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2025/OSDA Agent_ Leveraging Large Language Models for De Novo Design of Organic Structure Directing Agents/6c6b4170-5fdb-41f3-b334-fd49d893d405_origin.pdf

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2025/OSDA Agent_ Leveraging Large Language Models for De Novo Design of Organic Structure Directing Agents/full.md

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+# OSDA AGENT: LEVERAGING LARGE LANGUAGE MODELS FOR DE NOVO DESIGN OF ORGANIC STRUCTURE DIRECTING AGENTS
+
+Zhaolin Hu $^{1,2}$ , Yixiao Zhou $^{2,3}$ , Zhongan Wang $^{2}$ , Xin Li $^{4*}$ , Weimin Yang $^{4}$ , Hehe Fan $^{2}$ , Yi Yang $^{1,2*}$
+
+<sup>1</sup>The State Key Laboratory of Brain-Machine Intelligence, Zhejiang University, China
+
+$^{2}$ CCAI, Zhejiang University, China  $^{3}$ Shanghai Innovation Institute, China
+
+<sup>4</sup>The State Key Laboratory of Green Chemical Engineering and Industrial Catalysis, Sinopec Shanghai Research Institute of Petrochemical Technology, China
+
+{12321165,12421181,zhonganwang,hehefan,yangyics}@zju.edu.cn
+
+{lixin.sshy, yangwm.sshy}@sinopec.com
+
+# ABSTRACT
+
+Zeolites are crystalline porous materials that have been widely utilized in petrochemical industries as well as sustainable chemistry areas. Synthesis of zeolites often requires small molecules termed Organic Structure Directing Agents (OSDAs), which are critical in forming the porous structure. Molecule generation models can aid the design of OSDAs, but they are limited by single functionality and lack of interactivity. Meanwhile, large language models (LLMs) such as GPT-4, as general-purpose artificial intelligence systems, excel in instruction comprehension, logical reasoning, and interactive communication. However, LLMs lack in-depth chemistry knowledge and first-principle computation capabilities, resulting in uncontrollable outcomes even after fine-tuning. In this paper, we propose OSDA Agent, an interactive OSDA design framework that leverages LLMs as the brain, coupled with computational chemistry tools. The OSDA Agent consists of three main components: the Actor, responsible for generating potential OSDA structures; the Evaluator, which assesses and scores the generated OSDAs using computational chemistry tools; and the Self-reflector, which produces reflective summaries based on the Evaluator's feedback to refine the Actor's subsequent outputs. Experiments on representative zeolite frameworks show the generation-evaluation-reflection-refinement workflow can perform de novo design of OSDAs with superior generation quality than the pure LLM model, generating candidates consistent with experimentally validated OSDAs and optimizing known OSDAs.
+
+# 1 INTRODUCTION
+
+Zeolites, a class of microporous silicate-based materials, have been widely used as highly efficient catalysts and adsorbents, in petrochemical industries and sustainable chemistry processes (Davis, 2002). The unique properties of zeolites stem from their porous frameworks and the synthesis of zeolites often requires the use of small molecules named Organic Structure Directing Agents (OS-DAs) (Moliner et al., 2013). OSDAs act as templates influencing the size, shape, and connectivity of the pores of zeolites. The design of effective OSDAs is crucial for tailoring the properties of the target zeolite, but this process has traditionally been guided by empirical knowledge and labor-intensive trial-and-error methods.
+
+Recently, artificial intelligence has been extensively applied across various scientific disciplines, including chemistry (Liu et al., 2023; Jiang et al., 2024), biology (Fan et al., 2022; Corso et al.; Fan et al., 2023), and materials (Sriram et al., 2024; Gruver et al., 2024; Yang et al., 2024; Zeni et al., 2023). In particular, artificial intelligence has significantly influenced the design of OSDAs. Various heuristic algorithms, such as genetic algorithms (Pophale et al., 2013) and ant colony algo
+
+rithms (Muraoka et al., 2020), have been employed to explore the vast chemical space of potential OSDAs, identifying promising candidates. Machine learning has also made significant strides in this area (Chong et al., 2020; Xie et al., 2019; Manuel Serra et al., 2007; Moliner et al., 2019). With the development of large-scale zeolite synthesis datasets (Pan et al., 2024; Muraoka et al., 2019; Jensen et al., 2021), neural networks trained on these extensive datasets can predict the suitability of OSDAs for specific zeolite structures. These networks effectively learn from established synthesis recipes and experimental results, enabling the design of OSDA molecules (Jensen et al., 2021; Xu et al., 2023). Although these methods enable OSDA predictions, their inability to accept feedback from human experts limits the interactability of the model.
+
+Large language models (LLMs), such as GPT-4 developed by OpenAI (OpenAI, 2023), are capable of processing and generating human-like text and performing complex reasoning based on extensive datasets. These models have demonstrated remarkable proficiency in various natural language processing and multimodal tasks, including text generation, translation, summarization, and multimodal understanding (Raffel et al., 2020; Brown, 2020; Devlin, 2018; Li et al.; Zhang et al., 2024; Li et al., 2025). OSDA molecular design based on LLM can provide a more natural and seamless user experience that goes beyond traditional machine learning methods, with more aspects of the model accepting suggestions from chemical experts, leading to better generation (Ito et al., 2024b;a). While LLMs can process vast amounts of data and generate novel molecular structures, they often lack the domain-specific reasoning and ability to effectively integrate complex experimental constraints or optimization criteria (Burtsev et al., 2023).
+
+In this paper, we introduce the OSDA Agent, an innovative interactive framework for the design of OSDAs that harnesses the capabilities of large language models as the core intelligence, complemented by advanced computational chemistry tools. The OSDA Agent comprises three key components: the Actor, tasked with generating a diverse array of potential OSDA structures based on predefined criteria, utilizing the generative capabilities of LLMs to explore vast chemical space and synthesize novel molecular architectures; the Evaluator, which critically assesses and scores these OSDAs using a suite of computational chemistry tools (Landrum, 2013; Coley et al., 2018), employing various metrics to determine the feasibility and effectiveness of each proposed structure; and the Self-reflector, which plays a pivotal role in the iterative design process by producing comprehensive reflective summaries (Shinn et al., 2024) based on the Evaluator's feedback, guiding the Actor in refining its outputs to enhance the quality of subsequent designs. Through the integration of these components, the OSDA Agent not only streamlines the design process but also fosters a continuous learning environment, enabling the system to adapt and improve over time, representing a significant advancement in the automated design of OSDAs and facilitating a more efficient and effective approach to zeolite synthesis. In summary, our contributions are threefold:
+
+- We introduce the OSDA Agent, an innovative interactive framework that integrates LLMs with computational chemistry tools for the de novo design of OSDAs given target zeolites.  
+- To cope with the problem of uncontrollable results generated by LLMs, we introduced a reflection mechanism that consists of a novel binding-energy prediction module and chemical traditional tools.  
+- Our experimental results demonstrate that the OSDA Agent significantly improves the quality of OSDA candidates compared to traditional methods, yielding structures that align with experimentally validated OSDAs and effectively optimizing known OSDAs.
+
+# 2 RELATED WORK
+
+Designing OSDAs for Zeolites. The interaction between OSDAs and zeolites can be captured through atomistic simulations, such as density functional theory (DFT) and molecular dynamics simulations, though these methods require substantial computational resources (Schwalbe-Koda & Gomez-Bombarelli, 2021a). Jensen et al. (Jensen et al., 2021) utilized data-driven approaches to explore relationships between OSDAs, qualitative gel chemistry, and zeolite structures. Daeyaert et al. (Daeyaert et al., 2019) and Xu et al. (Xu et al., 2023) applied machine learning to train neural networks for designing OSDAs tailored to specific zeolite structures. Recent studies by Ito et al. (Ito et al., 2024b;a) investigated the use of LLMs in designing novel OSDA molecules, highlighting their potential for advancing molecular innovation. In contrast to previous approaches, our method combines the knowledge embedded in LLMs with chemical tools to improve design outcomes.
+
+LLM Agents for Science. Large language model (LLM) agents are now widely used across various scientific disciplines (Chiang et al., 2024; Huang et al., 2024; Skarlinski et al., 2024; Ma et al., 2024; Ghafarollahi & Buehler, 2024; Lala et al., 2023; Boiko et al., 2023; Darvish et al., 2024). ChemCrow (M. Bran et al., 2024) developed an LLM agent integrated with specialized tools for literature retrieval, molecular modification, and reaction execution, enabling the autonomous execution of chemical synthesis. Several studies (Buehler, 2024; Sprueill et al., 2024; Ansari & Moosavi, 2023; Kang & Kim, 2024) have explored the application of LLM agents in the field of materials science. Furthermore, MedAgents (Tang et al., 2023) and DRUGAGENT (Inoue et al., 2024) have explored the applications of LLM Agents in the fields of medicine and drug discovery. In this work, we investigate the application of an LLM Agent for OSDA molecule design.
+
+Text-based de novo Molecule Generation. In recent decades, AI-driven methods have become central to advancing molecule discovery (Hu et al., 2023; Fan & Yang, 2024). Text-based de novo molecule generation utilizes natural language processing techniques alongside chemical information to design novel molecular structures. Text2Mol (Edwards et al., 2021), constructs paired datasets of molecules and textual descriptions to learn a shared semantic embedding space for improved retrieval. MolT5 (Edwards et al., 2022) and BioT5 (Pei et al., 2023), built on the T5 (Raffel et al., 2020) architecture, are pre-trained on large corpora of text and molecular strings, enhancing their ability to generate molecules from textual input. MoMu (Su et al., 2022) uses contrastive learning to bridge molecular graphs and related text data. TGM-DLM (Gong et al., 2024) explores diffusion models for molecular generation. Moreover, FrontierX (Sakhinana & Runkana, 2023) and MolReGPT (Li et al., 2024) utilize the capabilities of large language models, presenting new paradigms for molecular generation. However, traditional text-to-molecule methods struggle to perform well when faced with the complexity of designing OSDA molecules. In this work, we address this challenge by utilizing the comprehensive capabilities of the LLM Agent.
+
+# 3 PRELIMINARIES AND DATA DESCRIPTION
+
+# 3.1 DATA
+
+We utilized the Zeolite Organic Structure Directing Agent Database (OSDB) (Schwalbe-Koda et al., 2021) that includes 112,400 OSDA-Zeolite pairs which comprises 549 OSDA molecules and 209 distinct zeolite frameworks extracted from the literature. The database provides the corresponding binding energy for each OSDA-Zeolite pair along with the OSDA-Zeolite complex structure data saved in CIF files. In addition, for each OSDA molecule, we calculated its synthetic complexity score (SCScore) (Coley et al., 2018).
+
+Furthermore, we employed the Jensen dataset (Jensen et al., 2021) extracted from the entire body of zeolite literature, spanning over 140 journals from more than 15 publishers between 1966 and 2020. It includes 5,663 synthesis routes extracted from 1,384 articles, encompassing OSDA molecules, synthesis gel compositions, and the resulting zeolite phases. The dataset provides details on 758 distinct OSDA molecules and 205 zeolite phases, from which we extracted OSDA molecules as real-world examples of existing OSDAs.
+
+# 3.2 THE STRUCTURE OF OSDA-ZEOLITE COMPLEX
+
+The OSDA-Zeolite complex is a variation of crystal structure that still adheres to the geometric constraints of crystals. Thus, we represent it as a crystal graph. In the unit cell  $U = (X,P)$ ,  $X = [x_{1},x_{2},\dots ,x_{n - 1},x_{n}]^{T}\in \mathbb{R}^{n\times 1}$ , where  $n$  represents the number of atoms, and  $x_{i}\in \mathbb{R}^{1}$  represents the type of atom  $i$  in the unit cell.  $P = [p_1,p_2,\dots ,p_{n - 1},p_n]^T\in \mathbb{R}^{n\times 3}$  is the atomic position matrix, where  $p_i\in \mathbb{R}^3$  represents the Cartesian coordinates of atom  $i$  in the unit cell in three-dimensional space. To further encode the periodic pattern, additional lattice vectors  $L = [l_{1},l_{2},l_{3}]^{T}\in \mathbb{R}^{3\times 3}$  are used to describe how the unit cell repeats in three directions. Thus, in 3D space, we represent the OSDA-Zeolite complex as  $(X,P,L)$ , see Figure 11 in Appendix E.
+
+# 3.3 BINDING ENERGY
+
+The affinity of an OSDA-Zeolite pair can be quantitatively reflected via the binding energy. A smaller binding energy value indicates that the OSDA is more compatible with the zeolite. The
+
+calculation formula of binding energy is:
+
+$$
+E _ {b} = E _ {\text {z e o - o s d a}} - E _ {\text {z e o}} - E _ {\text {o s d a}} \tag {1}
+$$
+
+where  $E_{b}$  represents the binding energy,  $E_{\mathrm{zeo - osda}}$  is the energy of the OSDA-Zeolite complex,  $E_{\mathrm{zeo}}$  is the energy of the zeolite, and  $E_{\mathrm{osda}}$  is the energy of the OSDA (Schwalbe-Koda & Gomez-Bombarelli, 2021a).
+
+# 4 METHODOLOGIES
+
+# 4.1 OSDA AGENT FRAMEWORK
+
+In developing the OSDA Agent framework, we followed the methodology described in (Shinn et al., 2024). Our framework integrates three interconnected models: the Actor, the Evaluator, and the Self-reflection model. The Actor is responsible for creating OSDAs by employing advanced algorithms that guide molecules into desired configurations. The Evaluator assesses the OSDA molecules generated by the Actor, utilizing various chemical tools for evaluation. The Self-reflection model leverages feedback from the Evaluator to refine the synthesis processes within the Actor, continuously improving the efficacy of the organic structure-directing agents. This cohesive approach provides a robust framework for generating and optimizing OSDAs. The overall framework architecture is illustrated in Figure 1 (a).
+
+More specifically, the Actor in this work assumes the role of a zeolite expert. The Actor is based on a LLM that generates SMILES sequences for OSDA molecules in response to specified prompts and requirements. In this context, we utilize In-Context Learning (Yoo et al., 2021) and Chain of Thought (Wei et al., 2022) techniques to guide the Actor's performance. The specific prompt strategies we employ are detailed in Section 4.2. Additionally, we incorporate a memory component, which provides supplementary context to enhance the Actor's capabilities.
+
+The Evaluator component of the OSDA Agent framework plays a critical role in assessing the quality of the OSDA molecules generated by the Actor. It takes the candidate SMILES sequences produced by the Actor as input and employs a range of chemical tools for evaluation. The chemical tools we use consist of a series of small models and predefined rules. First, RDKit is utilized to verify that the generated SMILES expressions adhere to chemical validity. Then, the Evaluator provides a comprehensive assessment by scoring the potential OSDA molecules based on their binding energy and SCScore (Coley et al., 2018). This approach ensures that the generated OSDA molecules meet the required specifications and are synthetically feasible, as detailed in Section 4.3.
+
+Although LLMs exhibit powerful intelligence, they occasionally make reasoning errors (Madaan et al., 2024). To mitigate such errors, we implemented a self-reflection module. Based on a LLM, this module provides valuable feedback for future iterations through verbal self-reflection. It generates detailed and specific feedback based on the assessments from the Evaluator, the current generated results, and the memory of the agent. In subsequent trials, the agent leverages its past experiences to optimize the generation of OSDA molecules. This iterative process of trial and error, self-reflection, and persistent memory enables the agent to rapidly enhance its decision-making abilities across various environments by effectively utilizing feedback signals.
+
+# 4.2 STRUCTURE DESIGN WITH COT STRATEGY
+
+To address the limitations of LLMs in handling complex reasoning tasks, we adopted a few-shot Chain of Thought (few-shot CoT) approach. By designing specialized prompt strategies, we aim to foster a more continuous and incremental reasoning process within the model, enabling it to more effectively generate valid OSDA molecules.
+
+The few-shot CoT prompt combines In-Context Learning (ICL) (Brown, 2020) with the Chain of Thought (CoT) (Wei et al., 2022) technique. By introducing a continuous, step-by-step reasoning process through the prompt "Let's think step by step," we effectively guide the model in solving complex tasks. CoT enables the model to break down tasks into their constituent parts, leading to clearer and more logical solutions (Wei et al., 2022). As a few-shot prompt, we also provide the LLM with an example to help it better understand and respond to the required format and content.
+
+![](images/ac76faa758bc7dcdc9a07cc82efacb489a5378e39638b4ed2a8f23a77a93bc64.jpg)  
+Figure 1: OSDA Agent Framework. (a) The OSDA Agent pipeline, integrating three models: Actor, Evaluator, and Self-reflector, responsible for generating molecules, performing chemical evaluations, and providing optimization feedback. (b) The Evaluator uses three chemical tools to assess molecules based on chemical principles, synthetic feasibility, and binding energy. (c) Example of OSDA Agent in action.
+
+In crafting the prompts for the generative large language model, we also employed role-play instructions, assigning the model the role of an expert in zeolite OSDAs. Additionally, we extracted several well-performing OSDA molecules from the OSDB database to serve as context. The complete prompt can be found in Figure 7 and Figure 8 in Appendix D.
+
+# 4.3 EVALUATION WITH CHEMICAL TOOLS
+
+The LLM Agent leverages its powerful natural language processing capabilities to intelligently generate molecular structures. However, due to the intricate chemical constraints and reasoning requirements, the output generated by the LLM alone may not fully meet the precision required in chemistry. Therefore, we introduced a series of chemical tools to ensure that the generated molecules adhere to chemical rules and possess synthetic feasibility. This combination provides us with a robust and flexible framework for exploring and optimizing new OSDA molecules.
+
+We incorporated three key chemical tools to help the agent improve the chemical validity and accuracy of the generated OSDA molecules. These tools include the RDKit (Landrum, 2013) toolkit, SCScore (Coley et al., 2018), and the binding energy estimation model. In the following sections, we will provide a detailed introduction to each of these tools.
+
+We also developed a memory component that stores the evaluation results from the chemical tools for the generated molecules. These stored evaluations are utilized by the Reflection Mechanism to inform and improve subsequent iterations.
+
+# 4.3.1 RDKIT TOOLKIT
+
+The LLMs occasionally generate OSDA molecules that do not meet the required standards, including violations of chemical rules and OSDA-specific empirical guidelines. To address this issue, we used the RDKit (Landrum, 2013) toolkit to evaluate and filter the generated molecules. The specific empirical criteria, similar to those outlined in (Ito et al., 2024b), are listed in Table 2 in Appendix C.
+
+# 4.3.2 SYNTHETIC COMPLEXITY SCORE
+
+The synthetic accessibility of OSDAs is a crucial factor in assessing their quality. In our work, we employed SCScore to assess synthetic complexity. SCScore is a scoring method based on reaction database learning, where a neural network model is trained on a large dataset of 12 million reactions from the Reaxys database to predict the number of reaction steps required to synthesize the target
+
+![](images/de3adbc2ff90ec54ecb437638ab73b151644c181ec190c3b0e9eebd21d0bd38c.jpg)  
+Figure 2: Details of the binding energy estimation model. (a) Process: We provide  $\langle$ OSDA, Zeolite\rangle pairs to the Void tool to generate docking poses. The predicted energy for each pose is then computed using a neural network model, and the lowest is selected as the estimated binding energy. (b) Neural network architecture: The input consists of three types of information relevant to determining the binding energy: OSDA, Zeolite, and Complex. We use different architectures to extract information from each and then integrate them to predict the binding energy.
+
+![](images/9cb79566dd5787c4a1edc16977f9d16cfdb47fafd115bb5f8545664501202a97.jpg)
+
+molecule (Coley et al., 2018; Lawson et al., 2014). The model establishes a nonlinear metric by comparing the synthetic complexities of reactants and products, effectively capturing the increasing complexity across multi-step synthesis routes. The SCScore ranges from 1 to 5, with lower values indicating that a molecule is easier to synthesize.
+
+# 4.3.3 BINDING ENERGY ESTIMATION MODEL
+
+Since each OSDA Agent design involves evaluating the quality of the OSDA design multiple times using binding energy, traditional atomic simulation methods for calculating binding energy often require computing the energy of dozens to hundreds of complexes, which is time-consuming (Schwalbe-Koda & Gomez-Bombarelli, 2021a). To address this issue, we proposed a deep learning-based method to estimate the energy of these complexes, significantly reducing the time required for binding energy calculations, as shown in Figure 2 (b).
+
+Our inspiration stems from the calculation formula for the OSDA-Zeolite binding energy. In Equation equation 1, the binding energy is related to the energy of complex, the energy of zeolite, and the energy of OSDA. We follow this principle in our approach. The inputs to our method include molecular SMILES, zeolite structure diagrams, and complex structure diagrams. The zeolite structure diagrams are obtained by parsing CIF files. We obtain the complex's CIF files using the Void (Voronoi Organic-Inorganic Docker package) (Schwalbe-Koda & Gomez-Bombarelli, 2021b) and parse them to extract the complex structure diagrams.
+
+We utilize a pre-trained SMILES-BERT (Wang et al., 2019) model to extract molecular features, while the complex and zeolite structure diagrams are processed using CGCNN (Xie & Grossman, 2018) networks to extract complex and structural features. Then, these features (molecular, structural, and complex) are concatenated, followed by feature fusion using a transformer, and finally, a fully connected neural network is used to predict the complex energy. Our model is trained on data from the OSDB database. The effectiveness of each module is shown in Table 3 in Appendix E.
+
+Throughout the entire process, candidate OSDA molecular SMILES are generated using Actor, and the 3D conformations of the molecules are optimized using the MMFF94 (Halgren, 1996) method. Possible binding poses between the 3D molecular conformations and zeolite structures are generated using the Void tool. We predict the energy for each pose and use the lowest energy as the estimated binding energy, as illustrated in Figure 2 (a).
+
+# 4.4 REFLECTION MECHANISMS
+
+The Reflection Mechanism reads the evaluations of the OSDA molecules generated by the Actor, stored in the memory component. Using a specially designed reflection prompt, the Self-reflection LLM provides constructive feedback on the Actor's output. This process repeats for a set number of iterations, guiding the Actor to improve the quality and success rate of the generated molecules.
+
+The reflection prompt utilizes a few-shot prompting technique, providing the Self-reflection LLM with several examples of constructive feedback. We instruct the Self-reflection LLM to critique and offer suggestions on the generated results based on the evaluations from the chemical tools. The feedback can focus on several aspects, including the chemical validity of the generated molecules, their synthetic feasibility, and the estimated binding energy with the given zeolite structure. The goal is to produce OSDA molecules that are both effective and easy to synthesize. The complete prompt and examples can be found in Figure 9 and Figure 10 in Appendix D.
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENT SETUP
+
+In our OSDA Agent framework, the Actor utilizes the GPT-4 model, while the Self-reflection is based on the GPT-4o model. The examples used in the few-shot CoT prompts are sourced from the OSDB database. We built our framework using Langchain, and the RDKit version employed is 2023.9.1. In the experiments, the literature OSDAs were obtained from the Jensen dataset (Jensen et al., 2021). Baseline methods include state-of-the-art text-based de novo molecule generation approaches, such as BioT5 (Pei et al., 2023), MolT5 (Edwards et al., 2022).
+
+The binding energy estimation model was trained on OSDB data. The model was trained for 15 epochs using the Adam optimizer, with a 1e-5 learning rate and batch size of 32. It was implemented in Python v3.10.4 with the PyTorch v1.12 framework and trained on an NVIDIA A6000 GPU. To ensure the independence of the training and test sets, the dataset was split based on SMILES, following an 8:1:1 ratio. This split ensured that the OSDA molecules in the training set were completely distinct from those in the test set. The details and results can be found in Appendix E.
+
+# 5.2 RESULTS OF OSDA MOLECULE DESIGN
+
+To validate the effectiveness of our approach, we selected two representative zeolite frameworks for testing: the small-pore LTA framework (Reed & Breck, 1956) with a cage-like structure and the large-pore one-dimensional AFI framework (Bennett et al., 1983). We applied our method to design OSDA molecules tailored for these two zeolite structures. In this section, we focus on presenting the results and analysis of our method in designing OSDAs for the cage-like small-pore LTA framework and the large-pore one-dimensional AFI framework. Additionally, we extended our method to test other zeolite frameworks, including LEV (Barrer & Kerr, 1959), AEI (Simmen et al., 1991), AFX (McGuire et al., 1995), ITE (Camblor et al., 1997) and MOR (Simoncic & Armbruster, 2004), with the results provided in Appendix I.
+
+# 5.2.1 SUGGESTED OSDA CANDIDATES
+
+Figure 3 presents several OSDAs predicted by our OSDA Agent using a de novo molecular design workflow. Compared to previous work (Ito et al., 2024b;a), our method incorporates synthetic feasibility into the model, resulting in OSDA candidates that not only match known OSDAs but also exhibit a high degree of synthesizeability. Additionally, consultations with experts in zeolite chemistry affirmed that the generated molecules exhibit significant potential to function as OSDAs.
+
+# 5.2.2 DIMENSIONALITY REDUCTION ANALYSIS
+
+An essential capability for any molecular generation model is to capture the molecular distribution and generate diverse and realistic molecules. Such capabilities are paramount when constructing virtual libraries to advance computer-aided drug discovery endeavors (van Hilen et al., 2019). To evaluate the performance of our model in generating OSDAs, we performed a comparative analysis
+
+![](images/0904787edd6f8bf585ca9376e8c1a6fb4ef2eaf776b05d2aaf45bd3c1cb64e50.jpg)  
+Figure 3: Examples of OSDAs for two representative zealite frameworks, large-pore AFI framework and small-pore LTA framework, generated using the OSDA Agent. The  $E_{b}$  is measured in kJ/mol Si.
+
+against previously reported OSDAs using Principal Component Analysis (PCA) based on Weighted Holistic Invariant Molecular (WHIM) (Todeschini & Gramatica, 1997) descriptors (detailed in Appendix C).
+
+![](images/4ff842a5020ff02a24aca9dc1c8d30e5e4ca0c86292fdf44f5f7c5091b9f71ad.jpg)  
+PCA Analysis: OSDAs for AFI
+
+![](images/d0c0e7c178cec450c9322fa4afaee6fdf52d080ddbee003a0c1e26d24a00e04e.jpg)  
+Figure 4: Principal Component Analysis (PCA) of WHIM vector representation for OSDA molecules used in AFI framework. PCA 1, 2, and 3 represent the first three principal axes. Red points show OSDAs generated by the OSDA Agent, blue points represent OSDAs of AFI framework from the literature, and gray points represent all OSDAs extracted from the literature.
+
+The dataset used for this analysis includes all OSDAs from the literature (shown in gray), OSDAs generated by our model (shown in red), and OSDAs of specific frameworks of zeolite previously reported in the Jensen dataset (shown in blue). WHIM descriptors, which capture molecular geometry based on principal components derived from molecular field methods, were calculated for both datasets. These descriptors were then standardized and processed via PCA to visualize and assess the overlap between the OSDAs generated by our model and those previously reported in the reduced-dimensionality space.
+
+Consistent with previous work(Jensen et al., 2021; Xu et al., 2023), we used PCA to reduce the WHIM descriptors to three dimensions. The explained variance of the principal components confirms that PCA 1, PCA 2, and PCA 3 capture the majority of the variance in the data (Figure 6 in Appendix C). We conducted tests on AFI and LTA frameworks, with the PCA results shown in Figures 4 and Figure 14 in Appendix F, displaying data points projected onto the first three principal components. The left plot (PCA 1 vs. PCA 2) illustrates the distribution of all OSDAs from the Jensen dataset (in gray), OSDAs generated by our model (in red), and OSDAs of AFI and LTA
+
+frameworks previously reported in the literature (in blue). The right plot (PCA 1 vs. PCA 3) offers a complementary view of the data distribution.
+
+This visualization demonstrates a notable overlap between the generated OSDAs and the reported OSDAs, which indicates that our model has successfully generated molecules that closely align with known, experimentally validated OSDAs. Furthermore, the red points suggest that our model is also capable of generating novel OSDAs that explore new regions of the molecular descriptor space, highlighting the generative model's potential for de novo OSDA design.
+
+# 5.2.3 PERFORMANCE COMPARISON
+
+To evaluate the effectiveness of our OSDA Agent framework, we compared the results of our approach with two text-based de novo molecule generation methods, MolT5 and BioT5, as well as the results generated by the pure GPT-4 model without the additional enhancements provided by our integrated method. To ensure a fair comparison, we supplied the GPT-4 model with the same OSDB data used for prompts in the OSDA Agent. We generated approximately 100 OSDA molecules each for both AFI and LTA frameworks using the OSDA Agent and GPT-4, while MolT5 and BioT5 produced around 500 OSDA molecules each for both AFI and LTA frameworks.
+
+In this study, we employed seven well-established metrics (detailed in Appendix C) to assess the capability of our approach to generate OSDA molecules that align with real-world distributions and empirical rules. The first metric is Validity, which evaluates whether the generated molecules comply with the empirical rules for OSDAs, as shown in Table 2 of Appendix C. Next, we utilized four commonly used text-to-molecule metrics BLEU (Papineni et al., 2002), Morgan, MACCS, and RDK to measure the similarity between the generated OSDA molecules and those reported in the literature. Finally, we applied the Energy Distance (ED) (Székely & Rizzo, 2013) and Kullback-Leibler (KL) divergence (Kullback & Leibler, 1951) to assess the distributional similarity between the generated OSDAs and the ones reported in the literature.
+
+As shown in Table 1, traditional text-to-molecule methods like MolT5 (Edwards et al., 2022) and BioT5 (Pei et al., 2023) struggle to generate molecules that comply with OSDA rules, largely due to their lack of specialized knowledge in zeolite OSDAs. The molecules they generate significantly differ from those reported in the literature. While large language models such as GPT-4 possess the relevant knowledge to produce more plausible OSDA molecules, they lack the guidance of specialized chemical tools. As a result, GPT-4 generated OSDA molecules exhibit significant shortcomings in terms of accuracy and diversity. Specifically, more than  $50\%$  of the OSDAs generated by the pure GPT-4 model do not meet OSDA requirements. Although certain similarity metrics (e.g., Morgan) show relatively high scores, the generated molecules exhibit significant differences in their overall distribution compared to the OSDAs reported in the literature. This issue is clearly illustrated in Figures 15 in Appendix F, where the distributions of molecular volume, molecular weight, and asphericity are visualized. Notably, the molecular volume and weight of the purely GPT-4 generated molecules deviate substantially from those of the OSDAs found in the literature.
+
+Our OSDA Agent integrates the GPT-4 model with targeted prompt strategies, chemical validation tools, and a reflection mechanism, consistently producing OSDAs with higher chemical relevance and synthesizability. The inclusion of the SCScore model and RDKit for chemical rule validation allows our framework to filter out unfeasible molecules early in the process. Additionally, the reflection mechanism iteratively refines the generated OSDAs, enhancing their alignment with known experimental data and synthesis routes. This capability enables the OSDA Agent to accurately generate molecules that conform to OSDA rules and exhibit greater similarity to those found in the literature. As shown in Table 1, our OSDA Agent outperforms other methods across the majority of similarity metrics. In terms of distribution, the OSDA molecules generated by the OSDA Agent exhibit lower KL divergence and Energy distance (ED) values. Therefore, we conclude that the OSDAs generated by the OSDA Agent are more representative of those found in real-world scenarios.
+
+# 5.3 RESULTS OF OPTIMIZING EXISTING OSDA MOLECULES
+
+In the previous sections, we have demonstrated the capability of our OSDA Agent in molecular design, and we hope that our approach can also be effective in optimizing existing OSDA molecules. To demonstrate the capability of our approach in refining complex and challenging synthesized molecules, we applied the model to several OSDAs previously reported in the literature. These
+
+Table 1: The performance of the OSDA Agent in the de novo design of OSDAs. The best scores are in bold, and the second-best scores are underlined.  
+
+<table><tr><td>Type</td><td>Method</td><td>Validity↑</td><td>BLEU ↑</td><td>Morgan↑</td><td>MACCS↑</td><td>RDK↑</td><td>ED↓</td><td>KL Divergence ↓</td></tr><tr><td rowspan="4">AFI</td><td>MolT5 (Edwards et al., 2022)</td><td>0.000</td><td>0.204</td><td>0.157</td><td>0.556</td><td>0.384</td><td>266.4</td><td>7.753</td></tr><tr><td>BioT5 (Pei et al., 2023)</td><td>0.008</td><td>0.193</td><td>0.147</td><td>0.384</td><td>0.162</td><td>120.7</td><td>3.128</td></tr><tr><td>GPT-4</td><td>0.440</td><td>0.554</td><td>0.332</td><td>0.750</td><td>0.591</td><td>4.061</td><td>0.653</td></tr><tr><td>OSDA Agent</td><td>1.000</td><td>0.577</td><td>0.308</td><td>0.781</td><td>0.608</td><td>2.567</td><td>0.588</td></tr><tr><td rowspan="4">LTA</td><td>MolT5 (Edwards et al., 2022)</td><td>0.000</td><td>0.266</td><td>0.151</td><td>0.520</td><td>0.297</td><td>189.4</td><td>2.985</td></tr><tr><td>BioT5 (Pei et al., 2023)</td><td>0.005</td><td>0.228</td><td>0.130</td><td>0.431</td><td>0.337</td><td>73.43</td><td>1.093</td></tr><tr><td>GPT-4</td><td>0.471</td><td>0.382</td><td>0.213</td><td>0.677</td><td>0.360</td><td>12.80</td><td>0.976</td></tr><tr><td>OSDA Agent</td><td>1.000</td><td>0.499</td><td>0.229</td><td>0.737</td><td>0.412</td><td>2.084</td><td>0.744</td></tr></table>
+
+molecules, selected from the OSDB database, were chosen specifically for their complexity and synthesis difficulty. The goal was to enhance synthetic feasibility while maintaining low binding energies.
+
+We tested the ability of our OSDA Agent to optimize existing OSDA molecules for AFI framework. We selected ten OSDA molecules from the OSDB database with relatively high SCScore, averaging around 3.45. Using our approach, we modified the prompt to "Can you start with this molecule and optimize this OSDA molecule...?" and set the reflection steps to 4 for each molecule. To ensure the reliability of the results, we repeated the optimization process five times for each of the molecules.
+
+We calculated the average SCScore and binding energy across the five experiments, as shown in Figure 5. The results indicate that the average SCScore decreased from 3.45 to 2.46, suggesting a lower synthesis difficulty. The optimized binding energy averaged -6.71 kcal/mol, which remains within the reasonable range (-3.38 kcal/mol to -9.00 kcal/mol)
+
+![](images/2a949a12589b309120d62a09c53eb9dfc021e382f7d10256f9354d23d62b8eeb.jpg)  
+Figure 5: During the molecular optimization process, the SCScore and Binding Energy of the OSDA molecules varied with the number of reflection iterations.
+
+based on the Jensen dataset. Additionally, the optimized molecules maintained structural similarity to the originals, preserving the functional properties necessary for zeolite templating. Detailed optimization results can be found in Appendix F, Figure 17.
+
+The results indicate that our OSDA Agent successfully generated optimized versions of these molecules, reducing the SCScore and thereby lowering the synthesis difficulty, while preserving the functional characteristics necessary for zeolite templating. This demonstrates that the OSDA Agent has the potential not only to design new molecules but also to improve existing ones.
+
+# 6 CONCLUSION
+
+In this work, we introduced the OSDA Agent, which effectively designs and optimizes OSDAs through iterative refinement with targeted prompts. Applied to small and large pore frameworks, including LTA and AFI, the OSDA Agent generated chemically valid and structurally compatible molecules. Comparisons with literature-reported OSDAs revealed competitive, and in some cases superior, designs. These results highlight the OSDA Agent's potential for molecular design and optimization. Future work will focus on expanding the applicability of our methods to a broader range of scientific fields and further integrating experimental constraints into the optimization process.
+
+# ACKNOWLEDGMENTS
+
+This work are supported by National Science and Technology Major Project (2023ZD0120801), National Natural Science Foundation of China (62472381) and Fundamental Research Funds for the Zhejiang Provincial Universities (226-2024-00208).
+
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+
+# A LINK TO DATASETS AND MODELS
+
+Jensen dataset: https://github.com/olivettigroup/OSDA_Generator
+
+OSDB database: https://zeodb.mit.edu/index
+
+Pre-trained SMILES-BERT: https://huggingface.co/unikei/bert-base-smiles
+
+GPT-4: https://openai.com/chatgpt/
+
+VOID: https://github.com/learningmatter-mit/VOID
+
+Mistral-Nemo-Instruct-2407:https://huggingface.co/mistralai/Mistral-Nemo-Instruct-2407
+
+Llama-3.1-70B-Instruct:https://huggingface.co/meta-llama/Llama-3.1-70B-Instruct
+
+# B SUPPLEMENTING THE BACKGROUND AND MOTIVATION
+
+# The Importance of OSDA in Zeolite Synthesis
+
+In zeolite synthesis, Organic Structure-Directing Agents (OSDAs) are essential in determining the final framework structure and morphology of the material. Zeolites are microporous materials with a wide range of applications in catalysis, adsorption, and ion exchange. The specific properties of zeolites, such as pore size and shape, depend on their crystalline structure, which is directly influenced by the presence of OSDAs. These organic molecules act as templates during the synthesis process, guiding the formation of the zeolite framework and ensuring the desired structural arrangement.
+
+OSDAs not only direct the crystallization of zeolite but also contribute to the stability of the resulting material. In some cases, OSDAs function as charge-balancing agents, helping to maintain the integrity of the aluminosilicate framework during synthesis. Without the appropriate OSDA, zeolite formation may either fail or occur or result in amorphous materials with poorly defined structures. Therefore, the choice of OSDA is critical in controlling the final zeolite structure, affecting both the framework topology and the material's performance.
+
+# Why do the design of OSDAs currently present significant challenges?
+
+OSDA plays a crucial role in determining the topology of zeolites, especially in the formation of their porous structures. Traditional methods mainly rely on two approaches: first, using empirical experience and experimentation to search for feasible OSDA molecules within a molecular library; and second, employing computational simulations of zeolite-OSDA interactions (such as Density Functional Theory, DFT) to assist in OSDA design. Both experience-based and simulation-based traditional methods are highly time-consuming and resource-intensive. In particular, the vast chemical space of OSDA (small molecules) presents significant challenges in discovering new OSDA candidates using conventional methods.
+
+Moreover, due to the difficulty in finding new OSDA molecules, the existing pool of OSDA candidates is very limited. For example, the Jensen dataset, which collects data from papers published between 1966 and 2020, contains only 758 different OSDA molecules. Traditional machine learning generation methods typically require large datasets, but the small number of OSDA molecules makes it challenging to train effective generative models. Several traditional generative model approaches (Jensen et al., 2021; Xu et al., 2023) are typically trained using synthetic routes. This means that when designing OSDAs, information related to the synthetic pathway, such as gel chemistry, must be provided. Moreover, the limited number of different OSDA molecules in the dataset negatively impacts the model's ability to search effectively within the chemical space. Currently, Large Language Models (LLMs) have broad knowledge across related fields, and we aim to leverage the domain expertise of LLMs to overcome the data scarcity issue.
+
+# Why is an iterative, feedback-based approach employed in this context?
+
+Due to the phenomenon of "hallucination" in large language models (LLMs), especially when performing complex tasks like molecular design, it is necessary to introduce additional chemical knowledge to help the LLMs accomplish the task more effectively. Through an interactive, feedback-driven approach, we professionally evaluate the molecules designed by the LLM, including factors
+
+such as OSDA empirical rules, molecular synthesis difficulty, safety, etc., and provide feedback to the LLM. This process is similar to how chemists validate through experiments and make improvements based on the experimental results.
+
+# C EXPERIMENTAL DETAILS AND METRICS
+
+We outline the metrics employed to evaluate the performance of the generative models in our experiments, encompassing:
+
+Validity. The proportion of generated molecules that follow the empirical rules for OSDAs can be expressed as:
+
+$$
+V = \frac {N _ {\text {v a l i d}}}{N _ {\text {t o t a l}}},
+$$
+
+where  $N_{\mathrm{valid}}$  is the number of valid molecules and  $N_{\mathrm{total}}$  is the total number of generated molecules (see Table 2).
+
+BLEU. To assess the difference between the generated molecules and OSDAs reported in the literature, the evaluation is conducted by calculating the average BLEU score  $B$  between the SMILES representations of the generated molecules and those of the closest OSDAs documented in the literature:
+
+$$
+B = \frac {1}{N} \sum_ {i = 1} ^ {N} \mathrm {B L E U} (S _ {\mathrm {g e n}, i}, S _ {\mathrm {O S D A}, i ^ {*}}),
+$$
+
+where  $N$  is the number of generated molecules,  $S_{\mathrm{gen},i}$  is the SMILES representation of the  $i$ -th generated molecule, and  $S_{\mathrm{OSDA},i^*} = S_{\mathrm{OSDA},j^*}$  with
+
+$$
+j ^ {*} = \arg \max  _ {j} \mathrm {B L E U} (S _ {\mathrm {g e n}, i}, S _ {\mathrm {O S D A}, j}),
+$$
+
+where  $j$  iterates over all OSDAs, ensuring  $S_{\mathrm{OSDA},i^*}$  is the SMILES representation of the OSDA most similar to the  $i$ -th generated molecule.
+
+Morgan. Morgan fingerprints, which generate feature vectors by calculating the local atomic environments in molecules, are used here to measure the difference between the generated molecules and the OSDAs reported in the literature. Specifically, this is done by calculating the similarity  $S_{\text{Morgan}}$  between each generated molecule and its closest OSDA documented in the literature, followed by averaging these scores:
+
+$$
+S _ {M o r g a n} = \frac {1}{N} \sum_ {i = 1} ^ {N} \operatorname {s i m} \left(F _ {\text {g e n}, i}, F _ {\text {O S D A}, i ^ {*}}\right),
+$$
+
+where  $N$  is the number of generated molecules,  $F_{\mathrm{gen},i}$  is the Morgan fingerprint of the  $i$ -th generated molecule, and  $F_{\mathrm{OSDA},i^*} = F_{\mathrm{OSDA},j^*}$  with
+
+$$
+j ^ {*} = \arg \max  _ {j} \operatorname {s i m} (F _ {\text {g e n}, i}, F _ {\text {O S D A}, j}),
+$$
+
+where  $j$  iterates over all OSDAs, ensuring  $F_{\mathrm{OSDA},i^{*}}$  is the Morgan fingerprint of the OSDA most similar to the  $i$ -th generated molecule.
+
+MACCS. MACCS fingerprints generate a fixed-length bit vector based on a predefined set of structural subgraphs or chemical features, with each bit representing the presence or absence of a specific chemical feature. Here, we use MACCS fingerprints to evaluate the difference between the generated molecules and the OSDAs reported in the literature. Specifically, this is done by calculating the
+
+average similarity  $S_{MACCS}$  between the MACCS fingerprints of the generated molecules and those of the closest OSDAs documented in the literature:
+
+$$
+S _ {M A C C S} = \frac {1}{N} \sum_ {i = 1} ^ {N} \operatorname {s i m} \left(F _ {\text {g e n}, i}, F _ {\text {O S D A}, i ^ {*}}\right),
+$$
+
+where  $N$  is the number of generated molecules,  $F_{\mathrm{gen},i}$  is the MACCS fingerprint of the  $i$ -th generated molecule, and  $F_{\mathrm{OSDA},i^*} = F_{\mathrm{OSDA},j^*}$  with:
+
+$$
+j ^ {*} = \operatorname * {a r g   m a x} _ {j} \operatorname * {s i m} (F _ {\text {g e n}, i}, F _ {\text {O S D A}, j}),
+$$
+
+with  $j$  iterating over all OSDAs, ensuring  $F_{\mathrm{OSDA},i^*}$  is the MACCS fingerprint of the OSDA most similar to the  $i$ -th generated molecule.
+
+RDK. RDK fingerprints generate feature bit vectors by traversing the bond paths in a molecule. Here, we use RDK fingerprints to evaluate the difference between the generated molecules and the OSDAs reported in the literature. Specifically, this is done by calculating the average similarity  $S_{RDK}$  between the RDK fingerprints of the generated molecules and those of the closest OSDAs documented in the literature:
+
+$$
+S _ {R D K} = \frac {1}{N} \sum_ {i = 1} ^ {N} \operatorname {s i m} \left(F _ {\text {g e n}, i}, F _ {\text {O S D A}, i ^ {*}}\right),
+$$
+
+where  $N$  is the number of generated molecules,  $F_{\mathrm{gen},i}$  is the RDK fingerprint of the  $i$ -th generated molecule, and  $F_{\mathrm{OSDA},i^*} = F_{\mathrm{OSDA},j^*}$  with:
+
+$$
+j ^ {*} = \arg \max  _ {j} \operatorname {s i m} (F _ {\text {g e n}, i}, F _ {\text {O S D A}, j}),
+$$
+
+with  $j$  iterating over all OSDAs, ensuring  $F_{\mathrm{OSDA},i^*}$  is the RDK fingerprint of the OSDA most similar to the  $i$ -th generated molecule.
+
+WHIM.WHIM descriptors are a class of molecular descriptors used to characterize the three-dimensional structure of molecules by considering global properties such as molecular geometry, mass distribution, polarity, and electron density. Here, we use WHIM descriptors to calculate the difference between the distribution of the generated molecules and the distribution of OSDAs reported in the literature.WHIM descriptors capture information about a molecule's three-dimensional conformation, including its size, shape, symmetry, and atomic distribution. Due to molecular flexibility, different conformations can result in significantly different WHIM representations. For example, a long linear molecule can either stretch out or fold, leading to two distinct 3D representations. To address this challenge, we calculated the average conformation WHIM descriptors using the geometries obtained from RDKit (Landrum, 2013). This approach allows us to capture the diverse 3D characteristics of each molecule across its various conformations.
+
+Energy Distance(ED). Energy Distance is a statistical measure used to quantify the difference between two probability distributions. It is based on the concept of energy statistics and captures the distance between distributions in terms of expected Euclidean distances between samples. Energy Distance is symmetric and particularly sensitive to both location and shape differences between distributions. In this study, we compute the Energy Distance between the distribution of the generated molecules and the distribution of OSDAs reported in the literature, using the WHIM descriptors. The formula for Energy Distance is given by:
+
+$$
+\operatorname {E D} (X, Y) = 2 \cdot \mathbb {E} [ d (X, Y) ] - \mathbb {E} [ d (X, X ^ {\prime}) ] - \mathbb {E} [ d (Y, Y ^ {\prime}) ],
+$$
+
+where  $X$  and  $Y$  represent independent samples from the generated molecules and OSDA distributions, respectively. Here,  $d(X,Y)$  denotes the Euclidean distance between samples, and  $\mathbb{E}$  denotes the expectation. The Energy Distance captures both the mean differences between the distributions and their internal variability, making it a robust measure for comparing the overall structure of the molecular distributions.
+
+KL-divergence. Kullback-Leibler (KL) divergence is an asymmetric measure used to quantify the difference between two probability distributions. It measures the information loss or gain when one distribution is compared to another. Here, we use the WHIM descriptors of the molecules, after dimensionality reduction via PCA, to calculate the difference between the distribution of the generated molecules and the distribution of OSDAs reported in the literature. The formula for KL-divergence is:
+
+$$
+D _ {\mathrm {K L}} (P \| Q) = \int P (x) \log \frac {P (x)}{Q (x)} d x,
+$$
+
+where  $P(x)$  is the probability density function of the generated molecules' distribution, and  $Q(x)$  is the probability density function of the OSDAs' distribution. The KL-divergence quantifies the amount of information lost when  $Q$  is used to approximate  $P$ , with a larger value indicating greater divergence between the two distributions.
+
+![](images/f08d9ac8dbc816c8aff52c1cabc04380f6a6d891b8911decec9ab9549d8d789a.jpg)  
+Scree Plot of PCA  
+Figure 6: Scree plot of PCA on WHIM descriptors.
+
+Table 2: Screening criteria  
+
+<table><tr><td>Property</td><td>Criteria</td></tr><tr><td>Number of rotatable bonds</td><td>Less than 5</td></tr><tr><td>Size of rings without bridge bonds</td><td>Smaller than 9-membered rings</td></tr><tr><td>Size of smallest rings</td><td>Larger than 4-membered rings</td></tr><tr><td>Distortion of N-rings</td><td>Must not contain triple bonds in N-rings</td></tr><tr><td>C to N ratio</td><td>More than 4 and less than 20</td></tr><tr><td rowspan="2">Presence of NH+and NH3+group</td><td>Must not be contained</td></tr><tr><td>Must not contain these atoms and bonds</td></tr><tr><td>Presence of unfavorable atoms and bonds</td><td>(O, Si, P, S, B, F, -O-C-O-, -C=O, and N-N)</td></tr></table>
+
+# D USING THE FEW-SHOT COT (CHAIN OF THOUGHT) PROMPT ENGINEERING METHOD FOR OSDA MOLECULE DESIGN CASES
+
+In this section, we present the complete prompt that we used.
+
+# Chain of Thought
+
+Let's think step by step to design an effective OSDA molecule for a given zeolite structure. We will follow these steps:
+
+(1)Analyze the Target Zeolite Structure: Begin by reviewing the key features of the target zeolite, including its pore size, topology, and framework type. Consider how these structural characteristics might influence the binding and stability of potential OSDA molecules.  
+(2)Identify Functional Groups: Based on the analysis of the zeolite structure, identify functional groups that are likely to enhance the interaction with the zeolite framework, Consider groups that can fit well within the pores and stabilize the structure.  
+(3)Analyze Example Molecules: I need to analyze the given examples to understand the relationship between the SMILES structures and their binding energies and synthesizeability scores. This analysis will help guide the selection of structural features in the design of new OSDA molecules.  
+(4)Generate SMILES for OSDA Candidates: Using the identified functional groups, construct a series of SMILES representations for potential OSDA molecules. Ensure that these molecules are chemically valid and capable of directing the assembly of the zeolite structure.
+
+Figure 7: Chain of Thought prompts in OSDA design.
+
+# Chain of Thought prompts
+
+Relevant Context: You are a chemist tasked with designing new Organic Structure Directing Agents (OSDAs) for AFI Zeolite. Your goal is to generate OSDAs with specific binding energies and synthesizeability scores. Below are several examples, including the SMILES strings of the OSDAs, their binding energies, and synthesizability scores.
+
+For instance, the SMILES string C[C@@H]1CCC[N+]12CCCCC2 has a binding energy of -7.101 kcal/mol and a synthesizeability score of 1.927.
+
+Another example, NCCCNCCNCCCN, has a binding energy of -7.500 kcal/mol and a synthesizeability score of 1.818.
+
+The SMILES string C[N+]1(C)CC[C@@H]2CCCC[C@@H]2CC1 has a binding energy of -6.961 kcal/mol and a synthesisizability score of 1.879.
+
+Finally, the structure C[C@@H]1[C@@H]2CC[C@@](C)(CC[N+]2(C)C)[C@@H]1C has a binding energy of -8.199 kcal/mol and a synthesizeability score of 2.544. Lower binding energies and synthesizeability scores are preferred.
+
+Question: Can you design a new OSDA of AFI Zeolite and provide its SMILES string? You just need to provide SMILES.
+
+Thought: Let's think step by step to design an effective OSDA molecule for a given zeolite structure. We will follow these steps:
+
+(1) Analyze the Target Zeolite Structure: Begin by reviewing the key features of the target zeolite, including its pore size, topology, and framework type. Consider how these structural characteristics might influence the binding and stability of potential OSDA molecules.  
+(2) Identify Functional Groups: Based on the analysis of the zeolite structure, identify functional groups that are likely to enhance the interaction with the zeolite framework. Consider groups that can fit well within the pores and stabilize the structure.  
+(3) Analyze Example Molecules: I need to analyze the given examples to understand the relationship between the SMILES structures, their binding energies, and synthesizeability scores. This analysis will help guide the selection of structural features in the design of new OSDA molecules.  
+(4) Generate SMILES for OSDA Candidates: Using the identified functional groups, construct a series of SMILES representations for potential OSDA molecules. Ensure that these molecules are chemically valid and capable of directing the assembly of the zeolite structure.
+
+Action: Finish[Cc1cc(C)[n+](C)n1C]
+
+Figure 8: Details of the few-shot CoT (Chain of Thought) prompts. The data in the Relevant Context section comes from the OSDB database.
+
+# Reflection prompts
+
+Relevant Context: You are a chemist tasked with designing new Organic Structure Directing Agents (OSDAs) for AFI Zeolite. Your goal is to generate OSDAs with specific binding energies and synthesizability scores. Below are several examples, including the SMILES strings of the OSDAs, their binding energies, and synthesizability scores.
+
+For instance, the SMILES string C[C@@H]1CCC[N+]12CCCCC2 has a binding energy of -7.101 kcal/mol and a synthesisizability score of 1.927.
+
+Another example, NCCCNCCNCCN, has a binding energy of -7.500 kcal/mol and a synthesizeability score of 1.818.
+
+The SMILES string C[N+]1(C)CC[C@@H]2CCCCC[C@@H]2CC1 has a binding energy of -6.961 kcal/mol and a synthesizeability score of 1.879.
+
+Finally, the structure C[C@@H]1[C@@H]2CC[C@@](C)(CC[N+]2(C)C)[C@@H]1C has a binding energy of -8.199 kcal/mol and a synthesizeability score of 2.544. Lower binding energies and synthesizeability scores are preferred.
+
+Question: Can you design a new OSDA of AFI Zeolite and provide its SMILES string? You just need to provide SMILES.
+
+Thought: Let's think step by step to design an effective OSDA molecule for a given zeolite structure. We will follow these steps:
+
+(1) Analyze the Target Zeolite Structure: Begin by reviewing the key features of the target zeolite, including its pore size, topology, and framework type. Consider how these structural characteristics might influence the binding and stability of potential OSDA molecules.  
+(2) Identify Functional Groups: Based on the analysis of the zeolite structure, identify functional groups that are likely to enhance the interaction with the zeolite framework. Consider groups that can fit well within the pores and stabilize the structure.  
+(3) Analyze Example Molecules: I need to analyze the given examples to understand the relationship between the SMILES structures, their binding energies, and synthesizability scores. This analysis will help guide the selection of structural features in the design of new OSDA molecules.  
+(4) Generate SMILES for OSDA Candidates: Using the identified functional groups, construct a series of SMILES representations for potential OSDA molecules. Ensure that these molecules are chemically valid and capable of directing the assembly of the zeolite structure.
+
+Action: Finish[Cc1cc(C)[n+](C)n1N]
+
+Reflection: My answer included a SMILES string that was not in the correct format. In the future, I should ensure that the SMILES strings are accurately represented and conform to standard notation to avoid similar issues.
+
+Figure 9: Details of the reflection prompts. The data in the Relevant Context section comes from the OSDB database.
+
+# Self-reflection
+
+You have attempted to answer following question before and failed. The following reflection(s) give a plan to avoid failing to answer the question in the same way you did previously. Use them to improve your strategy of correctly answering the given question.
+
+# Reflections:
+
+-My previous attempt to design a new OSDA for AFI zeolite failed because the synthesizeability score of the generated SMILES string was too high. This indicates that the molecule was too difficult to synthesize. Additionally, the binding energy was not optimized to be as low as possible.
+
+-My previous attempt failed because the generated SMILES string did not meet the empirical characteristics of OSDA molecules, specifically the nitrogen-to-carbon ratio and the structural requirements for rings and rotatable bonds.
+
+# Actor
+
+# Thought:
+
+Based on the reflections and the given context, it's clear that the previous attempts failed due to high synthesizability scores and not optimizing the binding energy. To avoid these issues, I need to design a molecule that is simpler to synthesize and has a lower binding energy. The given examples show that molecules with nitrogen and carbon atoms, as well as ring structures, tend to have lower binding energies and synthesizability scores. Therefore, I should aim to design a molecule with these characteristics. Additionally, I should aim to keep the molecule as simple as possible to ensure a lower synthesizability score.
+
+Figure 10: Example of the reflection.
+
+# E BINDING ENERGY ESTIMATION MODEL DETAILS
+
+This section covers the details and results of the Binding Energy Estimation Model, including our ablation study on the model.
+
+The OSDA-Zeolite complexes in the real world are quite complex, as OSDAs may adopt multiple conformations when interacting with zeolites, resulting in complexes that do not exhibit the strict periodicity of typical crystals. However, both the complexes from the OSDB database and those generated from voids are stored as CIF files and treated as unit cells, as illustrated in Figure 11. Therefore, we continue to use crystal structures and periodic graph neural networks (CGCNN) for modeling in this context.
+
+![](images/c221b2bf4ad927109b67ff2e7ea71bee0ee4091e14e04a98f2a0b51a8e8afed8.jpg)  
+(a) AFI
+
+![](images/666666c69dbb40aeb608944e02519c9c1a61eb2d6ef29f1ee2f5a6f07ddc7b39.jpg)  
+(b) CCN(CC)CC-AFI Complex  
+Figure 11: Dock illustration. (a) An AFI framework zeolite. (b) The combined complex with the 'CCN(CC)CC' OSDA.
+
+Table 3: Binding energy prediction error (MAE) on the OSDB database.  
+
+<table><tr><td>Model</td><td>Binding Energy (kJ/mol Si) (MAE ↓)</td></tr><tr><td>full model</td><td>0.384</td></tr><tr><td>Remove Zeolite encoder</td><td>0.402</td></tr><tr><td>Remove Smiles encoder</td><td>0.411</td></tr><tr><td>Only Complex encoder</td><td>0.469</td></tr></table>
+
+![](images/1bb414f5f9001373c3bf3a8a9d9009780ba24547daec3c8b7776856737273d84.jpg)  
+Figure 12: The results of the Binding Energy Estimation model on the test set.
+
+![](images/226635207d3f1b0057410bc42de5344b9ceedfb0c73d632b58907ce47ffa76e9.jpg)  
+Figure 13: Model loss trends.
+
+# F SUPPLEMENTARY EXPERIMENTAL RESULTS
+
+This section primarily provides supplementary results from the experiments.
+
+![](images/b8174b43874c87a490234465341dbd45f1d69b8b2b7e4f4546d94b2daea4e431.jpg)  
+Figure 14: Principal Component Analysis (PCA) of WHIM vector representation for OSDA molecules used in LTA framework. PCA 1, 2, and 3 represent the first three principal axes. Red points show OSDAs generated by the OSDA Agent, blue points represent OSDAs of LTA framework from the literature, and gray points represent all OSDAs extracted from the literature.
+
+![](images/3675fbeddf8e934c18047b8a5879f396d24aaad12d3cc601a10d5714e93c6c8a.jpg)  
+Figure 15: The distribution of molecular volume, molecular weight, and asphericity is compared between the OSDAs reported in the literature and those generated by the OSDA Agent in conjunction with GPT-4.
+
+![](images/ec5e0746e6ffa7c3d823d8cfcb01e1b057d8e16db586abdfb2ec88b7dcd882d0.jpg)
+
+![](images/2fd73e343cf81a4cce411d1df1067423cc0f01fe2c825602bd8f0a12ab07ab6c.jpg)  
+Figure 16: The distribution of molecular volume, molecular weight, and asphericity is compared between the OSDAs reported in the literature and those generated by the OSDA Agent in conjunction with GPT-4, BioT5, MolT5.
+
+![](images/8ffabf2be2ac39dc396b7e80cdf4d35e06d157f37fa67ec8aab2e9474df910c8.jpg)  
+Figure 17: The optimization results of OSDA molecules are presented, with the OSDA molecules from the OSDB database on the left and the molecules optimized by the OSDA Agent on the right.
+
+# G ABLATION STUDY
+
+In this section, we perform an ablation study on the proposed components, focusing on the reflection mechanism and the individual components within the Evaluator. To minimize randomness as much as possible, we use 15 identical sets of prompts to design OSDA molecules for AFI framework. The results obtained are as follows:
+
+Table 4: Performance metrics for the ablation study of different methods.  
+
+<table><tr><td>Method</td><td>Validity ↑</td><td>BLEU ↑</td><td>Morgan↑</td><td>MACCS ↑</td><td>RDK ↑</td><td>ED ↓</td><td>KL Divergence ↓</td><td>Avg Rank</td></tr><tr><td>OSDA Agent</td><td>1.000</td><td>0.601</td><td>0.368</td><td>0.816</td><td>0.623</td><td>0.934</td><td>0.825</td><td>1.28</td></tr><tr><td>w/o reflection mechanism</td><td>0.702</td><td>0.581</td><td>0.331</td><td>0.782</td><td>0.553</td><td>1.359</td><td>0.973</td><td>4.75</td></tr><tr><td>w/o RDKit</td><td>0.770</td><td>0.593</td><td>0.355</td><td>0.751</td><td>0.566</td><td>1.233</td><td>0.830</td><td>3.42</td></tr><tr><td>w/o SCSore</td><td>1.000</td><td>0.570</td><td>0.371</td><td>0.802</td><td>0.614</td><td>1.256</td><td>1.001</td><td>2.85</td></tr><tr><td>w/o blending energy</td><td>1.000</td><td>0.627</td><td>0.356</td><td>0.787</td><td>0.619</td><td>1.275</td><td>0.972</td><td>2.42</td></tr></table>
+
+Here, we observe that removing the reflection mechanism or any individual component within the Evaluator leads to worse results.
+
+# H EVALUATION OF ALTERNATIVE LLM COMPONENTS
+
+In this section, we explore the use of alternative large language models (LLMs) and compare their performance. Specifically, we tested two popular open-source LLMs, Mistral (Jiang et al., 2023) and LLaMA 3.1(Dubey et al., 2024), and observed that our method also resulted in substantial improvements in their ability to design OSDAs:
+
+Table 5: Performance metrics for different methods.  
+
+<table><tr><td>Method</td><td>BLEU ↑</td><td>Morgan ↑</td><td>MACCS ↑</td><td>RDK ↑</td></tr><tr><td>OSDA Agent</td><td>0.601</td><td>0.368</td><td>0.816</td><td>0.624</td></tr><tr><td>OSDA Agent*</td><td>0.571</td><td>0.317</td><td>0.772</td><td>0.601</td></tr><tr><td>llama</td><td>0.522</td><td>0.301</td><td>0.628</td><td>0.416</td></tr><tr><td>OSDA Agent(llama)</td><td>0.551</td><td>0.315</td><td>0.754</td><td>0.565</td></tr><tr><td>Mistral</td><td>0.338</td><td>0.177</td><td>0.411</td><td>0.224</td></tr><tr><td>OSDA Agent(Mistral)</td><td>0.512</td><td>0.306</td><td>0.740</td><td>0.541</td></tr></table>
+
+The OSDA Agent is our default model. The OSDA Agent* replaces the actor with GPT-4o while the OSDA Agent (Llama) is fully built on Llama, the OSDA Agent (Mistra) is fully built on Mistra. Our experimental results show that, regardless of the model used, our OSDA Agent significantly enhances the overall design results. However, currently, its performance is still slightly inferior to GPT-4, which remains the most effective model for this specific task.
+
+# I EXAMPLES OF OSDAS FOR AFI, AEI, LEV, AFX, MOR, LTA AND ITE ZEOLITES GENERATED USING THE OSDA AGENT
+
+We used the OSDA Agent to generate OSDA molecules for seven types of zeolites, including AFI, AEI, LEV, AFX, MOR, LTA, and ITE. After consulting experts in materials and chemistry, they confirmed that the generated molecules have potential as OSDAs.
+
+Table 6: Overview of the zeolites utilized.  
+
+<table><tr><td>Framework Type</td><td>Crystal system</td><td>Length a (Å)</td><td>Length b (Å)</td><td>Length c (Å)</td><td>Angle α (°)</td><td>Angle β (°)</td><td>Angle γ (°)</td><td>pore size (Å)</td></tr><tr><td>LTA</td><td>Cubic</td><td>11.92</td><td>11.92</td><td>11.92</td><td>90</td><td>90</td><td>90</td><td>4.21</td></tr><tr><td>AFI</td><td>Hexagonal</td><td>13.83</td><td>13.83</td><td>8.58</td><td>90</td><td>90</td><td>120</td><td>7.42</td></tr><tr><td>ITE</td><td>Orthorhombic</td><td>20.75</td><td>9.80</td><td>20.01</td><td>90</td><td>90</td><td>90</td><td>4.21</td></tr><tr><td>LEV</td><td>Trigonal</td><td>13.17</td><td>13.17</td><td>22.58</td><td>90</td><td>90</td><td>120</td><td>3.53</td></tr><tr><td>MOR</td><td>Orthorhombic</td><td>18.26</td><td>20.53</td><td>7.54</td><td>90</td><td>90</td><td>90</td><td>6.45</td></tr><tr><td>AFX</td><td>Hexagonal</td><td>13.67</td><td>13.67</td><td>19.70</td><td>90</td><td>90</td><td>120</td><td>3.73</td></tr><tr><td>AEI</td><td>Orthorhombic</td><td>13.68</td><td>12.61</td><td>18.50</td><td>90</td><td>90</td><td>90</td><td>3.84</td></tr></table>
+
+![](images/4f3a40170a288590388f35d3e0d5caef54c65643199afe4de8e50291af887997.jpg)  
+Figure 18: OSDAs for AFI framework
+
+![](images/c54460934093ba1a944bc2e917bd66aa0793ed66848d6f46c69ea1cd046c73a5.jpg)  
+Figure 19: OSDAs for AEI framework
+
+![](images/91d64d007615c99bca29d825a43bfd994a6ae8f3e4d1eeae21d11038f588ee75.jpg)  
+Figure 20: OSDAs for AFX framework
+
+![](images/9a2ac3e0df9c874bf9ad6d8b4263ecb5e2b9de3ff6a8358b57d9006223f0c34a.jpg)  
+Figure 21: OSDAs for ITE framework
+
+![](images/50821c0b82fd15fd807f3f8b3e4ed3e3507dfb5a6fce0ba2de854d2cb9eb92bb.jpg)  
+Figure 22: OSDAs for LEV framework
+
+![](images/44eb1439c2bdf7cdee28881d218d446a045685e32fc43ff72de6a9ad19aa5cfd.jpg)  
+Figure 23: OSDAs for LTA framework
+
+![](images/e16d8f1bdce7ecbeeafc010ac8035ce429ff25d119609b5bb09570a14404152b.jpg)  
+Figure 24: OSDAs for MOR framework
+
+# J DEFINITION AND EXPLANATION OF TERMS AND CONCEPTS IN THE PAPER
+
+# SMILES (Simplified Molecular Input Line Entry System)
+
+SMILES is a symbolic language used to represent the structure of chemical molecules. It encodes the atoms, bonds, and spatial arrangements of a molecule in a string of characters. SMILES is widely used in computational chemistry and AI for Science due to its simplicity in storing and exchanging chemical information and its ability to be parsed by computer programs.
+
+# In-Context Learning (ICL)
+
+In-Context Learning refers to the process where a pre-trained language model performs reasoning and generates responses based on provided examples or contextual information in the input. This is done without the need for additional model training or parameter updates, making it a form of immediate inference for a given task.
+
+# Chain of Thought
+
+Chain of Thought refers to a reasoning process where a model breaks down a complex problem into a sequence of logical steps to improve the accuracy and transparency of its reasoning. This approach helps in step-by-step problem-solving by guiding the model through intermediate steps.
+
+# RDKit
+
+RDKit is an open-source toolkit for cheminformatics that provides a wide range of functionalities for molecular manipulation, feature extraction, structure visualization, and drug design. It is widely used in chemistry and bioinformatics.
+
+# SCScore (Synthetic Complexity Score)
+
+SCScore is a metric used to assess the synthetic difficulty of a chemical molecule. It considers factors such as the number of synthetic steps, reagents, and reaction conditions required to synthesize the molecule, evaluating its complexity from laboratory synthesis to industrial production. A higher score indicates greater synthetic difficulty.
+
+# Zeolites
+
+Zeolites are a class of porous materials with a regular framework structure composed of silicon-oxygen  $(\mathrm{SiO}_4)$  and aluminum-oxygen  $(\mathrm{AlO}_4)$  tetrahedra linked by oxygen bridges. Their unique molecular sieve properties allow them to selectively adsorb and separate molecules based on size and shape. Zeolites are widely used in various industries for applications such as catalysis, adsorption, ion exchange, gas separation, and water treatment. Due to their tunable structure, the pore size and crystal morphology of zeolites can be optimized by using different organic structure-directing agents (OSDAs) to meet specific application requirements.
+
+# OSDA
+
+OSDA (Organic Structure-Directing Agent) refers to an organic compound, typically a quaternary ammonium salt or other organic molecules, that is used in the synthesis of zeolites to guide the formation of their crystalline structure. During the synthesis process, OSDAs interact with the aluminosilicate framework and help direct the arrangement of atoms, thereby influencing key characteristics of the zeolite such as pore size, pore shape, and crystal morphology. The use of different OSDAs allows for the tuning of the zeolite's properties to suit specific industrial applications, such as catalysis, adsorption, and gas separation.
+
+# VOID
+
+VOID (Voronoi Organic-Inorganic Docker) is a tool for molecular docking and materials design, especially for nanoporous materials. It uses Voronoi diagrams to identify docking sites in crystal structures and combines geometric fitness functions and Monte Carlo methods to optimize docking. Supporting batch docking with tensor operations for efficiency, VOID is widely used in materials science and molecular modeling.

