedding model. We fine-tuned the LMMs using LoRA, focusing on all linear layers. Training was conducted on 8 H100 GPUs, utilizing HuggingFace (Wolf, 2019) and DeepSpeed frameworks for efficient distributed training and optimization. Further details are provided in the appendix. We also provide our code here<sup>1</sup>.
+
+Baselines: Given the novel problem of generating interpretable answers without image-specific annotations, we design the following baselines for comparisons: (1) Base LMM: Assesses the base multimodal model's performance in the zero-shot setting. (2) Naive Label Fine-tuning (NL):
+
+Table 1: Our method achieves superior accuracy and explanation quality across diverse datasets.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Method</td><td colspan="4">Accuracy ↑ Per Iteration</td><td colspan="3">Explanation Quality</td><td>General Ability</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>EE ↑</td><td>CS ↑</td><td>FS ↓</td><td>MMMU ↑</td></tr><tr><td rowspan="4">CUB-200</td><td>Base</td><td>2.69</td><td>-</td><td>-</td><td>-</td><td>0.92</td><td>0.67</td><td>4.28</td><td>35.56</td></tr><tr><td>NL</td><td>73.42</td><td>78.25</td><td>79.94</td><td>82.21</td><td>0.00</td><td>-</td><td>-</td><td>35.67</td></tr><tr><td>L+GE</td><td>61.48</td><td>72.23</td><td>73.23</td><td>73.06</td><td>1.00</td><td>0.70</td><td>6.84</td><td>34.89</td></tr><tr><td>Ours</td><td>80.24</td><td>83.76</td><td>84.69</td><td>85.02</td><td>1.00</td><td>0.82</td><td>6.53</td><td>35.00</td></tr><tr><td rowspan="4">Stanford dogs</td><td>Base</td><td>12.2</td><td>-</td><td>-</td><td>-</td><td>0.94</td><td>0.69</td><td>5.47</td><td>35.56</td></tr><tr><td>NL</td><td>82.73</td><td>82.34</td><td>84.03</td><td>84.27</td><td>0.00</td><td>-</td><td>-</td><td>34.67</td></tr><tr><td>L+GE</td><td>73.45</td><td>77.89</td><td>78.15</td><td>76.55</td><td>1.00</td><td>0.77</td><td>7.50</td><td>34.56</td></tr><tr><td>Ours</td><td>85.29</td><td>86.75</td><td>86.86</td><td>86.91</td><td>1.00</td><td>0.86</td><td>7.41</td><td>34.56</td></tr><tr><td rowspan="4">FGVC-A</td><td>Base</td><td>3.00</td><td>-</td><td>-</td><td>-</td><td>0.97</td><td>0.42</td><td>5.39</td><td>35.56</td></tr><tr><td>NL</td><td>83.47</td><td>87.28</td><td>87.82</td><td>87.73</td><td>0.00</td><td>-</td><td>-</td><td>35.56</td></tr><tr><td>L+GE</td><td>72.13</td><td>79.87</td><td>82.45</td><td>82.69</td><td>1.0</td><td>0.76</td><td>8.59</td><td>35.56</td></tr><tr><td>Ours</td><td>88.78</td><td>90.91</td><td>91.42</td><td>91.99</td><td>1.0</td><td>0.79</td><td>7.00</td><td>37.33</td></tr><tr><td rowspan="4">PLD</td><td>Base</td><td>0.00</td><td>-</td><td>-</td><td>-</td><td>0.95</td><td>-</td><td>-</td><td>35.56</td></tr><tr><td>NL</td><td>89.38</td><td>94.52</td><td>94.29</td><td>93.95</td><td>0.00</td><td>-</td><td>-</td><td>34.78</td></tr><tr><td>L+GE</td><td>24.03</td><td>25.27</td><td>24.56</td><td>24.90</td><td>1.00</td><td>0.76</td><td>10.45</td><td>35.44</td></tr><tr><td>Ours</td><td>75.96</td><td>92.80</td><td>96.59</td><td>97.16</td><td>1.00</td><td>0.86</td><td>9.01</td><td>35.22</td></tr><tr><td rowspan="4">HAM10000</td><td>Base</td><td>1.62</td><td>-</td><td>-</td><td>-</td><td>0.98</td><td>0.63</td><td>3.93</td><td>35.56</td></tr><tr><td>NL</td><td>77.28</td><td>80.75</td><td>82.49</td><td>81.71</td><td>0.00</td><td>-</td><td>-</td><td>35.33</td></tr><tr><td>L+GE</td><td>7.47</td><td>8.83</td><td>9.35</td><td>8.45</td><td>1.00</td><td>0.94</td><td>9.68</td><td>35.22</td></tr><tr><td>Ours</td><td>79.37</td><td>82.29</td><td>83.69</td><td>85.06</td><td>1.00</td><td>0.87</td><td>7.43</td><td>35.89</td></tr><tr><td rowspan="4">Chest X-ray Pneumonia (LLaVA-Med)</td><td>Base</td><td>62.50</td><td>-</td><td>-</td><td>-</td><td>1.00</td><td>0.24</td><td>3.49</td><td>-</td></tr><tr><td>NL</td><td>85.58</td><td>89.10</td><td>85.90</td><td>89.58</td><td>0.00</td><td>-</td><td>-</td><td>-</td></tr><tr><td>L+GE</td><td>62.50</td><td>62.50</td><td>62.98</td><td>62.66</td><td>1.00</td><td>0.79</td><td>7.19</td><td>-</td></tr><tr><td>Ours</td><td>97.60</td><td>96.31</td><td>99.04</td><td>98.72</td><td>1.00</td><td>0.87</td><td>8.25</td><td>-</td></tr></table>
+
+Base: original model; NL: only train with labels; L+GE: train with labels and general explanations
+
+Fine-tunes the base model using only class labels and a simple template (e.g., "This is a picture of {label}"). (3) Label with General Explanations (L+GE): We adopt the data synthesis approach introduced in the LLaVA paper (Liu et al., 2024b) and (Kim & Ji, 2024), where training data is generated using a language-only model (e.g., GPT-4), based on class labels and their corresponding label-level knowledge. More details, including the prompts, are in the appendix.
+
+# 4.2 RQ1: TRAINING ON OUR SYNTHETIC DATA IMPROVES CLASSIFICATION
+
+We evaluated our model's classification capabilities using a multi-round, progressive rejection sampling training process comprising four iterations, each with two epochs. Classification accuracy was measured after each iteration, with success defined as the presence of the ground truth label in the model's response (Kim & Ji, 2024). All trainable baselines were trained and evaluated under identical settings to ours for a fair comparison. More details on training are provided in the appendix.
+
+Our proposed method achieves higher accuracy than both NL and L+GE baselines, as shown in Table 1. While the baselines, particularly NL, exhibit overfitting, our method improves accuracy with each iteration, demonstrating its effectiveness in fine-grained classification tasks. Our approach's resilience to overfitting stems from the use of rejection sampling, which generates a more diverse training dataset. This expanded data pool enhances the model's generalization capabilities. Moreover, training with our dataset, which includes specific visual features, helps the model learn more detailed visual knowledge, contributing to further gains in classification performance. In contrast, the L+GE baseline struggles with certain datasets, notably HAM10000, likely due to irrelevant information in general explanations impeding effective learning. Moreover, we assessed the general ability of the models using the MMMU metric (Yue et al., 2024) evaluated by the LMMs-eval tool (Zhang et al., 2024). The results indicate that all trained models maintain comparable general abilities to the base model, with negligible degradation. Note that the LMMs-eval tool does not support the LLaVA-Med model; therefore, the general ability metric is not reported. More experiment results can be found in the appendix.
+
+![](images/4dc17a8b10afa004b4bfb9a348b7858b5bd7fe41932662f0dc0302a64a3cf424.jpg)  
+Ours: The specific name of this bird is the Rusty Blackbird. The Rusty Blackbird is a medium-sized blackbird with a slightly curved, slender black bill. It is known for its rusty or brownish tones on its body, particularly in the non-breeding season. The bird is often seen in large flocks and is commonly found in wetland habitats.
+
+![](images/4b3d4e22d78cec35974c8e376f843714a852c6b6f89cc80630fdae06908e7573.jpg)  
+Ours: "The specific name of this dog is a Tzu. The Tzu is a small, compact dog with a square-like frame, short legs, and a short muzzle. The dog's hair is often tied up with a bow on top of the head, and it has a distinctive undershot bite."
+
+![](images/40556401730d72f05fc082222886ac7df772c8fbe8613bc6a872deb238b4fbd0.jpg)  
+Ours: The airplane shown in this image is a Cessna Citation. This conclusion is based on the visual features of the aircraft, such as the short, rounded wing roots connecting to the fuselage, the low-wing design, the distinctive nose with a slight downward curve, and the retractable tricycle landing gear. These features are characteristic of Cessna Citation aircraft, which are known for their performance, comfort, and reliability.
+
+Figure 4: Our generated answers contain detailed visual explanations.  
+![](images/85b1535191e079412d36b0ddda05338d165c9a1121ee0488bceec29798fb34fc.jpg)  
+Ours: The specific name of this bird is the Western Meadowlark. The Western Meadowlark is a small bird with a bright yellow underbelly, a black "V" shape on the chest, and a long, pointed bill. It is commonly found in the grasslands and meadows of the western United States.
+
+![](images/b67502a4fdb1188c8fffdbee46835813470d9a8e535e5c7393c68023b439e202.jpg)  
+Ours: "The specific name of this dog is a Tibetan Terrier. This breed is known for its distinctive appearance, which includes a wide range of colors, a pronounced beard and moustache, and a square proportioned body. The dog in the image has a black and white coat, which is a common color pattern for this breed."
+
+![](images/26d38e6ae894ca12bac2b4f9bae0ac70fbbe9c2d03e8a005a6beae013505a0c4.jpg)  
+Ours: The airplane shown in this image is a Canadian CRJ-900. This conclusion is based on the following visual features: $^{a1}$  n1. The presence of two engines mounted on the rear fuselage, which is a characteristic feature of the CRJ-900 $^{a2}$  n1. The slanted cockpit windshield with a modern, streamlined appearance, which is also a characteristic feature of the CRJ-900 $^{a3}$  n1. The long, narrow fuselage with a sleek, aerodynamic design, which is also a common feature of the CRJ-900 $^{a4}$
+
+# 4.3 RQ2: OUR METHOD PROVIDES HIGH-QUALITY EXPLANATION
+
+Assessing the quality of generated explanations is challenging, especially without case-by-case ground truth annotations (Ding et al., 2022; Schuff et al., 2022). Following Mohseni et al. (2021), we evaluate our explanation from three aspects: explanation existence, cognition level, and fluency.
+
+Explanation Existence (EE). This metric assesses a model's capability to generate explanations (Xu et al., 2023). Following Bills et al. (2023); Bricken et al. (2023), we employ an advanced proprietary LLM (GPT-4o) to determine whether each model-generated answer includes an explanation. Formally, for a set of generated answers  $Y = \{y_{1}, y_{2}, \ldots, y_{n}\}$  queried by prompt like "What is the bird name? Provide your reason," EE is defined as:  $\mathrm{EE} = \frac{1}{n} \sum_{i=1}^{n} e_{i}$ , where  $e_{i} = 1$  if the  $i^{\text{th}}$  answer  $y_{i}$  includes an explanation, and  $e_{i} = 0$  otherwise. We provide the evaluation prompt in the appendix.
+
+Results: As shown in Table 1, our method achieves an EE of 1.00 across all datasets, indicating that it consistently produces explanations for its predictions. The base model can generate explanations at most times but fails to do so in some cases. In contrast, the NL baseline records an EE of 0.00, reflecting its inability to generate explanations due to training solely on class labels. The L+GE method also attains an EE of 1.00 but falls short in other quality metrics compared to our approach.
+
+Cognition Score (CS). This metric evaluates the coherence and logical flow of generated explanations (Nourani et al., 2019; Fan et al., 2020). Following Liu et al. (2024e); Bills et al. (2023); Lieberum et al. (2024), we employ an advanced proprietary LLM (GPT-4o) to assess the rational integrity of explanations by analyzing their alignment with expert knowledge. For an answer  $y_{i}$  containing a label and explanation, we first extract label-level concepts  $Z$  corresponding to the label. We then use an evaluation prompt to obtain a cognition score from LLM:  $c s_{i} = \mathrm{LLM}(y_{i}, Z, \mathrm{Eval\_Prompt})$ . The Eval\_Prompt is detailed in the appendix. Scores range from 0 to 1, with higher scores indicating better alignment with expert-predefined knowledge. We only assess correct answers. The total CS is calculated as:  $\mathrm{CS} = \frac{1}{|Y_c|} \sum_{i=1}^{|Y_c|} c s_{i}$ , where  $Y_{c}$  is the set of correct answers.
+
+Results: Our method generally achieves higher CS values than the baselines, demonstrating superior coherence and logical reasoning in the explanations. This improvement reflects the effectiveness of our approach in generating explanations that are more useful and trustworthy to human users. The reason behind our better cognition is that we select image-level concepts to build the training dataset, rather than using all label-level features. This ensures that our model focuses on relevant features present in the image and effectively connects the image content with expert-defined concepts, avoiding distractions from irrelevant features not shown in the image.
+
+Fluency Score (FS). This metric evaluates the grammatical correctness and naturalness of the explanations (Liu et al., 2021). We use perplexity, a standard metric that measures how well a language model predicts the text, with lower values indicating better fluency.
+
+Results: Our method demonstrates good fluency in the generated explanations. The base model receives the best fluency. This is because training on expert-level knowledge inevitably increases the complexity of explanations, which can lead to higher perplexity scores. However, our method generally achieves lower perplexity than the L+GE baseline, indicating that our explanations are more natural, highlighting the robustness of our approach.
+
+In summary, our proposed method not only enhances classification accuracy but also consistently generates high-quality explanations that are coherent, logical, and fluent. We also show some quantitative results in our explanation in Figure 4. More results can be found in the appendix.
+
+# 4.4 RQ3: EVALUATION ON INTERPRETABLE CONCEPTS SELECTION
+
+To evaluate the efficacy of our proposed model in selecting expert-defined features from images, we conducted a comprehensive study with human experts. Experts were invited to identify and annotate the top-4 relevant concepts for each image from a dataset of six bird species, providing ground truth annotations for our evaluation. Our investigation examined two primary aspects: the effect of varying the number of descriptions on concept selection performance, and a comparative analysis against several baseline methods. These baselines include GPT40 for concept extraction, which relied on a carefully crafted prompt to guide the model in identifying and returning the four most probable concepts for each image. We applied the same approach to the LLaVA model for concept extraction. In contrast, for the CLIP model, concept selection was performed by identifying top-4 concepts with the highest CLIPScore (Hessel et al., 2021) relative to the target image.
+
+![](images/1dd981ee36fc8438052a548b5ae97629b445d0e41354e6c01419cada87746f14.jpg)  
+Figure 5: Our method demonstrates superior precision in concept selection compared to applying GPT-4o.
+
+We evaluated performance using precision, comparing model-selected concepts with human-annotated ground truth. Results demonstrated that our model outperforms strong baselines in the concept selection task. Its precision improves proportionally with the increased number of descriptions, peaking at  $72.89\%$  with 25 descriptions. In contrast, GPT-4o maintained a relatively high precision of  $63.95\%$ , while both LLaVA and CLIP showed weaker performances at approximately  $55\%$ . These findings highlight our model's superior ability to leverage repeated sampled descriptions for more accurate concept selection. We show some qualitative results in Figure 6, with more in the appendix.
+
+![](images/ab3b5aa7476f79c14f0432f3211e8970225a0f0222aa3c13899a2369c6df9709.jpg)  
+$\succ$  White crescent-shaped markings around the eyes  $\succ$  Dark sooty-brown plumage  
+$\succ$  Dusky, almost black head
+
+![](images/f7753cada7c17c0ca6261295c6d8f598a281a349265a193eb0633e85d7a16262.jpg)  
+$\succ$  Deep-set facial folds  
+> Underbite, often noticeable when the mouth is closed  
+> Wrinkled short-muzzled face  
+$\succ$  Distinctive facial expression, often characterized as "worried" or "thoughtful"  $\succ$  Round, prominent eyes
+
+![](images/e452e44128081c3cfa201ada787e0488ea930f4c4b3d22a6701b198b595c9891.jpg)  
+$\succ$  Yellowing of leaf margins and interveinal areas  
+> Overall reduced leaf size compared to healthy leaves  
+$\succ$  Curling and distortion of leaf edges  
+$\succ$  Thickening of leaf texture  $\succ$  Curling and distortion of leaf edge
+
+Sooty Albatross  
+![](images/e18131ac1f853232c33996ade4ca280d85240a949384bf81345c3941f41a2635.jpg)  
+> Long, slender wings  
+> Streamlined body  
+Often seen gliding with stiff wing posture  
+> Subtle light streaks on upper wings
+
+Pug  
+![](images/13b141e45a74cf3bfbafaef61863740513857cc0ba0916727948b506e6d5a0af.jpg)  
+> Short, glossy coat  
+> Small, floppy ears, either rose-shaped or button-shaped  
+> Short, straight limbs  
+> Underbite, often noticeable when the mouth is closed  
+> Wrinkled short-muzzled face  
+> Square, sturdy build
+
+Tomato Yellow Leaf Curl Virus Disease  
+Figure 6: Different image-level visual concepts for objects with the same label.  
+![](images/d2f28c49d45c207ce8648452eedb7992dba4004ae4d52520831513d64fb1c04e.jpg)  
+> Increased prominence of veins, sometimes appearing as a mosaic  
+$\succ$  Upward or downward cupping of leaves  
+> Leaves may appear more rigid and less flexible
+
+# 4.5 RQ4: VISUALIZATION OF VISUAL FOCUS OF LMMs
+
+To interpret our model's predictions, we visualize the outputs of LMMs using a gradient-based explanation method (Wu et al., 2023). This approach identifies the image regions most influential in generating the model's answer by computing gradients of answer token probabilities with respect to image pixels. Figure 7 presents case examples demonstrating that training on our synthetic answers with cognitive explanations can effectively prevent the model from learning spurious features, whereas directly linking images to naive labels may result in shortcut learning.
+
+# 4.6 RQ5: EVALUATION ON USABILITY
+
+Previous LMMs often struggle with specialized questions due to difficulty recognizing domain-specific objects. Our method enhances LMMs' ability to identify specialized objects and provide verifiable visual details, as demonstrated in Table 10. In Case 1, the LMM accurately identifies a bell pepper leaf disease, describing specific visual cues such as brown necrotic spots and chlorosis. It then recommends an appropriate treatment. In Case 2, the LMM correctly identifies a male Northern Cardinal, noting distinctive features like its bright red plumage and black mask. These improvements enable LMMs to serve as effective visual assistants in specialized domains, offering users accurate and interpretable responses that can be verified against the provided images.
+
+![](images/8fa723bffca109c3f03f38c49233c7bd1442cc87aadbf855ba28e9467174b4b9.jpg)  
+Figure 7: Visualization of heatmaps for different datasets. From left to right in each triplet: Original Image, Heatmap of NL model, Heatmap of Ours.
+
+![](images/6f2ecd8580333d89e520958342fee57a92d19f29b92047572e031dd24f11e863.jpg)
+
+![](images/428df14978721073a02548e38ccd342550bb4c65dbdeed62d78aa5ffc932a689.jpg)
+
+![](images/97b2f6eee6037839219441ebdc5d97b1bcf6270832b91311258ec6438d17b32e.jpg)
+
+![](images/7f62370a7c31f6a3d255ce8e48516ae489ed01155a66b7812b9cac862d2f446d.jpg)
+
+![](images/da1c409603346e813b5e1ef42f2e8814dbd478b41a38ab05ab805850e9ac54bb.jpg)
+
+![](images/61dd21cf677d04ba625fc526a81c6865b77292f57cd5cffb1c724d16cf2a552e.jpg)
+
+Table 2: LMMs fine-tuned with our method provide verifiable details, enhancing user trust.  
+
+<table><tr><td>User: LMM:</td><td rowspan="2">What is the name of this leaf disease? Can you give me your reason? The name of this leaf disease is Bell Pepper Leaf with Bacterial Spot Disease. The reason for this name is that the leaf has brown or black necrotic spots, which are characteristic of bacterial infections. These spots can coalesce, forming larger necrotic areas, and the leaf may exhibit chlorosis or yellowing of surrounding leaf tissue. The spots are often located on the underside of the leaf, and the leaf may become crinkled or deformed near affected areas. OK then, what chemical should I use to treat this disease? It is recommended to use a copper-based fungicide to treat this disease.</td></tr><tr><td>User: LMM:</td></tr><tr><td>User: LMM:</td><td>Identify this bird. What features led to your conclusion? The bird is a male Northern Cardinal with a bright red plumage on its head, throat, and chest. The presence of a crest on its head and a black &quot;mask&quot; around its eyes further supports the identification of this bird. Where can I find this bird? You can find this bird in the United States, specifically in the eastern and central parts of the country. The Northern Cardinal is a common resident bird in areas with dense woodlands, shrubs, and gardens.</td></tr></table>
+
+# 5 RELATED WORK: SELF-IMPROVEMENT AND DATA SYNTHESIS
+
+In the field of LLMs, self-generated data has become a powerful tool for enhancing model performance (Liu et al., 2024d). Researchers have explored various techniques, such as rejection sampling (Touvron et al., 2023; Dubey et al., 2024), self-rewarding (Yuan et al., 2024), and self-play (Chen et al., 2024c), enabling models to improve using synthetic data. Recent studies (Snell et al., 2024; Brown et al., 2024) have proposed the "inference-time scaling law", suggesting that increasing inference samples size boosts the likelihood of generating high-quality data. Data synthesis techniques have also been applied to LMMs, improving general vision tasks like visual question answering (Zhou et al., 2024a; Deng et al., 2024; Gao et al., 2023; Zhou et al., 2024b) and enhancing instruction-following capabilities (Luo et al., 2024). Our work extends this line of research by focusing on domain-specific visual classification to enable effective visual assistance in professional tasks. In contrast to existing methods, we address the unique challenges of data synthesis in specialized domains, thereby extending these techniques to support expert-driven applications.
+
+# 6 CONCLUSION
+
+In this work, we addressed LMMs' limitations in domain-specific visual classification tasks by introducing a novel framework that enhances their cognition and explainability through iterative fine-tuning with self-synthesized interpretable answers. By applying the IB principle to select relevant visual concepts without extensive image-specific annotations, our approach significantly improved classification accuracy and explanation quality across various datasets. By enabling LMMs to generate accurate, interpretable explanations grounded in domain-specific visual features, our framework advances their applicability in specialized domains, paving the way for more reliable multimodal models in knowledge-intensive applications. Future work may explore more complex tasks than classification tasks, and refinements to enhance scalability and generalizability.
+
+# ACKNOWLEDGMENTS
+
+The work is, in part, supported by NSF (#IIS-2223768). Dr. Xiang Li is supported by the Google Research Scholar Program. The views and conclusions in this paper are those of the authors and should not be interpreted as representing any funding agencies.
+
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+
+# A THEORETICAL PROOF ON THEOREM 1
+
+Theorem 3. Let  $X$  be the true image content with label  $c$  and  $D = \{d_1, d_2, \ldots, d_n\}$  be independent and identically distributed (i.i.d.) samples from  $P(D|X)$ . Let  $Z$  be an expert-defined concept list about label  $c$ . Under the assumptions of conditional independence and convergence (Assumptions 1 and 2), as  $n \to \infty$ , the mutual information  $I(D;Z)$  converges to  $I(X;Z)$ :
+
+$$
+\lim  _ {n \to \infty} I (D; Z) = I (X; Z).
+$$
+
+Proof. We proceed with the following steps:
+
+# A.1 ASSUMPTIONS
+
+Assumption 1. Conditional Independence:  $d_{1}, d_{2}, \ldots, d_{n}$  are conditionally independent of  $Z$  given  $X$ .
+
+Assumption 2. Convergence of Mutual Information: As  $n \to \infty$ , the mutual information between  $D$  and  $X$  converges to the entropy of  $X$ :
+
+$$
+\lim  _ {n \to \infty} I (D; X) = H (X)
+$$
+
+# A.2 PROOF STEPS
+
+1. Mutual Information Expression: We start with the definition of mutual information:
+
+$$
+I (D; Z) = H (D) - H (D | Z)
+$$
+
+2. Expanding  $H(D|Z)$  Using the Chain Rule: Apply the chain rule for entropy:
+
+$$
+H (D | Z) = H (D | Z, X) + H (X | Z) - H (X | D, Z)
+$$
+
+Substituting this back into the mutual information expression:
+
+$$
+I (D; Z) = H (D) - [ H (D | Z, X) + H (X | Z) - H (X | D, Z) ]
+$$
+
+3. Applying Conditional Independence: By Assumption 1, we have:
+
+$$
+H (D | Z, X) = H (D | X)
+$$
+
+Therefore,
+
+$$
+I (D; Z) = H (D) - H (D | X) - H (X | Z) + H (X | D, Z)
+$$
+
+Recognizing  $I(D;X) = H(D) - H(D|X)$ , we have:
+
+$$
+I (D; Z) = I (D; X) - H (X | Z) + H (X | D, Z)
+$$
+
+4. Taking the Limit as  $n\to \infty$  Apply the limit to both sides:
+
+$$
+\lim  _ {n \rightarrow \infty} I (D; Z) = \lim  _ {n \rightarrow \infty} \left[ I (D; X) - H (X | Z) + H (X | D, Z) \right]
+$$
+
+By Assumption 2, we have:
+
+$$
+\lim  _ {n \rightarrow \infty} I (D; X) = H (X)
+$$
+
+For the term  $H(X|D,Z)$ , we argue that:
+
+$$
+\lim  _ {n \rightarrow \infty} H (X | D, Z) = 0
+$$
+
+This follows from Assumption 2, as it implies that  $D$  becomes a sufficient statistic for  $X$  as  $n \to \infty$ . Therefore, conditioning on  $Z$  does not add any information about  $X$  once we have  $D$ .
+
+5. Conclusion: Substituting these limits into our equation:
+
+$$
+\lim  _ {n \rightarrow \infty} I (D; Z) = H (X) - H (X | Z) + 0 = I (X; Z)
+$$
+
+This demonstrates that as the number of sampled descriptions  $n$  increases indefinitely, the mutual information between the aggregated descriptions  $D$  and the human concepts  $Z$  converges to the mutual information between the true image content  $X$  and  $Z$ .
+
+# A.3 ADDITIONAL NOTES
+
+# A.3.1 CONVERGENCE OF  $I(D;X)$  TO  $H(X)$
+
+The statement:
+
+$$
+\lim _ {n \to \infty} I (D; X) = H (X)
+$$
+
+This is directly from Assumption 2. To understand its implications:
+
+- Recall that mutual information is defined as:  $I(D;X) = H(X) - H(X|D)$  
+- For this equality to hold as  $n \to \infty$ , it must be true that:
+
+$$
+\lim  _ {n \to \infty} H (X | D) = 0
+$$
+
+- This means that as we gather more samples  $(D)$ , we eliminate all uncertainty about  $X$ .  
+- In other words, with infinite samples,  $D$  contains all information about  $X$ .
+
+# A.3.2 CONVERGENCE OF  $H(X|D,Z)$  TO 0
+
+The statement:
+
+$$
+\lim  _ {n \to \infty} H (X | D, Z) = 0
+$$
+
+This follows from the previous point. Here's the reasoning:
+
+- We've established that as  $n \to \infty$ ,  $D$  contains all information about  $X$ .  
+- This means  $D$  becomes a sufficient statistic for  $X$ .  
+- A sufficient statistic contains all the information that the sample provides about the parameter (in this case,  $X$ ).  
+- Therefore, once we know  $D$ , knowing  $Z$  doesn't provide any additional information about  $X$ .  
+- Mathematically, this means:  $H(X|D,Z) = H(X|D)$ .  
+- But we know from the first part that  $\lim_{n\to \infty}H(X|D) = 0$  
+Thus,  $\lim_{n\to \infty}H(X|D,Z) = 0$
+
+# B THEORETICAL PROOF ON THEOREM 2
+
+Theorem 4. Let  $X, Y$ , and  $Z$  be discrete random variables. Define:
+
+$$
+Z^{*} = \operatorname *{arg  max}_{Z^{\prime}\subseteq Z}I(X;Z^{\prime}) - \beta I(Z^{\prime};Z)
+$$
+
+$$
+Y^{*} = \operatorname *{arg  max}_{\substack{Y^{\prime}\subseteq Y}}I(Y^{\prime};Z^{*})
+$$
+
+Then, the following inequality holds:
+
+$$
+I (X; Y; Z) \geq I \left(Y ^ {*}; Z ^ {*}\right) + I (X; Z ^ {*}) - I \left(Z ^ {*}; Z\right). \tag {8}
+$$
+
+where  $I(\cdot ;\cdot)$  denotes mutual information and  $H(\cdot)$  denotes entropy.
+
+Proof. We prove this theorem using fundamental principles of information theory:
+
+1. Recall the definition of multivariate mutual information:
+
+$$
+I (X; Y; Z) = I (X; Y) - I (X; Y | Z) \tag {9}
+$$
+
+2. By the chain rule of mutual information (Shi et al., 2023), we can rewrite this as:
+
+$$
+I (X; Y; Z) = I (X; Z) + I (Y; Z) - I (X, Y; Z) \tag {10}
+$$
+
+3. Consider our subsets  $Z^{*}$  and  $Y^{*}$ . By definition of mutual information and the data processing inequality:
+
+$$
+I \left(Y ^ {*}; Z ^ {*}\right) \leq I (Y; Z ^ {*}) \tag {11}
+$$
+
+$$
+I (X; Z ^ {*}) \leq I (X; Z) \tag {12}
+$$
+
+4. Substituting these into our equation:
+
+$$
+I (X; Y; Z) \geq I (X; Z ^ {*}) + I \left(Y ^ {*}; Z ^ {*}\right) - I (X, Y; Z) \tag {13}
+$$
+
+5. For any random variables  $A$  and  $B$ :
+
+$$
+I (A; B) \leq \min  (H (A), H (B)) \tag {14}
+$$
+
+Therefore:
+
+$$
+I (X, Y; Z) \leq \min  (H (X, Y), H (Z)) \leq I \left(Z ^ {*}; Z\right) \tag {15}
+$$
+
+6. Applying this to our inequality:
+
+$$
+I (X; Y; Z) \geq I (X; Z ^ {*}) + I \left(Y ^ {*}; Z ^ {*}\right) - I \left(Z ^ {*}; Z\right) \tag {16}
+$$
+
+7. Therefore, we can conclude:
+
+$$
+I (X; Y; Z) \geq I \left(Y ^ {*}; Z ^ {*}\right) + I (X; Z ^ {*}) - I \left(Z ^ {*}; Z\right) \tag {17}
+$$
+
+![](images/0f19598a36b7bedd87fb474fea1975c8a3438719cd5ba2511a685498c3ff4fe4.jpg)
+
+# C PROMPTS FOR DIFFERENT TASKS
+
+EXAMPLE PROMPT FOR BIRD IMAGE DESCRIPTION IN CUB-200
+
+<table><tr><td>Prompt 1</td><td>Focus solely on the bird shown in the image. Describe the bird&#x27;s appearance in detail, emphasizing its most prominent physical features. Avoid mentioning the background or other elements not related to the bird.</td></tr><tr><td>Prompt 2</td><td>Provide a focused analysis of the bird in this image, detailing its distinctive physical features. Concentrate exclusively on the bird and describe its appearance without referencing the surroundings or any extraneous details.</td></tr><tr><td>Prompt 3</td><td>Directly observe the bird depicted and offer a precise description of its visual attributes. Ensure your description is limited to the bird itself, detailing its primary features and omitting any unrelated background elements.</td></tr></table>
+
+EXAMPLE REWRITE PROMPTS FOR DIFFERENT DATASETS
+
+Table 3: Prompts for bird image analysis in CUB-200 dataset  
+
+<table><tr><td>Dataset</td><td>Prompt</td></tr><tr><td>cub-200 / stanford_dogs</td><td>This is a picture of a {label} with the following visual features: {concepts_str}. Based on the information provided, please answer the following question. 
+Question: ’{query}’</td></tr><tr><td>HAM10000</td><td>This is a dermatoscopic image of {label} disease with the following visual features: {concepts_str}. Based on the information provided, please answer the following question. 
+Question: ’{query}’</td></tr><tr><td>PLD / fgvc</td><td>This is a picture of {label} with the following visual features: {concepts_str}. Based on the information provided, please answer the following question. 
+Question: ’{query}’</td></tr><tr><td>chest-xray</td><td>This is a chest-xray of {label} with the following visual features: {concepts_str}. Based on the information provided, please answer the following question. 
+Question: ’{query}’</td></tr></table>
+
+L+GE SYNTHESIZE PROMPT EXAMPLE
+
+Table 4: Answer rewrite prompts for different datasets.  
+
+<table><tr><td>Dataset</td><td>Prompt</td></tr><tr><td>L+GE synthesize prompt</td><td>There is a picture of a {label}, which is known for the following characteristics: {concepts_str}.Act as if you can see the picture. Please answer the following question based on the above information. Make your answer concise.Question: ’{query}’Answer:</td></tr></table>
+
+Table 5: Prompt for answering questions based on image characteristics for general image datasets.
+
+EVALUATION PROMPTS FOR EXPLANATION EXISTENCE (EE) AND COGNITION SCORE (CS)  
+
+<table><tr><td>Prompt Type</td><td>Prompt</td></tr><tr><td>EE Prompt</td><td>Determine whether the following answer contains a valid explanation supporting its conclusion. Respond with only ‘true’ or ‘false’. 
+Answer: {answer} 
+Contains an explanation?</td></tr><tr><td>CS Prompt</td><td>Evaluate the coherence and logical alignment of the following explanation with the provided concepts. Please note: the explanation does not need to fully encompass all concepts. 
+Assign a consistency score between 0 and 1, where 1 indicates the explanation contains no irrelevant information to the listed concepts, and 0 indicates complete misalignment with entirely irrelevant information. Only give the score. 
+Concepts: {concepts_formatted} 
+Explanation: {explanation} 
+Consistency Score:</td></tr></table>
+
+# CONCEPT EXTRACTION PROMPTS
+
+To obtain label-level concepts, we used GPT-4o with prompts designed to elicit detailed visual features associated with each class label. An example prompt is:
+
+Table 6: Prompts for Explanation Existence (EE) and Cognition Score (CS).  
+
+<table><tr><td>Prompt Type</td><td>Prompt</td></tr><tr><td>Concept Extraction Prompt</td><td>Please provide a list of visual characteristics that are commonly associated with the bird species {Class Name}. Include features such as color patterns, shapes, and distinctive markings.</td></tr></table>
+
+Table 7: Prompts for Concept Extraction.
+
+# D EXPERIMENTS DETAILS
+
+# D.1 TRAINING FRAMEWORK IMPLEMENTATION
+
+Our experimental setup leverages state-of-the-art techniques for fine-tuning the LLaVA-1.5-7B model. We initialize the model using pre-trained weights obtained from the Hugging Face model  $\mathrm{hub}^2$ . Our training process uses the SFTTrainer for supervised fine-tuning and implement Low-Rank Adaptation (LoRA) to achieve efficient parameter fine-tuning. In our configuration, we set the LoRA rank  $(r)$  to 128, the scaling factor  $(\alpha)$  to 256, and the dropout rate to 0.1. We fine tune model on a mixture of our domain-specific synthetic data with LLaVA-1.5's original general instruction-following training data to stabilize training without introducing additional data. We enable gradient checkpointing to reduce memory usage, allowing us to process larger batch sizes by not storing all intermediate activations. In addition, we employ DeepSpeed ZeRO Stage 3 to fully utilize our multi-GPU configuration. We also utilize mixed precision training with the BF16 format, which further reduces memory requirements while maintaining computational performance. Gradient accumulation is implemented with 2 steps and a per-device train batch size of 16. Moreover, slight data augmentation are applied to the image data to create diversity and enhance training robustness.
+
+# D.2 DATASET OVERVIEW AND EXPERIMENTAL RUNNING TIMES
+
+Table 8: Summary of Selected Datasets  
+
+<table><tr><td>Dataset Name</td><td>Test / Training Cases</td><td>Label Categories</td><td>Running Time Per Iter</td></tr><tr><td>CUB-200-2011</td><td>5,794 / 5,994</td><td>200</td><td>143 minutes</td></tr><tr><td>FGVC-Aircraft</td><td>3,333 / 6,667</td><td>100</td><td>171 minutes</td></tr><tr><td>Stanford Dogs</td><td>8,580 / 12,000</td><td>120</td><td>310 minutes</td></tr><tr><td>Chest X-ray images</td><td>624 / 5,216</td><td>2</td><td>113 minutes</td></tr><tr><td>HAM1000</td><td>5,008 / 5,007</td><td>7</td><td>107 minutes</td></tr><tr><td>PLD</td><td>47,523 / 5,279</td><td>37</td><td>109 minutes</td></tr></table>
+
+We use the original labels and train/test split from the original datasets. Detailed train/test data statics are reported in Table 8. Regarding the experiments running time, each iteration of rejection sampling and fine-tuning took approximately 2.5 hours on a single NVIDIA H100 GPU for datasets like CUB-200 and FGVC-Aircraft. A more detailed report can be found in Table 8.
+
+# E VISUALIZATION RESULTS
+
+Table 9: Generated Answer Examples.  
+
+<table><tr><td>User:</td><td>Based on the visual content, what is the specific name of this bird? Provide an explanation.</td></tr><tr><td>LMM:</td><td>The specific name of this bird is the Yellow breasted Chat. This bird is known for its bright yellow throat and chest, olive-green back, wings, and tail, and short, rounded wings. It is a small bird that is commonly found in the Americas, particularly in the United States and Canada.</td></tr><tr><td>User:</td><td>Based on the visual content, what is the specific name of this bird? Provide an explanation.</td></tr><tr><td>LMM:</td><td>The specific name of this bird is the Blue Grosbeak. The Blue Grosbeak is a small, colorful bird with a deep blue overall plumage in males and a brown over-all coloration with subtle blue tinges and faint streaking in females. The bird has a slightly glossy sheen to its blue feathers and is known for its melodious songs.</td></tr><tr><td>User:</td><td>Based on the visual content, what is the specific name of this bird? Provide an explanation.</td></tr><tr><td>LMM:</td><td>The specific name of this bird is the Orchard Oriole. The bird has a black head, back, throat, and upper chest, with a rich chestnut-brown underparts. It has a black tail and a black throat patch developing. The bird&#x27;s wings are black with white wing bars, and it has a black bill. The Orchard Oriole is a small bird with a distinctive appearance, and it is known for its melodious songs.</td></tr><tr><td>User:</td><td>Based on the visual content, what is the specific name of this bird? Provide an explanation.</td></tr><tr><td>LMM:</td><td>The specific name of this bird is the Black throated Sparrow. This bird is known for its distinctive black throat patch and white stripe below its eyes. It is a small bird with a brown and white coloration, and it is often seen perched on branches or low vegetation.</td></tr><tr><td>User:</td><td>Based on the visual content, what is the specific name of this bird? Provide an explanation.</td></tr><tr><td>LMM:</td><td>The specific name of this bird is the Henslow Sparrow. This bird is known for its olive-colored head, short, pointed bill, and pale eye-ring. It is a small bird with a brown and white coloration, and it is often seen flitting low in dense grassy habitats.</td></tr></table>
+
+# F ADDITIONAL EXPERIMENTS
+
+# F.1 IMPACT ON GENERAL ABILITIES
+
+To assess the effect of our proposed training method on the general abilities of the models, we conducted evaluations on four widely used multimodal benchmarks: MMStar, SEED-Bench-2 Plus, MMBench, and MME (Cognition). Table 11 summarizes the results.
+
+- MMStar (Chen et al., 2024b): MMStar is a benchmark designed to evaluate vision-indispensable multimodal capabilities of MLLMs. It comprises 1,500 high-quality samples selected through a meticulous process from an initial pool of 22,401 samples. The benchmark assesses six core capabilities, each containing 250 samples, distributed across 18 detailed axes. This structure ensures a comprehensive evaluation of models' performance in tasks that require visual understanding.  
+- SEED-Bench-2 Plus (Li et al., 2024a): SEED-Bench-2 Plus is specifically designed to evaluate text-rich visual comprehension in MLLMs. It features 2,300 multiple-choice questions with precise human annotations, spanning three broad categories: Charts, Maps, and Webs. These categories encompass a wide spectrum of real-world text-rich scenarios, effectively simulating environments where visual and textual information are intertwined. The benchmark aims to assess models' proficiency in interpreting complex visual data embedded with textual content.
+
+Table 10: Selected Concepts Examples.  
+
+<table><tr><td></td><td>Concepts: &quot;Powerful, thick neck&quot;
+&quot;Broad and strong back&quot;
+&quot;Large and muscular build&quot;
+&quot;Well-sprung ribs&quot;</td></tr><tr><td></td><td>Concepts: &quot;Pure white or cream-colored fur&quot;
+&quot;Well-feathered tail that blends with the body fur&quot;
+&quot;Strong, straight legs&quot;
+&quot;Broad head with a slightly rounded skull&quot;</td></tr><tr><td></td><td>Concepts: &quot;Straight, arched tail that rests on the back&quot;
+&quot;Broad, flat skull&quot;
+&quot;Thick double coat, either rough or smooth&quot;</td></tr><tr><td></td><td>Concepts: &quot;Deep-set, almond-shaped eyes&quot;
+&quot;Erect, triangular ears&quot;
+&quot;Thick ruff of fur around the neck&quot;
+&quot;Thick double coat, either rough or smooth&quot;</td></tr><tr><td></td><td>Concepts: &quot;Ears: Large, bat-like ears that stand erect and move attentively.&quot;
+&quot;Distinctive facial features: Elongated muzzle with a moderate stop and express- 
+sive eyes.&quot;
+&quot;Whiskers: Short or absent due to the lack of hair follicles around the muzzle 
+area.&quot;</td></tr></table>
+
+- MMBench (Liu et al., 2024e): MMBench is a comprehensive benchmark that evaluates MLLMs across 20 ability dimensions. It includes approximately 3,000 multiple-choice questions, each with a single correct answer. MMBench addresses limitations of traditional benchmarks by focusing on fine-grained abilities and introducing robust evaluation strategies. The benchmark employs ChatGPT to match a model's prediction with the choices of a question, providing a more reliable assessment of model performance.  
+- MME (Cognition) (Fu et al., 2023): MME (Cognition) benchmark evaluates the cognitive abilities of MLLMs through tasks requiring reasoning and understanding of visual and textual inputs. The cognition part includes four subtasks: Commonsense Reasoning, Numerical Calculation, Text Translation, and Code Reasoning. These tasks test the model's ability to integrate multimodal information, such as interpreting visual scenes, performing arithmetic based on images, translating text in images, and reasoning about code snippets. The dataset consists of carefully curated images and instruction-answer pairs to ensure robust and fair assessment.
+
+As shown in Table 11, the fine-tuned models exhibit improved performance across multiple benchmarks. Notably, the model fine-tuned on HAM10000 achieves a significant overall improvement of  $12.9\%$ , indicating that our training method enhances domain-specific cognition without compromising and, in some cases, improves the models' general abilities.