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2025/OmniRe_ Omni Urban Scene Reconstruction/full.md

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+# OMNIRE: OMNI URBAN SCENE RECONSTRUCTION
+
+Ziyu Chen $^{1,6*}$  Jiawei Yang $^{6}$  Jiahui Huang $^{5}$  Riccardo de Lutio $^{5}$   
+Janick Martinez Esturo $^{5}$  Boris Ivanovic $^{5}$  Or Litany $^{2,5}$  Zan Gojcic $^{5}$   
+Sanja Fidler $^{3,5}$  Marco Pavone $^{4,5}$  Li Song $^{1}$  Yue Wang $^{5,6}$
+
+1Shanghai Jiao Tong University 2Technion 3University of Toronto 4Stanford University 5NVIDIA Research 6University of Southern California
+
+# ABSTRACT
+
+We introduce OmniRe, a comprehensive system for efficiently creating high-fidelity digital twins of dynamic real-world scenes from on-device logs. Recent methods using neural fields or Gaussian Splatting primarily focus on vehicles, hindering a holistic framework for all dynamic foregrounds demanded by downstream applications, e.g., the simulation of human behavior. OmniRe extends beyond vehicle modeling to enable accurate, full-length reconstruction of diverse dynamic objects in urban scenes. Our approach builds scene graphs on 3DGS and constructs multiple Gaussian representations in canonical spaces that model various dynamic actors, including vehicles, pedestrians, cyclists, and others. OmniRe allows holistically reconstructing any dynamic object in the scene, enabling advanced simulations ( $\sim60\mathrm{Hz}$ ) that include human-participated scenarios, such as pedestrian behavior simulation and human-vehicle interaction. This comprehensive simulation capability is unmatched by existing methods. Extensive evaluations on the Waymo dataset show that our approach outperforms prior state-of-the-art methods quantitatively and qualitatively by a large margin. We further extend our results to 5 additional popular driving datasets to demonstrate its generalizability on common urban scenes. Code and results are available at omnire.
+
+# 1 INTRODUCTION
+
+Creating photorealistic digital twins of 4D real-world is valuable for enabling high-fidelity simulation, robust algorithm training and evaluation. As autonomous driving algorithms increasingly adopt end-to-end models, the need for scalable and domain-gap-free simulation environments, where these systems can be evaluated in closed-loop, is becoming more evident. While traditional artist-generated assets are reaching their limits in scale, diversity, and realism, recent advances in data-driven methods offer a strong alternative by creating realistic digital twins directly from real-world sensor data. Indeed, neural radiance fields (NeRFs) (Mildenhall et al., 2020; Barron et al., 2021; Yang et al., 2023b; Guo et al., 2023; Yang et al., 2023a; Wu et al., 2023b) and Gaussian Splitting (GS) (Kerbl et al., 2023; Yan et al., 2024) have emerged as powerful tools for reconstructing 3D scenes with high levels of visual and geometric fidelity. However, accurately and holistically reconstructing dynamic urban scenes remains a significant challenge, especially due to the diverse dynamic actors and their complex rigid and non-rigid motions in real-world environments.
+
+Several works have already tried to tackle this challenge. Early methods typically ignore dynamic actors and reconstruct only static parts of the scene (Tancik et al., 2022; Martin-Brualla et al., 2021; Rematas et al., 2022; Guo et al., 2023). Subsequent works aim to reconstruct the dynamic scenes by either (i) modeling the scenes as a combination of a static and time-dependent dynamic neural field, where the static-dynamic decomposition is an emergent property (Yang et al., 2023a; Turki et al., 2023), or (ii) building a scene graph, in which dynamic actors and the static background are represented as nodes and reconstructed in their canonical frame. The nodes of the scene graph are connected with edges that encode relative transformation representing the motion of each actor through time (Ost et al., 2021; Kundu et al., 2022; Yang et al., 2023b; Wu et al., 2023b; Tondersski
+
+![](images/0c3a4c0cd8d3d331a8f1b3c9ef0d8dec5caa7a99c2640e0085eae88b48945d8b.jpg)
+
+![](images/434a5819753763c74171adab0fb7be682f4359b8463ad86b36064701f07dedf9.jpg)  
+(a) Scene Decomposition
+
+![](images/cab9eddbbaebeb8d063450d2ed0ea1b32a7bef27b66ecd8f0bb2862788e14523.jpg)
+
+![](images/91d4f36404ddd62815f0e3a73720db59e85cf0eccf8675fea8fc65f2a8ec0064.jpg)  
+(b) Corner Case Handling
+
+![](images/c817652272e3546e17e1ce480b9406923c64061cac962574209593f47668c292.jpg)  
+Figure 1: (a) Decomposition of different parts of a scene. (b) Out-of-distribution categories that are overlooked by previous methods can be accurately handled by OmniRe. (c) OmniRe enables diverse applications including vehicle editing (c1, c2), human-vehicle interaction (c3), human behavior simulation (c4), etc.
+
+![](images/bd52e9a72b11b2530a1a3ce932cec8c4773c3bdacc85d41af7f1b97877d49620.jpg)  
+(c) Scene Simulation
+
+![](images/0212de12a77b6178c3ac39a0425e88a27aaef0d4cca8e401bbcfb2ce25a5d871.jpg)
+
+![](images/18b7804af378dcb45b8a0ef0d3991c285a2a09d965ef2faa812831d522df3650.jpg)
+
+et al., 2024; Fischer et al., 2024b). But both approaches fall short of meeting the requirements for comprehensive and interactive digital twins: while providing a more general formulation, methods of (i) lack editability and cannot be directly controlled with classical behavior models. Previous methods following (ii) still focus primarily on vehicles, which can be represented as rigid bodies, thereby largely neglecting other vulnerable road users (VRUs) such as pedestrians and cyclists that are fundamental and critical in urban scene simulation.
+
+To fill this critical gap, our work aims to model all dynamic actors, including vehicles, pedestrians, and cyclists, and many others, in a manner that allows for interactive simulation. This leads to two primary challenges: (i) developing a unified approach for modeling diverse non-rigid dynamic actors, given the wide range of non-rigid categories in real-world scenes; (ii) giving specific focus on humans, as their behavior is critical for decision-making in scenarios like driving, where pedestrian actions directly impact safety. Thus, precise joint-level reconstruction (Lei et al., 2023; Jiang et al., 2022; Kocabas et al., 2024) is crucial for fine control of human behavior in the simulator. To address the specific challenge of modeling human actors, we must consider several additional complexities. First, in-the-wild scenarios present significant challenges for capturing human motion dynamics due to unfavorable sensor observations and cluttered environments with frequent occlusions (Wang et al., 2024; Yang et al., 2021; Wang et al., 2023). Furthermore, reconstructing high-fidelity human appearance from sparse sensor data beyond mere geometry dynamics adds additional complexity. Lastly, interactions with large equipment, such as wheelchairs or strollers, which cannot be represented by explicit templates (e.g., SMPL), further complicate both appearance and geometry modeling.
+
+To address these challenges, we propose an "omni" system capable of handling diverse actors for urban digital twins. Our method OmniRe efficiently reconstructs high-fidelity urban scenes that include static backgrounds, driving vehicles, and non-rigidly moving dynamic actors (see Fig. 1). Specifically, we construct a dynamic neural scene graph (Ost et al., 2021) based on 3D Gaussian Splatting (Kerbl et al., 2023), with dedicated Gaussian representations for different kinds of dynamic actors in their local canonical spaces. In our framework, backgrounds and vehicles are represented as static Gaussians, while vehicles undergo rigid body transformations to simulate their motion over time. For non-rigid actors, we incorporate the SMPL model to enable joint-level control for pedestrians using dynamic Gaussians, as SMPL provides a prior template geometry for 3DGS initialization and explicit control for modeling desired human behaviors, which is advantageous for downstream simulation applications. To extract SMPL parameters for human motion modeling, we
+
+designed a novel human body pose estimation pipeline dedicated to driving logs with multi-camera setups and severe in-the-wild occlusions. For other template-less dynamic actors, we propose a shared deformation field approach in a self-supervised manner. This framework enables a unified representation of all non-rigid categories and achieves specialized joint-level control for pedestrians. Thus, OmniRe allows for accurate representation and controllable reconstruction of most objects of interest in the scene. Notably, our representation is directly amenable to behavior and animation models that are commonly used in AV simulation (e.g., Fig. 1-(c)).
+
+To summarize, we make the following contributions:
+
+- We introduce OmniRe, a holistic framework for dynamic urban scene reconstruction that embodies the "omni" principle of dynamic category coverage and representation flexibility. OmniRe leverages dynamic neural scene graphs based on Gaussian representations to unify the reconstruction of static backgrounds, driving vehicles, and non-rigidly moving dynamic actors (§ 4). It enables high-fidelity scene reconstruction and human-centered simulations, such as pedestrian behavior and human-vehicle interaction—capabilities unmatched by existing methods. (§ 5).  
+- We address the challenges of modeling humans and other dynamic actors from driving logs such as occlusion, cluttered environments, and the limitations of existing human pose prediction models (§ 4.2). We demonstrate our method on six popular driving datasets to show its generalizability in urban driving scenes (project page). While our findings are based on AV scenarios, they can generalize to other domains.  
+- We perform extensive experiments and ablations to demonstrate the benefits of our holistic framework. OmniRe achieves state-of-the-art performance in scene reconstruction and novel view synthesis (NVS), significantly outperforming previous methods in terms of full image metrics (+1.88 PSNR for reconstruction and +2.38 PNSR for NVS). The differences are pronounced for dynamic actors, such as vehicles (+1.18 PSNR), and humans (+4.09 PSNR for reconstruction and +3.06 PSNR for NVS) (Tab. 1).
+
+# 2 RELATED WORK
+
+Dynamic Scene Modeling. Neural representations are dominating novel view synthesis (Mildenhall et al., 2020; Barron et al., 2022; 2021; Müller et al., 2022; Fridovich-Keil et al., 2022; Kerbl et al., 2023). These have been extended in different ways to enable dynamic scene reconstruction. Deformation-based approaches (Pumarola et al., 2020; Park et al., 2021a; Tretschk et al., 2021; Park et al., 2021b; Cai et al., 2022) and recently DeformableGS (Yang et al., 2023c) and (Wu et al., 2023a) propose to model dynamic scenes using a 3D neural representation for the canonical space, coupled with a deformation network mapping time-dependent observations to canonical deformations. These are generally limited to small scenes with limited movement, making them inadequate for challenging urban dynamic scenes. Modulation-based techniques operate by directly feeding the image timestamps (or latent codes) as an additional input to a neural representation (Xian et al., 2021; Li et al., 2021; Luiten et al., 2024). However, this generally results in an underconstrained formulation, therefore requiring additional supervision, such as depth (Li et al., 2021) and optical flow (Xian et al., 2021), or multi-view inputs captured from synchronized cameras (Li et al., 2022; Luiten et al., 2024).  $\mathrm{D}^2\mathrm{NeRF}$  (Wu et al., 2022) proposed to expand on this formulation by partitioning the scene into static and dynamic fields. Following this, SUDS (Turki et al., 2023) and EmerNeRF (Yang et al., 2023a) have shown impressive reconstruction ability for dynamic autonomous driving scenes. However, they model all dynamic elements using a single dynamic field, rather than modeling each separately, thus they lack controllability, limiting their practicality as sensor simulators. Explicit decomposition of the scene into separate agents enables controlling them individually. These agents can be represented as bounding boxes in a scene graph as in Neural Scene Graphs (NSG) (Ost et al., 2021) that is widely adopted in UniSim (Yang et al., 2023b), MARS (Wu et al., 2023b), NeuRAD (Tondersski et al., 2024), ML-NSG (Fischer et al., 2024b) and recent Gaussian-based works StreetGaussians (Yan et al., 2024), DrivingGaussians (Zhou et al., 2023), and HUGS (Zhou et al., 2024). However, these approaches handle only rigid objects due to limitations of time-independent representations (Ost et al., 2021; Wu et al., 2023b; Yang et al., 2023b; Zhou et al., 2023; Yan et al., 2024; Tondersski et al., 2024; Fischer et al., 2024b) or limitations of deformation-based techniques (Yang et al., 2023c; Huang et al., 2023). A recent concurrent work Fischer et al. (2024a) also considers non-rigid modeling using a deformation field, addressing a subset of the challenges in modeling holistic dynamics, but does not address fine-grained human models that allow flexible control. To address them, OmniRe proposes a
+
+Gaussian scene graph that incorporates various Gaussian representations for both rigid and non-rigid objects, providing extra flexibility and controllability for diverse actors.
+
+Human Modeling. Human bodies have variable appearance and complex motions, calling for dedicated modeling techniques. NeuMan (Jiang et al., 2022) proposes to employ the SMPL body model (Loper et al., 2015) to warp ray points to a canonical space. This approach enables the reconstruction of non-rigid human bodies and warrants fine control. Similarly, recent works such as GART (Lei et al., 2023), GauHuman (Hu & Liu, 2023) and HumanGaussians (Kocabas et al., 2024) have combined the Gaussian representation and the SMPL model. However, these methods are not directly applicable in-the-wild. As for recovering human dynamics in driving scenes, Yang et al. (2021) focuses on shape and pose reconstruction for LiDAR simulation, while Wang et al. (2023; 2024) aim to recreate natural and accurate human motion from partial observations. However, these methods focus solely on shape and pose estimation and are limited in appearance modeling. In contrast, our method not only models human appearance but also integrates this modeling within a holistic scene framework, to achieve a comprehensive solution. Urban scenes typically involve numerous pedestrians, with sparse observation, often accompanied by severe occlusion. We analyze these challenges in detail and address them in  $\S 4.2$ .
+
+# 3 PRELIMINARIES
+
+3D Gaussian Splatting. First introduced in Kerbl et al. (2023), 3D Gaussian Splatting (3DGS) represents scenes via a set of colored blobs  $\mathcal{G} = \{\pmb{g}\}$  whose intensity distribution is a Gaussian. Each Gaussian (blob)  $\pmb{g} = (o, \pmb{\mu}, \mathbf{q}, s, c)$  contains the following attributes: opacity  $o \in (0,1)$ , mean position  $\pmb{\mu} \in \mathbb{R}^3$ , rotation  $\mathbf{q} \in \mathbb{R}^4$  represented as a quaternion, anisotropic scaling factors  $s \in \mathbb{R}_+^3$ , and view-dependent colors  $c \in \mathbb{R}^F$  represented as spherical harmonics (SH) coefficients. To compute the color  $C$  of a pixel, Gaussians overlapping with this pixel are sorted by their distance to the camera center (sorted by  $i \in \mathcal{N}$ ) and  $\alpha$ -blended:  $C = \sum_{i \in \mathcal{N}} c_i \alpha_i \prod_{j=1}^{i-1} (1 - \alpha_j)$ , where  $\alpha_i$  is computed as  $\alpha_i = o_i \exp(-\frac{1}{2} (\mathbf{p} - \pmb{\mu}_i)^T \pmb{\Sigma}_i^{-1} (\mathbf{p} - \pmb{\mu}_i))$ ,  $\pmb{\Sigma}_i$  is the 2D projection covariance. We further define the application of a rigid (affine) transformation  $\mathbf{T} = (\mathbf{R}, t) \in \mathbb{S}\mathbb{E}(3)$  to all Gaussians in the set as:  $\mathbf{T} \otimes \mathcal{G} = (o, \mathbf{R}\pmb{\mu} + t, \mathrm{Rot}(\mathbf{R}, \mathbf{q}), s, c)$ , where  $\mathrm{Rot}(\cdot)$  denotes rotating the quaternion by the rotation matrix.
+
+Skinned Multi-Person Linear (SMPL) Model. SMPL (Loper et al., 2015) is a parametric human body model that combines the advantages of a parametric mesh with linear blending skinning (LBS) to manipulate body shape and pose. At its core, SMPL uses a template mesh  $\mathcal{M}_h = (\mathcal{V},\mathcal{F})$  defined in a canonical rest pose, parameterized by  $n_v$  vertices  $\mathcal{V}\in \mathbb{R}^{n_v\times 3}$ . The template mesh can be shaped and transformed using shape parameters  $\beta$  and pose parameters  $\theta$ :  $\mathcal{V}_S = \mathcal{V} + B_S(\beta) + B_P(\theta)$ , where  $B_{S}(\beta)\in \mathbb{R}^{n_v\times 3}$  and  $B_{P}(\theta)\in \mathbb{R}^{n_v\times 3}$  are the xyz offsets to individual vertices (Kocabas et al., 2024) and  $\nu_{S}$  are the vertex locations in the shaped space.
+
+To further deform the vertices  $\nu_{S}$  to achieve the desired pose  $\theta^{\prime}$ , SMPL utilizes pre-defined LBS weights  $W\in \mathbb{R}^{n_k\times n_v}$  and the joint transformations  $\pmb{G}$  to define the deformation of each vertex  $\pmb{v}_i$ :  $\pmb{v}_i' = (\sum_k\pmb{W}_{k,i}\pmb{G}_k)\pmb{v}_i$ , where  $n_k$  is the number of joints, and the joint transformations  $\pmb{G}$  are derived from the source pose  $\pmb{\theta}$ , the target pose  $\pmb{\theta}'$  and shape  $\beta$ . The pose parameters include the body pose component  $\pmb{\theta}_b\in \mathbb{R}^{23\times 3\times 3}$  and the global orientation component  $\pmb{\theta}_g\in \mathbb{R}^{3\times 3}$ . For more details of SMPL, we refer readers to Loper et al. (2015). Our method obtains pose parameters  $\pmb{\theta}$  for each pedestrian across all frames, as well as their individual shape parameters  $\beta \in \mathbb{R}^{10}$ , these pose sequences initialize the non-rigid dynamics of pedestrians. The detailed process is described in § 4.2.
+
+# 4 METHOD
+
+As overviewed in Fig. 2, we build a comprehensive 3DGS framework that holistically reconstructs both the static background and diverse movable entities. We discuss our systematic approach that represents different semantic classes with diverse Gaussian representations in § 4.1, highlighting that this complex yet efficient system-level framework is one of our primary contributions. Modeling humans in unconstrained environments is particularly challenging due to the complexity of human motions and the difficulty of accurately modeling geometry and appearance due to severe occlusions in the wild. We present our approach to this problem in § 4.2, which significantly expands our
+
+![](images/04af9dbd558e54f7a6e4bfb7a0c9e1807e05f1939a2a10b2d438b23d6bd6bc98.jpg)  
+Figure 2: Method Overview. Gaussians of all foreground models are defined in their local or canonical spaces. At a given time  $t$ , the Gaussians are deformed and transformed into the world space, forming a Gaussian scene graph together with background Gaussians to model the entire scene. The Gaussians in the scene graph are rasterized to render images and depth, and are jointly optimized using reconstruction losses. We utilize SMPL Gaussians to model non-rigid human bodies and deformable Gaussians to handle out-of-distribution non-rigid categories.
+
+effectiveness in common driving scenes. Lastly, we show how the scene representation is end-to-end optimized to obtain faithful and controllable reconstructions in  $\S 4.3$ .
+
+# 4.1 DYNAMIC GAUSSIAN SCENE GRAPH MODELING
+
+Gaussian Scene Graph. To allow for flexible control of diverse movable objects in the scene without sacrificing reconstruction quality, we opt for a Gaussian Scene Graph representation. Our scene graph is composed of the following nodes: (1) a Sky Node representing the sky that is far away from the ego-car, (2) a Background Node representing the static scene background such as buildings, roads, and vegetation, (3) a set of Rigid Nodes, each representing a rigidly movable object such as a vehicle, (4) a set of Non-rigid Nodes that model non-rigid individuals, e.g. pedestrians and cyclists. Nodes of type (2,3,4) can be converted directly into world-space Gaussians which we will introduce next. These Gaussians are concatenated and rendered using the rasterizer proposed in Kerbl et al. (2023). The Sky Node is represented by an estimizable environment texture map, similar to Chen et al. (2023), rendered separately, and composited with the rasterized Gaussian image with simple alpha blending.
+
+Background Node. The background node is represented by a set of static Gaussians  $\mathcal{G}^{\mathrm{bg}}$ . These Gaussians are initialized by accumulating the LiDAR points and additional points generated randomly in accordance with the strategy described in Chen et al. (2023).
+
+Rigid Nodes. Gaussians representing the vehicles (e.g. cars or trucks) are defined as  $\bar{\mathcal{G}}_v^{\mathrm{rigid}}$  in the object's local space (denoted by the upper bar), where  $v$  is the index of the vehicle/node. While the Gaussians within a vehicle will not change over time in the local space, the positions of Gaussians in world space will change according to the vehicle's pose  $\mathbf{T}_v\in \mathbb{S}\mathbb{E}(3)$ . At a given time  $t\in \mathbb{R}$ , the Gaussians are transformed into world space by simply applying the pose transformation:
+
+$$
+\mathcal {G} _ {v} ^ {\text {r i g i d}} (t) = \mathbf {T} _ {v} (t) \otimes \bar {\mathcal {G}} _ {v} ^ {\text {r i g i d}}. \tag {1}
+$$
+
+Non-Rigid Nodes. Non-rigid individuals are often overlooked by previous methods (Zhou et al., 2024; Yan et al., 2024; Zhou et al., 2023; Fischer et al., 2024b) due to the modeling complexity, despite their importance for human-centered simulation. Unlike rigid vehicles, non-rigid dynamic classes such as pedestrians and cyclists, require extra consideration of both their global movements in world space and their continuous deformations in local space to accurately reconstruct their dynamics. To enable a reconstruction that fully explains the underlying geometry, we further subdivide the non-rigid nodes into two categories: SMPL Nodes for walking or running pedestrians with SMPL templates that enable joint-level control and Deformable Nodes for out-of-distribution non-rigid instances (such as cyclists and other template-less dynamic entities).
+
+Non-Rigid SMPL Nodes. As introduced in § 3, SMPL provides a parametric way of representing human poses and deformations, and we hence use the model parameters  $(\theta(t), \beta)$  to drive the 3D
+
+Gaussians within the nodes. Here  $\theta(t) \in \mathbb{R}^{24 \times 3 \times 3}$  represents the human posture that changes over time  $t$ . For each node indexed by  $h$ , We tessellate the SMPL template mesh  $\mathcal{M}_h$  instantiated from the resting pose (the 'Da' pose) with 3D Gaussians  $\bar{\mathcal{G}}_h^{\mathrm{SMPL}}$  using a strategy similar to Lei et al. (2023), where each Gaussian is bound to its corresponding vertex of  $\mathcal{M}_h$ . The world-space Gaussians for each node can be then computed as:
+
+$$
+\mathcal {G} _ {h} ^ {\mathrm {S M P L}} (t) = \mathbf {T} _ {h} (t) \otimes \operatorname {L B S} (\boldsymbol {\theta} (t), \bar {\mathcal {G}} _ {h} ^ {\mathrm {S M P L}}). \tag {2}
+$$
+
+Here  $\mathbf{T}_h(t) \in \mathbb{SE}(3)$  is the global pose of the node at time  $t$ , and  $\mathrm{LBS}(\cdot)$  is the linear blend skinning operation that deforms the Gaussians according to the SMPL pose parameters. In order to compute the LBS operator, one first precomputes the skinning weights of each Gaussian in  $\bar{\mathcal{G}}_h^{\mathrm{SMPL}}$  w.r.t. the SMPL key joints. Once  $\theta$  changes over time, the key joints' transformations are updated and linearly interpolated onto the Gaussians to obtain the final deformed positions and rotations, while other attributes in the Gaussian remain unchanged. Crucially, it is highly challenging to accurately optimize the SMPL poses  $\theta(t)$  from scratch just with sensor observations, even for single-person or indoor scenarios (Jiang et al., 2022; Lei et al., 2023; Kocabas et al., 2024). Hence a rough initialization of  $\theta(t)$  is typically needed, whose details are deferred to a dedicated section § 4.2.
+
+Non-Rigid Deformable Nodes. These nodes act as a unified representation for other significant non-rigid instances, including those that fall beyond the scope of SMPL modeling, such as extremely faraway pedestrians for which even state-of-the-art SMPL estimators cannot provide accurate estimations, or out-of-distribution, template-less non-rigid individuals. Hence, we propose to use a general deformation network  $\mathcal{F}_{\varphi}$  with parameter  $\varphi$  to fit the non-rigid motions within the nodes. Specifically, for node  $h$ , the world-space Gaussians are defined as:
+
+$$
+\mathcal {G} _ {h} ^ {\text {d e f o r m}} (t) = \mathbf {T} _ {h} (t) \otimes \left(\bar {\mathcal {G}} _ {h} ^ {\text {d e f o r m}} \oplus \mathcal {F} _ {\varphi} \left(\bar {\mathcal {G}} _ {h} ^ {\text {d e f o r m}}, \boldsymbol {e} _ {h}, t\right)\right), \tag {3}
+$$
+
+where the deformation network generates the changes of the Gaussian attributes from time  $t$  to the canonical space Gaussians  $\bar{\mathcal{G}}_h^{\mathrm{deform}}$ , outputting the changes in position  $\delta \pmb{\mu}_h(t)$ , rotation  $\delta \mathbf{q}_h(t)$ , and the scaling factors  $\delta s_h(t)$ . The changes are applied back to  $\bar{\mathcal{G}}_h^{\mathrm{deform}}$  with the  $\oplus$  operator that internally performs a simple arithmetic addition that results in  $(o, \pmb{\mu} + \delta \pmb{\mu}(t), \mathbf{q} + \delta \mathbf{q}(t), s + \delta s(t), c)$ . Notably, previous approaches such as Yang et al. (2023c) utilizes a single deformation network for the entire scene, and usually fail in highly complex outdoor dynamic scenes with rapid movements. On the contrary, in our work, we define a per-node deformation field which has much more representation power. To maintain computational efficiency, the network weights  $\varphi$  are shared and the identities of the nodes are disambiguated via an instance embedding parameter  $e_h$ . Experimental results in § 5.2 show that deformable Gaussians are essential for achieving good reconstruction quality.
+
+Sky Node. We use a separate estimizable environmental map to fit the sky color from viewing directions. Compositing the sky image  $C_{\mathrm{sky}}$  with the rendered Gaussians  $C_{\mathcal{G}}$  consisting of  $(\mathcal{G}^{\mathrm{bg}}, \{\mathcal{G}_v^{\mathrm{rigid}}\}, \{\mathcal{G}_h^{\mathrm{SMPL}}\}, \{\mathcal{G}_h^{\mathrm{deform}}\})$ , we obtain the final rendering as:
+
+$$
+C = C _ {\mathcal {G}} + \left(1 - O _ {\mathcal {G}}\right) C _ {\text {s k y}}, \tag {4}
+$$
+
+where  $O_{\mathcal{G}} = \sum_{i=1}^{N} \alpha_i \prod_{j=1}^{i-1} (1 - \alpha_j)$  is the rendered opacity mask of Gaussians.
+
+# 4.2 RECONSTRUCTING IN-THE-WILD HUMANS
+
+Reconstructing humans from driving logs faces challenges as in-the-wild pose estimators (Goel et al., 2023; Rajasegaran et al., 2022) are typically designed for single video input and often miss predictions in occlusion cases. We designed a pipeline that addresses these limitations to predict accurate and temporally consistent human poses from multi-view videos with frequent occlusions.
+
+Formally, given a set of 3D tracklets for  $N$  pedestrians  $\mathbf{T} = \{\mathbf{T}_h\}_{h=0}^{N-1}$  from the dataset, our goal is to obtain the corresponding SMPL pose sets:  $\pmb{\theta} = \{\pmb{\theta}_h\}_{h=0}^{N-1}$ . Here,  $\mathbf{T}_h$  and  $\pmb{\theta}_h$  (For brevity,  $(t)$  is omitted) represent the boxes sequence and body pose sequence of the  $h$ -th human. We apply 4D-Humans (Goel et al., 2023) to each camera's video independently in our multi-camera setup. This yields separately processed results of human tracklets and poses:  $\hat{\mathbf{T}} = \bigcup_{c=0}^{C-1} \hat{\mathbf{T}}^c$  and  $\hat{\pmb{\theta}} = \bigcup_{c=0}^{C-1} \hat{\pmb{\theta}}^c$ , where  $\hat{\mathbf{T}}^c = \{\hat{\mathbf{T}}_j^c\}_{j \in \mathcal{D}^c}$  and  $\hat{\pmb{\theta}}^c = \{\hat{\pmb{\theta}}_j^c\}_{j \in \mathcal{D}^c}$  represent the predicted tracklets and poses from camera  $c$ , respectively. Here,  $\mathcal{D}^c$  is the set of detected human indices in camera  $c$ . Our task is to reconstruct  $\pmb{\theta}$  using  $\hat{\pmb{\theta}}$ . We achieve this through the following steps:
+
+![](images/0154016414f82b1b19fab751b1850603b97c8e7eac1ed81271a685c7b22f3718.jpg)  
+Figure 3: Human Pose Processing. (a) Human ID matching ensures consistent identification across cameras. (b) Missing pose completion to recover poses of occluded individuals.
+
+Tracklet Matching: We define a matching function  $\mathcal{M}$  that finds the most similar predicted tracklets for each ground truth tracklet by computing the maximum mean IoU of their 2D projections:
+
+$$
+\tilde {\boldsymbol {\theta}} _ {h} = \mathcal {M} (h, \hat {\boldsymbol {\theta}}, \mathbf {T}, \hat {\mathbf {T}}). \tag {5}
+$$
+
+This function learns a matching between ground truth tracklets and predicted tracklets, then outputs the corresponding matched pose sequences. Consider 3-camera setup as an example (Fig. 3(a)), if the  $h$ -th ground truth tracklet matches with detected tracklets  $j_0, j_1, j_2$  in cameras 0 to 2 respectively, then  $\tilde{\pmb{\theta}}_h = \{\hat{\pmb{\theta}}_{j_0}^0, \hat{\pmb{\theta}}_{j_1}^1, \hat{\pmb{\theta}}_{j_2}^2\}$ , where  $\hat{\pmb{\theta}}_{jk}^c$  is the pose sequence from camera  $c$  for the detected tracklet  $j_k$ .
+
+Pose Completion: As visualized in Fig. 3(b), 4D-Humans (Goel et al., 2023) fails to predict SMPL poses for occluded individuals in driving scenes, we design a process to recover missing poses:
+
+$$
+\boldsymbol {\theta} _ {h} = \mathcal {H} (\tilde {\boldsymbol {\theta}} _ {h}, \mathbf {T}, \hat {\mathbf {T}}). \tag {6}
+$$
+
+Here, function  $\mathcal{H}$  identifies missing detections by comparing the ground truth and predicted tracklets, and interpolates missing poses to complete  $\theta_h$  from  $\bar{\theta}_h$ .
+
+# 4.3 OPTIMIZATION
+
+We simultaneously optimize all the parameters as mentioned in § 4.1 in a single stage to reconstruct the entire scene. These parameters include: (1) all the Gaussian attributes (opacity, mean positions, scaling, rotation, and appearance) in their local spaces, namely  $\mathcal{G}^{\mathrm{bg}}$ ,  $\{\bar{\mathcal{G}}_v^{\mathrm{rigid}}\}$ ,  $\{\bar{\mathcal{G}}_h^{\mathrm{SMPL}}\}$ ,  $\{\bar{\mathcal{G}}_h^{\mathrm{deform}}\}$ , (2) the poses of both rigid and non-rigid nodes for each frame  $t$ , i.e.,  $\{\mathbf{T}_v(t)\}$ ,  $\{\mathbf{T}_h(t)\}$ , (3) the human poses of all the SMPL nodes for each frame  $t$ , i.e.,  $\{\pmb {\theta}(t)\}$ , and the corresponding skinning weights, (4) the weight  $\varphi$  of the deformation network  $\mathcal{F}$ , (5) the weight of the sky model.
+
+We use the following objective function for optimization:
+
+$$
+\mathcal {L} = \left(1 - \lambda_ {r}\right) \mathcal {L} _ {1} + \lambda_ {r} \mathcal {L} _ {\text {S S I M}} + \lambda_ {\text {d e p t h}} \mathcal {L} _ {\text {d e p t h}} + \lambda_ {\text {o p a c i t y}} \mathcal {L} _ {\text {o p a c i t y}} + \mathcal {L} _ {\text {r e g}}, \tag {7}
+$$
+
+where  $\mathcal{L}_1$  and  $\mathcal{L}_{\mathrm{SSIM}}$  are the L1 and SSIM losses on rendered images,  $\mathcal{L}_{\mathrm{depth}}$  compares the rendered depth of Gaussians with sparse depth signals from LiDAR,  $\mathcal{L}_{\mathrm{opacity}}$  encourages the opacity of the Gaussians to align with the non-sky mask, and  $\mathcal{L}_{\mathrm{reg}}$  represents various regularization terms applied to different Gaussian representations. Detailed descriptions of loss terms are provided in the Appendix.
+
+# 5 EXPERIMENTS
+
+Dataset. We conduct experiments on the Waymo Open Dataset (Sun et al., 2020), which comprises real-world driving logs. We tested up to 32 dynamic scenes in Waymo, including eight highly complex dynamic scenes that, in addition to typical vehicles, also contain diverse dynamic classes such as pedestrians and cyclists. Each selected segment contains approximately 150 frames. The segment IDs are listed in Tab. 12 and Tab. 6. To further demonstrate our effectiveness on common driving scenes, we extend our results to 5 additional popular driving datasets: NuScenes (Caesar et al., 2020), Argoverse2 (Wilson et al., 2023), PandaSet (Xiao et al., 2021), KITTI (Geiger et al., 2012), and NuPlan (Caesar et al., 2021).
+
+Baselines. We compare our method against several Gaussian Splatting approaches: 3DGS (Kerbl et al., 2023), DeformableGS (Yang et al., 2023c), StreetGS (Yan et al., 2024), HUGS (Zhou et al.,
+
+Table 1: Comparison on Waymo Open Dataset. We compute PSNR and SSIM for both the full image and dynamic regions. Vehicle indicates regions corresponding to vehicle-related classes, while Human indicates regions corresponding to human-related classes. Box indicates methods that utilize bounding boxes for dynamic modeling. LiDAR means method using LiDAR information.  
+
+<table><tr><td rowspan="3">Methods</td><td rowspan="3">Box</td><td rowspan="3">LiDAR</td><td colspan="5">Scene Reconstruction</td><td colspan="5">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td>Vehicle</td><td colspan="2">Full Image</td><td colspan="2">Human</td><td>Vehicle</td></tr><tr><td>PSNR†</td><td>SSIM†</td><td>PSNR†</td><td>SSIM†</td><td>PSNR†</td><td>PSNR†</td><td>SSIM†</td><td>PSNR†</td><td>SSIM†</td><td>PSNR†</td></tr><tr><td>EmerNeRF(Yang et al., 2023a)</td><td></td><td>✓</td><td>31.93</td><td>0.902</td><td>22.88</td><td>0.578</td><td>24.65</td><td>0.723</td><td>29.67</td><td>0.883</td><td>20.32</td><td>0.454</td></tr><tr><td>3DSG(Kerb et al., 2023)</td><td></td><td>✓</td><td>26.00</td><td>0.912</td><td>16.88</td><td>0.414</td><td>16.18</td><td>0.425</td><td>25.57</td><td>0.906</td><td>16.62</td><td>0.387</td></tr><tr><td>DeformGS(Yang et al., 2023c)</td><td></td><td>✓</td><td>28.40</td><td>0.929</td><td>17.80</td><td>0.460</td><td>19.53</td><td>0.570</td><td>27.72</td><td>0.922</td><td>17.30</td><td>0.426</td></tr><tr><td>PVG(Chen et al., 2023)</td><td></td><td>✓</td><td>32.37</td><td>0.937</td><td>24.06</td><td>0.703</td><td>25.02</td><td>0.787</td><td>30.19</td><td>0.919</td><td>21.30</td><td>0.567</td></tr><tr><td>HUGS(Zhou et al., 2024)</td><td>✓</td><td>✓</td><td>28.26</td><td>0.923</td><td>16.23</td><td>0.404</td><td>24.31</td><td>0.794</td><td>27.65</td><td>0.914</td><td>15.99</td><td>0.378</td></tr><tr><td>StreetGS(Yan et al., 2024)</td><td>✓</td><td>✓</td><td>29.08</td><td>0.936</td><td>16.83</td><td>0.420</td><td>27.73</td><td>0.880</td><td>28.54</td><td>0.928</td><td>16.55</td><td>0.393</td></tr><tr><td>Ours</td><td>✓</td><td>✓</td><td>34.25</td><td>0.954</td><td>28.15</td><td>0.845</td><td>28.91</td><td>0.892</td><td>32.57</td><td>0.942</td><td>24.36</td><td>0.727</td></tr></table>
+
+![](images/9b35d5b2cb9b1a9b79dd18319bc04fd0f7fd1e871446bde9b883a63d36055d01.jpg)  
+(a) Ground Truth
+
+![](images/8c1e8707a695c6446a92de5e472f23bd1ebec27137639c4f3ab5ef56e9061788.jpg)
+
+![](images/cc8769864b40cdfc69ea3d410714a9bc05ea459d4a71df321c28288bcd05807e.jpg)  
+(b) Ours
+
+![](images/914ca53b3c13957d3cafb26e3aaca01aa9644e4b0914704da1d23d00622310cc.jpg)
+
+![](images/b8c692d42bff9c3c38e0f0d6316f60950d597ee9caabdc2ca90addc2e6c3c048.jpg)
+
+![](images/a2425aa8ba437187ffe6ec74cd4813a47c20c1467cbe0ccf7a4fa0e50592930b.jpg)  
+(c) EmerNeRF
+
+![](images/dce324ef9d416e98fe1c62f35407e9a4e2c65c56b041bed5d0ec0da44c81872e.jpg)  
+(d) StreetGS
+
+![](images/417caa5b674c8417990d7d51f2988bda55d1ee9537cd04e5652dabfd5ad00ada.jpg)
+
+![](images/81f88dc4c59047ee23f27094ea2c308cc38e4742cfea7ecafa4044e79258955f.jpg)  
+(e) HUGS  
+Figure 4: Qualitative Comparison of Novel View Synthesis. The insets highlight the details of the reconstructed dynamic objects. OmniRe manages to recover very fine details, achieving high-quality reconstruction of various common dynamic objects, including vehicles, pedestrians, and cyclists.
+
+![](images/0a9c33b27107d9e1b6f374c2ea74964d2cdc4e75e737a01bb882191826560ebf.jpg)
+
+![](images/a5a0c061e031468c7f27ea31ca50893147c0afee804b51de9657072a143cb4ba.jpg)  
+(f) PVG
+
+![](images/8d57dee180a79156add7ff20e91c789e0563d4e0bcc7b0841b4da35342c18f7a.jpg)
+
+2024), and PVG (Chen et al., 2023). Additionally, we compare our method with NeRF-based approach EmerNeRF (Yang et al., 2023a). Among methods compared, for StreetGS (Yan et al., 2024), we use our own reimplementation. For 3DGS (Kerbl et al., 2023) and DeformableGS (Yang et al., 2023c), we use the implementation with LiDAR supervision to ensure the comparison fairness. For other methods, we use their official code. For training, we utilize data from the three front-facing cameras, resized to a resolution of  $640 \times 960$  for all methods, along with LiDAR data for supervision. We utilize the instance bounding boxes provided by the dataset to transform objects and refine them via pose optimization during training. For further implementation details, please refer to Appendix.
+
+# 5.1 MAIN RESULTS
+
+Appearance. We evaluate our method on scene reconstruction and novel view synthesis (NVS) tasks, using every 10th frame as the held-out test set for NVS. We report PSNR and SSIM scores for full images, as well as human-related and vehicle-related regions, to assess dynamic reconstruction capabilities. The quantitative results in Tab. 1 show that OmniRe outperforms all other methods, with a significant margin in human-related regions, validating our holistic modeling of dynamic actors. Additionally, while StreetGS (Yan et al., 2024) and our method model vehicles in a similar way, we observe that OmniRe is slightly better than StreetGS even in vehicle regions. This is due to the absence of human modeling in StreetGS, which allows supervision signals from human regions (e.g., colors, LiDAR depth) to incorrectly influence vehicle modeling. The issues StreetGS faces are one of our motivations for modeling almost everything in a scene holistically, aiming to eliminate erroneous supervision and unintended gradient propagation.
+
+In addition, we show visualizations in Fig. 4 to assess model performance qualitatively. Although PVG (Chen et al., 2023) performs well on the scene reconstruction task, it struggles with the
+
+novel view synthesis task in highly dynamic scenes, resulting in blurry dynamic objects in novel views (Fig. 4-(f)). HUGS (Zhou et al., 2024) (Fig. 4-(e)), StreetGS (Yan et al., 2024)(Fig. 4-(d)) and 3DGS (Kerbl et al., 2023) (Fig. 8-(h)) fail to recover the pedestrians because they are not capable of modeling non-rigid objects. DeformableGS (Yang et al., 2023c) (Fig. 8-(g)) suffers from extreme motion blur for outdoor dynamic scenes with rapid movements, despite achieving reasonable performance for indoor scenes and cases with small motion. EmerNeRF (Yang et al., 2023a) reconstructs coarse structures of moving humans and vehicles to a certain level, but struggles with fine-grained details(Fig. 4-(c)). In contrast to all these methods in comparison, our method faithfully reconstructs fine details for any object in the scene, handling occlusion, deformation, and extreme motion. Video comparisons are included in the project page.
+
+Geometry. In addition to appearance, we also investigate whether our method can reconstruct fine geometry of urban scenes. We evaluate the Root Mean Squared Error (RMSE) and two-way Chamfer Distances (CD) for LiDAR depth reconstruction on both training frames and novel frames. Details about evaluation procedures are provided in the Appendix. Tab. 3 reports the results. Our method outperforms others by a large margin. Fig. 5 illustrates the accurate reconstruction of dynamic actors achieved by our method in comparison to other approaches.
+
+![](images/8eed206d264e789ff83308c8d3f01d5e531e4a88a56faffea4a4f83f64ab9239.jpg)  
+Figure 5: Visualizations of Rendered LiDAR. Our method accurately reconstructs LiDAR data for humans and vehicles compared to other approaches.
+
+# 5.2 ABLATION STUDIES & APPLICATIONS
+
+SMPL Modeling. SMPL modeling is important to model the local, continuous movements of humans. We study its impact by disabling the human pose transformation enabled by SMPL and report the results in Tab. 2 ((a) v.s. (b)) and illustrate these effects in Fig. 7-(B). Without template-based modeling, the reconstructed human renderings appear highly blurred, particularly around the legs, thus failing to accurately reconstruct human body movements. This contrasts sharply with the precise leg reconstruction observed in our default setting. Moreover, SMPL modeling provides joint-level control, improving the controllability (Fig. 1-(c,3), (c,4)).
+
+Human Body Pose Refinement. The human body poses extracted as described in (§ 4.2) exhibit prediction errors and scale ambiguity, which subsequently lead to pose errors that degrade reconstruction quality, as shown in Fig. 6 (Noisy). We improve this by jointly optimizing the human poses and Gaussians via the same reconstruction losses. Tab. 2-(a) v.s. (c) ablates this design choice, and Fig. 6 showcases the refined poses. These results verify the effectiveness of our refinement strategy.
+
+Deformable Nodes. Deformable nodes are important for accurately reconstructing out-of-distribution or template-less actors. Our approach addresses this challenge by learning a self-supervised deformation field that transforms Gaussians from their canonical space to the shape space. Tab. 2 ((a) v.s. (d)) proves the importance of this component. In Fig. 7-(A) shows that without deformable nodes, some dynamic actors are either ignored or incorrectly blended into the background.
+
+Boxes Refinement. In practice, we observe that the instance bounding boxes provided by the dataset are imprecise. These noisy ground truth boxes can be harmful to rendering quality. To address this, we jointly refine the bounding box parameters during training. Tab. 4 and Fig. 12 show the practical
+
+Table 2: Ablation on Non-Rigid Modeling.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Full PSNR</td><td colspan="2">Human PSNR</td></tr><tr><td>Recon.</td><td>NVS</td><td>Recon.</td><td>NVS</td></tr><tr><td>(a) Ours default</td><td>34.25</td><td>32.57</td><td>28.15</td><td>24.36</td></tr><tr><td>(b) w/o SMPL actors</td><td>32.80</td><td>31.76</td><td>24.71</td><td>23.18</td></tr><tr><td>(c) w/o Body pose refine</td><td>33.84</td><td>32.44</td><td>26.97</td><td>24.04</td></tr><tr><td>(d) w/o Deformed actors</td><td>33.64</td><td>32.17</td><td>25.26</td><td>22.41</td></tr></table>
+
+Table 3: Evaluation of LiDAR Depth Accuracy.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">Training Frames</td><td colspan="2">Novel Frames</td></tr><tr><td>CD↓</td><td>RMSE↓</td><td>CD↓</td><td>RMSE↓</td></tr><tr><td>3DGS(Kerbl et al., 2023)</td><td>0.415</td><td>2.804</td><td>0.467</td><td>2.896</td></tr><tr><td>DeformableGS(Yang et al., 2023c)</td><td>0.384</td><td>2.965</td><td>0.383</td><td>2.990</td></tr><tr><td>StreetGS(Yan et al., 2024)</td><td>0.274</td><td>2.199</td><td>0.286</td><td>2.228</td></tr><tr><td>Ours</td><td>0.242</td><td>1.894</td><td>0.244</td><td>1.909</td></tr></table>
+
+![](images/3e98f79f8dc59e5b90e9c2cf975d28863f79d5e9004e616efaf585b4e115f0e0.jpg)  
+Noisy
+
+![](images/02a312416fc24c297fcbb9c93b4d49a4c27cc9337b6d5f894a6854d739db832d.jpg)  
+Refined
+
+![](images/30a9971d6705d513e6b60bd4475bb298ffeb6cafe74db3011914323461db8109.jpg)  
+Noisy
+
+![](images/ed93a39bfa9e2c8faf2e59366af9d8e2ca8fe5b4bd5d441f9c31516dd0590aba.jpg)  
+Refined
+
+![](images/ed3bad646a87dbc62ae684d0b8ae5de402b41d928f35c98ef0ca0ab478fded55.jpg)  
+Noisy
+
+![](images/f8d5cfbb4dc45403d2c379b2becac63675408af84c40d5dd20c467b2a205a1c3.jpg)  
+Refined
+
+![](images/fdda56145583199c09dbb3862bcbdbdc8222c1fc509273051413a8ddf27f8a5a.jpg)  
+Noisy
+
+![](images/497f2ba420120a6e9af5f20ea040785a6f3359a6a52f70c4992de2c7e2747e84.jpg)  
+Refined
+
+![](images/f64aff85fca49ab033deb502ec4b17b7cda596d91ffb1545da201f4e04e8582d.jpg)  
+w/o Deformed Actors  
+Figure 7: Ablation of Human Modeling.
+
+![](images/d7dbdb1d3a32f56f7d87e4388271a1cc40e9ccc6950715c41f08b9303c9b3638.jpg)  
+Figure 6: Ablation of Human Body Pose Refinement.  
+w/ Deformed Actors
+
+![](images/c6df041bf05d5e31a1933a9443d50e96edae6c602b212cfb704af71412642aa7.jpg)  
+w/o SMPL  
+w/ SMPL
+
+benefits of this simple yet effective step, which results in improved numeric metrics and reduced blurriness of foreground objects.
+
+Applications to Simulation. Thanks to the decomposition nature of OmniRe, each instance is modeled separately. After joint training, we obtain reconstructed assets that can be flexibly edited in terms of position and rotation. Beyond editing within a single
+
+scene, we can also transfer assets from one scene to another, adding variety and complexity to the reconstructed environments. Fig. 1(c,left) demonstrates a swap of the black vehicle originally in the scene (inset) with a reconstructed vehicle from another scene; and (c,right) an insertion of a pedestrian from the scene in the inset to a new scene. Additional car swap edits are shown in Fig. 11. Through explicit modeling of pedestrians and other non-rigid individuals, we achieve the simulation of reenacted scenarios involving detailed pedestrian-vehicle interaction. As demonstrated in Fig. 9, we simulate a moving vehicle stopping at a crossing, waiting for a pedestrian who slowly crosses. The pedestrian is reconstructed from another scene. This simulation of humans is extremely challenging for previous methods. This level of precise control over reconstructed photorealistic assets opens up possibilities for integration with previous simulators for automated simulation (Wang et al., 2022; Wei et al., 2024)
+
+Table 4: Ablation on GT Boxes Refinement.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Full PSNR</td><td colspan="2">Human PSNR</td><td colspan="2">Vehicle PSNR</td></tr><tr><td>Recon.</td><td>NVS</td><td>Recon.</td><td>NVS</td><td>Recon.</td><td>NVS</td></tr><tr><td>Complete Model</td><td>34.25</td><td>32.57</td><td>28.15</td><td>24.36</td><td>28.91</td><td>27.57</td></tr><tr><td>w/o Box Refine.</td><td>33.04</td><td>31.72</td><td>26.53</td><td>23.67</td><td>25.57</td><td>24.78</td></tr></table>
+
+# 6 CONCLUSION
+
+Our method, OmniRe, tackles comprehensive urban scene modeling using Gaussian Scene Graphs. It achieves fast, high-quality reconstruction and rendering, suggesting promise for driving and robotics simulation. We also present solutions for human modeling in complex environments. Future work includes self-supervised learning, improved scene representations, and safety/privacy considerations. To ensure reproducibility, the code is available at link.
+
+Broader impact. Our method aims to address a significant problem in autonomous driving—simulation. This approach has the potential to enhance the development and testing of autonomous vehicles, potentially leading to safer and more efficient AV systems. Simulation, in a safe and controllable manner, remains an open and challenging research question.
+
+Limitations. While enabling holistic scene modeling, OmniRe still has certain limitations. First, our method does not explicitly model lighting effects, which may lead to visual harmony issues during simulations, particularly when combining elements reconstructed under varying lighting conditions. Addressing this non-trivial challenge requires dedicated efforts beyond the scope of our current work. Further research into modeling light effects and enhancing simulation realism remains crucial for achieving more convincing and harmonious results. Second, similar to other per-scene optimization methods, OmniRe produces less satisfactory novel views when the camera deviates significantly from the training trajectories. Future works to address this issue include incorporating data-driven priors, such as image or video generative models, and optimizing camera poses jointly.
+
+# 7 ETHICS STATEMENT
+
+Our work does not involve the collection or annotation of new data. We utilize well-established public datasets that adhere to strict ethical guidelines. These datasets ensure that sensitive information, including identifiable human features, is blurred or anonymized to protect individual privacy. We are committed to ensuring that our method, as well as future applications, are employed responsibly and ethically to maintain safety and preserve privacy.
+
+# 8 ACKNOWLEDGEMENTS
+
+This work is supported by funding from Toyota Research Institute, Dolby, and Google DeepMind. Yue Wang is also supported by a Powell Faculty Research Award. We also thank Jiageng Mao, Junjie Ye, Ziyi Yang, Haozhe Lou, and Yifan Lu for their valuable discussions during the project, which helped us resolve issues and improve the methods.
+
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+Xiaoyu Zhou, Zhiwei Lin, Xiaojun Shan, Yongtao Wang, Deqing Sun, and Ming-Hsuan Yang. Drivinggaussian: Composite gaussian splatting for surrounding dynamic autonomous driving scenes. arXiv preprint arXiv:2312.07920, 2023.
+
+# Supplemental Material
+
+# A IMPLEMENTATION DETAILS
+
+Initialization: For the background model, we refer to PVG (Chen et al., 2023), combining  $6 \times 10^{5}$  LiDAR points with  $4 \times 10^{5}$  random samples, which are divided into  $2 \times 10^{5}$  near samples uniformly distributed by distance to the scene's origin and  $2 \times 10^{5}$  far samples uniformly distributed by inverse distance. To initialize the background, we filter out the LiDAR samples of dynamic objects. For rigid nodes and non-rigid deformable nodes, we utilize their bounding boxes to accumulate the LiDAR points, while for non-rigid SMPL nodes, we initialize their Gaussians on the template mesh in their canonical space. To determine the initial color of Gaussians, we project LiDAR points onto the image plane, whereas random samples are initialized with random colors. The initial human body pose sequences of non-rigid SMPL Nodes are obtained through the process described in § 4.2.
+
+Training: Our method trains for 30,000 iterations with all scene nodes optimized jointly. The learning rate for Gaussian properties aligns with the default settings of 3DGS (Kerbl et al., 2023), but varies slightly across different node types. Specifically, we set the learning rate for the rotation of Gaussians to  $5 \times 10^{-5}$  for non-rigid SMPL nodes and  $1 \times 10^{-5}$  for other nodes. The degrees of spherical harmonics are set to 3 for background nodes, rigid nodes, and non-rigid deformable nodes, while it is set to 1 for non-rigid SMPL nodes. The learning rate for the rotation of instance boxes is  $1 \times 10^{-5}$ , decreasing exponentially to  $5 \times 10^{-6}$ . The learning rate for the translation of instance boxes is  $5 \times 10^{-4}$ , decreasing exponentially to  $1 \times 10^{-4}$ . The learning rate for human body poses of non-rigid SMPL nodes is  $5 \times 10^{-5}$ , decreasing exponentially to  $1 \times 10^{-5}$ . For the Gaussian densification strategy, we utilize the absolute gradient of Gaussians introduced in Ye et al. (2024) to control memory usage. We set the densification threshold of position gradient to  $3 \times 10^{-4}$ . This use of absolute gradient has a minimal impact on performance, as discussed in detail in Appendix D.4. The densification threshold for scaling is  $3 \times 10^{-3}$ . Our method runs on a single NVIDIA RTX 4090 GPU, with training for each scene taking about 1 hour. Training time varies with different training settings.
+
+**Optimization:** We utilize the loss function introduced in Eq (7) to jointly optimize all learnable parameters. The image loss is computed as:
+
+$$
+\mathcal {L} _ {\text {i m a g e}} = \left(1 - \lambda_ {r}\right) \mathcal {L} _ {1} + \lambda_ {r} \mathcal {L} _ {\text {S S I M}} \tag {8}
+$$
+
+due to sparse temporal-spatial observation of the dynamic part, its supervision signal is insufficient. To address this, we apply a higher image loss weight to the dynamic regions identified by the rendered dynamic mask. This weight is set to 5. The depth map loss is computed as:
+
+$$
+\mathcal {L} _ {\text {d e p t h}} = \frac {1}{h w} \sum \left\| \mathcal {D} ^ {s} - \hat {\mathcal {D}} \right\| _ {1} \tag {9}
+$$
+
+where  $\mathcal{D}^s$  is the inverse of the sparse depth map. We project LiDAR points onto the image plane to generate the sparse LiDAR map, and  $\hat{\mathcal{D}}$  is the inverse of the predicted depth map.
+
+The mask loss  $\mathcal{L}_{\mathrm{opacity}}$  is computed as:
+
+$$
+\mathcal {L} _ {\text {o p a c i t y}} = - \frac {1}{h w} \sum O _ {\mathcal {G}} \cdot \log O _ {\mathcal {G}} - \frac {1}{h w} \sum M _ {\text {s k y}} \cdot \log (1 - O _ {\mathcal {G}}) \tag {10}
+$$
+
+where  $M_{\mathrm{sky}}$  is the sky mask, and  $O_{\mathcal{G}}$  is the rendered opacity map.
+
+In addition to the reconstruction losses, we introduce various regularization terms for different Gaussian representations to improve quality. Among these, an important regularization term is  $\mathcal{L}_{\mathrm{pose}}$  designed to ensure smooth human body poses  $\theta (t)$ . This term is defined as:
+
+$$
+\mathcal {L} _ {\text {p o s e}} = \frac {1}{2} \| \boldsymbol {\theta} (t - \delta) + \boldsymbol {\theta} (t + \delta) - 2 \boldsymbol {\theta} (t) \| _ {1} \tag {11}
+$$
+
+where  $\delta$  is a randomly chosen integer from  $\{1,2,3,4,5\}$ . We set the weight of the SSIM loss,  $\lambda_r$ , to 0.2, the depth loss,  $\lambda_{\mathrm{depth}}$ , to 0.1, the opacity loss,  $\lambda_{\mathrm{opacity}}$ , to 0.05, and the pose smoothness loss,  $\lambda_{\mathrm{pose}}$ , to 0.01.
+
+![](images/33b6e90b728ab459c1ab2124f396608a42c1c4d9824af0dac8d9f067e7990cee.jpg)  
+(g) DeformableGS
+
+![](images/593243a6bb7c8854c1525e4dc9f559ddcd21d46b57fa0743a352e5a99190b857.jpg)  
+Figure 8: Additional Qualitative Comparison of Novel View Synthesis.
+
+![](images/72110f49f6fb7351cdd51c5ce6667fafc7b5264d3040899af0d5a3cbca4ce63d.jpg)  
+(h) 3DGS
+
+![](images/63e830971a34968bc5b698f831a60cb939941884682c9827bb19adedf87fd8de.jpg)
+
+![](images/44f548eb7b839b401447d24b61d6e77f8c6a92463c351e3885923eefa02c513c.jpg)  
+Figure 9: A sample of human-vehicle interaction simulation in driving scenarios.
+
+![](images/db7252cda78df40ebb0ece0e3e1c4596104e5e1698ce9d1ae9433aae3bb27ba7.jpg)
+
+![](images/6b704825fc1a60346b32836321148715501f3495faa2d5547553b04ccc665dcb.jpg)
+
+# B BASELINES
+
+- EmerNeRF (Yang et al., 2023a) is a state-of-the-art NeRF-Based method for dynamic driving scene reconstruction. EmerNeRF uses a static field represented by a 3D Hash-Grid to model the static parts of the scene and a dynamic field with a 4D Hash-Grid to model the dynamic parts. Additionally, it employs a flow field to aggregate the dynamic features. This self-supervised decomposition approach yields good results on dynamic scene modeling and static-dynamic decomposition, with the scene flow emerging in the flow field.  
+- DeformableGS (Yang et al., 2023c) defines a canonical space to represent scenes with Gaussians. To model dynamics, it uses a deformation network to predict offsets of Gaussian properties. These offsets then deform the Gaussians to fit the scene dynamics. DeformableGS works well in synthetic and indoor datasets. We compare it to our method to evaluate its ability to model challenging out-door dynamic scenes.  
+- StreetGS (Yan et al., 2024) is a dynamic scene modeling method based on Gaussian Splatting for driving scenes. StreetGS models the components of dynamic scenes separately: the static background and the foreground vehicles. It utilizes boxes predicted by an off-the-shelf model to warp the Gaussians of foreground vehicles and refine them during training. StreetGS yields good results on driving scenes but ignores other non-rigid dynamic objects in the scene.  
+- HUGS (Zhou et al., 2024) is a GS-based method for driving scene modeling and understanding. It not only models the appearance of a scene but also distills 2D flow maps and semantic maps into the 3D scene to enable holistic urban scene understanding. Similar to StreetGaussian (Yan et al., 2024), HUGS uses object boxes for compositing dynamic elements. HUGS achieves good performance in both scene modeling and semantic modeling. However, it primarily focuses on rigid backgrounds and objects, without addressing non-rigid dynamics.  
+- PVG (Chen et al., 2023) introduces Periodic Vibration Gaussians that vibrate over time with estimizable vibration directions, life span, and life peak (the moment of highest opacity) to represent dynamic scenes. These Gaussians are optimized using a self-supervised approach. The method achieves static-dynamic decomposition by categorizing Gaussians based on their life spans. We compare PVG with our method to evaluate our capability in modeling highly complex dynamic scenes.
+
+Among all the compared methods, HUGS (Zhou et al., 2024) and StreetGaussians (Yan et al., 2024) require bounding boxes of foreground objects. PVG (Chen et al., 2023), StreetGaussians (Yan et al., 2024), and EmerNeRF (Yang et al., 2023a) utilize LiDAR data for depth supervision. While the original implementations of 3DGS (Kerbl et al., 2023) and DeformableGS (Yang et al., 2023c) do not include depth supervision, we added LiDAR depth supervision in experiments to ensure a fair comparison.
+
+# C EVALUATION
+
+Appearance. For the Novel View Synthesis task, we select every 10th frame from the original sequence as the test set. We use PSNR and SSIM to evaluate the quality of the rendered images. Since we focus on dynamic scenes, we also compute the PSNR and SSIM for regions with vehicles and humans. To identify regions of vehicles and humans, we use Segformer (Xie et al., 2021) to obtain semantic masks. We further identify the movable dynamic parts using projections of moving object bounding boxes, utilizing their velocity information. One example of dynamic masks can bee seen in Fig. 10.
+
+![](images/7f2300b0442052f5dfb42b8f09bdffa247eaf8f46d96a42b424c9fe627afa16f.jpg)  
+GT Image
+
+![](images/5af5275c7317dbb663c3f509b97bf611e1812a0b3fdf0699384dceb8082b1a83.jpg)  
+Human Region  
+Figure 10: An example of the dynamic masks for computing dynamic region metrics.
+
+![](images/3b5c2cb54f0c120f8649c1d9032d74cce3d965d24e5ed5edc91474cb29ae1f91.jpg)  
+Vehicle Region
+
+**Geometry.** Our method uses LiDAR data to initialize Gaussians and supervise scene depth by comparing the rendered depth map with the sparse LiDAR depth map. Post-training, Gaussians typically deviate from their initial state through densification or optimization, Therefore, comparing the LiDAR depth reconstruction is still a valid comparison. We follow the depth evaluation method of StreetSurf (Guo et al., 2023): render a depth map and match depth pixels to LiDAR rays. For Chamfer Distance, re-project the predicted depth to 3D using the LiDAR ray direction and origin. For RMSE, compare the GT and predicted ranges for LiDAR rays.
+
+# D ADDITIONAL RESULTS
+
+# D.1 QUALITATIVE COMPARISON
+
+We recommend readers to our project page for video comparisons of the methods.
+
+# D.2 QUANTITATIVE COMPARISON
+
+To further validate our method's effectiveness, we tested our method against StreetGS (Yan et al., 2024) and EmerNeRF (Yang et al., 2023a) on 32 dynamic scenes from the Waymo dataset, with results reported in Tab. 5.
+
+Table 5: We expanded our evaluation to 32 dynamic scenes from the Waymo dataset, comparing our method with StreetGS (Yan et al., 2024) and EmerNeRF (Yang et al., 2023a). The segment IDs are listed in Tab. 6.  
+
+<table><tr><td rowspan="3">Methods</td><td colspan="6">Scene Reconstruction</td><td colspan="6">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>EmerNeRF</td><td>31.29</td><td>0.877</td><td>23.14</td><td>0.581</td><td>24.47</td><td>0.709</td><td>29.04</td><td>0.851</td><td>20.76</td><td>0.467</td><td>21.80</td><td>0.582</td></tr><tr><td>StreetGS</td><td>29.93</td><td>0.931</td><td>19.63</td><td>0.524</td><td>27.48</td><td>0.871</td><td>28.73</td><td>0.910</td><td>18.77</td><td>0.470</td><td>26.18</td><td>0.825</td></tr><tr><td>Ours</td><td>33.73</td><td>0.946</td><td>28.28</td><td>0.855</td><td>28.02</td><td>0.880</td><td>31.71</td><td>0.924</td><td>24.57</td><td>0.730</td><td>26.55</td><td>0.833</td></tr></table>
+
+Table 6: Segment IDs of 32 dynamic scenes of Waymo Dataset used in the test for Tab. 5.  
+
+<table><tr><td>seg104554...</td><td>seg125050...</td><td>seg169514...</td><td>seg584622...</td><td>seg776165...</td><td>seg138251...</td><td>seg448767...</td><td>seg965324...</td></tr><tr><td>seg119252...</td><td>seg122514...</td><td>seg132544...</td><td>seg134024...</td><td>seg166004...</td><td>seg173881...</td><td>seg215148...</td><td>seg391164...</td></tr><tr><td>seg454855...</td><td>seg560223...</td><td>seg571325...</td><td>seg587066...</td><td>seg842457...</td><td>seg952165...</td><td>seg952995...</td><td>seg112365...</td></tr><tr><td>seg152664...</td><td>seg411445...</td><td>seg123218...</td><td>seg102252...</td><td>seg148106...</td><td>seg265611...</td><td>seg179934...</td><td>seg104859...</td></tr></table>
+
+# D.3 OMNIRE ON CHALLENGING SCENES
+
+To demonstrate the effectiveness and robustness of OmniRe, we tested our method alongside StreetGS (Yan et al., 2024), PVG (Chen et al., 2023), and DeformableGS (Yang et al., 2023c) on various challenging scenes. The results show that OmniRe maintains high reconstruction quality under most challenging conditions. This section provides the quantitative performance of the compared methods. Video visualizations and comparison are included in our project page.
+
+Super Crowded Scenes. Reconstructing scenes with extreme dynamic occlusions poses significant challenges. OmniRe is specifically designed to handle these scenes and performs well even in highly occluded scenes. We conducted experiments on three extremely crowded scenes from Waymo (seg112520..., seg104859..., seg152664...), including situations with a large crowd of people simultaneously crossing a street. Results in Tab. 7 show that OmniRe demonstrates strong performance in these scenes.
+
+Table 7: Comparison on super crowded scenes.  
+
+<table><tr><td rowspan="3">Methods</td><td colspan="6">Scene Reconstruction</td><td colspan="6">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>DeformGS</td><td>26.69</td><td>0.861</td><td>19.48</td><td>0.504</td><td>17.49</td><td>0.455</td><td>25.66</td><td>0.839</td><td>18.71</td><td>0.448</td><td>17.12</td><td>0.414</td></tr><tr><td>PVG</td><td>29.11</td><td>0.865</td><td>24.56</td><td>0.686</td><td>21.29</td><td>0.644</td><td>27.21</td><td>0.830</td><td>22.24</td><td>0.555</td><td>19.63</td><td>0.528</td></tr><tr><td>StreetGS</td><td>26.81</td><td>0.865</td><td>19.31</td><td>0.516</td><td>24.79</td><td>0.787</td><td>25.67</td><td>0.841</td><td>18.53</td><td>0.459</td><td>23.40</td><td>0.723</td></tr><tr><td>Ours</td><td>31.25</td><td>0.900</td><td>27.50</td><td>0.809</td><td>25.91</td><td>0.812</td><td>28.91</td><td>0.866</td><td>23.78</td><td>0.675</td><td>24.46</td><td>0.750</td></tr></table>
+
+Nighttime Scenes. We tested three nighttime scenes (seg129008..., seg102261..., seg128560...) from Waymo. Results in Tab. 8 indicate that OmniRe outperforms the compared methods, achieving superior reconstruction quality under low-light conditions.
+
+Table 8: Comparison on night time scenes.  
+
+<table><tr><td rowspan="3">Methods</td><td colspan="6">Scene Reconstruction</td><td colspan="6">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>DeformGS</td><td>30.06</td><td>0.767</td><td>27.40</td><td>0.665</td><td>20.62</td><td>0.612</td><td>28.88</td><td>0.740</td><td>22.30</td><td>0.530</td><td>19.20</td><td>0.535</td></tr><tr><td>PVG</td><td>30.55</td><td>0.768</td><td>30.74</td><td>0.777</td><td>22.77</td><td>0.713</td><td>29.32</td><td>0.740</td><td>25.22</td><td>0.634</td><td>20.90</td><td>0.611</td></tr><tr><td>StreetGS</td><td>30.60</td><td>0.775</td><td>27.54</td><td>0.667</td><td>25.68</td><td>0.772</td><td>29.38</td><td>0.741</td><td>21.88</td><td>0.523</td><td>24.61</td><td>0.729</td></tr><tr><td>Ours</td><td>31.14</td><td>0.778</td><td>31.06</td><td>0.790</td><td>25.87</td><td>0.774</td><td>29.79</td><td>0.744</td><td>25.71</td><td>0.689</td><td>24.29</td><td>0.724</td></tr></table>
+
+Adverse Weather Conditions. To evaluate performance under adverse weather conditions, we tested seven scenes with varying weather types: a. rainy (seg113555..., seg109277..., seg141339...) b. foggy (seg161022..., seg172163...) c. cloudy (seg144275..., seg157956...). OmniRe handled these challenging conditions robustly, maintaining high reconstruction fidelity across all weather conditions (Tab. 9).
+
+Table 9: Comparison on adverse weather scenes.  
+
+<table><tr><td rowspan="3">Methods</td><td colspan="6">Scene Reconstruction</td><td colspan="6">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>DeformGS</td><td>32.92</td><td>0.933</td><td>20.06</td><td>0.497</td><td>23.12</td><td>0.658</td><td>31.43</td><td>0.916</td><td>19.65</td><td>0.478</td><td>22.06</td><td>0.598</td></tr><tr><td>PVG</td><td>32.75</td><td>0.927</td><td>27.75</td><td>0.785</td><td>27.20</td><td>0.795</td><td>31.12</td><td>0.907</td><td>25.02</td><td>0.656</td><td>24.44</td><td>0.677</td></tr><tr><td>StreetGS</td><td>33.49</td><td>0.933</td><td>18.26</td><td>0.424</td><td>32.28</td><td>0.916</td><td>32.05</td><td>0.918</td><td>18.56</td><td>0.434</td><td>29.60</td><td>0.841</td></tr><tr><td>Ours</td><td>34.00</td><td>0.935</td><td>30.12</td><td>0.857</td><td>32.55</td><td>0.919</td><td>32.58</td><td>0.920</td><td>26.75</td><td>0.763</td><td>29.82</td><td>0.844</td></tr></table>
+
+High-Speed Scenes. For high-speed scenes such as highways, we tested three highway scenes from the Waymo dataset (seg109239..., seg113924..., seg118396). Results are provided in Tab. 10. As highways typically lack non-rigid objects (e.g., humans), human metrics were not applicable. Since non-rigid modeling was not required in these scenes, OmniRe and StreetGS demonstrated comparable performance.
+
+Table 10: Comparison on high speed scenes.  
+
+<table><tr><td rowspan="3">Methods</td><td colspan="6">Scene Reconstruction</td><td colspan="6">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>DeformGS</td><td>26.95</td><td>0.855</td><td>-</td><td>-</td><td>17.49</td><td>0.482</td><td>25.56</td><td>0.837</td><td>-</td><td>-</td><td>16.07</td><td>0.407</td></tr><tr><td>PVG</td><td>28.28</td><td>0.861</td><td>-</td><td>-</td><td>21.14</td><td>0.617</td><td>25.79</td><td>0.825</td><td>-</td><td>-</td><td>17.65</td><td>0.470</td></tr><tr><td>StreetGS</td><td>30.89</td><td>0.902</td><td>-</td><td>-</td><td>29.45</td><td>0.883</td><td>28.05</td><td>0.863</td><td>-</td><td>-</td><td>24.88</td><td>0.766</td></tr><tr><td>Ours</td><td>30.85</td><td>0.902</td><td>-</td><td>-</td><td>29.51</td><td>0.882</td><td>28.01</td><td>0.864</td><td>-</td><td>-</td><td>24.52</td><td>0.760</td></tr></table>
+
+# D.4 ABLATION STUDIES
+
+Absolute Gradient. In our implementation, we applied AbsGrad for 3DGS densification across all reproduced methods (StreetGS, DeformableGS, and 3DGS) as a standard practice. To quantify the impact of AbsGrad, we conducted a comparative study with and without its application. The results of this analysis are presented in Tab. 11. We see that disabling AbsGrad leads to a marginal performance decrease (about 0.1 PSNR) for all methods, proving that AbsGrad is not the key factor contributing to our performance advantage over others. Note that DeformableGS fails to run due to out-of-memory issues when AbsGrad was disabled. Based on these findings, we recommend the incorporation of AbsGrad as a standard practice in 3DGS densification and related methods.
+
+Table 11: Ablation on AbsGrad. By default, we apply AbsGrad to all GS-based approaches reproduced by us. We now disable it to analyze its impact. We mark methods with AbsGrad enabled with grey background. We observe that 1) DeformableGS fails under w/o. AbsGrad setting because of out of memory issue; 2) enabling AbsGrad is a good practice ( $+0.1$  PNSR for all methods) but not an enabling factor for our performance lead.  
+
+<table><tr><td rowspan="3">Methods</td><td colspan="6">Scene Reconstruction</td><td colspan="6">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>EnerNeRF</td><td>31.93</td><td>0.902</td><td>22.88</td><td>0.578</td><td>24.65</td><td>0.723</td><td>29.67</td><td>0.883</td><td>20.32</td><td>0.454</td><td>22.07</td><td>0.609</td></tr><tr><td>3DGS*</td><td>26.00</td><td>0.912</td><td>16.88</td><td>0.414</td><td>16.18</td><td>0.425</td><td>25.57</td><td>0.906</td><td>16.62</td><td>0.387</td><td>16.00</td><td>0.407</td></tr><tr><td>3DGS</td><td>25.84</td><td>0.910</td><td>16.69</td><td>0.405</td><td>16.02</td><td>0.415</td><td>25.61</td><td>0.905</td><td>16.52</td><td>0.383</td><td>15.97</td><td>0.405</td></tr><tr><td>DeformGS*</td><td>28.40</td><td>0.929</td><td>17.80</td><td>0.460</td><td>19.53</td><td>0.570</td><td>27.72</td><td>0.922</td><td>17.30</td><td>0.426</td><td>18.91</td><td>0.530</td></tr><tr><td>DeformGS</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>PVG</td><td>32.37</td><td>0.937</td><td>24.06</td><td>0.703</td><td>25.02</td><td>0.787</td><td>30.19</td><td>0.919</td><td>21.30</td><td>0.567</td><td>22.28</td><td>0.679</td></tr><tr><td>HUGS</td><td>28.26</td><td>0.923</td><td>16.23</td><td>0.404</td><td>24.31</td><td>0.794</td><td>27.65</td><td>0.914</td><td>15.99</td><td>0.378</td><td>23.27</td><td>0.748</td></tr><tr><td>StreetGS*</td><td>29.08</td><td>0.936</td><td>16.83</td><td>0.420</td><td>27.73</td><td>0.880</td><td>28.54</td><td>0.928</td><td>16.55</td><td>0.393</td><td>26.71</td><td>0.846</td></tr><tr><td>StreetGS</td><td>28.89</td><td>0.932</td><td>16.70</td><td>0.409</td><td>28.07</td><td>0.878</td><td>28.46</td><td>0.926</td><td>16.41</td><td>0.387</td><td>26.86</td><td>0.845</td></tr><tr><td>Ours*</td><td>34.25</td><td>0.954</td><td>28.15</td><td>0.845</td><td>28.91</td><td>0.892</td><td>32.57</td><td>0.942</td><td>24.36</td><td>0.727</td><td>27.57</td><td>0.858</td></tr><tr><td>Ours</td><td>34.11</td><td>0.953</td><td>28.00</td><td>0.842</td><td>28.83</td><td>0.890</td><td>32.46</td><td>0.941</td><td>24.28</td><td>0.726</td><td>27.55</td><td>0.857</td></tr></table>
+
+Table 12: Segment IDs of 8 dynamic scenes of Waymo Dataset used in the test for Tab. 1. and Tab. 11  
+
+<table><tr><td>seg104554...</td><td>seg125050...</td><td>seg169514...</td><td>seg584622...</td><td>seg776165...</td><td>seg138251...</td><td>seg448767...</td><td>seg965324...</td></tr></table>
+
+Additional Results The Tab. 14 is the full table of Tab. 2 that includes evaluation on SSIM. The Tab. 13 is the full table of Tab. 4 that includes evaluation on SSIM.
+
+Table 13: Ablation on GT Boxes Refinement.  
+
+<table><tr><td rowspan="3"></td><td colspan="6">Scene Reconstruction</td><td colspan="6">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>Complete Model</td><td>34.25</td><td>0.954</td><td>28.15</td><td>0.845</td><td>28.91</td><td>0.892</td><td>32.57</td><td>0.942</td><td>24.36</td><td>0.727</td><td>27.57</td><td>0.858</td></tr><tr><td>w/o Box Refine.</td><td>33.04</td><td>0.947</td><td>26.53</td><td>0.790</td><td>25.57</td><td>0.813</td><td>31.72</td><td>0.936</td><td>23.67</td><td>0.686</td><td>24.78</td><td>0.785</td></tr></table>
+
+# E OMNIRE IN PRACTICE
+
+Bounding Boxes. Similar to other scene-graph-based approaches (Ost et al., 2021; Yang et al., 2023b; Tondersski et al., 2024; Fischer et al., 2024b; Zhou et al., 2023; 2024; Yan et al., 2024), we utilize bounding boxes for driving scene reconstruction, as they are widely used for producing superior reconstruction results compared to methods that do not employ them. Additionally, bounding boxes offer significant controllability. It allows precise manipulation of both rigid objects like vehicles and non-rigid objects such as individual human body movements—an ability lacking in self-supervised methods like EmerNeRF (Yang et al., 2023a) and PVG (Chen et al., 2023) that do not use instance information. This level of controllability is crucial for tasks like scene simulation, which require the ability to manage the movement of all participating agents. Lastly, bounding box annotation is a standard and generally straightforward process in the autonomous driving field, with most popular datasets already providing these annotations via established auto-labeling tools, thereby minimizing manual effort and making the resource both efficient and accessible. For real-world driving logs, these auto-labeling tools can generate precise bounding boxes at little cost.
+
+Table 14: Ablation on Non-Rigid Modeling.  
+
+<table><tr><td rowspan="3"></td><td colspan="4">Scene Reconstruction</td><td colspan="4">Novel View Synthesis</td></tr><tr><td colspan="2">Full Image</td><td colspan="2">Human</td><td colspan="2">Full Image</td><td colspan="2">Human</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>(a) Ours default</td><td>34.25</td><td>0.954</td><td>28.15</td><td>0.845</td><td>32.57</td><td>0.942</td><td>24.36</td><td>0.727</td></tr><tr><td>(b) w/o SMPL actors</td><td>32.80</td><td>0.949</td><td>24.71</td><td>0.770</td><td>31.76</td><td>0.939</td><td>23.18</td><td>0.694</td></tr><tr><td>(c) w/o Body Pose Refine.</td><td>33.84</td><td>0.952</td><td>26.97</td><td>0.815</td><td>32.44</td><td>0.941</td><td>24.04</td><td>0.712</td></tr><tr><td>(d) w/o Deformed actors</td><td>33.64</td><td>0.953</td><td>25.26</td><td>0.766</td><td>32.17</td><td>0.941</td><td>22.41</td><td>0.653</td></tr></table>
+
+![](images/4521fd706c7a4c1c4b6b5a5e697dabf0a93711be939f55a30d421578341126f2.jpg)  
+Reconstructed Background  
+Figure 11: Our method allows for flexible editing of scene assets.
+
+![](images/6216dbf0e137bfeceb64c88979c07c95be5b88ae488ad06aa7110377c20a2195.jpg)  
+Original Car
+
+![](images/1eae1cb4fdfe5c72ebeb50d2aa44a418ef84627073a0263febe55dcac6de2d66.jpg)  
+Insert Car A
+
+![](images/d4d2ab269bcf42ebfe3ed1686e6e8bbfc332fef8a43813e94af3c9b3831f9049.jpg)  
+Insert Car B
+
+How to Determine Gaussian Representations for Humans? We categorize pedestrians into two groups for modeling. Near-range pedestrians, detected by our human pose processing module introduced in § 4.2, are modeled using SMPL nodes. Far-range pedestrians, typically undetected due to distance, are modeled using deformable nodes. This approach naturally distinguishes between near and far-range pedestrians based on human detection capabilities.
+
+Other individuals, such as those using wheelchairs, skateboards, or bicycles, are often labeled as "cyclists" in the datasets we study. However, these labels may be specific to the dataset used, and in some cases, the annotations might not be accurate. For instance, in the Waymo Dataset, a person on a motorcycle may be labeled as a "vehicle". This reliance on dataset-specific labels could potentially limit the generalization of our method to other scenarios with imperfect labels.
+
+To address this issue, we conducted preliminary experiments using GPT-4o (Achiam et al., 2023) to classify individuals (cropped by bounding boxes) into two categories: pedestrians and humans using personal transportation devices (e.g., wheelchairs, bicycles, motorcycles). Testing on 60 individuals (30 from each category), GPT-4o (Achiam et al., 2023) achieved  $100\%$  accuracy. This suggests that accurate labels can be obtained relatively easily, thanks to the development of vision-language models.
+
+![](images/63188dea8fbf884e556131d0460155d4dc8ed79ab50489aa44c64d779e38cf31.jpg)  
+Ground Truth  
+Figure 12: Ablation of Boxes Refinement.
+
+![](images/4c0da45115a501b908b8411408c6dabfc79f8140971841f7da35b3accb7cbcd0.jpg)  
+w/BoxRefine.
+
+![](images/8094715024b19ac5d3dbfc4b5df4eb2f0129a08a8f52ab671ff376043b3a0172.jpg)  
+w/o Box Refine.

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2025/On Disentangled Training for Nonlinear Transform in Learned Image Compression/99e05f2d-89ee-4325-bbe2-28387fa53a3b_origin.pdf

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2025/On Disentangled Training for Nonlinear Transform in Learned Image Compression/full.md