+
+Table 11: General ability evaluation across additional benchmarks. The results demonstrate that our fine-tuned models not only retain their general abilities but also achieve overall improvements compared to the base LLaVA-1.5 model.  
+
+<table><tr><td>Model</td><td>MMStar</td><td>SEED-Bench-2 Plus</td><td>MMBench</td><td>MME (Cognition)</td><td>Overall Improvement</td></tr><tr><td>LLaVA-1.5 (Base)</td><td>34.46</td><td>41.81</td><td>63.05</td><td>334.28</td><td>-</td></tr><tr><td>Trained on CUB-200</td><td>33.40</td><td>41.78</td><td>63.14</td><td>355.00</td><td>3.2% ↑</td></tr><tr><td>Trained on Stanford Dogs</td><td>34.93</td><td>40.97</td><td>63.06</td><td>365.71</td><td>8.3% ↑</td></tr><tr><td>Trained on FGVC-Aircraft</td><td>35.14</td><td>40.14</td><td>63.23</td><td>348.57</td><td>2.1% ↑</td></tr><tr><td>Trained on PLD</td><td>35.30</td><td>40.89</td><td>63.14</td><td>337.14</td><td>1.1% ↑</td></tr><tr><td>Trained on HAM10000</td><td>34.46</td><td>41.11</td><td>64.08</td><td>378.21</td><td>12.9% ↑</td></tr></table>
+
+# F.2 EFFECTIVENESS OF FILTERING STRATEGIES
+
+To evaluate the importance of our reward model-free rejection sampling method described in Section 3.3, we conducted an ablation study comparing our approach with a baseline that does not employ the filtering mechanism. In this baseline, the model generates the most probable responses during each iteration, which are used directly for training without any filtering. Table 12 presents the classification accuracy and cognition score (CS) across four iterations for both the baseline without filtering and our proposed method.
+
+Table 12: Comparison of accuracy and cognition scores (CS) for the baseline without filtering and our proposed method across iterations.  
+
+<table><tr><td>Dataset</td><td>Method</td><td>Accuracy (Iter 1)</td><td>Accuracy (Iter 2)</td><td>Accuracy (Iter 3)</td><td>Accuracy (Iter 4)</td><td>CS Score</td></tr><tr><td rowspan="2">CUB-200</td><td>w/o Filtering</td><td>68.90</td><td>70.11</td><td>70.85</td><td>70.45</td><td>0.71</td></tr><tr><td>Ours</td><td>80.24</td><td>83.76</td><td>84.69</td><td>85.02</td><td>0.82</td></tr><tr><td rowspan="2">FGVC-Aircraft</td><td>w/o Filtering</td><td>76.36</td><td>76.60</td><td>77.11</td><td>76.78</td><td>0.72</td></tr><tr><td>Ours</td><td>88.78</td><td>90.91</td><td>91.42</td><td>91.99</td><td>0.79</td></tr><tr><td rowspan="2">Stanford Dogs</td><td>w/o Filtering</td><td>76.60</td><td>78.53</td><td>78.61</td><td>78.26</td><td>0.74</td></tr><tr><td>Ours</td><td>85.29</td><td>86.75</td><td>86.86</td><td>86.91</td><td>0.86</td></tr><tr><td rowspan="2">PLD</td><td>w/o Filtering</td><td>61.19</td><td>63.74</td><td>62.32</td><td>62.59</td><td>0.66</td></tr><tr><td>Ours</td><td>75.96</td><td>92.80</td><td>96.59</td><td>97.16</td><td>0.86</td></tr><tr><td rowspan="2">HAM10000</td><td>w/o Filtering</td><td>71.79</td><td>72.10</td><td>72.34</td><td>72.29</td><td>0.75</td></tr><tr><td>Ours</td><td>79.37</td><td>82.29</td><td>83.69</td><td>85.06</td><td>0.87</td></tr></table>
+
+From Table 12, it is evident that our filtering strategy significantly enhances both classification accuracy and explanation quality, as measured by the CS score. The baseline without filtering shows marginal improvements initially but fails to achieve comparable performance to our method. This demonstrates the critical role of our filtering mechanism in refining synthetic data and improving the model's performance iteratively.
+
+# F.3 IMPACT OF TEXT ENCODER
+
+Our framework relies on a text embedding model for estimating mutual information during concept selection (Section 3.2). To assess the impact of different text encoders on concept selection accuracy, we compared three models: E5 (Wang et al., 2022), BERT-Large (Devlin, 2018), and BERT-Base (Devlin, 2018). The results are presented in Table 13.
+
+Table 13: Concept selection accuracy using different text encoders.  
+
+<table><tr><td>Text Encoder</td><td>Concept Selection Accuracy (%)</td></tr><tr><td>E5</td><td>72.9</td></tr><tr><td>BERT-Large</td><td>71.4</td></tr><tr><td>BERT-Base</td><td>69.7</td></tr></table>
+
+As shown in Table 13, the E5 model achieves the highest concept selection accuracy. This indicates that using a more powerful text embedding model improves the quality of concept selection, which in turn enhances the effectiveness of our overall framework.
+
+# F.4 CROSS-DATASET TRANSFER PERFORMANCE
+
+We conducted experiments to evaluate the cross-dataset transferability of our method. Specifically, we trained the model on the CUB-200 dataset and evaluated its performance on the Stanford Dogs dataset. The results are shown in Table 14.
+
+Table 14: Cross-dataset transfer performance when training on CUB-200 and evaluating on Stanford Dogs.  
+
+<table><tr><td>Training Dataset</td><td>Evaluation Dataset</td><td>Accuracy (%)</td></tr><tr><td>None</td><td>Stanford Dogs</td><td>12.20</td></tr><tr><td>CUB-200</td><td>Stanford Dogs</td><td>16.60</td></tr><tr><td>Stanford Dogs</td><td>Stanford Dogs</td><td>86.91</td></tr></table>
+
+The results in Table 14 indicate that training on CUB-200 provides a marginal improvement when evaluated on Stanford Dogs. However, the performance remains significantly lower than when the model is trained directly on Stanford Dogs. This suggests that while our method may improve general cognition to some extent, domain-specific fine-tuning is crucial for achieving high accuracy in specialized tasks.
+
+# F.5 EXPLANATION QUALITY IMPROVEMENT ACROSS ITERATIONS
+
+To assess the benefits of our iterative fine-tuning approach, we evaluated the Cognition Score (CS) of the models across four iterations for various datasets. The results are summarized in Table 15.
+
+Table 15: Cognition Scores (CS) across iterations for various datasets. The CS Improvement represents the percentage increase from Iteration 1 to Iteration 4.  
+
+<table><tr><td>Dataset</td><td>CS Value (Iter 1)</td><td>CS Value (Iter 2)</td><td>CS Value (Iter 3)</td><td>CS Value (Iter 4)</td><td>CS Improvement</td></tr><tr><td>CUB-200</td><td>0.77</td><td>0.76</td><td>0.78</td><td>0.82</td><td>6.5% ↑</td></tr><tr><td>Stanford Dogs</td><td>0.82</td><td>0.84</td><td>0.83</td><td>0.86</td><td>4.9% ↑</td></tr><tr><td>FGVC-Aircraft</td><td>0.78</td><td>0.78</td><td>0.78</td><td>0.79</td><td>1.3% ↑</td></tr><tr><td>PLD</td><td>0.84</td><td>0.85</td><td>0.85</td><td>0.86</td><td>2.4% ↑</td></tr><tr><td>HAM10000</td><td>0.77</td><td>0.84</td><td>0.83</td><td>0.87</td><td>13.0% ↑</td></tr><tr><td>Chest X-ray</td><td>0.67</td><td>0.80</td><td>0.81</td><td>0.87</td><td>29.9% ↑</td></tr></table>
+
+As observed in Table 15, the CS scores generally improve over iterations, indicating that our iterative fine-tuning process enhances the explanation quality of the models. The most significant improvements are seen in the HAM10000 and Chest X-ray datasets, with  $13.0\%$  and  $29.9\%$  reported.

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2025/Enhancing Document Understanding with Group Position Embedding_ A Novel Approach to Incorporate Layout Information/full.md

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+# ENHANCING DOCUMENT UNDERSTANDING WITH GROUP POSITION EMBEDDING: A NOVEL APPROACH TO INCORPORATE LAYOUT INFORMATION
+
+Yuke Zhu, Yue Zhang*, Dongdong Liu†, Chi Xie, Zihua Xiong, Bo Zheng†, Sheng Guo MYbank, Ant Group
+
+{felix.yk,guosheng.guosheng}@mybank.cn
+
+# ABSTRACT
+
+Recent advancements in document understanding have been dominated by Large Language Models (LLMs) and Large Multimodal Models. However, enabling LLMs to comprehend complex document layouts and structural information often necessitates intricate network modifications or costly pre-training, limiting their practical applicability. In this paper, we introduce Group Position Embedding (GPE), a novel and efficient technique to enhance the layout understanding capabilities of LLMs without architectural changes or additional pre-training. GPE achieves this by strategically grouping the attention heads and feeding each group with distinct positional embeddings, effectively encoding layout information relevant to document comprehension. This simple yet powerful method allows for effective integration of layout information within the existing LLM framework. We evaluate GPE against several competitive baselines across five mainstream document tasks. We also introduce a challenging benchmark called BLADE, specifically designed to assess the layout comprehension capabilities. Extensive experiments on both established benchmarks and BLADE confirm the efficacy of GPE in significantly advancing the state-of-the-art in document understanding. Our code is available at https://github.com/antgroup/GroupPositionEmbedding.git.
+
+# 1 INTRODUCTION
+
+Understanding complex document using AI has long been a popular topic in academia. Since the advent of Transformers (Vaswani et al., 2017), a series of pretrained models (Xu et al., 2020; 2021; Huang et al., 2022) based on the Transformer architecture have been proposed. For an extended period, these methods have been the mainstream solutions for document understanding. Recently, large language models(LLMs) like ChatGPT (OpenAI, 2022) have exhibited impressive capabilities across various tasks, particularly their Zero-Shot abilities, which have quickly established them as a new paradigm for future AI applications. In terms of document tasks, LLM-based methods (Perot et al., 2023; He et al., 2023; Zhang et al., 2023; Ye et al., 2023a; Bai et al., 2023) have also emerged as a hot research direction.
+
+The approaches to document processing based on LLMs have evolved into several paradigms. The first approach (Liu et al., 2023b; Zhang et al., 2023; Ye et al., 2023a; Bai et al., 2023; Shi et al., 2023) directly processes document images. It first encodes the image using a visual encoder and then inputs the encoded representation into the LLM. Such methods typically require substantial pretraining data to enable the model to learn how to understand documents based on visual information. The second approach (Perot et al., 2023; Luo et al., 2024; Liao et al., 2024; Wang et al., 2023) utilizes OCR (Optical Character Recognition) models to first convert the document into text, then fed them to LLM. This method does not require the use of costly Vision-Language Pretraining, thereby reducing the challenge of direct document image understanding for LLM. However, it struggles with preserving the original layout and structure information of the documents. To address this shortfall, recent efforts (Luo et al., 2024; Wang et al., 2023; Liao et al., 2024) have employed large-scale
+
+data pre-training to achieve this. In summary, when employing these methods to enable LLMs to understand complex documents, extensive pre-training is indispensable.
+
+In this paper, we introduce a novel and efficient method for modeling document layout information based on LLMs, characterized by its concise design that eliminates the need for costly pre-training. We design a new positional embedding scheme for layout information to help the LLM understand layout nuances. Notably, this approach does not rely on complex visual encoders or introduce any additional learnable parameters. Instead, we incorporate layout information solely by modifying the existing position embedding of LLM. Specifically, we view the layout information as a multidimensional vector and use multiple sets of position embeddings to represent it. Inspired by the independence of attention heads in the multi-head attention mechanism, we partition the attention heads into groups as needed. Each group incorporates its corresponding positional information. We refer to this proposed positional embedding as Group Position Embedding (GPE). Its conceptual foundation and implementation are remarkably straightforward. In the context of document-specific tasks, we employ GPE to encode the coordinates of textual elements, effectively modeling layout information. The large language model, modified with GPE, does not incur any additional computational overhead or alter the original model architecture; thus, it can achieve the goal of understanding complex documents with minimal fine-tuning.
+
+Some recent works (Luo et al., 2024; Wang et al., 2023; Liao et al., 2024; Fujitake, 2024; Lu et al., 2024a; Perot et al., 2023) also attempt to incorporate layout information into large language models. The simplest approach involves flattening the text along with its corresponding coordinate boxes (Perot et al., 2023) before inputting them into the model. However, this method not only increases the input length but also disrupts the original coherence of the text, and it does not necessarily guarantee that the model will learn the layout information embedded in the coordinate boxes. Alternatively, some methods treat coordinate box information as an additional modality (Wang et al., 2023; Fujitake, 2024; Liao et al., 2024), integrating it with the original text before feeding it to the model. The introduction of this new modality necessitates extensive pre-training to achieve modality alignment. Other approaches (Luo et al., 2024) have introduced more complex layout information encoding network, which increase computational load in order to inject layout information. In contrast, our method preserves the original input structure of the text and employs an extremely simple approach to model layout information.
+
+For verification, we conducted experiments on several different base models and benchmarked our method on five mainstream documental tasks. Additionally, we introduced a more challenging Benchmark for Layout Analysis and Document Evaluation (BLADE). It is meticulously collected and designed to ensure that layout information is a critical factor for correctly answering questions. Through a series of experiments, we observed that our method effectively enables various LLMs to learn layout information and demonstrates highly competitive results compared to similar methods. In summary, the contributions of our method include the following points:
+
+1. A universal and extremely simple positional embedding called Group Position Embedding, which helps LLMs understand document layout information.  
+2. A new and more challenging document evaluation benchmark, BLADE, designed to assess LLMs' comprehension of complex layouts.  
+3. Comprehensive experiments and valuable insights into model behavior.
+
+# 2 RELATED WORK
+
+MLLMs for document understanding. Recently, large language models represented by ChatGPT(OpenAI, 2022) have demonstrated astonishing performance in text tasks, particularly in Zero-Shot tasks. To enable LLMs to understand more complex modal information, various Large Multimodal Models (Liu et al., 2023a; Zhang et al., 2023; Ye et al., 2023b; Bai et al., 2023) have shown impressive performance in adapting to different modalities. In document-related tasks, multimodal LLMs can be categorized into two types. One type employs an end-to-end approach (Liu et al., 2023a; Zhang et al., 2023; Ye et al., 2023b; Bai et al., 2023), which directly accepts raw image inputs and processes document image inputs in an end-to-end manner. This approach requires the incorporation of an additional visual encoder and relies on costly pre-training processes to achieve fine-grained alignment between the image and text modalities. The other type (Luo et al., 2024;
+
+![](images/a59912a89a9400b5233f888b51bba998528583cd8837d5f0f0628ec3e1dcf39d.jpg)  
+Multi-Head Attention with previous PE
+
+![](images/ed787a15e059fd5e6a527bf0d4eca031b2374908171d89254aee194eda23c9b4.jpg)  
+Multi-Head Attention with our Group PE  
+Figure 1: The concept of Group Position Embedding. Left: The standard multi-head attention, in which identical position embedding is applied to all heads. Right: Proposed group position embedding, in which positional information differs across head groups.
+
+Wang et al., 2023; Liao et al., 2024; Lu et al., 2024a; Perot et al., 2023) utilizes traditional OCR models to first convert document images into text and text box information before using the LLM to process this information. This latter method was widely used before the advent of LLMs and has proven effective at learning complex document layout information. Exploring how to integrate LLMs with document layout information is a research direction worth delving into. Our proposed Group Position Embedding is based on the second approach.
+
+Layout-aware Position Embeddings. Positional embedding is a common method used in structures like Transformers to capture sequence positional information. Earlier sinusoidal positional encodings (Vaswani et al., 2017) were frequently used in tasks such as language modeling and machine translation. The original positional embedding was mainly designed to represent the positional information of one-dimensional sequences, but many tasks require two-dimensional or multidimensional positional information. In image-related tasks, it is necessary to capture positional information in both width and height dimensions. A series of works exemplified by ViT (Dosovitskiy, 2020) achieve this by concatenating positional embeddings from two dimensions to represent 2D information. This concatenation method is also very common in detection tasks; for instance, in works like Deformable-DETR (Zhu et al., 2020), the positional embeddings of the bounding boxes are generated by encoding information for each dimension separately and then concatenating them together. In document tasks, LayoutLM (Xu et al., 2020), as an early pioneer, represents the spatial layout of documents by encoding the information from the corresponding bounding boxes for the text. It separately encodes each dimension and then sums the positional vectors of different dimensions at the input stage. Subsequent improvements (Xu et al., 2020; 2021; Huang et al., 2022; Li et al., 2021b;a;c; Appalaraju et al., 2021; Gu et al., 2021; Wang et al., 2022; Gu et al., 2022; Hong et al., 2022; Yu et al., 2023; Peng et al., 2022; Luo et al., 2023; Da et al., 2023) based on LayoutLM have continued to utilize similar coordinate encoding schemes. Compared with these approaches, GPE is an entirely new method that can be combined with any positional encoding and quickly applied to models.
+
+# 3 METHOD
+
+In this section, we will first review the commonly used position embeddings in large language models. Later, we will illustrate the idea of Group Position Embedding(GPE) for LLMs and how to apply the grouping technique to the commonly used position embeddings. Lastly, focusing on the problem of document understanding, we will demonstrate how LLM can be facilitated to comprehend complex positional information.
+
+# 3.1 REVIEW OF POSITION EMBEDDINGS
+
+We review two representative position embeddings. The first is Sinusoidal Position Embedding, which serves as an exemplar of additive position embedding. The second is Rotary Position Embedding, representing the category of multiplicative position embedding.
+
+Sinusoidal Position Embeddings. Sinusoidal Position Embeddings (Vaswani et al., 2017) are constant vectors that are added to token embeddings on input to the first layer of the transformer. They are pre-generated using sinusoidal function and selected based on the token index during model inference. For a given index  $m$ , the calculation process is as follows
+
+$$
+P E (m) _ {2 i + 1} = \cos \left(m / 1 0 0 0 0 0 ^ {2 i / d}\right), P E (m) _ {2 i} = \sin \left(m / 1 0 0 0 0 ^ {2 i / d}\right) \tag {1}
+$$
+
+where  $PE(m)$  denotes the Sinusoidal Embedding function.  $2i$  or  $2i + 1$  denotes the index of the embedding channels and  $d$  is the dimension of the embedding. The calculated position embedding is then added to the input embedding  $\mathbf{x}$  as  $PE(\mathbf{x},m) = \mathbf{x} + PE(m)$ .
+
+Rotary Position Embeddings (RoPE). Rotary position Embedding (Su et al., 2024) is the most widely used position embedding in current LLMs. It ingeniously combines relative position encoding with absolute position encoding and excels in length normalization. Given an input  $\mathbf{x}$  and its corresponding position  $m$ , Rotary position Embedding (RoPE) is added as following steps. First, the embedding  $\mathbf{x}$  is viewed as a set of complex numbers by pairing consecutive elements of the vector:  $\mathbf{x} = (x_{1} + ix_{2}, x_{3} + ix_{4}, \dots, x_{d - 1} + ix_{d})$ . It can also be denoted as  $\mathbf{x} = \sum_{j = 1}^{d / 2}(x_{j} + ix_{j + 1})\vec{\mathbf{e}}_{j}$  where  $\vec{\mathbf{e}}_j$  is the  $j$ -th union direction vector. Then the RoPE for  $\mathbf{x}$  is applied by
+
+$$
+f (\mathbf {x}, m) = \sum_ {j = 1} ^ {d / 2} x _ {j} e ^ {i m \theta_ {j}} \vec {\mathbf {e}} _ {j} \tag {2}
+$$
+
+Based on RoPE, given query and key vector  $\mathbf{q},\mathbf{k}$ , the attention score with position embedding is calculated as  $< f(\mathbf{q},m),f(\mathbf{k},n)> = \sum_{j = 1}^{d / 2}q_jk_je^{i(m - n)\theta_j}$ .
+
+# 3.2 GROUP POSITION EMBEDDING
+
+Group Position Embedding is proposed in order to feed multi-dimensional position information to transformers. As illustrated in Figure-1, in the conventional multi-head attention mechanism, all heads share a common set of position embeddings. Following the intent of multi-head attention, GPE forces different heads to attend to different positional information. Given a multi-dimensional position  $\mathbf{P}$  with dimension  $l$  and the number of attention heads  $h$ , Denote  $x$  as the input embedding. The computation of GPE unfolds as two steps. First, group attention heads to match the dimension of position  $\mathbf{P}$ . We denote  $Gr(i)$  as the group mapping function, where  $i \in [1,2,\dots,h]$  and  $Gr(i) \in [1,2,\dots,l]$ . Second, using the selected position embedding to apply the position information to each head. For Sinusoidal Position Embedding, we use GroupPE(x,P) as the position embedding function. It writes as
+
+$$
+\operatorname {G r o u p P E} (\mathbf {x}, \mathbf {P}, i) = M L P _ {i} (\mathbf {x}) + P E \left(\mathbf {P} _ {G r (i)}\right) \tag {3}
+$$
+
+where  $i$  is the head index and  $MLP_{i}(\mathbf{x})$  projects the input embedding to  $i$ -th query vector or key vector. For Rotary Position Embedding, we use GroupRoPE(x, P) as the position embedding function. It writes as
+
+$$
+\operatorname {G r o u p R o P E} (\mathbf {x}, \mathbf {P}, i) = f \left(M L P _ {i} (\mathbf {x}), \mathbf {P} _ {G r (i)}\right) \tag {4}
+$$
+
+Grouping in a multi-head manner is a general technique and can be applied to any position embeddings. It ensures that positional information from different dimensions holds an entirely equal status, collectively participating in the inference process of the large language model. The grouping technique is completely compatible with the original one-dimensional position encoding. For a pure text input, we can just design  $Gr(i)$  as an identity mapping, indicating that all heads being mapped to identical position index.
+
+# 3.3 GROUP POSITION EMBEDDING FOR DOCUMENT UNDERSTANDING
+
+In this section, we introduce the application of the GPE to the task of understanding complex documents. Conventionally, to enable LLMs to comprehend a typical document, Optical Character
+
+![](images/ef111e4d946f957a8c7d282515a64681077fd7307a79cb7838001739274d67f7.jpg)  
+Figure 2: Group Position Embeddings for document understanding. Each token is equipped with a N-D positional information, which includes its reading order and corresponding bounding box.
+
+Recognition (OCR) techniques, among others, are first employed to extract text boxes and textual information from the document. This is followed by ordering this information according to certain rules before feeding it into the LLM for processing. However, in more generic document tasks, obtaining a one-dimensional sorting can be immensely challenging due to elements such as tables and intricate layouts, necessitating the design of an exceedingly complex processing system. With GPE, we don't need to restore the exact positional order for each token; instead, we can simply assign ordering information using some straightforward strategies. For instance, we could adopt a global left-to-right sequence to index tokens, or directly assign them indices corresponding to their positions within local bounding boxes. Both approaches have their pros and cons in practice, which we will delve into further in subsequent discussions.
+
+Specifically, given a token embedding  $\mathbf{x}$ , we consider its positional information from two perspectives. Firstly, in terms of reading order,  $\mathbf{x}$  corresponds to a position id  $m$ . Secondly, we consider the spatial coordinates  $\mathbf{B}$  of the bounding box where  $\mathbf{x}$  resides. Without loss of generality, we utilize the coordinates of the top-left and bottom-right corners of the bounding box to denote its spatial position, i.e.,  $\mathbf{B} = [x0, y0, x1, y1]$ . In practice, when handling spatial positions, we normalize the coordinates  $\mathbf{B}$ , resulting in  $\bar{x}_i = \lambda * \frac{x_i - \min(x)}{\max(x) - \min(x)}$ ,  $\bar{y}_i = \lambda * \frac{y_i - \min(y)}{\max(y) - \min(y)}$ , where  $\lambda$  is the scaling factor. It is set to 1000 in our implementation. Consequently, the positional information to be encoded for token embedding  $\mathbf{x}$  can be represented as  $\mathbf{P} = [m, \mathbf{B}]$ . For the grouping function, we map the majority of heads to the layout part, each dimension using equal heads, the remaining for the reading order. Assuming the total number of heads is 32, the  $Gr(i)$  is defined as
+
+$$
+G r (i) = \left\{ \begin{array}{l l} 1 & \text {i f} 0 <   i \leq 4, \\ \lceil (i - 4) / 7 + 1 \rceil & \text {i f} 5 \leq i \leq 3 2, \end{array} \right. \tag {5}
+$$
+
+For the instruction part, only its one-dimensional positional information is considered, and this portion is encoded following the conventional manner in which LLMs process text sequences. An illustration of our method for imposing positional encoding on inputs of complex documents is provided in Figure-2, using a document example.
+
+Notably, there are various encoding schemes for the layout information of documents. For instance, we could choose to encode all four corner points of the bounding box to accommodate more complex scenarios. It will be discussed in experiments.
+
+# 4 BLADE: A NEW BENCHMARK FOR COMPLEX LAYOUT ANALYSIS
+
+We construct a more challenging Benchmark for Layout Analysis and Document Evaluation, named BLADE, in response to the saturation trend seen in current common document evaluation benchmarks. BLADE focuses on complex layout issues, with the majority of documents manually curated to ensure richness in layout information, thereby increasing the difficulty of assessment. Furthermore, the questions formulated based on these documents have been meticulously designed to ensure
+
+![](images/e28ced4ea05aada49ada037a3e44a218e7cacc34697ecee952dd3787017b5716.jpg)
+
+![](images/bf7224a26b6287ead77d47c513978ddd5bd453393d97ab3658178e550dc2531b.jpg)  
+(a)
+
+![](images/06b5f7ea958600ae468897ec4af415ab741cb6e960d4b665c7df4094348dc173.jpg)  
+Figure 3: (a) Samples from 6 scenarios of BLADE. (b) An example from Forms, in which several questions are listed. Answering these questions necessitates the structural info of each element, making the dataset rather challenging.
+
+![](images/47d898cbb41dcc4f90066c03e604d9b9869e7b8a499f474df0b0789f0090c623.jpg)  
+(b)
+
+![](images/1cd27e67d53de5090756e89cce6a10dccb6a80fda213f0a9ca33c81192941524.jpg)
+
+that correct answers can only be inferred with the aid of spatial layout information. Examples from some of these evaluation sets are illustrated in Figure-3a and Figure-3b. It can be seen that they are quite challenging for large language models that lack spatial layout understanding.
+
+The data in BLADE can be divided into two major categories. The first category comprises real-world data, including newspapers, web pages, slides, and various forms. The second category consists of synthesized data generated through rendering, involving synthetic documents and tables. For the real data, we primarily rely on manual annotation methods combined with the utilization of large language models to create question-answer pairs. In contrast, for the synthetic data, question-answer pairs are constructed using template questions. For details of BLADE, refer to the appendix.
+
+# 5 EXPERIMENT
+
+# 5.1 DATASETS
+
+For the training process of LLM with GPE, we collect several publicly available document understanding datasets. They include two synthetic datasets: Document Dense Description (DDD) and Layout-aware SFT used in (Luo et al., 2024) and four VisualQA datasets:DocVQA(Mathew et al., 2021), InfoVQA (Mathew et al., 2022), ChartQA(Masry et al., 2022), VisualMRC (Tanaka et al., 2021) and three KIE datasets:SROIE (Huang et al., 2019), CORD(Park et al., 2019), and FUNSD (Jaume et al., 2019). Besides, we also incorporate a proportion of instruction data for finetuning large models to combat the decline in the model's inherent capabilities after being trained on document data. For details of the training set, refer to the appendix.
+
+# 5.2 IMPLEMENTATION DETAILS
+
+For implementation, we selected several representative base models to apply GPE, including the Llama2-7B (Touvron et al., 2023), qwen2-7B (Yang et al., 2024), and ChatGLM-6B (GLM et al., 2024). Although these models are all based on the transformer architecture, they exhibit variations in certain details. For instance, qwen2 (Yang et al., 2024) uses group query attention instead of standard multi-head attention. The ChatGLM (GLM et al., 2024) adopts a prefix structure and utilize a bidirectional attention mechanism for the historical tokens. Since these structural details are related to GPE, we select them to validate our approach, thereby demonstrating its generality. For the training recipe, we use a learning rate of 1e-5 with 750 warmup steps and constant learning rate schedule, and Adam optimizer (Kingma, 2014) with beta1=0.9, beta2=0.99 and weight decay of 0.1. The maximum sequence length is set to 4096 for all base models during the entire training process. The training also involves the bf16 precision on 880GB A100 GPUs using the deepspeed training framework (Rasley et al., 2020) to complete one epoch.
+
+# 5.3 RESULTS
+
+In this section, we compare our method based on GPE with several document-oriented approaches across several commonly used benchmarks. We use ANLS(Mathew et al., 2021) as the evaluation
+
+Table 1: Comparison with related methods on common document benchmarks. Note LLMs and ours methods are trained under identical setting. MLLMs' results are implemented using their provided weights. Results of LLMs with layout are copied from their papers. We use ANLS as the default evaluation metric for all benchmarks except VisualMRC, for which we use CIDEr.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Method</td><td rowspan="2">data amount</td><td colspan="3">Document VQA</td><td colspan="2">VIE</td></tr><tr><td>DocVQA</td><td>VisualMRC</td><td>FUNSD</td><td>CORD</td><td>SROIE</td></tr><tr><td>LLM</td><td>ChatGLM-6B</td><td></td><td>26.1</td><td>131.3</td><td>25.8*</td><td>68.0*</td><td>65.1*</td></tr><tr><td rowspan="2">Plain Text</td><td>Llama2-7B-Chat</td><td>0.5M</td><td>66.6</td><td>335.4</td><td>62.7*</td><td>75.6*</td><td>88.8*</td></tr><tr><td>Qwen2-7B</td><td></td><td>71.8</td><td>367.4</td><td>78.1*</td><td>83.4*</td><td>97.7*</td></tr><tr><td rowspan="4">MLLM</td><td>LLaVAR-7B†</td><td>1.2M</td><td>11.6</td><td>-</td><td>1.7</td><td>13.6</td><td>2.4</td></tr><tr><td>LLaVA-1.5-7B†</td><td>1.2M</td><td>13.3</td><td>-</td><td>1.9</td><td>18.1</td><td>3.8</td></tr><tr><td>Qwen-VL-7B‡</td><td>1.4B</td><td>65.1</td><td>-</td><td>47.1</td><td>30.0</td><td>58.6</td></tr><tr><td>TextMonkey+‡</td><td>2.6M</td><td>66.7</td><td>-</td><td>42.9</td><td>-</td><td>46.2</td></tr><tr><td rowspan="2">LLM</td><td>DocLayLLM(with image)†</td><td>3.1M</td><td>72.8</td><td>310.6</td><td>80.7</td><td>79.4</td><td>84.4</td></tr><tr><td>LayTextLLM-Llama2-7B</td><td>6M</td><td>77.2</td><td>277.8</td><td>81.0*</td><td>82.5*</td><td>96.1*</td></tr><tr><td rowspan="2">with Layout</td><td>LayoutLLM-Vicuna-7B†</td><td>6M</td><td>74.3</td><td>-</td><td>78.7*</td><td>62.2*</td><td>71.0*</td></tr><tr><td>DocLLM-Llama2-7B</td><td>3.8M</td><td>69.5</td><td>264.1</td><td>51.8</td><td>67.4</td><td>91.9</td></tr><tr><td rowspan="3">ours</td><td>GPE-GLM-7B</td><td></td><td>57.9</td><td>129.3</td><td>69.6*</td><td>83.0*</td><td>94.8*</td></tr><tr><td>GPE-Llama2-7B</td><td>0.5M</td><td>77.2</td><td>318.9</td><td>77.6*</td><td>84.7*</td><td>97.9*</td></tr><tr><td>GPE-Qwen2-7B</td><td></td><td>78.1</td><td>344.0</td><td>82.6*</td><td>86.9*</td><td>97.8*</td></tr></table>
+
+† fully zero-shot setting, ‡ only zero-shot on VIE. *test set from (Luo et al., 2024)
+
+metric for all benchmarks except VisualMRC, which we use CIDEr(Vedantam et al., 2015) as the metric. The comparative methods can be broadly categorized into three types: the first comprising pure LLMs, for which we selected ChatGLM6B(GLM et al., 2024), Llama2-7B(Touvron et al., 2023), and Qwen2-7B(Yang et al., 2024); the second type involves end-to-end Large Multimodal Models, including LLaVAR-7B(Zhang et al., 2023), LLaVA-1.5-7B(Liu et al., 2023a), Qwen-VL-7B(Bai et al., 2023) and TextMonkey+ (Liu et al., 2024); and the third category consists of LLMs that model layout information, exemplified by LayoutLLM(Luo et al., 2024), DocLLM(Wang et al., 2023), DocLayLLM(Liao et al., 2024) and LayTextLLM (Lu et al., 2024b). We implemented GPE on the above chosen pure LLMs, yielding GPE-GLM-6B, GPE-Llama2-7B, and GPE-Qwen2-7B. Note that for pure LLMs, they are tuned under the identical setting of GPE. Considering that these methods are based on different base models, trained with varying datasets and strategies, and some without publicly available model weights, the comparison is not entirely equitable. Nonetheless, valuable insights emerge from the comparative results presented in Table-1.
+
+Our findings reveal that, after integrating GPE, LLMs exhibit notable improvements across various benchmarks compared to their original counterparts, affirming the efficacy of our proposed approach. Remarkably, GPE achieves highly competitive performance with significantly less training data compared to other methods. Additionally, we observe that end-to-end approaches fare less impressively on these benchmarks, as these tasks are inherently more challenging for such models, requiring them to also extract textual information from images. A noteworthy observation is the exceptional performance of the Qwen2-7B, particularly in VisualMRC, outperforming all other methods. This suggests that most of the benchmarks assess a model's text comprehension capability. Our analysis of these benchmarks(refer to the appendix) also indicates that scores on these benchmarks do not readily measure a model's ability to understand complex layout information. It is under this context that we introduce a new benchmark, BLADE, designed to address these limitations. Another observation is that GPE gets inferior performance compared with Qwen2-7B, it means that GPE may slightly influence the LLM's text comprehension capability. This phenomenon is further discussed in the appendix.
+
+# 5.4 COMPARISON
+
+To ensure a fair comparison between GPE and similar methods, we conduct a comparative experiment in this section using the same base model, dataset, and training strategy. We compare GPE with three representative methods. 1. Text Box (Perot et al., 2023): This approach directly represents bounding boxes as text, inserts the coordinate text into the original information text, and feeds the resulting new text sequence into the large language model as input. 2. Add Box Embedding (Xu
+
+Table 2: Comparison with other layout-aware methods on common document benchmarks. Methods in this table are trained under identical setting. Qwen2-7B is selected as the base model. We use ANLS as the evaluation metric for all benchmarks except VisualMRC, for which we use CIDEr.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Document VQA</td><td colspan="2">VIE</td></tr><tr><td>DocVQA</td><td>VisualMRC</td><td>FUNSD</td><td>CORD</td><td>SROIE</td></tr><tr><td>Raw Text</td><td>71.8</td><td>367.4</td><td>78.1</td><td>83.4</td><td>97.7</td></tr><tr><td>Text Box</td><td>71.5</td><td>357.5</td><td>78.4</td><td>81.5</td><td>98.0</td></tr><tr><td>Insert Box Embedding</td><td>67.8</td><td>301.4</td><td>71.2</td><td>63.8</td><td>90.3</td></tr><tr><td>Add Box Embedding</td><td>71.5</td><td>350.1</td><td>78.4</td><td>79.5</td><td>87.9</td></tr><tr><td>GPE</td><td>78.1</td><td>344.0</td><td>82.6</td><td>86.9</td><td>97.8</td></tr></table>
+
+Table 3: Comparison with other layout-aware methods on BLADE. Methods in this table are trained under identical setting. Qwen2-7B is selected as the base model.  
+
+<table><tr><td>Method</td><td>SynthTables</td><td>Forms</td><td>Slides</td><td>Websites</td><td>SynthDocs</td><td>Newspapers</td></tr><tr><td>Raw Text</td><td>50.3</td><td>33.0</td><td>42.2</td><td>46.1</td><td>53.3</td><td>43.8</td></tr><tr><td>Text Box</td><td>25.0</td><td>24.9</td><td>25.9</td><td>39.6</td><td>29.0</td><td>31.9</td></tr><tr><td>Insert Box Embedding</td><td>43.4</td><td>30.9</td><td>36.5</td><td>41.6</td><td>31.7</td><td>31.8</td></tr><tr><td>Add Box Embedding</td><td>51.4</td><td>32.8</td><td>43.2</td><td>50.3</td><td>57.6</td><td>45.1</td></tr><tr><td>GPE-Sinusoidal</td><td>67.2</td><td>45.6</td><td>48.3</td><td>66.5</td><td>55.2</td><td>48.7</td></tr><tr><td>GPE-RoPE</td><td>77.3</td><td>57.2</td><td>64.5</td><td>79.3</td><td>62.2</td><td>52.2</td></tr></table>
+
+et al., 2020): A commonly employed strategy in conventional methods, it encodes bounding boxes into vectors and adds them to the token embeddings of their corresponding text as input. 3. Insert Box Embedding (Lu et al., 2024a): Similarly encoding bounding boxes into vectors, it differs from LayoutLM by inserting the encoded position vectors into the original token embedding sequence. For GPE, we also implement the Sinusoidal version GPE-Sinusoidal for comparison. We select Qwen2-7B as the base model. The results of this comparison are shown in Table-2 and Table-3. It can be observed that GPE demonstrates remarkably strong performance across these document tasks. Under our training setup, the Text box and Insert Box Embedding struggles to function effectively, primarily because they alter the input format of the LLM, hence, necessitating an expensive pre-training stage when applied practically. Besides, these two methods have to process a longer input sequence compared to GPE and the method of Add Box Embedding. Add Box Embedding achieved second-best performance, but it is clearly weaker than the GPE method. We argue that this is mainly because Add Box Embedding effectively changes the original input vector space, requiring the model to adapt to a new modality during training. In contrast, GPE directly reuses the original position embedding space, making it easier for the model to learn. For GPE-Sinusoidal, although it uses different type of position embedding with the base model, it still achieves second-best result. This implies that the grouping manner is a better way to incorporate layout information than the other methods.
+
+# 5.5 ANALYSIS
+
+For the following parts, we use Llama2-7B as the base model if not particularly mentioned. Influence of Reading Order. We investigate the impact of reading order part in GPE on model performance, i.e the design of  $m$  in the sequence representation  $\mathbf{P} = [m, B]$ . We compare several designs for reading order as follows. W/O: Solely utilize the positional information of text boxes without imposing any specific reading sequence. Left2Right: Follow the natural left-to-right, top-to-bottom order as humans typically read. XYCut: Following(Gu et al., 2022), obtain proper reading order for rich layout document. Random: Randomly shuffle the order of text boxes to arrange tokens in a random sequence. Local: Treat each text box equally in this method, with all tokens within a box encoded sequentially starting from 0, ignoring their spatial relationship with other boxes. Considering that the attention structure also influences how reading order is handled, we select two base models for experiments: GLM and qwen2. The results are shown in Table-4. Based on the results, we observe a general concordance in trends between models employing the GLM and Llama2 architectures. Approaches that omit position encoding or utilize random position encoding exhibit inferior performance. The two methods that excel are Local and Left2Right,
+
+Table 4: Ablation on the design of Reading Order representation.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Llama2</td><td colspan="4">GLM</td></tr><tr><td>SynthTables</td><td>Forms</td><td>SynthDocs</td><td>Newspapers</td><td>SynthTables</td><td>Forms</td><td>SynthDocs</td><td>Newspapers</td></tr><tr><td>W/O</td><td>2.4</td><td>1.7</td><td>16.5</td><td>2.5</td><td>46.6</td><td>8.6</td><td>39.9</td><td>12.6</td></tr><tr><td>Random</td><td>63.9</td><td>31.8</td><td>63.0</td><td>26.7</td><td>66.0</td><td>15.9</td><td>47.0</td><td>32.3</td></tr><tr><td>Local</td><td>77.3</td><td>41.6</td><td>70.7</td><td>28.4</td><td>71.7</td><td>32.2</td><td>54.1</td><td>29.3</td></tr><tr><td>Left2Right</td><td>76.4</td><td>38.7</td><td>72.7</td><td>35.7</td><td>69.8</td><td>21.2</td><td>56.2</td><td>40.3</td></tr><tr><td>XYCut</td><td>71.8</td><td>39.8</td><td>60.2</td><td>31.3</td><td>66.9</td><td>24.9</td><td>52.5</td><td>39.4</td></tr></table>
+
+Table 5: Ablation on the representation methods of layout information on BLADE.  
+
+<table><tr><td rowspan="2">Setting</td><td colspan="3">W/O Rotation</td><td colspan="3">W/ Rotation</td></tr><tr><td>SynthTables</td><td>Slides</td><td>Websites</td><td>SynthTables</td><td>Slides</td><td>Websites</td></tr><tr><td>LeftPoint</td><td>76.0</td><td>36.9</td><td>55.1</td><td>73.6</td><td>31.5</td><td>51.6</td></tr><tr><td>Rectangular</td><td>76.4</td><td>38.7</td><td>60.5</td><td>74.3</td><td>38.8</td><td>71.6</td></tr><tr><td>Quadrilateral</td><td>77.1</td><td>42.9</td><td>56.4</td><td>75.1</td><td>40.7</td><td>57.4</td></tr></table>
+
+with each demonstrating a contrasting superiority across different datasets. Specifically, Left2Right excels on SynthDocs and Newspapers, while the Local method outperforms on SynthTables and Forms. We think that these results are influenced by the distinct characteristics of the respective domains. SynthDocs and Newspapers predominantly comprise natural language paragraphs, in which the sequential order of words holds significant importance. Although Left2Right may still occasionally disrupt the reading sequence, it largely preserves sequential information, a feature crucial for understanding narrative coherence in these text-heavy contexts. Conversely, SynthTables and Forms are typified by tabular structures where the reading order among fields is less critical. Here, the Local approach proves advantageous as it enables the model to more effectively leverage spatial information, a key aspect in deciphering structured data arrangements common to tables and forms.