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+# ON DISENTANGLED TRAINING FOR NONLINEAR TRANSFORM IN LEARNED IMAGE COMPRESSION
+
+Han Li $^{1}$ , Shaohui Li $^{2*}$ , Wenrui Dai $^{1*}$ , Maida Cao $^{1}$ , Nuowen Kan $^{1}$ , Chenglin Li $^{1}$ , Junni Zou $^{1}$ , Hongkai Xiong $^{1}$ $^{1}$ Shanghai Jiao Tong University,  $^{2}$ Tsinghua Shenzhen International Graduate School, Tsinghua University  
+{qingshi9974, daiwenrui}@sjtu.edu, lishaohui@sz.tsinghua.edu.cn
+
+# ABSTRACT
+
+Learned image compression (LIC) has demonstrated superior rate-distortion (R-D) performance compared to traditional CODECs, but is challenged by training inefficiency that could incur more than two weeks to train a state-of-the-art model from scratch. Existing LIC methods overlook the slow convergence caused by compacting energy in learning nonlinear transforms. In this paper, we first reveal that such energy compaction consists of two components, i.e., feature decorrelation and uneven energy modulation. On such basis, we propose a linear auxiliary transform (AuxT) to disentangle energy compaction in training nonlinear transforms. The proposed AuxT obtains coarse approximation to achieve efficient energy compaction such that distribution fitting with the nonlinear transforms can be simplified to fine details. We then develop wavelet-based linear shortcuts (WLSs) for AuxT that leverages wavelet-based downsampling and orthogonal linear projection for feature decorrelation and subband-aware scaling for uneven energy modulation. AuxT is lightweight and plug-and-play to be integrated into diverse LIC models to address the slow convergence issue. Experimental results demonstrate that the proposed approach can accelerate training of LIC models by 2 times and simultaneously achieves an average  $1\%$  BD-rate reduction. To our best knowledge, this is one of the first successful attempt that can significantly improve the convergence of LIC with comparable or superior rate-distortion performance. Code will be released at https://github.com/qingshi9974/AuxT
+
+# 1 INTRODUCTION
+
+Recent advances in learned image compression (LIC) (Ballé et al., 2018; Minnen et al., 2018; Cheng et al., 2020; Liu et al., 2023; Li et al., 2024a) have been attracting increasing attention. LIC usually employs a pair of nonlinear analysis and synthesis transforms (Ballé et al., 2020) to achieve mapping between input images and their latent representations. The nonlinear transforms are jointly optimized along with uniform quantizer and entropy model to render superior rate-distortion (R-D) performance than traditional handcrafted image CODECs.
+
+Despite their promising R-D performance, existing LIC methods suffer from slow convergence on training. We take TCM (Liu et al., 2023), the recent state-of-the-art LIC model, as an example. TCM requires more than 2 million training iterations that take more than 15 days on an NVIDIA GeForce RTX 4090 GPU to train a model for a single R-D point, and needs 6 models at different rates to guarantee satisfactory R-D performance over the whole rate region. Moreover, low-efficiency training hinders fast fine-tuning for varying image datasets and downstream tasks or adaptation to variable target bitrates. This issue limits the practical application of LIC models.
+
+In fact, efficient training of LIC models has not been thoroughly explored, despite the properties of nonlinear transforms have been studied. Previous studies (Duan et al., 2022; He et al., 2022; Li et al., 2024c) find that the analysis transform in LIC exhibits energy compaction property (i.e., the energy of latent representation concentrates in several dominant channels), and further observe two characteristics of nonlinear transforms to realize energy compaction (i.e., feature decorrelation and uneven energy modulation). However, they ignore to consider the evolution of energy compaction during training. In this paper, we further investigate this problem, and reveal that LIC models struggle to improve the efficiency of energy compaction through the training process. As
+
+![](images/bcc5319983aa79d21de5e375a36b3b3996d886fc9ee67147ef3a68bdfa144496.jpg)  
+(a) Energy compaction vs. iterations
+
+![](images/9e31501da57de4efa3bad64d100b7e8c91160e1fea79e3bdf801580ed10b18e9.jpg)  
+(b) Test R-D loss vs. iterations  
+Figure 1: Illustration on the evolution of energy compaction and R-D loss during training of TCM-small with and without the proposed auxiliary transform (AuxT). The energy is computed by the  $L_{2}$  norm. (a) Energy compaction of the top  $10\%$  of channels with highest energy. The numbers on the curve is the corresponding energy ratio relative to the total energy of all channels. (b) Convergence curves of the R-D loss. Both TCM-small (Liu et al., 2023) and TCM-small+AuxT are trained with  $\lambda$  set to 0.0483, and the energy and test R-D loss are averaged over the Kodak test set.
+
+demonstrated in Figure 1a (blue line), during training, an LIC model could use only  $10\%$  channels of the latent representation to preserve  $87\%$  of its energy in the first 0.5 million iterations, but would require extra 1.5 million iterations to further increase the energy by  $6\%$ . This suggests the necessity to ease energy compaction in training nonlinear transform and thus improve training efficiency.
+
+Motivated by these facts, we propose to disentangle the learning process of energy compaction for nonlinear transform with a simple yet effective linear auxiliary transform (AuxT). AuxT works as a bypass transform in addition to the nonlinear transform to achieve coarse approximation of feature decorrelation and uneven energy modulation. Such a design enables the nonlinear transform to focus on fitting the distribution of fine details and consequently yield faster convergence. Specifically, AuxT contains several wavelet-based linear shortcuts (WLSs) with each comprising a wavelet-based downsampling and an orthogonal linear projection for feature decorrelation and a subband-aware scaling for uneven energy modulation.
+
+The auxiliary transform is lightweight and plug-and-play, which can be integrated seamlessly with existing LIC models. Figure 1 presents an instance of incorporating AuxT with TCM (specifically, the small version). Compared to vanilla TCM, TCM with AuxT rapidly reduces the channel-wise correlations of latent representations and evidently enhances energy compaction with identical training iterations. Consequently, the training process of LIC is significantly accelerated by introducing the lightweight AuxT.
+
+To our best knowledge, this is one of the first successful attempt for LIC that significantly accelerates the convergence of training while achieving comparable or superior R-D performance. The contributions of this paper are summarized as below.
+
+- We interpret the training process of LIC from the perspective of energy compaction property, and reveal the low efficiency of existing nonlinear transforms in achieving feature decorrelation and uneven energy modulation for energy compaction.  
+- We design a novel method to accelerate LIC training. Specifically, we propose a lightweight and plug-and-play auxiliary transform (AuxT) to achieve energy compaction with coarse feature decorrelation and uneven energy modulation.  
+- We demonstrate superior efficiency of AuxT on diverse LIC models, achieving approximately  $40\%$  to  $70\%$  reduction in training time with an average BD-rate reduction of  $1.3\%$ .
+
+# 2 PRELIMINARY: LEARNED IMAGE COMPRESSION
+
+Learned image compression models typically consist of two fundamental components: nonlinear transforms and an entropy model. The nonlinear transforms include a nonlinear analysis transform
+
+![](images/df7015751d73eb2576ea846284e28d0d593bd5079d8e2df9f444ac4a36160339.jpg)  
+(a)
+
+![](images/bf0530a3443a8b11c9a74ba594c3e09e5ea589dd9d8b8e0914051056263ba373.jpg)  
+(b)  
+Figure 2: Characteristics of energy distributions in latent representations for TCM-small (Liu et al., 2023), averaged over 24 images from the Kodak dataset. (a) Channel-wise energy distribution after convergence. We sort the channels of latent and split them into 10 groups. The energy is predominantly concentrated in the first group associated with low-frequency features, while other groups corresponding to high-frequency features carry significantly less energy. (b) Evolution of the energy distribution during the training process. The total energy of each group is plotted over the training process, with the dotted line representing the total energy of all channels. The numbers on the curve indicate the energy ratio of the group with the highest energy.
+
+$g_{a}(\cdot ;\pmb{\theta}_{a})$  with trainable parameters  $\pmb{\theta}_{a}$  at the encoder side and a nonlinear synthesis transform  $g_{s}(\cdot ;\pmb{\theta}_{s})$  with  $\pmb{\theta}_{s}$  at the decoder side.  $g_{a}(\cdot)$  transforms the input image  $\pmb{x}$  into a latent representation  $\pmb{y} = g_{a}(\pmb{x};\pmb{\theta}_{a})$ , which is then discretized to  $\hat{\pmb{y}} = Q(\pmb{y})$  using uniform quantization  $Q(\cdot)$ , whereas  $g_{s}(\cdot)$  obtains the reconstructed image  $\hat{\pmb{x}} = g_{s}(\hat{\pmb{y}};\pmb{\theta}_{s})$  from the quantized latent  $\hat{\pmb{y}}$ .
+
+The probability of  $\hat{\pmb{y}}$  is usually estimated with multi-dimensional Gaussians with mean  $\pmb{\mu}$  and scale  $\pmb{\sigma}$ , where the mean and scale are estimated by the entropy model using side information  $\hat{z}$  and contextual information  $\phi$ . The estimated probability is further employed in entropy coding that outputs a bitstream with an average length calculated by
+
+$$
+\mathcal {R} = \mathbb {E} _ {\boldsymbol {x} \sim p _ {x}} \left[ - \log_ {2} p _ {\hat {\boldsymbol {y}}} (\hat {\boldsymbol {y}}) \right] + \mathbb {E} _ {\boldsymbol {x} \sim p _ {x}} \left[ - \log_ {2} p _ {\hat {\boldsymbol {z}}} (\hat {\boldsymbol {z}}) \right]. \tag {1}
+$$
+
+Consequently, LIC models are optimized using the R-D loss, which incorporates a Lagrangian multiplier  $\lambda$  that controls the trade-off between the rate  $\mathcal{R}$  and the distortion  $\mathcal{D}$ , expressed as:
+
+$$
+\mathcal {L} _ {R D} = \mathcal {R} + \lambda \cdot \mathcal {D}. \tag {2}
+$$
+
+According to Equation 2, different bitrates are realized using different values of  $\lambda$ . Please refer to Appendix A.1 for more related works on LIC.
+
+# 3 ENERGY BASED INTERPRETATION OF LIC TRAINING
+
+Understanding the training process of LIC is crucial to addressing the slow convergence issue. In this section, we begin by re-examining the results of previous work on the energy-compaction property of nonlinear transforms (Duan et al., 2022; He et al., 2022; Li et al., 2024c), and then show the two reasons why the energy-compaction property is hard to learn.
+
+Traditional transform coding usually leverages orthogonal linear transforms such as Discrete Cosine Transform (DCT), Discrete Wavelet Transform (DWT), Karhunen-Loève Transform (KLT), and multiscale geometric analysis to aggregate most energy of the image into few components. And the energy-compaction property has a significant contribution to the compression performance. In an optimized LIC model, the majority of energy in the latent representation is concentrated in a few dominant channels, while the other channels contain considerably less energy (as illustrated in Figure 2). According to (Li et al., 2024c), the dominant channels holding high energy are low-frequency channels, whereas the others are high-frequency channels.
+
+For a clear demonstration, we categorize these latent representation channels into 10 groups according to their average energy, and visualize the evolution of the energy ratio (i.e., the ratio of energy in the total energy) for each group during the training process. Figure 2 shows that the LIC model is trained to gradually achieve energy compaction. Specifically, we obtain the following two observations.
+
+(i) High-energy channels exhibit a trend of increasing energy ratio, but the growth of both energy and energy ratio is slow.  
+(ii) Low-energy channels are not stable in energy ratios, and might adversely affect the training stability of LIC.
+
+These observations highlight the challenge of slow energy compaction in LIC training. To address this challenge, we further identify two concrete reasons behind it.
+
+Feature Decorrelation. Traditional transform coding inherently obtains energy-compacting representation through orthogonal linear transforms that are well designed for decorrelating frequency components in a structured manner. LIC learns nonlinear transforms to adapt to the source distribution of natural images and also achieves decorrelation in latent representation. In Figure 3, we estimate the average similarity between each two channels based on their guided backpropagation patterns (Springenberg et al., 2015) on the Kodak dataset throughout the whole training process. The pairwise channel similarity suggests the correlation of any two channels of latent representations is gradually reduced during the training process. However, unlike traditional transform coding, LIC requires numerous iterations in training to achieve ideal decorrelation, since orthogonality is not guaranteed for the analysis transform.
+
+![](images/b0517f594958e6960ecb5fa5663dac88b5d6fb1db5df0c5465ea034d738f2f5d.jpg)  
+Figure 3: Normalized histograms of pairwise channel similarities in the analysis transform for different training iterations. We rescale the  $x$ -axis for better visualization. Please Zoom in for a better view
+
+Uneven Energy Modulation. Traditional transform coding commonly applies different quantization steps on the transform coefficients of different frequencies. The low-frequency coefficients usually adopt smaller quantization steps than high-frequency coefficients to reduce average distortion. On the contrary, LIC models employ uniform quantizers with a fixed quantization step of 1 for all the latent representation. Low-frequency channels that inherently carry higher energy require additional amplitude amplification (i.e., uneven energy modulation) to achieve finer quantization. Thus, the analysis transform gradually learns to amplify low-frequency channels, and overall energy in the latent representation increases as training proceeds. Figure 4 shows that the max intensity of latent representations is significantly scaled up from a value of 1 in the source space to over 100 in the latent space. However, the
+
+![](images/9b9c33bad3ed4c875a6a2f9a6cf13d6b2f6b876c132219357aa24983d4530559.jpg)  
+Figure 4: Maximum output intensity for each layer of the analysis transform  $g_{a}$  of TCM-small (Liu et al., 2023). The blue area is subsampling layer.
+
+scaling-up primarily occurs in the last layer of neural network realizing the analysis transform. This causes unbalanced gradients that could slow down the training process.
+
+In light of these findings, we propose a method that disentangles the training process of the nonlinear transforms into feature decorrelation and uneven energy modulation to accelerate training.
+
+# 4 AUXILIARY TRANSFORM FOR DISENTANGLED TRAINING
+
+We introduce an auxiliary transform (AuxT) to disentangle the learning process of the nonlinear transform such that the difficulty in achieving energy compaction in training could be reduced and the convergence of training could be accelerated. The proposed AuxT works in parallel with
+
+![](images/8fa8965f33a02f87d16e435a0b65a09bec6f078cd04cb36603ab895811723910.jpg)  
+(a) Overview of proposed method
+
+![](images/47c9850c21845b34d8a65e6b3882ffb97d15e7c7afe94404fad60d3b4488ad04.jpg)  
+(b) WLS
+
+![](images/5e71c6b52f83783858754d7b1229fd3ea6e15b9bde24950a8c6ce53d2aeca729.jpg)  
+(c) iWLS  
+Figure 5: (a) Overview of the proposed method. Without loss of generality, we adopt the nonlinear transform from TCM (Liu et al., 2023) for illustration of our Auxiliary Transform (AuxT), while it can also integrate seamlessly with other LIC models. RBS and RBU denote the Residual Block with Stride and Residual Block Upsampling, respectively. Our method does not involve modifications to the entropy model, we omit the context model in this figure for simplicity. (b) Proposed wavelet-based linear shortcut (WLS) for analysis transform, where DWT, subband-aware scaling and orthogonal linear projection (OLP) are performed sequentially. (c) iWLS for synthesis transform, which implements the inverse operation of WLS.
+
+the nonlinear transform and efficiently provides a complementary latent representation with coarse decorrelation and energy modulation. This allows the nonlinear transform to pay more attention to learning latent representations for fine details, ultimately achieving higher compression performance with significantly reduced overall training effort. AuxT is achieved with multiple wavelet-based linear shortcuts (WLSs), where each WLS consists of three components in a sequence, i.e., wavelet-based downsampling, subband-aware scaling, and orthogonal linear projection (OLP).
+
+# 4.1 WAVELET-BASED LINEAR SHORTCUT (WLS)
+
+Wavelet-based down-sampling is employed to achieve feature decorrelation. Besides, it achieves spatial down-sampling to align with the common design of LIC models. Let  $P \in \mathbb{R}^{H \times W \times C}$  be the input of WLS, where  $H$ ,  $W$ , and  $C$  denote the height, width, and channel respectively. The input input  $P$  is down-sampled by the Discrete Wavelet Transform (DWT). For simplicity, we use the Haar wavelet to decompose  $P$  into four wavelet subbands  $P_{LL} \in \mathbb{R}^{\frac{H}{2} \times \frac{W}{2} \times C}$ ,  $P_{LH} \in \mathbb{R}^{\frac{H}{2} \times \frac{W}{2} \times C}$ ,  $P_{HL} \in \mathbb{R}^{\frac{H}{2} \times \frac{W}{2} \times C}$ , and  $P_{HH} \in \mathbb{R}^{\frac{H}{2} \times \frac{W}{2} \times C}$ . The filters to produce the four subbands are
+
+$$
+\boldsymbol {f} _ {L L} = \frac {1}{2} \left[ \begin{array}{l l} 1 & 1 \\ 1 & 1 \end{array} \right], \boldsymbol {f} _ {L H} = \frac {1}{2} \left[ \begin{array}{c c} 1 & 1 \\ - 1 & - 1 \end{array} \right], \boldsymbol {f} _ {H L} = \frac {1}{2} \left[ \begin{array}{c c} 1 & - 1 \\ 1 & - 1 \end{array} \right], \boldsymbol {f} _ {H H} = \frac {1}{2} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right], \tag {3}
+$$
+
+where  $\pmb{f}_{LL}$  is the low-frequency filter, and  $\pmb{f}_{LH},\pmb{f}_{HL}$ , and  $\pmb{f}_{HH}$  are the high-frequency filters.
+
+Subband-aware scaling. Considering lower-frequency channels require greater energy amplification (i.e., uneven energy modulation) than higher-frequency channels, we design a subband-aware scaling method to adaptively amplify the energy of wavelet subbands. Specifically, each WLS employs four learnable scaling vectors,  $\boldsymbol{s}_{ll}, \boldsymbol{s}_{lh}, \boldsymbol{s}_{hl}, \boldsymbol{s}_{hh} \in \mathbb{R}^{C}$  to scale the corresponding subbands  $P_{LL}, P_{LH}, P_{HL},$  and  $P_{HH}$ , respectively. The scaled subbands are then concatenated along the channel dimension to form  $\tilde{P} \in \mathbb{R}^{\frac{H}{2} \times \frac{W}{2} \times 4C}$ :
+
+$$
+\tilde {\boldsymbol {P}} = \operatorname {C o n c a t} \left(\boldsymbol {P} _ {l l} \odot e ^ {\boldsymbol {s} _ {l l}}, \boldsymbol {P} _ {l h} \odot e ^ {\boldsymbol {s} _ {l h}}, \boldsymbol {P} _ {h l} \odot e ^ {\boldsymbol {s} _ {h l}}, \boldsymbol {P} _ {h h} \odot e ^ {\boldsymbol {s} _ {h h}}\right), \tag {4}
+$$
+
+where  $\odot$  denotes channel-wise multiplication. The values of  $s_{ll}, s_{lh}, s_{hl}, s_{hh}$  are initialized with 1, 0.5, 0.5, 0, respectively, reflecting uneven energy modulation for different subbands.
+
+Orthogonal Linear Projection (OLP). Channel projection is considered to ensure that the output channels align with the intermediate features of the nonlinear transform. However, an unconstrained
+
+linear projection layer is inferior in channel-wise decorrelation and causes excessive training time, due to the lack of orthogonality guarantees. To address this issue, we develop OLP by applying an orthogonality constraint  $||\boldsymbol{W}^T\boldsymbol{W} - \boldsymbol{I}||_F^2$  to a  $1 \times 1$  convolution layer  $\mathbf{W} \in \mathbb{R}^{4C \times D}$ . OLP projects  $\tilde{\boldsymbol{P}}$  to obtain the final output of WLS, i.e.,  $\hat{\boldsymbol{P}} \in \mathbb{R}^{\frac{H}{2} \times \frac{W}{2} \times D}$ . Please refer to Appendix A.5 for more details about the orthogonality constraint.
+
+# 4.2 OVERALL ARCHITECTURE WITH MULTI-STAGE SHORTCUTS
+
+As illustrated in Figure 5, we apply four stacked WLS in the analysis transform and four stacked iWLS in the synthesis transform. For the analysis transform, the output of each WLS will be added to the output of each down-sampling layer of the nonlinear transform  $g_{a}$ . The output of the last WLS (i.e.,  $\hat{P}_{final}$ ) will be added to the output of analysis transform  $F$  to form the final latent representation  $y$  that is subsequently quantized and encoded. For the synthesis transform, iWLS implements the inverse operation of WLS in the analysis transform, where the DWT is replaced by the inverse DWT (IDWT) and the channel-wise multiplication in Equation 4 is replaced by a channel-wise division to achieve corresponding energy reduction.
+
+Progressive Energy Modulation. The orthogonality of the DWT and our OLP ensures that they act as energy-preserving transforms. In this structure, multiple subband-aware scaling operations in the multi-stage architecture can progressively and efficiently scales energy, which can avoid dramatic changes in energy and amplitude, leading to a more stable training process.
+
+Loss Function. The proposed architecture is optimized in an end-to-end fashion using a combination of the R-D loss  $\mathcal{L}_{RD}$  in Equation 2 and orthogonality regularization loss  $\mathcal{L}_{orth}$  for OLP. The overall loss function is formulated as
+
+$$
+\mathcal {L} _ {\text {o v e r a l l}} = \mathcal {L} _ {R D} + \lambda_ {\text {o r t h}} \mathcal {L} _ {\text {o r t h}}, \quad \text {w h e r e} \quad \mathcal {L} _ {\text {o r t h}} = \sum_ {\boldsymbol {W} \in \mathcal {W}} \| \boldsymbol {W} ^ {\top} \boldsymbol {W} - \boldsymbol {I} \| _ {F} ^ {2}, \tag {5}
+$$
+
+where  $\mathcal{W}$  is the set of the weight matrix for all OLPs and  $\lambda_{orth}$  is the regularization parameter.
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENTAL SETUP
+
+Training. We apply our AuxT to several mainstream LIC models to show its effectiveness, including mb2018 mean (Minnen et al., 2018), ELIC (He et al., 2022), STF (Zou et al., 2022), and TCM (Liu et al., 2023). All the models are trained on the ImageNet-1k (Deng et al., 2009) dataset and optimized using Adam optimizer (Kingma & Ba, 2015). We set the batch size to 16 for convolution-based LIC models (Minnen et al., 2018; He et al., 2022) and 8 for transformer-based LIC models Zou et al. (2022); Liu et al. (2023). We train the models without our AuxT for 0.6M and 2M iterations respectively, and train the models with our AuxT for 0.6M and 1M iterations, respectively. The learning rate is initialized as  $10^{-4}$  and is decayed by a factor of 10 after 0.55M iterations for 0.6M training iterations scenario, after 0.9M iterations for 1M training iterations scenario, and after 1.8M iterations for 2M training iterations scenario.
+
+Two kinds of quality metrics, i.e., mean square error (MSE) and multi-scale structural similarity (MS-SSIM), are used to measure the distortion  $\mathcal{D}$ . The Lagrangian multiplier  $\lambda$  in the R-D loss used for training MSE-optimized models are  $\{0.0025, 0.0035, 0.0067, 0.0130, 0.0250, 0.0483\}$ , and those for MS-SSIM-optimized models are  $\{2.40, 4.58, 8.73, 16.64, 31.73, 60.50\}$ . The orthogonal regularization weight  $\lambda_{orth}$  is 0.1. Experiments are performed on NVIDIA GeForce RTX 4090 GPU and Intel Xeon Platinum 8260 CPU.
+
+Evaluation. We adopt three benchmark datasets, i.e., Kodak image set (Kodak, 1993) with 24 images of  $768 \times 512$  pixels, Tecnick testset (Asuni & Giachetti, 2014) with 100 images of  $1200 \times 1200$  pixels, and CLIC Professional Validation dataset (CLIC, 2021) with 41 images of at most  $2\mathrm{K}$  resolution, for evaluations. We use both PSNR and MS-SSIM to measure the distortion, and bits per pixel (BPP) to evaluate bitrates. We present the PSNR-BPP results evaluated on Kodak in Table 1, and provide the results of other dataset and MS-SSIM metric in Appendix A.7.
+
+Table 1: Performance comparison of the proposed method applied to various LIC anchor models on Kodak. We report the GMACs calculated using  $768 \times 512$  input images for complexity comparison and GPU hours for training a single R-D point model for time comparison. BD-rate computed from PSNR-BPP curves by comparing with the anchor VTM-18.0 is adopted as the quantitative metric. The relative values represent the comparison with the anchor model trained for 2M iterations  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2"># of Iterations (M)</td><td colspan="2">GMACs</td><td rowspan="2">Training time (GPU hours)</td><td rowspan="2">#Params (M)</td><td rowspan="2">BD-rate (%)</td></tr><tr><td>Enc.</td><td>Dec.</td></tr><tr><td colspan="7">Convolution-based nonlinear transforms</td></tr><tr><td>mbt2018mean (Minnen et al., 2018)</td><td>0.6</td><td>44</td><td>43</td><td>15</td><td>17.6</td><td>34.3</td></tr><tr><td>mbt2018mean (Minnen et al., 2018)</td><td>2.0</td><td>44</td><td>43</td><td>50</td><td>17.6</td><td>25.5</td></tr><tr><td>mbt2018mean + AuxT</td><td>0.6</td><td>49</td><td>48</td><td>18 (-64%)</td><td>18.6</td><td>26.4 (+0.9)</td></tr><tr><td>mbt2018mean + AuxT</td><td>1.0</td><td>49</td><td>48</td><td>30 (-40%)</td><td>18.6</td><td>22.6 (-2.9)</td></tr><tr><td>ELIC (He et al., 2022)</td><td>0.6</td><td>132</td><td>130</td><td>43</td><td>33.8</td><td>-0.5</td></tr><tr><td>ELIC (He et al., 2022)</td><td>2.0</td><td>132</td><td>130</td><td>143</td><td>33.8</td><td>-4.5</td></tr><tr><td>ELIC + AuxT</td><td>0.6</td><td>137</td><td>135</td><td>46 (-68%)</td><td>34.8</td><td>-4.0 (+0.5)</td></tr><tr><td>ELIC + AuxT</td><td>1.0</td><td>137</td><td>135</td><td>76 (-47%)</td><td>34.8</td><td>-5.7 (-1.2)</td></tr><tr><td colspan="7">Transformer-based nonlinear transforms</td></tr><tr><td>STF (Zou et al., 2022)</td><td>0.6</td><td>143</td><td>161</td><td>35</td><td>99.8</td><td>4.6</td></tr><tr><td>STF (Zou et al., 2022)</td><td>2.0</td><td>143</td><td>161</td><td>116</td><td>99.8</td><td>-3.2</td></tr><tr><td>STF + AuxT</td><td>0.6</td><td>144</td><td>162</td><td>36 (-68%)</td><td>100.6</td><td>-3.2 (-0)</td></tr><tr><td>STF + AuxT</td><td>1.0</td><td>144</td><td>162</td><td>61 (-47%)</td><td>100.6</td><td>-5.7 (-2.5)</td></tr><tr><td>TCM-small (Liu et al., 2023)</td><td>0.6</td><td>112</td><td>148</td><td>72</td><td>45.2</td><td>-0.1</td></tr><tr><td>TCM-small (Liu et al., 2023)</td><td>2.0</td><td>112</td><td>148</td><td>240</td><td>45.2</td><td>-5.3</td></tr><tr><td>TCM-small + AuxT</td><td>0.6</td><td>114</td><td>150</td><td>75 (-68%)</td><td>45.8</td><td>-4.8 (+0.5)</td></tr><tr><td>TCM-small + AuxT</td><td>1.0</td><td>114</td><td>150</td><td>125 (-48%)</td><td>45.8</td><td>-6.0 (-0.7)</td></tr><tr><td>TCM-large (Liu et al., 2023)</td><td>0.6</td><td>315</td><td>449</td><td>100</td><td>76.6</td><td>-4.1</td></tr><tr><td>TCM-large (Liu et al., 2023)</td><td>2.0</td><td>315</td><td>449</td><td>330</td><td>76.6</td><td>-10.6</td></tr><tr><td>TCM-large + AuxT</td><td>0.6</td><td>324</td><td>457</td><td>105 (-68%)</td><td>78.2</td><td>-9.7 (+0.9)</td></tr><tr><td>TCM-large + AuxT</td><td>1.0</td><td>324</td><td>457</td><td>175 (-47%)</td><td>78.2</td><td>-11.3 (-0.7)</td></tr></table>
+
+# 5.2 EXPERIMENT RESULTS
+
+Table 1 presents a thorough comparison of the proposed AuxT applied to diverse LIC models. Each anchor LIC model can achieve good R-D performance when trained for long iterations. However, when trained for only 0.6M iterations, these anchor models perform substantially worse due to their slow convergence. By integrating AuxT, we consistently achieve a significant acceleration on training. For instance, after 0.6M iterations,  $\mathrm{STF + AuxT}$  achieves comparable BD-rate performance to STF trained for 2M iterations, reducing training time by  $68\%$ . Furthermore, training  $\mathrm{STF + AuxT}$  for 1M iterations yields a  $2.5\%$  BD-rate improvement over the anchor model trained for 2M iterations, with only half the training time. Since AuxT is lightweight, it only brings a few additional parameter complexity (about 1M) and has little increase on inference complexity ( $1\sim 10$  GMACs). In addition, Figure 1b shows the convergence curve for TCM-small (Liu et al., 2023), it clearly demonstrates that AuxT not only accelerates convergence but also enhances stability. We further compare the inference time in Appendix A.6 and provide the detailed R-D curves in Appendix A.8.
+
+# 5.3 ANALYSIS ON FEATURE DECORRELATION AND ENERGY MODULATION
+
+We analyze the energy and decorrelation behavior of our proposed auxiliary transform along with analysis transform to better understand the disentangling effect of proposed method.
+
+Energy distribution among channels of both branches. Figure 6a illustrates the energy distribution among channels for both the nonlinear transform and AuxT when the model converges. We observe the following key points:
+
+(1) The simple auxiliary transform can effectively capture the high-energy lower-frequency representations. In contrast, the complicated nonlinear transform focuses and contributes more on the intricate low-energy higher-frequency representations.  
+(2) By integrating AuxT, the nonlinear transform exhibits a more balanced energy distribution across channels, which facilitates a stable train when the architecture is sophisticated.
+
+![](images/46a06bdece3153c70324ff73e8207a889870cd512146b8a7f6d4941510ac6ab6.jpg)  
+(a)
+
+![](images/2936f97d2d6ff7c4535e38d0b8c257e1b940e3600bb410b275e0377aa6b2033f.jpg)  
+(b)  
+Figure 6: Visualization of the energy distribution of our proposed auxiliary transform and the nonlinear transform. (a) Energy distribution among channels of different branches, with the dotted line indicating the total energy for both branches. (b) Evolution of energy during training of the two transforms, where the dotted line represents the total energy for both branches over time. (c) Evolution of energy distribution during training, channels are grouped same as Figure 2. The dotted line presents the total energy for all channels. Please Zoom in for a better view.
+
+![](images/5dc187cf0606e17161dc86efa781d74615292f278a6c7260bc6027afe68caea9.jpg)  
+(c)
+
+Evolution of the energy of both branches. Figure 6b shows the evolution of the total energy for both branches during training. AuxT and the nonlinear transform have similar energy at the beginning of the training process. As the training progresses, the energy of the auxiliary transform gradually increases, ultimately exhibiting a significant  $31.1 \times$  energy amplification. In contrast, the energy of the nonlinear transform exhibits a more stable trend, with only a sudden energy reduction in the early stages. After this drop, it only gradually exhibits a  $1.5 \times$  energy amplification. This phenomenon further demonstrates that uneven energy modulation is primarily achieved by AuxT, which alleviates the training burden on the nonlinear transform.
+
+Evolution of the energy of different Channels. Figure 6c illustrates the evolution of total energy across different channels, where the grouping strategy is identical to Figure 2. We observe the following phenomena. (1) AuxT significantly enhances energy compaction. The top  $10\%$  high-energy channels account for  $96\%$  of the total energy in the latent representation after 1M training iterations, compared to  $93\%$  in the anchor model trained for 2M iterations (see Figure 2b). (2) Low-energy high-frequency channels exhibit a more stable energy evolution, showing almost no change in the latter half of training, compared to the unstable trend observed in the anchor model (see Figure 2b). Since these channels are primarily realized by the nonlinear transform, this further demonstrates that AuxT contributes to the stable training of the nonlinear transform.
+
+**Decorrelation efficiency of both branches.** Figure 7 shows the pairwise channel similarity for both AuxT and the nonlinear transform. We observe that the nonlinear transform exhibits a strong decorrelation ability due to its sophisticated design, with most pairwise channel similarities being less than 0.01. Additionally, AuxT alone also demonstrates a decent level of decorrelation. When integrating both branches, the overall decorrelation is further enhanced, leading to even greater decorrelation efficiency and compression performance.
+
+# 5.4 ABLATION STUDIES
+
+We perform ablation studies to further evaluate the effectiveness of our proposed AuxT. Experiments are conducted on TCM-small (Liu et al., 2023), and each model is trained for 0.6M training iterations.
+
+Effect of the components of WLS. We first evaluate the
+
+effectiveness of each WLS component through the following experiments: [A1] removing the orthogonal constraint for OLP; [A2] removing the subband-aware scaling; [A3] replacing DWT with an average pooling layer; and [A4] replacing DWT with a convolutional layer with a stride of 2. For [A3] and [A4], we also remove the subband-aware scaling since DWT is no longer used.
+
+![](images/ae3ba0f493cb5a7b2d5cef7f73cd3b33e7e2d7c34f0ab74006dfc8415cca389b.jpg)  
+Figure 7: Normalized histograms of pairwise channel similarities for different branches. We rescale the  $x$ -axis for better visualization.
+
+![](images/3702e3ccb96fb12fd68c54f4a3ab1531158e6f8339f3f27f27c0816becdfba84.jpg)
+
+![](images/e4a9d91a1663cbb2c84332c8ad940fbfa95297b2f0310ac3c6da33b16b6c63ec.jpg)  
+(c) Effect of AuxT in analysis and synthesis
+
+![](images/a62a0aaadf72dba695f2275f9eb23e235cc66fec5c233ca9b5a3a54a666b6881.jpg)  
+(b) Effect on the base of wavelet
+
+![](images/ece568b9de605a37cb89b4def7ec15565a2f3d531498072fd1f1182b7f8aafae.jpg)  
+(d) Effect of the linearity of AuxT  
+Figure 8: Ablation studies. We use the TCM-small as the anchor model and calculate the rate saving relatives to it. Please Zoom in for a better view.
+
+As shown in Figure 8a, removing the subband-aware scaling results in a notable degradation in performance. However, replacing OLP or DWT with non-orthogonal layers leads to an even more significant performance drop, highlighting the importance of orthogonal linear transforms in WLS for stable energy modulation and enhanced feature decorrelation. Additionally, we observe that the frequency decomposition provided by DWT is essential and cannot be replaced by a simple low-frequency filter such as pooling.
+
+Effect of the bases of wavelet. [A3] and [A4] highlight the importance of DWT in our WLS. We further evaluate the effect of different wavelet bases in DWT. As shown in Figure 8b, employing more complex wavelets like Daubechies wavelet db4 and biorthogonal wavelet bior2.2 leads to a performance degradation, indicating that the simple Haar wavelet is better.
+
+Effect of AuxT in analysis and synthesis. We evaluate the effectiveness of AuxT in the analysis transform  $g_{a}$  and the synthesis transform  $g_{s}$ . Figure 8c indicates that AuxT in  $g_{s}$  yields trivial gain on convergence. The primary benefits come from implementing AuxT in  $g_{a}$ , highlighting the importance of enhancing the feature decorrelation and uneven energy modulation for  $g_{a}$ .
+
+Effect of the linearity of AuxT. Unlike the nonlinear transform used in LIC, our AuxT is a purely linear transform without any nonlinear operators. We demonstrate the necessity of this linearity by introducing a nonlinear layer after the OLP. As shown in Figure 8d, adding either ReLU (Glorot et al., 2011) or GDN (Balle et al., 2016) significantly hinders the performance of AuxT. This is because the nonlinearity causes energy attenuation and disrupts orthogonality, which negatively impacts effective energy compaction. This demonstrates that a simple linear module can be more effective for energy compaction. More ablation studies can be found in Appendix A.3.
+
+# 6 CONCLUSION
+
+In this paper, we propose AuxT, a lightweight and plug-and-play linear auxiliary transform that works as a bypass transform alongside the nonlinear transform to address the slow convergence issue for existing learned image compression (LIC) models. AuxT leverages multiple wavelet-based linear shortcuts to achieve coarse approximations for feature decorrelation and uneven energy modulation, allowing the nonlinear transform to learn simplified distributions, thus accelerating the training without sacrificing performance. Empirical results demonstrate that AuxT is effectively integrated into several mainstream LIC models and speeds up their training process by 2 times, with an average BD-rate reduction of  $1\%$ .
+
+# ACKNOWLEDGEMENT
+
+This work was supported in part by the National Natural Science Foundation of China under Grant 62320106003, Grant 62371288, Grant 62301299, Grant 62401357, Grant 62431017, Grant 62401366, Grant 62120106007, Grant 62125109, Grant U24A20251, and in part by the Program of Shanghai Science and Technology Innovation Project under Grant 24BC3200800.
+
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+
+# A APPENDIX
+
+# A.1 RELATED WORK
+
+Learned Image Compression. Learned image compression has seen significant advancements in recent years due to its impressive rate-distortion (R-D) performance. Unlike traditional transform coding methods that use linear transforms to decorrelate input images, LIC employs nonlinear transforms to achieve a more compact latent representation. Early works (Ballé et al., 2017; 2018; Minnen et al., 2018; Minnen & Singh, 2020) utilized convolutional neural networks (CNNs) to develop these nonlinear transforms, focusing on capturing local patterns in images. Later approaches incorporated non-local blocks (Cheng et al., 2020) to enhance the ability to capture global correlations. More recently, advancements have leveraged attention-based mechanisms and transformer architectures (Lu et al., 2022; Zou et al., 2022; Liu et al., 2023; Li et al., 2024a) for LIC, underscoring the idea that increasing complexity and capabilities can lead to more expressive nonlinear transforms.
+
+Another key aspect of LIC research involves designing more effective entropy models. Earlier works (Balle et al., 2017; 2018) introduced factorized and hyperprior models to boost compression efficiency. More recent efforts (Minnen et al., 2018; Minnen & Singh, 2020; Koyuncu et al., 2022; Qian et al., 2022) have incorporated spatial or channel-wise autoregression into entropy models, further enhancing their performance.
+
+Recent works (Liu et al., 2022; 2024; Li et al., 2024b) in LIC have focused on developing image codec tailored for machine perception rather than solely for human perception, aiming to enhance the performance of downstream machine vision tasks while reducing storage and transmission costs.
+
+Despite these advancements, many existing LIC methods suffer from long training times, requiring millions of iterations to converge effectively. This prolonged training duration poses significant challenges, hindering the real-world applications.
+
+Energy Compaction Property in LIC. Energy compaction is a fundamental characteristic in traditional transform coding, indicating that most of the signal's energy is concentrated into a few components. In the field of LIC, several studies have explored this concept. For instance, Cheng et al. (2019b) and Cheng et al. (2019a) introduced a spatial energy compaction-based penalty in the loss function, encouraging energy concentration within a few channels. He et al. (2022) observed that the analysis transform in LIC inherently exhibits an information compaction property, where a limited number of channels carry significantly more average energy. Additionally, Li et al. (2024c) further demonstrated that higher-energy channels are typically characterized by low-frequency components, while lower-energy channels tend to carry high-frequency information. In our work, we find that the energy compaction process is closely linked to the slow training issue of LIC and propose an auxiliary transform to address this issue.
+
+**Decorrelation for LIC.** Existing LIC methods often struggle with achieving better decorrelation ability to further remove the redundancies in the latent representation. The main solution is to leverage more complex neural network modules to enhance the overall representation ability of nonlinear transforms (Balle et al., 2017; 2018; Minnen et al., 2018; Minnen & Singh, 2020; Lu et al., 2022; Zou et al., 2022; Liu et al., 2023; Li et al., 2024a). In addition, Balle et al. (2016) proposed the Generalized Divisive Normalization (GDN) layer to helps to decorrelate information across channels. Guo et al. (2021) further explored the cross-channel relationships of the latents and achieve better channel-wise decorrelation by the proposed causal context prediction module.
+
+Training Acceleration for Neural Networks. The acceleration of neural network training has been a heated topic in deep learning research.
+
+One key approach to speeding up training is the development of advanced optimizers. In addition to widely-used adaptive learning rate methods like Adam (Kingma & Ba, 2015) and accelerated schemes such as Nesterov momentum Nesterov (1983), several recent approaches have emerged to further enhance training efficiency. For example, Goyal et al. (2017) introduced a scaling rule to adjust the learning rate, enabling faster training when using large mini-batches. Furthermore, You et al. (2017) and You et al. (2020) proposed layer-wise adaptive learning rates, which allow for the scaling of batch size, ultimately reducing training time.
+
+Other research has focused on training neural networks in lower-dimensional subspaces and employing batch selection to improve efficiency. For example, Li et al. (2018) proposed the idea of training in a reduced random subspace to measure the intrinsic dimension of the loss objective. Later work by Gressmann et al. (2020) enhanced this approach by considering the layer-wise structure and re-drawing the random bases at each step, leading to improved training performance. Li et al. (2022) proposed a low-dimensional trajectory hypothesis, which extracts subspaces from historical training dynamics, significantly improving the dimensionality efficiency of neural network training. In addition, Hong et al. (2024) introduced Diversified Batch Selection (DivBS), a reference-model-free method that efficiently selects diverse and representative samples for machine learning training.
+
+In the filed of learned image compression, a concurrent work (Anonymous, 2024) also focuses on training acceleration. This work is inspired by prior research on subspace training (Li et al., 2018; Gressmann et al., 2020; Li et al., 2022) and focuses on modeling the training dynamics of the parameters of LIC model. It reduces the training space dimension and decreases the number of active trainable parameters over time, thereby achieving lower training complexity for LIC models. However, Anonymous (2024) does not thoroughly investigate the underlying reasons of slow convergence in LIC models, nor does it incorporate the specific characteristics of LIC. In contrast, our method leverages these unique characteristics (i.e., energy compaction and uneven energy modulation) and, for the first time, analyzes the training process of LIC from the perspective of energy compaction. We believe that our approach, in combination with the methods presented in Anonymous (2024), can complement each other and further accelerate convergence.
+
+# A.2 DISCUSSION ON THE DWT IN LEARNED IMAGE COMPRESSION.
+
+In end-to-end learned image compression, many works have introduced DWT into the nonlinear transform to enhance its decorrelation ability. Mishra et al. (2020) use DWT to preprocess the input image and separately encode its wavelet subbands. Ma et al. (2020) design a wavelet-like nonlinear transform using the lifting scheme. Fu et al. (2024) replace the down-sampling and up-sampling layers of LIC with DWT and IDWT, respectively, to better remove frequency-domain correlation. We highlight that our method differs from these methods as we do not modify the nonlinear transform itself, but instead introduce a bypass auxiliary transform to address the convergence issue. To further demonstrate the superiority of our method, we conduct experiments to test whether adopting DWT directly in the nonlinear transform can accelerate convergence. Specifically, we replace the sub-sampling and up-sampling layers of TCM with DWT and IDWT, respectively and append a Residual Block before and after the DWT and IDWT layers to maintain the parameter count.
+
+![](images/81e537ba642a7addd825138cb35683599af9a62b8b4ec9c2d2c0420228869db3.jpg)  
+Figure 9: Convergence curves of the test R-D loss.  $\lambda$  is set as 0.0483.
+
+Figure 9 shows the convergence curves of different methods. We observe that replacing all the sub-sampling and up-sampling layers with wavelet yields a slight performance gain compared to
+
+![](images/bf7c5e1786c5ec6c9e1515f97bfdc0ab32b63980445574a73177fd19ff6436be.jpg)  
+Figure 10: Convergence curves of the test R-D loss.  $\lambda$  is set as 0.0483.
+
+Table 2: Effect of the orthogonality regularization weight  $(\lambda_{orth})$  on the R-D performance. The TCM-small+AuxT model is trained for 0.6M iterations by  $\lambda$  set as 0.0483, and the R-D losses are evaluated on the Kodak dataset.  
+
+<table><tr><td>λorth</td><td>Rate (bpp)</td><td>Distortion (PSNR)</td><td>Test R-D Loss</td></tr><tr><td>0</td><td>0.8958</td><td>37.726</td><td>1.447</td></tr><tr><td>0.01</td><td>0.8890</td><td>37.816</td><td>1.429</td></tr><tr><td>0.1</td><td>0.8865</td><td>37.834</td><td>1.424</td></tr><tr><td>1.0</td><td>0.8950</td><td>37.742</td><td>1.445</td></tr></table>
+
+anchor model, but it also results in higher parameter complexity (51.5M vs. the original 45.2M). In addition, replacing all the sub-sampling and up-sampling layers with proposed WLS and iWLS leads to a performance drop. This may be because these additional modules (subband-aware scaling, OLP) and orthogonal constraints hinder the nonlinear transforms from effectively learning latent representations for fine details. This further highlights the importance of the additional parallel branch design in accelerating convergence.
+
+# A.3 ADDITIONAL ABLATION STUDIES
+
+Compared with Energy Compaction Penalty. Cheng et al. (2019b) and Cheng et al. (2019a) introduce a spatial energy compaction-based penalty to encourage the nonlinear transform of Learned Image Compression (LIC) to exhibit strong energy concentration properties. We also apply this penalty in the TCM-small (Liu et al., 2023), with the convergence curve shown in Figure 10. Our observations indicate that this spatial energy compaction-based penalty does enhance energy compaction, with the top  $10\%$  of energy channels accounting for  $99\%$  of the total energy, compared to  $96\%$  for TCM-small + AuxT and  $90\%$  for TCM-small. However, this strong penalty leads to a degradation in rate-distortion (R-D) performance and the training progress is still unstable. In contrast, our AuxT enhances energy compaction in a structured manner, which benefits both convergence and overall R-D performance.
+
+Effect of  $\lambda_{\text{orth}}$ . We further evaluate the effect of the weight of orthogonality regularization loss. We train the TCM-small+AuxT with different value of  $\lambda_{\text{orth}}$  for 0.6M iterations and the R-D losses evaluated on Kodak dataset are listed in the Table 2. The results indicate the best choice of  $\lambda_{\text{orth}}$  is 0.1.
+
+![](images/bad8a7d2a82bce1dce871154f9f401f0f21c2a37a9bc46307a840f6bb94758f1.jpg)  
+(a) TCM-small
+
+![](images/eb3813a629e70506436a4f38da1c255a20fb728551e00e0673f7f06ac5391172.jpg)  
+(b) TCM-small
+
+![](images/8dbc53d6fbd3f25a8819f1280e0f50a093280f65ac7f039eaa65d9bec64566e0.jpg)  
+(c) TCM-small+ AuxT
+
+![](images/284aeff9a3dcb072418e68492e4ea152f39836549b412267ea28b1eb19bce761.jpg)  
+(d) TCM-small  $^+$  AuxT  
+Figure 11: (a) Maximum output intensity for each layer of the analysis transform  $g_{a}$  of TCM-small (b) Average energy for each layer of the analysis transform  $g_{a}$  of TCM-small (c) Maximum output intensity for each layer of the analysis transform  $g_{a}$  and auxiliary transform of TCM-small+Aux (d) Average energy for each layer of the analysis transform  $g_{a}$  and auxiliary transform of TCM-small+Aux.
+
+# A.4 ENERGY MODULATION BEHAVIOR
+
+Figure 11 shows the effect of our subband-aware scaling which can progressively achieve energy modulation, thus avoiding dramatic changes in energy and amplitude, leading to a more stable training prccss.
+
+# A.5 DETAILS FOR ORTHOGONAL CONSTRAINT.
+
+The orthogonality concept has been widely explored in the filed of deep learning (Le et al., 2011; Vorontsov et al., 2017; Bansal et al., 2018; Wang et al., 2020), as it implies energy preservation and encourages filter diversity. In our work, we use the orthogonal kernel regularization to force our orthogonal linear projection  $\mathbf{W} \in \mathbb{R}^{4C \times D}$  as orthogonal layer. Specifically, if  $4C > D$ ,  $\mathbf{W}$  is an undercomplete matrix, and we use a row orthogonal regularizer  $\| \mathbf{W}^T \mathbf{W} - \mathbf{I} \|_F^2$ , where  $\mathbf{I} \in \mathbb{R}^{D \times D}$  is the identity matrix. Conversely, if  $4C < D$ ,  $\mathbf{W}$  becomes an overcomplete matrix, and column orthogonality is encouraged using the penalty  $\| \mathbf{W} \mathbf{W}^T - \mathbf{I}' \|_F^2$ . In our practical implementation, we use the row orthogonal regularizer for both cases, since the row orthogonality and column orthogonality are equivalent in the mean squared error (MSE).
+
+Lemma A.1 The row orthogonality and column orthogonality are equivalent in the mean squared error (MSE), i.e.,  $||\pmb{W}^T\pmb{W} - \pmb{I}||_F^2 = ||\pmb{W}\pmb{W}^T - \pmb{I}'||_F^2 + U$ , where  $U$  is a constant.
+
+Table 3: Comparison on the coding complexity of the proposed method applied to various LIC anchor models. The entropy coding time is excluded.  
+
+<table><tr><td>Model</td><td>Enc. Inference Time (ms)</td><td>Dec. Inference Time (ms)</td></tr><tr><td colspan="3">Convolution-based nonlinear transforms</td></tr><tr><td>mbt2018mean</td><td>35.3</td><td>11.8</td></tr><tr><td>mbt2018mean + AuxT</td><td>35.4</td><td>13.2</td></tr><tr><td>ELIC</td><td>74.2</td><td>53.5</td></tr><tr><td>ELIC + AuxT</td><td>76.7</td><td>55.6</td></tr><tr><td colspan="3">Transformer-based nonlinear transforms</td></tr><tr><td>STF</td><td>104.4</td><td>110.3</td></tr><tr><td>STF + AuxT</td><td>106.0</td><td>111.6</td></tr><tr><td>TCM-small</td><td>185.2</td><td>183.8</td></tr><tr><td>TCM-smal + AuxT</td><td>188.6</td><td>186.4</td></tr><tr><td>TCM-large</td><td>216.4</td><td>215.6</td></tr><tr><td>TCM-large + AuxT</td><td>222.1</td><td>219.6</td></tr></table>
+
+The following proof is provided in the supplementary material of Le et al. (2011). We would like to present it here for the reader's convenience.
+
+Proof. For  $\pmb{W} \in \mathbb{R}^{M \times N}$  as an arbitrary matrix, we denote  $\| \pmb{W}^T \pmb{W} - \pmb{I}_N \|_F^2$  as  $L_r$  and  $\| \pmb{W} \pmb{W}^T - \pmb{I}_M \|_F^2$  as  $L_c$ .
+
+$$
+\begin{array}{l} L _ {r} = \left\| \boldsymbol {W} ^ {T} \boldsymbol {W} - \boldsymbol {I} _ {N} \right\| _ {F} ^ {2} \\ = \operatorname {t r} \left[ \left(\boldsymbol {W} ^ {T} \boldsymbol {W} - \boldsymbol {I} _ {N}\right) ^ {T} \left(\boldsymbol {W} ^ {T} \boldsymbol {W} - \boldsymbol {I} _ {N}\right) \right] \\ = \operatorname {t r} \left(\boldsymbol {W} ^ {T} \boldsymbol {W} \boldsymbol {W} ^ {T} \boldsymbol {W}\right) - 2 \operatorname {t r} \left(\boldsymbol {W} ^ {T} \boldsymbol {W}\right) + \operatorname {t r} \left(\boldsymbol {I} _ {N}\right) \\ = \operatorname {t r} \left(\boldsymbol {W} \boldsymbol {W} ^ {T} \boldsymbol {W} \boldsymbol {W} ^ {T}\right) - 2 \operatorname {t r} \left(\boldsymbol {W} \boldsymbol {W} ^ {T}\right) + \operatorname {t r} \left(\boldsymbol {I} _ {M}\right) + N - M \tag {6} \\ = \operatorname {t r} \left[ \boldsymbol {W} \boldsymbol {W} ^ {T} \boldsymbol {W} \boldsymbol {W} ^ {T} - 2 \boldsymbol {W} \boldsymbol {W} ^ {T} + \boldsymbol {I} _ {M} \right] + N - M \\ = \operatorname {t r} \left[ \left(\boldsymbol {W} \boldsymbol {W} ^ {T} - \boldsymbol {I} _ {M}\right) \left(\boldsymbol {W} \boldsymbol {W} ^ {T} - \boldsymbol {I} _ {M}\right) \right] + N - M \\ = \| \boldsymbol {W} \boldsymbol {W} ^ {T} - \boldsymbol {I} _ {M} \| _ {F} ^ {2} + N - M \\ = L _ {c} + U \\ \end{array}
+$$
+
+where  $U = N - M$ .
+
+# A.6 COMPARISON ON THE CODING COMPLEXITY
+
+We compare the coding complexity of the proposed AuxT the proposed method applied to various LIC anchor models. The coding complexity is measured by inference latency during encoding and decoding process, where the entropy coding time is excluded. The experiments show that the increase on the additional latency time caused by our AuxT can be ignored.
+
+# A.7 ADDITIONAL BD-RATE RESULTS
+
+Table 4 provide the BD-rate results for MS-SSIM metric evaluated on Kodak dataset (Kodak, 1993) and Table 5 provide the BD-rate results for PSNR metric evaluated on the CLIC (CLIC, 2021) and Tecnick (Asuni & Giachetti, 2014) dataset.
+
+# A.8 ADDITIONAL R-D CURVES.
+
+As shown in Figure 12, Figure 13, Figure 14, Figure 15, and Figure 16, we present the detailed rate-distortion (R-D) curves obtained by integrating our proposed auxiliary transform (AuxT) with various LIC anchor models at different training iterations.
+
+Table 4: Comparison on the performance of the proposed method applied to TCM (Liu et al., 2023). The BD-rate is computed from MS-SSIM-BPP curves evaluated on the Kodak (Kodak, 1993) dataset as the quantitative metric with VTM-18.0 as the anchor.  
+
+<table><tr><td>Model</td><td># of Iterations (M)</td><td>BD-rate (%) with MS-SSIM Kodak (Kodak, 1993)</td></tr><tr><td colspan="3">Convolution-based nonlinear transforms</td></tr><tr><td>TCM-small</td><td>2.0</td><td>-44.6</td></tr><tr><td>TCM-small + AuxT</td><td>1.0</td><td>-45.7 (-1.1)</td></tr><tr><td>TCM-large</td><td>2.0</td><td>-48.3</td></tr><tr><td>TCM-large + AuxT</td><td>1.0</td><td>-49.0 (-0.7)</td></tr></table>
+
+Table 5: Comparison on the performance of the proposed method applied to various LIC anchor models. The BD-rate is computed from PSNR-BPP curves evaluated on the CLIC (CLIC, 2021) and Tecnick (Asuni & Giachetti, 2014) dataset as the quantitative metric with VTM-18.0 as the anchor.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2"># of Iterations (M)</td><td colspan="2">BD-rate (%) with PSNR</td></tr><tr><td>Tecnick (Asuni &amp; Giachetti, 2014)</td><td>CLIC (CLIC, 2021)</td></tr><tr><td colspan="4">Convolution-based nonlinear transforms</td></tr><tr><td>mbt2018mean</td><td>2.0</td><td>21.9</td><td>26.4</td></tr><tr><td>mbt2018mean + AuxT</td><td>1.0</td><td>20.6 (-1.3)</td><td>24.2(-2.2)</td></tr><tr><td>ELIC</td><td>2.0</td><td>-7.6</td><td>-2.0</td></tr><tr><td>ELIC + AuxT</td><td>1.0</td><td>-8.2 (-0.6)</td><td>-3.2 (-1.2)</td></tr><tr><td colspan="4">Transformer-based nonlinear transforms</td></tr><tr><td>STF</td><td>2.0</td><td>-5.6</td><td>-1.6</td></tr><tr><td>STF + AuxT</td><td>1.0</td><td>-6.6 (-1.0)</td><td>-3.7 (-2.1)</td></tr><tr><td>TCM-small</td><td>2.0</td><td>-5.9</td><td>-3.2</td></tr><tr><td>TCM-small + AuxT</td><td>1.0</td><td>-6.6 (-0.7)</td><td>-4.0 (-0.8)</td></tr><tr><td>TCM-large</td><td>2.0</td><td>-11.4</td><td>-8.0</td></tr><tr><td>TCM-large + AuxT</td><td>1.0</td><td>-11.9 (-0.5)</td><td>-8.4 (-0.4)</td></tr></table>
+
+![](images/ec48195b2a66a7fc1a1f02fb45f78dc5094dee1f186b03a969a3370a91fc98d7.jpg)  
+Figure 12: R-D curve for mbt2018mean
+
+![](images/c93950f9c241d3b4b8f297fb0d87e1109e2f3fcad3ebf195592c42f9ba0f0e39.jpg)  
+Figure 13: R-D curve for ELIC
+
+![](images/caba62ac68aac89ad6e0659f3cd6c1ec0bb34f3d63d9da6e96b2becd0962095c.jpg)  
+Figure 14: R-D curve for STF
+
+![](images/88b514096db491042189339ea709570aa00233e0dbda1228e0db538f82259555.jpg)  
+Figure 15: R-D curve for TCM-small
+
+![](images/cf1c9519e35b7e57a47613a55e6d5e96ca1a069fbb45bcd6a1325b98bc77dd86.jpg)  
+Figure 16: R-D curve for TCM-large

2025/On Disentangled Training for Nonlinear Transform in Learned Image Compression/images.zip

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2025/On Quantizing Neural Representation for Variable-Rate Video Coding/d7081a60-493e-4148-b3bd-5434e86e0673_origin.pdf

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2025/On Quantizing Neural Representation for Variable-Rate Video Coding/full.md

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+# ON QUANTIZING NEURAL REPRESENTATION FOR VARIABLE-RATE VIDEO CODING
+
+Junqi Shi, Zhujia Chen, Hanfei Li, Qi Zhao, Ming Lu\*Tong Chen, Zhan Ma  
+School of Electronic Science and Engineering, Nanjing University  
+{junqishi, zhujiachen, hanfei.li, qizhao}@smail.nju.edu.cn,  
+{minglu, chentong, mazhan}@nju.edu.cn
+
+# ABSTRACT
+
+This work introduces NeuroQuant, a novel post-training quantization (PTQ) approach tailored to non-generalized Implicit Neural Representations for variable-rate Video Coding (INR-VC). Unlike existing methods that require extensive weight retraining for each target bitrate, we hypothesize that variable-rate coding can be achieved by adjusting quantization parameters (QPs) of pre-trained weights. Our study reveals that traditional quantization methods, which assume inter-layer independence, are ineffective for non-generalized INR-VC models due to significant dependencies across layers. To address this, we redefine variable-rate INR-VC as a mixed-precision quantization problem and establish a theoretical framework for sensitivity criteria aimed at simplified, fine-grained rate control. Additionally, we propose network-wise calibration and channel-wise quantization strategies to minimize quantization-induced errors, arriving at a unified formula for representation-oriented PTQ calibration. Our experimental evaluations demonstrate that NeuroQuant significantly outperforms existing techniques in varying bitwidth quantization and compression efficiency, accelerating encoding by up to eight times and enabling quantization down to INT2 with minimal reconstruction loss. This work introduces variable-rate INR-VC for the first time and lays a theoretical foundation for future research in rate-distortion optimization, advancing the field of video coding technology. The materials will be available at https://github.com/Eric-qi/NeuroQuant.
+
+# 1 INTRODUCTION
+
+Implicit Neural Representations (INRs) (Sitzmann et al., 2020; Chen et al., 2021a) have recently introduced a new approach to video coding. They focus on learning a mapping from coordinates, like frame indices, to pixel values, such as colors. This represents a significant departure from the widely used variational autoencoder (VAE)-based frameworks (Lu et al., 2019; Li et al., 2021a; Lu et al., 2024), which rely on generalized models trained on large datasets to create compact representations for various input signals. Instead, INR-based video coding (INR-VC) encodes each video as a unique neural network through end-to-end training, removing the need for extensive datasets. By using specific, non-generalized network weights for each video, INR-VC provides a tailored video coding method that has shown promising results (Chen et al., 2023; Kwan et al., 2024a).
+
+INR-VC typically focuses on two main objectives: 1) Representation, where a neural network models the target video with a minimized distortion, and 2) Compression, where the network's weights are compressed to lower the bitrate. Many prominent methods adopt a consistent precision (quantization bitwidth) for all weights before lossless entropy coding, meaning the video bitrate depends solely on the number of learnable weights. Consequently, independent weight training is needed for each target bitrate, making the process very time-consuming. For example, encoding a 1080p video with 600 frames at a specific bitrate can take up to 10 hours.
+
+To address this inefficiency, we consider how bitrate is managed in pretrained INR-VC model, which is proportional to the sum of the bitwidth of each weight. Inspired by generalized CODECs (Sullivan
+
+![](images/79bfc59c1230b32a08da0d0d4092b7b8939364f93071cbfcc02fe34f9ec6066d.jpg)  
+Figure 1: Left: Typical INR-VCs assume a consistent bitwidth and require separate weight training with varying quantities for each target rate. Right: The proposed NeuroQuant achieves variable rate by modifying the corresponding QPs, significantly reducing training costs.
+
+et al., 2012; Li et al., 2023) that adjust quantization parameters (QPs) (Wang & Kwong, 2008) to control bitrate, we pose the hypothesis: Can variable-rate INR-VC be achieved by modifying the QP of post-training weights, thus eliminating the need for repeated model training for each target rate? In the context of weight quantization, this can be approached by: 1) allocating quantization bitwidth to match the target bitrate, and 2) calibrating QPs to preserve reconstruction fidelity.
+
+However, directly adopting a consistent quantization bitwidth cannot support fine-grained rate control, e.g., only seven options from INT2 to INT8 are available. Additionally, existing mixed-precision quantization methods (Nagel et al., 2021; Chen et al., 2021b), primarily designed for general-purpose neural networks, encounter two key problems when applied to non-generalized INR-VCs. First, mixed-precision algorithms (Dong et al., 2019; 2020; Chen et al., 2021b) typically assume inter-layer independence with tolerable approximation errors. This assumption breaks down in non-generalized INR-VCs, where layers exhibit significant dependencies. Second, popular layerwise calibration methods<sup>1</sup> (Nagel et al., 2020; Li et al., 2021b) also rely on inter-layer independence and aims at generalizing the network, making them unsuitable for INR-VC. Therefore, a dedicated quantization methodology tailored for variable-rate INR-VC is necessary.
+
+In this work, we explore, for the first time, the post-training quantization (PTQ) of weights in nongeneralized INR-VCs. Building on both empirical and theoretical insights, we propose NeuroQuant, a state-of-the-art PTQ approach for INR-VC that enables variable-rate coding without complex retraining. Our contributions tackle key challenges through the following research questions:
+
+1. How to realize variable bitrate (Sec. 3.1): We redefine variable-rate coding as a mixed-precision quantization problem. By theoretically demonstrating that the assumption of inter-layer independence (Dong et al., 2020; Guan et al., 2024) does not apply to non-generalized models, we highlight the necessity of incorporating weight perturbation directionality and off-diagonal Hessian information for sensitivity assessment in quantizing INR-VC. Additionally, we introduce the Hessian-Vector product to simplify computations by eliminating the need for explicit Hessian calculations.  
+2. How to ensure reconstruction quality (Sec. 3.2): We enhance reconstruction quality by calibrating the QPs on the corresponding video-specific weights. Through second-order analysis, we derive a unified formula for MSE-oriented calibration across varying granularities. By considering significant cross-layer dependencies and the diverse distribution of weights, we conduct network-wise calibration and channel-wise quantization to minimize reconstruction loss.  
+3. How NeuroQuant performs (Sec. 4): We benchmark proposed NeuroQuant across various architectures against existing quantization techniques, achieving state-of-the-art results. For variable-rate coding, NeuroQuant outperforms competitors while reducing encoding time by  $80\%$ . Moreover, NeuroQuant is able to quantize weights down to INT2 without notable performance degradation.  
+4. How to advance INR-VC (Sec. 3.3): We revisit INR-VC through the lens of variational inference, proposing that the success of NeuroQuant stems from resolving the mismatch between the representation and compression. We also suggest that rate-distortion (R-D) optimization is also applicable to INR-VC and has the potential to achieve improved performance.
+
+# 2 PRELIMINARIES
+
+Basic Notations. We follow popular notations used in neural network. Vectors are denoted by lowercase bold letters, while matrices (or tensors) are denoted by uppercase bold letters. For instance,  $\mathbf{W}$  refers to the weight tensor, and  $\mathbf{w}$  is its flattened version. The superscript of  $\mathbf{w}^{(l)}$  indicates the layer index. For a convolutional or a fully-connected layer, we mark input and output vectors by  $\mathbf{x}$  and  $\mathbf{z}$ . Given a feedforward neural network with  $n$  layers, the forward process is expressed as
+
+$$
+\boldsymbol {x} ^ {(l + 1)} = h (\boldsymbol {z} ^ {(l)}) = h \left(\boldsymbol {w} ^ {(l)} \boldsymbol {x} ^ {(l)}\right), 1 \leq l \leq n, \tag {1}
+$$
+
+where  $h(\cdot)$  denotes the activation function. For simplicity, we omit the additive bias, merging it into the activation. In the following, the notation  $||\cdot||$  represents the Frobenius norm. Suppose  $\pmb{x}$  is sampled from the dataset  $\mathcal{X}$ , then the overall task loss is expressed as  $\mathbb{E}_{\pmb{x}\sim \mathcal{X}}[\mathcal{L}(\pmb{w},\pmb{x})]$ .
+
+INR-based Video Coding. INR-VC operates on the principle that a target video can be encoded into learned weights through end-to-end training. For each frame  $V_{t}$  in an RGB video sequence  $\mathbb{V} = \{V_t\}_{t=1}^T \in \mathbb{R}^{T \times 3 \times H \times W}$ , INR-VC assumes the existence of an implicit continuous mapping  $\mathcal{F} : [0,1]^{d_{in}} \to \mathbb{R}^{d_{out}}$  in the real-world system such that  $V_{t} = \mathcal{F} \circ t$ . According to the Universal Approximation Theorem (Hanin, 2019; Park et al., 2021), the unknown  $\mathcal{F}$  can be approximated by a neural network  $\mathcal{D}$  of finite length  $L_{\mathcal{D}}$ . The estimated  $\hat{V}_t$  is then expressed as:
+
+$$
+\hat {V} _ {t} = \mathcal {D} \circ \mathcal {E} (t) = U _ {L} \circ h \circ U _ {L - 1} \circ \dots \circ h \circ U _ {1} \circ \mathcal {E} (t), \tag {2}
+$$
+
+where  $\mathcal{D}$  consists of cascaded upsampling layers  $U$ , and  $\mathcal{E}(\cdot)$  is an embedding of the timestamp  $t$ . Typically, index-based INR-VCs (Chen et al., 2021a) employ a fixed Positional Encoding function or a learnable grid (Lee et al., 2023) as  $\mathcal{E}(\cdot)$ , while content-based INR-VCs (Chen et al., 2023; Zhao et al., 2023) utilize a learnable encoder. The encoding of INR-VC involves training the learnable weights  $\boldsymbol{w}$  and subsequently compressing  $\boldsymbol{w}$  into a bitstream using quantization and entropy coding techniques. While existing INR-VC works primarily focus on minimizing distortion during the training stage, video coding is fundamentally a R-D trade-off.
+
+Post-Training Quantization. PTQ offers a push-button solution to quantize pretrained models without weights training. It contrasts with Quantization-Aware Training (QAT), which involves both weight optimization and quantization during training, leading to huge training cost. PTQ is generally a two-step process: 1) initializing QPs (e.g., steps) with allocated bitwidth and weight distribution statistics; 2) calibrating QPs to reduce quantization-induced loss. PTQ typically employs uniform affine transformation to map continuous  $w \in \mathbb{R}$  to fixed-point integers  $\hat{w}$ . Traditional methods aim to minimize quantization error  $||\hat{w} - w||$ . However, an increasing number of explorations (Stock et al., 2020; Nagel et al., 2020; Hubara et al., 2021) suggest that this approach can yield sub-optimal results, as the parameter space error does not equivalently reflect task loss. To analyze quantization-induced loss degradation, AdaRound (Nagel et al., 2020) interprets quantization error as weight perturbation, i.e.,  $\hat{w} = w + \Delta w$ . The loss degradation can be approximated using Taylor series:
+
+$$
+\mathbb {E} \left[ \mathcal {L} (\boldsymbol {w} + \Delta \boldsymbol {w}, \boldsymbol {x}) - \mathcal {L} (\boldsymbol {w}, \boldsymbol {x}) \right] \approx \Delta \boldsymbol {w} ^ {T} \cdot \boldsymbol {g} ^ {(\boldsymbol {w})} + \frac {1}{2} \Delta \boldsymbol {w} ^ {T} \cdot \boldsymbol {H} ^ {(w)} \cdot \Delta \boldsymbol {w}, \tag {3}
+$$
+
+where  $g^{(w)} = \mathbb{E}[\nabla_w\mathcal{L}]$  and  $H^{(w)} = \mathbb{E}[\nabla_w^2\mathcal{L}]$  represent expected gradient and the second-order Hessian matrix, respectively. For well-converged weights, gradients tend to be close to 0. AdaRound further assumes inter-layer independence, leading to a diagonal Hessian matrix optimization. BRECQ (Li et al., 2021b) extends AdaRound's layer-wise calibration to block granularity based on inter-block independence. However, these methods can significantly degrade the performance of non-generalized INR-VCs, which exhibit significant dependencies among layers.
+
+Mixed-Precision Quantization. Mixed-precision quantization facilitates fine-grained rate control in INR-VCs, with bit allocation being crucial due to the varying levels of redundancy across layers and their different contributions to overall performance. However, determining optimal bitwidth assignments presents a significant challenge because of the extensive search space. For a network with  $N$  layers and  $M$  candidate bitwidths per layer, exhaustive combinatorial searches exhibit exponential time complexity of  $\mathcal{O}(M^N)$ . To address it, various strategies have been explored, including search-based reinforcement learning Wang et al. (2019); Lou et al. (2019), neural architecture search (Wu et al., 2016), and Hessian-based criteria (Dong et al., 2019; 2020). Despite these efforts, they often prove impractical for INR-VCs, as the search costs may surpass those of retraining a model. Furthermore, many existing criteria lack a robust theoretical basis for their optimality, rendering them less reliable in INR-VC systems.
+
+![](images/c62f9d654a5f5e71b9d36869256e12fe6388837d765db30574a6113be405c766.jpg)  
+(a) Layer-wise sensitivity  $\Omega$
+
+![](images/865b9130d1a9b5f4095750548fbb6b8ff20e21428db21302f88c1d65e207092d.jpg)  
+(b) 2-th loss landscape  
+Figure 2: Examples of quantizing layers in sequence. (a) Different layers exhibits varying sensitivities. (b) Lower  $\Omega$  means flatter loss landscape. (c) Higher  $\Omega$  is otherwise, and the loss landscape shows pronounced directivity, indicating the necessity of considering the direction of  $\Delta w$ .
+
+![](images/e11c3b1bf4dbd51339eb5ae08c94a6fa8bba31b6323a67477334481bbd6469f6.jpg)  
+(c) 6-th loss landscpae
+
+# 3 METHODOLOGY
+
+We introduce the proposed NeuroQuant for high-performance variable-rate INR-VC as follows:
+
+Problem 1 (NeuroQuant). Given learned video-specific weights, the objective of NeuroQuant is to achieve different  $R-D$  trade-offs by quantizing post-training weights with variable QPs. This can be formulated as a rate-constrained optimization process:
+
+$$
+\arg \min  \mathbb {E} [ \mathcal {L} (\mathcal {Q} (\boldsymbol {w}), \mathcal {Q} (\boldsymbol {e})) - \mathcal {L} (\boldsymbol {w}, \boldsymbol {e}) ] \tag {4}
+$$
+
+$$
+s. t. \sum_ {l = 1} ^ {L} \operatorname {P a r a m} \left(\boldsymbol {w} ^ {(l)}\right) \cdot b _ {\boldsymbol {w}} ^ {(l)} + \sum_ {t = 1} ^ {T} \operatorname {P a r a m} \left(\boldsymbol {e} ^ {(t)}\right) \cdot b _ {\boldsymbol {e}} = \mathcal {R} \pm \epsilon , \tag {5}
+$$
+
+where  $\mathcal{R}$  represents the target bitrate,  $e$  denotes the embedding,  $\operatorname{Param}(\cdot)$  indicates the number of parameters, and  $b$  denotes the bitwidth.
+
+We decouple this problem into three sub-problems: 1) Sec. 3.1: The rate-constrained term in Eq. 5 is defined as a mixed-precision bit assignment problem, accounting for fine-grained rate control and varying layer sensitivity; 2) Sec. 3.2: The objective in Eq. 4 is interpreted as QP calibration problem, focusing on calibration and quantization granularity of non-generalized INR-VC; 3) Sec. 3.3: We revisit the entire problem from the perspective of variational inference to provide a broader theoretical grounding.
+
+# 3.1 HOW TO REALIZE VARIABLE BITRATE
+
+Sensitivity Criterion. The core concept of mixed-precision quantization is to allocate higher precision (e.g., greater bitwidth) to sensitive layers while reducing precision in insensitive ones. Sensitivity can be intuitively understood through the flatness of the loss landscape (Li et al., 2018), as illustrated in Fig. 2. A flatter landscape, indicating lower sensitivity, corresponds to smaller loss changes with weight perturbations, whereas a sharper landscape indicates otherwise. Sensitivity essentially captures the curvature of the loss function, often described using second-order information, particularly the Hessian matrix  $\pmb{H}^{(w)}$ .  $\pmb{H}^{(w)}$  defines how perturbations in weights affect task loss. For instance, HAWQ (Dong et al., 2019) uses the top Hessian eigenvalue as a sensitivity criterion, while HAWQ-V2 (Dong et al., 2020) demonstrates that the trace offers a better measure. However, these criteria rely on two key assumptions: 1) Layer Independence: Layers are mutually independent, allowing  $\pmb{H}^{(w)}$  to be treated as diagonal. 2) Isotropy: The loss function is directionally uniform under weight perturbations  $\Delta w$ , meaning only  $\pmb{H}^{(w)}$  is considered, ignoring  $\Delta w$ .
+
+While these assumptions may hold for general-purpose networks, they break down in the context of non-generalized INR-VC, where significant inter-layer dependencies (Fig. 3(c)) and anisotropic behavior (Fig. 2(c)) exist. The following toy examples demonstrate why relying solely on diagonal information from  $H$  is suboptimal.
+
+Example 1 (Inter-Layer Dependencies). Consider three functions,  $\mathcal{F}_1 = 4x^2 + y^2$ ,  $\mathcal{F}_2 = 4x^2 + 2y^2$ , and  $\mathcal{F}_3 = 4x^2 + 2y^2 + 5xy$ . Their corresponding Hessians are given as:
+
+$$
+\boldsymbol {H} ^ {\left(\mathcal {F} _ {1}\right)} = \left[ \begin{array}{l l} 8 & 0 \\ 0 & 2 \end{array} \right], \quad \boldsymbol {H} ^ {\left(\mathcal {F} _ {2}\right)} = \left[ \begin{array}{l l} 8 & 0 \\ 0 & 4 \end{array} \right], \quad \boldsymbol {H} ^ {\left(\mathcal {F} _ {3}\right)} = \left[ \begin{array}{l l} 8 & 5 \\ 5 & 4 \end{array} \right]. \tag {6}
+$$
+
+All three functions share the same top eigenvalue (8), yet  $\mathcal{F}_2$  and  $\mathcal{F}_3$  are clearly more sensitive than  $\mathcal{F}_1$ . Although  $\mathcal{F}_2$  and  $\mathcal{F}_3$  have the same trace (12),  $\mathcal{F}_3$  exhibits greater sensitivity due to the presence of off-diagonal terms (i.e.,  $5xy$ ).
+
+This demonstrates that inter-layer dependencies are overlooked when relying solely on diagonal information (e.g., eigenvalues or traces). Off-diagonal terms are essential to accurately capture sensitivity, highlighting the need to consider the full Hessian matrix. The story does not end there.
+
+Example 2 (Weight Perturbation Directions). Assuming a perturbation  $[\Delta x, \Delta y]$  applied to  $\mathcal{H}^{(\mathcal{F}_3)}$  from above, the increase in loss is approximately proportional to
+
+$$
+\mathcal {F} _ {3} (x + \Delta x, y + \Delta y) - \mathcal {F} _ {3} (x, y) \approx [ \Delta x, \Delta y ] \boldsymbol {H} [ \Delta x, \Delta y ] ^ {T} = 8 \Delta x ^ {2} + 4 \Delta y ^ {2} + 1 0 \Delta x \Delta y. \tag {7}
+$$
+
+Now, consider two cases: 1) Lower perturbation:  $[\Delta x, \Delta y] = [0.1, 0.1]$ ; 2) Higher perturbation:  $[\Delta x, \Delta y] = [0.2, -0.2]$ . The increases in task loss are 0.22 and 0.08, respectively. Surprisingly, the higher perturbation results in a smaller task loss.
+
+This counterintuitive behavior is also observed in practice, where quantizing layers with higher  $H$  sensitivity to a lower bitwidth does not necessarily lead to significant performance degradation. We argue that allocating higher bitwidth to layers primarily reduces  $||\Delta w||$ . However, this does not always guarantee a lower task loss, as  $\mathcal{L}$  is anisotropy under  $\Delta w$  in INR-VC. The key insight is that task loss also depends on the direction of  $\Delta w$ , not just its magnitude  $||\Delta w||$ .
+
+In conclusion, the sensitivity criterion of INR-VC must account for both the full Hessian matrix  $\pmb{H}^{(w)}$  and the direction of weight perturbations  $\Delta w$ . This leads to the following theorem:
+
+Theorem 1. Assuming the INR-VC weights are twice differentiable and have converged to a local minima such that the first and second order optimality conditions are satisfied (i.e., the gradients are zero and the Hessian is positive semi-definite), the optimal sensitivity criteria for mixed-precision INR-VC is given by weighted Hessian information  $\Omega = \Delta \pmb{w}^T \cdot \pmb{H}^{(\pmb{w})} \cdot \Delta \pmb{w}$ .
+
+The criteria  $\Omega$ , formed by Hessian-Vector product, can essentially be interpreted as a linear transformation on  $H^{(w)}$ , accounting for  $H^{(w)}$  along the weight perturbation directions. Existing Hessian-based criteria can be viewed as a degraded version of the proposed  $\Omega$  that neglects the off-diagonal terms. For instance, Eq. 7 would degrade to  $8\Delta x^2 + 4\Delta y^2$ , and thus, loss is independent of inter-variable dependencies and perturbation direction.
+
+Approximating Hessian-Vector Product. The Hessian matrix is challenging to explicitly compute and store as its quadratic complexity relative to the number of weights. Instead of forming  $\pmb{H}^{(w)}$  explicitly, we focus on the sensitivity criterion  $\Omega = \Delta \pmb{w}^T \cdot \pmb{H}^{(w)} \cdot \Delta \pmb{w}$ . Let's construct a function of the form  $\mathcal{G} = g\Delta w$ , where  $g$  is the gradient of  $\mathcal{L}$  with respect to  $\pmb{w}$ . The gradient of  $\mathcal{G}$  can be expressed as:
+
+$$
+\nabla_ {\boldsymbol {w}} \mathcal {G} = \frac {\partial \boldsymbol {g} \Delta \boldsymbol {w}}{\partial \boldsymbol {w}} = \frac {\partial \boldsymbol {g}}{\partial \boldsymbol {w}} \Delta \boldsymbol {w} + \boldsymbol {g} \frac {\partial \Delta \boldsymbol {w}}{\partial \boldsymbol {w}} = \frac {\partial^ {2} \mathcal {L}}{\partial \boldsymbol {w} ^ {2}} \Delta \boldsymbol {w} + \boldsymbol {g} \frac {\partial \Delta \boldsymbol {w}}{\partial \boldsymbol {w}} = \boldsymbol {H} ^ {(\boldsymbol {w})} \Delta \boldsymbol {w} + \boldsymbol {g} \frac {\partial \Delta \boldsymbol {w}}{\partial \boldsymbol {w}}. \tag {8}
+$$
+
+In a converged model,  $g$  approaches 0. Moreover, quantization error can be modeled as a random vector, with its component sampled independently from a Uniform distribution:  $\Delta w \sim U(-0.5, 0.5)$  (Balle et al., 2017). Thus, the second term in Eq. 8 can be ignored. This approximation is also akin to straight-through estimator (STE) (Liu et al., 2022), where  $\frac{\partial \hat{w}}{\partial w} = \frac{\partial w}{\partial w}$  leads to  $\frac{\partial \Delta w}{\partial w} = 0$ . Consequently, we arrive at the final formulation for  $\Omega$ :
+
+$$
+\Omega = \mathbb {E} [ \Delta \boldsymbol {w} ^ {T} \nabla_ {\boldsymbol {w}} \mathcal {G} ], \quad \text {w h e r e} \quad \mathcal {G} = \boldsymbol {g} \Delta \boldsymbol {w} = \mathbb {E} [ \nabla_ {\boldsymbol {w}} \mathcal {L} \Delta \boldsymbol {w} ], \quad \mathcal {G} \in \mathbb {R} ^ {1}. \tag {9}
+$$
+
+In Eq. 9,  $\Delta w$  is treated as a perturbation around  $w$ , allowing us to compute  $g$  centered at  $w$ . For each potential bitwidth configuration, we only need to compute  $\Delta w$  and the gradient of  $\mathcal{G}$  in linear time. Notably, different from using  $\mathcal{L}$  directly, such criteria-based methods do not require supervised labels or forward inference over the entire full datasets for each potential bitwidth candidate, enabling efficient mixed-precision search using techniques like integer programming, genetic algorithms (Guo et al., 2020), or iterative approaches. So far, we have realized bit allocation for a target bitrate. The next step involves calibrating QPs to minimize the reconstruction distortion.
+
+![](images/b286f219b945d0c383a7e6c65e37e82d3c92f269d83c66a8cc5c4720d45db63e.jpg)  
+(a) Layer statistics
+
+![](images/f76bacd4b9407d4b121615000772061cf142406dd28e28a5a6cc27db3dfa4975.jpg)  
+(b) Channel statistics
+
+![](images/7d829a6f44811c564d585f6a698ecf8cb8ec5a3e813c055b2c447d1b7d852f71.jpg)  
+(c) Hessian statistics  
+Figure 3: Statistic of the weight distribution among (a) layers and (b) channels. Other layers/channel exhibits similar distributions. (c) Non-generalized representation shows obvious layer/block dependencies, implying that layer/block-wise calibration is not suitable for INR-VC. Exemplified statistic information is based on HNeRV (1M) in Beauty sequence.
+
+# 3.2 HOW TO ENSURE RECONSTRUCTION QUALITY
+
+We follow the principle of PTQ to calibrate QPs (including steps and rounding variables in Eq. 16) without optimizing the underlying weights. PTQ allows us to preserve reconstruction quality by only calibrating QPs, instead of engaging in complex weight training as QAT.
+
+Unified Calibration Objective. We begin by investigating the unified calibration objective. The quantization-induced task loss degradation of a well-converged model can be approximated using the second-order Taylor expansion:
+
+$$
+\mathbb {E} \left[ \mathcal {L} (\boldsymbol {w} + \Delta \boldsymbol {w}, \boldsymbol {x}) - \mathcal {L} (\boldsymbol {w}, \boldsymbol {x}) \right] \approx \frac {1}{2} \Delta \boldsymbol {w} ^ {T} \cdot \boldsymbol {H} ^ {(w)} \cdot \Delta \boldsymbol {w}. \tag {10}
+$$
+
+From this estimation, we aim to calibrate QPs by minimizing the proxy loss, defined as:  $\min \Delta \boldsymbol{w}^T \cdot \boldsymbol{H}^{(\boldsymbol{w})} \cdot \Delta \boldsymbol{w}$ . Denote the neural network output as  $z^{(n)}$ . Using the chain rule, we can compute the Hessian matrix  $\boldsymbol{H}^{(\boldsymbol{w})}$  as follows:
+
+$$
+\frac {\partial \mathcal {L} ^ {2}}{\partial \boldsymbol {w} _ {i} \partial \boldsymbol {w} _ {j}} = \frac {\partial}{\partial \boldsymbol {w} _ {j}} \sum_ {k = 1} ^ {m} \frac {\partial \mathcal {L}}{\partial \boldsymbol {z} _ {k} ^ {(n)}} \frac {\partial \boldsymbol {z} _ {k} ^ {(n)}}{\partial \boldsymbol {w} _ {i}} = \sum_ {k = 1} ^ {m} \frac {\partial \mathcal {L}}{\partial \boldsymbol {z} _ {k} ^ {(n)}} \frac {\partial^ {2} \boldsymbol {z} _ {k} ^ {(n)}}{\partial \boldsymbol {w} _ {i} \partial \boldsymbol {w} _ {j}} + \sum_ {k, l = 1} ^ {m} \frac {\partial \boldsymbol {z} _ {k} ^ {(n)}}{\partial \boldsymbol {w} _ {i}} \frac {\partial^ {2} \mathcal {L}}{\partial \boldsymbol {z} _ {k} ^ {(n)} \partial \boldsymbol {z} _ {l} ^ {(n)}} \frac {\partial \boldsymbol {z} _ {l} ^ {(n)}}{\partial \boldsymbol {w} _ {j}}. \tag {11}
+$$
+
+Since the post-training model converges to a local minimum, we can assume the Hessian is positive-semi-definite (PSD). With  $\nabla_{\boldsymbol{z}^{(n)}}\mathcal{L}$  close to 0, the first term in Eq. 11 is negligible (Martens, 2010), yielding Gaussian-Newton form  $G^{(\boldsymbol{w})}$ :
+
+$$
+\boldsymbol {H} ^ {(w)} \approx \boldsymbol {G} ^ {(w)} = \boldsymbol {J} _ {\boldsymbol {z} (n)} ^ {T} \boldsymbol {H} ^ {(\boldsymbol {z} ^ {(n)})} \boldsymbol {J} _ {\boldsymbol {z} (n)}, \tag {12}
+$$
+
+where  $J_{z(n)}$  is the Jacobian matrix of the network output  $z^{(n)}$  with respect to the weights  $w$ .
+
+Considering  $\mathcal{L}$  is given by a commonly used mean squared error (MSE), we have:
+
+$$
+\boldsymbol {H} ^ {\left(\boldsymbol {z} ^ {(n)}\right)} = 2 \boldsymbol {I} _ {m} \quad \text {s . t .} \quad \mathcal {L} = \sum_ {k = 1} ^ {m} \left(\boldsymbol {z} _ {k} ^ {(n)} - x _ {k}\right) ^ {2}. \tag {13}
+$$
+
+By substituting the Eq. 12 and Eq. 13 into the Eq. 10, we get the following minimization objective:
+
+$$
+\min  \Delta \boldsymbol {w} ^ {T} \cdot \boldsymbol {H} ^ {(\boldsymbol {w})} \cdot \Delta \boldsymbol {w} \approx \min  [ \boldsymbol {J} _ {\boldsymbol {z} (n)} \Delta \boldsymbol {w} ] ^ {T} [ \boldsymbol {J} _ {\boldsymbol {z} (n)} \Delta \boldsymbol {w} ] \approx \min  \mathbb {E} [ \| \Delta \boldsymbol {z} ^ {(n)} \| ^ {2} ]. \tag {14}
+$$
+
+Remark 1. This results in a unified form of MSE-oriented calibration. Other distortion metrics may have analogous form, but this is not guaranteed, as not all metrics necessarily correspond to an analytical Hessian form. This can be addressed through approximated Hessian (Appendix C).
+
+Calibration Granularity. Based on the above objective, we further investigate the optimal granularity for calibration. Following Li et al. (2021b); Wei et al. (2022), we can define:
+
+1. Layer-wise Calibration: This assumes layers are mutually independent and calibrates each layers individually, resulting in a layer-diagonal Hessian matrix (blue diagonal in Fig. 3(c)), where  $\Delta z$  in Eq. 14 reflects the output error of individual layers.  
+2. Block-wise Calibration: A block is defined as a collection of several layers (e.g., residual blocks). This calibration considers intra-block dependencies while assuming inter-block independence. Calibration is performed block by block, leading to a block-diagonal Hessian matrix (orange diagonal in Fig. 3(c)), with  $\Delta z$  representing the output error of the block as a whole.  
+3. Network-wise Calibration: This granularity calibrates entire quantized network by considering the global Hessian (full matrix in Fig. 3(c)), where  $\Delta z$  reflects the entire network's output error.
+
+In generalized networks, layer/block-wise calibration is widely accepted, as dependencies are primarily found within layers/blocks, allowing inter-layer/block dependencies to be neglected (Li et al., 2021b). However, their network-wise calibration may lead to poor performance due to high generalization error. In the context of INR-VC, the situation differs: inter-layer/block dependencies cannot be ignored (Fig. 3(c)), and INR-VC typically prioritizes non-generalized representation over generalization. Therefore, we advocate for network-wise calibration as our preferred approach, as it better captures the dependencies across the network layers.
+
+Quantization Granularity. Next, we consider the granularity for quantization steps. Weight distributions vary significantly across different layers of a given INR-VC weights (Fig. 3(a)), and even among channels within a specific layer (Fig. 3(b)). This variability suggests that weights should be modeled on a channel-wise basis, where all weights in a channel share the same quantization step.
+
+Calibration. Once the granularity of both calibration and quantization are determined, we next calibrate QPs to minimize distortion. While solving Eq. 14 circumvents the complexity issues associated with the Hessian, it remains an discrete optimization problem. Inspired by Nagel et al. (2020), we reformulate it into a continuous optimization framework using soft weight variables:
+
+$$
+\arg \min  _ {\boldsymbol {s}, \boldsymbol {v}} \underbrace {| | \mathcal {F} (\boldsymbol {x} , \boldsymbol {w}) - \mathcal {F} (\boldsymbol {x} , \tilde {\boldsymbol {w}}) | | ^ {2}} _ {\text {D i s t o r t i o n t e r m} \mathcal {L} _ {D} = | | \Delta \boldsymbol {z} ^ {(n)} | | ^ {2}} + \lambda \underbrace {\sum_ {\text {R e g u l a r i z a t i o n t e r m} \mathcal {L} _ {R e g}}} _ {\text {R e g u l a r i z a t i o n t e r m}}. \tag {15}
+$$
+
+$$
+s. t. \tilde {\boldsymbol {w}} = \boldsymbol {s} \cdot c l i p \left(\left\lfloor \frac {\boldsymbol {w}}{\boldsymbol {s}} \right\rfloor + h (\boldsymbol {v}), - 2 ^ {b - 1}, 2 ^ {b - 1} - 1\right), \tag {16}
+$$
+
+where  $\mathbf{V}_{i,j}$  represents the continuous variable to optimize, and  $h(\mathbf{V}_{i,j})$  is any differentiable function constrained between 0 and 1, i.e.,  $h(\mathbf{V}_{i,g}) \in [0,1]$ .  $\mathcal{L}_{Reg}$  acts as a differentiable regularizer, guiding  $h(\mathbf{V}_{i,j})$  to converge towards either 0 or 1, ensuring at convergence  $h(\mathbf{V}_{i,j}) \in \{0,1\}$ . We also anneal  $\beta$  in  $\mathcal{L}_{reg}$  to facilitate stable convergence. This approach yields an intriguing observation: by minimizing the discrepancy between a high-precision teacher model and an initialized student model, we can effectively calibrate the quantized network. However, it's crucial to emphasize that we are different from knowledge distillation (Polino et al., 2018) that requires similar computational and data resources as naive training, making it impractical for our variable-rate coding.
+
+# 3.3 HOW TO ADVANCE INR-VC
+
+Our NeuroQuant adjusts QP to manage R-D trade-off where higher rates lead to lower distortion and vice versa. This trade-off can be formally interpreted through variational inference (Kingma, 2013; Balle et al., 2017), aiming to approximate the true posterior  $p_{\tilde{w} |x}(\tilde{w} |x)$  with a variational density  $q(\tilde{w} |x)$  by minimizing the Kullback-Leibler (KL) divergence over the data distribution  $p_x$ :
+
+$$
+\mathbb {E} _ {\boldsymbol {x}} D _ {K L} [ q | | p _ {\tilde {\boldsymbol {w}} | \boldsymbol {x}} ] = \mathbb {E} _ {\boldsymbol {x} \sim p _ {\boldsymbol {x}}} \mathbb {E} _ {\tilde {\boldsymbol {w}} \sim q} \left[ \underbrace {\log q (\tilde {\boldsymbol {w}} | \boldsymbol {x})} _ {= 0} - \underbrace {\log p _ {\boldsymbol {x} | \tilde {\boldsymbol {w}}} (\boldsymbol {x} | \tilde {\boldsymbol {w}})} _ {\text {D i s t o r t i o n} \mathcal {L} _ {D}} - \underbrace {\log p _ {\tilde {\boldsymbol {w}}} (\tilde {\boldsymbol {w}})} _ {\text {R a t e} \mathcal {L} _ {R}} + \underbrace {\log p _ {\boldsymbol {x}} (\boldsymbol {x})} _ {\text {c o n s t}} \right]. \tag {17}
+$$
+
+This leads to the R-D trade-off for INR-VC (Appendix B):
+
+$$
+\mathcal {L} = \mathcal {L} _ {R} + \lambda \mathcal {L} _ {D} = \log p _ {\tilde {\boldsymbol {w}}} (\tilde {\boldsymbol {w}}) + \frac {1}{2 \sigma^ {2}} \| \boldsymbol {x} - \hat {\boldsymbol {x}} \| ^ {2}, \text {s . t .} p _ {\boldsymbol {x} | \tilde {\boldsymbol {w}}} (\boldsymbol {x} | \tilde {\boldsymbol {w}}) = \mathcal {N} (\boldsymbol {x} | \tilde {\boldsymbol {x}}, \sigma^ {2}), \tag {18}
+$$
+
+In coding,  $\tilde{w}$  is replaced by the discrete symbol  $\hat{w}$ , which can be losslessly compressed using entropy coding techniques, such as arithmetic coding (Rissanen & Langdon, 1981).
+
+Remark 2 (NeuroQuant vs. INR-VC). Popular INR-VCs (Chen et al., 2021a; Li et al., 2022b; He et al., 2023; Zhao et al., 2023; 2024) focus on  $\min \log p_{\boldsymbol{x}|\boldsymbol{w}}(\boldsymbol{x}|\boldsymbol{w})$  without considering the impact of weight quantization, creating a mismatch between representation and compression objectives. i.e.,  $p(\boldsymbol{x}|\boldsymbol{w})$  and  $p(\boldsymbol{x}|\tilde{\boldsymbol{w}})$ , which degrades performance after quantization. In contrast, the proposed NeuroQuant directly optimizes  $\log p_{\boldsymbol{x}|\tilde{\boldsymbol{w}}}(\boldsymbol{x}|\tilde{\boldsymbol{w}})$ , bridging this mismatch and yielding superior results.
+
+While QAT (Ladune et al., 2023; Kim et al., 2024) can also optimize the same objective, NeuroQuant's strength lies in transforming R-D optimization into a post-training process, particularly advantageous for efficient variable-rate video coding, thereby reducing encoding costs. Despite NeuroQuant enables a more flexible R-D trade-off than existing approaches, it is not yet a fully joint optimization framework. Recent works (Gomes et al., 2023; Zhang et al., 2024) have begun to recognize the necessity of joint R-D optimization in INR-VC, but they lack principled explanations and typically underperform compared to generalized codecs. For example, they often assume a Gaussian prior or i.i.d. context model (Minnen et al., 2018; He et al., 2021), which is not entirely correct for INR-VC. Future advancements will require the development of customized context models to better capture weight characteristics, enabling true joint R-D optimization for INR-VC systems.
+
+# 4 EXPERIMENTS
+
+In this section, we conduct thorough experiments to verify the effectiveness of NeuroQuant. Detailed implementation and additional experiments are available in Appendix E.
+
+INR-VC Baselines. We select three representative INR-VCs as baselines to evaluate various quantization methods. NeRV (Li et al., 2022b) is a pioneering model that first maps frame indices to video frames. HNeRV (Chen et al., 2023) first introduces an encoder to generate learnable embeddings instead of the positional encoding used in NeRV. HiNeRV (Kwan et al., 2024a) further optimizes the network architecture, achieving start-of-the-art performance.
+
+Quantization Benchmarks. We first introduce naive PTQ in HNeRV, a widely adopted approach across various INR-VCs. We also include four leading task-oriented PTQs: AdaRound (Nagel et al., 2020), BRECQ (Li et al., 2021b), QDrop (Wei et al., 2022), as well as RDO-PTQ (Shi et al., 2023), which is specifically designed for R-D optimization. Two QATs tailored for INR-VCs derived from FFNeRV (Lee et al., 2023) and HiNeRV (Kwan et al., 2024a) are also evaluated.
+
+# 4.1 QUANTIZATION
+
+We summarize the quantization performance in Table 1. Starting with direct quantization in HNeRV, the results confirm Remark 2: optimizing reconstructive representation without considering weight quantization (through training awareness or post-calibration) significantly degrades performance. When compared with leading task-oriented PTQs—AdaRound, BRECQ, QDrop, and RDOPTQ—NeuroQuant consistently demonstrates superior performance, with the advantage growing as bitwidth decreases. The results support our analysis: non-generalized INR-VC exhibits significant inter-layer dependencies, making network-wise calibration necessary. Our NeuroQuant also outperforms QATs designed for INR-VC (i.e., FFNeRV and HiNeRV) across all listed baselines, particularly at lower bitwidths, achieving gains of more than 3dB. Its adaptability to varying precision highlights superior bitrate flexibility, indicating its potential for variable-rate video coding.
+
+# 4.2 VARIABLE-RATE CODING PERFORMANCE
+
+Figure 4 depicts the R-D curves for the evaluated methods. Compared to NeRV and HNeRV equipped with direct 8-bit quantization, NeuroQuant demonstrates significant compression efficiency gains of  $27.8\%$  and  $25.5\%$ , respectively. This improvement primarily stems from their lack of optimization for the compression objective. In contrast, HiNeRV achieves superior compression performance by incorporating QAT, as discussed in Sec. 3.3. Despite this, NeuroQuant also brings another  $4.8\%$  gains by replacing the built-in QAT. A key advantage of NeuroQuant is its ability to support variable-rate coding without training separate weights for each target bitrate, offering both flexibility and efficiency in practice. Moreover, NeuroQuant outperforms the network coding tool DeepCABAC (Wiedemann et al., 2020), highlighting the benefits of task-oriented QPs. Incorporating its advanced entropy coding with NeuroQuant is an exciting avenue for future research.
+
+Table 1: Reconstruction quality comparison of different quantization methods (vertical) across various INR-VCs with different model sizes (horizontal) in terms of PSNR on UVG (Mercat et al., 2020). Bold values indicate the best results.  $\dagger$  denotes QAT strategies and  $*$  represents mixed precision. All implementations are based on open-source codes.  
+