+
+Design of Layout Information. We focus on the impact of design of  $\mathbf{B}$  in  $\mathbf{P} = [m,\mathbf{B}]$ . We compare three design approaches. LeftPoint: Utilize the coordinates of the top-left corner, where  $B = [x_0,y_0]$ , with  $(x_0,y_0)$  representing the position coordinates of the top-left corner. Rectangular: Employ the coordinates of both the top-left and bottom-right corners, expressed as  $B = [x_0,y_0,x_1,y_1]$ , where  $x_i,y_i$  denote the position coordinates of either the top-left ( $i = 0$ ) or bottom-right ( $i = 1$ ) corner. Quadrilateral: Involve the coordinates of all four corner points, formatted as  $B = [x_0,y_0,x_1,y_1,x_2,y_2,x_3,y_3]$ , arranged in a clockwise order, with  $(x_i,y_i)$  being the coordinates of each corner point. In each configuration, normalization of coordinates is consistently applied. We employed two conditions, with and without visual rotation, to investigate the impact of coordinate representation. Under the rotated condition, we applied random rotations to the entire evaluation set to emulate the effects of real-world shooting angles. The results are depicted in Table-5. Our findings reveal that the LeftPoint method yields the lowest performance across all evaluation benchmarks, although not significantly underperforming compared to other approaches. This suggests that a substantial portion of layout information within documents is indeed encapsulated by the top-left corner point. The Rectangular and Quadrilateral methods are largely comparable. Notably, we observed in our experiments that the Rectangular approach converges faster. Regardless of the rotational condition, Quadrilateral marginally outperforms Rectangular, though this advantage is not pronounced. These observations imply that, for the majority of document-related tasks, adopting the Rectangular method is sufficiently effective in achieving commendable outcomes.
+
+Design of Grouping Function As previously mentioned, the dimension of  $\mathbf{P} = [m,\mathbf{B}]$  and the number of heads in the model typically do not align. Consequently, a grouping function is required to map multiple heads to their corresponding position information. In this part, we explore the effects of varying grouping methodologies. Several straightforward Grouping Functions are: PadZero First expand the position vector  $\mathbf{P}$  by appending zeros till its dimension match the number of attention heads, then establish a one-to-one correspondence between them; More Reading Order Map the majority of heads to the reading order, the rest heads to the layout infomation; More Layout Map the majority of heads to layout information, the rest heads to reading order. The results are depicted in Table-6. In terms of results, the strategy of allocating more layout information to the group turns
+
+Table 6: Ablation on the selection of group function on BLADE.  
+
+<table><tr><td>Group Function</td><td>SynthTables</td><td>Forms</td><td>Slides</td><td>Websites</td><td>SynthDocs</td><td>Newspapers</td></tr><tr><td>Baseline</td><td>24.5</td><td>11.1</td><td>18.9</td><td>23.8</td><td>72.5</td><td>31.3</td></tr><tr><td>PadZero</td><td>2.8</td><td>10.7</td><td>1.4</td><td>2.0</td><td>11.8</td><td>1.6</td></tr><tr><td>More Reading Order</td><td>76.3</td><td>33.4</td><td>50.7</td><td>65.6</td><td>74.0</td><td>35.7</td></tr><tr><td>More Layout</td><td>76.4</td><td>38.7</td><td>56.2</td><td>70.5</td><td>72.7</td><td>35.7</td></tr></table>
+
+![](images/a24389808f593209a12127ca9c696fce29bdde39d53fc60bd4997c2bcbc43dde.jpg)  
+(a)
+
+![](images/136ab0320f30fa949b98ac58b5e57d4b02f5ed208127ea6b58a381a8af57b6ef.jpg)  
+(b)  
+Figure 4: Qualitative results on BLADE from Forms and Newspapers. (a) The question requires to list the whole column corresponding to Candidate, only GPE outputs the correct answer. (b) To correct answer the question, model needs to focus on the correct paragraph. Only GPE achieves this.
+
+out to be more advantageous, performing better across nearly all evaluation sets. Coupled with the conclusions in Table-4, Reading Order information is also identified as indispensable.
+
+Qualitative Results. We selected two samples from BLADE, and the output results from GPE and other methods are depicted in Figure-4. In the example from Forms, where a complete response to the "Candidate" column is required. Other methods either provide additional irrelevant information or miss out on details, whereas only GPE accurately furnishes the answer. In the Newspapers instance, comprehension of the content within a specified area is necessary to answer the question, and again, it is solely GPE that provides the correct response. These two examples illustrate that, with the aid of GPE, LLMs can precisely locate the relevant sections, disregarding distractions from unrelated information, a feat unachievable without the assistance of layout information.
+
+# 6 LIMITATION
+
+While GPE shows promise, several areas remain unexplored. First, although GPE is applicable to any task requiring multidimensional positional encoding, this paper focuses only on document-centric tasks. Graph-structured data could also benefit from similar grouping strategies. Second, most methods, including ours, face a decline in inherent capability, possibly due to structural changes or the quality of document-specific training data. Our approach minimizes alterations to the base model, and we believe future iterations of GPE will further address these challenges.
+
+# 7 CONCLUSION
+
+We introduce Group Position Embedding (GPE), which enhances LLMs' ability to understand multidimensional positional information, particularly in document-related tasks. GPE allows large models to comprehend complex document layouts by injecting independent positional information into grouped attention heads without altering the model architecture or input format. This method outperforms similar approaches. To evaluate large models' understanding of complex documents, we developed BLADE, a challenging benchmark that emphasizes layout information in both data sources and question design. Extensive validation on mainstream document tasks and BLADE demonstrates the effectiveness of our approach.
+
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+
+# A APPENDIX
+
+# A.1 DATASETS DESCRIPTION
+
+The training set in this work is composed of the following several parts.
+
+LayoutLLM-aware SFT Data(Luo et al., 2024) The layout-aware SFT data for LayoutLLM is generated by GPT-3.5 Turbo and converted from existing text-based Machine Reading Comprehension (MRC) datasets.
+
+ChartQA Dataset(Masry et al., 2022) This dataset encompasses 9,600 manually crafted questions along with 23,100 questions generated from manually written chart summaries, aiming to address complex problems involving visual and logical reasoning.
+
+InfographicVQA Dataset(Mathew et al., 2022) Designed for the Visual Question Answering (VQA) task on infographics, this dataset comprises around 10,000 images, with an average of about 10 questions per image, each question associated with approximately 4 to 5 answers. Infographics are graphical representations of information, data, or knowledge designed to make complex information more comprehensible, including various types such as charts, flowcharts, and maps.
+
+DocVQA Dataset(Mathew et al., 2021) Specialized for document visual question answering tasks, this dataset focuses on textual and graphic elements within documents, assessing machine learning models' performance in understanding and answering questions related to document content. It includes roughly 11,000 images and 40,000 questions.
+
+VisualMRC Dataset(Tanaka et al., 2021) Given a question and a document image, this dataset requires machines to read and comprehend the text within the image and answer in natural language. Compared to existing VQA datasets that involve image text, VisualMRC emphasizes nurturing natural language understanding and generation abilities more. It consists of over 30,000 question-answer pairs abstracted from more than 10,000 document images sourced from diverse web domains.
+
+FUNSD Dataset(Jaume et al., 2019) Comprising 199 fully annotated real scanned forms with 9,707 semantic entities and 31,485 words, this dataset organizes tables by semantic entity lists. Each entity is identified uniquely, tagged (as question, answer, title, or other), bounded by a box, accompanied by a list of relations to other entities, and a word list. The dataset is split into 149 training and 50 testing samples.
+
+SROIE Dataset(Huang et al., 2019) This dataset contains 626 receipts for training and 347 for testing. Each receipt is organized as a list of bounding-boxed text lines. Four types of entities are labeled on each receipt: company, date, address, and total.
+
+CORD Dataset(Park et al., 2019) Created for post-OCR parsing, this is the first public dataset containing 11,000 receipt images from Indonesian shops and restaurants. Collected through crowdsourcing, each image was initially annotated using a web-based tool and then reviewed for accuracy and adherence to annotation guidelines. To avoid unintentional disclosure of sensitive personal data, sensitive information like credit card numbers or full names was blurred in the final receipt dataset.
+
+In addition to the document-related public datasets mentioned above, we have also gathered extra data for fine-tuning LLM instructions. This additional data includes BELLE-CN (3.5 million entries)(BELLEGroup, 2023), FireFly (1.1 million entries)(Yang, 2023), OpenOrca(Yang, 2023), and ShareGPT (90k entries from public). We start by combining these instructions and then randomly select a portion of them to mix with the document data. Through experimentation, we have found that a mixing ratio of document data to text instruction data at 2:1 achieves a well-balanced effect.
+
+# A.2 DETAILS OF BLADE
+
+Overview. BLADE comprises six scenarios, SynthTables, Forms, Slides, Websites, SynthDocs, and Newspapers, totaling 3,007 documents and 7,924 question-answer pairs. The specific data volume for each scenario is illustrated in Table-7. The construction of BLADE involves both manual creation and leveraging large language models. The images for Forms are sourced from the (Nan et al., 2022), and those for Newspapers come from the (Cheng et al., 2023). The images for Slides and Websites are obtained through web crawling. The corpus for SynthTables originates from entity names and values extracted from Chinese entity extraction datasets, including (Li et al., 2020),(Tianchi, 2021).
+
+The corpus for SynthDocs is derived from public reading comprehension datasets, including (He et al., 2017),(Cui et al., 2018),(Rajpurkar et al., 2018),(Wang et al., 2020).
+
+<table><tr><td>Year</td><td>Series</td><td>Role</td><td>Genre</td></tr><tr><td>2013</td><td>Dirt Cheap</td><td>Ron Furick</td><td>Comedy</td></tr><tr><td>2012</td><td>Any Questions for Ben?</td><td>Brand Manager</td><td>Comedy</td></tr><tr><td>2011</td><td>It&#x27;s Awkward (short)</td><td>Hero</td><td>Comedy</td></tr><tr><td>2006</td><td>Alex and Alexa (short)</td><td>Alex</td><td>Drama</td></tr></table>
+
+Scheme 1: Ask for all values corresponding to a header A.
+
+Q: What is the corresponding content of 'Genre' in the Table?
+
+A: Comedy\nComedy\nComedy\nDrama
+
+Scheme 2: Ask for the value of header B when the value of header A is x.
+
+Q: When the 'Year' column is '2011', what is the corresponding value in the 'Role'?
+
+A: Hero
+
+Scheme 3: Ask for the content of the nth row.
+
+Q: "In the table, excluding the header row, what is the content of the 2nd row?"
+
+A: 2012\nAny Questions for Ben? \nBrand Manager\nComedy
+
+Scheme 4: Ask for the value at the nth row and mth column or the mth column and nth row.
+
+Q: "What is the 2nd 'Role'?" or "What is the 2nd value corresponding to 'Role'?"
+
+A: Brand Manager
+
+Figure 5: Types of questions from SynthTables.
+
+Data construction. For the data scenarios of Forms, Slides, and Websites, the predominant method employed was manual annotation, where annotators would select question and answer pairs based on image information, followed by a manual screening process to filter out challenging question-answer pairs. In the case of Newspapers and SynthDocs, given their characteristic of extensive text blocks, a different approach was adopted, utilizing the power of large language models for generation. Specifically, this involved first identifying a key segment within the text, then tasking the large model with constructing question-answer pairs based on that key segment using predefined templates. This way, question-answer pairs that met the required criteria are synthesized.
+
+The distinction between Newspapers and SynthDocs lies in the former's need for an initial manual selection of a text segment from the document, whereas SynthDocs, being synthetically generated, allows for an automated process of segment selection, which is then fed into the large model's pipeline for generating answer pairs. This is because a given key segment text is required in advance for LLM to generate QA pairs. Newspapers are typically documents with complex layouts, making it difficult to obtain the correct reading order through simple OCR sorting. Therefore, we first have annotators to select one key segment, usually a small paragraph, from which the reading order within this small paragraph can be obtained through simple OCR sorting. SynthDocs uses synthetic documents, for which the reading order of the text is already known, thus eliminating the need for manual annotation. The generation pipeline of SynthDocs is shown in Figure-6.
+
+For SynthTables, we also designed an automated pipeline to generate table-question-answer pairs. Specifically, we begin by constructing a set of synthetic tables based on the collected entity data. Following this, SQL is employed to generate a series of question templates on these tables. The questions can be categorized into 4 types, shown in Figure-5. Lastly, we execute the question templates to obtain answers. Through this process, we get a substantial number of table-question-answer pairs.
+
+# Coupus From MRC
+
+然而，水蛭及其近亲的身体结构在这个群体中是非常一致的……个体腔相隔甚远，贯穿整个身体。它们是主要的血管，虽然它们是并排的，而不是上下的……
+
+However, the body structure of leeches and their dose relatives is very consistent in this group... individual cavities are far apart and run through the entire body. They are the main blood vessels, although they are side by side, not up and down....
+
+![](images/5a4e6a5f49f717d8f3233bb5a46b21ed57a4dc2bb7b876881057e1b59433d8de.jpg)
+
+# Image with complex layout generation
+
+![](images/881e942e5c1a4b08a414a761524085f10a362e1a1764afe4ef5d7752807e886c.jpg)  
+Figure 6: The data generation flowchart of SynthDocs.
+
+![](images/d6aa1162917ebfc139b8d4f03de56162ac78c628e686d3707e5f9b5210c2cd2b.jpg)
+
+![](images/3ef03af56aa6c9a537b94ae34e859e1947d46b32970c37ea3789e9b07ce0484a.jpg)
+
+![](images/792cfe33ab366e976119938f8e42204a7f18be9c162e0542d026c70cbd51aa10.jpg)
+
+![](images/5baa0152d4f7cf7864f04517141bdea4f9f81ef4afd2606e47154c69d3fba52c.jpg)
+
+![](images/64c16ab0947e9acbb4eadc1770a1838dd4b33b141af572a09675e58f3b7f1ecb.jpg)
+
+![](images/a6582140848adcb996d5437190152b36005fd39b2e1b9fb0adae0178014dac0d.jpg)
+
+Sorted OCR From Top Left To Bottom Right
+
+<bbox><x01><y01><x02><x02></bbox> text0
+
+<bbox><x11><y11><x12><x12></bbox> text1
+
+<bbox><x21><y21><x22><x22></bbox> text2
+
+# QA Generation By LLM Self-Instruct
+
+Prompt Example
+
+给定文本如下：{Corpus} 请你根据这段文本提出2个问题，答案不少于10个字。
+
+Prompt: The given text is as follows: {corpus} Please propose two questions based on this text, with answers no less than 10 words.
+
+LLM Answer
+
+Query: "水蛭及其近亲的身体结构有何特点，与其它环节动物有何不同？"
+
+Anaswer: "水蛭及其近亲没有隔，……，这与其它环节动物不同。"
+
+Query: "What are the characteristics of the body structure of leeches and their close relatives, and how are they different from other link animals?"
+
+Answer: "Leeches and their close relatives are not separated, ..., which is different from other link animals."
+
+# Final Data
+
+Instruction: ["参考下面的段落，回答下列问题。“然而，水蛭及其”，“个体腔相隔甚远，贯穿整个身体。它们是主要的血”，“动物既没有下颚”，“近亲的身体结构”，“管，虽然它们是并排的，而不是上下的，然而，它们是”，“也没有不可逆”，“问题：水蛭及其近亲的身体结构有何特点，与其它环节动物有���不同？”]Bbox: [[0,0,0,0], [32,6,242,45], ..., [0,0,0,0]]
+
+Instruction: ["Referring to the following paragraphs, answer the following questions." However, leeches and their individual cavities are far apart and run through the entire body. They are the main blood "", animals have neither jaws "", the body structure of their dose relatives "", tubes, although they are side by side, not up and down, however, they are "", and there is no irreversibility ", ....," Question: What are the characteristics of the body structure of leeches and their dose relatives, and how are they different from other link animals?"] Bbox: [[0,0,0,0], [32,6,242,45], ..., [0,0,0,0]]
+
+Table 7: Statistics of BLADE.  
+
+<table><tr><td>params</td><td>SynthTables</td><td>Forms</td><td>Slides</td><td>Websites</td><td>SynthDocs</td><td>Newspapers</td></tr><tr><td>#docs</td><td>1110</td><td>162</td><td>111</td><td>174</td><td>1000</td><td>450</td></tr><tr><td>#qas</td><td>2287</td><td>296</td><td>518</td><td>732</td><td>2000</td><td>2091</td></tr></table>
+
+These synthetic tables are subsequently rendered into images, with the detailed procedure outlined in Figure-7.
+
+Quality Verification. For the SynthDocs data, given that the QA pairs are generated by a LLM, there may be instances of hallucination. To ensure the quality of these generated questions, we utilize LLMs for a screening process during this phase. The quality check primarily focuses on assessing the relevance between the generated questions and answers in relation to the provided text
+
+![](images/eaf8deb1da9bd2d009a2a50a99cdcc606dd67420d9e8aef24c4412c0afd90eb1.jpg)  
+Figure 7: The data generation flowchart of SynthTables.
+
+content, as well as determining whether any hallucinations have occurred. The specific prompt used for quality screening is as follows:
+
+# Prompt for quality verification
+
+Given a text: {text}, there is a question and its corresponding answer, question: {question}, answer: {answer}. Please make judgments on the following aspects:
+
+1. Can the question be answered solely based on the content of the material? Please only respond with "Yes" or "No".  
+2. Is the answer given based on the provided text? Please only respond with "Yes" or "No".  
+3. Is the answer correct? Please only respond with "Yes" or "No".  
+Return in the format:"""{ {1: "Yes/No",2:"Yes/No",3:"Yes/No"}}"
+
+# A.3 ANALYSIS OF COMMON DOCUMENT BENCHMARKS
+
+In this section, we discuss the characteristics of several commonly used document evaluation benchmarks, specifically two Document Visual Question Answering (VQA) datasets, DocVQA and VisualMRC, as well as three Visual Information Extraction (VIE) datasets, FUNSD, CORD, and SROIE. Based on our previous experiments, the majority of these evaluation tasks fail to assess a model's capability to understand layout information. A robust text large language model, such as qwen2, can generally achieve high scores on these tasks. Our proposed method, GPE, offers limited improvement on top of such a model, and in the case of VisualMRC, including our approach, various methods we implemented to augment layout information actually led to performance degradation, which we attribute to potential impairment of the model's text comprehension abilities.
+
+To fully comprehend the implications behind the numerical variations across these five benchmark datasets, we first examine from the sample level the demands they place on model capabilities. As illustrated in Figure-8 and Figure-9, DocVQA encompasses a variety of photographed documents, where we find that while some questions may necessitate the integration of layout information, the majority can be answered without it, relying solely on the understanding of text paragraphs, as exemplified in samples Figure-8(a). VisualMRC is a collection of various multimedia documents, where accurately answering questions does not require layout information. FUNSD consists of form-like data, similar to DocVQA, yet still presents instances where questions and answers are so straightforward that layout understanding is unnecessary for correct responses. CORD and SROIE involve diverse receipt types, with questions predominantly focused on recognizing entities within receipts.
+
+Evidently, intuitively analyzing these five benchmark sets reveals that except for a minority of questions in DocVQA and FUNSD that might benefit from layout comprehension, the vast majority of tasks, in principle, can be readily tackled with sufficiently powerful text understanding capabilities. Particularly, VisualMRC is fundamentally a reading comprehension task based on large segments of natural text, assessing a model's ability to extract and summarize information from paragraphs. This underscores the motivation behind introducing BLADE, to provide a more nuanced evaluation that goes beyond simple text comprehension and truly examines a model's capacity to understand the layout and structure inherent in document data.
+
+# A.4 TYPE OF POSITION EMBEDDING
+
+In this section, we investigate the impact of the difference between the positional encoding used by GPE and the model's native positional encoding on the performance of downstream tasks. In previous experiments, we implemented two types of encoding: GPE-RoPE and GPE-Sinusoidal. While GPE-Sinusoidal also brings improvements in document tasks, it does not match the performance boost achieved when GPE-RoPE employs the same encoding scheme as the model. Here, we experiment with altering the base scale of GPE-RoPE and observe its performance on BLADE. The results, summarized in Table-8, show that a scale of 1e6 is what the model originally utilizes.
+
+We observe that, on benchmarks akin to reading comprehension question-answering tasks such as Newspapers and SynthDocs, settings close to the original scale yield better performance. For several other categories of tasks, scales approximately around the base scale, like 1e5 and 1e7, exhibit comparable performance. However, significantly deviating from this range, as seen with 1e4, leads to noticeable declines across all tasks. Our experiments have consistently revealed that the design of GPE has differential effects on two types of tasks: those involving extensive natural paragraphs, which emphasize understanding natural language, and those filled with tabular data, which empha
+
+![](images/ab4865e9012a4860a1f12e16cebb9beafa850b50ddd14045fb2ad0ab092524f8.jpg)  
+Question: What is code 4 for?
+
+![](images/0a65654d7e552e17a340a26195b8564864bbfbfbd5b606e9000066225b517319.jpg)  
+Answer: Above Average  
+(a)
+
+# Matthew Lesko
+
+Matthew Lesko is one of the most authentic and courageous humans who has ever lived. So much so that I have started referring to his attitude about living unapologetically the life you are meant to lead "Lesko's Law."
+
+# Lesko's Law
+
+He might not like my term for one specific quote from Lesko during an interview, but I will never forget it.
+
+"See, to me, this is me inside. And it took so long to bring it out. And I said, well I'm on my third business I'm gonna have fun. You think this is crazy? We're saving each other lots of time. I'm probably not gonna like you either. This is who I really am." (Matthew Lesko)
+
+![](images/4551e09bc878f71fa0242a5f8786d5c091715361877cc485f934b76e24fc0314.jpg)  
+Question: What is "Lesko's Law" referring to?  
+(b)  
+Figure 8: Samples from DocVQA and VisualMRC. (a) DocVQA comprises a variety of photographed documents. Some questions may necessitate the integration of layout information (the upper case), the majority can be answered without it (the bottom case). (b) VisualMRC is a collection of various multimedia documents. Most of the designed question does not need the layout information.
+
+![](images/c45af2ed6372049f9445577f9a1b83a858b9a1589e5f4bf7558f93f76c83ad04.jpg)  
+(a)  
+Question: What is in the filter production field?  
+Answer: MFG
+
+![](images/bda5e6641bfdc9fbfd23f269a13e79b893a2998655c4891ff7b5176228a60355.jpg)
+
+![](images/c30a9e390fd18b4fc76dce77018793772a0d6795acf4606348363628812dc0d3.jpg)  
+Question: What is the total price?  
+Answer: 31.0
+
+![](images/1068e57de4452c1ad3d1003485a44021be2f2c5adf8ad344961928436c1d8ad0.jpg)  
+(b)  
+Question: What is the "company" in the given document?  
+Answer: TSH POWER HARDWARE TRADING  
+(c)  
+Figure 9: Samples from FUND, CORD and SROIE. (a) FUNSD consists of form-like data. Some questions may need layout information, others not. (b) CORD involves diverse receipt types. Most questions are designed like entity recognition. Layout information is unimportant for accomplishing this task.(c) SROIE is similar to CORD.
+
+Table 8: Ablation on type of position embedding. Qwen2-7B is selected as the base model.  
+
+<table><tr><td>setting</td><td>SynthTables</td><td>Forms</td><td>Slides</td><td>Websites</td><td>SynthDocs</td><td>Newspapers</td></tr><tr><td>Baseline</td><td>50.3</td><td>33.0</td><td>42.2</td><td>46.1</td><td>53.3</td><td>43.8</td></tr><tr><td>GPE-Sinusodal</td><td>67.2</td><td>45.6</td><td>48.3</td><td>66.5</td><td>55.2</td><td>48.7</td></tr><tr><td>GPE-RoPE with base scale 1e4</td><td>77.4</td><td>54.4</td><td>59.8</td><td>79.1</td><td>58.9</td><td>48.1</td></tr><tr><td>GPE-RoPE with base scale 1e5</td><td>80.9</td><td>60.2</td><td>63.4</td><td>75.3</td><td>61.4</td><td>49.7</td></tr><tr><td>GPE-RoPE with base scale 1e6</td><td>77.3</td><td>57.2</td><td>64.5</td><td>79.3</td><td>62.2</td><td>52.2</td></tr><tr><td>GPE-RoPE with base scale 1e7</td><td>77.1</td><td>58.7</td><td>63.0</td><td>76.6</td><td>61.6</td><td>49.0</td></tr></table>
+
+size understanding layout relationships. This suite of experimental outcomes suggests that GPE can be optimized for these two types of tasks.
+
+In this work, we adopt a uniform encoding scheme across all heads by default. Exploring more suitable encoding designs separately for the reading sequence component and the coordinate box component, aiming for an even more advantageous GPE configuration, is left as a direction for future research.
+
+# A.5 EFFICIENT TUNING USING LORA
+
+In previous experiments, we employed full parameter training for our model. Under the setting of full parameter updates, GPE outperforms comparable methods with significantly less data, achieving superior results within a mere few tuning steps. Nonetheless, conducting full parameter updates on large models remains highly resource-intensive. Currently, there exist lightweight training approaches in industry that are based on LoRA, which aim to conserve GPU memory and expedite training by updating only a portion of the model's parameters. The trade-off is that the final performance of the model may not match that of full parameter training; however, this discrepancy varies across different scenarios. In this section, we integrate the training methodologies of GPE with LoRA and examine the model's performance at different steps on BLADE, as illustrated in Figure-10. Our findings reveal that, compared to full parameter training, the LoRA approach requires a longer training duration to reach an equivalent level across all scenarios. Nevertheless, considering that LoRA training for a 7-billion-parameter model can be executed on a single 80GB A100 GPU, the combination of GPE with LoRA presents an efficiently applicable solution in most practical scenarios.
+
+![](images/7d302feeace62dfbf26d0ec7c8f75162e8ba5b5e75fc027635c5fe25ff52c831.jpg)  
+Figure 10: The comparison between full-parameter tuning and LoRA-Tuning on BLADE.
+
+# A.6 INFLUENCE ON BASE MODEL CAPABILITY
+
+We observe that the capabilities of foundation language models are significantly influenced by the document training data they are subjected to. Through experimentation, we have found that this is primarily due to the quality of the document data. Using Qwen-7B as our base model, we mixed textual instruction data with document-type data and adjusted the blending ratio to observe the model's performance on MMLU-Pro(Wang et al., 2024). We discovered that the untrained original Qwen model achieved the best score of 39.55 on MMLU-Pro, but once document-type data was incorporated, the model's text capability was notably impaired. Furthermore, as the proportion of mixed document tasks increased, the decline became more pronounced, aligning closely with the trend observed in GPE. Additionally, under the same data mixing ratio, the performance degradation induced by GPE compared to the original Qwen was not substantial, suggesting that the changes introduced by GPE in position encoding have a minor impact on the model's fundamental abilities. This experiment also revealed that the training data from various document tasks currently integrated can affect the model's inherent capabilities, an aspect that seems to have escaped widespread attention. We believe that this represents a highly promising direction for further research.
+
+Table 9: Influence of data mix ratio on base model's capabilities. The data ratio is organized as textual instruction data vs document data  
+
+<table><tr><td>Setting</td><td>MMLU-Pro</td><td>DocVQA</td></tr><tr><td>GPE with mixed data ratio 1:2</td><td>26.8</td><td>76.0</td></tr><tr><td>GPE with mixed data ratio 1:1</td><td>28.9</td><td>72.2</td></tr><tr><td>GPE with mixed data ratio 2:1</td><td>31.2</td><td>70.8</td></tr><tr><td>Qwen with mixed data ratio 1:2</td><td>27.1</td><td>-</td></tr><tr><td>Qwen with mixed data ratio 1:1</td><td>28.4</td><td>-</td></tr><tr><td>Qwen with mixed data ratio 2:1</td><td>31.6</td><td>-</td></tr><tr><td>Qwen-7B-Instruct</td><td>39.6</td><td>-</td></tr></table>
+
+# A.7 IMPACT OF SCALING FACTOR
+
+In this section, we explore the influence of the scaling factor lambda. We use 1000 in our default implementation. The results are shown in Table-10. It is observed that, with the exception of SynthTables, the variations in performance across the other datasets are relatively minor. However, for SynthTables, higher lambda leads to a significant improvement in performance. We find that this enhancement can be attributed to the dense arrangement of text boxes within SynthTables. In these tables, up to 10 text boxes are often placed in a single row, and their slight positional differences are crucial for determining their column assignments.
+
+By increasing the scaling factor, these subtle positional distinctions become more pronounced, thereby aiding the model in more accurately distinguishing between different cells. This finding underscores the importance of fine-tuning the scaling factor, especially when dealing with datasets that have densely packed and closely positioned elements.
+
+Table 10: The influence of scaling factor.  
+
+<table><tr><td>λ</td><td>SynthTables</td><td>Forms</td><td>Slides</td><td>Websites</td><td>SynthDocs</td><td>Newspapers</td></tr><tr><td>100</td><td>53.3</td><td>55.7</td><td>59.1</td><td>64.3</td><td>62.1</td><td>51.5</td></tr><tr><td>500</td><td>72.1</td><td>56.6</td><td>66.3</td><td>75.7</td><td>61.3</td><td>52.9</td></tr><tr><td>1000(default)</td><td>77.3</td><td>57.2</td><td>64.5</td><td>79.3</td><td>62.2</td><td>52.2</td></tr><tr><td>5000</td><td>86.3</td><td>56.6</td><td>66.6</td><td>79.5</td><td>61.8</td><td>53.6</td></tr></table>
+
+# A.8 INTUITIVE ANALYSIS OF GPE
+
+The key technique that enhances the effectiveness of GPE is its **head-specific design**. This design ensures that each dimension of layout information is processed by each attention head. The benefits of this approach are akin to those of multi-head attention, as it encourages the model to learn positional relationships from various perspectives. By distributing the layout relations across multiple attention scores, rather than relying on a single score, the model can more effectively capture and utilize different aspects of the layout. Intuitively, some heads might focus on horizontal (left-to-right) relationships, while others concentrate on vertical (up-to-down) relationships.
+
+To intuitively illustrate the impact of GPE on different attention heads, we use heatmaps to visualize the influence of positional information across various attention heads. Specifically, we construct a 2D grid sequence of size  $256 \times 256$ , where each grid point corresponds to a token input into the LLM, and its spatial position is represented by the grid coordinates  $[x, y]$ . We define the reading order of the grid points using a left-to-right, top-to-bottom scanline sequence. This way, we establish the positional information  $\mathbf{P} = [m, x, y]$  for the grid points. We then encode  $\mathbf{P}$  using GPE-RoPE with an embedding dimension of 128. We use the grid point at the bottom right corner as the query and observe its attention scores with all other grid points based on their positional information. The results are shown in Figure-11. The visualization clearly demonstrates the contribution of different positional information to the attention scores of different heads.
+
+![](images/8e84a5cee03e06f0220949edc1d97a99977d25300fe63c4f9a1bf97e4f127363.jpg)  
+(a)
+
+![](images/bda763019e2acaf514352860cf0486800439511e79e52045b36b0d53fc773710.jpg)  
+(b)  
+Figure 11: Visualization of attention heatmaps of each head. (a) Attention map with reading order. (b) Attention map with coordinate  $x$ . (c) Attention map with coordinate  $y$ .
+
+![](images/3a2916bdb9b05c8b167ff9a6c25734008a56fc27b52d8cf05ffd6409f5b5490d.jpg)  
+(c)

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+    {
+        "type": "text",
+        "text": "ENHANCING END-TO-END AUTONOMOUS DRIVING WITH LATENT WORLD MODEL",
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+        "text": "Yingyan Li $^{1,2,3,4}$",
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+    },
+    {
+        "type": "text",
+        "text": "Lue Fan $^{1,2,3}$",
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+    {
+        "type": "text",
+        "text": "Jiawei He1,2,3",
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+        "text": "Yuqi Wang1,2,3",
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+        "text": "Yuntao Chen $^{1,2,3}$",
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+        "type": "text",
+        "text": "Zhaoxiang Zhang $^{1,2,3,4}$",
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+    {
+        "type": "text",
+        "text": "Tieniu Tan1,2,3",
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+        "type": "list",
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+        "list_items": [
+            "$^{1}$  Institute of Automation, Chinese Academy of Sciences (CASIA)",
+            "$^{2}$  New Laboratory of Pattern Recognition (NLPR)",
+            "3 State Key Laboratory of Multimodal Artificial Intelligence Systems (MAIS)",
+            "$^{4}$  School of Future Technology, University of Chinese Academy of Sciences (UCAS)"
+        ],
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+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
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+        ],
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+        "text": "In autonomous driving, end-to-end planners directly utilize raw sensor data, enabling them to extract richer scene features and reduce information loss compared to traditional planners. This raises a crucial research question: how can we develop better scene feature representations to fully leverage sensor data in end-to-end driving? Self-supervised learning methods show great success in learning rich feature representations in NLP and computer vision. Inspired by this, we propose a novel self-supervised learning approach using the LAtent World model (LAW) for end-to-end driving. LAW predicts future scene features based on current features and ego trajectories. This self-supervised task can be seamlessly integrated into perception-free and perception-based frameworks, improving scene feature learning and optimizing trajectory prediction. LAW achieves state-of-the-art performance across multiple benchmarks, including real-world open-loop benchmark nuScenes, NAVSIM, and simulator-based closed-loop benchmark CARLA. The code is released at https://github.com/BraveGroup/LAW.",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
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+    },
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+        "text": "End-to-end planners (Hu et al., 2022c; Jiang et al., 2023; Prakash et al., 2021; Wu et al., 2022; Hu et al., 2022b; Zhang et al., 2022; Wu et al., 2023) have garnered significant attention due to their distinct advantages over traditional planners. Traditional planners operate on pre-processed outputs from perception modules, such as bounding boxes and trajectories. In contrast, end-to-end planners directly utilize raw sensor data to extract scene features, minimizing information loss. This direct use of sensor data raises an important research question: how can we develop more effective scene feature representations to fully leverage the richness of sensor data in end-to-end driving?",
+        "bbox": [
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+        "text": "In recent years, self-supervised learning has emerged as a powerful method for extracting comprehensive feature representations from large-scale datasets, particularly in fields like NLP (Devlin, 2018) and computer vision (He et al., 2022). Building on this success, we aim to enrich scene feature learning and further improve end-to-end driving performance through self-supervised learning. Traditional self-supervised methods in computer vision(He et al., 2022; Chen et al., 2020b) often focus on static, single-frame images. However, autonomous driving relies on continuous video input, so effectively using temporal data is crucial. Temporal self-supervised tasks, such as future prediction (Han et al., 2019; 2020), have shown promise. Traditional future prediction tasks often overlook the impact of ego actions, which play a crucial role in shaping the future in autonomous driving.",
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+        "type": "text",
+        "text": "Considering the critical role of ego actions, we propose utilizing a world model to predict future states based on the current state and ego actions. While several image-based driving world models (Wang et al., 2023b; Hu et al., 2023a; Jia et al., 2023a) exist, they exhibit inefficiencies in enhancing scene",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>Email: liyingyan2021@ia.ac.cn",
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+        "text": "1",
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+    {
+        "type": "image",
+        "img_path": "images/0dd07acd177b029ce279dea1e097861ec2d30efbd67094b69c3e58c8301f9b67.jpg",
+        "image_caption": [
+            "(a) Annotation-based Perception Tasks",
+            "(b) Our Self-supervised Task",
+            "Figure 1: The illustration of our self-supervised method. Traditional methods utilize perception annotations to assist with scene feature learning. In contrast, our self-supervised approach uses temporal information to guide feature learning. Pred.: predicted. Seg.: segmentation. The blue part indicates the pipeline of an end-to-end planner."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "feature representations due to their reliance on diffusion models, which may take several seconds to generate images of a future scene. To address this limitation, we introduce a latent world model designed to predict future latent features directly from the current latent features and ego actions, as depicted in Fig. 1. Specifically, given images, a visual encoder extracts scene features (current state), which are then fed into an action decoder to predict ego trajectory. Based on the current state and action, the latent world model predicts the scene feature of the future frame. During training, the predicted future features are supervised using the extracted features from the future frame. By supervising the predicted future feature, this self-supervised method jointly optimizes the current scene feature learning and ego trajectory prediction.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "After introducing the concept of the latent world model, we turn our attention to its universality across various end-to-end autonomous driving frameworks. In end-to-end autonomous driving, the frameworks can generally be categorized into two types: perception-free and perception-based. Perception-free approaches (Toromanoff et al., 2020; Chen et al., 2020a; Wu et al., 2022) bypass explicit perception tasks, relying solely on trajectory supervision. Prior work Wu et al. (2022) in this category typically extracts perspective-view features to predict future trajectory. In contrast, perception-based approaches (Prakash et al., 2021; Hu et al., 2022c; Jiang et al., 2023; Hu et al., 2022b) incorporate perception tasks, such as detection, tracking, and map segmentation, to guide scene feature learning. These methods generally use BEV feature maps as a unified representation for these perception tasks. Our latent world model accommodates both frameworks. It can either predict perspective-view features in the perception-free setting or predict BEV features in the perception-based setting, showcasing its universality across different autonomous driving paradigms.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experiments show that our latent world model enhances performance in both perception-free and perception-based frameworks. Furthermore, we achieve state-of-the-art performance on multiple benchmarks, including the real-world open-loop datasets nuScenes (Caesar et al., 2020) and NAVSIM (Dauner et al., 2024) (based on nuPlan (Caesar et al., 2021)), as well as the simulator-based closed-loop CARLA benchmark (Dosovitskiy et al., 2017). These results underscore the efficacy of our approach and highlight the potential of self-supervised learning to advance end-to-end autonomous driving research. In summary, our contributions are threefold:",
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+    {
+        "type": "text",
+        "text": "- Future prediction by latent world model: We introduce the LAtent World model (LAW) to predict future scene latents from current scene latents and ego trajectories. This self-",
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+        "type": "text",
+        "text": "supervised task jointly enhances scene representation learning and trajectory prediction in end-to-end driving.",
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+            "- Cross-framework universality:  $LAW$  demonstrates universality across various common autonomous driving paradigms. It can either predict perspective-view features in the perception-free framework or predict BEV features in the perception-based framework.",
+            "- State-of-the-art performance: Our self-supervised approach achieves state-of-the-art results on the real-world open-loop nuScenes, NAVSIM, and the simulator-based close-loop CARLA benchmark."
+        ],
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+    {
+        "type": "text",
+        "text": "2 RELATED WORKS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 END-TO-END AUTONOMOUS DRIVING",
+        "text_level": 1,
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+        "text": "We divide end-to-end autonomous driving methods (Hu et al., 2022c; Jiang et al., 2023; Renz et al., 2022; Toromanoff et al., 2020; Tian et al., 2024; Hwang et al., 2024; Pan et al., 2024; Wang et al., 2024a;b) into two categories, perception-based methods and perception-free methods, depending on whether performing perception tasks. Perception-based methods (Casas et al., 2021; Prakash et al., 2021; Jaeger et al., 2023; Shao et al., 2023; Hu et al., 2022b; Sadat et al., 2020) perform multiple perception tasks simultaneously, such as detection (Li et al., 2022b; Huang et al., 2021; Li et al., 2022a; 2024a), tracking (Zhou et al., 2020; Wang et al., 2021), map segmentation (Hu et al., 2022c; Jiang et al., 2023) and occupancy prediction (Wang et al., 2023a; Huang et al., 2023). As a representative, UniAD (Hu et al., 2022c) integrates multiple modules to support goal-driven planning. VAD (Jiang et al., 2023) explores vectorized scene representation for planning purposes.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Perception-free end-to-end methods (Toromanoff et al., 2020; Chen et al., 2020a; Zhang et al., 2021; Wu et al., 2022) present a promising direction as they avoid utilizing a large number of perception annotations. Early perception-free end-to-end methods (Zhang et al., 2021; Toromanoff et al., 2020) primarily relied on reinforcement learning. For instance, MaRLn (Toromanoff et al., 2020) designed a reinforcement learning algorithm based on implicit affordances, while LBC (Chen et al., 2020a) trained a reinforcement learning expert using privileged (ground-truth perception) information. Using trajectory data generated by the reinforcement learning expert, TCP (Wu et al., 2022) combined a trajectory waypoint branch with a direct control branch, achieving good performance. However, perception-free end-to-end methods often suffer from inadequate scene representation capabilities. Our work aims to address this issue through the latent world model.",
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+    {
+        "type": "text",
+        "text": "2.2 WORLD MODEL IN AUTONOMOUS DRIVING",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Existing world models in autonomous driving can be categorized into two types: image-based world models and occupancy-based world models. Image-based world models (Hu et al., 2022a; Wang et al., 2023b; Hu et al., 2023a) aim to enrich the autonomous driving dataset through generative approaches. GAIA-1 (Hu et al., 2023a) is a generative world model that utilizes video, text, and action inputs to create realistic driving scenarios. MILE (Hu et al., 2022a) produces urban driving videos by leveraging 3D geometry as an inductive bias. Drive-WM (Wang et al., 2023b) utilizes a diffusion model to predict future images and then plans based on these predicted images. Copilot4D (Zhang et al., 2023) tokenizes sensor observations with VQVAE (Van Den Oord et al., 2017) and then predicts the future via discrete diffusion. Another category involves occupancy-based world models (Zheng et al., 2023; Min et al., 2024). OccWorld (Zheng et al., 2023) and DriveWorld (Min et al., 2024) use the world model to predict the occupancy, which requires occupancy annotations. On the contrary, our proposed latent world model requires no manual annotations.",
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+    {
+        "type": "text",
+        "text": "3 PRELIMINARY",
+        "text_level": 1,
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+        "text": "Vision-based End-to-end Autonomous Driving In the task of end-to-end autonomous driving, the objective is to estimate the future trajectory of the ego vehicle in the form of waypoints. Formally, let  $\\mathbf{I}_t = \\{\\mathbf{I}_t^1,\\mathbf{I}_t^2,\\ldots ,\\mathbf{I}_t^N\\}$  be the set of  $N$  surrounding multi-view images captured at time step  $t$ . We expect the model to predict a sequence of waypoints  $\\mathbf{W}_t = \\{\\mathbf{w}_t^1,\\mathbf{w}_t^2,\\dots ,\\mathbf{w}_t^M\\}$ , where each",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
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+        "page_idx": 2
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+    {
+        "type": "image",
+        "img_path": "images/5d6c4d3fa6b2cb77cfcb5b3c9a02f11c9b613b4fd73e5361e311d446ddf76ec2.jpg",
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+        "image_caption": [
+            "Perception-free with Perspective-view Features"
+        ],
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+        "image_caption": [
+            "Perception-based with BEV Features",
+            "Figure 2: The Overall Framework. The encoder extracts visual latents from the images, while the decoder predicts waypoints based on these latents. Our latent world model predicts future visual latents using the visual latents and waypoints of the current frame. During training, the predicted visual latents are supervised by the extracted latents from the future frame. The latent world model is compatible with both perception-free and perception-based frameworks, which differ in their encoder and decoder structures. In the perception-based framework, the supervision icon indicates that map annotations supervise the output of the map construction module, while the agent's future trajectory supervises the output of the motion prediction module. Pred.: predicted."
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "waypoint  $\\mathbf{w}_t^i = (x_t^i, y_t^i)$  represents the predicted BEV position of the ego vehicle at time step  $t + i$ .  $M$  represents the number of future positions of the ego vehicle that the model aims to predict.",
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+    },
+    {
+        "type": "text",
+        "text": "World Model A world model aims to predict future states based on the current state and action. In the autonomous driving task, let  $\\mathbf{S}_t$  represent the states at time step  $t$ ,  $\\mathbf{W}_t = \\{\\mathbf{w}_t^1, \\mathbf{w}_t^2, \\dots, \\mathbf{w}_t^M\\}$  denote the sequence of predicted waypoints by the planner, the world model predicts state  $\\mathbf{S}_{t+1}$  at time step  $t+1$  using  $\\mathbf{S}_t$  and  $\\mathbf{W}_t$ .",
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+    {
+        "type": "text",
+        "text": "4 METHODOLOGY",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our methodology is composed of three key components: i) Latent World Model: We utilize the latent world model to realize the self-supervised task. This model takes two inputs: latent features extracted by the visual encoder and waypoints predicted by the waypoint decoder. This task is compatible with two common frameworks. ii) Perception-free framework with perspective-view latents: it consists of the perspective-view encoder and perception-free decoder within this framework. iii) Perception-based framework with BEV latents: it contains the BEV encoder and perception-based decoder within this framework.",
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+        "type": "text",
+        "text": "4.1 LATENT WORLD MODEL",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we utilize the latent world model to predict the visual latents of the future frame based on the current visual latents and waypoints.",
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+    {
+        "type": "text",
+        "text": "Visual Latents and Waypoints Extraction The visual encoder processes the images from the current  $t$  time step to produce the corresponding visual latent feature set",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} _ {t} = \\{\\mathbf {v} _ {t} ^ {1}, \\mathbf {v} _ {t} ^ {2}, \\dots , \\mathbf {v} _ {t} ^ {L} \\},\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $L$  denotes the number of feature vectors, and each vector  $\\mathbf{v}_t^i\\in \\mathbb{R}^D$ , with  $D$  representing the number of feature channels. These feature vectors can be derived from various sources, such as a flattened image feature map or a flattened BEV feature map. Based on the  $\\mathbf{V}_t$ , the waypoint decoder",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "4",
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+    {
+        "type": "text",
+        "text": "predicts the waypoints  $\\mathbf{W}_t = \\{\\mathbf{w}_t^1, \\mathbf{w}_t^2, \\dots, \\mathbf{w}_t^M\\}$ .  $M$  represents the number of waypoints, where each waypoint  $\\mathbf{w}_t^i = (x_t^i, y_t^i)$ .",
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+    {
+        "type": "text",
+        "text": "Action-aware Latents Construction We produce action-aware latents by integrating visual latents and waypoints. The action-aware latents are then used as input to the latent world model. Let  $M$  represent the number of waypoints, with each waypoint  $\\mathbf{w}_t^i = (x_t^i, y_t^i)$ . We first reshape  $\\mathbf{W}_t$  which has the shape  $[M, 2]$ , into a one-dimensional vector  $\\widetilde{\\mathbf{w}}_t \\in \\mathbb{R}^{2M}$ . Then, we concatenate each visual latent  $\\mathbf{v}_t^i$  with the waypoint vector  $\\widetilde{\\mathbf{w}}_t$  along the feature channel dimension. The resulting concatenated vector is passed through an MLP to produce the action-aware latent  $\\mathbf{a}_t^i$ , which has the same shape as  $\\mathbf{v}_t^i$ . Formally, the action-aware latent for the  $i$ -th feature vector is expressed as",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {a} _ {t} ^ {i} = \\operatorname {M L P} \\left(\\left[ \\mathbf {v} _ {t} ^ {i}, \\widetilde {\\mathbf {w}} _ {t} \\right]\\right), \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $[\\cdot, \\cdot]$  denotes the concatenating operation. The full set of action-aware latents is denoted as  $\\mathbf{A}_t = \\{\\mathbf{a}_t^1, \\mathbf{a}_t^2, \\ldots, \\mathbf{a}_t^L\\}$ , where  $L$  denotes the number of feature vectors.",
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+    {
+        "type": "text",
+        "text": "Future Latent Prediction The latent world model utilizes the action-aware latents to predict the future visual latents as follows. Given  $\\mathbf{A}_t$ , we predict visual latents  $\\hat{\\mathbf{V}}_{t+1}$  of the frame  $t+1$  by the latent world model:",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {\\mathbf {V}} _ {t + 1} = \\operatorname {L a t e n t W o r l d M o d e l} \\left(\\mathbf {A} _ {t}\\right). \\tag {2}\n$$\n",
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+    {
+        "type": "text",
+        "text": "The network architecture of the latent world model consists of transformer blocks. Each block contains a self-attention and feed-forward module. The self-attention is performed across the latent feature vectors.",
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+        "type": "text",
+        "text": "Future Latent Supervision During training, we extract the visual latents  $\\mathbf{V}_{t + 1} = \\{\\mathbf{v}_{t + 1}^{1},\\dots ,\\mathbf{v}_{t + 1}^{L}$  from the images of frame  $t + 1$  , which are used as the ground truth to supervise the  $\\hat{\\mathbf{V}}_{t + 1}$  using a Mean Squared Error (MSE) loss function as",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\text {l a t e n t}} = \\frac {1}{L} \\sum_ {i = 1} ^ {L} \\| \\hat {\\mathbf {v}} _ {t + 1} ^ {i} - \\mathbf {v} _ {t + 1} ^ {i} \\| _ {2}. \\tag {3}\n$$\n",
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+    {
+        "type": "text",
+        "text": "The latent world model is compatible with both perception-free and perception-based frameworks. In the following sections, we detail the implementation of these two frameworks.",
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+        "type": "text",
+        "text": "4.2 PERCEPTION-FREE FRAMEWORK WITH PERSPECTIVE-VIEW LATENTS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "First, we introduce our perception-free framework. Previous perception-free frameworks (Wu et al., 2022) typically employ a perspective-view encoder for visual latent extraction and a perception-free decoder for waypoint prediction. Our framework is built upon this established paradigm.",
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+        "type": "text",
+        "text": "Perspective-view Encoder In the perspective-view encoder, we produce visual latents based on multi-view images. Initially, multi-view images are processed by an image backbone to obtain their corresponding image features. Following PETR (Liu et al., 2022), we generate 3D position embeddings for these image features, which are then added to the image features to uniquely distinguish each feature vector. The enriched image features are denoted as  $\\mathbf{F} = \\{\\mathbf{f}^1, \\mathbf{f}^2, \\dots, \\mathbf{f}^N\\}$ , where  $N$  represents the number of views. The shape of  $\\mathbf{f}^i$  is  $[H, W, D]$ , where  $H, W$  represents the height and width of the image feature map and  $D$  is the number of feature channels.",
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+    {
+        "type": "text",
+        "text": "To encode the image features into high-level visual latents suitable for planning, we apply a view attention mechanism. To be specific, for  $N$  views, there are  $N$  corresponding learnable view queries  $\\mathbf{Q}_{\\mathrm{view}} = \\{\\mathbf{q}_{\\mathrm{view}}^{1}, \\mathbf{q}_{\\mathrm{view}}^{2}, \\dots, \\mathbf{q}_{\\mathrm{view}}^{N}\\}$ . Each view query  $\\mathbf{q}_{\\mathrm{view}}^{i}$  undergoes a cross-attention with its corresponding image feature  $\\mathbf{f}^{i}$ , resulting in  $N$  visual latent  $\\mathbf{V}_{\\mathrm{pf}} = \\{\\mathbf{v}_{\\mathrm{pf}}^{1}, \\mathbf{v}_{\\mathrm{pf}}^{2}, \\dots, \\mathbf{v}_{\\mathrm{pf}}^{N}\\}$ , where the subscript \"pf\" stands for perception-free. Formally,",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {v} _ {\\mathrm {p f}} ^ {i} = \\operatorname {C r o s s A t t e n t i o n} \\left(\\mathbf {q} _ {\\text {v i e w}} ^ {i}, \\mathbf {f} ^ {i}, \\mathbf {f} ^ {i}\\right), \\tag {4}\n$$\n",
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+    {
+        "type": "text",
+        "text": "where  $\\mathbf{f}^i$  serves as the key and value of the cross attention.  $\\mathbf{V}_{\\mathrm{pf}}$  are then used as input for the perception-free decoder, which we describe next.",
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+    {
+        "type": "text",
+        "text": "Perception-free Decoder The perception-free decoder deocodes waypoint from  $\\mathbf{V}_{\\mathrm{pf}}$ . Specifically, we initialize  $M$  waypoint queries,  $\\mathbf{Q}_{\\mathrm{wp}} = \\{\\mathbf{q}_{\\mathrm{wp}}^{1}, \\mathbf{q}_{\\mathrm{wp}}^{2}, \\ldots, \\mathbf{q}_{\\mathrm{wp}}^{M}\\}$ , where each query is a learnable embedding. These waypoint queries interact with  $\\mathbf{V}_{\\mathrm{pf}}$  through a cross-attention mechanism. The",
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+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
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+    {
+        "type": "text",
+        "text": "updated waypoint queries are then passed through an MLP head to predict the waypoints  $\\mathbf{W} = \\{\\mathbf{w}^1, \\mathbf{w}^2, \\dots, \\mathbf{w}^M\\}$ , which is formulated as",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {W} = \\operatorname {M L P} \\left(\\text {C r o s s A t t e n t i o n} \\left(\\mathbf {Q} _ {\\mathrm {w p}}, \\mathbf {V} _ {\\mathrm {p f}}, \\mathbf {V} _ {\\mathrm {p f}}\\right)\\right). \\tag {5}\n$$\n",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Perception-free Supervision In the perception-free framework, we rely solely on ground truth waypoints for supervision, as no additional perception annotations are provided. We employ an L1 loss to measure the discrepancy between the predicted waypoints  $\\mathbf{W}$  and the ground truth waypoints  $\\tilde{\\mathbf{W}} = \\{\\tilde{\\mathbf{w}}^1,\\tilde{\\mathbf{w}}^2,\\dots ,\\tilde{\\mathbf{w}}^M\\}$  as",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\text {w a y p o i n t}} = \\frac {1}{M} \\sum_ {j = 1} ^ {M} \\| \\mathbf {w} _ {t} ^ {j} - \\tilde {\\mathbf {w}} _ {t} ^ {j} \\| _ {1}, \\tag {6}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thus, the final loss for the perception-free framework is",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {p f}} = \\mathcal {L} _ {\\text {l a t e n t}} + \\mathcal {L} _ {\\text {w a y p o i n t}}. \\tag {7}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4.3 PERCEPTION-BASED FRAMEWORK WITH BEV LATENTS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our latent world model is also compatible with perception-based frameworks, which commonly utilize BEV feature maps for perception tasks. We adhere to this paradigm and the perception-based framework is composed of two key components: a BEV encoder and a perception-based decoder. The BEV encoder generates BEV feature maps from images, while the perception-based decoder uses these maps for perception tasks such as motion prediction and map construction. The final waypoints are then predicted based on the outputs of these perception tasks.",
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+        "type": "text",
+        "text": "BEV Encoder We follow Li et al. (2022b) to encode the BEV feature map. First, we encode the image features using a backbone network. Then, a set of BEV queries projects these image features into BEV features. The resulting BEV feature map is flattened into a shape of  $[K, D]$ , where  $K$  represents the number of feature vectors in the BEV feature map and  $D$  is the number of feature channels. The flattened features are denoted as  $\\mathbf{V}_{\\mathrm{pb}} = \\{\\mathbf{v}_{\\mathrm{pb}}^{1}, \\mathbf{v}_{\\mathrm{pb}}^{2}, \\ldots, \\mathbf{v}_{\\mathrm{pb}}^{K}\\}$ , where the subscript \"pb\" refers to perception-based.",
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+        "type": "text",
+        "text": "Perception-based Decoder Following Jiang et al. (2023), the decoder predicts waypoints with the help of perception tasks, namely motion prediction and map construction. For motion prediction,  $N_{\\text{agent}}$  learnable queries interact with  $\\mathbf{V}_{\\text{pb}}$  via cross-attention to generate agent features  $\\mathbf{F}_{\\text{agent}}$ .  $\\mathbf{F}_{\\text{agent}}$  are then used to predict agent trajectories. Similarly, for map construction,  $N_{\\text{map}}$  learnable queries perform cross-attention with  $\\mathbf{V}_{\\text{pb}}$  to extract map features  $\\mathbf{F}_{\\text{map}}$ .  $\\mathbf{F}_{\\text{map}}$  are then used to predict map vectors. Finally, the learnable waypoint queries perform cross-attention with  $\\mathbf{F}_{\\text{agent}}$  and  $\\mathbf{F}_{\\text{map}}$ . The output is then passed through an MLP head to predict the waypoints.",
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+        "type": "text",
+        "text": "Perception-based Supervision The perception-based framework uses the same waypoint supervision as in equation 6. In addition, it includes losses from perception tasks as",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\text {p e r c e p t i o n}} = \\mathcal {L} _ {\\text {a g e n t}} + \\mathcal {L} _ {\\text {m a p}}. \\tag {8}\n$$\n",
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+    {
+        "type": "text",
+        "text": "Here,  $\\mathcal{L}_{\\mathrm{agent}}$  is the loss for motion prediction and  $\\mathcal{L}_{\\mathrm{map}}$  is the loss for map construction, as defined in (Jiang et al., 2023). The final loss for the perception-based framework is",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {p b}} = \\mathcal {L} _ {\\text {l a t e n t}} + \\mathcal {L} _ {\\text {w a y p o i n t}} + \\mathcal {L} _ {\\text {p e r c e p t i o n}}. \\tag {9}\n$$\n",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "5.1 BENCHMARKS",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "nuScenes (Caesar et al., 2020) The nuScenes dataset contains 1,000 driving scenes. In line with previous works (Hu et al., 2022b; 2023b; Jiang et al., 2023), we use L2 displacement error and collision rate as comprehensive metrics to evaluate planning performance. L2 displacement error measures the L2 distance between the predicted and ground truth trajectories, while collision rate quantifies the frequency of collisions with other objects along the predicted trajectory.",
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+    {
+        "type": "text",
+        "text": "NAVSIM (Dauner et al., 2024) We conducted further experiments using the NAVSIM benchmark, as the nuScenes dataset proved to be overly simplistic. The NAVSIM dataset (Dauner et al., 2024)",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "page_number",
+        "text": "6",
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+    },
+    {
+        "type": "text",
+        "text": "is built on OpenScene (Contributors, 2023), which provides 120 hours of driving logs condensed from the nuPlan dataset (Caesar et al., 2021). NAVSIM enhances OpenScene by resampling the data to reduce the occurrence of simple scenarios, such as straight-line driving. As a result, traditional ego status modeling becomes inadequate under the NAVSIM benchmark. NAVSIM evaluates model performance using the predictive driver model score (PDMS), which is calculated based on five factors: no at-fault collision (NC), drivable area compliance (DAC), time-to-collision (TTC), comfort (Comf.) and ego progress (EP).",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "CARLA (Dosovitskiy et al., 2017) Closed-loop evaluation is essential to autonomous driving as it constantly updates the sensor inputs based on the driving actions. For the closed-loop benchmark, the training dataset is collected from the CARLA (Dosovitskiy et al., 2017) simulator (version 0.9.10.1) using the teacher model Roach (Zhang et al., 2021) following (Wu et al., 2022; Jia et al., 2023b), resulting in 189K frames. We use the widely-used Town05 Long benchmark (Jia et al., 2023b; Shao et al., 2022; Hu et al., 2022a) to assess the closed-loop driving performance. For metric, Route Completion (RC) represents the percentage of the route completed by the autonomous driving model. Infraction Score (IS) quantifies the number of infractions as well as violations of traffic rules. A higher Infraction Score indicates better adherence to safe driving practices. Driving Score (DS) is the primary metric used to evaluate overall performance. It is calculated as the product of Route Completion and Infraction Score.",
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+    {
+        "type": "text",
+        "text": "5.2 IMPLEMENTATION DETAILS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "nuScenes Benchmark We implement both perception-free and perception-based frameworks. In the perception-free framework, Swin-Transformer-Tiny (Swin-T)(Liu et al., 2021) is used as the backbone. Input images are resized to  $800 \\times 320$ . We adopt a Cosine Annealing learning rate schedule(Loshchilov & Hutter, 2016), starting at 5e-5. The model is trained using the AdamW optimizer (Loshchilov & Hutter, 2017) with a weight decay of 0.01, batch size 8, and 12 epochs across 8 A6000 GPUs. For the perception-based framework, following Jiang et al. (2023), we train the model in two stages. In the first stage, we train the encoder and perception head using only perception loss for 48 epochs. In the second stage, we introduce waypoint and latent prediction losses for training another 12 epochs. The network architecture of the latent world model utilizes deformable self-attention for improved convergence.",
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+    {
+        "type": "text",
+        "text": "NAVSIM Benchmark The perception-free framework is implemented on NAVSIM. Specifically, We employ a ResNet-34 backbone, training for 20 epochs in line with Prakash et al. (2021) to ensure a fair comparison. Input images are resized to  $640 \\times 320$ . The Adam optimizer is used with a learning rate of 1e-4 and a batch size of 32.",
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+    {
+        "type": "text",
+        "text": "CARLA Benchmark We follow Wu et al. (2022) to implement a perception-free framework on CARLA. To be specific, we use ResNet-34 as the backbone and employ the TCP head (Wu et al., 2022) as in Jia et al. (2023b). Input images are resized to  $900 \\times 256$ . The Adam optimizer is used with a learning rate of 1e-4 and weight decay of 1e-7. The model is trained for 60 epochs with a batch size of 128. After 30 epochs, the learning rate is halved.",
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+    {
+        "type": "text",
+        "text": "5.3 COMPARISON WITH STATE-OF-THE-ART METHODS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "For the nuScenes benchmark, we compare our proposed framework with several state-of-the-art methods, including BEV-Planner (Li et al., 2024b) and VAD (Jiang et al., 2023). The results are summarized in Table 1. Our perception-free framework demonstrates competitive performance, while the perception-based framework achieves state-of-the-art results in both L2 displacement and collision rates. For the NAVSIM benchmark, detailed in Table 2, our method achieves state-of-the-art results in overall PDMS. For the CARLA benchmark, as shown in Table 3, our proposed method outperforms all existing methods. Notably, our perception-free approach surpasses previous leading methods such as ThinkTwice (Jia et al., 2023c) and DriveAdapter (Jia et al., 2023b), which incorporate extensive supervision from depth estimation, semantic segmentation, and map segmentation.",
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+    {
+        "type": "text",
+        "text": "5.4 ABLATION STUDY",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "All experiments are conducted within the perception-free framework unless otherwise specified.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
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+    {
+        "type": "table",
+        "img_path": "images/49a52481cde90538817047f9ecd0976a965bd641253ed4556faf7ecbcc8ff135.jpg",
+        "table_caption": [
+            "Table 1: Performance on the nuScenes (Caesar et al., 2020). The overall collision results are computed by the traditional computation way used in Jiang et al. (2023).  $\\ddagger$ : The collision results are computed by the way in Li et al. (2024b). We do not utilize the historical ego status information."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">L2 (m) ↓</td><td colspan=\"4\">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>NMP (Zeng et al., 2019)</td><td>-</td><td>-</td><td>2.31</td><td>-</td><td>-</td><td>-</td><td>1.92</td><td>-</td></tr><tr><td>SA-NMP (Zeng et al., 2019)</td><td>-</td><td>-</td><td>2.05</td><td>-</td><td>-</td><td>-</td><td>1.59</td><td>-</td></tr><tr><td>FF (Hu et al., 2021)</td><td>0.55</td><td>1.20</td><td>2.54</td><td>1.43</td><td>0.06</td><td>0.17</td><td>1.07</td><td>0.43</td></tr><tr><td>EO (Khurana et al., 2022)</td><td>0.67</td><td>1.36</td><td>2.78</td><td>1.60</td><td>0.04</td><td>0.09</td><td>0.88</td><td>0.33</td></tr><tr><td>ST-P3 (Hu et al., 2022b)</td><td>1.33</td><td>2.11</td><td>2.90</td><td>2.11</td><td>0.23</td><td>0.62</td><td>1.27</td><td>0.71</td></tr><tr><td>UniAD (Hu et al., 2022c)</td><td>0.48</td><td>0.96</td><td>1.65</td><td>1.03</td><td>0.05</td><td>0.17</td><td>0.71</td><td>0.31</td></tr><tr><td>VAD (Jiang et al., 2023)</td><td>0.41</td><td>0.70</td><td>1.05</td><td>0.72</td><td>0.07</td><td>0.17</td><td>0.41</td><td>0.22</td></tr><tr><td>BEV-Planner (Li et al., 2024b)</td><td>0.30</td><td>0.52</td><td>0.83</td><td>0.55</td><td>0.10‡</td><td>0.37‡</td><td>1.30‡</td><td>0.59‡</td></tr><tr><td>LAW (perception-free)</td><td>0.26</td><td>0.57</td><td>1.01</td><td>0.61</td><td>0.14</td><td>0.21</td><td>0.54</td><td>0.30</td></tr><tr><td>LAW (perception-based)</td><td>0.24</td><td>0.46</td><td>0.76</td><td>0.49</td><td>0.08</td><td>0.10</td><td>0.39</td><td>0.19</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7d8c93ab357251dcd22532058c9827e38eefd32f374e3e06fde7a9c7a7c55387.jpg",
+        "table_caption": [
+            "Table 2: Performance on NAVSIM test set. NC: no at-fault collision. DAC: drivable area compliance. TTC: time-to-collision. Conf.: comfort. EP: ego progress. PDMS: the predictive driver model score. LAW is in the perception-free setting."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>NC ↑</td><td>DAC ↑</td><td>TTC ↑</td><td>Comf. ↑</td><td>EP ↑</td><td>PDMS ↑</td></tr><tr><td>Human</td><td>100</td><td>100</td><td>100</td><td>99.9</td><td>87.5</td><td>94.8</td></tr><tr><td>Constant Velocity</td><td>69.9</td><td>58.8</td><td>49.3</td><td>100</td><td>49.3</td><td>21.6</td></tr><tr><td>Ego Status MLP</td><td>93.0</td><td>77.3</td><td>83.6</td><td>100</td><td>62.8</td><td>65.6</td></tr><tr><td>TransFuser (Prakash et al., 2021)</td><td>97.7</td><td>92.8</td><td>92.8</td><td>100</td><td>79.2</td><td>84.0</td></tr><tr><td>UniAD (Hu et al., 2022c)</td><td>97.8</td><td>91.9</td><td>92.9</td><td>100</td><td>78.8</td><td>83.4</td></tr><tr><td>PARA-Drive (Weng et al., 2024)</td><td>97.9</td><td>92.4</td><td>93.0</td><td>99.8</td><td>79.3</td><td>84.0</td></tr><tr><td>LAW</td><td>96.4</td><td>95.4</td><td>88.7</td><td>99.9</td><td>81.7</td><td>84.6</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d715ef5e412793ed069997c4f0e3157cec069422d8eb0f22c90fbc4bd5d4f4b8.jpg",
+        "table_caption": [
+            "Table 3: Performance on Town05 Long benchmark on CARLA. Expert: Imitation learning from the driving trajectories of a privileged expert. Seg.: semantic segmentation. Map.: BEV map segmentation. Dep.: depth estimation. Det.: 3D object detection. Latent Prediction: our proposed self-supervised task. RC: route completion. IS: infraction score. DS: driving score. LAW is in the perception-free setting."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Supervision</td><td>RC↑</td><td>IS↑</td><td>DS↑</td></tr><tr><td>CILRS (Codevilla et al., 2019)</td><td>Expert</td><td>10.3±0.0</td><td>0.75±0.05</td><td>7.8±0.3</td></tr><tr><td>LBC (Chen et al., 2020a)</td><td>Expert</td><td>31.9±2.2</td><td>0.66±0.02</td><td>12.3±2.0</td></tr><tr><td>Transfuser (Prakash et al., 2021)</td><td>Expert, Dep., Seg., Map., Det.</td><td>47.5±5.3</td><td>0.77±0.04</td><td>31.0±3.6</td></tr><tr><td>Roach (Zhang et al., 2021)</td><td>Expert</td><td>96.4±2.1</td><td>0.43±0.03</td><td>41.6±1.8</td></tr><tr><td>LAV (Chen &amp; Krähenbuhl, 2022)</td><td>Expert, Seg., Map., Det.</td><td>69.8±2.3</td><td>0.73±0.02</td><td>46.5±2.3</td></tr><tr><td>TCP (Wu et al., 2022)</td><td>Expert</td><td>80.4±1.5</td><td>0.73±0.02</td><td>57.2±1.5</td></tr><tr><td>MILE (Hu et al., 2022a)</td><td>Expert, Map., Det.</td><td>97.4±0.8</td><td>0.63±0.03</td><td>61.1±3.2</td></tr><tr><td>ThinkTwice (Jia et al., 2023c)</td><td>Expert, Dep., Seg., Det.</td><td>95.5±2.0</td><td>0.69±0.05</td><td>65.0±1.7</td></tr><tr><td>DriveAdapter (Jia et al., 2023b)</td><td>Expert, Map., Det.</td><td>94.4±-</td><td>0.72±-</td><td>65.9±-</td></tr><tr><td>Interfuser (Shao et al., 2022)</td><td>Expert, Map., Det.</td><td>95.0±2.9</td><td>-</td><td>68.3±1.9</td></tr><tr><td>LAW</td><td>Expert, Latent Prediction</td><td>97.8±0.9</td><td>0.72±0.03</td><td>70.1±2.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Ablation Study on Latent World Model In this ablation study, we assess the effectiveness of our proposed latent world model. For the nuScenes benchmark, the results are shown in Table 1. We ablate the latent prediction task in both the perception-free and perception-based frameworks, and further investigate the contribution of each input to the latent world model. The findings demonstrate that accurate future latent predictions depend on incorporating driving actions, supporting the validity of the latent world model. We also present ablation studies on NAVSIM and CARLA, as detailed in Table 5. In NAVSIM, we observed a significant improvement in PDMS, mainly driven by the",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+    {
+        "type": "table",
+        "img_path": "images/6e17aea36011aa51f3125346cfe2460cc1cc41e5f70a45e64beb4eb71f159c66.jpg",
+        "table_caption": [
+            "Table 4: Effectiveness of latent prediction on nuScenes benchmark. The latent world model receives two types of inputs: visual latents and predicted trajectory. No input refers to not utilizing the world model. Pred.: predicted. Traj.: trajectory. Avg.: average."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Framework</td><td colspan=\"2\">Input of Latent World Model</td><td colspan=\"4\">L2 (m) ↓</td><td colspan=\"4\">Collision (%) ↓</td></tr><tr><td>Visual Latent</td><td>Pred. Traj.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td rowspan=\"3\">Perception-free</td><td>-</td><td>-</td><td>0.32</td><td>0.67</td><td>1.14</td><td>0.71</td><td>0.20</td><td>0.30</td><td>0.73</td><td>0.41</td></tr><tr><td>✓</td><td>-</td><td>0.30</td><td>0.64</td><td>1.12</td><td>0.68</td><td>0.18</td><td>0.27</td><td>0.66</td><td>0.37</td></tr><tr><td>✓</td><td>✓</td><td>0.26</td><td>0.57</td><td>1.01</td><td>0.61</td><td>0.14</td><td>0.21</td><td>0.54</td><td>0.30</td></tr><tr><td rowspan=\"3\">Perception-based</td><td>-</td><td>-</td><td>0.30</td><td>0.52</td><td>0.80</td><td>0.54</td><td>0.09</td><td>0.17</td><td>0.48</td><td>0.25</td></tr><tr><td>✓</td><td>-</td><td>0.27</td><td>0.49</td><td>0.80</td><td>0.52</td><td>0.08</td><td>0.12</td><td>0.42</td><td>0.21</td></tr><tr><td>✓</td><td>✓</td><td>0.24</td><td>0.46</td><td>0.76</td><td>0.49</td><td>0.08</td><td>0.10</td><td>0.39</td><td>0.19</td></tr></table>",
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+        "table_caption": [
+            "Table 5: Ablation study on latent prediction on NAVSIM and CARLA benchmark. NC: no at-fault collision. DAC: drivable area compliance. TTC: time-to-collision. Conf.: comfort. EP: ego progress. PDMS: the predictive driver model score. RC: route Completion. IS: infraction Score. DS: driving Score. LAW is in the perception-free setting."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Latent Prediction</td><td colspan=\"6\">NAVSIM</td><td colspan=\"3\">CARLA</td></tr><tr><td>NC ↑</td><td>DAC ↑</td><td>TTC ↑</td><td>Comf. ↑</td><td>EP ↑</td><td>PDMS ↑</td><td>RC ↑</td><td>IS ↑</td><td>DS ↑</td></tr><tr><td>×</td><td>94.4</td><td>89.4</td><td>84.8</td><td>100.0</td><td>75.1</td><td>77.5</td><td>98.6±0.8</td><td>0.68±0.02</td><td>67.9±2.1</td></tr><tr><td>✓</td><td>96.4</td><td>95.4</td><td>88.7</td><td>99.9</td><td>81.7</td><td>84.6</td><td>97.8±0.9</td><td>0.72±0.03</td><td>70.1±2.6</td></tr></table>",
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+        "type": "text",
+        "text": "enhancements in drivable area compliance (DAC) and Ego progress (EP) metrics. This suggests that our self-supervised task effectively enhances the quality of the driving trajectory. Similarly, in CARLA, we observed notable improvements in the Driving Score.",
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+        "text": "The Time Horizon of Latent World Model In this experiment, the world model predicts latent features at three distinct future time horizons: 0.5 seconds, 1.5 seconds, and 3.0 seconds. This corresponds to the first, third, and sixth future frames from the current frame, given that keyframes occur every 0.5 seconds in the nuScenes dataset. The results, displayed in Table 6, show that the model achieves the best performance at the 1.5-second horizon. In comparison, the 0.5-second interval typically presents scenes with minimal changes, providing insufficient dynamic content to improve feature learning. In contrast, the 3.0-second interval often presents scenes that may significantly differ from the current frame, making accurate future predictions more challenging. Moreover, we observe that predicting latents 10 seconds into the future completely diminishes the gains provided by the world model, suggesting that predicting features too far into the future is ineffective. This conclusion aligns with observations from MAE (He et al., 2022), where both excessively low and high mask ratios negatively impact the ability of the network.",
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+        "table_caption": [
+            "Table 6: Different time horizons for latent prediction. The time intervals of 0.5, 1.5, 3.0, and 10.0 seconds correspond to the first, third, sixth, and twentieth future frames from the current frame, as keyframes occur every 0.5 seconds in the nuScenes dataset."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Time Horizon</td><td colspan=\"4\">L2 (m) ↓</td><td colspan=\"4\">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>0.5s</td><td>0.26</td><td>0.57</td><td>1.01</td><td>0.61</td><td>0.14</td><td>0.21</td><td>0.54</td><td>0.30</td></tr><tr><td>1.5s</td><td>0.26</td><td>0.54</td><td>0.93</td><td>0.58</td><td>0.14</td><td>0.17</td><td>0.45</td><td>0.25</td></tr><tr><td>3.0s</td><td>0.28</td><td>0.59</td><td>1.01</td><td>0.63</td><td>0.13</td><td>0.20</td><td>0.48</td><td>0.27</td></tr><tr><td>10.0s</td><td>0.33</td><td>0.67</td><td>1.16</td><td>0.72</td><td>0.24</td><td>0.28</td><td>0.76</td><td>0.43</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Network Architecture of Latent World Model To validate the impact of the network architecture of the latent world model, we conduct experiments as shown in Table 7. Firstly, it is evident that a single-layer neural network, represented as Linear Projection, is not adequate for fulfilling the functions of the world model, resulting in significantly degraded performance. The two-layer MLP shows considerable improvement in performance. However, it lacks the capability to facilitate interactions among different latent vectors. Therefore, we use the stacked transformer blocks as our",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "page_number",
+        "text": "9",
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+    {
+        "type": "image",
+        "img_path": "images/76a981eb0677f98364b9c871a1f817307b0b7e3728d86b9438b9f94d27f5065e.jpg",
+        "image_caption": [
+            "Figure 3: Visualization. This figure compares LAW in the perception-based setting with VAD (Jiang et al., 2023). On the right side of this figure, we display the results of ego trajectory prediction, agent motion prediction, and map construction in BEV. As indicated by the red circles, our method captures more crucial scene information, which VAD overlooks. Consequently, VAD predicts a forward trajectory that results in a rear-end collision, as highlighted by the yellow circles."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "default network architecture, which achieves the best results among the tested architectures. This indicates that interactions between feature vectors from different positions are important.",
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+    {
+        "type": "table",
+        "img_path": "images/691233c36a09744023086b50168ca0f6d50e0ada51851f0f99f930a919666c83.jpg",
+        "table_caption": [
+            "Table 7: Different network architecture of the latent world model. Linear Projection means a single-layer network."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Architecture</td><td colspan=\"4\">L2 (m) ↓</td><td colspan=\"4\">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>Linear Projection</td><td>0.31</td><td>0.65</td><td>1.14</td><td>0.70</td><td>0.26</td><td>0.34</td><td>0.66</td><td>0.42</td></tr><tr><td>Two-layer MLP</td><td>0.27</td><td>0.58</td><td>1.07</td><td>0.64</td><td>0.17</td><td>0.23</td><td>0.59</td><td>0.33</td></tr><tr><td>Transformer Blocks</td><td>0.26</td><td>0.57</td><td>1.01</td><td>0.61</td><td>0.14</td><td>0.21</td><td>0.54</td><td>0.30</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "5.5 VISUALIZATION",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Figure 3 compares the results of LAW in the perception-based setting with VAD (Jiang et al., 2023). Leveraging our latent world model, our approach acquires more comprehensive scene representations.",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In conclusion, we present the latent world model to predict future features from current features and ego trajectories, which is a novel self-supervised learning method for end-to-end autonomous driving. This method jointly enhances scene representation learning and ego trajectory prediction. Our approach demonstrates universality by accommodating both perception-free and perception-based frameworks, predicting perspective-view features and BEV features respectively. We achieve state-of-the-art results on benchmarks like nuScenes, NAVSIM, and CARLA.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "10",
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+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGMENTS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This work was supported by National Science and Technology Major Project (2022ZD0116500). This work was also supported in part by the National Key R&D Program of China (No. 2022ZD0116500), the National Natural Science Foundation of China (No. U21B2042, No. 62320106010), and in part by the 2035 Innovation Program of CAS.",
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+    {
+        "type": "text",
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+        ],
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+            102,
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+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    },
+    {
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+    },
+    {
+        "type": "list",
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+        ],
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+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "text",
+        "text": "A APPENDIX",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A.1 PREDICTING MULTIPLE FEATURES USING MULTIPLE INPUT FRAMES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Predicting multiple future features To better investigate the ability of our latent world model, we utilize the latent world model to predict multiple future frame latents, with the results presented in Table 8. We conduct this experiment using only the front-view camera to facilitate fast training. In detail, the future frame latents are predicted in an auto-regressive manner. For example, we first predicted the latent for 1.5 seconds into the future, then used this predicted latent to further predict the latent for 3 seconds into the future. The latent world model shares the same weights throughout this process. The results demonstrate that predicting multiple future latents further improves performance.",
+        "bbox": [
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+        "type": "table",
+        "img_path": "images/bc8c43e445c50f2f0a3676533ec86a09eb0fcb89fc49074d1371205b96f4b47e.jpg",
+        "table_caption": [
+            "Table 8: Predicting multiple future latents in an auto-regressive manner."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Predicted Future</td><td colspan=\"4\">L2 (m) ↓</td><td colspan=\"4\">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>1.5s</td><td>0.34</td><td>0.69</td><td>1.17</td><td>0.73</td><td>0.12</td><td>0.22</td><td>0.63</td><td>0.32</td></tr><tr><td>1.5s → 3s</td><td>0.31</td><td>0.65</td><td>1.12</td><td>0.69</td><td>0.11</td><td>0.19</td><td>0.57</td><td>0.29</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Predicting multiple future features using multiple input frames Building on the previous section, we further conduct experiments to predict multiple future frame latents while incorporating multiple input frame latents to leverage temporal information more effectively. Specifically, we adopted a two-stage training paradigm for improved convergence, inspired by SOLOFusion (Park et al., 2022). In the first stage, we trained the model using single input frame latents for 12 epochs. This corresponds to the model denoted as \"1.5s  $\\rightarrow$  3s\" in Table 8. In the second stage, we fine-tuned the model for an additional 6 epochs, now using two input frame latents, from 0s and 1.5s ago. The results are summarized in Table 9. The baseline (first row) represents the model fine-tuned using only single input frame latents. In contrast, the second row corresponds to the model fine-tuned with two input frame latents. The latter achieves significantly better performance. This highlights the crucial role of temporal information in autonomous driving.",
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+        "type": "table",
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+        "table_caption": [
+            "Table 9: Predicting future latents with multiple history frame inputs."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Predicted Future</td><td rowspan=\"2\">Input Frames</td><td colspan=\"4\">L2 (m) ↓</td><td colspan=\"4\">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>1.5s → 3s</td><td>0s</td><td>0.30</td><td>0.64</td><td>1.09</td><td>0.68</td><td>0.14</td><td>0.23</td><td>0.62</td><td>0.33</td></tr><tr><td>1.5s → 3s</td><td>-1.5s, 0s</td><td>0.26</td><td>0.51</td><td>0.87</td><td>0.55</td><td>0.08</td><td>0.09</td><td>0.33</td><td>0.17</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "A.2 MORE VISUALIZATION",
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+        "text": "In the appendix, we provide more visualization figures. We also provide a demo based on the CARLA simulator in the supplementary materials.",
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+        "type": "header",
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+        "image_caption": [
+            "Figure 4: Visualization. As shown in the red circle, our map construction results are noticeably better than those of VAD."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            "Figure 5: Visualization. As shown in the red circle, our agent motion prediction results are noticeably better than those of VAD."
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2025/Enhancing End-to-End Autonomous Driving with Latent World Model/4b6daa3f-8e0b-4638-98d5-1604235f6011_origin.pdf

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2025/Enhancing End-to-End Autonomous Driving with Latent World Model/full.md

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+# ENHANCING END-TO-END AUTONOMOUS DRIVING WITH LATENT WORLD MODEL
+
+Yingyan Li $^{1,2,3,4}$
+
+Lue Fan $^{1,2,3}$
+
+Jiawei He1,2,3
+
+Yuqi Wang1,2,3
+
+Yuntao Chen $^{1,2,3}$
+
+Zhaoxiang Zhang $^{1,2,3,4}$
+
+Tieniu Tan1,2,3
+
+$^{1}$  Institute of Automation, Chinese Academy of Sciences (CASIA)  
+$^{2}$  New Laboratory of Pattern Recognition (NLPR)  
+3 State Key Laboratory of Multimodal Artificial Intelligence Systems (MAIS)  
+$^{4}$  School of Future Technology, University of Chinese Academy of Sciences (UCAS)
+
+# ABSTRACT
+
+In autonomous driving, end-to-end planners directly utilize raw sensor data, enabling them to extract richer scene features and reduce information loss compared to traditional planners. This raises a crucial research question: how can we develop better scene feature representations to fully leverage sensor data in end-to-end driving? Self-supervised learning methods show great success in learning rich feature representations in NLP and computer vision. Inspired by this, we propose a novel self-supervised learning approach using the LAtent World model (LAW) for end-to-end driving. LAW predicts future scene features based on current features and ego trajectories. This self-supervised task can be seamlessly integrated into perception-free and perception-based frameworks, improving scene feature learning and optimizing trajectory prediction. LAW achieves state-of-the-art performance across multiple benchmarks, including real-world open-loop benchmark nuScenes, NAVSIM, and simulator-based closed-loop benchmark CARLA. The code is released at https://github.com/BraveGroup/LAW.
+
+# 1 INTRODUCTION
+
+End-to-end planners (Hu et al., 2022c; Jiang et al., 2023; Prakash et al., 2021; Wu et al., 2022; Hu et al., 2022b; Zhang et al., 2022; Wu et al., 2023) have garnered significant attention due to their distinct advantages over traditional planners. Traditional planners operate on pre-processed outputs from perception modules, such as bounding boxes and trajectories. In contrast, end-to-end planners directly utilize raw sensor data to extract scene features, minimizing information loss. This direct use of sensor data raises an important research question: how can we develop more effective scene feature representations to fully leverage the richness of sensor data in end-to-end driving?
+
+In recent years, self-supervised learning has emerged as a powerful method for extracting comprehensive feature representations from large-scale datasets, particularly in fields like NLP (Devlin, 2018) and computer vision (He et al., 2022). Building on this success, we aim to enrich scene feature learning and further improve end-to-end driving performance through self-supervised learning. Traditional self-supervised methods in computer vision(He et al., 2022; Chen et al., 2020b) often focus on static, single-frame images. However, autonomous driving relies on continuous video input, so effectively using temporal data is crucial. Temporal self-supervised tasks, such as future prediction (Han et al., 2019; 2020), have shown promise. Traditional future prediction tasks often overlook the impact of ego actions, which play a crucial role in shaping the future in autonomous driving.
+
+Considering the critical role of ego actions, we propose utilizing a world model to predict future states based on the current state and ego actions. While several image-based driving world models (Wang et al., 2023b; Hu et al., 2023a; Jia et al., 2023a) exist, they exhibit inefficiencies in enhancing scene
+
+![](images/0dd07acd177b029ce279dea1e097861ec2d30efbd67094b69c3e58c8301f9b67.jpg)  
+(a) Annotation-based Perception Tasks  
+(b) Our Self-supervised Task  
+Figure 1: The illustration of our self-supervised method. Traditional methods utilize perception annotations to assist with scene feature learning. In contrast, our self-supervised approach uses temporal information to guide feature learning. Pred.: predicted. Seg.: segmentation. The blue part indicates the pipeline of an end-to-end planner.
+
+feature representations due to their reliance on diffusion models, which may take several seconds to generate images of a future scene. To address this limitation, we introduce a latent world model designed to predict future latent features directly from the current latent features and ego actions, as depicted in Fig. 1. Specifically, given images, a visual encoder extracts scene features (current state), which are then fed into an action decoder to predict ego trajectory. Based on the current state and action, the latent world model predicts the scene feature of the future frame. During training, the predicted future features are supervised using the extracted features from the future frame. By supervising the predicted future feature, this self-supervised method jointly optimizes the current scene feature learning and ego trajectory prediction.
+
+After introducing the concept of the latent world model, we turn our attention to its universality across various end-to-end autonomous driving frameworks. In end-to-end autonomous driving, the frameworks can generally be categorized into two types: perception-free and perception-based. Perception-free approaches (Toromanoff et al., 2020; Chen et al., 2020a; Wu et al., 2022) bypass explicit perception tasks, relying solely on trajectory supervision. Prior work Wu et al. (2022) in this category typically extracts perspective-view features to predict future trajectory. In contrast, perception-based approaches (Prakash et al., 2021; Hu et al., 2022c; Jiang et al., 2023; Hu et al., 2022b) incorporate perception tasks, such as detection, tracking, and map segmentation, to guide scene feature learning. These methods generally use BEV feature maps as a unified representation for these perception tasks. Our latent world model accommodates both frameworks. It can either predict perspective-view features in the perception-free setting or predict BEV features in the perception-based setting, showcasing its universality across different autonomous driving paradigms.