+<table><tr><td>Methods</td><td>W-bits</td><td>NeRV</td><td>NeRV</td><td>HNeRV</td><td>HNeRV</td><td>HiNeRV</td><td>Avg</td></tr><tr><td>Full Prec. (dB)</td><td>32</td><td>31.39</td><td>32.30</td><td>32.49</td><td>33.80</td><td>35.09</td><td>33.01</td></tr><tr><td>Param.</td><td>-</td><td>3.1M</td><td>6.2M</td><td>3.0M</td><td>6.2M</td><td>3.1M</td><td>-</td></tr><tr><td>HNeRV (Chen et al., 2023)</td><td>6</td><td>30.68</td><td>31.56</td><td>32.05</td><td>33.29</td><td>32.10</td><td>31.94</td></tr><tr><td>FFNeRV (Lee et al., 2023)†</td><td>6</td><td>31.10</td><td>32.02</td><td>32.15</td><td>33.34</td><td>34.03</td><td>32.53</td></tr><tr><td>HiNeRV (Kwan et al., 2024a)†</td><td>6</td><td>31.20</td><td>32.09</td><td>32.25</td><td>33.48</td><td>34.54</td><td>32.71</td></tr><tr><td>AdaRound (Nagel et al., 2020)</td><td>6</td><td>31.03</td><td>31.96</td><td>32.10</td><td>33.26</td><td>33.92</td><td>32.45</td></tr><tr><td>BRECQ (Li et al., 2021b)</td><td>6</td><td>31.11</td><td>32.05</td><td>32.18</td><td>33.42</td><td>34.10</td><td>32.57</td></tr><tr><td>QDrop (Wei et al., 2022)</td><td>6</td><td>31.15</td><td>32.10</td><td>32.20</td><td>33.44</td><td>34.27</td><td>32.63</td></tr><tr><td>RDOPTQ (Shi et al., 2023)</td><td>6</td><td>31.15</td><td>32.06</td><td>32.16</td><td>33.39</td><td>34.23</td><td>32.60</td></tr><tr><td>NeuroQuant (Ours)</td><td>6</td><td>31.31</td><td>32.22</td><td>32.38</td><td>33.61</td><td>34.67</td><td>32.84</td></tr><tr><td>HNeRV (Chen et al., 2023)</td><td>4</td><td>27.02</td><td>27.86</td><td>28.14</td><td>28.60</td><td>24.30</td><td>27.18</td></tr><tr><td>FFNeRV (Lee et al., 2023)†</td><td>4</td><td>30.14</td><td>30.90</td><td>31.11</td><td>32.13</td><td>32.37</td><td>31.33</td></tr><tr><td>HiNeRV (Kwan et al., 2024a)†</td><td>4</td><td>30.37</td><td>31.32</td><td>31.46</td><td>32.67</td><td>32.95</td><td>31.75</td></tr><tr><td>AdaRound (Nagel et al., 2020)</td><td>4</td><td>30.12</td><td>30.65</td><td>31.02</td><td>31.98</td><td>32.10</td><td>31.17</td></tr><tr><td>BRECQ (Li et al., 2021b)</td><td>4</td><td>30.22</td><td>30.93</td><td>31.26</td><td>32.24</td><td>32.56</td><td>31.44</td></tr><tr><td>QDrop (Wei et al., 2022)</td><td>4</td><td>30.28</td><td>31.05</td><td>31.33</td><td>32.45</td><td>32.68</td><td>31.56</td></tr><tr><td>RDOPTQ (Shi et al., 2023)</td><td>4</td><td>30.25</td><td>30.96</td><td>31.24</td><td>32.25</td><td>32.60</td><td>31.46</td></tr><tr><td>NeuroQuant (Ours)</td><td>4</td><td>30.85</td><td>31.77</td><td>31.64</td><td>32.81</td><td>33.33</td><td>32.08</td></tr><tr><td>HNeRV (Chen et al., 2023)</td><td>2</td><td>14.81</td><td>15.50</td><td>13.32</td><td>13.06</td><td>13.30</td><td>14.00</td></tr><tr><td>FFNeRV (Lee et al., 2023)†</td><td>2</td><td>22.52</td><td>22.89</td><td>23.84</td><td>24.14</td><td>21.25</td><td>22.93</td></tr><tr><td>HiNeRV (Kwan et al., 2024a)†</td><td>2</td><td>24.08</td><td>25.30</td><td>25.11</td><td>26.51</td><td>23.89</td><td>24.98</td></tr><tr><td>AdaRound (Nagel et al., 2020)</td><td>2</td><td>22.51</td><td>23.44</td><td>23.70</td><td>24.56</td><td>22.10</td><td>23.26</td></tr><tr><td>BRECQ (Li et al., 2021b)*</td><td>2</td><td>24.05</td><td>25.17</td><td>25.40</td><td>26.32</td><td>25.87</td><td>25.36</td></tr><tr><td>QDrop (Wei et al., 2022)*</td><td>2</td><td>25.32</td><td>26.14</td><td>25.94</td><td>26.85</td><td>26.60</td><td>26.17</td></tr><tr><td>RDOPTQ (Shi et al., 2023)*</td><td>2</td><td>24.33</td><td>25.75</td><td>25.57</td><td>26.50</td><td>26.14</td><td>25.66</td></tr><tr><td>NeuroQuant (Ours)*</td><td>2</td><td>27.39</td><td>28.48</td><td>28.02</td><td>29.05</td><td>28.92</td><td>28.37</td></tr></table>
+
+# 4.3 DIVING INTO NEUROQUANT
+
+In this subsection, we perform a series of evaluations to gain deeper insights into how our NeuroQuant functions.
+
+Encoding Complexity. Table 2 presents the encoding time evaluation for  $960 \times 1920$  videos with approximate 3M parameters. From NeRV to HiNeRV, compression performance is improved at the cost of increased encoding complexity. Without support for variable bitrates, generating a new bitrate point typically requires retraining the model, leading to an encoding time that is generally unacceptable (e.g., exceeding 22 hours for HiNeRV). In this context, NeuroQuant provides a practical solution by leveraging a pretrained model, enabling adaptation to variable bitrates while ensuring efficient encoding, achieving speedups of up to 7.9 times.
+
+Furthermore, we select the Jockey sequence from UVG as a representative example to have an analysis of variable-rate coding in Fig 5. The left subplot presents the available points of mixed precision versus unified precision. The middle sub
+
+![](images/dce7e9aefb0fe8fca5ff2a674f26cf1a554ddf0c413bbf12d619edb9a7a9bc3c.jpg)  
+Figure 4: Compression efficiency comparison. Variable-rate models are labeled with -VR suffix using the solid line. * is INR-VC using QAT.
+
+Table 2: Encoding time required to support a new bitrate. Note that our pretrained model is shared for all bitrates in range.  
+
+<table><tr><td>Baselines</td><td>Naive</td><td>NeuroQuant</td><td>Pretrain</td></tr><tr><td>NeRV</td><td>1.8 h</td><td>0.4 h (×4.5)</td><td>1.8 h</td></tr><tr><td>HNeRV</td><td>4.7 h</td><td>1.0 h (×4.7)</td><td>4.7 h</td></tr><tr><td>HiNeRV</td><td>22.2 h</td><td>2.8 h (×7.9)</td><td>18.9 h</td></tr></table>
+
+![](images/8903afc88fb1bb20ba836c9c5efbbc5a73cdd2b98e2a95d3cbfcc98a9c0bfe73.jpg)
+
+![](images/3e1090e112f0a34fa3443ed63b99e0ac60604673184aa89a07fcb0f76539f449.jpg)
+
+![](images/5f03bb2e7f06082f13b68405d8e602723e230cf58a98d33bf98e68dbc350f85e.jpg)
+
+![](images/94319330413059ce20a127d6edee7255243bf1ed83c0c5d059e190cf9396eb03.jpg)  
+Figure 5: Quantitative comparison of NeRV (up) and HNeRV (bottom).
+
+![](images/99bec23a47a308b76fd986035603626a7d6d73f9c8a630958a6a440ad1acaf91.jpg)
+
+![](images/38e96fb323028b77b909bf51f0bcca987a1742dce2c50288e2b7a860b7ae4a60.jpg)
+
+plot depicts the RD curves for different quantization bitwidths, while the right subplot displays the variable-rate range for models of varying sizes. More results can be found in Appendix E.3. We have the following conclusions:
+
+Rate Control and Mixed Precision. Mixed precision supports finer-grained rate control compared to unified precision, making NeuroQuant more flexible for various bitrate. Additionally, it enables better bit allocation, resulting in improved coding efficiency.
+
+Unified R-D Curves across Bits. The coarseness of quantization affects both rate and distortion, leading to a R-D trade-off. Within a certain range, different bitwidths exhibit unified R-D characteristics, providing a foundation for variable-rate coding through adjusting quantization parameters. While we demonstrate state-of-the-art performance, the degradation is still noticeable below INT3. As a result, we do not recommend INT3 and below for variable-rate scenarios.
+
+Variable-Rate Range. While NeuroQuant significantly reduces the need for exhaustive individual weights training, achieving infinite rate range through quantization alone is impossible due to the inherent lower bound of bitwidth. We believe this limitation can be addressed by introducing pruning, which can be interpreted as a special case of NeuroQuant with bitwidth  $= 0$ . This approach opens up new possibilities for further exploration based on the foundations laid by this work.
+
+# 5 CONCLUSIONS
+
+Conclusion. In this work, we introduce NeuroQuant, a novel approach for variable-rate INR-VC that adjusts the QPs of post-training weights to control bitrate efficiently. Our key insight is that non-generalized INR-VC exhibits distinct characteristics for quantization. Through empirical and theoretical analysis, we establish the state-of-the-art weight quantization for INR-VC. NeuroQuant significantly reduces encoding complexity while maintaining leading compression performance, providing an effective solution for variable-rate video coding in neural representation.
+
+Limitations and Future Work. Despite the success of NeuroQuant, there are some limitations. The variable-rate range is inherently constrained by the available bitwidths. A promising solution is integrating weight pruning techniques to endow the model weights with greater variability. Additionally, as discussed in Sec. 3.3, NeuroQuant enables variable R-D trade-offs than existing methods, but it still fall short of achieving true joint R-D optimization. Future work will explore customized probabilistic modeling for INR-VC, aiming to enable a fully joint R-D optimization process.
+
+# ACKNOWLEDGMENTS
+
+This work was supported in part by National Key Research and Development Project of China (Grant No. 2022YFF0902402), Natural Science Foundation of Jiangsu Province (Grant No. BK20241226) and Natural Science Foundation of China (Grant No. 62401251, 62431011). The authors would like to express their sincere gratitude to the Interdisciplinary Research Center for Future Intelligent Chips (Chip-X) and Yachen Foundation for their invaluable support.
+
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+
+# A VARIABLE-RATEBACKGROUND
+
+In real-world communication scenarios, practical video codecs pursue variable-rate coding to adapt to complex network environments while reducing storage and computational overhead.
+
+In traditional video codec (Wiegand et al., 2003; Sullivan et al., 2012), variable-rate coding is typically achieved by adjusting quantization parameters (QPs) (Liu et al., 2008; Li et al., 2020). For learned generalized video codec, current variable-rate methods can be categorized into two main approaches: feature modulation and multi-granularity quantization. Feature modulation focuses on modifying encoding and decoding features to achieve representations with varying entropy. For
+
+instance, Lin et al. (2021) proposed a scaling network that modulates the internal feature maps of motion and residual encoders. Similarly, Rippel et al. (2021) converted discrete rate levels into one-hot vectors, which were then fed into various codec sub-modules. Duan et al. (2024) introduced an adaptive conditional convolution method that applies affine transformations to features based on input Lagrangian multipliers  $(\lambda)$ . In contrast, multi-granularity quantization aims to control bitrate by adjusting the quantization levels of feature maps (also referred to as latents). For example, Li et al. (2022a) proposed a coarse-to-fine quantization strategy using three learnable quantization granularity parameters: global levels, channel-wise levels, and spatial-channel-wise levels.
+
+However, INR-based video codecs (INR-VCs) take a fundamentally different approach, converting video signal compression into model weight compression. This diverges from the conventional focus on extracting compact feature representations, making tailored variable-rate coding strategies for INR-VCs essential. While preliminary studies have begun exploring variable-rate coding in INR-VCs, their performance remains limited (detailed in Sec. E.2).
+
+In this paper, we reduce this gap by framing variable-rate coding for INR-VCs as a mixed-precision bit allocation problem. Our approach achieves variable-rate coding through QP adjustments and demonstrates state-of-the-art performance.
+
+# B VARIATIONAL INFERENCE PERSPECTIVE
+
+In INR-VC, the variational inference framework interprets the rate-distortion (R-D) trade-off as an optimization problem, aiming to approximate the true posterior distribution  $p_{\tilde{w} |x}(\tilde{w} |x)$  with a variational density  $q(\tilde{w} |x)$  by minimizing the expected Kullback-Leibler (KL) divergence over the data distribution  $p_x$ . Starting with the KL divergence, we have:
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {x} \sim p _ {\boldsymbol {x}}} D _ {K L} [ q | | p _ {\tilde {\boldsymbol {w}} | \boldsymbol {x}} ] = \mathbb {E} _ {\boldsymbol {x} \sim p _ {\boldsymbol {x}}} \mathbb {E} _ {\tilde {\boldsymbol {w}} \sim q} \left[ \log \frac {q (\tilde {\boldsymbol {w}} | \boldsymbol {x})}{p _ {\tilde {\boldsymbol {w}} | \boldsymbol {x}} (\tilde {\boldsymbol {w}} | \boldsymbol {x})} \right] (19) \\ = \mathbb {E} _ {\boldsymbol {x} \sim p _ {\boldsymbol {x}}} \mathbb {E} _ {\tilde {\boldsymbol {w}} \sim q} \left[ \log q (\tilde {\boldsymbol {w}} | \boldsymbol {x}) - \log p _ {\tilde {\boldsymbol {w}} | \boldsymbol {x}} (\tilde {\boldsymbol {w}} | \boldsymbol {x}) \right]. (20) \\ \end{array}
+$$
+
+Based on the Bayes' theorem, the posterior  $p_{\tilde{w} |x}(\tilde{w} |x)$  can be expressed as:
+
+$$
+p _ {\tilde {w} | \boldsymbol {x}} (\tilde {\boldsymbol {w}} | \boldsymbol {x}) = \frac {p _ {\boldsymbol {x} | \tilde {\boldsymbol {w}}} (\boldsymbol {x} | \tilde {\boldsymbol {w}}) p _ {\tilde {\boldsymbol {w}}} (\tilde {\boldsymbol {w}})}{p _ {\boldsymbol {x}} (\boldsymbol {x})}. \tag {21}
+$$
+
+Substituting this expression into Eq. 20, we get:
+
+$$
+\mathbb {E} _ {\boldsymbol {x} \sim p _ {\boldsymbol {x}}} D _ {K L} [ q | | p _ {\tilde {\boldsymbol {w}} | \boldsymbol {x}} ] = \mathbb {E} _ {\boldsymbol {x} \sim p _ {\boldsymbol {x}}} \mathbb {E} _ {\tilde {\boldsymbol {w}} \sim q} [ \log q (\tilde {\boldsymbol {w}} | \boldsymbol {x}) - \log p _ {\boldsymbol {x} | \tilde {\boldsymbol {w}}} (\boldsymbol {x} | \tilde {\boldsymbol {w}}) - \log p _ {\tilde {\boldsymbol {w}}} (\tilde {\boldsymbol {w}}) + \log p _ {\boldsymbol {x}} (\boldsymbol {x}) ]. \tag {22}
+$$
+
+Here, we arrive at a form consistent with Eq.17. Let's examine each of terms individually.
+
+First Term. For unknown mapping from timestamps  $t$  to video data  $\boldsymbol{x} = \mathcal{F}(\boldsymbol{w}, t)$ , the inference refers to computing the inverse transformation from input  $\boldsymbol{x}$  (Balle et al., 2017). Using a soft variable similar to Eq. 16, we have:
+
+$$
+q (\tilde {\boldsymbol {w}} | \boldsymbol {x}) = \prod_ {i} \mathcal {U} (\tilde {\boldsymbol {w}} _ {i} | \boldsymbol {w} _ {i} - \frac {1}{2}, \boldsymbol {w} _ {i} + \frac {1}{2}), \quad \text {w i t h} \boldsymbol {w} = \mathcal {F} ^ {- 1} \circ \boldsymbol {x}, \tag {23}
+$$
+
+where  $\mathcal{U}$  denotes a uniform distribution centered around  $\boldsymbol{w}_i$ . Then:
+
+$$
+\mathbb {E} _ {\tilde {\boldsymbol {w}} \sim q} [ \log q (\tilde {\boldsymbol {w}} | \boldsymbol {x}) ] = \mathbb {E} _ {\tilde {\boldsymbol {w}} \sim q} \left[ \log \mathcal {U} \left(- \frac {1}{2}, \frac {1}{2}\right) \right] = 0. \tag {24}
+$$
+
+Second Term. For the second term, suppose  $p_{\boldsymbol{x}|\tilde{\boldsymbol{w}}}(\boldsymbol{x}|\tilde{\boldsymbol{w}})$  follows a normal distribution:
+
+$$
+p _ {\boldsymbol {x} | \tilde {\boldsymbol {w}}} (\boldsymbol {x} | \tilde {\boldsymbol {w}}) = \mathcal {N} (\boldsymbol {x} | \tilde {\boldsymbol {x}}, \sigma^ {2}), \quad \tilde {\boldsymbol {x}} = \mathcal {F} (\tilde {\boldsymbol {w}}, t). \tag {25}
+$$
+
+Maximizing  $\log p_{\pmb{x}|\tilde{\pmb{w}}}(\pmb{x}|\tilde{\pmb{w}})$  is equivalent to minimizing the squared error term:
+
+$$
+\begin{array}{l} \max  \log p _ {\boldsymbol {x} | \tilde {\boldsymbol {w}}} (\boldsymbol {x} | \tilde {\boldsymbol {w}}) = - \min  \log \mathcal {N} (\boldsymbol {x} | \tilde {\boldsymbol {x}}, \sigma^ {2}) (26) \\ = - \min  \log \frac {1}{\sigma \sqrt {2 \pi}} \exp \left(- \frac {1}{2 \sigma^ {2}} | | \boldsymbol {x} - \tilde {\boldsymbol {x}} | | ^ {2}\right) (27) \\ = \min  \frac {1}{2 \sigma^ {2}} | | \boldsymbol {x} - \tilde {\boldsymbol {x}} | | ^ {2} (28) \\ \end{array}
+$$
+
+Third Term. The third term reflects the cost of encoding the latent variables  $\tilde{w}$ , representing the model complexity.
+
+Forth Term. For a specific video data, the true distribution  $p(\pmb{x})$  is constant.
+
+In conclusion, we obtain the objective function as derived in Eq. 17:
+
+$$
+\mathcal {L} = \mathcal {L} _ {R} + \lambda \mathcal {L} _ {D} = \log p _ {\tilde {\boldsymbol {w}}} (\tilde {\boldsymbol {w}}) + \frac {1}{2 \sigma^ {2}} \| \boldsymbol {x} - \hat {\boldsymbol {x}} \| ^ {2}. \tag {29}
+$$
+
+By matching the variational density with the INR-based video coding framework, we observe that minimizing the KL divergence corresponds to optimizing weights for the rate-distortion performance.
+
+# C APPROXIMATING HESSIAN
+
+While MSE is the most commonly used loss function in INR-VC, it is impractical to assume that all future scenarios encountered by NeuroQuant will use MSE as their objective. To address this limitation, we consider an alternative approach based on the variation inference discussed earlier. The loss function of INR-VC can be framed as the likelihood estimation for  $p(\boldsymbol{x}|\tilde{\boldsymbol{w}})$ , allowing us to define the Hessian as:
+
+$$
+\boldsymbol {H} = \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} \mathcal {L} = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x})} \left[ - \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \right]. \tag {30}
+$$
+
+The negative expected Hessian of the log-likelihood function is equivalent to the Fisher information matrix (FIM) (LeCun et al., 2002). We define the FIM, denoted by  $\mathbf{F}$ , as follows:
+
+$$
+\begin{array}{l} \boldsymbol {F} = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \left[ - \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \right] (31) \\ = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \left[ - \frac {1}{p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \right] + \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \left[ \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) ^ {T} \right]. (32) \\ \end{array}
+$$
+
+The first term on the right side of Eq. 32 is zero, as shown in Chen et al. (2021b):
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \left[ - \frac {1}{p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \right] (33) \\ = \int \frac {1}{p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) d x (34) \\ = \int \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) d x (35) \\ = \nabla_ {\tilde {w}} ^ {2} \int p (\boldsymbol {x} | \tilde {w}) d x (36) \\ = 0. (37) \\ \end{array}
+$$
+
+Therefore,
+
+$$
+\begin{array}{l} \boldsymbol {F} = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \left[ - \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \right] (38) \\ = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \left[ \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) ^ {T} \right]. (39) \\ \end{array}
+$$
+
+In Eq. 11, the first term is ignored, and we approximate  $H$  using Gauss-Newton matrix form  $G$ . Similarly, in the context of likelihood estimation, this can be represented as:
+
+$$
+\boldsymbol {G} = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x})} \left[ \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) ^ {T} \right]. \tag {40}
+$$
+
+To summarize, we have the following relationships:
+
+$$
+\boldsymbol {H} = \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} \mathcal {L} = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x})} \left[ - \nabla_ {\tilde {\boldsymbol {w}}} ^ {2} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \right], \tag {41}
+$$
+
+$$
+\boldsymbol {G} = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x})} \left[ \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) ^ {T} \right], \tag {42}
+$$
+
+$$
+\begin{array}{l} \boldsymbol {F} = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x} | \boldsymbol {\bar {w}})} \left[ - \nabla_ {\boldsymbol {\bar {w}}} ^ {2} \log p (\boldsymbol {x} | \boldsymbol {\bar {w}}) \right] (43) \\ = \mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x} | \tilde {\boldsymbol {w}})} \left[ \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) \nabla_ {\tilde {\boldsymbol {w}}} \log p (\boldsymbol {x} | \tilde {\boldsymbol {w}}) ^ {T} \right]. (44) \\ \end{array}
+$$
+
+In INR-VC, the observed data is consistent with the true data. Hence, when the target data distribution  $p(\pmb{x})$  equals to the fitted distribution  $p(\pmb{x}|\tilde{\pmb{w}})$ , we derive:
+
+$$
+\boldsymbol {H} = \boldsymbol {G} = \boldsymbol {F}. \tag {45}
+$$
+
+This approximation, constrained by network capacity (e.g., bitrate) without data distribution, reveals a unique advantage compared to generalized CODECs. Therefore, we can get an equivalent objective under the FIM form, though not exactly the same as the Hessian form
+
+$$
+\min  \mathbb {E} \left[ \Delta \boldsymbol {z} ^ {T} \operatorname {d i a g} \left(\left(\frac {\partial \mathcal {L}}{\partial \boldsymbol {z} _ {1}}\right) ^ {2}, \dots , \left(\frac {\partial \mathcal {L}}{\partial \boldsymbol {z} _ {n}}\right) ^ {2}\right) \Delta \boldsymbol {z} \right] \tag {46}
+$$
+
+For some loss functions, such as perceptual loss (Rad et al., 2019), which lack an analytical Hessian form, the FIM can serve as a suitable substitute.
+
+# D NEUROQUANT ALGORITHM
+
+In our approach,  $s$  represents the quantization steps. Once each layer is assigned a bitwidth  $b^{l}$ , the initial value of  $s$  is computed channel-wisely using
+
+$$
+s ^ {l, k} = \frac {\operatorname* {m a x} \left(w ^ {l , k}\right) - \operatorname* {m i n} \left(x ^ {l , k}\right)}{2 ^ {b ^ {l}} - 1}, \tag {47}
+$$
+
+where  $w^{l,k}$  is the weights in the  $k$ -th channel of the  $l$ -th layer. During the calibration process,  $s$  is further optimized to minimize task loss.
+
+Algorithm 1 Bit Allocation for Target Bitrate (Mixed Precision, Sec. 3.1)  
+Input: Pretrained weights  $W$  (FP32); Initial bitwidth  $b$  (e.g., 8 bit); Potential mixed-precision con  
+figures set  $S$  under target rate  $R$    
+Output: Quantization steps  $s$ , Quantized weights  $\widehat{W}$    
+1: Quantize  $W$  to initial bitwidth  $b$  and get  $W_{0}$  .   
+2: Compute the gradient of  $W_{0}$  with backward propagation  $\pmb {g} = \frac{\partial\mathcal{L}}{\partial W_0}$  .   
+3: for  $S_{i}$  in  $S$  do   
+4: Get layer-wise bitwidth  $b^{l}$  from  $S_{i}$  .   
+5: for  $l = 1,2,\dots ,N$  th layer in  $W_{0}$  do   
+6: Compute channel-wise quantization parameter  $\{s^{l,k}\}_{k = 1}^{k = c}$  .  $\triangleright$  Eq. 47   
+7: Compute de-quantized  $\widehat{W}^l = s^l\cdot \mathrm{Round}(W_0 / s^l)$  .   
+8: end for   
+9: Compute weight perturbation  $\Delta W = W - \widehat{W}$  .   
+10: Compute Hessian-Vector Product  $H\Delta W$  with  $\nabla \mathcal{G} = \nabla (g\Delta W)$  .  $\triangleright$  Eq. 8   
+11: Compute criteria  $\Omega_{i} = \Delta W^{T}H\Delta W$  .  $\triangleright$  Eq. 9   
+12: end for   
+13:  $s,\widehat{W} = \arg \min \Omega$
+
+# EXPERIMENTS
+
+# E.1 IMPLEMENTATION DETAILS
+
+Evaluation. We conducted experiments on the UVG dataset<sup>2</sup>, which consists of 7 videos, each with a resolution of  $1920 \times 1080$  and recorded at 120 FPS over 5 or 2.5 seconds. We applied a center crop to achieve a resolution of  $1920 \times 960$ , similar to the preprocessing used in HNeRV and NeRV. For evaluation metrics, we employed PSNR to measure reconstruction distortion and bits per pixel (bpp) to access bitrate. Additionally, the Bjøntegaard Delta Rate (BD-Rate) (Bjontegaard, 2001) was calculated for each baseline codec.
+
+Algorithm 2 Encoding for Target Rate (Calibration, Sec. 3.2)  
+Input: Pretrained weights  $W$  (FP32); Potential mixed-precision configures set  $S$  under target rate  $R$ ; iteration T  
+Output: Bitstream, bpp  
+1: Search optimal bit configure and get  $s$ ,  $\widehat{W}$ ; ▷ Algorithm 1  
+2: Forward propagation and collect the FP network output  $z$ ;  
+3: for  $i = 1, 2, \dots, T$  do ▷ Calibration  
+4: Forward propagation and collect the quantized network output  $\hat{z}$ ;  
+5: Compute  $\mathcal{L}$  and gradient descent; ▷ eq. 16  
+6: Update  $s$  first with a few iterations and then update rounding parameters  $\pmb{v}$   
+7: end for  
+8: Quantize  $W$  to integer with calibrated QPs, i.e.,  $s, v$ ;  
+9: Get bitstream with lossless entropy coding.  
+10: Compute bpp ▷ eq. 48
+
+Here, the bpp calculation includes both the quantized network parameters  $\hat{w}$  and the quantization parameters  $s$ :
+
+$$
+b p p = b p p _ {w} + b p p _ {s} = \frac {\sum E (\hat {w})}{H \times W \times T} + \frac {\sum s \cdot b _ {s}}{H \times W \times T}, \tag {48}
+$$
+
+where  $E$  denotes lossless entropy coding. For example, on a 1080p video sequence with INT4  $\hat{w}$  and FP16  $s$ , the bpp for HNeRV-3M is approximately:  $bpp_{w} \approx 0.01$ ,  $bpp_{s} \approx 0.00004$ . As shown, the contribution of  $s$  to the overall bpp is negligible, but it is still included in all calculations for fairness.
+
+Baselines. All baselines were implemented using open-source codes, including NeRV (Chen et al., 2021a), HNeRV(Chen et al., 2023), and HiNeRV (Kwan et al., 2024a).
+
+NeuroQuant. In the mixed-precision bit allocation, we targeted the final weights size instead of the bpp, as the actual bpp cannot be accurately estimated after entropy coding. Specifically, the bitwidth was constrained within the range of [3bit, 8bit]. The weights compression ratio (FP32 weights size/quantized weights size) was maintained within an average range of [4.5, 10], where different baselines and weights size had small difference. We calculated potential bitwidth configurations while allowing a  $5\%$  error tolerance to avoid empty solution. In current implementation, we conducted search in a group of predefined bitwidth configures for a given target size, which can be finished in one minute. Integer Programming (Hubara et al., 2021) also can achieve same objective.
+
+Once the bits are allocated, we employed the Adam optimizer (Kingma, 2014) to calibrate quantization parameters (e.g., quantization steps, weight rounding) to minimize distortion. For frame-wise INR-VC systems like NeRV and HNeRV, the batchsize was set to 2, while for patch-wise INR-VC systems like HiNeRV, the batchsize was set to 144. The learning rate was set to  $3e - 3$  with a cosine annealing strategy. QP were be optimized for  $2.1\times 10^{4}$  iterations, although most cases converged in fewer than  $1.5\times 10^{4}$  iterations. All experiments were conducted using Pytorch with Nvidia RTX 3090 GPUs.
+
+Our code will be made open-source upon the release of the paper.
+
+Task-Oriented PTQs. Benchmarks were based on open-source implementations of AdaRound, BRECQ, QDrop, and RDOPTQ. These methods were reproduced for INR-VC systems, using the same batchsize and learning rate as NeuroQuant, which provided better results than naive learning rate. Each layer/block was optimized for  $2.1 \times 10^{4}$  iterations.
+
+INR-Oriented QATs. For FFNeRV, the QAT process is summarised as follows:
+
+$$
+\text {F o r w a r d}: \hat {\boldsymbol {w}} = \operatorname {s i g n} (\boldsymbol {w}) \cdot \frac {\left\lfloor (2 ^ {b} - 1) \cdot \tanh  (| \boldsymbol {w} |) \right\rfloor}{2 ^ {b} - 1}, \tag {49}
+$$
+
+$$
+\text {B a c k w a r d}: \frac {\partial \mathcal {L}}{\partial \hat {w}} \approx \frac {\partial \mathcal {L}}{\partial w}. \tag {50}
+$$
+
+For HiNeRV, the QAT process is based on QuantNoise (Stock et al., 2021) but forbidding the naive Straight-Through Estimator (STE):
+
+$$
+\text {F o r w a r d}: \tilde {\boldsymbol {w}} = \boldsymbol {w} \cdot \operatorname {m a s k} + \hat {\boldsymbol {w}} \cdot (1 - \operatorname {m a s k}), \tag {51}
+$$
+
+$$
+w h e r e \hat {\boldsymbol {w}} = \left\lfloor \frac {\boldsymbol {w} \cdot \left(2 ^ {b} - 1\right)}{2 \cdot \max (\boldsymbol {w})} \right\rfloor \cdot \frac {2 \cdot \max (\boldsymbol {w})}{2 ^ {b} - 1}, \tag {52}
+$$
+
+$$
+\text {B a c k w a r d}: \frac {\partial \mathcal {L}}{\partial \tilde {w}} \approx \frac {\partial \mathcal {L}}{\partial w} \cdot \operatorname {m a s k}, \tag {53}
+$$
+
+$$
+\text {I n f e r e n c e :} \quad \hat {\boldsymbol {w}} = \left\lfloor \frac {\boldsymbol {w} \cdot (2 ^ {b} - 1)}{2 \cdot \max (\boldsymbol {w})} \right\rceil \cdot \frac {2 \cdot \max (\boldsymbol {w})}{2 ^ {b} - 1} \tag {54}
+$$
+
+where mask is a random binary tensor with the same shape as  $\pmb{w}$ . This random dropping is also similar to QDrop.
+
+Generalized Neural Video Codes. We also included two representative generalized Neural Video Coding Systems, DCVC (Li et al., 2021a) and DCVC-DC (Li et al., 2023), to compare with the existing non-generalized INR-VC systems. Pretrained models were used to test the UVG dataset, where all frames of each video were evaluated. The group of pictures (GOP) was set to 32, consistent with other learned video coding methods.
+
+Generalized Traditional Codes. The command to encode using x264 in our paper is:
+
+```tcl
+1 ffmpeg  
+2 -s {width} x {height}  
+3 -pix_fmt yuv44p10le  
+4 -framerate {frame rate}  
+5 -i {input yuv name}  
+6 -c:v libx264  
+7 -preset veryslow  
+8 -g 32  
+9 -qp {qp}  
+10 {bitstream file name}
+```
+
+The command to encode using x265 in our paper is:
+
+```txt
+1 ffmpeg  
+2 -s {width} x {height}  
+3 -pix_fmt yuv444p10le  
+4 -framerate {frame rate}  
+5 -i {input yuv name}  
+6 -c:v libx265  
+7 -preset veryslow  
+8 -x265-params  
+9 `qp= {qp}:keyint=32'  
+10 {bitstream file name}
+```
+
+The pre-process followed the suggestions in Sheng et al. (2022); Li et al. (2024), where we used BT.601 color range to convert between YUV and RGB.
+
+# E.2 VARIABLE-RATE COMPARISON
+
+Here we further compare NeuroQuant with two typical variable-rate techniques: (1) Neural Network Coding (NNC) techniques (Wiedemann et al., 2020) and Entropy Regularization (EM) techniques (Gomes et al., 2023; Kwan et al., 2024b; Zhang et al., 2021). NNC uses video codec to compress neural network, while EM introduces additional weight entropy regularization. We employ Deep-CABAC (Wiedemann et al., 2020) and Gomes et al. (2023) as the typical represented methods, respectively. We use the NeRV across three video sequences: Beauty, Jockey, and ReadySetGo to compare. The results are summarized in the following table:
+
+As shown, NeuroQuant consistently outperforms DeepCABAC and Gomes et al. across different sequences and bitrates. These results demonstrate the effectiveness of our method in improving rate-distortion performance. For one training iteration, Gomes et al. (2023) involves additional overhead due to entropy estimation, resulting in approximately  $\times 1.4$  encoding time compared to NeuroQuant.
+
+Table 3: Variable-Rate comparison.  
+
+<table><tr><td>Methods</td><td>Bpp</td><td>Beauty</td><td>Jockey</td><td>ReadyS</td><td>Avg.</td></tr><tr><td>Full Prec. (dB)</td><td>-</td><td>33.08</td><td>31.15</td><td>24.36</td><td>29.53</td></tr><tr><td>DeepCABAC (Wiedemann et al., 2020)</td><td>0.016</td><td>32.98</td><td>30.94</td><td>24.24</td><td>29.39</td></tr><tr><td>Gomes et al. (Gomes et al., 2023)</td><td>0.016</td><td>32.91</td><td>30.66</td><td>23.92</td><td>29.16</td></tr><tr><td>NeuroQuant (Ours)</td><td>0.016</td><td>33.04</td><td>31.09</td><td>24.31</td><td>29.48</td></tr><tr><td>DeepCABAC (Wiedemann et al., 2020)</td><td>0.013</td><td>32.43</td><td>30.23</td><td>23.92</td><td>28.86</td></tr><tr><td>Gomes et al. (Gomes et al., 2023)</td><td>0.013</td><td>32.78</td><td>30.29</td><td>23.61</td><td>28.89</td></tr><tr><td>NeuroQuant (Ours)</td><td>0.013</td><td>32.97</td><td>30.96</td><td>24.18</td><td>29.37</td></tr><tr><td>DeepCABAC (Wiedemann et al., 2020)</td><td>0.011</td><td>31.59</td><td>28.70</td><td>22.85</td><td>27.71</td></tr><tr><td>Gomes et al. (Gomes et al., 2023)</td><td>0.011</td><td>32.63</td><td>29.89</td><td>23.26</td><td>28.59</td></tr><tr><td>NeuroQuant (Ours)</td><td>0.011</td><td>32.83</td><td>30.67</td><td>23.85</td><td>29.12</td></tr></table>
+
+Besides, NeuroQuant achieves precise bitrate control by adjusting quantization parameters, as bitrate is directly proportional to the number of parameters and their bitwidth. In contrast, entropy-based methods can not directly estimate compressed bitrate from the Lagrangian multiplier  $\lambda$ . Instead, it requires multiple encoding runs to fit a mapping from  $\lambda$  to bitrate. This mapping is sequence-dependent, reducing its universality and reliability.
+
+Additionally, we acknowledge that the lossless entropy coding used in NeuroQuant currently is less advanced compared to CABAC-based techniques used in DeepCABAC. However, as a quantization method, NeuroQuant is compatible with various entropy coding techniques. In future work, we aim to incorporate CABAC and EM techniques into NeuroQuant as discussed in Sec. 3.3.
+
+# E.3 DIVING INTO NEUROQUANT
+
+To analyze the efficiency of NeuroQuant, we conducted a series of deeper studies on the three sequences (Beauty, Jockey, and ReadySetGo) using NeRV-3M and 4-bit precision. Below, we summarize our findings.
+
+# E.3.1 IS INTER-LAYER DEPENDENCE A GOOD PROPERTY FOR INR?
+
+Benefits of Inter-Layer Dependencies: INR models are inherently non-generalized and tailored to represent specific video data. This specialization often leads to strong inter-layer dependencies, where the contribution of each layer to the overall representation is tightly coupled. For video representation, such strong coupling can reduce redundancy across the network, leading to better rate-distortion performance in INR-VCs. In contrast, excessive independence could indicate poor utilization of the network's capacity.
+
+Challenges of Inter-Layer Dependencies: Strong dependencies complicate quantization, as perturbations in one layer can propagate across the network. This is where network-wise calibration, as proposed in NeuroQuant, becomes critical. Additionally, dependencies can limit scalability and robustness, as architectural modifications can disrupt the internal balance of the network.
+
+Simple Experiment: For INR-VCs, the overfitting (dependence) degree increases with the training iterations growing. To measure dependence, we quantized the fourth layer using vanilla MinMax quantization without any calibration, and observed its impact on final loss. Results for varying training epochs are shown below:
+
+The results indicate that stronger dependence means larger loss changes with quantization but provides better representation performance.
+
+For future work, exploring a middle ground—where moderate independence is encouraged to balance representation fidelity and quantization robustness—might be an interesting direction.
+
+![](images/5c4eaf9b9cd38d56955ea35bb3533fbcf1d9aef430e11855f15b5d32cad305ab.jpg)  
+(a) Representation Capcaity  
+Figure 6: The influence of inter-layer dependence.
+
+![](images/239ff51a805a00fd2050ae5aff3d41a8ed4fdcca98cf98035904a8b26dce4834.jpg)  
+(b) Quantization Error
+
+# E.3.2 ABLATION STUDIES
+
+Batch Size. We examined the impact of varying batch sizes on compression performance. As shown in Table 4, increasing the batch size results in diminishing returns in PSNR, with larger batch sizes also introducing greater complexity. For NeuroQuant, a batch size of 2 was chosen as the default setting.
+
+Table 4: Batch Size Ablation  
+
+<table><tr><td></td><td>1</td><td>2</td><td>4</td><td>8</td></tr><tr><td>Beauty</td><td>32.78</td><td>32.79</td><td>32.79</td><td>32.80</td></tr><tr><td>Jockey</td><td>30.54</td><td>30.58</td><td>30.60</td><td>30.62</td></tr><tr><td>ReadyS</td><td>23.75</td><td>23.79</td><td>23.82</td><td>23.84</td></tr><tr><td>Avg</td><td>29.02</td><td>29.05</td><td>29.07</td><td>29.09</td></tr></table>
+
+Learning Rate. We experimented with different learning rates, as shown in Table 5. A learning rate of  $3e - 3$  was selected as the default NeuroQuant setting.
+
+Table 5: Learning Rate Ablation  
+
+<table><tr><td></td><td>1.5e-3</td><td>2e-3</td><td>3e-3</td><td>5e-3</td></tr><tr><td>Beauty</td><td>32.86</td><td>32.81</td><td>32.83</td><td>32.79</td></tr><tr><td>Jockey</td><td>30.58</td><td>30.62</td><td>30.67</td><td>30.53</td></tr><tr><td>ReadyS</td><td>23.88</td><td>23.82</td><td>23.85</td><td>23.79</td></tr><tr><td>Avg</td><td>29.11</td><td>29.08</td><td>29.12</td><td>29.04</td></tr></table>
+
+Objective Trade-off. Eq. 14 describes the final calibration objective, which includes both the distortion term  $\mathcal{L}_D$  and the regularization term  $\mathcal{L}_{Reg}$ . We explored the trade-off between these two objectives by varying the regularization weight  $\lambda$ , as shown in Table 6. The results indicate that NeuroQuant maintains robust performance across different regularization weights.
+
+Quantization Granularity. We compared different quantization granularities, focusing on channel-wise (CW) and layer-wise (LW) quantization strategies. As shown in Table 7, while layer-wise granularity significantly degrades performance, NeuroQuant still yields satisfactory results in this context. Channel-wise quantization delivers superior performance, supporting the correctness of the proposed method.
+
+Table 6: Objective Trade-off Ablation  
+
+<table><tr><td></td><td>λ = 0.1</td><td>λ = 0.01</td><td>λ = 0.001</td></tr><tr><td>Beauty</td><td>32.79</td><td>32.78</td><td>32.78</td></tr><tr><td>Jockey</td><td>30.58</td><td>30.58</td><td>30.56</td></tr><tr><td>Ready</td><td>23.80</td><td>23.77</td><td>23.76</td></tr><tr><td>Avg</td><td>29.06</td><td>29.04</td><td>29.03</td></tr></table>
+
+Table 7: Quantization Granularity Ablation  
+
+<table><tr><td></td><td>LW w/o NeuroQuant</td><td>LW w/ NeuroQuant</td><td>CW w/ NeuroQuant</td></tr><tr><td>Beauty</td><td>28.90</td><td>32.12</td><td>32.79</td></tr><tr><td>Jockey</td><td>22.49</td><td>29.08</td><td>30.58</td></tr><tr><td>ReadyS</td><td>19.50</td><td>22.51</td><td>23.80</td></tr><tr><td>Avg</td><td>23.63</td><td>27.90</td><td>29.06</td></tr></table>
+
+Iterations. We analyzed the performance gains over different numbers of iterations, as illustrated in Fig. 7. NeuroQuant achieves significant improvements in the first  $10^{3}$  iterations, indicating its rapid convergence during calibration.
+
+![](images/ddee350828610a567cf56a5991bc326a68ee9f0ca7616152c69b8ee36b67d750.jpg)  
+(a) Beauty
+
+![](images/395b01aebd88054699533d9209b2ac110a51f208d627ec1bb6b9a07ca9959d1e.jpg)  
+(b) Jockey
+
+![](images/69e519fec8db1f80750364f348e16075432efaaad3b634b588c75b78e8d7b83a.jpg)  
+(c) ReadyS
+
+![](images/d027686d82877bc763925a3eff775302e45f923e3a10325542843e419a8d39f9.jpg)  
+(d) Beauty (log scale)
+
+![](images/c4973ac7dd05a1d2c1414a796fd79efe34d0ed84e1c55bab118f1a5bd3316973.jpg)  
+(e) Jockey (log scale)  
+Figure 7: Iterations Ablation.
+
+![](images/804b258b84edba9ea1449ec4adf861d45f49651fa92e2564ba09160319634e9e.jpg)  
+(f) ReadyS (log scale)
+
+Due to the tremendous encoding complexity of HiNeRV ( $10 \times$  compared to NeRV, e.g., more than 21GB memory requirement and nearly one GPU day for 3M weights), we exclude HiNeRV in next.
+
+Unified R-D Characteristics. Fig. 8 presents the Rate-Distortion (R-D) curves for three different video sequences: Jockey, Beauty, and ReadySetGo. In addition to the commonly used Jockey sequence, we include the less dynamic Beauty sequence, where baseline models tend to exhibit saturation (showing less than 1 dB PSNR improvement across the entire bitrate range). We also consider the highly dynamic ReadySetGo sequence, where all baselines show the worst R-D performance within the UVG dataset.
+
+NeuroQuant significantly outperforms naive quantization techniques in terms of compression efficiency. For the Beauty sequence, both NeRV and HNeRV show a saturation effect, where increasing the number of weights does not yield noticeable gains in distortion reduction. In this case, NeuroQuant performs worse compared to its performance on other sequences..
+
+On the other hand, NeuroQuant exhibits slight difference among different beseliens and video sequences. As highlighted in the zoomed-in areas of the R-D curves, NeuroQuant-4bit achieves higher PSNR in the NeRV on ReadySetGo. When it comes to the NeRV on Beauty, NeuroQuant-5bit achieves higher PSNR. Despite these differences, various precision levels (bitwidth) display nearly unified R-D characteristics.
+
+Visualization. We further visualize the lowest bitrate (0.01bpp) for all three sequences in Fig. 9, 10 and 11. Compared with naive 32bit floating-point HNeRV (FP32), NeuroQuant can compress weights to INT4 without visual information loss.
+
+![](images/ab0abea162a956b6c6b32511abc661a6409811ab2f8ed4ed0a323252bd3aef1b.jpg)
+
+![](images/d7bd4a11b3c4c0c785310c9ce1d2555f3c8747db51a442bb463ced56fc876b17.jpg)
+
+![](images/af31d1e234681580e29238df95f0e2ee2a18dce6e8ec5e2ee23f18a5de593da4.jpg)
+
+![](images/e7b6bd5b64ba632e595c7884b1461e0cb8c2c195407d4edb39159dd1bed32114.jpg)
+
+![](images/5ae932b831a75d0125ab7b3da69b54742f221ff74641130d2739b927612e2dee.jpg)  
+Figure 8: Quantitative comparison of NeRV (left) and HNeRV (right).
+
+![](images/75fcfd61915c3be08133b3a52165224b58e6f2c38f61957014f8f0f1eafc284c.jpg)
+
+![](images/03ebd533867308fc4d919f20f01acd6a3b33aaa6183d2d0e65cbb62288b1728f.jpg)  
+FP32  
+PSNR: 33.33dB  
+Model Size: 12.4 MB  
+Figure 9: Comparison around 0.01bpp with Beauty sequence in HNeRV.
+
+![](images/b52b4341dd3c4d403b54d571ddec32ffc336e6d3261851d5a98466ec09ca980b.jpg)  
+NeuroQuant  
+PSNR: 32.99dB  
+Model Size: 1.55 MB
+
+![](images/e60db0872491b56f1ae3348a16c38fc229351df35de497cb0c9ea82d951c0931.jpg)  
+HNeRV INT4  
+PSNR: 32.19dB  
+Model Size: 1.55 MB
+
+![](images/4822acfdede0bfcd5f23908a49660d7b615bfb0763164498b9cc87af91e790f2.jpg)  
+FP32  
+PSNR: 29.73dB  
+Model Size: 12.4 MB
+
+![](images/f6aab853d07f07fb181366db770bb8ed19afee66b7a4f37c529b9cd8559745d7.jpg)  
+NeuroQuant  
+PSNR: 28.56dB  
+Model Size: 1.55 MB
+
+![](images/8304070571c7d32b99d3ec6ff1961bcd880a42cd03cd0871e5279ef1ae0419e4.jpg)  
+HNeRV INT4  
+PSNR: 26.84dB  
+Model Size: 1.55 MB
+
+![](images/5c0af2a33116c1e156934c4987a6c0717ca91c2dbc05377941a7f751be0f136b.jpg)  
+Figure 10: Comparison around  $0.01bpp$  with Jockey sequence in HNeRV.  
+FP32  
+PSNR: 26.39dB  
+Model Size: 12.4 MB  
+Figure 11: Comparison around  $0.01bpp$  with ReadySetGo sequence in HNeRV.
+
+![](images/18a0cbec2e11c1d4655b2a1488c45adfdcd31603b30b768c7175f7795235534e.jpg)  
+NeuroQuant INT4  
+PSNR: 25.3dB  
+Model Size: 1.55 MB
+
+![](images/1b41612a2561329407292cb219dc321939032c469f4e6ea264bfb657ad65a51f.jpg)  
+HNeRV INT4  
+PSNR: 22.43dB  
+Model Size: 1.55 MB