+
+Experiments show that our latent world model enhances performance in both perception-free and perception-based frameworks. Furthermore, we achieve state-of-the-art performance on multiple benchmarks, including the real-world open-loop datasets nuScenes (Caesar et al., 2020) and NAVSIM (Dauner et al., 2024) (based on nuPlan (Caesar et al., 2021)), as well as the simulator-based closed-loop CARLA benchmark (Dosovitskiy et al., 2017). These results underscore the efficacy of our approach and highlight the potential of self-supervised learning to advance end-to-end autonomous driving research. In summary, our contributions are threefold:
+
+- Future prediction by latent world model: We introduce the LAtent World model (LAW) to predict future scene latents from current scene latents and ego trajectories. This self-
+
+supervised task jointly enhances scene representation learning and trajectory prediction in end-to-end driving.
+
+- Cross-framework universality:  $LAW$  demonstrates universality across various common autonomous driving paradigms. It can either predict perspective-view features in the perception-free framework or predict BEV features in the perception-based framework.  
+- State-of-the-art performance: Our self-supervised approach achieves state-of-the-art results on the real-world open-loop nuScenes, NAVSIM, and the simulator-based close-loop CARLA benchmark.
+
+# 2 RELATED WORKS
+
+# 2.1 END-TO-END AUTONOMOUS DRIVING
+
+We divide end-to-end autonomous driving methods (Hu et al., 2022c; Jiang et al., 2023; Renz et al., 2022; Toromanoff et al., 2020; Tian et al., 2024; Hwang et al., 2024; Pan et al., 2024; Wang et al., 2024a;b) into two categories, perception-based methods and perception-free methods, depending on whether performing perception tasks. Perception-based methods (Casas et al., 2021; Prakash et al., 2021; Jaeger et al., 2023; Shao et al., 2023; Hu et al., 2022b; Sadat et al., 2020) perform multiple perception tasks simultaneously, such as detection (Li et al., 2022b; Huang et al., 2021; Li et al., 2022a; 2024a), tracking (Zhou et al., 2020; Wang et al., 2021), map segmentation (Hu et al., 2022c; Jiang et al., 2023) and occupancy prediction (Wang et al., 2023a; Huang et al., 2023). As a representative, UniAD (Hu et al., 2022c) integrates multiple modules to support goal-driven planning. VAD (Jiang et al., 2023) explores vectorized scene representation for planning purposes.
+
+Perception-free end-to-end methods (Toromanoff et al., 2020; Chen et al., 2020a; Zhang et al., 2021; Wu et al., 2022) present a promising direction as they avoid utilizing a large number of perception annotations. Early perception-free end-to-end methods (Zhang et al., 2021; Toromanoff et al., 2020) primarily relied on reinforcement learning. For instance, MaRLn (Toromanoff et al., 2020) designed a reinforcement learning algorithm based on implicit affordances, while LBC (Chen et al., 2020a) trained a reinforcement learning expert using privileged (ground-truth perception) information. Using trajectory data generated by the reinforcement learning expert, TCP (Wu et al., 2022) combined a trajectory waypoint branch with a direct control branch, achieving good performance. However, perception-free end-to-end methods often suffer from inadequate scene representation capabilities. Our work aims to address this issue through the latent world model.
+
+# 2.2 WORLD MODEL IN AUTONOMOUS DRIVING
+
+Existing world models in autonomous driving can be categorized into two types: image-based world models and occupancy-based world models. Image-based world models (Hu et al., 2022a; Wang et al., 2023b; Hu et al., 2023a) aim to enrich the autonomous driving dataset through generative approaches. GAIA-1 (Hu et al., 2023a) is a generative world model that utilizes video, text, and action inputs to create realistic driving scenarios. MILE (Hu et al., 2022a) produces urban driving videos by leveraging 3D geometry as an inductive bias. Drive-WM (Wang et al., 2023b) utilizes a diffusion model to predict future images and then plans based on these predicted images. Copilot4D (Zhang et al., 2023) tokenizes sensor observations with VQVAE (Van Den Oord et al., 2017) and then predicts the future via discrete diffusion. Another category involves occupancy-based world models (Zheng et al., 2023; Min et al., 2024). OccWorld (Zheng et al., 2023) and DriveWorld (Min et al., 2024) use the world model to predict the occupancy, which requires occupancy annotations. On the contrary, our proposed latent world model requires no manual annotations.
+
+# 3 PRELIMINARY
+
+Vision-based End-to-end Autonomous Driving In the task of end-to-end autonomous driving, the objective is to estimate the future trajectory of the ego vehicle in the form of waypoints. Formally, let  $\mathbf{I}_t = \{\mathbf{I}_t^1,\mathbf{I}_t^2,\ldots ,\mathbf{I}_t^N\}$  be the set of  $N$  surrounding multi-view images captured at time step  $t$ . We expect the model to predict a sequence of waypoints  $\mathbf{W}_t = \{\mathbf{w}_t^1,\mathbf{w}_t^2,\dots ,\mathbf{w}_t^M\}$ , where each
+
+![](images/5d6c4d3fa6b2cb77cfcb5b3c9a02f11c9b613b4fd73e5361e311d446ddf76ec2.jpg)
+
+![](images/8071a5667a6856259826ecfe7f146aa6a133ae71873a4e3b801e4d2b3bd54ed2.jpg)  
+Perception-free with Perspective-view Features
+
+![](images/c98a8e28063ad922280b323a46c35372d763f1552ec701d7fb7d32b0db6ca6d8.jpg)  
+Perception-based with BEV Features  
+Figure 2: The Overall Framework. The encoder extracts visual latents from the images, while the decoder predicts waypoints based on these latents. Our latent world model predicts future visual latents using the visual latents and waypoints of the current frame. During training, the predicted visual latents are supervised by the extracted latents from the future frame. The latent world model is compatible with both perception-free and perception-based frameworks, which differ in their encoder and decoder structures. In the perception-based framework, the supervision icon indicates that map annotations supervise the output of the map construction module, while the agent's future trajectory supervises the output of the motion prediction module. Pred.: predicted.
+
+waypoint  $\mathbf{w}_t^i = (x_t^i, y_t^i)$  represents the predicted BEV position of the ego vehicle at time step  $t + i$ .  $M$  represents the number of future positions of the ego vehicle that the model aims to predict.
+
+World Model A world model aims to predict future states based on the current state and action. In the autonomous driving task, let  $\mathbf{S}_t$  represent the states at time step  $t$ ,  $\mathbf{W}_t = \{\mathbf{w}_t^1, \mathbf{w}_t^2, \dots, \mathbf{w}_t^M\}$  denote the sequence of predicted waypoints by the planner, the world model predicts state  $\mathbf{S}_{t+1}$  at time step  $t+1$  using  $\mathbf{S}_t$  and  $\mathbf{W}_t$ .
+
+# 4 METHODOLOGY
+
+Our methodology is composed of three key components: i) Latent World Model: We utilize the latent world model to realize the self-supervised task. This model takes two inputs: latent features extracted by the visual encoder and waypoints predicted by the waypoint decoder. This task is compatible with two common frameworks. ii) Perception-free framework with perspective-view latents: it consists of the perspective-view encoder and perception-free decoder within this framework. iii) Perception-based framework with BEV latents: it contains the BEV encoder and perception-based decoder within this framework.
+
+# 4.1 LATENT WORLD MODEL
+
+In this section, we utilize the latent world model to predict the visual latents of the future frame based on the current visual latents and waypoints.
+
+Visual Latents and Waypoints Extraction The visual encoder processes the images from the current  $t$  time step to produce the corresponding visual latent feature set
+
+$$
+\mathbf {V} _ {t} = \{\mathbf {v} _ {t} ^ {1}, \mathbf {v} _ {t} ^ {2}, \dots , \mathbf {v} _ {t} ^ {L} \},
+$$
+
+where  $L$  denotes the number of feature vectors, and each vector  $\mathbf{v}_t^i\in \mathbb{R}^D$ , with  $D$  representing the number of feature channels. These feature vectors can be derived from various sources, such as a flattened image feature map or a flattened BEV feature map. Based on the  $\mathbf{V}_t$ , the waypoint decoder
+
+predicts the waypoints  $\mathbf{W}_t = \{\mathbf{w}_t^1, \mathbf{w}_t^2, \dots, \mathbf{w}_t^M\}$ .  $M$  represents the number of waypoints, where each waypoint  $\mathbf{w}_t^i = (x_t^i, y_t^i)$ .
+
+Action-aware Latents Construction We produce action-aware latents by integrating visual latents and waypoints. The action-aware latents are then used as input to the latent world model. Let  $M$  represent the number of waypoints, with each waypoint  $\mathbf{w}_t^i = (x_t^i, y_t^i)$ . We first reshape  $\mathbf{W}_t$  which has the shape  $[M, 2]$ , into a one-dimensional vector  $\widetilde{\mathbf{w}}_t \in \mathbb{R}^{2M}$ . Then, we concatenate each visual latent  $\mathbf{v}_t^i$  with the waypoint vector  $\widetilde{\mathbf{w}}_t$  along the feature channel dimension. The resulting concatenated vector is passed through an MLP to produce the action-aware latent  $\mathbf{a}_t^i$ , which has the same shape as  $\mathbf{v}_t^i$ . Formally, the action-aware latent for the  $i$ -th feature vector is expressed as
+
+$$
+\mathbf {a} _ {t} ^ {i} = \operatorname {M L P} \left(\left[ \mathbf {v} _ {t} ^ {i}, \widetilde {\mathbf {w}} _ {t} \right]\right), \tag {1}
+$$
+
+where  $[\cdot, \cdot]$  denotes the concatenating operation. The full set of action-aware latents is denoted as  $\mathbf{A}_t = \{\mathbf{a}_t^1, \mathbf{a}_t^2, \ldots, \mathbf{a}_t^L\}$ , where  $L$  denotes the number of feature vectors.
+
+Future Latent Prediction The latent world model utilizes the action-aware latents to predict the future visual latents as follows. Given  $\mathbf{A}_t$ , we predict visual latents  $\hat{\mathbf{V}}_{t+1}$  of the frame  $t+1$  by the latent world model:
+
+$$
+\hat {\mathbf {V}} _ {t + 1} = \operatorname {L a t e n t W o r l d M o d e l} \left(\mathbf {A} _ {t}\right). \tag {2}
+$$
+
+The network architecture of the latent world model consists of transformer blocks. Each block contains a self-attention and feed-forward module. The self-attention is performed across the latent feature vectors.
+
+Future Latent Supervision During training, we extract the visual latents  $\mathbf{V}_{t + 1} = \{\mathbf{v}_{t + 1}^{1},\dots ,\mathbf{v}_{t + 1}^{L}$  from the images of frame  $t + 1$  , which are used as the ground truth to supervise the  $\hat{\mathbf{V}}_{t + 1}$  using a Mean Squared Error (MSE) loss function as
+
+$$
+\mathcal {L} _ {\text {l a t e n t}} = \frac {1}{L} \sum_ {i = 1} ^ {L} \| \hat {\mathbf {v}} _ {t + 1} ^ {i} - \mathbf {v} _ {t + 1} ^ {i} \| _ {2}. \tag {3}
+$$
+
+The latent world model is compatible with both perception-free and perception-based frameworks. In the following sections, we detail the implementation of these two frameworks.
+
+# 4.2 PERCEPTION-FREE FRAMEWORK WITH PERSPECTIVE-VIEW LATENTS
+
+First, we introduce our perception-free framework. Previous perception-free frameworks (Wu et al., 2022) typically employ a perspective-view encoder for visual latent extraction and a perception-free decoder for waypoint prediction. Our framework is built upon this established paradigm.
+
+Perspective-view Encoder In the perspective-view encoder, we produce visual latents based on multi-view images. Initially, multi-view images are processed by an image backbone to obtain their corresponding image features. Following PETR (Liu et al., 2022), we generate 3D position embeddings for these image features, which are then added to the image features to uniquely distinguish each feature vector. The enriched image features are denoted as  $\mathbf{F} = \{\mathbf{f}^1, \mathbf{f}^2, \dots, \mathbf{f}^N\}$ , where  $N$  represents the number of views. The shape of  $\mathbf{f}^i$  is  $[H, W, D]$ , where  $H, W$  represents the height and width of the image feature map and  $D$  is the number of feature channels.
+
+To encode the image features into high-level visual latents suitable for planning, we apply a view attention mechanism. To be specific, for  $N$  views, there are  $N$  corresponding learnable view queries  $\mathbf{Q}_{\mathrm{view}} = \{\mathbf{q}_{\mathrm{view}}^{1}, \mathbf{q}_{\mathrm{view}}^{2}, \dots, \mathbf{q}_{\mathrm{view}}^{N}\}$ . Each view query  $\mathbf{q}_{\mathrm{view}}^{i}$  undergoes a cross-attention with its corresponding image feature  $\mathbf{f}^{i}$ , resulting in  $N$  visual latent  $\mathbf{V}_{\mathrm{pf}} = \{\mathbf{v}_{\mathrm{pf}}^{1}, \mathbf{v}_{\mathrm{pf}}^{2}, \dots, \mathbf{v}_{\mathrm{pf}}^{N}\}$ , where the subscript "pf" stands for perception-free. Formally,
+
+$$
+\mathbf {v} _ {\mathrm {p f}} ^ {i} = \operatorname {C r o s s A t t e n t i o n} \left(\mathbf {q} _ {\text {v i e w}} ^ {i}, \mathbf {f} ^ {i}, \mathbf {f} ^ {i}\right), \tag {4}
+$$
+
+where  $\mathbf{f}^i$  serves as the key and value of the cross attention.  $\mathbf{V}_{\mathrm{pf}}$  are then used as input for the perception-free decoder, which we describe next.
+
+Perception-free Decoder The perception-free decoder deocodes waypoint from  $\mathbf{V}_{\mathrm{pf}}$ . Specifically, we initialize  $M$  waypoint queries,  $\mathbf{Q}_{\mathrm{wp}} = \{\mathbf{q}_{\mathrm{wp}}^{1}, \mathbf{q}_{\mathrm{wp}}^{2}, \ldots, \mathbf{q}_{\mathrm{wp}}^{M}\}$ , where each query is a learnable embedding. These waypoint queries interact with  $\mathbf{V}_{\mathrm{pf}}$  through a cross-attention mechanism. The
+
+updated waypoint queries are then passed through an MLP head to predict the waypoints  $\mathbf{W} = \{\mathbf{w}^1, \mathbf{w}^2, \dots, \mathbf{w}^M\}$ , which is formulated as
+
+$$
+\mathbf {W} = \operatorname {M L P} \left(\text {C r o s s A t t e n t i o n} \left(\mathbf {Q} _ {\mathrm {w p}}, \mathbf {V} _ {\mathrm {p f}}, \mathbf {V} _ {\mathrm {p f}}\right)\right). \tag {5}
+$$
+
+Perception-free Supervision In the perception-free framework, we rely solely on ground truth waypoints for supervision, as no additional perception annotations are provided. We employ an L1 loss to measure the discrepancy between the predicted waypoints  $\mathbf{W}$  and the ground truth waypoints  $\tilde{\mathbf{W}} = \{\tilde{\mathbf{w}}^1,\tilde{\mathbf{w}}^2,\dots ,\tilde{\mathbf{w}}^M\}$  as
+
+$$
+\mathcal {L} _ {\text {w a y p o i n t}} = \frac {1}{M} \sum_ {j = 1} ^ {M} \| \mathbf {w} _ {t} ^ {j} - \tilde {\mathbf {w}} _ {t} ^ {j} \| _ {1}, \tag {6}
+$$
+
+Thus, the final loss for the perception-free framework is
+
+$$
+\mathcal {L} _ {\mathrm {p f}} = \mathcal {L} _ {\text {l a t e n t}} + \mathcal {L} _ {\text {w a y p o i n t}}. \tag {7}
+$$
+
+# 4.3 PERCEPTION-BASED FRAMEWORK WITH BEV LATENTS
+
+Our latent world model is also compatible with perception-based frameworks, which commonly utilize BEV feature maps for perception tasks. We adhere to this paradigm and the perception-based framework is composed of two key components: a BEV encoder and a perception-based decoder. The BEV encoder generates BEV feature maps from images, while the perception-based decoder uses these maps for perception tasks such as motion prediction and map construction. The final waypoints are then predicted based on the outputs of these perception tasks.
+
+BEV Encoder We follow Li et al. (2022b) to encode the BEV feature map. First, we encode the image features using a backbone network. Then, a set of BEV queries projects these image features into BEV features. The resulting BEV feature map is flattened into a shape of  $[K, D]$ , where  $K$  represents the number of feature vectors in the BEV feature map and  $D$  is the number of feature channels. The flattened features are denoted as  $\mathbf{V}_{\mathrm{pb}} = \{\mathbf{v}_{\mathrm{pb}}^{1}, \mathbf{v}_{\mathrm{pb}}^{2}, \ldots, \mathbf{v}_{\mathrm{pb}}^{K}\}$ , where the subscript "pb" refers to perception-based.
+
+Perception-based Decoder Following Jiang et al. (2023), the decoder predicts waypoints with the help of perception tasks, namely motion prediction and map construction. For motion prediction,  $N_{\text{agent}}$  learnable queries interact with  $\mathbf{V}_{\text{pb}}$  via cross-attention to generate agent features  $\mathbf{F}_{\text{agent}}$ .  $\mathbf{F}_{\text{agent}}$  are then used to predict agent trajectories. Similarly, for map construction,  $N_{\text{map}}$  learnable queries perform cross-attention with  $\mathbf{V}_{\text{pb}}$  to extract map features  $\mathbf{F}_{\text{map}}$ .  $\mathbf{F}_{\text{map}}$  are then used to predict map vectors. Finally, the learnable waypoint queries perform cross-attention with  $\mathbf{F}_{\text{agent}}$  and  $\mathbf{F}_{\text{map}}$ . The output is then passed through an MLP head to predict the waypoints.
+
+Perception-based Supervision The perception-based framework uses the same waypoint supervision as in equation 6. In addition, it includes losses from perception tasks as
+
+$$
+\mathcal {L} _ {\text {p e r c e p t i o n}} = \mathcal {L} _ {\text {a g e n t}} + \mathcal {L} _ {\text {m a p}}. \tag {8}
+$$
+
+Here,  $\mathcal{L}_{\mathrm{agent}}$  is the loss for motion prediction and  $\mathcal{L}_{\mathrm{map}}$  is the loss for map construction, as defined in (Jiang et al., 2023). The final loss for the perception-based framework is
+
+$$
+\mathcal {L} _ {\mathrm {p b}} = \mathcal {L} _ {\text {l a t e n t}} + \mathcal {L} _ {\text {w a y p o i n t}} + \mathcal {L} _ {\text {p e r c e p t i o n}}. \tag {9}
+$$
+
+# 5 EXPERIMENTS
+
+# 5.1 BENCHMARKS
+
+nuScenes (Caesar et al., 2020) The nuScenes dataset contains 1,000 driving scenes. In line with previous works (Hu et al., 2022b; 2023b; Jiang et al., 2023), we use L2 displacement error and collision rate as comprehensive metrics to evaluate planning performance. L2 displacement error measures the L2 distance between the predicted and ground truth trajectories, while collision rate quantifies the frequency of collisions with other objects along the predicted trajectory.
+
+NAVSIM (Dauner et al., 2024) We conducted further experiments using the NAVSIM benchmark, as the nuScenes dataset proved to be overly simplistic. The NAVSIM dataset (Dauner et al., 2024)
+
+is built on OpenScene (Contributors, 2023), which provides 120 hours of driving logs condensed from the nuPlan dataset (Caesar et al., 2021). NAVSIM enhances OpenScene by resampling the data to reduce the occurrence of simple scenarios, such as straight-line driving. As a result, traditional ego status modeling becomes inadequate under the NAVSIM benchmark. NAVSIM evaluates model performance using the predictive driver model score (PDMS), which is calculated based on five factors: no at-fault collision (NC), drivable area compliance (DAC), time-to-collision (TTC), comfort (Comf.) and ego progress (EP).
+
+CARLA (Dosovitskiy et al., 2017) Closed-loop evaluation is essential to autonomous driving as it constantly updates the sensor inputs based on the driving actions. For the closed-loop benchmark, the training dataset is collected from the CARLA (Dosovitskiy et al., 2017) simulator (version 0.9.10.1) using the teacher model Roach (Zhang et al., 2021) following (Wu et al., 2022; Jia et al., 2023b), resulting in 189K frames. We use the widely-used Town05 Long benchmark (Jia et al., 2023b; Shao et al., 2022; Hu et al., 2022a) to assess the closed-loop driving performance. For metric, Route Completion (RC) represents the percentage of the route completed by the autonomous driving model. Infraction Score (IS) quantifies the number of infractions as well as violations of traffic rules. A higher Infraction Score indicates better adherence to safe driving practices. Driving Score (DS) is the primary metric used to evaluate overall performance. It is calculated as the product of Route Completion and Infraction Score.
+
+# 5.2 IMPLEMENTATION DETAILS
+
+nuScenes Benchmark We implement both perception-free and perception-based frameworks. In the perception-free framework, Swin-Transformer-Tiny (Swin-T)(Liu et al., 2021) is used as the backbone. Input images are resized to  $800 \times 320$ . We adopt a Cosine Annealing learning rate schedule(Loshchilov & Hutter, 2016), starting at 5e-5. The model is trained using the AdamW optimizer (Loshchilov & Hutter, 2017) with a weight decay of 0.01, batch size 8, and 12 epochs across 8 A6000 GPUs. For the perception-based framework, following Jiang et al. (2023), we train the model in two stages. In the first stage, we train the encoder and perception head using only perception loss for 48 epochs. In the second stage, we introduce waypoint and latent prediction losses for training another 12 epochs. The network architecture of the latent world model utilizes deformable self-attention for improved convergence.
+
+NAVSIM Benchmark The perception-free framework is implemented on NAVSIM. Specifically, We employ a ResNet-34 backbone, training for 20 epochs in line with Prakash et al. (2021) to ensure a fair comparison. Input images are resized to  $640 \times 320$ . The Adam optimizer is used with a learning rate of 1e-4 and a batch size of 32.
+
+CARLA Benchmark We follow Wu et al. (2022) to implement a perception-free framework on CARLA. To be specific, we use ResNet-34 as the backbone and employ the TCP head (Wu et al., 2022) as in Jia et al. (2023b). Input images are resized to  $900 \times 256$ . The Adam optimizer is used with a learning rate of 1e-4 and weight decay of 1e-7. The model is trained for 60 epochs with a batch size of 128. After 30 epochs, the learning rate is halved.
+
+# 5.3 COMPARISON WITH STATE-OF-THE-ART METHODS
+
+For the nuScenes benchmark, we compare our proposed framework with several state-of-the-art methods, including BEV-Planner (Li et al., 2024b) and VAD (Jiang et al., 2023). The results are summarized in Table 1. Our perception-free framework demonstrates competitive performance, while the perception-based framework achieves state-of-the-art results in both L2 displacement and collision rates. For the NAVSIM benchmark, detailed in Table 2, our method achieves state-of-the-art results in overall PDMS. For the CARLA benchmark, as shown in Table 3, our proposed method outperforms all existing methods. Notably, our perception-free approach surpasses previous leading methods such as ThinkTwice (Jia et al., 2023c) and DriveAdapter (Jia et al., 2023b), which incorporate extensive supervision from depth estimation, semantic segmentation, and map segmentation.
+
+# 5.4 ABLATION STUDY
+
+All experiments are conducted within the perception-free framework unless otherwise specified.
+
+Table 1: Performance on the nuScenes (Caesar et al., 2020). The overall collision results are computed by the traditional computation way used in Jiang et al. (2023).  $\ddagger$ : The collision results are computed by the way in Li et al. (2024b). We do not utilize the historical ego status information.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">L2 (m) ↓</td><td colspan="4">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>NMP (Zeng et al., 2019)</td><td>-</td><td>-</td><td>2.31</td><td>-</td><td>-</td><td>-</td><td>1.92</td><td>-</td></tr><tr><td>SA-NMP (Zeng et al., 2019)</td><td>-</td><td>-</td><td>2.05</td><td>-</td><td>-</td><td>-</td><td>1.59</td><td>-</td></tr><tr><td>FF (Hu et al., 2021)</td><td>0.55</td><td>1.20</td><td>2.54</td><td>1.43</td><td>0.06</td><td>0.17</td><td>1.07</td><td>0.43</td></tr><tr><td>EO (Khurana et al., 2022)</td><td>0.67</td><td>1.36</td><td>2.78</td><td>1.60</td><td>0.04</td><td>0.09</td><td>0.88</td><td>0.33</td></tr><tr><td>ST-P3 (Hu et al., 2022b)</td><td>1.33</td><td>2.11</td><td>2.90</td><td>2.11</td><td>0.23</td><td>0.62</td><td>1.27</td><td>0.71</td></tr><tr><td>UniAD (Hu et al., 2022c)</td><td>0.48</td><td>0.96</td><td>1.65</td><td>1.03</td><td>0.05</td><td>0.17</td><td>0.71</td><td>0.31</td></tr><tr><td>VAD (Jiang et al., 2023)</td><td>0.41</td><td>0.70</td><td>1.05</td><td>0.72</td><td>0.07</td><td>0.17</td><td>0.41</td><td>0.22</td></tr><tr><td>BEV-Planner (Li et al., 2024b)</td><td>0.30</td><td>0.52</td><td>0.83</td><td>0.55</td><td>0.10‡</td><td>0.37‡</td><td>1.30‡</td><td>0.59‡</td></tr><tr><td>LAW (perception-free)</td><td>0.26</td><td>0.57</td><td>1.01</td><td>0.61</td><td>0.14</td><td>0.21</td><td>0.54</td><td>0.30</td></tr><tr><td>LAW (perception-based)</td><td>0.24</td><td>0.46</td><td>0.76</td><td>0.49</td><td>0.08</td><td>0.10</td><td>0.39</td><td>0.19</td></tr></table>
+
+Table 2: Performance on NAVSIM test set. NC: no at-fault collision. DAC: drivable area compliance. TTC: time-to-collision. Conf.: comfort. EP: ego progress. PDMS: the predictive driver model score. LAW is in the perception-free setting.  
+
+<table><tr><td>Method</td><td>NC ↑</td><td>DAC ↑</td><td>TTC ↑</td><td>Comf. ↑</td><td>EP ↑</td><td>PDMS ↑</td></tr><tr><td>Human</td><td>100</td><td>100</td><td>100</td><td>99.9</td><td>87.5</td><td>94.8</td></tr><tr><td>Constant Velocity</td><td>69.9</td><td>58.8</td><td>49.3</td><td>100</td><td>49.3</td><td>21.6</td></tr><tr><td>Ego Status MLP</td><td>93.0</td><td>77.3</td><td>83.6</td><td>100</td><td>62.8</td><td>65.6</td></tr><tr><td>TransFuser (Prakash et al., 2021)</td><td>97.7</td><td>92.8</td><td>92.8</td><td>100</td><td>79.2</td><td>84.0</td></tr><tr><td>UniAD (Hu et al., 2022c)</td><td>97.8</td><td>91.9</td><td>92.9</td><td>100</td><td>78.8</td><td>83.4</td></tr><tr><td>PARA-Drive (Weng et al., 2024)</td><td>97.9</td><td>92.4</td><td>93.0</td><td>99.8</td><td>79.3</td><td>84.0</td></tr><tr><td>LAW</td><td>96.4</td><td>95.4</td><td>88.7</td><td>99.9</td><td>81.7</td><td>84.6</td></tr></table>
+
+Table 3: Performance on Town05 Long benchmark on CARLA. Expert: Imitation learning from the driving trajectories of a privileged expert. Seg.: semantic segmentation. Map.: BEV map segmentation. Dep.: depth estimation. Det.: 3D object detection. Latent Prediction: our proposed self-supervised task. RC: route completion. IS: infraction score. DS: driving score. LAW is in the perception-free setting.  
+
+<table><tr><td>Method</td><td>Supervision</td><td>RC↑</td><td>IS↑</td><td>DS↑</td></tr><tr><td>CILRS (Codevilla et al., 2019)</td><td>Expert</td><td>10.3±0.0</td><td>0.75±0.05</td><td>7.8±0.3</td></tr><tr><td>LBC (Chen et al., 2020a)</td><td>Expert</td><td>31.9±2.2</td><td>0.66±0.02</td><td>12.3±2.0</td></tr><tr><td>Transfuser (Prakash et al., 2021)</td><td>Expert, Dep., Seg., Map., Det.</td><td>47.5±5.3</td><td>0.77±0.04</td><td>31.0±3.6</td></tr><tr><td>Roach (Zhang et al., 2021)</td><td>Expert</td><td>96.4±2.1</td><td>0.43±0.03</td><td>41.6±1.8</td></tr><tr><td>LAV (Chen &amp; Krähenbuhl, 2022)</td><td>Expert, Seg., Map., Det.</td><td>69.8±2.3</td><td>0.73±0.02</td><td>46.5±2.3</td></tr><tr><td>TCP (Wu et al., 2022)</td><td>Expert</td><td>80.4±1.5</td><td>0.73±0.02</td><td>57.2±1.5</td></tr><tr><td>MILE (Hu et al., 2022a)</td><td>Expert, Map., Det.</td><td>97.4±0.8</td><td>0.63±0.03</td><td>61.1±3.2</td></tr><tr><td>ThinkTwice (Jia et al., 2023c)</td><td>Expert, Dep., Seg., Det.</td><td>95.5±2.0</td><td>0.69±0.05</td><td>65.0±1.7</td></tr><tr><td>DriveAdapter (Jia et al., 2023b)</td><td>Expert, Map., Det.</td><td>94.4±-</td><td>0.72±-</td><td>65.9±-</td></tr><tr><td>Interfuser (Shao et al., 2022)</td><td>Expert, Map., Det.</td><td>95.0±2.9</td><td>-</td><td>68.3±1.9</td></tr><tr><td>LAW</td><td>Expert, Latent Prediction</td><td>97.8±0.9</td><td>0.72±0.03</td><td>70.1±2.6</td></tr></table>
+
+Ablation Study on Latent World Model In this ablation study, we assess the effectiveness of our proposed latent world model. For the nuScenes benchmark, the results are shown in Table 1. We ablate the latent prediction task in both the perception-free and perception-based frameworks, and further investigate the contribution of each input to the latent world model. The findings demonstrate that accurate future latent predictions depend on incorporating driving actions, supporting the validity of the latent world model. We also present ablation studies on NAVSIM and CARLA, as detailed in Table 5. In NAVSIM, we observed a significant improvement in PDMS, mainly driven by the
+
+Table 4: Effectiveness of latent prediction on nuScenes benchmark. The latent world model receives two types of inputs: visual latents and predicted trajectory. No input refers to not utilizing the world model. Pred.: predicted. Traj.: trajectory. Avg.: average.  
+
+<table><tr><td rowspan="2">Framework</td><td colspan="2">Input of Latent World Model</td><td colspan="4">L2 (m) ↓</td><td colspan="4">Collision (%) ↓</td></tr><tr><td>Visual Latent</td><td>Pred. Traj.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td rowspan="3">Perception-free</td><td>-</td><td>-</td><td>0.32</td><td>0.67</td><td>1.14</td><td>0.71</td><td>0.20</td><td>0.30</td><td>0.73</td><td>0.41</td></tr><tr><td>✓</td><td>-</td><td>0.30</td><td>0.64</td><td>1.12</td><td>0.68</td><td>0.18</td><td>0.27</td><td>0.66</td><td>0.37</td></tr><tr><td>✓</td><td>✓</td><td>0.26</td><td>0.57</td><td>1.01</td><td>0.61</td><td>0.14</td><td>0.21</td><td>0.54</td><td>0.30</td></tr><tr><td rowspan="3">Perception-based</td><td>-</td><td>-</td><td>0.30</td><td>0.52</td><td>0.80</td><td>0.54</td><td>0.09</td><td>0.17</td><td>0.48</td><td>0.25</td></tr><tr><td>✓</td><td>-</td><td>0.27</td><td>0.49</td><td>0.80</td><td>0.52</td><td>0.08</td><td>0.12</td><td>0.42</td><td>0.21</td></tr><tr><td>✓</td><td>✓</td><td>0.24</td><td>0.46</td><td>0.76</td><td>0.49</td><td>0.08</td><td>0.10</td><td>0.39</td><td>0.19</td></tr></table>
+
+Table 5: Ablation study on latent prediction on NAVSIM and CARLA benchmark. NC: no at-fault collision. DAC: drivable area compliance. TTC: time-to-collision. Conf.: comfort. EP: ego progress. PDMS: the predictive driver model score. RC: route Completion. IS: infraction Score. DS: driving Score. LAW is in the perception-free setting.  
+
+<table><tr><td rowspan="2">Latent Prediction</td><td colspan="6">NAVSIM</td><td colspan="3">CARLA</td></tr><tr><td>NC ↑</td><td>DAC ↑</td><td>TTC ↑</td><td>Comf. ↑</td><td>EP ↑</td><td>PDMS ↑</td><td>RC ↑</td><td>IS ↑</td><td>DS ↑</td></tr><tr><td>×</td><td>94.4</td><td>89.4</td><td>84.8</td><td>100.0</td><td>75.1</td><td>77.5</td><td>98.6±0.8</td><td>0.68±0.02</td><td>67.9±2.1</td></tr><tr><td>✓</td><td>96.4</td><td>95.4</td><td>88.7</td><td>99.9</td><td>81.7</td><td>84.6</td><td>97.8±0.9</td><td>0.72±0.03</td><td>70.1±2.6</td></tr></table>
+
+enhancements in drivable area compliance (DAC) and Ego progress (EP) metrics. This suggests that our self-supervised task effectively enhances the quality of the driving trajectory. Similarly, in CARLA, we observed notable improvements in the Driving Score.
+
+The Time Horizon of Latent World Model In this experiment, the world model predicts latent features at three distinct future time horizons: 0.5 seconds, 1.5 seconds, and 3.0 seconds. This corresponds to the first, third, and sixth future frames from the current frame, given that keyframes occur every 0.5 seconds in the nuScenes dataset. The results, displayed in Table 6, show that the model achieves the best performance at the 1.5-second horizon. In comparison, the 0.5-second interval typically presents scenes with minimal changes, providing insufficient dynamic content to improve feature learning. In contrast, the 3.0-second interval often presents scenes that may significantly differ from the current frame, making accurate future predictions more challenging. Moreover, we observe that predicting latents 10 seconds into the future completely diminishes the gains provided by the world model, suggesting that predicting features too far into the future is ineffective. This conclusion aligns with observations from MAE (He et al., 2022), where both excessively low and high mask ratios negatively impact the ability of the network.
+
+Table 6: Different time horizons for latent prediction. The time intervals of 0.5, 1.5, 3.0, and 10.0 seconds correspond to the first, third, sixth, and twentieth future frames from the current frame, as keyframes occur every 0.5 seconds in the nuScenes dataset.  
+
+<table><tr><td rowspan="2">Time Horizon</td><td colspan="4">L2 (m) ↓</td><td colspan="4">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>0.5s</td><td>0.26</td><td>0.57</td><td>1.01</td><td>0.61</td><td>0.14</td><td>0.21</td><td>0.54</td><td>0.30</td></tr><tr><td>1.5s</td><td>0.26</td><td>0.54</td><td>0.93</td><td>0.58</td><td>0.14</td><td>0.17</td><td>0.45</td><td>0.25</td></tr><tr><td>3.0s</td><td>0.28</td><td>0.59</td><td>1.01</td><td>0.63</td><td>0.13</td><td>0.20</td><td>0.48</td><td>0.27</td></tr><tr><td>10.0s</td><td>0.33</td><td>0.67</td><td>1.16</td><td>0.72</td><td>0.24</td><td>0.28</td><td>0.76</td><td>0.43</td></tr></table>
+
+Network Architecture of Latent World Model To validate the impact of the network architecture of the latent world model, we conduct experiments as shown in Table 7. Firstly, it is evident that a single-layer neural network, represented as Linear Projection, is not adequate for fulfilling the functions of the world model, resulting in significantly degraded performance. The two-layer MLP shows considerable improvement in performance. However, it lacks the capability to facilitate interactions among different latent vectors. Therefore, we use the stacked transformer blocks as our
+
+![](images/76a981eb0677f98364b9c871a1f817307b0b7e3728d86b9438b9f94d27f5065e.jpg)  
+Figure 3: Visualization. This figure compares LAW in the perception-based setting with VAD (Jiang et al., 2023). On the right side of this figure, we display the results of ego trajectory prediction, agent motion prediction, and map construction in BEV. As indicated by the red circles, our method captures more crucial scene information, which VAD overlooks. Consequently, VAD predicts a forward trajectory that results in a rear-end collision, as highlighted by the yellow circles.
+
+default network architecture, which achieves the best results among the tested architectures. This indicates that interactions between feature vectors from different positions are important.
+
+Table 7: Different network architecture of the latent world model. Linear Projection means a single-layer network.  
+
+<table><tr><td rowspan="2">Architecture</td><td colspan="4">L2 (m) ↓</td><td colspan="4">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>Linear Projection</td><td>0.31</td><td>0.65</td><td>1.14</td><td>0.70</td><td>0.26</td><td>0.34</td><td>0.66</td><td>0.42</td></tr><tr><td>Two-layer MLP</td><td>0.27</td><td>0.58</td><td>1.07</td><td>0.64</td><td>0.17</td><td>0.23</td><td>0.59</td><td>0.33</td></tr><tr><td>Transformer Blocks</td><td>0.26</td><td>0.57</td><td>1.01</td><td>0.61</td><td>0.14</td><td>0.21</td><td>0.54</td><td>0.30</td></tr></table>
+
+# 5.5 VISUALIZATION
+
+Figure 3 compares the results of LAW in the perception-based setting with VAD (Jiang et al., 2023). Leveraging our latent world model, our approach acquires more comprehensive scene representations.
+
+# 6 CONCLUSION
+
+In conclusion, we present the latent world model to predict future features from current features and ego trajectories, which is a novel self-supervised learning method for end-to-end autonomous driving. This method jointly enhances scene representation learning and ego trajectory prediction. Our approach demonstrates universality by accommodating both perception-free and perception-based frameworks, predicting perspective-view features and BEV features respectively. We achieve state-of-the-art results on benchmarks like nuScenes, NAVSIM, and CARLA.
+
+# 7 ACKNOWLEDGMENTS
+
+This work was supported by National Science and Technology Major Project (2022ZD0116500). This work was also supported in part by the National Key R&D Program of China (No. 2022ZD0116500), the National Natural Science Foundation of China (No. U21B2042, No. 62320106010), and in part by the 2035 Innovation Program of CAS.
+
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+
+# A APPENDIX
+
+# A.1 PREDICTING MULTIPLE FEATURES USING MULTIPLE INPUT FRAMES
+
+Predicting multiple future features To better investigate the ability of our latent world model, we utilize the latent world model to predict multiple future frame latents, with the results presented in Table 8. We conduct this experiment using only the front-view camera to facilitate fast training. In detail, the future frame latents are predicted in an auto-regressive manner. For example, we first predicted the latent for 1.5 seconds into the future, then used this predicted latent to further predict the latent for 3 seconds into the future. The latent world model shares the same weights throughout this process. The results demonstrate that predicting multiple future latents further improves performance.
+
+Table 8: Predicting multiple future latents in an auto-regressive manner.  
+
+<table><tr><td rowspan="2">Predicted Future</td><td colspan="4">L2 (m) ↓</td><td colspan="4">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>1.5s</td><td>0.34</td><td>0.69</td><td>1.17</td><td>0.73</td><td>0.12</td><td>0.22</td><td>0.63</td><td>0.32</td></tr><tr><td>1.5s → 3s</td><td>0.31</td><td>0.65</td><td>1.12</td><td>0.69</td><td>0.11</td><td>0.19</td><td>0.57</td><td>0.29</td></tr></table>
+
+Predicting multiple future features using multiple input frames Building on the previous section, we further conduct experiments to predict multiple future frame latents while incorporating multiple input frame latents to leverage temporal information more effectively. Specifically, we adopted a two-stage training paradigm for improved convergence, inspired by SOLOFusion (Park et al., 2022). In the first stage, we trained the model using single input frame latents for 12 epochs. This corresponds to the model denoted as "1.5s  $\rightarrow$  3s" in Table 8. In the second stage, we fine-tuned the model for an additional 6 epochs, now using two input frame latents, from 0s and 1.5s ago. The results are summarized in Table 9. The baseline (first row) represents the model fine-tuned using only single input frame latents. In contrast, the second row corresponds to the model fine-tuned with two input frame latents. The latter achieves significantly better performance. This highlights the crucial role of temporal information in autonomous driving.
+
+Table 9: Predicting future latents with multiple history frame inputs.  
+
+<table><tr><td rowspan="2">Predicted Future</td><td rowspan="2">Input Frames</td><td colspan="4">L2 (m) ↓</td><td colspan="4">Collision (%) ↓</td></tr><tr><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td><td>1s</td><td>2s</td><td>3s</td><td>Avg.</td></tr><tr><td>1.5s → 3s</td><td>0s</td><td>0.30</td><td>0.64</td><td>1.09</td><td>0.68</td><td>0.14</td><td>0.23</td><td>0.62</td><td>0.33</td></tr><tr><td>1.5s → 3s</td><td>-1.5s, 0s</td><td>0.26</td><td>0.51</td><td>0.87</td><td>0.55</td><td>0.08</td><td>0.09</td><td>0.33</td><td>0.17</td></tr></table>
+
+# A.2 MORE VISUALIZATION
+
+In the appendix, we provide more visualization figures. We also provide a demo based on the CARLA simulator in the supplementary materials.
+
+![](images/4cedefaf97c57543bd7d95a525e5b4cc4b1e0d2326413221f288a791361db991.jpg)  
+Figure 4: Visualization. As shown in the red circle, our map construction results are noticeably better than those of VAD.
+
+![](images/5b49ba068d63f1e1f7e29980d07628f06db1010492cc86f4d0e9f02f832184b6.jpg)  
+Figure 5: Visualization. As shown in the red circle, our agent motion prediction results are noticeably better than those of VAD.
+
+![](images/de03c75e6f4299604cc9d636ea6f944750f8b7a5fed8fc6f8985ae1344dc7d8c.jpg)
+
+![](images/a3085091ac31009d9cd7553864db6af394ab3a8d0cfa4d1ad12952784e85d4b9.jpg)
+
+![](images/e9c16ef8f769484e5895ce13a123f0c403a31355a370c95f03ddf945ad242e90.jpg)
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+![](images/c8c1192758ddf0ef274d13ff61c2bba0160839b8f23024c4cc29d6b569691fa2.jpg)
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+![](images/4edf67e2f8bc9e118e9175fd18f675515f3df2248e121a302b45a1b5c30354cf.jpg)
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+![](images/890366d452cfa7a11b33ecf81cd0c754b374805bcb3c14bbf4dadc4a02f653a8.jpg)
+
+![](images/c1fa25378c3170f8d6d30bacd34a02ae191d881f8d4637da590de561ca9be688.jpg)  
+Figure 6: Visualization. As shown in the red circle, our map construction and agent motion prediction results are noticeably better than those of VAD, especially in heavily occluded and crowded conditions.
+
+![](images/f0912d2089aba0acf341fe6a8b32f214c9c79747c183a03a37a900d6f12a02eb.jpg)

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2025/Enhancing Federated Domain Adaptation with Multi-Domain Prototype-Based Federated Fine-Tuning/f85d9870-7312-4371-8cb3-a83313635b00_origin.pdf

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2025/Enhancing Federated Domain Adaptation with Multi-Domain Prototype-Based Federated Fine-Tuning/full.md

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+# ENHANCING FERERATED DOMAIN ADAPTATION WITH MULTI-DOMAIN PROTOTYPE-BASED FERERATED FINETUNING
+
+Jingyuan Zhang $^{1}$ , Yiyang Duan $^{1}$ , Shuaicheng Niu $^{1}$ , Yang Cao $^{2}$ , Wei Yang Bryan Lim $^{1*}$ $^{1}$  College of Computing and Data Science, Nanyang Technological University  
+ $^{2}$  Department of Computer Science, Institute of Science Tokyo  
+{jzhang149, yiyang007}@e.ntu.edu.sg, shuaicheng.niu@ntu.edu.sg  
+cao@c.titech.ac.jp, bryan.limwy@ntu.edu.sg
+
+# ABSTRACT
+
+Federated Domain Adaptation (FDA) is a Federated Learning (FL) scenario where models are trained across multiple clients with unique data domains but a shared category space, without transmitting private data. The primary challenge in FDA is data heterogeneity, which causes significant divergences in gradient updates when using conventional averaging-based aggregation methods, reducing the efficacy of the global model. This further undermines both in-domain and out-of-domain performance (within the same federated system but outside the local client). To address this, we propose a novel framework called Multi-domain Prototype-based Federated Fine-Tuning (MPFT). MPFT fine-tunes a pre-trained model using multi-domain prototypes, i.e., pretrained representations enriched with domain-specific information from category-specific local data. This enables supervised learning on the server to derive a globally optimized adapter that is subsequently distributed to local clients, without the intrusion of data privacy. Empirical results show that MPFT significantly improves both in-domain and out-of-domain accuracy over conventional methods, enhancing knowledge preservation and adaptation in FDA. Notably, MPFT achieves convergence within a single communication round, greatly reducing computation and communication costs. To ensure privacy, MPFT applies differential privacy to protect the prototypes. Additionally, we develop a prototype-based feature space hijacking attack to evaluate robustness, confirming that raw data samples remain unrecoverable even after extensive training epochs. The complete implementation of MPFL is available at https://ntu-zjy.github.io/DomainFL/.
+
+# 1 INTRODUCTION
+
+Federated Learning (FL) is a privacy-preserving distributed machine learning paradigm designed to protect the data of participating clients (McMahan et al., 2017b). In FL, only models trained on local data are shared between clients and servers, rather than the raw data itself, mitigating the risk of data leaks. Mainstream FL research primarily focuses on optimizing each client's performance within its local data domain (in-domain performance). However, in Federated Domain Adaptation (FDA) scenarios, clients need to perform well on the collective data domains shared by all participants to meet certain business requirements. For instance, consider a consortium of banks training a model to detect fraudulent transactions. Each bank has distinct customer bases and transaction patterns, leading to variations in their data (domains). A good out-of-domain performance is essential for anticipating new risks that a bank may not be exposed to yet. The challenge is to ensure the global model performs well across all banks, achieving good in-domain accuracy (within each bank's typical use cases) and out-of-domain accuracy (generalizing across the federated financial system).
+
+Studies have shown that FL achieves results similar to centralized training only when the datasets among clients are independently and identically distributed (i.i.d) and share similar domain char
+
+![](images/0804367e8802a8e5642a68b19cc9a405c3c83a0e992bf13adb59ca3385b33d00.jpg)  
+colors:domain
+
+![](images/6089c39b3308c4259bfaa59f89a47f5463e6e01ea581be110045058707ae9210.jpg)
+
+![](images/de1ac543b68a8db8db23bc493647261f460a4756b8dc97e09efa52651ad8643b.jpg)
+
+![](images/d4eafe0cd4ebb0634983cb863fd82a451266a6e13b71db2da296e35490a95328.jpg)  
+aggregated model
+
+![](images/d2013a516e2d977452e2e20011deceb7a3e9c1f289926e84090e38edf4e06814.jpg)  
+local model
+
+![](images/ca9d7e23dec8624a90c11fca05c484e1710765ee7ad0e7130964358a8143e832.jpg)  
+(a) centralized learning
+
+![](images/2301ee4673c5d21cde5be769f06e82cdf67f3ceacee52386b4fee7ca5fd0623e.jpg)  
+(b) FL with averaging-based aggregation
+
+![](images/be7d7178e160f10c3e4827ed86a93359b92a461a544ca416cd7a3a02c5163aa1.jpg)  
+(c) MPFT with average sampling  
+Figure 1: Comparison of MPFT to centralized learning and previous averaging-based FL approaches.
+
+![](images/9e8f47d274ca49517cebb410ab2ad2a6d2f9e05781f5bf984b9edd1ce155f781.jpg)  
+(d) MPFT with cluster sampling
+
+![](images/30a0af9cd037d635a35a210fadaa2f900f13eeae90bc9cc53b9ba366115a43e2.jpg)  
+(e) MPFT with random sampling
+
+acteristics (Liu et al., 2023; Seol & Kim, 2023; Shaheen et al., 2022). In typical scenarios where all clients' models are trained on local data from the same domains, update directions are aligned, making it feasible to use averaging-based aggregation algorithms such as FedAvg (McMahan et al., 2017b). However, in FDA, in which clients have unique domains but a shared category space, the significant dissimilarities of feature space among local model updates mean that averaging of weights may not yield an optimal global model (Su et al., 2024; Sun et al., 2021).
+
+To address this issue, we propose a novel framework called Multi-domain Prototype-based Federated Fine-Tuning (MPFT). MPFT tackles this issue by having each client generate a specific proportion of data embeddings (i.e., prototypes) to be transmitted to the server to create a prototype training dataset. This allows us to simulate a centralized learning approach without transferring raw data, as prototypes encapsulate sufficient domain-specific features to represent the entire data domain. The server then fine-tunes a global adapter using this comprehensive set of prototypes, with the goal of approximating the performance of centralized learning without relying on conventional averaging approaches to derive a global model. Figure 1 illustrates how our aggregation method compares to centralized learning and traditional averaging-based FL approaches.
+
+Since only a specific proportion of local data embeddings are sampled as prototypes, and MPFT requires only a single round of global communication to converge, our framework incurs significantly lower computation and communication costs compared to other multi-round FL frameworks. MPFT also incorporates a differential privacy mechanism to mitigate the risk that the original data of specific prototypes are exposed during the prototype transmission process. Furthermore, simulations of feature space hijacking attacks on MPFT demonstrate that attackers cannot reconstruct the original data from the uploaded prototypes, even when the pretrained prototype encoder is known.
+
+Our contributions are as follows: a) We propose MPFT, a one-round federated fine-tuning framework with convergence guarantees that outperforms previous methods on multi-domain environments. b) We introduce a novel metric to evaluate the performance of the FDA, specifically assessing out-of-domain and in-domain accuracy to consider the trade-off between knowledge preservation and adaptation. c) We demonstrate empirically that MPFT incurs lower computational and communication overheads as compared to other FL methods while ensuring privacy through differential privacy protection of the prototypes and maintaining robustness against feature space hijacking attacks.
+
+# 2 RELATED WORK
+
+Efforts to enhance performance in FL across heterogeneous datasets (FDA) have been extensive. Regularization methods during training, such as FedProx (Li et al., 2020) and FedDyn (Jin et al., 2023), or knowledge distillation techniques like FedGen (Venkateswaran et al., 2023) and FedNTD (Lee et al., 2022), aim to align local models more closely with the global model. Such approaches help mitigate the divergence among local models but rely on the assumption that the averaged global model is well-suited for the entire data distribution which often does not hold in FDA.
+
+Updating only parts of the model during aggregation is another strategy, exemplified by personalized FL (PFL) (Tan et al., 2022a), which aims to customize local models to enhance in-domain perfor
+
+mance. Some methods involve using a personalized aggregation base to select specific portions of global information for model aggregation, or updating only parts of the model during the aggregation phase, e.g., APFL (Deng et al., 2020), FedFomo (Zhang et al., 2020), FedAMP (Huang et al., 2021), FedPHP (Li et al., 2021c), APPLE (Luo & Wu, 2022), and FedALA (Zhang et al., 2023a). Additionally, some methods split model layers to segregate global and local components, such as in FedPer (Arivazhagan et al., 2019), LG-FedAvg (Liang et al., 2020), FedRep (Husnoo et al., 2022), FedRoD (Chen & Chao, 2021), FedBABU (Oh et al., 2021), FedCP (Zhang et al., 2023b), FedGH (Yi et al., 2023), and DBE (Zhang et al., 2024). However, a key drawback is these methods still rely on averaging-based aggregation, which results in poor out-of-domain adaptation performance, despite achieving some in-domain improvements.
+
+Some FL methods incorporate prototype learning. For instance, FedProto (Tan et al., 2022b) utilizes averaged local prototypes to train as the global prototype on local clients. FPL (Huang et al., 2023) transmit all the prototypes (data embeddings) to the server during the initial phase for clustering and averaging, which is then sent back to the clients for further optimization. This method introduces significant communication overhead and potential privacy leakage during the first phase. FedNH (Dai et al., 2023) leverages class prototype transmission to address class imbalance across clients, rather than tackling the more complex issue of domain heterogeneity.
+
+Unlike the above methods, we do not assume that the average aggregation of different client models or prototypes can represent the global model or global prototype in scenarios involving heterogeneous client data. MPFT uses prototypes from different clients as basic units that replace data distribution for centralized training on the server, thus obtaining an approximate global model capable of fitting all heterogeneous client domains distribution, as demonstrated in Appendix A.
+
+# 3 PROBLEM STATEMENT
+
+Scenario assumption. Consider a scenario involving  $N$  clients, where each client possesses a private training dataset  $\mathbb{D}_1,\ldots ,\mathbb{D}_N$ , each drawn from a unique data domain and shared category space<sup>1</sup>. Our goal is to train a model that balances domain knowledge preservation and domain knowledge adaptation. Domain knowledge preservation refers to the ability of the model to retain each client's unique domain insights. Domain knowledge adaptation is defined as the model's ability to indirectly extract and transfer knowledge from each participating client's domain to others, even if a client  $i$  cannot directly train on another client  $j$ 's data.
+
+Optimization goal. Building on this, we define the Domain Knowledge Preservation loss  $\mathcal{L}^P$  and the Domain Knowledge Adaptation loss  $\mathcal{L}^A$  as follows:
+
+$$
+\mathcal {L} ^ {P} = \sum_ {i = 1} ^ {N} \mathcal {L} _ {i} \left(\Theta_ {i} ^ {\mathcal {P}}; \mathbb {D} _ {i}; \Theta^ {\mathcal {G}}\right), \quad \mathcal {L} ^ {A} = \sum_ {i = 1} ^ {N} \sum_ {j = 1, j \neq i} ^ {N} \mathcal {L} _ {i} \left(\Theta_ {i} ^ {\mathcal {P}}; \mathbb {D} _ {j}; \Theta^ {\mathcal {G}}\right), \tag {1}
+$$
+
+where  $\mathcal{L}(\cdot)$  denotes the loss function,  $\Theta_i^{\mathcal{P}}$  denotes the local (personalized) model parameters for client  $i$ , and  $\Theta^{\mathcal{G}}$  denotes the global model parameters. In conventional FL, the local models  $\Theta_1^{\mathcal{P}},\ldots ,\Theta_N^{\mathcal{P}}$  are periodically synchronized with the global model  $\Theta^{\mathcal{G}}$ .
+
+The optimization goal is to decrease both the  $\mathcal{L}^P$  and  $\mathcal{L}^A$ . Thus, we formalize the FDA optimization goal as follows:
+
+$$
+\left\{\Theta_ {1} ^ {\mathcal {P}}, \dots , \Theta_ {N} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}} \right\} = \arg \min  \left(\alpha_ {i} ^ {1} \mathcal {L} ^ {P} + \alpha_ {i} ^ {2} \mathcal {L} ^ {A}\right), \tag {2}
+$$
+
+where  $\alpha_{i}^{1}$  and  $\alpha_{i}^{2}$  are client-defined weight parameters which balance the trade-off between domain knowledge preservation and domain knowledge adaptation, in line with the "no free lunch" theorem.
+
+Evaluation metrics. To quantify the effectiveness of the optimization, we propose two metrics: in-domain accuracy (ind acc) and out-of-domain accuracy (ood acc). Denoting  $\mathrm{ACC}_i^{(j)}$  to be the accuracy for a client  $i$  when tested on domain  $j$ , ind acc andood acc are defined as follows:
+
+$$
+\operatorname {i n d} \operatorname {a c c} = \frac {\sum_ {i = 1} ^ {N} \mathbf {A C C} _ {i} ^ {(i)} n _ {i}}{\sum_ {i = 1} ^ {N} n _ {i}}, \quad \text {o o d} \operatorname {a c c} = \frac {\sum_ {i = 1} ^ {N} \sum_ {j \neq i} \mathbf {A C C} _ {i} ^ {(j)} n _ {j}}{\sum_ {i = 1} ^ {N} \sum_ {j \neq i} n _ {j}}, \tag {3}
+$$
+
+where  $n_i$  is the number of test samples for client  $i$ . The ind acc measures each client's performance on its own domain data, while the ood acc evaluates adaptation performance when tested on data from other domains.
+
+# 4 METHODOLOGY
+
+Overview. We introduce Multi-domain Prototype-based Federated Fine-Tuning (MPFT), which consists of three main components: Prototype Generation, Global Adapter Initialization, and Few-shot Local Adaptation, as depicted in Figure 2. During the Prototype Generation phase, we generate domain-specific prototypes for each client based on specific sampling methods and ratios. The clients subsequently transmit their local prototypes to the server. In the Global Adapter Initialization phase, we utilize these prototypes to train a global adapter designed to handle the multi-domain distribution of all clients, thereby improving the ood performance of the global adapter across clients. The global adapter is then sent back to the clients for local inference. While the global adapter performs well in ood accuracy, some clients may require better performance. In such cases, they can proceed to the Few-shot Local Adaptation phase, where a few-shot dataset is sampled locally to further fine-tune the local adapter. Knowledge distillation is employed to mitigate catastrophic forgetting of global knowledge during this phase.
+
+![](images/59958043d17c08f4544aeba2c88419f12f519d6c7714ce3cdf3b1884079e7c5a.jpg)  
+Figure 2: An overview of MPFT.
+
+Prototype Generation. A prototype is a compact representation of a specific class feature that is unique to each domain. To synchronize the consistency of the prototype's embedding space across domains, each client utilizes the same pretrained image encoder to generate the prototype set  $\{\mathbb{P}^{(1)},\dots ,\mathbb{P}^{(\mathcal{K})}\}$ , where  $\kappa$  represents the total number of classes. Although all clients share the same label space, each label manifests uniquely within its respective feature domain.
+
+To generate these prototypes, different sampling methods can be applied, including Mean Sampling, Cluster Sampling, and Random Sampling, as described in Algorithm 1. The choice of sampling method depends on the desired trade-off between computational efficiency and prototype representational robustness. In Mean Sampling, each client  $i$  generates a prototype for class  $k$  by calculating the mean of the pretrained embeddings for that class. In Cluster Sampling, clustering (e.g., k-means) is performed on pretrained embeddings of each class, and a certain number of cluster centers are then selected based on a predefined sampling rate to form the prototype set. In Random Sampling, a fixed number of pretrained embeddings of each class are randomly selected according to the sampling rate, and these selected embeddings constitute the prototype set.
+
+Once each client has generated their prototypes, they transmit these to the server. Consequently, the server accumulates  $N$  domain-specific representations subset (prototypes subset) for each class  $k$ , which are collectively represented as  $\mathbb{D}^{\mathcal{P}} = \left\{\bigcup_{i=1}^{N}\left\{\mathbb{P}_{i}^{(1)}, \ldots, \mathbb{P}_{i}^{(\mathcal{K})}\right\}\right\}$ .
+
+Algorithm 1 Different Sampling Methods  
+1: function MEAN SAMPLING  $(\mathbb{D}_i)$    
+2: for class  $k$  in 1,..,K do   
+3:  $P_{i}^{(k)}\gets \frac{1}{|\mathbb{D}_{i}^{(k)}|}\sum_{(x,y)\in \mathbb{D}_{i}^{(k)},y = k}f(\phi ;x)$ $\triangleright$  Compute mean embeddings for each class   
+4: end for   
+5: return  $\mathbb{P}_i\gets \{P_i^{(1)},\ldots ,P_i^{(\mathcal{K})}\}$    
+6: end function   
+7: function CLUSTER SAMPLING  $(\mathbb{D}_i,r)$    
+8: for class  $k$  in 1,...,K do   
+9:  $C^k\gets \left[r\times |\mathbb{D}_i^{(k)}|\right]$ $\triangleright$  Set the number of cluster centers   
+10:  $\mathbb{P}_i^{(k)}\gets \text{Clusterr}\left(f(\phi ;\mathbb{D}_i^{(k)}),C^k\right)$ $\triangleright$  Perform cluster sampling   
+11: end for   
+12: return  $\mathbb{P}_i\gets \bigcup_{j = 1}^{\mathcal{K}}\mathbb{P}_i^{(j)}$    
+13: end function   
+14: function RANDOM SAMPLING  $(\mathbb{D}_i,r)$    
+15: for class  $k$  in 1,...,K do   
+16:  $C^k\gets \left[r\times |\mathbb{D}_i^{(k)}|\right]$ $\triangleright$  Set the number of randomly selected embeddings   
+17:  $\mathbb{P}_i^{(k)}\gets \text{RandomlySelect}\left(f(\phi ;\mathbb{D}_i^{(k)}),C^k\right)$ $\triangleright$  Perform random sampling   
+18: end for   
+19: return  $\mathbb{P}_i\gets \bigcup_{j = 1}^{\mathcal{K}}\mathbb{P}_i^{(j)}$    
+20: end function
+
+Global adapter initialization. Algorithm 2 outlines the global adapter initialization process. Note that we avoid averaging-based aggregation within each class across different clients, as this would distort the global distribution. Utilizing  $\mathbb{D}^{\mathcal{P}}$ , we train the global adapter  $A^G$  to adapt to the entire system's data distribution with cross-entropy loss  $\mathcal{L}$ :
+
+$$
+\left\{\Theta^ {\mathcal {G}}, A ^ {\mathcal {G}} \right\} = \arg \min  \mathcal {L} \left(\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}; A ^ {\mathcal {G}}\right). \tag {4}
+$$
+
+Upon successful training, the global adapter  $A^{\mathcal{G}}$  is sent to the clients, replacing their local adapters.
+
+Algorithm 2 Federated Learning with Global Adapter Initialization  
+Input:  $N$  clients,  $\mathcal{L}$  : loss function,  $\Theta^0\{f(\phi),g(\phi)\}$  : pretrained CLIP model  $\Theta^0$  with image encoder  $f(\phi)$  and text encoder  $g(\phi)$ $A^0$  : random initialized adapter,  $\eta$  : learning rate,  $\kappa$  : number of data classes,  $\mathbb{D}_i$  : client  $i$  training data,  $(x,y)$  : data sample, method: sampling method (e.g., mean, cluster, random),  $r$  : sampling rate.   
+Output: Reasonable global adapter  $A^G$    
+1: Generate linear probe classification head  $H$  by labels and pretrained text encoder  $g(\phi)$    
+2: Server sends  $f(\phi)$ $A^0$  and  $H$  to all clients to initialize local models.   
+3: for client  $i$  in 1,...,  $N$  in parallel do   
+4:  $\mathbb{P}_i\gets$  method  $(\mathbb{D}_i,r)$    
+5: end for   
+6: Clients send the prototype to the server.   
+7: Server constructs the prototype training dataset  $\mathbb{D}^{\mathcal{P}}$  by  $\mathbb{D}^{\mathcal{P}}\gets \bigcup_{i = 1}^{N}\mathbb{P}_{i}$    
+8: while  $A^G$  does not converge do   
+9: Server optimizes  $A^G$  by  $A^G\gets A^G -\eta \nabla_{A^G}\mathcal{L}(\mathbb{D}^{\mathcal{P}};\Theta^{\mathcal{G}};A^{\mathcal{G}})$    
+10: end while   
+11: return  $A^G$
+
+Few-shot local adaptation. While global adapter  $A^G$  performs well in ood accuracy, it may not be sufficient for ind accuracy. To address this, clients can use their local few-shot data  $\mathbb{D}_i^{\mathcal{F}}$  to further fine-tune  $A^G$ , adapting it to their local domain and improving ind accuracy, as shown in Algorithm 3.
+
+# Algorithm 3 Few-shot Local Adaptation
+
+Input:  $N$  joining clients,  $\mathcal{L}$ : loss function,  $\Theta^0\{f(\phi),g(\phi)\}$ : pretrained CLIP model  $\Theta^0$  with image encoder  $f(\phi)$  and text encoder  $g(\phi)$ ,  $A^G$ : global initialized adapter,  $\alpha$ : local learning rate,  $\mathcal{K}$ : number of data classes,  $\mathcal{F}$ : few-shot number,  $\mathbb{D}_i$ : client  $i$  training data,  $(x,y)$ : data sample.
+
+Output: Reasonable local adapters  $\{A_1^{\mathcal{L}},\ldots ,A_N^{\mathcal{L}}\}$
+
+1: Server sends  $A^G$  to all clients to initialize local adapter  $A^\mathcal{L}$ .  
+2: for client  $i$  in  $1,\ldots ,N$  in parallel do  
+3:  $\mathbb{D}_i^{\mathcal{F}}\gets \left\{\bigcup_{j = 1}^{\mathcal{K}}\left\{(x_m,y_m)\in \mathbb{D}_i\mid y_m = j,m = 1,\ldots ,\mathcal{F}\right\} \right\}$  Generate few-shot data  
+4: Client  $i$  obtains  $A_{i}^{\mathcal{L}}$  by  $A_{i}^{\mathcal{L}}\gets A_{i}^{\mathcal{L}} - \eta \nabla_{A_{i}^{\mathcal{L}}} \mathcal{L}(\mathbb{D}_{i}^{\mathcal{F}};\Theta^{\mathcal{G}};A_{i}^{\mathcal{L}};A^{\mathcal{G}})$ . ▷ Local adaptation
+
+5: end for
+
+6: return  $\{A_1^{\mathcal{L}},\ldots ,A_N^{\mathcal{L}}\}$
+
+To avoid catastrophic forgetting (French, 1999) of global knowledge during the local adaptation process, knowledge distillation (KD) method is employed to regularize the locally trained adapter  $A^{\mathcal{L}}$ , ensuring it does not deviate too much from the global adapter  $A^{\mathcal{G}}$ . The loss function for local adaptation is defined as:
+
+$$
+\mathcal {L} = \mathcal {L} _ {\mathrm {C E}} + \beta \mathcal {L} _ {\mathrm {K D}}, \tag {5}
+$$
+
+where  $\mathcal{L}_{\mathrm{CE}}$  represents the cross-entropy loss, and  $\mathcal{L}_{\mathrm{KD}}$  denotes the KD loss. The hyperparameter  $\beta$  balances the influence of the KD loss in the overall objective. In Section 5.4, we compare the effects of different KD weights  $\beta$  on balancing ind accuracy and ood accuracy.
+
+Convergence. We analyze the convergence of MPFT, with detailed proofs provided in Appendix B.
+
+Theorem 1 (Convergence of fine-tuning with prototypes) For a smooth, non-convex loss function  $\mathcal{L}$  with a Lipschitz continuous gradient with constant  $L$ , the global fine-tuning using prototypes  $\mathbb{D}^{\mathcal{P}}$  converges. The sequence of updates for the global adapter  $A^{\mathcal{G}}$  achieves a monotonic decrease in the loss function  $L(\mathbb{D}^{\mathcal{P}}; \Theta^{\mathcal{G}}, A^{\mathcal{G}})$ . Specifically, choosing a learning rate  $\eta$  such that  $0 < \eta < \frac{2}{L}$  ensures:
+
+$$
+\mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t + 1} ^ {\mathcal {G}}) \leq \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) - c \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} + \frac {\Delta}{N},
+$$
+
+where  $c = \eta - \frac{L\eta^2}{2}$  is a positive constant, ensuring that the step size is appropriately bounded for convergence. Here,  $\Delta$  is the maximum prototype divergence across clients.
+
+Corollary 1.1 (Convergence to stationary point and rate) As  $T$  increases, the average gradient norm decreases, indicating convergence to a stationary point:
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} \leq \frac {\mathcal {L} (\mathbb {D} ^ {\mathcal {P}} ; \Theta^ {\mathcal {G}} , A _ {1} ^ {\mathcal {G}}) - \mathcal {L} (\mathbb {D} ^ {\mathcal {P}} ; \Theta^ {\mathcal {G}} , A _ {T} ^ {\mathcal {G}}) + \frac {T \Delta}{N}}{c T}.
+$$
+
+This indicates that as  $T$  increases, the right-hand side approaches zero, confirming that the gradient norm diminishes and the updates converge to a stationary point.
+
+# 5 EXPERIMENT
+
+Datasets and Models. To simulate a FDA environment, We use the DomainNet (Peng et al., 2019) and PACS (Li et al., 2017) datasets which are widely used in multi-domain data adaptation. For these datasets, we employ pretrained CLIP models from OpenCLIP (Cherti et al., 2023; Radford et al., 2021; Schuhmann et al., 2022). The image encoder of CLIP for DomainNet is a ViT-B-32 pretrained on the LAION-2B dataset, while for PACS, we use a ConvNeXT-Base pretrained on the LAION-400M dataset as the image encoder.
+
+Implementation details. We implement various representative FL algorithms as baselines, including FedAvg (McMahan et al., 2017a), FedProx (Li et al., 2020), Ditto (Li et al., 2021b), MOON (Li et al., 2021a), FedProto (Tan et al., 2022b), and DBE (Zhang et al., 2024), using the PyTorch library
+
+and based on the integrated FL library PFLlib (Zhang et al., 2023c). To simulate the common FL scenario where data resides only on clients, we split the data for each client into a training set  $(70\%)$ , a test set  $(20\%)$ , and a validation set  $(10\%)$ . We evaluate in-domain (ind) and out-of-domain (ood) accuracy following Equation 3. To demonstrate the scalability of our method, we partition the DomainNet dataset, which includes 345 categories, into subsets containing 50, 100, and 150 classes, respectively. To more conveniently compute the convergence time for each FL method and compare the computational and communication costs of them, we introduce an early stopping strategy during training. More details about our experimental setup and baselines can be found in Appendix C.1 and Appendix C.2, with details on the early stopping strategy provided in Appendix C.3.
+
+Diverse FDA Scenarios. We conducted experiments and analyses from different perspectives on various potential FDA scenarios. Section 5.1 presents the performance of different FL methods in a basic scenario, where each client is assigned a unique data domain, with minimal overlap between domains. Section 5.2 provides a detailed analysis of the global model's performance on each client, exploring the fairness of different methods in FDA. In Section 5.3, we design a more realistic scenario where each client may hold data from multiple domains, reducing the domain heterogeneity between clients. Appendix F sets up another realistic scenario where multiple clients belong to the same data domain, significantly increasing the number of clients compared to the original setup. In Section 5.4, we investigate the role of local adaptation. Section 5.5 compares the computational and communication costs of different methods. Finally, in Section 5.6 and Appendix G, we explore the differential privacy mechanism in MPFT and its robustness against feature space hijacking attacks.
+
+# 5.1 PERFORMANCE ON MULTI-DOMAIN
+
+We evaluate our method alongside other FL approaches in Table 1, including local training (i.e., each client fine-tunes the pretrained model separately). Empirical results show that local training excels in ind accuracy but performs poorly in ood accuracy. A reason is that local fine-tuning results in catastrophic forgetting (Luo et al., 2023). Personalized FL methods such as FedProto and DBE, which generally maintain a personalized local model for each client, have higher ind accuracy but compromise ood accuracy. In contrast, methods like FedAvg, MOON, and Ditto demonstrate more balanced improvements in both ind and ood accuracies. FedProx, which introduces a regularization term between the global and local models, improves ood accuracy at the expense of ind accuracy. In comparison, our method consistently achieves superior performance in both ood and ind accuracy across all DomainNet subsets. As the subset size of DomainNet increases, we observe variable convergence stability across methods such as FedAvg, FedProx, and Ditto, while DBE demonstrates accelerated convergence. In contrast, our method requires only one global communication round, which significantly reduces both computational and communication costs. This benefit is further elaborated in section 5.5.
+
+Table 1: Test accuracy and communication rounds for different FL methods on DomainNet subsets and PACS. The communication rounds are determined using an early stopping strategy, where fewer rounds indicate faster convergence. Additionally, we compare the sensitivity of MPFT to different global convergence thresholds to ensure the robustness of the results across various hyperparameters, as refer to Appendix D.  
+
+<table><tr><td rowspan="2"></td><td colspan="3">DomainNet: Subset-50</td><td colspan="3">DomainNet: Subset-100</td><td colspan="3">DomainNet: Subset-150</td><td colspan="3">PACS</td></tr><tr><td>od acc</td><td>ind acc</td><td>rounds</td><td>od acc</td><td>ind acc</td><td>rounds</td><td>od acc</td><td>ind acc</td><td>rounds</td><td>od acc</td><td>ind acc</td><td>rounds</td></tr><tr><td>local</td><td>0.7361</td><td>0.8609</td><td>0</td><td>0.6554</td><td>0.8310</td><td>0</td><td>0.6003</td><td>0.8067</td><td>0</td><td>0.6547</td><td>0.9984</td><td>0</td></tr><tr><td>FedAvg (McMahan et al., 2017a)</td><td>0.7902</td><td>0.7345</td><td>24</td><td>0.7628</td><td>0.6966</td><td>49</td><td>0.7263</td><td>0.6709</td><td>17</td><td>0.9725</td><td>0.9887</td><td>32</td></tr><tr><td>FedProx (Li et al., 2020)</td><td>0.7752</td><td>0.7178</td><td>10</td><td>0.7499</td><td>0.6827</td><td>9</td><td>0.7131</td><td>0.6569</td><td>5</td><td>0.9219</td><td>0.9659</td><td>13</td></tr><tr><td>Ditto (Li et al., 2021b)</td><td>0.7811</td><td>0.7624</td><td>20</td><td>0.7511</td><td>0.7182</td><td>30</td><td>0.7149</td><td>0.6904</td><td>13</td><td>0.9172</td><td>0.9930</td><td>35</td></tr><tr><td>MOON (Li et al., 2021a)</td><td>0.7902</td><td>0.7344</td><td>28</td><td>0.7623</td><td>0.6952</td><td>16</td><td>0.7267</td><td>0.6715</td><td>31</td><td>0.9763</td><td>0.9888</td><td>42</td></tr><tr><td>FedProto (Tan et al., 2022b)</td><td>0.7296</td><td>0.7696</td><td>5</td><td>0.6732</td><td>0.7385</td><td>8</td><td>0.6321</td><td>0.7073</td><td>7</td><td>0.8627</td><td>0.9963</td><td>32</td></tr><tr><td>DBE (Zhang et al., 2024)</td><td>0.7421</td><td>0.7622</td><td>22</td><td>0.7179</td><td>0.7233</td><td>6</td><td>0.6820</td><td>0.6956</td><td>5</td><td>0.971</td><td>0.984</td><td>12</td></tr><tr><td>MPFT (Average)</td><td>0.8077</td><td>0.7813</td><td>1</td><td>0.7674</td><td>0.7399</td><td>1</td><td>0.7294</td><td>0.7099</td><td>1</td><td>0.9486</td><td>0.9703</td><td>1</td></tr><tr><td>MPFT (Cluster, rate=0.1)</td><td>0.7951</td><td>0.7957</td><td>1</td><td>0.7641</td><td>0.7692</td><td>1</td><td>0.7171</td><td>0.7256</td><td>1</td><td>0.9808</td><td>0.9880</td><td>1</td></tr><tr><td>MPFT (Cluster, rate=0.3)</td><td>0.8204</td><td>0.8294</td><td>1</td><td>0.7766</td><td>0.7791</td><td>1</td><td>0.7430</td><td>0.7514</td><td>1</td><td>0.9841</td><td>0.9896</td><td>1</td></tr><tr><td>MPFT (Random, rate=0.1)</td><td>0.7953</td><td>0.7899</td><td>1</td><td>0.7566</td><td>0.7509</td><td>1</td><td>0.7194</td><td>0.7233</td><td>1</td><td>0.9829</td><td>0.9888</td><td>1</td></tr><tr><td>MPFT (Random, rate=0.3)</td><td>0.8236</td><td>0.8294</td><td>1</td><td>0.7803</td><td>0.7811</td><td>1</td><td>0.7469</td><td>0.7542</td><td>1</td><td>0.9887</td><td>0.9919</td><td>1</td></tr></table>
+
+# 5.2 DETAILS ABOUT RESULTS ON EACH DOMAIN
+
+To further explain why MPFT achieves better ind and ood accuracy compared to the baselines, we visualize the performance of each domain in Figure 3. Each axis of the radar chart represents a specific data domain (e.g., Real or Painting), with the shape and coverage area of the curves illustrating the global model's performance across these domains. Empirically, the roundness of the curve could reflect the fairness of the model across different clients (domains)—the rounder the curve, the more fair the method is in the global distribution, leading to better ood accuracy.
+
+Compared to other FL baselines across different DomainNet subset sizes, MPFT with average sampling method performs exceptionally well in the Quickdraw domain, with a more balanced curve shape. Additionally, MPFT maintains strong performance across other domains relative to the baselines, thereby achieving better overall fairness. For more details about the effects of random and cluster sampling compared to average sampling on each domain, please refer to Appendix E.
+
+![](images/5577987f9bd762e8cb5e02c373e2486c37a4b27c1fbdb90f96350afa58ce1e6f.jpg)  
+FedAvg
+
+![](images/09b5d8326ce98c7c1848caf8fca9545a5d62e4658a18a6e61c24662b9b5014ba.jpg)  
+edProx
+
+![](images/d3065e819f21fa846ad748cd7fd4348f9be929702458f4712abd3dde35e6db53.jpg)  
+Ditto
+
+![](images/eff2d45a8218ec129430449a47766f762ededfd87885cdf99697b258b2316502.jpg)  
+MOON
+
+![](images/e93b0cca141836e5165275b53cf9f8651e16715cf016e154ec10bfb9a186f287.jpg)  
+FedProto
+
+![](images/331fe0d3f12a9f9dd00ca774be03de359c7c996693828931e304d16e02b5170c.jpg)  
+DBE
+
+![](images/7f05d5cf3bf9373b8da49eb00880b5f940d82f5bbc9c972fc8cf2fe98cbc28ba.jpg)  
+MPFT (average)
+
+![](images/6e257bb01b0beb6f922a03890787ab59ab0b5ef2cd53186242fd5dc72788a7ce.jpg)  
+DomainNet,Subset-50  
+Figure 3: Comparison of different FL methods across various DomainNet subset sizes.
+
+![](images/0bc161937742e823e7c4218c6ed532278055ebc14ce799b9bbf273b3c68e356e.jpg)  
+DomainNet,Subset-100
+
+![](images/920977124e5b64c7558ec6bf12d6eadd0c742b803a6e811befec2c36b5da6933.jpg)  
+DomainNet,Subset-150
+
+# 5.3 IMPACT OF MULTI-DOMAIN DIFFERENCES ON PERFORMANCE
+
+In real-world scenarios, a client may contain data from multiple domains rather than a single specific domain $^2$ . We simulate a situation where each client contains  $1 - mr$  percent of data from its original domain, mixed with  $mr$  percent of data from another domain. Here,  $mr$  represents the mixed ratio, indicating the level of domain diversity on the client side. We evaluate our method alongside other FL approaches under this scenario, as shown in Table 2, where DomainNet subset-50 is used. We observe a reduction in the performance advantage of our method compared to others, as the mixed ratio increases. This decline is due to the reduced heterogeneity within the FL system, which diminishes the strengths of our approach. However, it is still evident that our method outperforms most FL algorithms, particularly when using Random and Cluster sampling strategies.
+
+Table 2: Results of mixed multi-domain situation on DomainNet Subset-50.  
+
+<table><tr><td rowspan="2"></td><td colspan="3">original</td><td colspan="3">mixed ratio = 0.3</td><td colspan="3">mixed ratio = 0.4</td><td colspan="3">mixed ratio = 0.5</td></tr><tr><td>od acc</td><td>ind acc</td><td>rounds</td><td>od acc</td><td>ind acc</td><td>rounds</td><td>od acc</td><td>ind acc</td><td>rounds</td><td>od acc</td><td>ind acc</td><td>rounds</td></tr><tr><td>local</td><td>0.7361</td><td>0.8609</td><td>0</td><td>0.7316</td><td>0.8477</td><td>0</td><td>0.7339</td><td>0.844</td><td>0</td><td>0.7408</td><td>0.8356</td><td>0</td></tr><tr><td>FedAvg (McMahan et al., 2017a)</td><td>0.7902</td><td>0.7345</td><td>24</td><td>0.7975</td><td>0.7639</td><td>19</td><td>0.7996</td><td>0.7691</td><td>8</td><td>0.8090</td><td>0.7845</td><td>7</td></tr><tr><td>FedProx (Li et al., 2020)</td><td>0.7752</td><td>0.7178</td><td>10</td><td>0.7798</td><td>0.7386</td><td>67</td><td>0.7813</td><td>0.7448</td><td>34</td><td>0.7903</td><td>0.7588</td><td>27</td></tr><tr><td>Ditto (Li et al., 2021b)</td><td>0.7811</td><td>0.7624</td><td>20</td><td>0.7966</td><td>0.8027</td><td>13</td><td>0.7811</td><td>0.7808</td><td>7</td><td>0.7777</td><td>0.7757</td><td>6</td></tr><tr><td>MOON (Li et al., 2021a)</td><td>0.7902</td><td>0.7344</td><td>28</td><td>0.7984</td><td>0.7639</td><td>13</td><td>0.8043</td><td>0.7763</td><td>8</td><td>0.8109</td><td>0.7865</td><td>8</td></tr><tr><td>FedProto (Tan et al., 2022b)</td><td>0.7296</td><td>0.7696</td><td>5</td><td>0.7110</td><td>0.7395</td><td>6</td><td>0.7075</td><td>0.7421</td><td>7</td><td>0.7019</td><td>0.7293</td><td>6</td></tr><tr><td>DBE (Zhang et al., 2024)</td><td>0.7421</td><td>0.7622</td><td>22</td><td>0.7348</td><td>0.7371</td><td>3</td><td>0.7293</td><td>0.7349</td><td>14</td><td>0.7286</td><td>0.7331</td><td>11</td></tr><tr><td>MPFT (Average)</td><td>0.8077</td><td>0.7813</td><td>1</td><td>0.7879</td><td>0.7610</td><td>1</td><td>0.7817</td><td>0.7577</td><td>1</td><td>0.7783</td><td>0.7533</td><td>1</td></tr><tr><td>MPFT (Cluster, rate=0.1)</td><td>0.7951</td><td>0.7957</td><td>1</td><td>0.8213</td><td>0.8169</td><td>1</td><td>0.7887</td><td>0.7897</td><td>1</td><td>0.8007</td><td>0.7912</td><td>1</td></tr><tr><td>MPFT (Cluster, rate=0.3)</td><td>0.8204</td><td>0.8294</td><td>1</td><td>0.8276</td><td>0.8253</td><td>1</td><td>0.8209</td><td>0.8142</td><td>1</td><td>0.8142</td><td>0.8071</td><td>1</td></tr><tr><td>MPFT (Random, rate=0.1)</td><td>0.7953</td><td>0.7899</td><td>1</td><td>0.7928</td><td>0.7856</td><td>1</td><td>0.8040</td><td>0.7958</td><td>1</td><td>0.7919</td><td>0.7801</td><td>1</td></tr><tr><td>MPFT (Random, rate=0.3)</td><td>0.8236</td><td>0.8294</td><td>1</td><td>0.8184</td><td>0.8141</td><td>1</td><td>0.8108</td><td>0.8065</td><td>1</td><td>0.8137</td><td>0.8050</td><td>1</td></tr></table>
+
+# 5.4 PERFORMANCE WITH LOCAL ADAPTATION
+
+We compare the few-shot performance of local adaptation with different knowledge distillation (KD) weights in Figure 4. As the KD weight increases, there is less out-of-domain knowledge forgetting but worse in-domain knowledge alignment. With an increase in the number of few-shot samples, the odd and ind accuracy show a similar trend. We provide more details about the experiment implementation of KD in local adaptation in Appendix C.4.
+
+![](images/6da036d64b9cbb56a88997174a8366c095e8ca8f1861bac8c9cc28b6e5ead733.jpg)  
+Figure 4: Few-shot performance comparison of local adaptation with different KD weights.
+
+# 5.5 COMPUTATION COST AND COMMUNICATION COST
+
+We further analyze the computation cost and communication cost in Table 3. Computation cost is the total training time of FL, which is related to the number of communication rounds and the computational complexity within each round. Among the methods compared, Ditto is the most time-consuming due to the additional local training epoch it requires. In contrast, the MPFT, which converges in just one global round, significantly reduces training time, particularly when the sampling method is set to average. For other sampling methods, such as Random and Cluster, our approach trades a modest increase in training time for substantial improvements in both ind and ood accuracy. Communication cost is the number of parameters transmitted, which is theoretically influenced by the number of communication rounds  $\mathcal{R}$ , the model (adapter) parameters  $\sum$ , and the prototypes  $\prod$ . The number of communication rounds  $\mathcal{R}$  directly contributes to a linear increase in the overall communication cost. Furthermore, the model adapter parameters  $\sum$  and the size of the prototypes  $\prod$  determine the communication cost per round in these specific algorithms. In our empirical results, MPFT with average sampling achieved the lowest communication cost across all experiments. However, while random or cluster sampling slightly increases communication overhead, it also significantly improves MPFT's performance (see Table 1).
+
+Table 3: Computation cost and communication cost of FL methods.  $\mathcal{R}$  represents the convergence round,  $\sum$  represents the number of model (adapter) parameters,  $\prod$  represents the size of prototypes,  $N$  represents the number of clients. The subsets are derived from the DomainNet dataset.  