2025/On Quantizing Neural Representation for Variable-Rate Video Coding/images.zip

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2025/On the Expressiveness of Rational ReLU Neural Networks With Bounded Depth/4f247875-e206-4eb2-a383-3c61f428153d_origin.pdf

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2025/On the Expressiveness of Rational ReLU Neural Networks With Bounded Depth/full.md

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+# ON THE EXPRESSIVENESS OF RATIONAL RELU NEURAL NETWORKS WITH BOUNDED DEPTH
+
+Gennadiy Averkov
+
+BTU Cottbus-Senftenberg  
+averkov@b-tu.de
+
+Christopher Hojny
+
+TU Eindhoven c.hojny@tue.nl
+
+Maximilian Merkert
+
+TU Braunschweig m.merkert@tu-braunschweig.de
+
+# ABSTRACT
+
+To confirm that the expressive power of ReLU neural networks grows with their depth, the function  $F_{n} = \max \{0,x_{1},\ldots ,x_{n}\}$  has been considered in the literature. A conjecture by Hertrich, Basu, Di Summa, and Skutella [NeurIPS 2021] states that any ReLU network that exactly represents  $F_{n}$  has at least  $\lceil \log_2(n + 1)\rceil$  hidden layers. The conjecture has recently been confirmed for networks with integer weights by Haase, Hertrich, and Loho [ICLR 2023].
+
+We follow up on this line of research and show that, within ReLU networks whose weights are decimal fractions,  $F_{n}$  can only be represented by networks with at least  $\lceil \log_3(n + 1) \rceil$  hidden layers. Moreover, if all weights are  $N$ -ary fractions, then  $F_{n}$  can only be represented by networks with at least  $\Omega \left( \frac{\ln n}{\ln \ln N} \right)$  layers. These results are a partial confirmation of the above conjecture for rational ReLU networks, and provide the first non-constant lower bound on the depth of practically relevant ReLU networks.
+
+# 1 INTRODUCTION
+
+An important aspect of designing neural network architectures is to understand which functions can be exactly represented by a specific architecture. Here, we say that a neural network, transforming  $n$  input values into a single output value, (exactly) represents a function  $f\colon \mathbb{R}^n\to \mathbb{R}$  if, for every input  $x\in \mathbb{R}^n$ , the neural network reports output  $f(x)$ . Understanding the expressiveness of neural network architectures can help to, among others, derive algorithms (Arora et al., 2018; Khalife et al., 2024; Hertrich & Sering, 2024) and complexity results (Goel et al., 2021; Froese et al., 2022; Bertschinger et al., 2023; Froese & Hertrich, 2023) for training networks.
+
+One of the most popular classes of neural networks are feedforward neural networks with ReLU activation (Goodfellow et al., 2016). Their capabilities to approximate functions is well-studied and led to several so-called universal approximation theorems, e.g., see (Cybenko, 1989; Hornik, 1991). For example, from a result by Leshno et al. (1993) it follows that any continuous function can be approximated arbitrarily well by ReLU networks with a single hidden layer. In contrast to approximating functions, the understanding of which functions can be exactly represented by a neural network is much less mature. A central result by Arora et al. (2018) states that the class of functions that are exactly representable by ReLU networks is the class of continuous piecewise linear (CPWL) functions. In particular, they show that every CPWL function with  $n$  inputs can be represented by a ReLU network with  $\lceil \log_2(n + 1) \rceil$  hidden layers. It is an open question though for which functions this number of hidden layers is also necessary.
+
+An active research field is therefore to derive lower bounds on the number of required hidden layers. Arora et al. (2018) show that two hidden layers are necessary and sufficient to represent  $\max \{0, x_1, x_2\}$  by a ReLU network. However, there is no single function which is known to require more than two hidden layers in an exact representation. In fact, Hertrich et al. (2021) formulate the following conjecture.
+
+Conjecture 1. For every integer  $k$  with  $1 \leq k \leq \lceil \log_2(n + 1) \rceil$ , there exists a function  $f \colon \mathbb{R}^n \to \mathbb{R}$  that can be represented by a ReLU network with  $k$  hidden layers, but not with  $k - 1$  hidden layers.
+
+Hertrich et al. (2021) also show that this conjecture is equivalent to the statement that any ReLU network representing  $\max \{0, x_1, \ldots, x_{2^k}\}$  requires  $k + 1$  hidden layers. That is, if the conjecture
+
+holds true, the lower bound of  $\lceil \log_2(n + 1)\rceil$  by Arora et al. (2018) is tight. While Conjecture 1 is open in general, it has been confirmed for two subclasses of ReLU networks, namely networks all of whose weights only take integer values (Haase et al., 2023) and, for  $n = 4$ , so-called  $H$ -conforming neural networks (Hertrich et al., 2021).
+
+In this article, we follow this line of research by deriving a non-constant lower bound on the number of hidden layers in ReLU networks all of whose weights are  $N$ -ary fractions. Recall that a rational number is an  $N$ -ary fraction if it can be written as  $\frac{z}{N^t}$  for some integer  $z$  and non-negative integer  $t$ .
+
+Theorem 2. Let  $n$  and  $N$  be positive integers, and let  $p$  be a prime number that does not divide  $N$ . Every ReLU network with weights being  $N$ -ary fractions requires at least  $\lceil \log_p(n + 1) \rceil$  hidden layers to exactly represent the function  $\max \{0, x_1, \ldots, x_n\}$ .
+
+Corollary 3. Every ReLU network all of whose weights are decimal fractions requires at least  $\lceil \log_3(n + 1)\rceil$  hidden layers to exactly represent  $\max \{0,x_1,\ldots ,x_n\}$
+
+While Theorem 2 does not resolve Conjecture 1 because it makes no statement about general real weights, note that in most applications floating point arithmetic is used (IEEE, 2019). That is, in neural network architectures used in practice, one is actually restricted to weights being  $N$ -ary fractions. Moreover, when quantization, see, e.g., (Gholami et al., 2022) is used to make neural networks more efficient in terms of memory and speed, weights can become low-precision decimal numbers, cf., e.g., (Nagel et al., 2020). Consequently, Theorem 2 provides, to the best of our knowledge, the first non-constant lower bound on the depth of practically relevant ReLU networks.
+
+Relying on Theorem 2, we also derive the following lower bound.
+
+Theorem 4. There is a constant  $C > 0$  such that, for all integers  $n, N \geq 3$ , every ReLU network with weights being  $N$ -ary fractions that represents  $\max \{0, x_1, \ldots, x_n\}$  has depth at least  $C \cdot \frac{\ln n}{\ln \ln N}$ .
+
+Theorem 4, in particular, shows that there is no constant-depth ReLU network that exactly represents  $\max \{0,x_1,\ldots ,x_n\}$  if all weights are rational numbers all having a common denominator  $N$
+
+In view of the integral networks considered by Haase et al. (2023), we stress that our results do not simply follow by scaling integer weights to rationals, which has already been discussed in Haase et al. (2023, Sec. 1.3). We therefore extend the techniques by Haase et al. (2023) to make use of number theory and polyhedral combinatorics to prove our results that cover standard number representations of rationals on a computer.
+
+Outline To prove our main results, Theorems 2 and 4, the rest of the paper is structured as follows. First, we provide some basic definitions regarding neural networks that we use throughout the article, and we provide a brief overview of related literature. Section 2 then provides a short summary of our overall strategy to prove Theorems 2 and 4 as well as some basic notation. The different concepts of polyhedral theory and volumes needed in our proof strategy are detailed in Section 2.1, whereas Section 2.2 recalls a characterization of functions representable by a ReLU neural network from the literature, which forms the basis of our proofs. In Section 3, we derive various properties of polytopes associated with functions representable by a ReLU neural network, which ultimately allows us to prove our main results in Section 3.3. The paper is concluded in Section 4.
+
+Basic Notation for ReLU Networks To describe the neural networks considered in this article, we introduce some notation. We denote by  $\mathbb{Z}$ ,  $\mathbb{N}$ , and  $\mathbb{R}$  the sets of integer, positive integer, and real numbers, respectively. Moreover,  $\mathbb{Z}_{+}$  and  $\mathbb{R}_{+}$  denote the sets of non-negative integers and reals, respectively.
+
+Let  $k \in \mathbb{Z}_+$ . A feedforward neural network with rectified linear units (ReLU) (or simply ReLU network in the following) with  $k + 1$  layers can be described by  $k + 1$  affine transformations  $t^{(1)} \colon \mathbb{R}^{n_0} \to \mathbb{R}^{n_1}, \ldots, t^{(k + 1)} \colon \mathbb{R}^{n_k} \to \mathbb{R}^{n_{k + 1}}$ . It exactly represents a function  $f \colon \mathbb{R}^n \to \mathbb{R}$  if and only if  $n_0 = n$ ,  $n_{k + 1} = 1$ , and the alternating composition
+
+$$
+t ^ {(k + 1)} \circ \sigma \circ t ^ {(k)} \circ \sigma \circ \dots \circ t ^ {(2)} \circ \sigma \circ t ^ {(1)}
+$$
+
+coincides with  $f$ , where, by slightly overloading notation,  $\sigma$  denotes the component-wise application of the ReLU activation function  $\sigma \colon \mathbb{R} \to \mathbb{R}$ ,  $\sigma(x) = \max \{0, x\}$  to vectors in any dimension. For each  $i \in \{1, \ldots, k + 1\}$  and  $x \in \mathbb{R}^{n_i - 1}$ , let  $t^{(i)}(x) = A^{(i)}x + b^{(i)}$  for some  $A^{(i)} \in \mathbb{R}^{n_i \times n_i - 1}$  and  $b^{(i)} \in \mathbb{R}^{n_i}$ . The entries of  $A^{(i)}$  are called weights and those of  $b^{(i)}$  are called biases of the network. The network's depth is  $k + 1$ , and the number of hidden layers is  $k$ .
+
+The set of all functions  $\mathbb{R}^n\to \mathbb{R}$  that can be represented exactly by a ReLU network of depth  $k + 1$  is denoted by  $\mathrm{ReLU}_n(k)$ . Moreover, if  $R\subseteq \mathbb{R}$  is a ring, we denote by  $\mathrm{ReLU}_n^R (k)$  the set of all functions  $\mathbb{R}^n\rightarrow \mathbb{R}$  that can be represented exactly by a ReLU network of depth  $k + 1$  all of whose weights are contained in  $R$ . Throughout this paper, we will mainly work with the rings  $\mathbb{Z}$ ,  $\mathbb{R}$ , or the ring of  $N$ -ary fractions.
+
+The set  $\mathrm{ReLU}_n^R (0)$  is the set of affine functions  $f(x_{1},\ldots ,x_{n}) = b + a_{1}x_{1} + \dots +a_{n}x_{n}$  with  $b\in \mathbb{R}$ , and  $a_1,\ldots ,a_n\in R$ . It can be directly seen from the definition of  $\mathrm{ReLU}$  networks that, for  $k\in \mathbb{N}$ , one has  $f\in \mathrm{ReLU}_n^R (k)$  if and only if  $f(x) = u_0 + u_1\max \{0,g_1(x)\} +\dots +u_m\max \{0,g_m(x)\}$  holds for some  $m\in \mathbb{N}$ ,  $u_0\in \mathbb{R}$ ,  $u_{1},\ldots ,u_{m}\in R$ , and functions  $g_{1},\ldots ,g_{m}\in \mathrm{ReLU}_{n}^{R}(k - 1)$ .
+
+Related Literature Regarding the expressiveness of ReLU networks, Hertrich et al. (2021) show that four layers are needed to exactly represent  $\max \{0, x_1, \ldots, x_4\}$  if the network satisfies the technical condition of being  $H$ -conforming. By restricting the weights of a ReLU network to be integer, Haase et al. (2023) prove that  $\mathrm{ReLU}_n^{\mathbb{Z}}(k - 1) \subsetneq \mathrm{ReLU}_n^{\mathbb{Z}}(k)$  for every  $k \leq \lceil \log_2(n + 1) \rceil$ . In particular,  $\max \{0, x_1, \ldots, x_{2^k}\} \notin \mathrm{ReLU}_{2^k}^{\mathbb{Z}}(k)$ . If the activation function is changed from ReLU to  $x \mapsto \mathbb{1}_{\{x > 0\}}$ , Khalife et al. (2024) show that two hidden layers are both necessary and sufficient for all functions representable by such a network.
+
+If one is only interested in approximating a function, Safran et al. (2024) show that  $\max \{0, x_1, \ldots, x_n\}$  can be approximated arbitrarily well by  $\mathrm{ReLU}_n^{\mathbb{Z}}(2)$ -networks of width  $n(n + 1)$  with respect to the  $L_2$  norm for continuous distributions. By increasing the depth of these networks, they also derive upper bounds on the required width in such an approximation. The results by Safran et al. (2024) belong to the class of so-called universal approximation theorems, which describe the ability to approximate classes of functions by specific types of neural networks, see, e.g., (Cybenko, 1989; Hornik, 1991; Barron, 1993; Pinkus, 1999; Kidger & Lyons, 2020). However, Vardi & Shamir (2020) show that there are significant theoretical barriers for depth-separation results for polynomially-sized  $\mathrm{ReLU}_n(k)$ -networks for  $k \geq 3$ , by establishing links to the separation of threshold circuits as well as to so-called natural-proof barriers. When taking specific data into account, Lee et al. (2024) derive lower and upper bounds on both the depth and width of a neural network that correctly classifies a given data set. More general investigations of the relation between the width and depth of a neural network are discussed, among others, by Arora et al. (2018); Eldan & Shamir (2016); Hanin (2019); Raghu et al. (2017); Safran & Shamir (2017); Telgarsky (2016).
+
+# 2 PROOF STRATEGY AND THEORETICAL CONCEPTS
+
+To prove Theorems 2 and 4, we extend the ideas of Haase et al. (2023). We therefore provide a very concise summary of the arguments of Haase et al. (2023). Afterwards, we briefly mention the main ingredients needed in our proofs, which are detailed in the following subsections.
+
+A central ingredient for the results by Haase et al. (2023) is a polyhedral characterization of all functions in  $\mathrm{ReLU}_n(k)$ , which has been derived by Hertrich (2022). This characterization links functions representable by a ReLU network and so-called support functions of polytopes  $P \subseteq \mathbb{R}^n$  all of whose vertices belong to  $\mathbb{Z}^n$ , so-called lattice polytopes. It turns out that the function  $\max \{0, x_1, \ldots, x_n\}$  in Theorems 2 and 4 can be expressed as the support function of a particular lattice polytope  $P_n \subseteq \mathbb{R}^n$ . By using a suitably scaled version  $\mathrm{Vol}_n$  of the classical Euclidean volume in  $\mathbb{R}^n$ , one can achieve  $\mathrm{Vol}_n(P) \in \mathbb{Z}$  for all lattice polytopes  $P \subseteq \mathbb{R}^n$ . Haase et al. (2023) then show that, if the support function  $h_P$  of a lattice polytope  $P \subseteq \mathbb{R}^n$  can be exactly represented by a ReLU network with  $k$  hidden layers, all faces of  $P$  of dimension at least  $2^k$  have an even normalized volume. For  $n = 2^k$ , however,  $\mathrm{Vol}_n(P_n)$  is odd. Hence, its support function cannot be represented by a ReLU network with  $k$  hidden layers.
+
+We show that the arguments of Haase et al. (2023) can be adapted by replacing the divisor 2 with an arbitrary prime number  $p$ . Another crucial insight is that the theory of mixed volumes can be used to analyze the behavior of  $\mathrm{Vol}_n(A + B)$  for the Minkowski sum  $A + B \coloneqq \{a + b : a \in A, b \in B\}$  of lattice polytopes  $A, B \subset \mathbb{R}^n$ . The Minkowski-sum operation is also involved in the polyhedral characterization of Hertrich (2022), and so it is also used by Haase et al. (2023), who provide a version of Theorem 2 for integer weights. They, however, do not directly use mixed volumes. A key observation used in our proofs, and obtained by a direct application of mixed volumes, is that the
+
+map associating to a lattice polytope  $P$  the coset of  $\operatorname{Vol}_n(P)$  modulo a prime number  $p$  is additive when  $n$  is a power of  $p$ . Combining these ingredients yields Theorems 2 and 4.
+
+Some Basic Notation The standard basis vectors in  $\mathbb{R}^n$  are denoted by  $e_1, \ldots, e_n$ , whereas 0 denotes the null vector in  $\mathbb{R}^n$ . Throughout the article, all vectors  $x \in \mathbb{R}^n$  are column vectors, and we denote the transposed vector by  $x^\top$ . If  $x$  is contained in the integer lattice  $\mathbb{Z}^n$ , we call it a lattice point. For vectors  $x, y \in \mathbb{R}^n$ , their scalar product is given by  $x^\top y$ . For  $m \in \mathbb{N}$ , we will write  $[m]$  for the set  $\{1, \ldots, m\}$ . The convex-hull operator is denoted by  $\operatorname{conv}$ , and the base- $b$  logarithm by  $\log_b$ , while the natural logarithm is denoted  $\ln$ .
+
+The central function of this article is  $\max \{0, x_1, \ldots, x_n\}$ , which we abbreviate by  $F_{n}$ .
+
+# 2.1 BASIC PROPERTIES OF POLYTOPES AND LATTICE POLYTOPES
+
+As outlined above, the main tools needed to prove Theorems 2 and 4 are polyhedral theory and different concepts of volumes. This section summarizes the main concepts and properties that we need in our argumentation in Section 3. For more background, we refer the reader to the monographs (Beck & Robins, 2020; Hug & Weil, 2020; Schneider, 2014).
+
+Polyhedra, Lattice Polyhedra, and Their Normalized Volume A polytope  $P \subseteq \mathbb{R}^n$  is the convex hull  $\operatorname{conv}(p_1, \ldots, p_m)$  of finitely many points  $p_1, \ldots, p_m \in \mathbb{R}^n$ . We introduce the family
+
+$$
+\mathcal {P} (S) := \left\{\operatorname {c o n v} \left(p _ {1}, \dots , p _ {m}\right): m \in \mathbb {N}, p _ {1}, \dots , p _ {m} \in S \right\}
+$$
+
+of all non-empty polytopes with vertices in  $S \subseteq \mathbb{R}^n$ . Thus,  $\mathcal{P}(\mathbb{R}^n)$  is the family of all polytopes in  $\mathbb{R}^n$  and  $\mathcal{P}(\mathbb{Z}^n)$  is the family of all lattice polytopes in  $\mathbb{R}^n$ . For  $d \in \{0, \dots, n\}$ , we also introduce the family
+
+$$
+\mathcal {P} _ {d} (S) := \{P \in \mathcal {P} (S) \colon \dim (P) \leq d \}.
+$$
+
+of polytopes of dimension at most  $d$ , where the dimension of a polytope  $P$  is defined as the dimension of its affine hull, i.e., the smallest affine subspace of  $\mathbb{R}^n$  containing  $P$ . The Euclidean volume  $\mathrm{vol}_n$  on  $\mathbb{R}^n$  is the  $n$ -dimensional Lebesgue measure, scaled so that  $\mathrm{vol}_n$  is equal to 1 on the unit cube  $[0,1]^d$ . Note that measure-theoretic subtleties play no role in our context since we restrict the use of  $\mathrm{vol}_n$  to  $\mathcal{P}(\mathbb{R}^n)$ . The normalized volume  $\mathrm{Vol}_n$  in  $\mathbb{R}^n$  is the  $n$ -dimensional Lebesgue measure normalized so that  $\mathrm{Vol}_n$  is equal to 1 on the standard simplex  $\Delta_n \coloneqq \operatorname{conv}(0, e_1, \ldots, e_n)$ . Clearly,  $\mathrm{Vol}_n = n! \cdot \mathrm{vol}_n$  and  $\mathrm{Vol}_n$  takes non-negative integer values on lattice polytopes.
+
+Support Functions For a polytope  $P = \mathrm{conv}(p_1,\ldots ,p_m)\subseteq \mathbb{R}^n$  , its support function is
+
+$$
+h _ {P} (x) := \max  \{x ^ {\top} y: y \in P \},
+$$
+
+and it is well-known that  $h_P(x) = \max \{p_1^\top x, \ldots, p_m^\top x\}$ . Consequently,  $\max \{0, x_1, \ldots, x_n\}$  from Theorems 2 and 4 is the support function of  $\Delta_n$ .
+
+Mixed Volumes For sets  $A, B \subseteq \mathbb{R}^n$ , we introduce the Minkowski sum
+
+$$
+A + B := \{a + b: a \in A, b \in B \}
+$$
+
+and the multiplication
+
+$$
+\lambda A = \{\lambda a: a \in A \}
+$$
+
+of  $A$  by a non-negative factor  $\lambda \in \mathbb{R}_+$ . For an illustration of the Minkowski sum, we refer to Figure 2. Note that, if  $S \in \{\mathbb{R}^n, \mathbb{Z}^n\}$  and  $A, B \in \mathcal{P}(S)$ , then  $A + B \in \mathcal{P}(S)$ , too. If  $A$  and  $B$  are (lattice) polytopes, then  $A + B$  is also a (lattice) polytope, and the support functions of  $A, B$  and  $A + B$  are related by  $h_{A + B} = h_A + h_B$ .
+
+If  $(G, +)$  is an Abelian semi-group (i.e., a set with an associative and commutative binary operation), we call a map  $\phi \colon \mathcal{P}(\mathbb{R}^n) \to G$  Minkowski additive if the Minkowski addition on  $\mathcal{P}(\mathbb{R}^n)$  gets preserved by  $\phi$  in the sense that  $\phi(A + B) = \phi(A) + \phi(B)$  holds for all  $A, B \in \mathcal{P}(\mathbb{R}^n)$ .
+
+The following is a classical result of Minkowski.
+
+Theorem 5 (see, e.g., (Schneider, 2014, Ch. 5)). There exists a unique functional, called the mixed volume,
+
+$$
+\mathrm {V}: \mathcal {P} (\mathbb {R} ^ {n}) ^ {n} \to \mathbb {R},
+$$
+
+with the following properties valid for all  $P_{1},\ldots ,P_{n},A,B\in \mathcal{P}(\mathbb{R}^{n})$  and  $\alpha ,\beta \in \mathbb{R}_{+}$
+
+(a)  $\mathrm{V}$  is invariant under permutations, i.e.  $\mathrm{V}(P_1, \ldots, P_n) = \mathrm{V}(P_{\sigma(1)}, \ldots, P_{\sigma(n)})$  for every permutation  $\sigma$  on  $[n]$ .  
+(b)  $\mathrm{V}$  is Minkowski linear in all input parameters, i.e., for all  $i \in [n]$ , it holds that
+
+$$
+\begin{array}{l} \mathrm {V} \left(P _ {1}, \dots P _ {i - 1}, \alpha A + \beta B, P _ {i + 1}, \dots , P _ {n}\right) = \alpha \mathrm {V} \left(P _ {1}, \dots P _ {i - 1}, A, P _ {i + 1}, \dots , P _ {n}\right) \\ + \beta \mathrm {V} \left(P _ {1}, \dots P _ {i - 1}, B, P _ {i + 1}, \dots , P _ {n}\right) \\ \end{array}
+$$
+
+(c)  $\mathrm{V}$  is equal to  $\operatorname{Vol}_n$  on the diagonal, i.e.,  $\mathrm{V}(A, \ldots, A) = \operatorname{Vol}_n(A)$ .
+
+We refer to Chapter 5 of the monograph by Schneider (2014) on the Brunn-Minkowski theory for more information on mixed volumes, where also an explicit formula for the mixed volume is presented. Our definition of the mixed volume differs by a factor of  $n!$  from the definition in Schneider (2014) since we use the normalized volume  $\mathrm{Vol}_n$  rather than the Euclidean volume  $\mathrm{vol}_n$  to fix  $\mathrm{V}(P_1,\ldots ,P_n)$  in the case  $P_{1} = \dots = P_{n}$ . Our way of introducing mixed volumes is customary in the context of algebraic geometry. It is known that, for this normalization,  $\mathrm{V}(P_1,\ldots ,P_n)\in \mathbb{Z}_+$  when  $P_{1},\ldots ,P_{n}$  are lattice polytopes; see, for example, (Maclagan & Sturmfels, 2015, Ch. 4.6). From the defining properties one can immediately see that, for  $A,B\in \mathcal{P}(\mathbb{R}^n)$ , one has the analogue of the binomial formula, which we will prove in Appendix A.2 for the sake of completeness:
+
+$$
+\operatorname {V o l} _ {n} (A + B) = \sum_ {i = 0} ^ {n} \binom {n} {i} \mathrm {V} (\underbrace {A , \dots , A} _ {i}, \underbrace {B , \dots , B} _ {n - i}). \tag {1}
+$$
+
+Normalized Volume of Non-Full-Dimensional Polytopes So far, we have introduced the normalized volume  $\mathrm{Vol}_n\colon \mathcal{P}(\mathbb{R}^n)\to \mathbb{R}_+$ , i.e., if  $P\in \mathcal{P}(\mathbb{R}^n)$  is not full-dimensional, then  $\mathrm{Vol}_n(P) = 0$ . We also associate with a polytope  $P\in \mathcal{P}_d(\mathbb{Z}^n)$  of dimension at most  $d$  an appropriately normalized  $d$ -dimensional volume by extending the use of  $\mathrm{Vol}_d\colon \mathcal{P}(\mathbb{Z}^d)\to \mathbb{Z}_+$  to  $\mathrm{Vol}_d\colon \mathcal{P}_d(\mathbb{Z}^n)\to \mathbb{Z}_+$ . In the case  $\dim (P) < d$ , we define  $\mathrm{Vol}_d(P) = 0$ . If  $d = 0$ , let  $\mathrm{Vol}_d(P) = 1$ . In the non-degenerate case  $d = \dim (P)\in \mathbb{N}$ , we fix  $Y$  to be the affine hull of  $P$  and consider a bijective affine map  $T\colon \mathbb{R}^d\to Y$  satisfying  $T(\mathbb{Z}^d) = Y\cap \mathbb{Z}^n$ . For such choice of  $T$ , we have  $T^{-1}(P)\in \mathcal{P}(\mathbb{Z}^d)$ . It turns out that the  $d$ -dimensional volume of  $T^{-1}(P)$  depends only on  $P$  and not on  $T$  so that we define  $\mathrm{Vol}_d(P)\coloneqq \mathrm{Vol}_d(T^{-1}(P))$ . Based on (Beck & Robins, 2020, Corollary 3.17 and §5.4), there is the following intrinsic way of introducing  $\mathrm{Vol}_d(P)$ . Let  $G(P)$  denote the number of lattice points in  $P$ . Then, for  $t\in \mathbb{Z}_{+}$ , one has  $\mathrm{Vol}_d(P)\coloneqq d!\cdot \lim_{t\to \infty}\frac{1}{t^d} G(tP)$ .
+
+Remark 6. For every  $d$ -dimensional affine subspace  $Y \subseteq \mathbb{R}^n$  which is affinely spanned by  $d + 1$  lattice points, we can define  $\mathrm{Vol}_d$  for every polytope  $P \in \mathcal{P}(Y)$ , which is not necessarily a lattice polytope, by the same formula  $\mathrm{Vol}_d(P) \coloneqq \mathrm{Vol}_d(T^{-1}(P))$ , using an auxiliary map  $T \colon \mathbb{R}^d \to Y$  described above. Consequently, by replacing the dimension  $n$  with  $d$  and the family of polytopes  $\mathcal{P}(\mathbb{R}^n)$  with the family  $\mathcal{P}(Y)$  in Minkowski's Theorem 5, we can introduce the notion of mixed volumes for polytopes in  $\mathcal{P}(Y)$ . More specifically, we will make use of the mixed volumes of lattice polytopes in  $\mathcal{P}(Y \cap \mathbb{Z}^n)$ .
+
+Normalized Volume of the Affine Join The following proposition, borrowed from Haase et al. (2023), addresses the divisibility properties of the convex hull of the union of lattice polytopes that lie in skew affine subspaces.
+
+Proposition 7 (Haase et al. 2023, Lemma 6). Let  $A, B \in \mathcal{P}(\mathbb{Z}^n)$  be polytopes of dimensions  $i \in \mathbb{Z}_+$  and  $j \in \mathbb{Z}_+$ , respectively, such that  $P \coloneqq \operatorname{conv}(A \cup B)$  is of dimension  $i + j + 1$ . Then  $\operatorname{Vol}_{i+j}(P)$  is divisible by  $\operatorname{Vol}_i(A) \operatorname{Vol}_j(B)$ . In particular, if  $i = 0$ , then  $P$  is a pyramid over  $B$  whose normalized volume  $\operatorname{Vol}_{1+j}(B)$  is divisible by the normalized volume  $\operatorname{Vol}_j(B)$  of its base  $B$ .
+
+For an example illustration, see Figure 1. Since  $P_{1}$  and  $P_{2}$  lie in skew affine subspaces, Proposition 7 applies. Indeed,  $\mathrm{Vol}_3(\mathrm{conv}(P_1 \cup P_2)) = 12$  is divisible by  $\mathrm{Vol}_2(P_1) = 6$  (and  $\mathrm{Vol}_0(P_2) = 1$ ).
+
+# 2.2 A POLYHEDRAL CRITERION FOR FUNCTIONS REPRESENTABLE WITH  $k$  HIDDEN LAYERS
+
+Next to the geometric concepts that we discussed before, the second main building block of our proofs is the polyhedral characterization of  $\mathrm{ReLU}_n(k)$  by Hertrich (2022). In the following, we introduce the necessary concepts and present Hertrich's characterization.
+
+![](images/6cad8cb056a0c9a0023580552cf94ccf9efe68e6ade23c3231dd97566eacdcfa.jpg)  
+Figure 1: Illustration of the convex hull of a polytope and a point, relating to Proposition 7.
+
+![](images/30fed49d00781b981236a8c40d96094ee07b6f495e455eee7f977b02f0ce0269.jpg)  
+Figure 2: Illustration of the Minkowski sum of two polytopes, relating to Example 12.
+
+Note that  $F_{n}$  is positively homogeneous, i.e., for all  $\lambda \in \mathbb{R}_{+}$  and  $x\in \mathbb{R}^n$ , one has  $F_{n}(\lambda x) = \lambda F_{n}(x)$ . For positively homogeneous functions  $f$ , Hertrich et al. (2021) show that  $f\in \mathrm{ReLU}_n(k)$  if and only if there exists a ReLU network of depth  $k + 1$  all of whose biases are 0. This result easily generalizes to ReLU networks with weights being restricted to a ring  $R$ . We therefore denote by  $\mathrm{ReLU}_n^{R,0}(k)$  the set of all  $n$ -variate positively homogeneous functions representable by ReLU networks with  $k$  hidden layers, weights in  $R$ , and all biases being 0. Moreover,  $\mathrm{ReLU}_n^{R,0} := \bigcup_{k = 0}^{\infty}\mathrm{ReLU}_n^{R,0}(k)$ .
+
+To formulate the characterization by Hertrich (2022), we define the sum-union closure for a family of polytopes  $\mathcal{X}$  in  $\mathbb{R}^n$  as
+
+$$
+\operatorname {S U} (\mathcal {X}) := \left\{\sum_ {i = 1} ^ {m} \operatorname {c o n v} (A _ {i} \cup B _ {i}) \colon m \in \mathbb {N}, A _ {i}, B _ {i} \in \mathcal {X}, i \in [ m ] \right\}.
+$$
+
+The  $k$ -fold application of the operation gives the  $k$ -fold sum-union closure  $\mathrm{SU}^k (\mathcal{X})$ . In other words,  $\mathrm{SU}^0 (\mathcal{X}) = \mathcal{X}$  and  $\mathrm{SU}^k (\mathcal{X}) = \mathrm{SU}(\mathrm{SU}^{k - 1}(\mathcal{X}))$  for  $k\in \mathbb{N}$ . We will apply the  $k$ -fold sum-union closure to  $\mathcal{P}_0(S)$ , the set of all 0-dimensional polytopes of the form  $\{s\}$ , with  $s\in S$ .
+
+The set  $\mathrm{SU}^k (\mathcal{X})$  forms a semi-group with respect to Minkowski-addition since, directly from the representation of elements of  $\mathrm{SU}^k (\mathcal{X})$  as sums with arbitrarily many summands, one sees that  $\mathrm{SU}^k (\mathcal{X})$  is closed under Minkowski addition.
+
+Theorem 8 ((Hertrich, 2022, Thm. 3.35) for  $R = \mathbb{R}$  and (Haase et al., 2023, Thm. 8) for  $R = \mathbb{Z}$ ). Let  $R$  be  $\mathbb{R}$  or  $\mathbb{Z}$ . Then
+
+$$
+\operatorname {R e L U} _ {n} ^ {R, 0} (k) = \left\{h _ {A} - h _ {B} \colon A, B \in \mathrm {S U} ^ {k} \left(\mathcal {P} _ {0} \left(R ^ {n}\right)\right) \right\}.
+$$
+
+Corollary 9. Let  $k \in \mathbb{Z}_{+}$  and  $R$  be  $\mathbb{R}$  or  $\mathbb{Z}$ . Let  $P \in \mathcal{P}(R^n)$ . Then, the support function  $h_P$  of  $P$  satisfies  $h_P \in \mathrm{ReLU}_n^R(k)$  if and only if  $P + A = B$  for some  $A, B \in \mathrm{SU}^k(\mathcal{P}_0(R^n))$ .
+
+Proof. By Theorem 8, we have that  $h_P \in \mathrm{ReLU}_n^R(k)$  if and only if  $h_P = h_B - h_A$  for some  $A, B \in \mathrm{SU}^k(\mathcal{P}_0(R^n))$ . The equation  $h_P = h_B - h_A$  can be rewritten as  $h_B = h_P + h_A = h_{P + A}$ , as support functions are Minkowski additive. Furthermore, every polytope is uniquely determined by its support function, see (Hug & Weil, 2020), so  $h_{P + A} = h_B$  is equivalent to  $P + A = B$ .
+
+The characterization of  $\mathrm{ReLU}_n^{R,0}(k)$  via  $\mathrm{SU}^k (\mathcal{P}_0(R^n))$  as well as the geometric concepts of volumes will allow us to prove Theorem 2. The core step of our proof is to show that  $F_{n}$ , which is the support function of  $\Delta_{n}$ , is not contained in  $\mathrm{ReLU}_n^{\mathbb{Z},0}(k)$  for small  $k$ . As we will see later, it turns out to be useful to not work exclusively with full-dimensional polytopes in  $\mathrm{SU}^k (\mathcal{P}_0(\mathbb{Z}^n))$ , but with some of their lower-dimensional faces. The next lemma provides the formal mechanism that we use, namely if  $P\in \mathrm{SU}^k (\mathcal{P}_0(\mathbb{Z}^n))$  and  $F$  is a face of  $P$ , then  $h_F\notin \mathrm{ReLU}_n^{\mathbb{Z}}(k)$  implies also  $h_P\notin \mathrm{ReLU}_n^{\mathbb{Z}}(k)$ . We defer the lemma's proof to Appendix A.1.1.
+
+Lemma 10. Let  $k \in \mathbb{Z}_+$ . Then, for all  $P \in \mathrm{SU}^k(\mathcal{P}_0(\mathbb{Z}^n))$  and  $u \in \mathbb{R}^n$ , the face of  $P$  in direction  $u$ , given by
+
+$$
+P ^ {u} := \{x \in P \colon u ^ {\top} x = h _ {P} (u) \},
+$$
+
+belongs to  $\mathrm{SU}^k (\mathcal{P}_0(\mathbb{Z}^n))$ . In other words,  $\mathrm{SU}^k (\mathcal{P}_0(\mathbb{Z}^n))$  is closed under taking non-empty faces.
+
+# 3 RESULTS AND PROOFS
+
+The goal of this section is to prove Theorems 2 and 4 for the ring  $R$  of  $N$ -ary fractions. To this end, we will rescale  $F_{n}$  by a suitable scalar  $\lambda \in \mathbb{N}$  such that the containment  $F_{n} \in \mathrm{ReLU}_{n}^{R}(k)$  is equivalent to  $\lambda F_{n} \in \mathrm{ReLU}_{n}^{\mathbb{Z}}(k)$ . To show that  $\lambda F_{n} \notin \mathrm{ReLU}_{n}^{\mathbb{Z}}(k)$  if  $k$  is too small, we use a volume-based argument. More precisely, we show that, for lattice polytopes  $P \subseteq \mathbb{R}^n$  whose support functions  $h_P$  are contained in  $\mathrm{ReLU}_{n}^{\mathbb{Z}}(k)$  and suitably defined dimensions  $d$  and prime numbers  $p$ , their volumes  $\mathrm{Vol}_d(P)$  are divisible by  $p$ . In contrast,  $\mathrm{Vol}_d(\lambda \Delta_n)$  is not divisible by  $p$ , and thus,  $\lambda F_{n} \notin \mathrm{ReLU}_{n}^{\mathbb{Z}}(k)$ . This strategy is inspired by the proof of Haase et al. (2023) for  $F_{n} \notin \mathrm{ReLU}_{n}^{\mathbb{Z}}(k)$ , where related results are shown for the special case  $p = 2$ . Our results, however, are more general and do not follow directly from their results.
+
+To pursue this strategy, Sections 3.1 and 3.2 derive novel insights into volumes  $\mathrm{Vol}_d(P)$  of lattice polytopes  $P$  whose support functions  $h_p$  are contained in  $\mathrm{ReLU}_n^{\mathbb{Z}}(k)$ . These insights are then used in Section 3.3 to prove Theorems 2 and 4.
+
+# 3.1 DIVISIBILITY OF NORMALIZED VOLUMES BY A PRIME
+
+To understand the divisibility of  $\mathrm{Vol}_d$  by a prime number mentioned above, we investigate cases in which  $\mathrm{Vol}_d\colon \mathcal{P}_d(\mathbb{Z}^n)\to \mathbb{Z}$  modulo a prime is Minkowski additive. To make this precise, we introduce some notation.
+
+For  $a, b \in \mathbb{Z}$  and  $m \in \mathbb{N}$  we write  $a \equiv_{m} b$  if  $a - b$  is divisible by  $m$ . This is called the congruence of  $a$  and  $b$  modulo  $m$ . The coset  $[z]_m$  of  $z \in \mathbb{Z}$  modulo  $m$  is the set of all integers congruent to  $z$  modulo  $m$ , and we denote the set of all such cosets by  $\mathbb{Z}_m$ . The addition of cosets is defined by  $[a]_m + [b]_m := [a + b]_m$  for  $a, b \in \mathbb{Z}$ . Endowing  $\mathbb{Z}_m$  with the addition operation  $+$  yields a group of order  $m$ .
+
+The following is an easy-to-prove but crucial observation. It states that when we consider lattice polytopes in a  $d$ -dimensional subspace  $Y \subseteq \mathbb{R}^n$  spanned by  $d$  lattice points, the volume  $\mathrm{Vol}_d$ , taken modulo a prime number  $p$ , is an additive functional when  $d$  is a power of  $p$ .
+
+Proposition 11. Let  $d = p^t \leq n$  be a power of a prime number  $p$ , with  $t \in \mathbb{N}$ . Let  $P_1, \ldots, P_m \in \mathcal{P}_d(\mathbb{Z}^n)$  be such that  $\sum_{i=1}^{m} P_i \in \mathcal{P}_d(\mathbb{Z}^n)$ . Then,
+
+$$
+\operatorname {V o l} _ {d} \left(\sum_ {i = 1} ^ {m} P _ {i}\right) \equiv_ {p} \sum_ {i = 1} ^ {m} \operatorname {V o l} _ {d} (P _ {i}).
+$$
+
+Proof. Note that by the assumption  $\sum_{i=1}^{m} P_i \in \mathcal{P}_d(\mathbb{Z}^d)$  all of the  $P_i$ 's lie, up to appropriate translation, in a  $d$ -dimensional vector subspace  $Y$  of  $\mathbb{R}^d$ , which is spanned by  $d$  lattice points. There is no loss of generality in assuming that  $P_i \subseteq Y$  and, in view of Remark 6, we can use the mixed volume functional on  $d$ -tuples of polytopes from  $\mathcal{P}(Y)$ , which will give an integer value for polytopes in  $\mathcal{P}(Y \cap \mathbb{Z}^n)$ . By an inductive argument, it is sufficient to consider the case  $m = 2$ . It is well known that if  $d$  is a power of  $p$ , the binomial coefficients  $\binom{d}{1}, \ldots, \binom{d}{d-1}$  in (1) are divisible by  $p$ , see, e.g., Mihet (2010, Cor. 2.9). Thus, (1) implies  $\operatorname{Vol}_d(P_1 + P_2) \equiv_p \operatorname{Vol}_d(P_1) + \operatorname{Vol}_d(P_2)$  for  $P_1, P_2 \in \mathcal{P}(Y \cap \mathbb{Z}^n)$ .
+
+Example 12. Consider the polytope  $P_{1} + P_{2} \in \mathcal{P}_{2}(\mathbb{Z}^{2})$  for the rectangle  $P_{1} = [2,5] \times [0,1] \in \mathcal{P}_{2}(\mathbb{Z}^{2})$  and the shifted standard simplex  $P_{2} = \Delta_{2} + \{(0,2)^{\top}\} \in \mathcal{P}_{2}(\mathbb{Z}^{2})$  as depicted in Figure 2. In the picture,  $P_{1} + P_{2}$  is decomposed into regions in such a way that the volume of the mixed area  $\mathrm{V}(P_1,P_2)$  can be read off. In view of the equality  $\mathrm{Vol}_2(P_1 + P_2) = \mathrm{V}(P_1 + P_2,P_1 + P_2) = \mathrm{V}(P_1,P_1) + 2\mathrm{V}(P_1,P_2) + \mathrm{V}(P_2,P_2) = \mathrm{Vol}_2(P_1) + 2\mathrm{V}(P_1,P_2) + \mathrm{Vol}_2(P_2)$ , see (1), the total volume of the unshaded part of  $P_{1} + P_{2}$  must be exactly  $2\mathrm{V}(P_1,P_2)$ . For  $p = 2$  we have  $\mathrm{Vol}_2(P_1 + P_2) = 15 \equiv_2 6 + 1 = \mathrm{Vol}_2(P_1) + \mathrm{Vol}_2(P_2)$ , i.e., the parity of  $\mathrm{Vol}_2(P_1 + P_2)$  is indeed that of  $\mathrm{Vol}_2(P_1) + \mathrm{Vol}_2(P_2)$ . In contrast, divisibility by  $p = 3$  does not match, as  $15 \neq_3 7$ . However, this does not contradict Proposition 11, as  $d = 2$  is not a power of  $p = 3$ .
+
+To derive divisibility properties of  $\mathrm{Vol}_d(P)$  for lattice polytopes  $P$  with  $h_P \in \mathrm{ReLU}_n^{\mathbb{Z}}(k)$ , we make use of the characterization of  $\mathrm{ReLU}_n^{\mathbb{Z}}(k)$  via the SU-operator. Recall that one of the two defining
+
+operations of SU is  $\mathrm{conv}(A\cup B)$  for suitable polytopes  $A$  and  $B$ . A crucial observation is that for certain dimensions  $d$ , the divisibility of  $\mathrm{Vol}_d(\mathrm{conv}(A\cup B))$  by a prime number is inherited from particular lower-dimensional faces of  $A$  and  $B$ .
+
+Proposition 13. Let  $d = p^t \leq n$  be a power of a prime number  $p$ , with  $t \in \mathbb{N}$ . Moreover, let  $P = \operatorname{conv}(A \cup B) \in \mathcal{P}_d(\mathbb{Z}^n)$  for  $A, B \in \mathcal{P}_d(\mathbb{Z}^n)$ . If  $\operatorname{Vol}_{p^{t-1}}(F) \equiv_p 0$  for all  $p^{t-1}$ -dimensional faces  $F$  of  $A$  and  $B$ , then  $\operatorname{Vol}_{p^t}(P) \equiv_p 0$ .
+
+Note that this result also holds trivially if no face of dimension  $p^{t - 1}$  exists. We defer the proof of this result to Appendix A.1.2.
+
+# 3.2 MODULAR OBSTRUCTION ON VOLUME FOR REALIZABILITY WITH  $k$  HIDDEN LAYERS
+
+Equipped with the previously derived results, we have all ingredients together to prove the aforementioned results on the divisibility of  $\mathrm{Vol}_d(P)$  for lattice polytopes  $P$  with  $h_P \in \mathrm{ReLU}_n^{\mathbb{Z}}(k)$ .
+
+Theorem 14. Let  $d = p^t \leq n$  be a power of a prime number  $p$ , with  $t \in \mathbb{N}$ . Let  $k \in [t]$  and  $P \in \mathrm{SU}^k(\mathcal{P}_0(\mathbb{Z}^n))$ . Then  $\operatorname{Vol}_{p^k}(F) \equiv_p 0$  for all  $p^k$ -dimensional faces  $F$  of  $P$ .
+
+Proof. We argue by induction on  $k$ . If  $k = 1$ , then  $\mathrm{SU}^1(\mathcal{P}_0(\mathbb{Z}^n))$  consists of lattice zonotopes. These are polytopes of the form  $P = S_1 + \dots + S_m$ , where  $S_1, \ldots, S_m$  are line segments joining a pair of lattice points. One has  $\operatorname{Vol}_d(P) \equiv_p \operatorname{Vol}_d(\sum_{i=1}^m S_i) \equiv_p \sum_{i=1}^m \operatorname{Vol}_d(S_i)$ , by Proposition 11, arriving at  $\operatorname{Vol}_d(P) \equiv_p 0$ , since  $\operatorname{Vol}_d(S_i) = 0$  for all  $i$  as  $d > 1 \geq \dim(S_i)$ .
+
+In the inductive step, assume  $k \geq 2$  and that the assertion has been verified for  $\mathrm{SU}^{k-1}(\mathcal{P}_0(\mathbb{Z}^n))$ . Recall that every  $P \in \mathrm{SU}^k(\mathcal{P}_0(\mathbb{Z}^n))$  can be written as  $P = \sum_{i=1}^m \operatorname{conv}(A_i \cup B_i)$  for some polytopes  $A_i, B_i \in \mathrm{SU}^{k-1}(\mathcal{P}_0(\mathbb{Z}^n))$ . By the induction hypothesis, the  $p^{k-1}$ -dimensional normalized volumes of the  $p^{k-1}$ -dimensional faces of  $A_i$  and  $B_i$  are divisible by  $p$ . Consequently, by Proposition 13, the  $p^k$ -dimensional normalized volumes of the  $p^k$ -dimensional faces of  $\operatorname{conv}(A_i \cup B_i)$  are divisible by  $p$ . Since  $\mathrm{SU}^k(\mathcal{P}_0(\mathbb{Z}^n))$  is closed under taking faces (see Lemma 10), Proposition 11 applied to the  $p^k$ -dimensional faces of  $P$  implies that the  $p^k$ -dimensional normalized volume of the  $p^k$ -dimensional faces of  $P$  is divisible by  $p$ .
+
+Theorem 15. Let  $d = p^t \leq n$  be a power of a prime number  $p$ , with  $t \in \mathbb{N}$ . Let  $P$  be a lattice polytope in  $\mathcal{P}_d(\mathbb{R}^n)$ . If  $h_P \in \mathrm{ReLU}_n^{\mathbb{Z}}(k)$ ,  $k \in [t]$ , then  $\mathrm{Vol}_d(P)$  is divisible by  $p$ .
+
+Proof. By Corollary 9, we have  $P + A = B$  for some  $A, B \in \mathrm{SU}^k(\mathcal{P}_0(\mathbb{Z}^n))$ . Then, by Proposition 11, one has  $\operatorname{Vol}_d(P + A) \equiv_p \operatorname{Vol}_d(P) + \operatorname{Vol}_d(A) \equiv_p \operatorname{Vol}_d(B)$ , which means that  $\operatorname{Vol}_d(P) \equiv_p \operatorname{Vol}_d(A) - \operatorname{Vol}_d(B)$ . By Theorem 14, both  $\operatorname{Vol}_d(A)$  and  $\operatorname{Vol}_d(B)$  are divisible by  $p$ . This shows that  $\operatorname{Vol}_d(P)$  is divisible by  $p$ .
+
+# 3.3 PROOFS OF MAIN RESULTS
+
+We now turn to the proofs of Theorems 2 and 4. Let  $N \in \mathbb{N}$  and recall that a rational number is an  $N$ -ary fraction if it is of the form  $\frac{z}{N^t}$  with  $z \in \mathbb{Z}$  and  $t \in \mathbb{Z}_+$ . For  $N = 2$  and  $N = 10$ , one has binary and decimal fractions, used in practice in floating point calculations. Clearly, every binary fraction is also a decimal fraction, because  $\frac{z}{2^t} = \frac{5^t z}{10^t}$ .
+
+In order to relate ReLU networks with fractional weights to ReLU networks with integer weights, we can simply clear denominators, as made precise in the following lemma.
+
+Lemma 16. Let  $f\colon \mathbb{R}^n\to \mathbb{R}$  be exactly representable by a ReLU network with  $k$  hidden layers and with rational weights all having  $M$  as common denominator. Then  $M^{k + 1}f\in \mathrm{ReLU}_n^{\mathbb{Z}}(k)$ .
+
+Proof. We proceed by induction on  $k$ . For the base case  $k = 0$ ,  $f$  is an affine function  $f(x_{1},\ldots ,x_{n}) = b + a_{1}x_{1} + \dots +a_{n}x_{n}$  with  $b\in \mathbb{R}$  and  $M a_{1},\ldots ,M a_{n}\in \mathbb{Z}$ , from which the claim is immediately evident. Now let  $k\geq 1$  and consider a  $k$ -layer network with rational weights with common denominator  $M$  representing  $f$ . Then  $f$  is of the form  $f(x) = u_0 + u_1\max \{0,g_1(x)\} +\dots +u_m\{0,g_m(x)\}$  with  $m\in \mathbb{N}$ , where all  $g_{1},\ldots ,g_{m}$  are functions representable with  $k - 1$  hidden layers and all the weights, i.e.,  $u_{1},\ldots ,u_{m}$  and the ones used in
+
+expressions for  $g_{1}, \ldots, g_{m}$ , are rational numbers with  $M$  as a common denominator. Multiplying with  $M^{k+1}$  we obtain
+
+$$
+M ^ {k + 1} f (x) = M ^ {k + 1} u _ {0} + M u _ {1} \cdot \max  \{0, M ^ {k} g _ {1} (x) \} + \dots + M u _ {m} \cdot \max  \{0, M ^ {k} g _ {m} (x) \},
+$$
+
+where the weights  $M u_{1},\ldots ,M u_{m}$  are integer. By the induction hypothesis, for every  $i\in [m]$ , we have  $M^{k}g_{i}\in \mathrm{ReLU}_{n}^{\mathbb{Z}}(k - 1)$ , and hence  $M^{k + 1}f\in \mathrm{ReLU}_n^{\mathbb{Z}}(k)$ .
+
+We are now ready to prove our main results.
+
+Proof of Theorem 2. Let  $k = \lceil \log_p(n + 1) \rceil - 1$ , i.e.,  $k$  is the unique non-negative integer satisfying  $p^k < n + 1 \leq p^{k + 1}$ . If  $F_{n}$  was representable by a ReLU network with  $k$  hidden layers and  $N$ -ary fractions as weights,  $\max \{0, x_1, \ldots, x_{p^k}\} = F_n(x_1, \ldots, x_{p^k}, 0, \ldots, 0)$  would also be representable in this way. Thus, it suffices to consider the case  $n = p^k$ .
+
+Recall that  $F_{n}$  is the support function  $h_{\Delta_n}$  of the standard simplex. Suppose, for the sake of contradiction, that  $F_{n}$  can be represented by a ReLU network with  $k$  hidden layers and weights being  $N$ -ary fractions. Let  $t \in \mathbb{N}$  be large enough such that all weights are representable as  $\frac{z}{N^t}$  for some  $z \in \mathbb{Z}$ . We use Lemma 16 with  $M = N^t$  to clear denominators. That is,  $N^{t(k + 1)}F_{n}$  is representable by an integer-weight ReLU network with  $k$  hidden layers. Since  $F_{n}$  is homogeneous, we can assume that the network is homogeneous, too (Hertrich et al., 2021, Proposition 2.3). Observe that  $N^{t(k + 1)}F_{n}$  is the support function of  $N^{t(k + 1)}\Delta_{n}$ , whose normalized volume satisfies  $\mathrm{Vol}_n(N^{t(k + 1)}\Delta_n) \equiv_p N^{nt(k + 1)}\mathrm{Vol}_n(\Delta_n) = N^{nt(k + 1)}\cdot 1 \neq_p 0$ . Hence,  $N^{t(k + 1)}\Delta_{n}$  is a polytope in  $\mathbb{R}^{p^k}$  whose normalized volume is not divisible by  $p$ , but whose support function is represented by an integer-weight ReLU network with  $k$  hidden layers. This contradicts Theorem 15. Hence,  $F_{n}$  is not representable by a ReLU network with  $k$  hidden layers and weights being  $N$ -ary fractions.
+
+If  $N = 10$ , we can use  $p = 3$  in Theorem 2, so Corollary 3 is an immediate consequence. The bound  $\lceil \log_3(n + 1) \rceil$  in Corollary 3 is optimal up to a constant factor, as  $F_{n}$  is representable through a network with integral weights and  $\lceil \log_2(n + 1) \rceil$  hidden layers (Arora et al., 2018). A major open question raised by Hertrich et al. (2021) is whether this kind of result can be extended to networks whose weights belong to a larger domain like the rational numbers or, ideally, the real numbers.
+
+We can also show that the expressive power of ReLU networks with weights being decimal fractions grows gradually when the depth  $k$  is increasing in the range from 1 to  $\mathcal{O}(\log n)$ .
+
+Corollary 17. For each  $n \in \mathbb{N}$  and each integer  $k \in \{1, \dots, \lceil \log_3 n \rceil\}$ , within  $n$ -variate functions that are described by ReLU networks with weights being decimal fractions, there are functions representable using  $2k$  but not using  $k$  hidden layers.
+
+Proof. Function  $F_{3^k}$  is not representable through  $k$  hidden layers and weights being decimal fractions. Since  $3^k \leq 2^{2k}$ ,  $F_{3^k}$  is representable with  $2k$  hidden layers (and integer weights).
+
+By making use of Theorem 2, we now present the proof of Theorem 4.
+
+Proof of Theorem 4. To make use of Theorem 2, we need to find a prime number  $p$  that does not divide  $N$ . Let  $p_i$  denote the  $i$ -th prime number, i.e.,  $p_1 = 2, p_2 = 3, p_3 = 5$  etc. Moreover, assume that the prime number decomposition of  $N$  consists of  $t$  distinct primes, i.e.,  $N = p_{i_1}^{m_1} \cdots p_{i_t}^{m_t}$  where  $m_1, \ldots, m_t \in \mathbb{N}$  and  $i_1 < \cdots < i_t$ . Choose the minimal prime  $p$  that is not contained in  $\{p_{i_1}, \ldots, p_{i_t}\}$ , that is, the minimal prime not dividing  $N$ . Since  $\{p_1, \ldots, p_{t+1}\}$  has a prime not contained in  $\{p_{i_1}, \ldots, p_{i_t}\}$ , we see that  $p \leq p_{t+1}$ .
+
+To get a more concrete upper bound on  $p$ , we make use of the prime number theorem (Hardy & Wright, 2008, Ch. XXII), which is a fundamental result in number theory. The theorem states that  $\lim_{i\to \infty}\frac{p_i}{i\ln i} = 1$ . Consequently,  $p \leq p_{t + 1} \leq 2t\ln t$  when  $t \geq T$ , where  $T \in \mathbb{N}$  is large enough. We first stick to the case  $t \geq T$  and then handle the border case  $t < T$ .
+
+For  $\ln N$  we have
+
+$$
+\ln N = \sum_ {j = 1} ^ {t} m _ {j} \ln p _ {i _ {j}} \geq \sum_ {j = 1} ^ {t} \ln p _ {i _ {j}} \geq \sum_ {j = 1} ^ {t} \ln (j + 1) \geq \int_ {1} ^ {t + 1} \ln x d x = (t + 1) \ln (t + 1) - t
+$$
+
+for all  $t \geq T$ . Thus,  $\ln N \geq 1 / 2t\ln t$ . This implies  $\ln \ln N \geq \ln t + \ln \ln t - \ln 2$ . Compare this to  $\ln p \leq \ln 2 + \ln t + \ln \ln t$ . So, we see that  $\ln \ln N \geq C\ln p$  with an absolute constant  $C > 0$ . Hence, we can invoke Theorem 2 for  $p$ , getting that the number of layers needed to represent  $F_{n}$  with integer weights is at least  $\log_p n$ , where  $\log_p n \geq \frac{\ln n}{\ln p} \geq C \cdot \frac{\ln n}{\ln \ln N}$ . In the case  $t < T$ , we observe that  $p \leq p_T$  and obtain the lower bound  $\log_p n = \frac{\ln n}{\ln p} \geq \frac{\ln n}{\ln p_T}$ . Since  $T$  is fixed, the factor  $\ln p_T$  in the denominator is an absolute constant.
+
+# 4 CONCLUSIONS
+
+In summary, we proved that a lower bound for the number of hidden layers needed to exactly represent the function  $\max \{0, x_1, \ldots, x_n\}$  with a ReLU network with weights being  $N$ -ary fractions is  $\lceil \log_p(n + 1) \rceil$ , where  $p$  is a prime number that does not divide  $N$ . For  $p = 3$ , this covers the cases of binary fractions as well as decimal fractions, two of the most common practical settings. Moreover, it shows that the expressive power of ReLU networks grows for every  $N$  up to  $\mathcal{O}(\log n)$ . In the case of rational weights that are  $N$ -ary fractions for any fixed  $N$ , even allowing arbitrarily large denominators and arbitrary width does not facilitate exact representations of low constant depth.
+
+Theorem 4 can be viewed as a partial confirmation of Conjecture 1 for rational weights, as the term  $\ln \ln N$  is growing extremely slowly in  $N$ . If one could replace  $\ln \ln N$  by a constant, the conjecture would be confirmed for rational weights, up to a constant multiple. As already highlighted in Haase et al. (2023), confirmation of the conjecture would theoretically explain the significance of max-pooling in the context of ReLU networks: It seems that the expressive power of ReLU is not enough to model the maximum of a large number of input variables unless network architectures of high-enough depth are used. So, enhancing ReLU networks with max-pooling layers could be a way to reach higher expressive power with shallow networks.
+
+Methodologically, algebraic invariants - such as the  $d$ -dimensional volume  $\mathrm{Vol}_d$  modulo a prime number  $p$  when  $d$  is a power of  $p$  - play a key role in showing lower bounds for the depth of neural networks. Our proof approach provides an algebraic template for a general "separation strategy" to tackle problems on separation by depth. In the ambient Abelian group  $(G, +)$  of all possible functions that can be represented within a given model, one has a nested sequence of subgroups  $G_0 \subseteq G_1 \subseteq G_2 \subseteq \dots$ , with  $G_k$  consisting of functions representable by  $k$  layers. To demonstrate that an inclusion  $G_k \subseteq G_{k+1}$  is strict, one could look for an invariant that can distinguish  $G_k$  from  $G_{k+1}$  - this is a group homomorphism  $\phi$  on  $G$  that is zero on  $G_k$  but not zero on some  $f \in G_{k+1}$ . Most likely, the invariant needs to be "global" in the sense that, if  $\phi(f)$  is computed from the NN representation of  $f$ , then it would accumulate the contribution of all the nodes of the NN in one single value and would not keep track of the number of the nodes and, by this, disregard the widths of the layers. In the concrete case we handled in this contribution, the group  $G$  we choose is  $\mathrm{ReLU}^{\mathbb{Z},0}$ , whereas the invariant that we employ has values in  $\mathbb{Z}_p$  and is based on the computation of the volume of lattice polytopes. In the original setting of Conjecture 1, one has to deal with the nested chain of subspaces  $G_k = \mathrm{ReLU}^{\mathbb{R},0}(k)$  of the infinite-dimensional vector space  $G = \mathrm{ReLU}^{\mathbb{R},0}$ , which makes it natural to choose as an invariant a linear functional  $\phi: G \to \mathbb{R}$ . To make further progress, it is therefore worthwhile continuing to investigate the connection between ReLU networks and discrete polyhedral geometry, algebra, and number theory in order to settle Conjecture 1 in general.
+
+Finally, we raise a question on the role of the field of real numbers in Conjecture 1. Does the choice of a subfield of  $\mathbb{R}$  matter in terms of the expressiveness? More formally, we phrase
+
+Question 18. Let  $S$  be a subfield of  $\mathbb{R}$  and  $k \in \mathbb{N}$  and let  $f \colon \mathbb{R}^n \to \mathbb{R}$  be a function expressible via a ReLU network with weights in  $S$ . If  $f$  is expressible by a ReLU network with  $k$  hidden layers and weights in  $\mathbb{R}$ , is it also expressible by a ReLU network with  $k$  hidden layers and weights in  $S$ ? What is the answer for  $S = \mathbb{Q}$ ?
+
+If, for  $S = \mathbb{Q}$ , the answer to the above question is positive, then the version of Conjecture 1 with rational weights is equivalent to the original conjecture with real weights, which might be a helpful insight towards solving Conjecture 1. If the answer is negative, then the conjecture would have a subtle dependence on the underlying field of weights.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Sergey Grosman from Siemens (Konstanz, Germany) for his feedback on the practical role of quantization. The research of GA was partially supported by the DFG Project 539867386 "Extremal bodies with respect to lattice functionals", which is carried out within the DFG Priority Program 2458 "Combinatorial Synergies".
+
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+
+# A APPENDIX
+
+# A.1 DEFERRED PROOFS
+
+In this appendix, we provide the proofs missing in the main part of the article. For convenience of reading, we restate the corresponding statements.
+
+# A.1.1 PROOF OF LEMMA 10
+
+This appendix provides the missing proof of the following lemma.
+
+Lemma 10. Let  $k \in \mathbb{Z}_+$ . Then, for all  $P \in \mathrm{SU}^k(\mathcal{P}_0(\mathbb{Z}^n))$  and  $u \in \mathbb{R}^n$ , the face of  $P$  in direction  $u$ , given by
+
+$$
+P ^ {u} := \{x \in P \colon u ^ {\top} x = h _ {P} (u) \},
+$$
+
+belongs to  $\mathrm{SU}^k (\mathcal{P}_0(\mathbb{Z}^n))$ . In other words,  $\mathrm{SU}^k (\mathcal{P}_0(\mathbb{Z}^n))$  is closed under taking non-empty faces.
+
+Proof. Throughout the proof, let  $\mathcal{X} = \mathcal{P}_0(\mathbb{Z}^n)$ . The proof is by induction on  $k$ . For  $k = 0$ , we have  $\mathrm{SU}^0 (\mathcal{X}) = \mathcal{X}$ . Since every polytope in  $\mathcal{P}_0(\mathbb{Z}^n)$  consists of a single point  $s$ , every non-empty face of such a polytope also just consists of  $s$ , and is therefore contained in  $\mathcal{P}_0(\mathbb{Z}^n)$ . Thus, the claim holds.
+
+Now let  $k \geq 1$  and assume the assertion holds for  $k - 1$ . Furthermore, let  $u \in \mathbb{R}^n$  and  $P \in \mathrm{SU}^k(\mathcal{X})$  with  $P = \sum_{i=1}^{m} \operatorname{conv}(A_i \cup B_i)$  for some  $m \in \mathbb{N}$ ,  $A_i, B_i \in \mathrm{SU}^{k-1}(\mathcal{X}), i \in [m]$ . By definition and Minkowski additivity of the support function, we have  $P^u = (\sum_{i=1}^{m} \operatorname{conv}(A_i \cup B_i))^u = \sum_{i=1}^{m} (\operatorname{conv}(A_i \cup B_i))^u$ . Moreover, for each  $i \in [m]$ ,  $\operatorname{conv}(A_i \cup B_i)^u$  is equal to  $A_i^u, B_i^u$ , or  $\operatorname{conv}(A_i^u \cup B_i^u)$  depending on whether  $h_{A_i}(u) > h_{B_i}(u), h_{A_i}(u) < h_{B_i}(u)$ , or  $h_{A_i}(u) = h_{B_i}(u)$ , respectively. In any case, we obtain a representation of  $P^u$  that shows its membership in  $\mathrm{SU}^k(\mathcal{X})$ , since  $A_i, B_i \in \mathrm{SU}^{k-1}(\mathcal{X})$  for all  $i \in [m]$  by the induction hypothesis.
+
+# A.1.2 PROOF OF PROPOSITION 13
+
+The goal of this section is to prove the following statement.
+
+Proposition 13. Let  $d = p^t \leq n$  be a power of a prime number  $p$ , with  $t \in \mathbb{N}$ . Moreover, let  $P = \operatorname{conv}(A \cup B) \in \mathcal{P}_d(\mathbb{Z}^n)$  for  $A, B \in \mathcal{P}_d(\mathbb{Z}^n)$ . If  $\operatorname{Vol}_{p^{t-1}}(F) \equiv_p 0$  for all  $p^{t-1}$ -dimensional faces  $F$  of  $A$  and  $B$ , then  $\operatorname{Vol}_{p^t}(P) \equiv_p 0$ .
+
+To prove this result, we need two auxiliary results that we provide next.
+
+Proposition 19. Let  $m, s, d \in \mathbb{N}$  and  $s < d \leq n$ . If  $P \in \mathcal{P}_d(\mathbb{Z}^n)$  such that  $\mathrm{Vol}_s(F) \equiv_m 0$  for all  $s$ -dimensional faces  $F$  of  $P$ , then  $\mathrm{Vol}_d(P) \equiv_m 0$ .
+
+Proof. Note that we can restrict our attention to the case  $d = s + 1$ : Once the case  $d = s + 1$  is settled, it follows that the divisibility of  $\operatorname{Vol}_i(F)$  by  $m$  for  $i$ -dimensional faces  $F$  of  $P$  implies divisibility of  $\operatorname{Vol}_{i + 1}(G)$  by  $m$  for all  $(i + 1)$ -dimensional faces  $G$  of  $P$ . Hence, iterating from  $i = s$  to  $i = d - 1$ , we obtain the desired assertion. So, assume  $d = s + 1$ .
+
+Let  $P$  be a  $d$ -dimensional lattice polytope with facets having a normalized  $(d - 1)$ -dimensional volume divisible by  $m$ . We pick a vertex  $a$  of  $P$  and subdivide  $P$  into the union of the non-overlapping pyramids of the form  $\operatorname{conv}(\{a\} \cup F)$ , where  $F$  is a facet of  $P$ . By Proposition 7, the normalized  $d$ -dimensional volume of  $\operatorname{conv}(\{a\} \cup F)$  is divisible by  $\operatorname{Vol}_{d - 1}(F)$ . Since by assumption  $\operatorname{Vol}_{d - 1}(F)$  is divisible by  $m$ , we conclude that also  $\operatorname{Vol}_d(P)$  is divisible by  $m$ , because  $\operatorname{Vol}_d$  is additive as it is based on a Lebesgue measure.
+
+The second result analyzes the structure of  $\mathrm{conv}(A\cup B)$  in terms of a particular subdivision. Given a polytope  $P\in \mathcal{P}(\mathbb{R}^n)$  of dimension  $d$ , a subdivision of  $P$  is a finite collection  $\mathcal{C}\subseteq \mathcal{P}(\mathbb{R}^n)$  such that (i)  $P = \bigcup_{C\in \mathcal{C}}C$ ; (ii) for each  $C\in \mathcal{C}$ , the polytope  $C$  has dimension  $d$ ; (iii) for any two distinct  $C,C^{\prime}\in \mathcal{C}$ , the polytope  $C\cap C^{\prime}$  is a proper face of both  $C$  and  $C^\prime$ . The elements  $C\in \mathcal{C}$  are called the cells of subdivision  $\mathcal{C}$ , cf. (Lee & Santos, 2017).
+
+Proposition 20 (Haase et al. 2023, Prop. 10). For two polytopes  $A, B \in \mathcal{P}(\mathbb{R}^n)$ , there exists a subdivision of  $\operatorname{conv}(A \cup B)$  such that each full-dimensional cell is of the form  $\operatorname{conv}(F \cup G)$ , where  $F$  and  $G$  are faces of  $A$  and  $B$ , respectively, such that  $\dim(F) + \dim(G) + 1 = d$ .
+
+The term "full-dimensional" in Proposition 20 as well as in the original formulation of Haase et al. (2023, Prop. 10) refers to faces that have the same dimension as  $\mathrm{conv}(A\cup B)$ , while its authors make no assumption on whether that dimension is equal to  $n$  (but Haase et al. (2023) note in their proof that such an assumption would be without loss of generality).
+
+We are now able to prove Proposition 13.
+
+Proof of Proposition 13. Let  $P = \operatorname{conv}(A \cup B)$ . We apply Proposition 20 for obtaining a subdivision of  $P$  into  $d$ -dimensional polytopes  $P_{1} = \operatorname{conv}(F_{1} \cup G_{1}), \ldots, P_{m} = \operatorname{conv}(F_{m} \cup G_{m})$ , where for each  $s \in [m]$ ,  $F_{s}$  and  $G_{s}$  are faces of  $A$  and  $B$ , respectively, and  $\dim(F_{s}) + \dim(G_{s}) + 1 = d$ . That is,  $P$  is the union of polytopes whose relative interiors are disjoint. Consequently,  $\operatorname{Vol}_{d}(P) = \operatorname{Vol}_{d}(P_{1}) + \dots + \operatorname{Vol}_{d}(P_{m})$ . It therefore suffices to show that  $\operatorname{Vol}_{d}(P_{s}) \equiv_{p} 0$  for every such polytope  $P_{s}$  with  $s \in [m]$ .
+
+For given  $s \in [m]$  and faces  $F_{s}$  and  $G_{s}$  of  $A$  and  $B$ , respectively, denote their dimensions as  $i$  resp.  $j$ . Since  $i + j = d - 1 = p^{t} - 1$ , the dimension of  $F_{s}$  or  $G_{s}$  is at least  $p^{t - 1}$  (if this was not the case, we would have  $i + j \leq 2(p^{t - 1} - 1) < p^{t} - 1$ , which is a contradiction). By symmetry reasons, we assume without loss of generality that  $i \geq p^{t - 1}$ . Then, by Proposition 19,  $\operatorname{Vol}_i(F_s)$  is divisible by  $p$ . Consequently, by Proposition 7, the normalized volume of  $\operatorname{conv}(F_s \cup G_s)$  is also divisible by  $p$ .
+
+# A.2 PROOF OF BINOMIAL FORMULA FOR MIXED VOLUMES
+
+The symmetry and multilinearity of the mixed-volume functional makes computations with it similar in nature to calculations with an  $n$ -term product. Say, the identity  $(x + y)^2 = x^2 + 2xy + y^2$  over reals corresponds to the identity  $\mathrm{Vol}_2(A + B) = \mathrm{V}(A + B, A + B) = \mathrm{V}(A, A) + 2\mathrm{V}(A, B) + \mathrm{V}(B, B) = \mathrm{Vol}_2(A) + 2\mathrm{V}(A, B) + \mathrm{Vol}_2(B)$  for planar polytopes  $A, B$  and the way of deriving the latter identity
+
+is completely analogous to deriving the identity for  $(x + y)^2$  by expanding brackets. Very much in the same way, the binomial identity  $(x + y)^n = \sum_{i=0}^n \binom{n}{i} x^i y^{n-i}$  corresponds to the identity (1). Here is a formal proof:
+
+We use the notation  $P_0 = B$  and  $P_{1} = A$ . Then
+
+$$
+\operatorname {V o l} _ {n} \left(P _ {0} + P _ {1}\right) = \operatorname {V} \left(P _ {0} + P _ {1}, \dots , P _ {0} + P _ {1}\right)
+$$
+
+by Property (c) in Theorem 5. Using Property (b) in Theorem 5 for each of the  $n$  inputs of the mixed-volume functional, we obtain
+
+$$
+\operatorname {V o l} _ {n} (P _ {0} + P _ {1}) = \sum_ {i _ {1} \in \{0, 1 \}} \dots \sum_ {i _ {n} \in \{0, 1 \}} \mathrm {V} (P _ {i _ {1}}, \dots , P _ {i _ {n}}),
+$$
+
+where the right-had side is a sum with  $2^{n}$  terms. However, many of the terms are actually repeated, because  $\mathrm{V}(P_{i_1},\ldots ,P_{i_n})$  does not depend on the order of the polytopes in the input: this mixed volume contains  $i_1 + \dots +i_n$  copies of  $P_{1}$  and  $n - (i_1 + \dots +i_n)$  copies of  $P_0$ . Hence,
+
+$$
+\mathrm {V} (P _ {i _ {1}}, \ldots , P _ {i _ {n}}) = \mathrm {V} (\underbrace {P _ {0} , \ldots , P _ {0}} _ {n - (i _ {1} + \dots i _ {n})}, \underbrace {P _ {1} , \ldots , P _ {1}} _ {i _ {1} + \dots + i _ {n}}).
+$$
+
+In order to convert our  $2^{n}$ -term sum into an  $(n + 1)$ -term sum, for each choice of  $i = i_{1} + \dots + i_{n} \in \{0, \ldots, n\}$ , we can determine the number of choices of  $i_{1}, \ldots, i_{n} \in \{0, 1\}$  that satisfy  $i = i_{1} + \dots + i_{n}$ . This corresponds to choosing an  $i$ -element subset  $\{t \in [n] : i_{t} = 1\}$  in the  $n$ -element set  $\{1, \ldots, n\}$ . That is, the number of such choices is the binomial coefficient  $\binom{n}{i}$ . Hence, our representation with  $2^{n}$  terms amounts to
+
+$$
+\operatorname {V o l} _ {n} (P _ {0} + P _ {1}) = \sum_ {i = 0} ^ {n} \binom {n} {i} \operatorname {V} (\underbrace {P _ {0} , \ldots , P _ {0}} _ {n - i}, \underbrace {P _ {1} , \ldots , P _ {1}} _ {i}).
+$$