+
+<table><tr><td rowspan="2"></td><td colspan="4">Computation Cost (total training time)</td><td colspan="5">Communication Cost (total parameter transmission)</td></tr><tr><td>Subset-50</td><td>Subset-100</td><td>Subset-150</td><td>PACS</td><td>Subset-50</td><td>Subset-100</td><td>Subset-150</td><td>PACS</td><td>Theoretical</td></tr><tr><td>local</td><td>240.8s</td><td>834.8s</td><td>1575.1s</td><td>311.2s</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>FedAvg</td><td>1933.5s</td><td>5812.4s</td><td>3824.6s</td><td>663.9s</td><td>144MB</td><td>294MB</td><td>102MB</td><td>128MB</td><td>R × N × 2Σ</td></tr><tr><td>FedProx</td><td>803.9s</td><td>1253.5s</td><td>1120.1s</td><td>270.0s</td><td>60MB</td><td>54MB</td><td>30MB</td><td>52MB</td><td>R × N × 2Σ</td></tr><tr><td>Ditto</td><td>3095.7s</td><td>8365.8s</td><td>5862.3s</td><td>1455.6s</td><td>120MB</td><td>180MB</td><td>78MB</td><td>140MB</td><td>R × N × 2Σ</td></tr><tr><td>MOON</td><td>2164.2s</td><td>2237.4s</td><td>7027.4s</td><td>872.7s</td><td>168MB</td><td>96MB</td><td>186MB</td><td>168MB</td><td>R × N × 2Σ</td></tr><tr><td>FedProto</td><td>393.1s</td><td>1153.4s</td><td>1608.0s</td><td>683.3s</td><td>5.9MB</td><td>18.8MB</td><td>24.6MB</td><td>3.5MB</td><td>R × N × 2Π</td></tr><tr><td>DBE</td><td>1693.6s</td><td>837.6s</td><td>1160.3s</td><td>248.8s</td><td>132MB</td><td>36MB</td><td>30MB</td><td>48MB</td><td>R × N × 2Σ</td></tr><tr><td>MPFT (Average)</td><td>1.9s</td><td>7.3s</td><td>10.7s</td><td>0.1s</td><td>3.6MB</td><td>4.2MB</td><td>4.8MB</td><td>2MB</td><td>N × (Π + Σ)</td></tr><tr><td>MPFT (Cluster, rate=0.1)</td><td>44.7s</td><td>302.4s</td><td>208.0s</td><td>0.6s</td><td>12.5MB</td><td>20.9MB</td><td>30.7MB</td><td>3.1MB</td><td>N × (Π + Σ)</td></tr><tr><td>MPFT (Cluster, rate=0.3)</td><td>525.9s</td><td>624.1s</td><td>1344.4s</td><td>2.2s</td><td>30.9MB</td><td>55.8MB</td><td>84.8MB</td><td>5.3MB</td><td>N × (Π + Σ)</td></tr><tr><td>MPFT (Random, rate=0.1)</td><td>33.1s</td><td>99.9s</td><td>451.9s</td><td>0.4s</td><td>12.5MB</td><td>20.9MB</td><td>30.7MB</td><td>3.1MB</td><td>N × (Π + Σ)</td></tr><tr><td>MPFT (Random, rate=0.3)</td><td>454.5s</td><td>478.9s</td><td>1341.5s</td><td>2.4s</td><td>30.9MB</td><td>55.8MB</td><td>84.8MB</td><td>5.3MB</td><td>N × (Π + Σ)</td></tr></table>
+
+# 5.6 PRIVACY PRESERVATION ANALYSIS
+
+Following DBE (Zhang et al., 2024), we add Gaussian noise  $\mathcal{N}$  to client prototypes  $p_1, \ldots, p_N$  with a perturbation coefficient  $q$  for the noise and a scale parameter  $s$  for the noise distribution, the perturbed prototype  $\tilde{p}_i$  of client  $i$  can be defined as  $\tilde{p}_i = p_i + q \cdot \mathcal{N}(0, s^2)$ , where  $p_i$  is the original prototype of client  $i$ . The relationship between noise and privacy budget can be found in Appendix H.
+
+Table 4 shows the results of applying this differential privacy method on the DomainNet subset-50, with the sampling method set to average under various noise parameter combinations. This approach effectively mitigates attackers from inferring individual data points even when they possess the pretrained model and most of prototypes. Furthermore, we observe that specific noise configurations can reduce bias across heterogeneous datasets, enhancing the robustness of prototype data. In some cases, this even leads to improved performance compared to models without noise. For instance, the combinations of  $(q = 0.5, s = 0.1)$ ,  $(q = 0.1, s = 0.05)$ , and  $(q = 0.5, s = 0.05)$  exhibit such effects. According to DBE, setting  $q = 0.2$  and  $s = 0.05$  is sufficient to ensure privacy protection. However, excessively large noise can degrade model performance.
+
+Table 4: Performance of differential privacy with varying noise parameters configuration  
+
+<table><tr><td colspan="2"></td><td colspan="4">q = 0.1</td><td colspan="4">s = 0.05</td></tr><tr><td colspan="2">original</td><td>s = 0.1</td><td>s = 0.5</td><td>s = 1</td><td>s = 5</td><td>q = 0.1</td><td>q = 0.2</td><td>q = 0.5</td><td>q = 0.8</td></tr><tr><td>ood acc</td><td>0.8077</td><td>0.8064</td><td>0.8083</td><td>0.8065</td><td>0.7898</td><td>0.8078</td><td>0.8064</td><td>0.8083</td><td>0.8055</td></tr><tr><td>ind acc</td><td>0.7813</td><td>0.7806</td><td>0.7806</td><td>0.7747</td><td>0.7437</td><td>0.7824</td><td>0.7806</td><td>0.7820</td><td>0.7782</td></tr></table>
+
+To further evaluate the robustness of MPFT against adversarial attacks, we simulate a feature space hijacking attack (Vepakomma et al., 2021) on MPFT, please refer to Appendix G.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+We propose an adaptive and lightweight FDA framework, MPFT, designed to align a global model with heterogeneous domains by fitting prototypes from different domains. Extensive experiments demonstrate the effectiveness, low cost, and robustness of MPFT. This study may inspire further research in FDA that focuses on generalizing across heterogeneous domain prototypes, rather than relying on model parameter averaging for aggregation.
+
+While MPFT achieves strong performance, it has some limitations. First, the quality of the prototypes is highly dependent on the pretrained model's ability to extract meaningful features. Second, although attackers cannot reconstruct specific raw data from the prototypes, they may still be able to perform membership inference attacks (Shokri et al., 2017) or attribution inference attacks (Fredrikson et al., 2015) by exploiting statistical information contained within the prototypes. Addressing the aforementioned limitations could further enhance the viability and effectiveness of this approach in practical FL applications, making this method a viable alternative of averaging-based FL methods.
+
+# ACKNOWLEDGMENTS
+
+This research is supported by the NTU startup grant and the RIE2025 Industry Alignment Fund - Industry Collaboration Projects (IAF-ICP) (Award I2301E0026), administered by A*STAR, as well as supported by Alibaba Group and NTU Singapore through Alibaba-NTU Global e-Sustainability CorpLab (ANGEL). This research / project is supported by A*STAR under its Japan-Singapore Joint Call: JST-A*STAR 2024 (Project ID: R24I6IR139). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of the A*STAR. The authors would also like to thank our lab members Tiantong Wang, Tiantong Wu, Yurong Hao, Mengpu Liu and Wenjie Li for their valuable feedback.
+
+# REFERENCES
+
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+
+# A VISUALIZATION OF PROTOTYPE
+
+In order to allow readers to intuitively understand the relationship between prototypes and original data, we visualize the data embeddings (all prototypes) and averaged prototypes in two-dimensional coordinates using the t-SNE (Van der Maaten & Hinton, 2008) algorithm, as shown in Figure 5. Different colors represent different domains: for example, blue indicates the "Painting" domain, while orange signifies the "Real" domain. Different markers represent various categories of data samples. The darker markers located within each sample cluster represent the prototypes of the "domain-class". It is evident that each prototype effectively reflects the distribution of its specific domain-class information. However, the mean prototype, represented by the large red star, is distant from each individual prototype. This observation underscores why we do not average the prototypes, as it fails to accurately reflect the overall data distribution, particularly in FDA scenarios.
+
+![](images/446cd3cbc2efe6c12af2a786ba6735b7434d49397d506487d00b8537a0b8b318.jpg)  
+Figure 5: t-SNE visualization of multiple datasets with their corresponding prototypes.
+
+The Universal Approximation Theorem (Hornik et al., 1989) suggests that neural networks act as "universal" data distribution fitters, effectively fitting the distribution of given data samples. However, this also leads to parameter space deviations between different client models in heterogeneous FL. To address these issues, we introduce a more reasonable aggregation method rather than averaging-based aggregation. We treat the same category of data in different domains with significant differences as independent data prototypes. We then use the collection of these data prototypes, which form a more general data distribution that covers the entire FL system, to train a global model that better fits the global distribution.
+
+# B CONVERGENCE ANALYSIS
+
+Setup: The global loss function for fine-tuning using prototypes is:
+
+$$
+\mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A ^ {\mathcal {G}}) = \frac {1}{N} \sum_ {i = 1} ^ {N} \mathcal {L} _ {i} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A ^ {\mathcal {G}}),
+$$
+
+where  $\mathbb{D}^{\mathcal{P}}$  is the set of prototypes aggregated from all clients.
+
+Assumptions: The standard assumptions follow those of (Li et al., 2020; Tan et al., 2022b): 1) The loss function  $\mathcal{L}$  is non-convex but smooth, 2) The gradient of the loss function is Lipschitz continuous with constant  $L$ . 3) The divergence between the prototypes of different clients for a given class is bounded. Let  $\Delta_i^{(k)}$  be the divergence between client  $i$ 's prototype for class  $k$  and the average prototype across all clients:
+
+$$
+\Delta_ {i} ^ {(k)} = \| p _ {i} ^ {(k)} - \bar {p} ^ {(k)} \|, \quad \bar {p} ^ {(k)} = \frac {1}{N} \sum_ {i = 1} ^ {N} p _ {i} ^ {(k)},
+$$
+
+and assume that  $\Delta_i^{(k)}\leq \Delta$  for all  $i$  and  $k$ . Since the prototypes are derived from pretrained image embeddings, this assumption likely holds due to the consistency provided by the pretrained encoder in synchronizing the embedding space across domains.
+
+Proof: Given the update rule  $A_{t + 1}^{\mathcal{G}} = A_t^{\mathcal{G}} - \eta \nabla_{A^\mathcal{G}}\mathcal{L}(\mathbb{D}^\mathcal{P};\Theta^{\mathcal{G}},A_t^{\mathcal{G}})$ , for a smooth, non-convex loss function  $\mathcal{L}$ , the Lipschitz continuity implies:
+
+$$
+\mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t + 1} ^ {\mathcal {G}}) \leq \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) - \eta \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} + \frac {L \eta^ {2}}{2} \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} + \frac {\Delta}{N}.
+$$
+
+Rearranging the terms, we obtain:
+
+$$
+\mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t + 1} ^ {\mathcal {G}}) \leq \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) - \left(\eta - \frac {L \eta^ {2}}{2}\right) \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} + \frac {\Delta}{N}.
+$$
+
+Choosing  $\eta$  such that  $0 < \eta < \frac{2}{L}$ , we have:
+
+$$
+\mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t + 1} ^ {\mathcal {G}}) \leq \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) - c \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} + \frac {\Delta}{N},
+$$
+
+where  $c = \eta - \frac{L\eta^2}{2}$  is a positive constant. This ensures a decrease in the loss function at each step, leading to convergence, with an additional term accounting for prototype divergence.
+
+# B.1 COROLLARY 1: ON CONVERGENCE TO STATIONARY POINT AND CONVERGENCE RATE
+
+To analyze the convergence rate, we sum both sides of the inequality from  $t = 1$  to  $T$ :
+
+$$
+\sum_ {t = 1} ^ {T} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t + 1} ^ {\mathcal {G}}) \leq \sum_ {t = 1} ^ {T} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) - c \sum_ {t = 1} ^ {T} \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} + \frac {T \Delta}{N}.
+$$
+
+This simplifies to:
+
+$$
+\mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {1} ^ {\mathcal {G}}) - \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {T} ^ {\mathcal {G}}) \geq c \sum_ {t = 1} ^ {T} \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} - \frac {T \Delta}{N}.
+$$
+
+Rearranging the terms and dividing by  $cT$ , we obtain the convergence rate:
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} \leq \frac {\mathcal {L} (\mathbb {D} ^ {\mathcal {P}} ; \Theta^ {\mathcal {G}} , A _ {1} ^ {\mathcal {G}}) - \mathcal {L} (\mathbb {D} ^ {\mathcal {P}} ; \Theta^ {\mathcal {G}} , A _ {T} ^ {\mathcal {G}}) + \frac {T \Delta}{N}}{c T}.
+$$
+
+As  $T$  increases, the average gradient norm decreases, indicating convergence to a stationary point, adjusted for prototype divergence. Note that the size of the divergence  $\Delta$  affects the rate of convergence. A smaller  $\Delta$  implies that the prototypes across clients are more similar, leading to faster convergence. Conversely, a larger  $\Delta$  suggests greater variability between client data, which may slow down the convergence rate.
+
+# B.2 REMARK: ON ERROR BOUNDS
+
+By incorporating the average prototype divergence, the error bound for client  $i$  is:
+
+$$
+\epsilon_ {i} ^ {\mathrm {i n d}} \leq \epsilon_ {i} ^ {\mathrm {l o c a l}} + \alpha \Delta_ {i} ^ {\mathrm {a v g}},
+$$
+
+where  $\epsilon_i^{\mathrm{local}}$  is the local model error without federation,  $\alpha$  is a constant that measures the sensitivity of the error to the prototype divergence, and  $\Delta_i^{\mathrm{avg}} = \frac{1}{K}\sum_{k=1}^{K}\Delta_i^{(k)}$  is the average prototype divergence for client  $i$ . Thus, the in-domain error bound can be written as:
+
+$$
+\epsilon^ {\text {i n d}} \leq \frac {\sum_ {i = 1} ^ {N} \left(\epsilon_ {i} ^ {\text {l o c a l}} + \alpha \Delta_ {i} ^ {\text {a v g}}\right) n _ {i}}{\sum_ {i = 1} ^ {N} n _ {i}}.
+$$
+
+For out-of-domain accuracy (ood acc), we are interested in how well the global model generalizes across different client domains. The maximum divergence of prototypes captures the worst-case divergence between any two domains.
+
+Given the metric:
+
+$$
+\operatorname {o o d} \operatorname {a c c} = \frac {\sum_ {i = 1} ^ {N} \sum_ {j \neq i} \mathbf {A C C} _ {i} ^ {(j)} n _ {j}}{\sum_ {i = 1} ^ {N} \sum_ {j \neq i} n _ {j}},
+$$
+
+By incorporating the maximum prototype divergence, the error bound for client  $i$  on domain  $j$  is:
+
+$$
+\epsilon_ {i j} ^ {\mathrm {o o d}} \leq \epsilon_ {i} ^ {\mathrm {l o c a l}} + \beta \Delta_ {i j} ^ {\max },
+$$
+
+where  $\epsilon_i^{\mathrm{local}}$  is the local model error without federation,  $\beta$  is a constant that measures the sensitivity of the error to the prototype divergence, and  $\Delta_{ij}^{\mathrm{max}} = \max_k\| p_i^{(k)} - p_j^{(k)}\|$ .
+
+Thus, the overall out-of-domain error bound is:
+
+$$
+\epsilon^ {\mathrm {o o d}} \leq \frac {\sum_ {i = 1} ^ {N} \sum_ {j \neq i} \left(\epsilon_ {i j} ^ {\mathrm {l o c a l}} + \beta \Delta_ {i j} ^ {\max}\right) n _ {j}}{\sum_ {i = 1} ^ {N} \sum_ {j \neq i} n _ {j}}.
+$$
+
+We note that the in- and out-of-domain error bounds are directly influenced by average prototype divergence and maximum prototype divergence (worst-case scenario), respectively.
+
+- Impact of higher sampling rate: A higher sampling rate generally leads to better accuracy because more prototypes are generated per class, providing a richer representation of the feature distribution across domains. This helps the global model better generalize to out-of-domain data by capturing a diverse set of variations.  
+- Sampling methods: Cluster sampling is particularly effective for OOD accuracy because it captures the underlying structure of the data distribution within each class by selecting multiple representative prototypes (e.g., cluster centers). Mean sampling, while computationally efficient, oversimplifies the data distribution by averaging all data points, leading to a loss of critical information needed for robust adaptation. Random sampling performs almost as well as cluster sampling in some scenarios. This may be due to the fact that random sampling, by chance, captures sufficient variations within each class, providing a diverse enough representation to improve OOD accuracy. However, it may not be as reliable as cluster sampling because it lacks systematic selection of prototypes and could miss important subgroups within the class distribution.
+
+# B.3 CONVERGENCE ANALYSIS WITH DIFFERENTIAL PRIVACY NOISE
+
+The convergence analysis changes slightly when differential privacy (DP) noise is added. The key difference is the adjustment in the divergence term:
+
+The divergence between client prototypes for class  $k$  is given by:
+
+$$
+\Delta_ {i} ^ {(k)} = \| p _ {i} ^ {(k)} - \bar {p} ^ {(k)} \|, \quad \bar {p} ^ {(k)} = \frac {1}{N} \sum_ {i = 1} ^ {N} p _ {i} ^ {(k)}.
+$$
+
+Suppose Gaussian noise  $\beta(p_i^{(k)})$ , with an upper bound of  $\beta$ , is added to each prototype  $p_i^{(k)}$ . Since the Gaussian noise has a mean of zero, by the law of large numbers, we can assume that  $\bar{p}^{(k)}$  remains approximately unchanged when  $N$  is large. The new divergence term becomes:
+
+$$
+\tilde {\Delta} _ {i} ^ {(k)} \leq \Delta_ {i} ^ {(k)} + \beta ,
+$$
+
+leading to an updated upper bound:
+
+$$
+\Delta \leq \Delta + \beta .
+$$
+
+The new convergence inequality, adapted from Theorem 1, is:
+
+$$
+\mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t + 1} ^ {\mathcal {G}}) \leq \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) - c \| \nabla_ {A ^ {\mathcal {G}}} \mathcal {L} (\mathbb {D} ^ {\mathcal {P}}; \Theta^ {\mathcal {G}}, A _ {t} ^ {\mathcal {G}}) \| ^ {2} + \frac {\Delta + \beta}{N}.
+$$
+
+This indicates that the constant term of convergence increases linearly with the noise level. However, since the coefficient of  $\beta$  is  $\frac{1}{N}$ , the impact of DP noise becomes negligible as  $N$  grows larger, meaning that the addition of DP noise does not significantly hinder convergence in theory.
+
+# C MORE DETAILS ABOUT EXPERIMENT IMPLEMENTATION
+
+# C.1 BASIC SETUP
+
+Global classification head generation. We generate the global classification head by utilizing the manually designed prompts and class names of the dataset, calculated by the pretrained text encoder in the CLIP model used in our experiments. The global classification head is obtained by averaging the 'prompt+label' embeddings from all different domains. Following common fine-tune settings, we only train the adapter, while freezing the entire image encoder and global classification head.
+
+Training. Our simulations are conducted on a Google Compute Platform (GCP) equipped with 47 Intel Xeon CPUs and 4 NVIDIA L4 GPUs. For global adapter training, we employ cross-entropy loss with an AdamW optimizer, setting the learning rate to 0.001. We set the maximum global rounds to 200 and implement an early stopping strategy to evaluate the convergence rounds. It is important to note that our method achieves convergence in just one global round, rendering the early stopping strategy primarily applicable to other FL methods. For simplicity, we assume that all clients can participate in every communication round in all experiments.
+
+# C.2 DETAILS ABOUT IMPLEMENTATION OF BASELINES
+
+For the baseline models used in the experiments, we identified the best parameters for our dataset within the recommended parameter ranges provided in their original texts. For FedAvg, we followed the settings in the original article and used the dataset sizes of different clients as the basis for the weighted average. In FedProx, we set the regularization coefficient to 5, which is lower than the usual settings of 10, 100, or 1000. This adjustment was made because a higher regularization coefficient made it difficult for the model to converge to the global equilibrium point due to data heterogeneity. In Ditto, we used a local round number of 1 and set the regularization term to 2. For MOON, we set the coefficient  $\mu$  to 0.001 and the temperature coefficient  $\tau$  to 1, both within the recommended ranges of the original text. For FedProto, we used 50 as the regularization coefficient. In DBE, we adopted 0.01 as the momentum coefficient and 1 as the regular Xiang coefficient, both within the recommended ranges of the paper. For all baselines, we use the same set of hyperparameters, as shown in Table 5.
+
+To ensure fairness in the comparison, all baseline methods, including FedAvg and FedProx, were trained using the same pre-trained feature extractor as MPFT. Furthermore, the training setup for all
+
+Table 5: The hyperparameters used in all baselines.  
+
+<table><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>Learning rate</td><td>0.001</td></tr><tr><td>Batch size</td><td>32</td></tr><tr><td>Gradient clip</td><td>1</td></tr><tr><td>Local epoch</td><td>1</td></tr><tr><td>Maximum global rounds</td><td>200</td></tr><tr><td>Warm-up global rounds</td><td>10</td></tr><tr><td>Patience global rounds</td><td>10</td></tr></table>
+
+basielines was aligned with MPFT's training paradigm: the pre-trained feature extractor and the global head were frozen, and only the adapter was trained. For consistency, operations in other baselines (e.g., average aggregation in FedAvg) were applied specifically to the adapter parameters instead of the entire model parameters. This adaptation ensures a fair comparison of the performance of MPFT and the baselines under the same conditions.
+
+# C.3 DETAILS ABOUT EARLY STOPPING STRATEGY
+
+Each client completes one local epoch per global round. We set the total number of global rounds to 200 and implement an early stopping strategy to evaluate the convergence rounds of each algorithm. The criterion for early stopping is based on validation loss; specifically, we select the results from the round that achieves the best validation loss as the final outcome. The patience parameter is set to 10 rounds, meaning that if the validation loss does not decrease below the best recorded loss within a span of 10 consecutive rounds, the training process is terminated. By implementing the early stopping strategy, we can more easily test the convergence round of each method and use this strategy to find the round with the best result.
+
+# C.4 DETAILS ABOUT KNOWLEDGE DISTILLATION IN LOCAL ADAPTATION
+
+We use the most basic form of knowledge distillation strategy in our framework (Hinton et al., 2015), which is response-based knowledge distillation:
+
+$$
+A _ {i} ^ {\mathcal {L}} = \arg \min  \mathcal {L} _ {\mathrm {K D}} \left(\mathbb {D} _ {i} ^ {\mathcal {F}}; \Theta^ {\mathcal {G}}; A _ {i} ^ {\mathcal {L}}; A ^ {\mathcal {G}}\right). \tag {6}
+$$
+
+Here,  $A_{i}^{\mathcal{L}}$  represents the local adapter for client  $i$ ,  $\mathbb{D}_i^{\mathcal{F}}$  is the local dataset,  $\Theta^{\mathcal{G}}$  are the global model parameters, and  $A^{\mathcal{G}}$  represents the global adapter. Then, we have:
+
+$$
+I ^ {\mathcal {G}} = \left\{A ^ {\mathcal {G}} \left(f \left(\phi ; x _ {1}\right), \dots , A ^ {\mathcal {G}} \left(f \left(\phi ; x _ {n}\right)\right) \right\}, \quad \left\{\left(x _ {1}, y _ {1}\right), \dots , \left(x _ {n}, y _ {n}\right) \right\} \in \mathbb {D} _ {i} ^ {\mathcal {F}}. \right. \tag {7}
+$$
+
+Here,  $I^{\mathcal{G}}$  denotes the set of outputs from the global adapter for the local dataset  $\mathbb{D}_i^{\mathcal{F}}$ .
+
+$$
+I _ {i} ^ {\mathcal {L}} = \left\{A _ {i} ^ {\mathcal {L}} \left(f \left(\phi ; x _ {1}\right), \dots , A _ {i} ^ {\mathcal {L}} \left(f \left(\phi ; x _ {n}\right)\right) \right\}, \quad \left\{\left(x _ {1}, y _ {1}\right), \dots , \left(x _ {n}, y _ {n}\right) \right\} \in \mathbb {D} _ {i} ^ {\mathcal {F}}. \right. \tag {8}
+$$
+
+Similarly,  $I_i^{\mathcal{L}}$  denotes the set of outputs from the local adapter  $A_i^{\mathcal{L}}$  for the local dataset  $\mathbb{D}_i^{\mathcal{F}}$ .
+
+$$
+\mathcal {L} _ {\mathrm {K D}} = \mathrm {K L} \left(I ^ {\mathcal {G}} \| I _ {i} ^ {\mathcal {L}}\right) \tag {9}
+$$
+
+The knowledge distillation loss  $\mathcal{L}_{\mathrm{KD}}$  is computed as the Kullback-Leibler (KL) divergence between the outputs of the global adapter and the local adapter.
+
+$$
+\operatorname {K L} (p \| q) = \sum_ {i} p _ {i} \log \left(\frac {p _ {i}}{q _ {i}}\right) \tag {10}
+$$
+
+Here,  $p$  and  $q$  represent the probability distributions output by the global and local adapters, respectively, and  $\mathrm{KL}(p\parallel q)$  denotes the KL divergence.
+
+# D SENSITIVITY ANALYSIS OF GLOBAL CONVERGENCE THRESHOLD
+
+During the global adapter initialization phase, we set a threshold to ensure the model stops training the prototypes when the variance of loss over multiple rounds decreases to a low value, indicating that the global adapter has converged. To test the sensitivity of the experimental results to the threshold, we test different thresholds for the global adapter initialization process, specifically 0.1, 0.01, 0.001, and 0.0001. Figure 6 shows the impact of these thresholds<sup>3</sup>. As the threshold decreases, the out-of-distribution (ood) and in-distribution (ind) performance initially increase and then decrease. In contrast, the convergence epochs and training time consistently increase. This trend is intuitive because a lower threshold requires more rounds for the model to converge. Overall, we recommend using a threshold of 0.01 or 0.001 to minimize training time and reduce the risk of overfitting.
+
+![](images/7dd317c8e703ce3ad2b8822a44fee900f54fb57f1475a1fd0d663cf25f0b7e77.jpg)
+
+![](images/843ad3922fc2a2ccdb44fc3d1006b6251a5b2d0636b80cf5dc0cf0c43438c020.jpg)
+
+![](images/7eca6e691cbe03bde07d330ae6b93f446131c39e556a28522c0fbbb62e42938e.jpg)
+
+![](images/1564981746b6c601d2b40f22c8f1477166a16c3829a03aa9b7717d8b0c0cf669.jpg)
+
+![](images/5f1a8bf9d8ba5b404754f2eb0283d6935110cc5aa022cc227d7175a4a90c1a12.jpg)
+
+![](images/8d26cf7744600e31b9f1f34f0ac5d4bb1f83161e0d7e64fedcc4a0a35255df30.jpg)
+
+![](images/ae4a198a256784ebdfe92251e3a1792f305885c5352d1a65f7f66028cb894bcd.jpg)
+
+![](images/f975971f9e9ec9e344ed61de48fb0d3972e6aaa7ea2e11315b60ba48ce7bcdc3.jpg)
+
+![](images/9bf56138d6d5720d49b5929a0a44d6cd09e15cf8677dad63133be3757410e2d5.jpg)  
+Figure 6: Impact of different convergence thresholds on global adapter initialization.
+
+![](images/429846672cf6f5e5cdc3cda6d914b4fe00df854c576cbffa7dbd5f41e92a0326.jpg)
+
+![](images/4bd92af7df0f6ad6cc2ff9b89b4999b6fd7b71f917bfd5d6b4180449392f121f.jpg)
+
+![](images/edf620c8d5fd134ceb188858bb5bd4b985301e54a0a58ac40fe018f8e63c277d.jpg)
+
+# E MORE DETAILS ABOUT RESULTS ON EACH DOMAIN
+
+Figure 7 and Figure 8 further illustrate the effects of random sampling and cluster sampling compared to average sampling within the MPFT framework across different sizes of the DomainNet subset. They reveal similar trends: random sampling and cluster sampling achieve more balanced performance in the Quickdraw domain, with curves approaching circular shapes and covering larger areas. This suggests that these improved sampling methods enhance the model's ability to handle balance and diversity across various data domains.
+
+We also observe that as the sampling rate increases in the random or cluster sampling methods, the model's performance in the Quickdraw domain improves, leading to more globally optimized results. However, the increase in sampling results in more training convergence time consumption and higher data transmission between the server and clients, which raises both computation and communication costs, as shown in Table 3. This trade-off needs to be considered in practical applications.
+
+![](images/6674a91ead5dcdfbd3dad29195b3241975f9d81fd749be5ff15c4713e0d2d1cc.jpg)  
+Figure 7: Comparison of cluster and average sampling in MPFT framework.
+
+![](images/8d45101cf96aa4553685141d17e4abd49fb03880bbae2d948717b8f1afe08830.jpg)  
+Figure 8: Comparison of random and average sampling in MPFT framework.
+
+# F RESULTS OF MULTIPLE CLIENTS WITH SAME DOMAIN
+
+In this section, we present the results of experiments where multiple clients belong to the same data domain, a scenario commonly encountered in FL. Specifically, we partition the data from the same domain into multiple subsets, each maintaining the same class labels, and distribute these subsets across multiple clients. This setup results in a far greater number of clients than in the experiments discussed in Section 5.1. As the number of clients increases, the data distribution among clients becomes more similar, potentially reducing domain heterogeneity.
+
+In particular, we compare the results for scenarios with 6, 18, 24, and 30 clients on the DomainNet Subset-50 dataset. Table 6 shows the ind acc, ood acc and the number of communication rounds required for different FL methods.
+
+As the number of clients increases, we observe that methods such as FedAvg, FedProx, and MOON experience a decline in ood accuracy. This could be due to the reduced heterogeneity among clients, leading to less domain-specific knowledge being aggregated into the global model. Conversely, Ditto, FedProto and DBE show a reduction in ind accuracy, because these more personalized methods are less effective at preserving in-domain knowledge when client heterogeneity decreases.
+
+In contrast, MPFT manages to maintain both ind acc and ood acc, similar to the original experimental setup, even as the number of clients increases. Notably, in certain cases, MPFT even improves accuracy. For example, with cluster sampling at a rate of 0.1, MPFT achieves a significant performance boost with 18 clients compared to the original experiment. This suggests that MPFT's adaptive aggregation mechanism is robust to changes in client numbers and data distribution, making it more scalable and effective in scenarios with multiple clients from the same domain.
+
+Table 6: Results of multi clients with same domain on DomainNet Subset-50 dataset.  
+
+<table><tr><td rowspan="2"></td><td colspan="3">6 clients (original)</td><td colspan="3">18 clients</td><td colspan="3">24 clients</td><td colspan="3">30 clients</td></tr><tr><td>ood acc</td><td>ind acc</td><td>rounds</td><td>ood acc</td><td>ind acc</td><td>rounds</td><td>ood acc</td><td>ind acc</td><td>rounds</td><td>ood acc</td><td>ind acc</td><td>rounds</td></tr><tr><td>local</td><td>0.7361</td><td>0.8609</td><td>0</td><td>0.7330</td><td>0.8557</td><td>0</td><td>0.7437</td><td>0.8479</td><td>0</td><td>0.7434</td><td>0.8451</td><td>0</td></tr><tr><td>FedAvg</td><td>0.7902</td><td>0.7345</td><td>24</td><td>0.7708</td><td>0.7197</td><td>25</td><td>0.7693</td><td>0.7186</td><td>24</td><td>0.7685</td><td>0.7192</td><td>25</td></tr><tr><td>FedProx</td><td>0.7752</td><td>0.7178</td><td>10</td><td>0.7685</td><td>0.7165</td><td>0</td><td>0.7675</td><td>0.7163</td><td>0</td><td>0.7667</td><td>0.7172</td><td>1</td></tr><tr><td>Ditto</td><td>0.7811</td><td>0.7624</td><td>20</td><td>0.7643</td><td>0.7386</td><td>15</td><td>0.7639</td><td>0.7397</td><td>37</td><td>0.7619</td><td>0.7357</td><td>14</td></tr><tr><td>MOON</td><td>0.7902</td><td>0.7344</td><td>28</td><td>0.7709</td><td>0.7197</td><td>27</td><td>0.7691</td><td>0.7186</td><td>20</td><td>0.7682</td><td>0.7188</td><td>6</td></tr><tr><td>FedProto</td><td>0.7296</td><td>0.7696</td><td>5</td><td>0.7299</td><td>0.7530</td><td>9</td><td>0.7305</td><td>0.7476</td><td>9</td><td>0.7308</td><td>0.7562</td><td>11</td></tr><tr><td>DBE</td><td>0.7421</td><td>0.7622</td><td>22</td><td>0.7504</td><td>0.7602</td><td>8</td><td>0.7621</td><td>0.7511</td><td>4</td><td>0.7449</td><td>0.7580</td><td>24</td></tr><tr><td>MPFT (Average)</td><td>0.8077</td><td>0.7813</td><td>1</td><td>0.8062</td><td>0.7833</td><td>1</td><td>0.8032</td><td>0.7820</td><td>1</td><td>0.8032</td><td>0.7839</td><td>1</td></tr><tr><td>MPFT (Cluster, rate=0.1)</td><td>0.7951</td><td>0.7957</td><td>1</td><td>0.8091</td><td>0.8128</td><td>1</td><td>0.8002</td><td>0.8011</td><td>1</td><td>0.8217</td><td>0.8232</td><td>1</td></tr><tr><td>MPFT (Cluster, rate=0.3)</td><td>0.8204</td><td>0.8294</td><td>1</td><td>0.8139</td><td>0.8188</td><td>1</td><td>0.8103</td><td>0.8157</td><td>1</td><td>0.8126</td><td>0.8183</td><td>1</td></tr><tr><td>MPFT (Random, rate=0.1)</td><td>0.7953</td><td>0.7899</td><td>1</td><td>0.7911</td><td>0.7880</td><td>1</td><td>0.7909</td><td>0.7838</td><td>1</td><td>0.7975</td><td>0.7971</td><td>1</td></tr><tr><td>MPFT (Random, rate=0.3)</td><td>0.8236</td><td>0.8294</td><td>1</td><td>0.8218</td><td>0.8267</td><td>1</td><td>0.8119</td><td>0.8138</td><td>1</td><td>0.8219</td><td>0.8257</td><td>1</td></tr></table>
+
+# G FEATURE SPACE HIJACKING ATTACK
+
+![](images/e0a4d4046185aaf04cf50f9af95491c9a409d519161cb6727df4ba4c9682bc72.jpg)  
+Figure 9: Results of the feature space hijacking attack on our model. Each prototype was constructed from a single image. Even after one million iterations of training, the original image could not be recovered, demonstrating the security of using prototypes.
+
+In the MPFT framework, the communication of local clients' prototypes and a trained global adapter may present potential security vulnerabilities. We hypothesize an attack vector utilizing the architecture of our approach, to design a feature space hijacking attack. The attacker could leverage the pretrained model's image encoder  $f(\phi)$  and the uploaded prototype  $p^{(k)}$  to attempt restoration of the original training data  $x$ :
+
+1. An estimated input  $x^{*}$  is constructed and processed through the pretrained image encoder  $f(\phi)$  to obtain an estimated prototype  $p^{*}$ .
+
+2. The mean squared error (MSE) loss is used to iteratively refine  $x^{*}$ , aiming to minimize the discrepancy between  $p^{*}$  and the actual prototype  $p$ , thereby approximating  $p_{i}^{(k)}$ .
+
+To demonstrate the resistance of our method to feature space hijacking attacks, we randomly selected a picture from each client to form a prototype and attempted to simulate an attacker trying to restore the picture. One picture is chosen as the prototype because if a prototype attack composed of a single picture cannot be restored, a prototype composed of multiple average representations of the same category will be even more challenging for an attacker to restore and exploit.
+
+Figure 9 illustrates the process of restoring a single image by multiple clients. Intuitively, we observe that even after one million gradient descent iterations, the attacker still cannot restore the salient features of the original image. It is important to note that on our device, it takes nearly 8 hours to complete such iterative training for each image, imposing a significant time cost on attackers attempting large-scale attacks.
+
+# H RELATIONSHIP BETWEEN NOISE AND PRIVACY BUDGET
+
+We perform differential privacy (DP) analysis in the average prototype sampling stage:
+
+Proposition 1 (Post-Processing) Let  $f: \mathbb{N}^{|\mathcal{X}|} \to R$  be a randomized algorithm that is  $(\varepsilon, \delta)$ -differentially private. Let  $g: R \to R'$  be an arbitrary randomized mapping. Then  $g \circ f: \mathbb{N}^{|\mathcal{X}|} \to R'$  is  $(\varepsilon, \delta)$ -differentially private.
+
+In MPFT, we can regard sampling as  $f$  and learning from the prototype as  $g$ , which output is the learned model. Then the whole learning process is  $(\varepsilon, \delta)$ -differentially private provided that the sampling function  $f$  is  $(\varepsilon, \delta)$ -differentially private.
+
+Definition 1 (Gaussian Mechanism) Let  $f: \mathbb{N}^{|\mathcal{X}|} \to \mathbb{R}^d$  be an arbitrary  $d$ -dimensional function, and define its  $\ell_2$  sensitivity to be:
+
+$$
+\Delta_{2}f = \max_{adjacent  x,y}\| f(x) - f(y)\|_{2}.
+$$
+
+The Gaussian Mechanism with parameter  $\sigma$  adds noise scaled to  $\mathcal{N}(0, \sigma^2)$  to each of the  $d$  components of the output.
+
+Theorem 2 (Relationship between Gaussian Mechanism and privacy budget) Let  $\varepsilon \in (0,1)$  be arbitrary. For  $c^2 > 2\ln(1.25/\delta)$ , the Gaussian Mechanism with parameter
+
+$$
+\sigma \geq c \frac {\Delta_ {2} f}{\varepsilon}
+$$
+
+is  $(\varepsilon, \delta)$ -differentially private.
+
+For average sampling, the sampling procedure for class  $i$  involves the data embeddings  $D_{i} = \{d_{k}^{(i)}\}_{k=1}^{n_{i}}$ , where  $n_{i}$  is the number of data points in class  $i$ . For cluster sampling, simply replace  $D_{i}$  by data embeddings inside the cluster, with other analysis unchanged.
+
+The average sampling function  $f$  is
+
+$$
+f (D _ {i}) = \frac {\sum_ {k = 1} ^ {n _ {i}} d _ {k} ^ {(i)}}{n _ {i}}.
+$$
+
+We define  $f_{i}$  as the restriction of  $f$  taking only data points in class  $i$  as input. Two adjacent datasets differ by exactly one data point. For two adjacent datasets  $D_{p}, D_{q}$ , suppose they differ at datapoints  $d_{p}^{(i)} \in D_{p}, d_{q}^{(i)} \in D_{q}$ , then:
+
+$$
+\| f (D _ {p}) - f (D _ {q}) \| _ {2} = \left\| \frac {\sum_ {d _ {k} ^ {(i)} \in D _ {p}} d _ {k} ^ {(i)}}{n _ {i}} - \frac {\sum_ {d _ {k} ^ {(i)} \in D _ {q}} d _ {k} ^ {(i)}}{n _ {i}} \right\| _ {2} = \frac {\left\| d _ {p} ^ {(i)} - d _ {q} ^ {(i)} \right\| _ {2}}{n _ {i}},
+$$
+
+$$
+\Delta_ {2} f _ {i} = \max _ {p, q \in \{1, \ldots , n _ {i} \}} \frac {\left\| d _ {p} ^ {(i)} - d _ {q} ^ {(i)} \right\| _ {2}}{n _ {i}}.
+$$
+
+For  $(\delta_i,\varepsilon_i)$ -differentially private guarantee:
+
+$$
+\sigma \geq \frac {\sqrt {2 \ln (1 . 2 5 / \delta_ {i})} \cdot \Delta_ {2} f _ {i}}{\varepsilon_ {i}}.
+$$
+
+A strong differentially private guarantee is achieved when  $\varepsilon_{i} < 1$ ,  $\delta_{i} \leq \frac{1}{n_{i}}$  according to the theorem in (Dwork & Roth, 2014). In federated learning settings such as (Wei et al., 2020), a medium differentially private guarantee is achieved when  $\varepsilon_{i}$  is around 10.
+
+We calculate our differentially private guarantee as
+
+$$
+\varepsilon_ {i} = \frac {\sqrt {2 \ln (1 . 2 5 n _ {i})} \cdot \Delta_ {2} f _ {i}}{\sigma}.
+$$
+
+For our differential privacy experiment in Section 5.6, the  $\sigma$  value used is the multiplication  $qs$ . Table 7 shows empirical results of average privacy budget  $\bar{\varepsilon}$  with different  $\sigma$  values, where we take  $\delta_{i} = \frac{1}{n_{i}}$ :
+
+Table 7: Average privacy budget  $\bar{\varepsilon }$  for different  $\sigma$  values.  
+
+<table><tr><td></td><td>σ = 0.01</td><td>σ = 0.05</td><td>σ = 0.1</td><td>σ = 0.5</td></tr><tr><td>DomainNet, Subset=50</td><td>54.76</td><td>10.95</td><td>5.48</td><td>1.10</td></tr><tr><td>DomainNet, Subset=100</td><td>65.48</td><td>13.10</td><td>6.55</td><td>1.31</td></tr><tr><td>DomainNet, Subset=150</td><td>60.73</td><td>12.15</td><td>6.07</td><td>1.21</td></tr></table>
+
+The table demonstrates that, as the noise scale  $\sigma$  increases, the average privacy budget  $\bar{\varepsilon}$  decreases, indicating stronger privacy protection. When  $\sigma$  is above 0.05, the privacy budget is sufficient to provide robust privacy guarantees. Furthermore, as shown in Table 4, the performance of MPFT experiences minimal decline when  $\sigma$  is in the range of 0.05-0.1. These results, both theoretical and empirical, demonstrate that MPFT can effectively utilize differential privacy mechanisms to protect prototypes without significant performance loss.

2025/Enhancing Federated Domain Adaptation with Multi-Domain Prototype-Based Federated Fine-Tuning/images.zip

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2025/Enhancing Graph Of Thought_ Enhancing Prompts with LLM Rationales and Dynamic Temperature Control/72a53f44-c74c-472f-a096-4192621c6d9e_content_list.json

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+    {
+        "type": "text",
+        "text": "ENHANCING GRAPH OF THOUGHT: ENHANCING PROMPTS WITH LLM RATIONALES AND DYNAMIC TEMPERATURE CONTROL",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Sunguk Shin and Youngjoon Kim*",
+        "bbox": [
+            179,
+            194,
+            426,
+            210
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Korea University",
+        "bbox": [
+            179,
+            210,
+            326,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Seoul, Republic of Korea",
+        "bbox": [
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+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$\\{ssw1419,acorn421\\} @korea.ac.kr$",
+        "bbox": [
+            179,
+            237,
+            485,
+            252
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            287,
+            545,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce Enhancing Graph of Thoughts (EGoT), a method designed to enhance the performance of large language models (LLMs) on complex reasoning tasks. EGoT automates the process of generating accurate responses using given data and a base prompt. The process consists of several steps: It obtains an initial response from the answering node using the base prompt. Evaluation node evaluates the response and generates reasoning for it, utilizing the score's probabilities to enhance evaluation accuracy. The reasoning from both the answering node and the evaluation node is aggregated to identify the problem in the response. This aggregated reasoning is incorporated into the base prompt to obtain an enhanced response. These steps are organized in a graph architecture, where the final leaf nodes are merged to produce a final response. As the graph descends, the temperature is lowered using Cosine Annealing and scoring, to explore diverse responses with earlier nodes and to focus on precise responses with later nodes. The minimum temperature in Cosine Annealing is adjusted based on scoring, ensuring that nodes with low scores continue to explore diverse responses, while those with high scores confirm accurate responses. In sorting 256 elements using GPT-4o mini, EGoT performs  $88.31\\%$  accuracy, while GoT (Graph of Thoughts) achieves  $84.37\\%$  accuracy. In the frozen lake problem using GPT-4o, EGoT averages 0.55 jumps or falls into the hole, while ToT (Tree of Thoughts) averages 0.89.",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            607,
+            336,
+            622
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent research, the performance of large language models (LLMs) has evolved incredibly rapidly, with applications in a variety of fields, including math problem (Shao et al., 2024), robotics (Park et al., 2023), medicine (Lee et al., 2024b; Kwon et al., 2024), and even programming (Wang et al., 2023a; Duong & Meng, 2024; McAleese et al., 2024). To further improve the performance of LLMs, researchers are now actively exploring methods to significantly scale up the architecture of models, or optimize models through distillation (Qu et al., 2024) and fine-tuning (Singh et al., 2024). These efforts are broadening the scope of LLMs and enabling more innovative applications.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Training LLMs directly requires significant time and GPU resources. To address such limitations, Prompt engineering, which involves designing effective prompts rather than training the model directly, stands out. Prompt engineering is a technique that can improve the performance of LLMs on specific tasks without requiring additional training. Examples of prompt engineering include Chain of Thought (CoT) (Wei et al., 2022), Chain of Thought with Self-Consistency (CoT-SC) (Wang et al., 2023b), Tree of Thoughts (ToT) (Long, 2023; Yao et al., 2024), Exchange of Thought (EoT) (Yin et al., 2023), and Graph of Thoughts (GoT) (Besta et al., 2024). These approaches help LLMs generate more accurate and useful results.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, complex problems often impair the reasoning ability of LLMs. When an LLM provides a correct answer, its rationale steps are not always reliable (Hao et al., 2024). In addition, most architectures utilize external tools (Stechly et al., 2023; Gou et al., 2024) to improve performance,",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Corresponding author",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
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+    {
+        "type": "text",
+        "text": "and prompts often require specific examples (Lee et al., 2024a). Since obtaining the valid rationale makes LLM's performance highly contributing (Yin et al., 2024), the technique of prompting LLMs with a score to evaluate the performance (Valmeekam et al., 2023; Ren et al., 2023) is an ongoing research area. There is also research exploring dynamic temperature control techniques (Cai et al., 2024; Nasir et al., 2024; Zhang et al., 2024; Zhu et al., 2024) to further enhance the reasoning ability of LLMs.",
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+    {
+        "type": "text",
+        "text": "Our approach, EGoT, is an architecture that can automatically generate the prompts and answers from the LLM by only initializing the base prompt. During this process, log probability is utilized to evaluate the LLM's responses, increasing their confidence. We also propose dynamically adjusting the temperature based on the progress and score of the answer, applying the cosine annealing (Loshchilov & Hutter, 2016) to set a high temperature at the beginning of the graph and a low temperature at the end. The minimum temperature is set as the inverse of the score, so that nodes with high scores consistently provide correct answers, while nodes with low scores explore a wide range of answers. This approach has the advantage of showing constant and consistent performance without the evaluation metric, and it does not need additional examples to avoid bias in the results. Note that our study proposes a framework that strategically resolves conflicts from naively merging prior methods through trade-offs, thereby ensuring high performance.",
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+    {
+        "type": "text",
+        "text": "To summarize, EGoT provides the following advantages:",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Dynamic temperature control using Cosine Annealing to propagate more accurate rationales to child node prompts.",
+            "- Continuously appending of rationales to the base prompt in graph architecture to generate a high-quality final response.",
+            "- Enhanced confidence by utilizing the probability of LLM answers for scoring, while avoiding bias by excluding specific examples.",
+            "- Direct repetition of the input question in its original form to improve LLM comprehension, followed by integration of prior repetitions into the rationale."
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "2 EGOT ARCHITECTURE",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "2.1 OVERVIEW",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/3f120404029e6202881c60140c6cd4ac051a7bf193a06b360fe616dd0f6a5a04.jpg",
+        "image_caption": [
+            "Figure 1: Framework of EGoT. The left side illustrates the overall graph architecture and dynamic temperature. The right side illustrates the internals of each Node. Each Node contains ANSWERINGNODE, EVALUATIONNODE, and AGGREGATERATIONALENODE as sub-nodes. The temperature parameter updates its child nodes within the tree, propagating the rationale information to deeper levels. As the graph progresses, the temperature decreases and propagates the rationale information."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The EGoT graph structure is shown in Figure 1. Each node consists of three stages: Stage 1 (ANSWERINGNODE) obtains the answer to resolve the problem from the LLM; Stage 2",
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(EVALUATIONNODE) asks the LLM to evaluate its response and assign a score; and Stage 3 (AGGREGATERATIONALENODE) collects the LLM's rationales from both Stage 1 and Stage 2, forwarding them to the next nodes. METHODNODE is executed only once at the beginning of the overall structure, and this step can be replaced with an expert's problem-solving approach.",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 METHOD NODE",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The METHODNode inquires about the method for solving the problem and the methods for evaluating the answer. Although these methodologies can be formulated by human experts, in this paper, heuristic methods are requested from the LLM and utilized.  $m_{a}$  denotes the method for obtaining the answer to the question, and  $m_{e}$  denotes the method for evaluating the answer.  $t$  denotes the temperature of the LLM.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nm _ {a}, m _ {e} = \\text {M E T H O D N O D E} (\\text {P r o m p t}, t = 0) \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 ANSWERING NODE",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "ANSWERINGNODE finds the answer to the problem. The top root node solves the problem with the rules. The child node solves the problem using the rationale from the previous nodes. ANSWERINGNODE outputs the answer to the problem and the rationale for the answer.  $a$  and  $r_a$  are the answer and the rationale regarding the response provided by the LLM, and  $r_{pr}$  denotes the rationales of the previous nodes. In this study, the temperature  $t$  is fixed at 1 for the root node, while for all other nodes it is determined by the parent's temperature using cosine annealing. We denote the updated temperature as  $t_u$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\na, r _ {a} = \\left\\{ \\begin{array}{l l} \\text {A N S W E R I N G N O D E} (P r o m p t (m _ {a}, \\cdot), t = 1), & \\text {i f} \\quad \\text {N o d e} = \\text {R o o t N o d e} \\\\ \\text {A N S W E R I N G N O D E} (P r o m p t (m _ {a}, r _ {p r}), t = t _ {u}), & \\text {e l s e} \\quad \\text {N o d e} \\neq \\text {R o o t N o d e} \\end{array} \\right. \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.4 EVALUATION NODE",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "EVALUATIONNODE evaluates the answer provided by ANSWERINGNODE. The LLM outputs both the accuracy of the answer and the rationale for that accuracy score. If the probability of the score provided by the LLM is lower than the threshold, EVALUATIONNODE is executed again.  $s$  and  $r_s$  are the score and the rationale regarding the LLM's response, and  $\\operatorname{Pr}(s)$  is the probability of the score. We request a score range of 0 to 100 from the LLM to better represent the scores as percentages.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns, r _ {s}, \\Pr (s) = \\text {E V A L U A T I O N N O D E} (\\text {P r o m p t} (m _ {e}, a), t = 0) \\tag {3}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.5 AGGREGATE RATIONALE NODE",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "AGGREGATERATIONALENODE integrates the rationales provided from ANSWERINGNODE and EVALUATIONNODE. The LLM outputs the aggregated rationale along with the information considered inaccurate. AGGREGATERATIONALENODE aggregates the information from the answer rationale and the evaluation rationale, emphasizing the incorrect encountered during the reasoning while omitting details related to successful outputs. This concept is similar to the state evaluator in ToT (Yao et al., 2024); however, our approach provides a rationale for identifying flaws without the answer. The inaccurate information refers to elements that the LLM needs to recheck when conflicts occur between the two input rationales. It arises from the LLM's misinterpretation of the problem and can lead to hallucinations and incorrect reasoning. This information, derived from AGGREGATERATIONALENODE, is subsequently incorporated into the prompt of the child's ANSWERINGNODE.  $r_{pr}$  denotes the aggregated rationale and the inaccurate information.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nr _ {p r} = \\text {A G G R E G A T E R A T I O N A L E N O D e} \\left(\\text {P r o m p t} \\left(r _ {a}, r _ {s}\\right), t = 0\\right) \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+            47
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHODOLOGY",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 ENHANCING RESPONSE",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This section describes the methods to obtain enhancing responses from the LLM. Two main approaches are used: exploring varied answers for obtaining enriched responses and utilizing the probability of the answers for more accurate scoring.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.1 EXPLORING VARIORED ANSWER",
+        "text_level": 1,
+        "bbox": [
+            171,
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+    },
+    {
+        "type": "text",
+        "text": "To explore different answers, multiple root nodes are utilized in the architecture. Since the temperature decreases along the node depth, multiple graphs are used to generate different answers. In some cases, a node provides the same answer as its parent. To address this, if a child ANSWERINGNODE gives the same answer as its parent ANSWERINGNODE, a question is only asked once more. This is because it cannot be determined exactly whether it is the correct answer during the entire process of the graph.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.2 ENHANCING SCORE",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To enhance the scoring process, the probability that the LLM predicts the score token is used to answer the score. If the probability does not exceed a predefined threshold, the LLM is prompted for the score again. If the LLM outputs extreme score values, such as 0 or 100, a higher threshold is applied because these extreme scores are considered reliable only when the LLM is highly confident. For scores ranging from 1 to 99, the threshold is set lower to filter out nonsensical answers. It is important to consider the order in which the LLM is asked for the score and the rationale for the score. If the LLM is asked for the rationale first and then the score, the LLM thinks that it has a basis in the previous rationale. Therefore, a score of 0 or 100 is often returned regardless of whether the answer is correct or not, with a probability close to 1. For this reason, the score is asked for before the rationale, and the score is obtained with a variety of scores. Detailed explanations of the threshold settings are provided in each experiment.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 TEMPERATURE CONTROL",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The temperature in LLMs is typically set to 1.0 when generating creative answers. Whereas when creativity is not required, the temperature is set closer to 0 for consistent answers. However, setting the temperature to 0 from the start can lead to fixed answers and errors.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To gradually decrease temperature as the graph progresses, we employ cosine annealing. When a high-quality answer is generated, the temperature is reduced to produce a fixed response, whereas when the answer is uncertain, the temperature is kept high to explore different answers. The purpose for evaluating answers in EVALUATIONNODE is not only to generate a rationale but also to control the temperature. If the score is high, it indicates that the rationale of that ANSWERINGNODE is correct. Therefore, this rationale is forwarded to the child nodes, which are expected to generate good answers. On the other hand, if the score is low, the answer needs to be revised, and the rationale of ANSWERINGNODE also needs to improve, requiring various explorations until it is correct.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In cosine annealing, the maximum temperature  $(t_{max})$  is fixed at 0.7 and the minimum temperature is set to the inverse of the accuracy. This means that higher accuracy results in a lower temperature. The total epoch is set to the total number of nodes  $(N_{t})$  and the current epoch is defined as the progress of the nodes  $(N_{c})$ .",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nt _ {u} = t _ {m i n} + \\frac {1}{2} (t _ {m a x} - t _ {m i n}) (1 + c o s (\\frac {N _ {c}}{N _ {t}})), t _ {m i n} = 1 - \\sqrt {1 - (c - 1) ^ {2}}, c = s \\cdot \\operatorname * {P r} (s) ^ {\\frac {1}{e}} \\quad (5)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Here,  $c$  represents the confidence of ANSWERINGNODE. If an answer receives a high score and the probability assigned to that score by the LLM is also high, the confidence is high. Conversely, if an answer receives a low score or the probability assigned to the score is low, the confidence is low.  $c$  and  $t_{min}$  are between 0 and 1. The probability is used in  $t_{min}$  to differentiate between high and low probability cases when the LLM answers the score.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 EXAMPLE USE CASE",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/21a0b8648212aaa1c399d9fdc973b63dd711b9edb53733cf1536ed5c977bd41b.jpg",
+        "image_caption": [
+            "Figure 2: In the Frozen Lake example, the temperature decreases as it progresses down the graph, various positions are explored and the graph finds the correct answer using the information."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This section uses a practical example to illustrate the approaches presented in Sections 2 and 3. Figure 2 shows the results of the Frozen Lake experiment, one of the experimental results that demonstrate the advantages of EGoT. The blue background represents the hole and the light blue represents the frozen tile. The two black points on the top left (0, 0) and bottom right (4, 4) represent the start and end points. The green line indicates the route that the LLM predicts as the answer, the orange square marks what EVALUATIONNODE rationale explains as incorrect because it is a hole. The brown triangle represents the position that AGGREGATERATIONALENODE aggregates because the rationale from ANSWERINGNODE and EVALUATIONNODE conflict with each other.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Before the graph starts, METHODNode is invoked once. The information provided by the METHODNode is utilized by all subsequent nodes in Figure 2, from  $Node_{1,1}$  to the Final Node. In this experiment, the graph starts with 3 root nodes. In ANSWERINGNODE,  $Node_{1,1}$  passes through the holes (2, 1), (3, 1), and  $Node_{2,1}$  and  $Node_{3,1}$  pass through the holes (2, 4), (3, 4). At EVALUATIONNODE,  $Node_{1,1}$  observes the hole at (2, 1) and  $Node_{2,1}$  observes the hole at (2, 4). As a result, ANSWERINGNODE states that the answer is incorrect, lowering the confidence of  $Node_{1,1}$",
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+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and  $\\text{Node}_{2,1}$ . Conversely,  $\\text{Node}_{3,1}$  has high confidence in EVALUATIONNODE, because it does not find anything wrong. Since it is the first round, the temperature remains close to 0.7, regardless of confidence. The node updates the temperature of its two child nodes.  $\\text{Node}_{1,2}$  and  $\\text{Node}_{1,3}$  update the temperature by  $\\text{Node}_{1,1}$ .",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Because depth 0 informs that the coordinates (2, 1) and (2, 4) are holes, depth 1 nodes recognize them as holes and do not traverse these coordinates. Still,  $\\text{Node}_{1,3}$ ,  $\\text{Node}_{2,2}$ , and  $\\text{Node}_{3,3}$  are unsure of the correct answer because the propagated rationale confuses the information about frozen tile and hole. The nodes in depth 1 also cannot make a confident decision and incorrectly state that (3, 3) is a hole. Since one depth has passed, nodes with higher confidence have a lower temperature to update to their child. In the middle of the process (the omitted part of the figure), if a node gives an incorrect answer, the temperature increases again, and it explores the coordinate (3, 2). When the final node responds to the answer by incorporating aggregate rationales from the leaf nodes, the LLM explores the correct answer, avoiding (2, 1) and (3, 2).",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use the LangChain (Chase, 2022) library to construct the graph. The graph structure starts with three root nodes, and when solving a problem, the LLM responds with prompts that include all rationale information from the previous depth. At the end of the graph, the answer is aggregated into one by using the response from ANSWERINGNODE with the prompt that incorporates all the aggregated rationales from the leaf nodes.",
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+    },
+    {
+        "type": "text",
+        "text": "EGoT is evaluated through three experiments: document merging, number sorting, and Frozen Lake. In the document merging and number sorting experiments, we use the graphs with a depth of 3, and we use a graph with a depth of 4 in the Frozen Lake experiment. We experiment with ToT (Long, 2023) which appends the incorrect answer rather than evaluating and exploring each element. This is due to, in the experiments, the number of nodes increases exponentially to explore each case. In the original paper, GoT selects the best-performing node to evaluate the graph; however, it has been modified to select a median value to compare structural performance alone. Solving problems with evaluation metrics is not considered a structural advantage. Therefore, to fully automate the LLM process, the evaluation of nodes is assumed to be randomized and the median value is used as the expectation. Experiments are conducted multiple times with the same data. To compare the impact of temperature, the experiment is conducted with a temperature fixed to 1, referred to as EGoT*.",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 DOCUMENT MERGING",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We conducted an experiment with the dataset and the evaluation prompt provided by GoT for document merging. The evaluation compares non-redundancy and retained harmonic mean. The performance scores for each method are as follows: IO (75.96%), CoT (77.79%), ToT (76.74%), GoT (76.43%), EGoT (76.01%), and EGoT* (74.98%). This experiment suggests that scoring with an LLM should not simply be evaluated. The experiment shows that autonomous evaluation by an LLM does not have the logical and structural advantages of well-known CoT and ToT. Furthermore, it supports the idea that scoring should be evaluated more rigorously.",
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+    {
+        "type": "text",
+        "text": "4.2 NUMBER SORTING",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This experiment involves a sorting problem with random numbers as input. The LLM is able to sort short lists successfully. However, its performance decreases when sorting longer lists of numbers. To evaluate the sorting problem, two metrics are utilized: accuracy and number of errors (NOE). Accuracy is calculated as the intersection divided by the union to measure how similar the final result is to the ground truth. The number of errors represents the number of elements that ascend rather than descend. The higher the accuracy, the better, the lower the number of errors, the better. All nodes except ANSWERINGNODE set the temperature to 0. The threshold for score probability is set to 0.99 for 100 and 0, and 0.5 for others.",
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+    {
+        "type": "text",
+        "text": "The experiment uses 100 lists of 128 elements and 100 lists of 256 elements. For the 128-element lists, numbers are randomly selected from the range 1 to 1000, allowing for duplicates. For the 256-element lists, numbers are randomly selected from the range 1 to 1500, also allowing for du",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    },
+    {
+        "type": "page_number",
+        "text": "6",
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+    {
+        "type": "text",
+        "text": "plicates, because GPT-4o's tokenizer splits numbers over 1000 into two tokens. In this experiment, to demonstrate the effectiveness of repeating the question, CoT is performed in two ways. CoT1 utilizes the rationale to sort the entire list in three steps: divide the list into four parts, sort each part, and then combine them. CoT2 involves rewriting the input to ensure better understanding before sorting the corresponding numbers.",
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+    {
+        "type": "text",
+        "text": "4.3 FROZEN LAKE",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A Frozen Lake is a problem of finding a route to a destination while avoiding the holes. To find the correct route in a frozen lake, it is necessary to know the exact locations of the holes and understand the rules of the Frozen Lake. To evaluate the Frozen Lake problem, two metrics are utilized: accuracy and number of errors (NOE). Accuracy is the number of successful routes found correctly divided by the total number of attempts. The number of errors is defined as the sum of the number of times the agent falls into a hole and the distance of the jump, which is not valid in the problem setting. All nodes except ANSWERINGNODE set the temperature to 0. The threshold for score probability is set to 0.95 for 100 and 0, and 0.5 for other scores. This experiment is conducted on a 5 by 5 size lake with 20 test cases containing 8 holes and 20 test cases containing 10 holes. Both GPT-4o and GPT-4o mini are used in the experiment.",
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+    {
+        "type": "text",
+        "text": "5 EVALUATION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 NUMBER SORTING",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 1 presents the experimental result of number sorting. ToT achieves the best performance when sorting 128 elements, followed by the proposed EGoT. When sorting 256 elements, the proposed EGoT outperforms the other architectures. EGoT also achieves performance similar to that of EGoT, although slightly lower. Note that five experiments were conducted to verify the consistent performance of EGoT for 128 elements and 256 elements. The results are shown in Figure 3, which demonstrates generally consistent performance.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The result of CoT1 and CoT2 is the one to focus on here. While there is a relatively slight performance difference when sorting 128 numbers, the performance gap is significantly larger when sorting 256 numbers. The reason for the difference is that in the first step of CoT1's rationale, when dividing the list into 4 lists, many numbers are missing, and in the last step of the rationale, when merging the 4 lists, it sometimes returns only the numbers from the first list without merging. For this reason, the performance of CoT1 is significantly lower compared to the other experiments. Conversely, CoT2's first step of rationale, which is to repeat elements once more, is relatively simple for the LLM, leading to fewer missing numbers. Subsequently, when prompting for sorting with the previously mentioned numbers, the LLM performs the sorting without difficulty. The tradeoff is an increase in both processing time and the number of output tokens due to the additional rationale steps requiring more outputs.",
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+    {
+        "type": "text",
+        "text": "We also compared the performance of various LLMs instead of GPT. Since EGoT requires the probability to evaluate the answer, we utilize the Llama 3.1 405B model and the Mixtral  $8 \\times 22\\mathrm{B}$  model. The Claude 3 Haiku model does not provide the probability of the answer, therefore, we fix the probability to 1. The experiments were conducted using 10 samples for sorting 256 data. During the evaluation, both Llama and Mixtral, in contrast to GPT-4o mini, consistently assign a score of",
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+    {
+        "type": "table",
+        "img_path": "images/590d0dfe37a1368e8a5df9001a8186be5645979c26e172754ccf5cec7c2ebfe3.jpg",
+        "table_caption": [
+            "Table 1: Results of the Number Sorting experiment (GPT-4o mini)"
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>128 Elements</td><td>IO</td><td>CoT1</td><td>CoT2</td><td>ToT</td><td>GoT</td><td>EGoT</td><td>EGoT*</td></tr><tr><td>Accuracy</td><td>90.25%</td><td>72.13%</td><td>90.41%</td><td>92.28%</td><td>90.98%</td><td>92.09%</td><td>91.70%</td></tr><tr><td>Number of Errors</td><td>14.07</td><td>38.79</td><td>13.87</td><td>10.87</td><td>11.88</td><td>10.94</td><td>11.45</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/9f82430c713120018ec6624ebff5ca6c85f260a4b7692dfcb236bea86c342955.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">256 Elements</td></tr><tr><td>Accuracy</td><td>70.71%</td><td>49.50%</td><td>83.17%</td><td>75.58%</td><td>84.37%</td><td>88.31%</td><td>87.94%</td></tr><tr><td>Number of Errors</td><td>119.51</td><td>154.57</td><td>49.25</td><td>65.74</td><td>40.93</td><td>34.54</td><td>35.19</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/9a7045bc8029bd4d09dc11588067805df0299c854565ab4a8f12d64d3f7864fc.jpg",
+        "image_caption": [
+            "Figure 3: Figure shows Min, Max, and Average for multiple experiments. The blue line on the left of each graph represents accuracy and the green line on the right represents a number of errors. The bars represent the maximum and minimum values, and the darker color in the middle represents the average. In the sorting problem, IO, CoT2, ToT, and GoT architectures were included as comparative models, and the experiment was performed only once. The higher the ACC, the better, the lower the NOE, the better."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/8b9fc8606ad76fdc418837226198cfa8d20a3d58e7a645839adb1774b830d7a8.jpg",
+        "table_caption": [
+            "Table 2: Results of the 256 Number Sorting experiment using various LLMs"
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Llama 3.1 405B</td><td>CoT</td><td>ToT</td><td>GoT</td><td>EGoT</td></tr><tr><td>Accuracy</td><td>91.59%</td><td>92.05%</td><td>94.09%</td><td>95.85%</td></tr><tr><td>Number of Errors</td><td>22.53</td><td>21.3</td><td>16.4</td><td>11.5</td></tr><tr><td colspan=\"5\">Mixtral 8×22B</td></tr><tr><td>Accuracy</td><td>82.91%</td><td>71.91%</td><td>83.85%</td><td>89.05%</td></tr><tr><td>Number of Errors</td><td>73.63</td><td>83.6</td><td>44.6</td><td>30.67</td></tr><tr><td colspan=\"5\">Claude 3 Haiku</td></tr><tr><td>Accuracy</td><td>92.10%</td><td>97.62%</td><td>94.38%</td><td>95.00%</td></tr><tr><td>Number of Errors</td><td>20.4</td><td>6.2</td><td>14.6</td><td>12.9</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "100 in EVALUATIONNODE. In such cases, we request the LLM for the score again. The results of these experiments are presented in Table 2.",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 FROZEN LAKE",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 3 and Table 4 present the experimental results for the Frozen Lake problem. In the experiment, EGoT and EGoT* outperform the other architectures. To evaluate the consistent performance, five experiments were conducted using GPT-4o mini, and three experiments were conducted using GPT-4o. The results are shown in Figure 3. GoT is applicable only when the problem can be divided into sub-problems, whereas Frozen Lake cannot be broken down into smaller parts. Therefore, we cannot compare GoT in this experiment. When the rationale simply instructs the model to understand the positions of holes and tiles, the LLM often becomes confused. However, when the LLM explicitly writes the coordinates next to the input before attempting to understand the positions of the holes and tiles, its performance improves.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.3 EGOT'S ADVANTAGES",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "EGoT shows the benefits of utilizing rationale information in prompt engineering instead of focusing only on the LLM's answer. It also proposes a structure to improve prompt engineering performance by leveraging the effect of LLM's temperature. Therefore, EGoT has two main advantages.",
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+    {
+        "type": "text",
+        "text": "First, EGoT generalizes the problem by generating the prompts to enhance the basis prompt. The basis prompt contains only the rule and rationale step of the problem, and the child node enhances",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a720225e03697a07ffc2d76f4c5fdc1665f3308532e55b069cb81ec5d5176b1d.jpg",
+        "table_caption": [
+            "Table 3: Results of the Frozen Lake experiment (GPT-40 mini)"
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>5 by 5 with 8 holes</td><td>CoT</td><td>ToT</td><td>EGoT</td><td>EGoT*</td></tr><tr><td>Accuracy</td><td>36%</td><td>28.1%</td><td>43%</td><td>41%</td></tr><tr><td>Number of Errors</td><td>1.33</td><td>1.38</td><td>1.13</td><td>1.14</td></tr><tr><td colspan=\"5\">5 by 5 with 10 holes</td></tr><tr><td>Accuracy</td><td>36.3%</td><td>27.6%</td><td>41.0%</td><td>34.0%</td></tr><tr><td>Number of Errors</td><td>1.28</td><td>1.54</td><td>1.15</td><td>1.43</td></tr><tr><td colspan=\"5\">Table 4: Results of the Frozen Lake experiment (GPT-4o)</td></tr><tr><td>5 by 5 with 8 holes</td><td>CoT</td><td>ToT</td><td>EGoT</td><td>EGoT*</td></tr><tr><td>Accuracy</td><td>50.8%</td><td>39.7%</td><td>58.8%</td><td>53.3%</td></tr><tr><td>Number of Errors</td><td>0.83</td><td>1.03</td><td>0.64</td><td>0.62</td></tr><tr><td colspan=\"5\">5 by 5 with 10 holes</td></tr><tr><td>Accuracy</td><td>51.7%</td><td>44.4%</td><td>59.0%</td><td>60.3%</td></tr><tr><td>Number of Errors</td><td>0.80</td><td>0.89</td><td>0.55</td><td>0.60</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "the prompt by appending only the parent's rationale output. In all experiments, EGoT demonstrates high performance, showing that the enhancing prompt is effective.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Second, EGoT dynamically adjusts the temperature and requests a confidence score from the LLM based on both the score itself and the probability of the corresponding token. Cosine annealing is used to control the temperature, enabling the exploration of diverse answers and rationales in the early stages. Obtaining a variety of rationales helps identify issues in problem formulation and refine prompt engineering more effectively. In the end, the low temperature allows us to focus on more accurate answers rather than diversity.",
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+    },
+    {
+        "type": "text",
+        "text": "5.4 DIFFERENCES BETWEEN EGOT AND EGOT*",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "EGoT* does not include dynamic temperature control, which leads to continual exploration of diverse solutions. This exploration helps maintain greater diversity in the responses. In tasks where the LLM performs well, EGoT can identify the correct answer, rather than focusing on maintaining response diversity. However, in tasks where the LLM performance is lower, exploration may lead to better answers. EGoT can sometimes exhibit lower performance when it utilizes only a portion of the provided rationale instead of considering all of it. For this reason, EGoT* performs similarly to EGoT on average when solving Frozen Lake problems. However, when solving sorting problems, EGoT demonstrates better performance.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.5 DIFFERENCES WITH OTHER ARCHITECTURES",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since EGoT relies on an LLM for evaluation, it does not require external tools to verify the correctness of an answer. Mathematical problems can be easily evaluated for correctness using tools, however, general questions are more challenging to assess in this manner. EGoT does not require problem decomposition. GoT is a useful architecture if the problem can be divided hierarchically, however, it is difficult to apply to general problems where the problem cannot be partitioned. ToT functions similarly to BFS or A* in LLM-based reasoning. However, BFS or A* becomes inefficient when evaluating a large number of elements, as seen in tasks like number sorting. CoT-SC focuses solely on the answer, not the rationale, when voting for the final answer, which is efficient if the answer is a scalar. However, when the answer is a list or vector, such as experiments like number sorting or Frozen Lake, it is not as applicable as ToT. EGoT emphasizes the importance of rationale and proposes that rationale aggregation can serve a similar role to voting by continuously integrating valid rationales while discarding incorrect ones. The disadvantage of EGoT compared to the other architectures is that it requires more computational time and resources due to its larger number of nodes. Since EGoT utilizes three nodes (Answering, Evaluation, and Aggregate Rationale) to obtain",
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+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "a single answer, it requires approximately three times the time and computational cost to generate the same number of answers.",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORK",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.1 CHAINING ARCHITECTURE AND RATIONALE STEP",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "There are several Prompt engineering architectures, including CoT (Wei et al., 2022), CoT-SC (Wang et al., 2023b), ToT (Long, 2023; Yao et al., 2024), EoT (Yin et al., 2023), and GoT (Besta et al., 2024). Various methods for evolving CoT and voting on the results of CoT have been proposed. Some papers emphasize the correct answer, while others emphasize the rationale. EGoT utilizes a method to construct its architecture based on EoT and Determinlr (Sun et al., 2024).",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "CoT emphasizes the importance of rationale and CoT-SC, in contrast, focuses on the correct answer rather than the rationale. The importance of providing rationale steps in prompts is widely recognized, and this leads to research on which rationale steps should be included (Xu et al., 2024). Generally, this involves summarizing the input (Zhang et al., 2023), separating the steps, providing the feedback in the input (Yuan et al., 2024; Madaan et al., 2024), and providing an explanation of the input (Yugeswardeenoo et al., 2024). Villarreal-Haro et al. (2024) and Yin et al. (2024) demonstrate the effectiveness of two strategies: incorporating negative information into the rationale and evaluating the rationale along with its probability, both of which enhance rationale performance. These findings support the validity of the rationale step in EGoT.",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.2 TEMPERATURE CONTROL AND EVALUATION LLM RESPONSE",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Temperature increases LLM's response diversity, and it also affects the performance of answers. Zhu et al. (2024) show the performance increase by adapting temperature with token confidence. To evaluate LLM responses, voting (Li et al., 2022; Du et al., 2024), debating (Liang et al., 2023; Xiong et al., 2023) and scoring (Lee et al., 2024a) are utilized. Since evaluating LLM response affects the performance of the architecture significantly, external tools (Gou et al., 2024) are used to evaluate the confidence level of LLM responses (Zhu et al., 2023). Motivated by these methods, we utilize debating to obtain the answer by providing the rationale of the parent node to the LLM for inferring the correct answer. Additionally, we perform self-evaluation by requesting the score for a single token from the LLM, rather than utilizing the entire set of responses, such as rationales, in the EVALUATIONNODE. We define confidence by utilizing the score and the probability of the token responded by the LLM to self-evaluate.",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            638,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Prompt engineering is an area of study that is key to effectively utilizing LLMs, maximizing the advantage of LLMs: the applicability of the model to a wide variety of problems without training. Fine-tuning is essential when an LLM needs to acquire specialized skills for certain tasks, however, it can reduce generalization capabilities and tends to be costly and resource-intensive. Chain of Thought (CoT) approach enhanced the ability to reason in general situations, recently various architectures evolved methodologies that are more effective for special cases.",
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+        "text": "We emphasize that the performance of LLMs is already enough to enable automated solutions for intuitive problems. While reasoning strategies may vary based on individual needs and problem requirements, the EGoT architecture demonstrates broad applicability and consistently improves performance. Our work reemphasizes the importance of rationale, and its concise architecture suggests the possibility of prompt engineering for a wide variety of problems.",
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+        "text": "Improving the performance of the LLM is also important, obviously. We tried to compare chess puzzles to verify the performance of EGoT architecture. However, despite adding a rule in the prompts that no piece except the knight can jump, GPT-4o mini thinks it can jump over a piece in the middle of a move. As a result, no architectures can find a move that captures the opponent's piece and checkmates, and the performance is not enough to compare results. Therefore, we hope that prompt engineering techniques improve with LLM performance improvement.",
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+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "text",
+        "text": "REFERENCES",
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+        "bbox": [
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+        "page_idx": 10
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+        "type": "header",
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+        ],
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+    {
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+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "text",
+        "text": "A APPENDIX",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A.1 EVALUATING COSTS: A COMPARISON TO GOT",
+        "text_level": 1,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d804175080f68e236ed087e401a48f36e59bf7248b58c1f1a5eeda28dc11b266.jpg",
+        "table_caption": [
+            "Table 5: Cost Comparison for solving one problem (GPT-4o mini) in USD"
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>128 Elements</td><td>Average</td><td>Minimum</td><td>Maximum</td></tr><tr><td>EGoT</td><td>0.03844</td><td>0.02922</td><td>0.04404</td></tr><tr><td>GoT</td><td>0.03246</td><td>0.02815</td><td>0.03658</td></tr><tr><td colspan=\"4\">256 Elements</td></tr><tr><td>EGoT</td><td>0.04302</td><td>0.03484</td><td>0.06855</td></tr><tr><td>GoT</td><td>0.05289</td><td>0.03353</td><td>0.05852</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The number of nodes in the experiment is adjusted through iterative testing to ensure that the cost difference from GoT remains minimal. As a result, while the number of subnodes and tokens per node in EGoT is higher compared to GoT, the total number of nodes in the graph is reduced, ensuring that the overall cost difference is not substantial. For the sorting 128 elements, GoT generally exhibits approximately  $20\\%$  lower cost; however, it does not hold for the sorting 256 elements. This discrepancy can be attributed to EGoT placing greater emphasis on rationale within the prompt and requesting more detailed responses. Consequently, even as the problem and correct answer length increases, the total input/output tokens increase only marginally. In contrast, GoT focuses primarily on the answer, leading to a proportional increase in input/output tokens as the problem length grows.",
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+    {
+        "type": "text",
+        "text": "Mathematically, the total output tokens can be represented as  $A + R$ , where  $A$  denotes the number of answer tokens and  $R$  represents the number of rationale tokens. EGoT generally has  $R > A$ , so even with an increase in  $A$ , the rate of increase in the total token count  $A + R$  is relatively smaller. On the other hand, GoT typically has  $R < A$ , meaning that as  $A$  increases, the rate of increase in the total token count is greater. For sorting 256 elements, EGoT has a lower average cost, while GoT has lower costs in both minimum and maximum. When the LLM solves difficult problems, it tends to generate uncertain or additional responses, which increases the total number of tokens. Conversely, for simpler problems, a more concise answer is produced, resulting in fewer tokens.",
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+    {
+        "type": "text",
+        "text": "The reason for the cost difference in maximum is that EGoT has conditional statements. EGoT queries again for uncertain answers to increase the confidence of the answer at the ANSWERINGNODE and EVALUATIONNODE. If this process occurs at every node, it would require a higher cost. There is a higher cost increase for problems that are difficult for LLM to solve. However, in general, EGoT performs better than GoT even though its cost is lower.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    },
+    {
+        "type": "page_number",
+        "text": "14",
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+    {
+        "type": "text",
+        "text": "A.2 METHOD NODE PROMPT EXAMPLE (SORTING PROBLEM)",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d3964ddb7b1d023c5c6183ae165fb9f207e8234377dc8e31ca0dc7481c9dc4fb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Human Message</td></tr><tr><td>&lt;Goal&gt; \n- When a list of numbers is entered, answer the question about how to sort them. \n- Answer the questions about how to evaluate whether the sorted numbers found by this method is correct. \n&lt;/Goal&gt;</td></tr><tr><td>&lt;Instruction&gt; \n- Answers the method based on keywords. \n- The evaluation methods should be done by LLM, so only print heuristic answers. \n- Answer without being abstract, only considering the method with the best performance. \n&lt;/Instruction&gt;</td></tr><tr><td>LLM Response</td></tr><tr><td>Method: \n- QuickSort \nEvaluation Method: \n- Check if the sorted list is in ascending order by comparing each element with the next one. \n- Count the number of inversions in the original list and the sorted list; the number of inversions should decrease. \n- Verify that the sorted list contains the same elements as the original list by checking for duplicates and ensuring all elements are present.</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 6: METHODNode Prompt Example.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 6 is an example of a real prompt we used. The method and evaluation method may not be effective, however, we used it to minimize human intervention.",
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+    },
+    {
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+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "15",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.3 ANSWERING NODE PROMPT EXAMPLE (SORTING PROBLEM)",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/137991f0210eb4ac9428f68a346b00f73def45967810a0aa3d718538a3aaf9e4.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Human Message</td></tr><tr><td>...[Provide the method and data to resolve the problem.]</td></tr><tr><td>...</td></tr><tr><td>&lt;RationaleSTEP&gt; - Rationalize through 4 processes and must provide a step-by-step explanation in detail. - First, accurately determine and write the all numbers in Input_Data. - Be sure to sort the numbers and write down the rationale in detail using the information provided in the previous step. - Make sure you explain why the sorting is correct with a rationale for the entire process. - Finally, answer the process of deducing the correct answer by synthesizing the previous process.&lt;/RationaleSTEP&gt;</td></tr><tr><td>...[Provide the information from the upper nodes.]</td></tr><tr><td>...</td></tr><tr><td>{% if aggregate_rationale %}</td></tr><tr><td>&lt;Hint&gt; - Refer to the hint, but they are not always accurate, so verify the hint by checking the input data and using them. [% for idx in range(aggregate_rationale|length) %] - Hint_{{idx}}: {aggregate_rationale[{{idx}}}] [% endfor %] &lt;/Hint&gt;</td></tr><tr><td>{%endif %}</td></tr><tr><td>LLM Response</td></tr><tr><td>Answer:- 6, 20, 20, 25, ...Answer Rationale:- First, I will identify all the numbers in the input list, which are: [1348, 147, 1454, 1111, ...Next, using the QuickSort algorithm, I will sort this list in ascending order. Using QuickSort, I will choose a pivot and divide the list into smaller sub-lists of numbers less than or greater than the pivot, then further sort those sub-lists until the entire list is sorted. The resulting sorted list is: [6, 20, 20, 25 ...</td></tr></table>",
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+        "page_idx": 15
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+    {
+        "type": "text",
+        "text": "Table 7: ANSWERINGNODE Prompt Example. (Jinja2 template)",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Table 7 is the prompt that requests the LLM to answer the question. Because we emphasize the importance of rationale, we set four rationale steps and gather effective rationale. The top node does not have aggregate rationale information, while the lower nodes have aggregate rationale information received from the upper nodes. Therefore, the prompt utilizes jinja2 template and forwards this information to the lower nodes at the bottom of the prompt with the Hint tag.",
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+    {
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+        "text": "Published as a conference paper at ICLR 2025",
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+        "text": "16",
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+    {
+        "type": "text",
+        "text": "A.4 EVALUATION NODE PROMPT EXAMPLE (SORTING PROBLEM)",
+        "text_level": 1,
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+    {
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+        "img_path": "images/e28a358525a017d5bcf55acf4b97adc4d84ca71b187ac020c87a864955520116.jpg",
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+        "table_body": "<table><tr><td>Human Message</td></tr><tr><td>...[Provide the question, the answer from ANSWERINGNODE, and request a score.]...</td></tr><tr><td>Evaluated Rationale_Step&gt; - Rationalize through 4 processes and must provide a step-by-step explanation in detail. - First, accurately determine and write all the numbers in Input_Data. - Be sure to score each number included in the sorted data and write down the rationale in detail using the information provided in the previous step. - Make sure you explain why the sorting is correct with a rationale for the entire process. - Finally, answer the process of deducing the correct answer by synthesizing the previous process.</td></tr><tr><td>Evaluated Rationale_Step&gt;</td></tr><tr><td>Evaluated Methods&gt; {% for idx in range(evaluate_methods|length) %} - {evaluate_methods[{idx}]}) {% endfor %}</td></tr><tr><td>Evaluated Methods&gt;</td></tr><tr><td>LLM Response</td></tr><tr><td>Accuracy: - 80 Evaluation Rationale: ...For instance, the number &#x27;680&#x27; appears in the sorted data but is not present in the input data. Additionally, the number &#x27;994&#x27; is also present in the sorted data but is missing from the input. ...Therefore, while the majority of the numbers are sorted correctly, the presence of these discrepancies warrants a score below 90.</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 8: EVALUATIONNODE Prompt Example. (Jinja2 template)",
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+        "text": "Table 8 is the prompt that requests the LLM to evaluate the response. The Evaluation Rationale Step is almost identical to table 7. However, only the content of step 2 changes slightly because it needs to evaluate the score. By parsing the log probability of the number \"80\" in the LLM response, the confidence is calculated as 0.607.",
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+        "text": "Published as a conference paper at ICLR 2025",
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+        "text": "17",
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+    {
+        "type": "text",
+        "text": "A.5 AGGREGATE RATIONALE NODE PROMPT EXAMPLE (SORTING PROBLEM)",
+        "text_level": 1,
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+        "img_path": "images/13f96e972714286fa308683f01b605aa5c2073e1a9da82faf19011141413cb01.jpg",
+        "table_caption": [],
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+        "table_body": "<table><tr><td>Human Message</td></tr><tr><td>&lt;Input_Rationales&gt; \n- Evaluation_Rationale: {evaluation_rationale} \n- Answer_Rationale: {answer_rationale} \n&lt;/Input_Rationales&gt;</td></tr><tr><td>&lt;Instruction&gt; \n&lt;Aggregate_Rationale&gt; \n- Don’t print the successes or valid, just print the incorrectly missing numbers. \n- Integrate the flaws that are problematic or invalid. \n- Be sure to consider the previous two steps, aggregate the Input_Rationales. \n&lt;/Aggregate_Rationale&gt; \n&lt;Inaccurate Info&gt; \n- Inaccurate Info is information to recheck regarding the missing numbers, as it seems to be inaccurate. \n- If the number information in two rationales is different, the number should be included in the inaccurate information. \n&lt;/Inaccurate Info&gt; \n&lt;/Instruction&gt;</td></tr><tr><td>LLM Response</td></tr><tr><td>Aggregate Rationale: \n- The sorted list contains discrepancies, including numbers that are not present in the original input data, such as ‘680’ and ‘994’. ... \nInaccurate Info: \n- The numbers ‘680’ and ‘994’ are present in the sorted data but are missing from the input list. Additionally, the number ‘1496’ appears twice in the input list but is only represented once in the sorted list, indicating a potential error in the sorting process.</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 9: AGGREGATERATIONALENODE Prompt Example. (Jinja2 template)",
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+        "text": "Table 9 is the prompt that summarizes the rationale generated by the two nodes ANSWERINGNODE and EVALUATIONNODE. AGGREGATERATIONALENODE provides the rationale information by summarizing the key points. Additionally, it extracts negative information and propagates this to the lower nodes. In table 8, LLM informs that 680 and 994 are present in the input, however, the sorted result doesn't contain these numbers, therefore table 9 aggregates this information. Misinformation like 1496 also propagates, though the misinformation gradually vanishes as the graph progresses.",
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+        "text": "Published as a conference paper at ICLR 2025",
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+        "text": "18",
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