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+# ONE-PROMPT-ONE-STORY: FREE-LUNCH CONSISTENT TEXT-TO-IMAGE GENERATION USING A SINGLE PROMPT
+
+Tao Liu $^{1}$ , Kai Wang $^{2*}$ , Senmao Li $^{1}$ , Joost van de Weijer $^{2}$ , Fahad Shahbaz Khan $^{3,4}$ , Shiqi Yang $^{5}$ , Yaxing Wang $^{6}$ , Jian Yang $^{1}$ , Ming-Ming Cheng $^{1}$
+
+$^{1}$ VCIP, CS, Nankai University,  $^{2}$ Computer Vision Center, Universitat Autònoma de Barcelona  
+ $^{3}$ Mohamed bin Zayed University of AI,  $^{4}$ Linkoping University,  $^{5}$ SB Intuitions, SoftBank  
+ $^{6}$ Nankai International Advanced Research Institute (Shenzhen Futian), Nankai University  
+{ltolcy0, senmaonk, shiqi.yang147.jp}@gmail.com, {kwang, joost}@cvc.uab.es  
+fahad.khan@liu.se, {yaxing, csjyang, cmm}@nankai.edu.cn
+
+# ABSTRACT
+
+Text-to-image generation models can create high-quality images from input prompts. However, they struggle to support the consistent generation of identity-preserving requirements for storytelling. Existing approaches to this problem typically require extensive training in large datasets or additional modifications to the original model architectures. This limits their applicability across different domains and diverse diffusion model configurations. In this paper, we first observe the inherent capability of language models, coined context consistency, to comprehend identity through context with a single prompt. Drawing inspiration from the inherent context consistency, we propose a novel training-free method for consistent text-to-image (T2I) generation, termed "One-Prompt-One-Story" (1Prompt1Story). Our approach 1Prompt1Story concatenates all prompts into a single input for T2I diffusion models, initially preserving character identities. We then refine the generation process using two novel techniques: Singular-Value Reweighting and Identity-Preserving Cross-Attention, ensuring better alignment with the input description for each frame. In our experiments, we compare our method against various existing consistent T2I generation approaches to demonstrate its effectiveness through quantitative metrics and qualitative assessments. Code is available at https://github.com/byliutao/1Prompt1Story.
+
+# 1 INTRODUCTION
+
+Text-based image generation (T2I) (Ramesh et al., 2022; Sahara et al., 2022; Rombach et al., 2022; Ma et al., 2024; Su et al., 2023) aims to generate high-quality images from textual prompts, depicting various subjects in various scenes. The ability of T2I diffusion models to maintain subject consistency across a wide range of scenes is crucial for applications such as animation (Hu, 2024; Guo et al., 2024), storytelling (Yang et al., 2024; Gong et al., 2023; Cheng et al., 2024), video generation models (Khachatryan et al., 2023; Blattmann et al., 2023) and other narrative-driven visual applications. However, achieving consistent T2I generation remains a challenge for existing models, as shown in Fig. 1 (up).
+
+Recent studies tackle the challenge of maintaining subject consistency through diverse approaches. Most methods require time-consuming training on large datasets for clustering identities (Avrahami et al., 2023), learning large mapping encoders (Gal et al., 2023b; Ruiz et al., 2024), or performing fine-tuning (Ryu, 2023; Kopiczko et al., 2024), which carries the risk of inducing language drift (Heng & Soh, 2024; Wu et al., 2024a; Huang et al., 2024), etc. Several recent training-free approaches (Tewel et al., 2024; Zhou et al., 2024) demonstrate remarkable results in generating images with consistent subjects by leveraging shared internal activations from the pre-trained models. These methods require extensive memory resources or complex module designs to strengthen
+
+"A hyper-realistic digital painting of a fairy, dressed in a cloak of spider silk, wearing a garland of fireflies."
+
+![](images/e4d669cc7346c65ed433603e543c7064454f074c6924ebe18780ecd190639fed.jpg)
+
+![](images/f67938fc266865b9b311371aa6cce41a214534ef679fb2bc2f0d9ba61ba9415a.jpg)
+
+![](images/4467c32173a7f5128e7bb37a6dcedb99ae2e70a94494df8096d9a5750ed90d97.jpg)
+
+![](images/1c399db59d7e4ecfd5942a35cbee7f74b2535f5fd02dce575ad30f29cf338e8a.jpg)
+
+![](images/dfe20e867f5edc42fe5b8086ab0ee06564e6c78435e3505ae71ce4babcc675f9.jpg)
+
+![](images/0790e06314e1b67d5237855e8d1593144481569183a9bba0ba8e7c8017493383.jpg)
+
+![](images/aae8792bb43994d0c4a2b17f6a6542f2ffc99b2bc9f0d3143d6a740c0fb0c67e.jpg)
+
+![](images/48cdf9abc86198fc07bbe33333a1ec9f0b97fa124376e27aa93ba0058b89002f.jpg)  
+SDXL
+
+![](images/852e6cb6b04ea78f68c76be1b03281500647bf12abc041252715b82582fb11a3.jpg)  
+Juggernaut-X-v10
+
+![](images/37e5c9a62c3fe9170caddb06b4f21d40db6e45481adf703675658a6fd7e09379.jpg)  
+Texture-Inversion
+
+![](images/afb5089419046660ed7ac4d7c338ccbd62cbe23c5ac2116b29f5225b87139249.jpg)  
+Ip-Adapter
+
+![](images/4942e2effb1017f934cba5d0967ce30ba65f4de0da878442596f4a7d1db9f002.jpg)  
+ConsiStory
+
+![](images/66c201701f1d8e112cb9118fbaed7ed489eab9a1902a2ae05ba9bd2de794c7bc.jpg)  
+NPR
+
+![](images/316fad0143c3533f274ffe39eea4802c437f6b04ec24ed5842ebe1e64f07d117.jpg)  
+Ours
+
+"A photo of a woman with a slender figure, in a vintage kitchen", reading a novel, stand next to a landscape, shopping at a market, sipping coffee at a quaint cafe, tending to a small urban garden, walking a dog in the park.
+
+![](images/7123146ff577576021cb51c40c0dbe96a9b0bcadbd8286407fe8ab7179a8af2f.jpg)  
+Figure 1: Existing methods (up) encounter challenges in consistent T2I generation. T2I models such as SDXL (Podell et al., 2023) and Juggernaut-X-v10 (RunDiffusion, 2024) often exhibit noticeable identity inconsistency across generated images. Although recent methods including IP-Adapter and ConsiStory have improved identity consistency, they lost the alignment between the generated images and corresponding input prompts. Additional results of our 1Prompt1Story (down) demonstrate superior consistency without compromising the alignment between text and images.
+
+![](images/6c282305116adadc75c673960245766b323d35ee9032511221e739167fe73a65.jpg)
+
+![](images/1135b601a4cfc1ef032704a0b6f478848fc16a16243e13dabf7edaf726bbf04f.jpg)
+
+![](images/222ef00acbed99ea827ecd9092b0c034af977e9c67d46e62f1434502635a0362.jpg)
+
+![](images/46a1f27c1badc39f2eb5c2b098d28c50f1ee07f2f579575359878182d8b61689.jpg)
+
+![](images/ea60fce05a60f11c792ab17b3dd10f1f7d07f90e9254141fd2ee0f9d4e418915.jpg)
+
+![](images/48b84839829372fd819e064da7f8eb86f5825585c7e699995c0b73ec6cc01fa2.jpg)
+
+the T2I diffusion model to generate satisfactory consistent images. However, they all neglect the inherent property of long prompts that identity information is implicitly maintained by context understanding, which we refer to as the context consistency of language models. For example, the dog object in "A dog is watching the movie. Afterward, the dog is lying in the garden." can be easily understood as the same without any confusion since it appears in the same paragraph and is connected by the context. We take advantage of this inherent feature to eliminate the requirement of additional finetuning or complicated module design.
+
+Observing the inherent context consistency of language models, we propose a novel approach to generate images with consistent characters using a single prompt, termed One-Prompt-One-Story (1Prompt1Story). Specifically, 1Prompt1Story consolidates all desired prompts into a single longer sentence, which starts with an identity prompt that describes the corresponding identity attributes and continues with subsequent frame prompts describing the desired scenarios in each frame. We denote this first step as prompts consolidation. By reweighting the consolidated prompt embeddings, we can easily implement a basic method Naive Prompt Reweighting to adjust the T2I generation performance, and this approach inherently achieves excellent identity consistency. Fig. 1 (up, the 6th column) illustrates two examples, each featuring an image generated with different frame descriptions within a single prompt by reweighting the frame prompt embeddings. These examples demonstrate that Naive Prompt Reweighting is able to maintain identity consistency with various scenario prompts. However, this basic approach does not guarantee strong text-image alignment for each frame, as the semantics of each frame prompt are usually intertwined within the consolidated prompt embedding (Radford et al., 2021). To further enhance text-image alignment and identity consistency of the T2I generative models, we introduce two additional techniques: Singular-Value Reweighting (SVR) and Identity-Preserving Cross-Attention (IPCA). The Singular-Value Reweighting aims to refine the expression of the prompt of the current frame while attenuating the information from the other frames. Meanwhile, the strategy Identity-Preserving Cross-Attention strengthens the consistency of the subject in the cross-attention layers. By applying our proposed techniques, 1Prompt1Story achieves more consistent T2I generation results compared to existing approaches.
+
+In the experiments, we extend an existing consistent T2I generation benchmark as ConsiStory+ and compare it with several state-of-the-art methods, including ConsiStory (Tewel et al., 2024), Story-Diffusion (Zhou et al., 2024), IP-Adapter (Ye et al., 2023), etc. Both qualitative and quantitative
+
+performance demonstrate the effectiveness of our method 1Prompt1Story. In summary, the main contributions of this paper are:
+
+- To the best of our knowledge, we are the first to analyze the overlooked ability of language models to maintain inherent context consistency, where multiple frame descriptions within a single prompt inherently refer to the same subject identity.  
+- Based on the context consistency property, we propose One-Prompt-One-Story as a novel training-free method for consistent T2I generation. More specifically, we further propose Singular-Value Reweighting and Identity-Preserving Cross-Attention techniques to improve text-image alignment and subject consistency, allowing each frame prompt to be individually expressed within a single prompt while maintaining a consistent identity along with the identity prompt.  
+- Through extensive comparisons with existing consistent T2I generation approaches, we confirm the effectiveness of 1Prompt1Story in generating images that consistently maintain identity throughout a lengthy narrative over our extended ConsiStory+ benchmark.
+
+# 2 RELATED WORK
+
+T2I personalized generation. T2I personalization is also referred to T2I model adaptation. This aims to adapt a given model to a new concept by providing a few images and binding the new concept to a unique token. As a result, the adaptation model can generate various renditions of the new concept. One of the most representative methods is DreamBooth (Ruiz et al., 2023), where the pre-trained T2I model learns to bind a modified unique identifier to a specific subject given a few images, while it also updates the T2I model parameters. Recent approaches (Kumari et al., 2023; Han et al., 2023b; Shi et al., 2023) follow this pipeline and further improve the quality of the generation. Another representative, Textual Inversion (Gal et al., 2023a), focuses on learning new concept tokens instead of fine-tuning the T2I generative models. Textual Inversion finds new pseudo-words by conducting personalization in the text embedding space. The coming works (Dong et al., 2022; Voynov et al., 2023; Han et al., 2023a; Zeng et al., 2024) follow similar techniques.
+
+Consistent T2I generation. Despite recent advances, T2I personalization methods often require extensive training to effectively learn modifier tokens. This training process can be time-consuming, which limits their practical impact. More recently, there has been a shift towards developing consistent T2I generation approaches (Wang et al., 2024b;a), which can be considered a specialized form of T2I personalization. These methods mainly focus on generating human faces that possess semantically similar attributes to the input images. Importantly, they aim to achieve this identity-preserving T2I generation without the need for additional fine-tuning. They mainly take advantage of PEFT techniques (Ryu, 2023; Kopiczko et al., 2024) or pre-training with large datasets (Ruiz et al., 2024; Xiao et al., 2023) to learn the image encoder to be customized in the semantic space. For example, PhotoMaker (Li et al., 2023b) enhances its ability to extract identity embeddings by fine-tuning part of the transformer layers in the image encoder and merging the class and image embeddings. The Chosen One (Avrahami et al., 2023) utilizes an identity clustering method to iteratively identify images with a similar appearance from a set of images generated by identical prompts.
+
+However, most consistent T2I generation methods (Akdemir & Yanardag, 2024; Wang et al., 2024a) still require training the parameters of the T2I models, sacrificing compatibility with existing pretrained community models, or fail to ensure high face fidelity. Additionally, as most of these systems (Li et al., 2023b; Gal et al., 2023b; Ruiz et al., 2024) are designed specifically for human faces, they encounter limitations when applied to non-human subjects. Even for the state-of-the-art approaches, including StoryDiffusion (Zhou et al., 2024), The Chosen One (Avrahami et al., 2023) and ConsiStory (Tewel et al., 2024), they either require time-consuming iterative clustering or high memory demand in generation to achieve identity consistency.
+
+Storytelling. Story generation (Li et al., 2019; Maharana et al., 2021), also referred to as storytelling, is one of the active research directions that is highly related to character consistency. Recent researches (Tao et al., 2024; Wang et al., 2023) have integrated the prominent pre-trained T2I diffusion models (Rombach et al., 2022; Ramesh et al., 2022) and the majority of these approaches require intense training over storytelling datasets. For example, Make-a-Story (Rahman et al., 2023) introduces a visual memory module designed to capture and leverage contextual in
+
+formation throughout the storytelling process. StoryDALL-E (Maharana et al., 2022) extends the story generation paradigm to story continuation, using DALL-E capabilities to achieve substantial improvements over previous GAN-based methodologies. Note that the story continuation shares similarities with consistent Text-to-Image generation by using reference images. However, current consistent T2I generation methods prioritize preserving human face identities, whereas story continuation involves supporting various subjects or even multiple subjects within the generated images.
+
+In this paper, our proposed consistent T2I framework, 1Prompt1Story, diverges significantly from previous approaches in storytelling and consistent T2I generation methods. We explore the inherent context consistency property in language models instead of finetuning large models or designing complex modules. Importantly, it is compatible with various T2I generative models, since the properties of the text model are independent of the specific generation model used as the backbone.
+
+# 3 METHOD
+
+Consistent T2I generation aims to generate a set of images depicting consistent subjects in different scenarios using a set of prompts. These prompts start with an identity prompt, followed by the frame prompts for each subsequent visualization frame. In this section, we first empirically show that different frame descriptions included in a concatenated prompt can maintain identity consistency due to the inherent context consistency property of language models. We examine this observation through comprehensive analyses in Sec. 3.1 and propose the basic Naive Prompt Reweighting pipeline of our method  $1Prompt1Story$ . Following that, to ensure that each frame description within the prompt is expressed individually while diminishing the impact of other frame prompts, we introduce Singular-Value Reweighting and Identity-Preserving Cross-Attention in Sec. 3.2. The illustration of  $1Prompt1Story$  is shown in Fig. 4 and Algorithm 1 in the Appendix.
+
+# 3.1 CONTEXT CONSISTENCY
+
+Latent Diffusion Models. We build our approach on the SDXL (Podell et al., 2023) model, a latent diffusion model that contains two main components: an autoencoder (i.e., an encoder  $\mathcal{E}$  and a decoder  $\mathcal{D}$ ) and a diffusion model (i.e.,  $\epsilon_{\theta}$  parameterized by  $\theta$ ). The model  $\epsilon_{\theta}$  is trained with the following loss function:
+
+$$
+L _ {L D M} := \mathbb {E} _ {z _ {0} \sim \mathcal {E} (x), \epsilon \sim \mathcal {N} (0, 1), t \sim \operatorname {U n i f o r m} (1, T)} \left[ \| \epsilon - \epsilon_ {\theta} \left(z _ {t}, t, \tau_ {\xi} (\mathcal {P})\right) \| _ {2} ^ {2} \right], \tag {1}
+$$
+
+where  $\epsilon_{\theta}$  is a UNet that conditions on the latent variable  $z_{t}$ , a timestep  $t \sim \mathrm{Uniform}(1, T)$ , and a text embedding  $\tau_{\xi}(\mathcal{P})$ . In text-guided diffusion models, images are generated based on a textual condition, with  $\mathcal{C} = \tau_{\xi}(\mathcal{P}) \in \mathbb{R}^{M \times D}$ , where  $M$  is the number of tokens,  $D$  is the feature dimension of each token, and  $\tau_{\xi}$  is the CLIP text encoder (Radford et al., 2021)<sup>1</sup>. For a given input, the model  $\epsilon_{\theta}(z_{t}, t, \mathcal{C})$  produces a cross-attention map. Let  $f_{z_{t}}$  denote the feature map output from  $\epsilon_{\theta}$ . We can obtain a query matrix  $Q = l_{Q}(f_{z_{t}})$  using the projection network  $l_{Q}$ . Similarly, the key matrix  $\mathcal{K}$  is computed from the text embedding  $\mathcal{C}$  using another projection network  $l_{\mathcal{K}}$ , such that  $\mathcal{K} = l_{\mathcal{K}}(\mathcal{C})$ . The cross-attention map  $\mathcal{A}_{t}$  is then calculated as:  $\mathcal{A}_{t} = \text{softmax}(Q \cdot \mathcal{K}^{T} / \sqrt{d})$ , where  $d$  is the dimension of the query and key matrices. The entry  $[\mathcal{A}_{t}]_{ij}$  represents the attention weight of the  $j$ -th token to the  $i$ -th token.
+
+Problem Setups. In the T2I diffusion models, the text embedding  $\mathcal{C} = \tau_{\xi}(\mathcal{P}) \in \mathbb{R}^{M \times D}$  is with  $M$  tokens. The  $M$  tokens contain a start token [SOT], followed by  $|\mathcal{P}|$  tokens corresponding to the prompt, and  $M - |\mathcal{P}| - 1$  padding end tokens [EOT]. Previous consistent T2I generation works (Avrahami et al., 2023; Tewel et al., 2024; Zhou et al., 2024) generate images from a set of  $N$  prompts. This set of prompts starts with an identity prompt  $\mathcal{P}_0$  that describes the relevant attribute of the subject and continues with multiple frame prompt  $\mathcal{P}_i$ , where  $i = 1, \dots, N$  describes each frame scenario. However, this separate generation pipeline ignores the inherent language property, i.e., the context consistency, by which identity is consistently ensured by the context information inherent in language models. This property stems from the self-attention mechanism within Transformer-based text encoders (Radford et al., 2021; Vaswani et al., 2017), which allows learning the interaction between phrases in the text embedding space.
+
+In the following, we analyze the context consistency under different prompt configurations in both textual space and image space. Specifically, we refer to the conventional prompt setups as multi-prompt generation, which is commonly used in existing consistent T2I generation methods. The multi-prompt generation uses  $N$  prompts separately for each generated frame, each sharing the same identity prompt and the corresponding frame prompt as  $[\mathcal{P}_0; \mathcal{P}_i], i \in [1, N]$ . In contrast, our single-prompt generation concatenates all the prompts as  $[\mathcal{P}_0; \mathcal{P}_1; \ldots; \mathcal{P}_N]$  for each frame generation, which we refer as the Prompt Consolidation (PCoN).
+
+# 3.1.1 CONTEXT CONSISTENCY IN TEXT EMBEDDINGS
+
+Empirically, we find that the frame prompt  $\{\mathcal{P}_i\mid i = 1,\ldots ,N\}$  in the single-prompt generation setup have relatively small semantic distances among each other in the textual embedding space, whereas those across multi-prompt generation have comparatively larger distances. For instance, we set the identity frame  $\mathcal{P}_0 =$  "A watercolor of a cute kitten" as an example. We then create  $N = 5$  frame prompts  $\{\mathcal{P}_i,i\in [1,N]\}$  as "in a garden", "dressed in a superhero cape", "wearing a collar with a bell", "sitting in a basket", and "dressed in a cute sweater", respectively. Under the multi-prompt setup, each frame is generated by the text embedding defined as  $\mathcal{C}_i = \tau_{\xi}([\mathcal{P}_0;\mathcal{P}_i]) =$ $[c^{SOT},c^{\mathcal{P}_0},c^{\mathcal{P}_i},c^{EOT}]$ ,  $(i = 1,\dots ,N)$ , while the text embedding of the Prompt Consolidation in the single-prompt case is  $\mathcal{C} = \tau_{\xi}([\mathcal{P}_0;\mathcal{P}_1;\dots ;\mathcal{P}_N]) = [\pmb {c}^{SOT},\pmb {c}^{\mathcal{P}_0},\pmb {c}^{\mathcal{P}_1},\dots ,\pmb {c}^{\mathcal{P}_N},\pmb {c}^{EOT}]$ .
+
+To analyze the distances among the frame prompts, we extract  $c^{\mathcal{P}_i}$  from  $\mathcal{C}_i$  for multi-prompt setup and apply t-SNE for 2D visualization (Fig. 2-left). Similarly, we extract all  $c^{\mathcal{P}_i}$  from  $\mathcal{C}$  for the single-prompt setup (Fig. 2-left). As can be observed, the text embeddings of frame prompts under the multi-prompt setup are widely distributed in the text representation space (red dots) with an average Euclidean  $L_2$  distance of 71.25. In contrast, the embeddings in the single-prompt case exhibit more compact distributions (blue dots), with a much smaller average  $L_2$  distance of 46.42. We also performed a similar distance analysis on all prompt sets in our benchmark ConsiStory+. As shown in Fig.2-right, we can conclude a similar observation that the frame prompts share more similar semantic information and identity consistency within the single-prompt setup.
+
+![](images/e9c4614eaeb6fcb770aed7918856dcb6e79ec1698f4b4f2fcb87b2d0b61dc369.jpg)  
+Figure 2: t-SNE visualization of text embeddings (Left):  $c^{\mathcal{P}_i}$  from single-prompt generation are closer together compared to those from multi-prompt generation. Statistical results (Right): We evaluated the average distances between the corresponding point sets of all prompt sets on the ConsiStory+ benchmark after dimensionality reduction. The average distance between text embeddings from single-prompt generation is smaller than that from multi-prompt generation.
+
+# 3.1.2 CONTEXT CONSISTENCY IN IMAGE GENERATION
+
+To demonstrate that context consistency is also maintained in the image space, we further conducted image generation experiments using the prompt example above. The images generated by the SDXL model with the multi-prompt configuration, as illustrated in Fig. 3 (left, the first row), show various characters that lack identity consistency. Instead, we use our proposed concatenated prompt  $\mathcal{P} = [\mathcal{P}_0; \mathcal{P}_1; \dots; \mathcal{P}_N]$ . To generate the  $i$ -th frame ( $i = 1, \dots, N$ ), we reweight the  $c^{\mathcal{P}_i}$  corresponding to the desired frame prompt  $\mathcal{P}_i$  by a magnification factor while rescaling the embeddings of the other frame prompts by a reduction factor. This modified text embedding is then imported to the T2I model to generate the frame image. We refer to this simplistic reweighting approach as Naive Prompt Reweighting (NPR). By this means, the T2I model synthesizes frame images with the same subject identity. However, the backgrounds get blended among these frames, as shown in Fig. 3 (left, the second row). By contrast, our full model  $IPrompt1Story$  introduced in Sec. 3.2 generates images with better consistent identity and text-image alignment for each frame prompt, as shown in Fig. 3 (left, the last row).
+
+To visualize identity similarity among images, we removed backgrounds using CarveKit (Selin, 2023) and extracted visual features with DINO-v2 (Oquab et al., 2023; Darcet et al., 2023). These features are then projected into the 2D space by t-SNE (Hinton & Roweis, 2002) (as shown in Fig. 3 (mid)). Our complete approach 1Prompt1Story obviously obtains better identity consistency than
+
+![](images/f537ed021bedfb2617d975bbf39d49683e50ae40019c3011d954dc567cc4f313.jpg)  
+Figure 3: (Left): SDXL generates frame images using multi-prompt generation, while Naive Prompt Reweighting (NPR) and our method utilize the single-prompt setup. (Mid): Image features are extracted by DINOv2 (Oquab et al., 2023) and visualized by the t-SNE reduction. Naive Prompt Reweighting and 1Prompt1Story show more consistent identity generations than the SDXL model. (Right): Statistics of the average feature distances among generated images from the prompts in our extended ConsiStory+ benchmark, which further confirms that 1Prompt1Story produces better identity consistency.
+
+the other two comparison methods, while Naive Prompt Reweighting shows improvements over the SDXL baseline. We also applied the analysis across our extended benchmark ConsiStory+ and calculated the average pairwise distance, as shown in Fig. 3 (right). These results further consolidate our conclusion that the frame prompts in a single-prompt setup share more identity consistency than the multi-prompt case.
+
+# 3.2 ONE-PROMPT-ONE-STORY
+
+As also observed from the above section, simply concatenating the prompts as Naive Prompt Reweighting cannot guarantee that the generated images accurately reflect the frame prompt descriptions, for which we assume that the T2I model cannot accurately capture the correct partition of the concatenated prompt embeddings. Furthermore, the various semantics within the consolidated descriptions interact with each other (Chefer et al., 2023; Rassin et al., 2024). To mitigate this issue, we propose additional techniques based on the Prompt Consolidation (PCon), namely Singular-Value Reweighting (SVR) and Identity-Preserving Cross-Attention (IPCA).
+
+Singular-Value Reweighting. After the Prompt Consolidation as  $\mathcal{C} = \tau_{\xi}([\mathcal{P}_0;\mathcal{P}_1;\dots ;\mathcal{P}_N]) = [c^{SOT},c^{\mathcal{P}_0},c^{\mathcal{P}_1},\ldots ,c^{\mathcal{P}_N},c^{EOT}]$ , we require the current frame prompt to be better expressed in the T2I generation, which we denote as  $\mathcal{P}^{exp} = \mathcal{P}_j$ $(j = 1,\dots,N)$ . We also expect the remaining frames to be suppressed in the generation, which we denote as  $\mathcal{P}^{sup} = \mathcal{P}_k$ $k\in [1,N]\backslash \{j\}$ . Thus, the  $N$  frame prompts of the subject description can be written as  $\{\mathcal{P}^{exp},\mathcal{P}^{sup}\}$ . As the [EOT] token contains significant semantic information (Li et al., 2023a; Wu et al., 2024b), the semantic information corresponding to  $\mathcal{P}^{exp}$ , in both  $\mathcal{P}_j$  and [EOT], needs to be enhanced, while the semantic information corresponding to  $\mathcal{P}^{sup}$ , in  $\mathcal{P}_k$ $k\neq j$  and [EOT], need to be suppressed. We extract the token embeddings for both express and suppress sets as  $\mathcal{X}^{exp} = [\pmb {c}^{\mathcal{P}_j},\pmb{c}^{EOT}]$  and  $\mathcal{X}^{sup} = [c^{\mathcal{P}_1},\dots ,c^{\mathcal{P}_{j - 1}},c^{\mathcal{P}_{j + 1}},\dots ,c^{\mathcal{P}_N},c^{EOT}]$ .
+
+Inspired by (Gu et al., 2014; Li et al., 2023a), we assume that the main singular values of  $\mathcal{X}^{exp}$  correspond to the fundamental information of  $\mathcal{P}^{exp}$ . We then perform SVD decomposition as:  $\mathcal{X}^{exp} = \boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^T$ , where  $\boldsymbol{\Sigma} = \text{diag}(\sigma_0, \sigma_1, \dots, \sigma_{n_j})$ , the singular values  $\sigma_0 \geq \dots \geq \sigma_{n_j}$ . To enhance the expression of the frame  $\mathcal{P}_j$ , we introduce the augmentation for each singular value, termed as  $\mathbf{SVR}+$  and formulated as:
+
+$$
+\hat {\sigma} = \beta e ^ {\alpha \sigma} * \sigma . \tag {2}
+$$
+
+where the symbol  $e$  is the exponential,  $\alpha$  and  $\beta$  are parameters with positive numbers. We recover the tokens as  $\hat{\mathcal{X}}^{exp} = \pmb{U}\hat{\pmb{\Sigma}}\pmb{V}^T$ , with the updated  $\hat{\pmb{\Sigma}} = \text{diag}(\hat{\sigma}_0,\hat{\sigma}_1,\dots,\hat{\sigma}_{n_j})$ . The new prompt embedding is defined as  $\hat{\mathcal{X}}^{exp} = [\hat{c}^{\mathcal{P}_j},\hat{c}^{EOT}]$ , and  $\hat{\mathcal{C}} = [c^{SOT},c^{\mathcal{P}_0},\dots,\hat{c}^{\mathcal{P}_j},\dots,c^{\mathcal{P}_N},\hat{c}^{EOT}]$ . Note that there is an updated  $\hat{\mathcal{X}}^{sup} = [c^{\mathcal{P}_1},\ldots ,c^{\mathcal{P}_{j - 1}},c^{\mathcal{P}_{j + 1}},\ldots ,c^{\mathcal{P}_N},\hat{c}^{EOT}]$ .
+
+![](images/a46f8b1658d10ef634e6352572d8b75411e872f8ffecf4aadb1141d1f0bbf316.jpg)  
+Figure 4: (a): The overall pipeline of  $1Prompt1Story$ . We combine the identity prompt and frame prompts into a single prompt, then we apply both Singular-Value Reweighting (SVR) and Identity-Preserving Cross-Attention (IPCA) to generate identity-consistent images. (b): During SVR, we first enhance the semantic information of the express set  $\mathcal{X}^{exp}$  (red arrow), then iteratively weaken the semantics for the suppress set  $\mathcal{X}^{sup}$  (blue arrow). (c): In IPCA, we concatenate  $\tilde{\mathcal{K}}$  with  $\bar{\mathcal{K}}$  and  $\tilde{\nu}$  with  $\bar{\nu}$  to improve identity consistency.
+
+Similarly, we suppress the expression of the remaining frames. Since  $\hat{\mathcal{X}}^{sup}$  contains information related to multiple frames, the main singular values of SVD in  $\hat{\mathcal{X}}^{sup}$  only capture a small portion of these descriptions, which may lead to insufficient weakening of such semantics (as shown in the Appendix of Fig. 11-right). Therefore, we propose to weaken each frame prompt in  $\hat{\mathcal{X}}^{sup}$  separately. We construct the matrix as  $\hat{\mathcal{X}}_k^{sup} = [c^{\mathcal{P}_k},\hat{c}^{EOT}],k\neq j$  to perform SVD with the singular values  $\hat{\sigma}_0\geq \dots \geq \hat{\sigma}_{n_k}$ . Then, each singular value is weakened as follows, termed as SVR-:
+
+$$
+\tilde {\sigma} = \beta^ {\prime} e ^ {- \alpha^ {\prime} \hat {\sigma}} * \hat {\sigma}. \tag {3}
+$$
+
+where  $\alpha'$  and  $\beta'$  are parameters with positive numbers. The recovered structure is  $\tilde{\mathcal{X}}_k^{sup} = [\tilde{c}^{\mathcal{P}_k},\tilde{c}^{EOT}]$ , After reducing the expression of each suppress token, we finally obtain the new text embedding  $\tilde{\mathcal{C}} = [c^{SOT},c^{\mathcal{P}_0},\tilde{c}^{\mathcal{P}_1},\dots ,\hat{c}^{\mathcal{P}_j},\dots ,\tilde{c}^{\mathcal{P}_N},\tilde{c}^{EOT}]$ .
+
+Identity-Preserving Cross-Attention. The use of Singular-Value Reweighting can reduce the blending of frame descriptions in single-prompt generation. However, we observed that it could also impact context consistency within the single prompt, leading to images generated slightly less similar in identity (as shown in the ablation study of Fig. 7). Recent work (Liu et al., 2024) demonstrated that cross-attention maps capture the characteristic information of the token, while self-attention preserves the layout information and the shape details of the image. Inspired by this, we propose Identity-Preserving Cross-Attention to further enhance the identity similarity between images generated from the concatenated prompt of our proposed Prompt Consolidation.
+
+For a specific timestep  $t$ , after applying Singular-Value Reweighting, we have the updated text embedding  $\tilde{\mathcal{C}}$ . During a denoising pass through the diffusion model, we obtain the corresponding  $\tilde{\mathcal{Q}}, \tilde{\mathcal{K}}, \tilde{\mathcal{V}}$  in the cross-attention layer. Here, we aim to strengthen the identity consistency among the images and mitigate the impact of irrelevant prompts. We set the token features in  $\tilde{\mathcal{K}}$  corresponding to  $\bar{P}_i, i \in [1, N]$  to zero, resulting in  $\tilde{\mathcal{K}}$ . Here, only the identity prompt remains to augment the identity semantics. Similarly, we can get  $\bar{\mathcal{V}}$ . We form a new version of  $\tilde{\mathcal{K}}$  by concatenating it with  $\tilde{\mathcal{K}}$ , dubbed  $\tilde{\mathcal{K}} = \mathrm{Concat}(\tilde{\mathcal{K}}^{\top}, \bar{\mathcal{K}}^{\top})^{\top}$ . The new cross-attention map is then given by:
+
+$$
+\tilde {\mathcal {A}} = \operatorname {s o f t m a x} \left(\tilde {\mathcal {Q}} \tilde {\mathcal {K}} ^ {\top} / \sqrt {d}\right) \tag {4}
+$$
+
+where  $d$  is the dimension of  $\tilde{\mathcal{Q}}$  and  $\tilde{\mathcal{K}}$ . Similarly, we update  $\tilde{\mathcal{V}} = \mathrm{Concat}(\tilde{\mathcal{V}}^{\top},\bar{\mathcal{V}}^{\top})^{\top}$ . The final output feature of the cross-attention layer is  $\tilde{\mathcal{A}}\times \tilde{\mathcal{V}}$ . This output is a reweighted version that strengthens identity consistency using filtered features, which only contain the identity prompt semantics.
+
+![](images/32a3c08e035c8045dfe501f33a7e69d051a1c7feae4161872011fe99e40c8deb.jpg)  
+Figure 5: Qualitative results. We compare our method with PhotoMaker, IP-Adapter, ConsiStory, and StoryDiffsion. Among them, Texture Inversion, PhotoMaker, ConsiStory, and StoryDiffsion struggled to maintain identity consistency for the dragon object while IP-Adapter produced images with relatively similar poses and backgrounds. See Comparison with the remaining methods in Fig. 22 of the Appendix.
+
+![](images/64f9d458789774b63e70cc28ff9d7a98c32f6199b81e2cfa439192e61f9061e7.jpg)
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUPS
+
+Comparison Methods and Benchmark. We compare our method with the following consistent T2I generation approaches: BLIP-Diffusion (Li et al., 2024), Textual Inversion (TI)(Gal et al., 2023a), IP-Adapter(Ye et al., 2023), PhotoMaker (Li et al., 2023b), The Chosen One (Avrahami et al., 2023), ConsiStory (Tewel et al., 2024), and StoryDiffusion (Zhou et al., 2024). We follow the default configurations in their papers or open-source implementations.
+
+To evaluate their performance, we introduce ConsiStory+, an extension of the original ConsiStory (Tewel et al., 2024) benchmark. This new benchmark incorporates a wider range of subjects, descriptions, and styles. Following the evaluation protocol outlined in ConsiStory, we evaluated both prompt alignment and subject consistency across ConsiStory+, generating up to 1500 images on 200 prompt sets. Additional details on the construction of our benchmark and the implementation of the methods are provided in Appendix B.2 and Appendix B.3.
+
+Evaluation Metrics. To assess prompt alignment performance, we compute the average CLIP-Score (Hessel et al., 2021) for each generated image in relation to its corresponding prompt, which we denote as CLIP-T. For the identity consistency evaluation, we measure image similarity using
+
+Table 1: Quantitative comparison. The best and second best results are highlighted in bold and underlined, respectively. Vanilla SD1.5 and Vanilla SDXL are shown as references and excluded from this comparison.  
+
+<table><tr><td>Method</td><td>Base Model</td><td>Train-Free</td><td>CLIP-T↑</td><td>CLIP-I↑</td><td>DreamSim↓</td><td>Steps</td><td>Memory (GB)↓</td><td>Inference Time (s)↓</td></tr><tr><td>Vanilla SD1.5</td><td>-</td><td>-</td><td>0.8353</td><td>0.7474</td><td>0.5873</td><td>50</td><td>4.73</td><td>2.4657</td></tr><tr><td>Vanilla SDXL</td><td>-</td><td>-</td><td>0.9074</td><td>0.8165</td><td>0.5292</td><td>50</td><td>16.04</td><td>13.0890</td></tr><tr><td>BLIP-Diffusion</td><td>SD1.5</td><td>X</td><td>0.7607</td><td>0.8863</td><td>0.2830</td><td>26</td><td>7.75</td><td>1.9284</td></tr><tr><td>Textual Inversion</td><td></td><td>X</td><td>0.8378</td><td>0.8229</td><td>0.4268</td><td>40</td><td>32.94</td><td>282.507</td></tr><tr><td>The Chosen One</td><td>SDXL</td><td>X</td><td>0.7614</td><td>0.7831</td><td>0.4929</td><td>35</td><td>10.93</td><td>11.2073</td></tr><tr><td>PhotoMaker</td><td></td><td>X</td><td>0.8651</td><td>0.8465</td><td>0.3996</td><td>50</td><td>23.79</td><td>18.0259</td></tr><tr><td>IP-Adapter</td><td></td><td>X</td><td>0.8458</td><td>0.9429</td><td>0.1462</td><td>30</td><td>19.39</td><td>13.4594</td></tr><tr><td>ConsiStory</td><td></td><td>✓</td><td>0.8769</td><td>0.8737</td><td>0.3188</td><td>50</td><td>34.55</td><td>34.5894</td></tr><tr><td>StoryDiffusion</td><td rowspan="2">SDXL</td><td>✓</td><td>0.8877</td><td>0.8755</td><td>0.3212</td><td>50</td><td>45.61</td><td>25.6928</td></tr><tr><td>Naive Prompt Reweighting (NPR)</td><td>✓</td><td>0.8411</td><td>0.8916</td><td>0.2548</td><td>50</td><td>16.04</td><td>17.2413</td></tr><tr><td>1Prompt1Story (Ours)</td><td></td><td>✓</td><td>0.8942</td><td>0.9117</td><td>0.1993</td><td>50</td><td>18.70</td><td>23.2088</td></tr></table>
+
+both DreamSim (Fu et al., 2023), which has been shown to closely reflect human judgment in evaluating visual similarity, and CLIP-I (Hessel et al., 2021), calculated by the cosine distance between image embeddings. In line with the methodology proposed in DreamSim (Fu et al., 2023), we remove image backgrounds using CarveKit (Selin, 2023) and replace them with random noise to ensure that similarity measurements focus solely on the identities of subjects.
+
+# 4.2 EXPERIMENTAL RESULTS
+
+Qualitative Comparison. In Fig. 5, we present the qualitative comparison results. Our method  $IPrompt1Story$  demonstrates well-balanced performance in several key aspects, including identity preservation, accurate frame descriptions, and diversity in the pose of objects. In contrast, other methods exhibit shortcomings in one or more of these aspects. Specifically, PhotoMaker, ConsiStory, and StoryDiffusion all produce inconsistent identities for the subject "dragon" in the examples on the left. Additionally, IP-Adapter tends to generate images with repetitive poses and similar backgrounds, often neglecting frame prompt descriptions. ConsiStory also displays duplicated background generation in the consistent T2I generation.
+
+Quantitative Comparison. In Table 1, we illustrate the quantitative comparison with other approaches. In all evaluation metrics,  $1Prompt1Story$  ranks first among the training-free methods, and second when including training-required methods. Furthermore, compared to other training-free methods, our approach demonstrates a reasonable fast inference speed while achieving excellent performance. More specifically, our method  $1Prompt1Story$  achieves the CLIP-T score closely aligned with the vanilla SDXL model. In terms of identity similarity, measured by CLIP-I and DreamSim, our method ranks just below IP-Adapter. However, the high identity similarity of IP-Adapter mainly stems from its tendency to generate images with characters depicted in similar poses and layouts. To further explore this potential bias, we conducted a user study to investigate human preferences. Following ConsiStory, we also visualized our quantitative results using a chart, as shown in Fig. 6. Training-based methods, such as IP-Adapter and Textual Inversion, often overfit character identity and perform poorly on prompt alignment. In contrast, among training-free methods, our approach achieves the best balance in both prompt alignment and identity consistency.
+
+User Study. In the user study, we compare our method with several state-of-the-art approaches, including IP-Adapter, ConsiStory, and StoryDiffusion. From our benchmark, we randomly selected 30 sets of prompts, each comprising four fixed-length prompts, to generate test images. Twenty participants were asked to select the image that best demonstrated overall performance in terms of identity consistency, prompt alignment, and image diversity. As shown in Table 2, the results indicated that our method  $1Prompt1Story$  aligns best with human preference. More details of the user study are shown in Appendix F.
+
+Ablation study. We performed an ablation study to analyze each component, as illustrated both qualitatively and quantitatively in Fig. 7 and Table 3. When using Singular-Value Reweighting exclusively with improving the express set as  $\mathbf{SVR}+$  (that is, Eq. 2), the generated images blend with other descriptions, as can be seen in Fig. 7 (left, first row). Similarly, when Singular-Value Reweighting is only to weaken the suppress set as SVR- (i.e., Eq.3), the same issue appears in Fig. 7
+
+Table 2: User study with 37 people to vote for the best consistent T2I generation method according to human preference.  
+
+<table><tr><td>Method</td><td>IP-Adapter</td><td>ConsiStory</td><td>StoryDiffusion</td><td>Ours</td></tr><tr><td>Percent (%)↑</td><td>8.60</td><td>13.00</td><td>29.80</td><td>48.60</td></tr></table>
+
+Table 3: Ablation study. We evaluated the influence of each component in 1Prompt1Story, including the Singular-Value Reweighting (SVR+ and SVR-), and Identity-Preserving Cross-Attention (IPCA).  
+
+<table><tr><td>Method</td><td>CLIP-T↑</td><td>CLIP-I↑</td><td>DreamSim↓</td></tr><tr><td>PCon; SVR+</td><td>0.8774</td><td>0.8886</td><td>0.2560</td></tr><tr><td>PCon; SVR-</td><td>0.8910</td><td>0.8904</td><td>0.2605</td></tr><tr><td>PCon; SVR+; SVR-</td><td>0.8989</td><td>0.8849</td><td>0.2538</td></tr><tr><td>PCon; SVR+; SVR-; IPA (Ours)</td><td>0.8942</td><td>0.9117</td><td>0.1993</td></tr></table>
+
+![](images/17db244b79d3f681db8f90c43a08db6ab148b6652a00dce97d9a7b0cbf9f7ca0.jpg)  
+Figure 6: Prompt alignment vs. identity consistency. Our method 1Prompt1Story is positioned in the upper right corner.
+
+"A photo of a dog<sup>0</sup>, chasing a frisbee in a colorful park<sup>1</sup>, dancing to music at a vibrant street festival<sup>2</sup>, jumping through a hoop at a circus performance<sup>3</sup>, posing for a photoshoot in a modern art gallery<sup>4</sup)."
+
+![](images/10b1d193a09d2c58bbb1c86b1c932cc661f1f818557f5ff07458b0cb54a57c84.jpg)  
+Figure 7: Qualitative ablation study. All ablated cases with incomplete components of  $1Prompt1Story$  struggle to achieve both prompt alignment and identity consistency as effectively as our full method.
+
+(left, second row). In contrast, integrating both SVR+ and SVR- in Singular-Value Reweighting can effectively mitigate blending in generated images (Fig. 7 (right, first row)). Although Singular-Value Reweighting can effectively resolve frame prompt blending issues, without Identity-Preserving Cross-Attention, there remains a weak inconsistency among the generated images. As shown in Fig. 7 (right, second row), the results indicate that using Singular-Value Reweighting and Identity-Preserving Cross-Attention achieves the best performance, as also evident in Table 3 (the last row). Additional results of ablation analysis and visualization are presented in the Appendix. C.
+
+Additional applications. 1Prompt1Story can also achieve spatial controls, integrating with existing control-based generative methods such as ControlNet (Zhang & Agrawala, 2023). As shown in Fig. 8 (left), our method effectively generates consistent characters with human pose control. Furthermore, our method can be combined with other approaches, such as PhotoMaker (Li et al., 2023b), to improve the consistency of identity with real images. By applying our method, the generated images more closely resemble the real identities, as demonstrated in Fig. 8 (right).
+
+# 5 CONCLUSION
+
+In this paper, we addressed the critical challenge of maintaining subject consistency in text-to-image (T2I) generation by leveraging the inherent property of context consistency found in natural language. Our proposed method, One-Prompt-One-Story (1Prompt1Story), effectively utilizes a single extended prompt to ensure consistent identity representation across diverse scenes. By integrating techniques such as Singular-Value Reweighting and Identity-Preserving Cross-Attention, our approach not only refines frame descriptions but also strengthens the consistency at the attention level. The experimental results on the ConsiStory+ benchmark demonstrated the superiority of 1Prompt1Story over state-of-the-art techniques, showcasing its potential for applications in animation, interactive storytelling, and video generation. Ultimately, our contributions highlight the importance of understanding context in T2I diffusion models, paving the way for more coherent and narrative-consistent visual output.
+
+# ACKNOWLEDGEMENTS
+
+This work was supported by NSFC (NO. 62225604) and Youth Foundation (62202243). We acknowledge the support of the project PID2022-143257NB-I00, funded by the Spanish Government through MCIN/AEI/10.13039/501100011033 and FEDER. Additionally, we recognize the "Science and Technology Yongjiang 2035" key technology breakthrough plan project (2024Z120). The computations for this paper were facilitated by the resources provided by the Supercomputing Center of Nankai University (NKSC).
+
+We would like to extend our gratitude to all the co-authors for their invaluable assistance and insightful suggestions throughout this work. In particular, we wish to thank Kai Wang, a postdoctoral researcher at the Computer Vision Center, Universitat Autònoma de Barcelona. His meticulous advice and guidance were instrumental in the completion of this project.
+
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+
+# APPENDIX
+
+# A BOARDER IMPACTS AND LIMITATIONS
+
+Boarder Impacts. The application of T2I models in consistent image generation offers extensive potential for various downstream applications, enabling the adaptation of images to different contexts. In particular, synthesizing consistent characters has diverse applications, however, it is a challenging task for diffusion models. Our 1Prompt1Story can help the users customize their desired characters in different story scenarios, resulting in significant time and resource savings. Notably, current methods have inherent limitations, as discussed in this paper. However, our model can serve as an intermediary solution while offering valuable insights for further advancements.
+
+Limitations. While our method 1Prompt1Story can achieve high-fidelity consistent T2I generation, it is not free of limitations. Firstly, we have to know all the prompts in advance. Additionally, the length of the input prompt is constrained by the maximum capacity of the text encoder. Although we proposed a sliding window technique that facilitates infinite-length story generation in Appendix D.2, this approach may encounter issues where the identity of the generated images gradually diverges and becomes less consistent.
+
+# B IMPLEMENTATION DETAILS
+
+# B.1 MODEL CONFIGURATIONS
+
+We generate subject-consistent images by modifying text embeddings and cross-attention modules at inference time, without any training or optimization processes. Our primary base model is the pretrained Stable Diffusion XL (SDXL) $^3$ . SDXL has two text encoders: the CLIP L/14 encoder (Radford et al., 2021) and the OpenCLIP bigG/14 encoder (Cherti et al., 2023). We separately update the text embeddings produced by each encoder. For Naive Prompt Reweighting, we multiply the text embedding corresponding to the frame prompt that needs to be expressed by a factor of 2, while the text embedding corresponding to the frame prompts that need to be suppressed is multiplied by a factor of 0.5, keeping the  $c^{EOT}$  unchanged.
+
+In our method, 1Prompt1Story, we set the parameters as follows:  $\alpha = 0.01$ ,  $\beta = 0.05$  in Eq.2, and  $\alpha' = 0.01$ ,  $\beta' = 1.0$  in Eq.3. During the generation process, we initialize all frames with the same noise and apply a dropout rate of 0.5 to the token features in  $\tilde{\mathcal{K}}$  corresponding to  $\mathcal{P}_0$ . In the implementation of IPCA, the concatenated  $\tilde{\mathcal{K}}$  and  $\tilde{\mathcal{V}}$  are derived from the original text embeddings prior to applying SVR. We design an attention mask where all values in the column corresponding to  $\mathcal{P}_i$ ,  $i \in [1, N]$  are set to zero, while all other positions are set to one. The natural logarithm of this mask is then added to the original attention map. Our full algorithm is presented in Algorithm 1. Following (Tewel et al., 2024; Alaluf et al., 2024; Luo et al., 2023), we use Free-U (Si et al., 2024) to enhance the generation quality. All generated images based on SDXL are produced at a resolution of  $1024 \times 1024$  using a Quadro RTX 3090 GPU with 24GB VRAM.
+
+# B.2 BENCHMARK DETAILS
+
+To evaluate the effectiveness of our method, we developed ConsiStory+, an extended prompt benchmark based on ConsiStory (Tewel et al., 2024). We enhanced both the diversity and size of the original benchmark, which only comprised 100 sets of 5 prompts across 4 superclasses. Our expansion resulted in 200 sets, with each set containing between 5 and 10 prompts, categorized into 8 superclasses: humans, animals, fantasy, inanimate, fairy tales, nature, technology, and foods. The extended prompt benchmark was generated using ChatGPT 4.0-turbo<sup>4</sup>, involving two main steps. First, we expanded the 100 prompt sets from the original benchmark, increasing each to a length of 5 to 10 prompts, as shown in Fig. 9 (left). Then, we generated new prompt sets for each of the new superclasses, as illustrated in Fig. 9 (right). The prompt sets collected through these two steps were combined to form our benchmark, ConsiStory+.
+
+![](images/b4c08a0a7c09d5503f82fd5c32ed8f4163aaedd04bff09eeda1d147f77c1716b.jpg)  
+Figure 8: (Left): Our method 1Prompt1Story can integrate with ControlNet to enable spatial control for consistent character generation. (Right): Additionally, our method can also combine with other methods, such as PhotoMaker, to achieve real-image personalization with improved identity consistency.
+
+Algorithm 1 1Prompt1Story  
+```latex
+Input: A text embedding  $\mathcal{C} = [c^{SOT}, c^{\mathcal{P}_0}, c^{\mathcal{P}_1}, \dots, c^{\mathcal{P}_N}, c^{EOT}]$  and latent vector  $z_t$ . Output: The subject consistency images  $\mathcal{I}_1, \dots, \mathcal{I}_N$ .
+```
+
+```txt
+for  $j = 1,\dots ,N$  do   
+// Singular-Value Reweighting   
+ $\hat{\mathcal{X}}^{exp} = [\hat{c}^{\mathcal{P}_j},\hat{c}^{EOT}]\gets \mathcal{X}^{exp} = [c^{\mathcal{P}_j},c^{EOT}]$  (Eq.2);   
+for  $k = [1,N]\setminus \{j\}$  do   
+ $\tilde{\mathcal{X}}^{sup} = [\tilde{c}_k^\mathcal{P},\tilde{c}^{EOT}]\gets [c_k^\mathcal{P},\tilde{c}^{EOT}]$  (Eq.3);   
+end   
+ $\tilde{\mathcal{C}} = [\pmb {c}^{SOT},\pmb {c}^{\mathcal{P}_0},\tilde{\pmb{c}}^{\mathcal{P}_1},\dots ,\hat{\pmb{c}}^{\mathcal{P}_j},\dots ,\tilde{\pmb{c}}^{\mathcal{P}_N},\tilde{\pmb{c}}^{EOT}];$    
+// Identity-Preserving Cross-Attention   
+for  $t = T,\ldots ,1$  do   
+ $\tilde{\mathcal{K}},\tilde{\mathcal{V}}\gets \tilde{\mathcal{C}};$ $\tilde{\mathcal{K}},\tilde{\mathcal{V}}\gets \tilde{\mathcal{K}},\tilde{\mathcal{V}};$ $\tilde{\mathcal{K}} = \mathrm{Concat}(\tilde{\mathcal{K}}^{\top},\bar{\mathcal{K}}^{\top})^{\top},\tilde{\mathcal{V}} = \mathrm{Concat}(\tilde{\mathcal{V}}^{\top},\bar{\mathcal{V}}^{\top})^{\top};$ $\tilde{\mathcal{A}}\gets \tilde{\mathcal{Q}},\tilde{\mathcal{K}}$  (Eq.4);   
+ $z_{t - 1}\gets \epsilon_{\theta}(z_t,t,\tilde{\mathcal{C}})$  with  $\tilde{\mathcal{A}},\tilde{\mathcal{V}}$    
+end   
+ $\mathcal{I}_j = D(z_0)$    
+end   
+Return  $\mathcal{I}_1,\dots ,\mathcal{I}_N$
+```
+
+# B.3 COMPARISON METHOD IMPLEMENTATIONS
+
+We compare our method with all other approaches based on Stable Diffusion XL, except for BLIP-Diffusion (Li et al., 2024), which is based on Stable Diffusion v1.5<sup>5</sup>. The DDIM steps is set to the default value in the open-source code of each method. Below are the third-party packages we used for method implementations:
+
+- The unofficial implementation of Textual Inversion (TI) (Gal et al., 2023a) at https://github.com/oss-roettger/XL-Textual-Inversion.  
+- The unofficial implementation of The Chosen One (Avrahami et al., 2023) at https://github.com/ZichengDuan/TheChosenOne.  
+- The official implementation of IP-Adapter (Ye et al., 2023) at https://github.com/ tencent-ailab/IP-Adapter.  
+- The official implementation of PhotoMaker (Li et al., 2023b) at https://github.com/TencentARC/PhotoMaker.
+
+![](images/f7aca8c3da755907d86fc888df8c481fe817d1ddb93362b12c8cc6f5dc01470a.jpg)  
+Figure 9: (Left): We expand the length of the original prompt sets to a random number between 5 and 10. (Right): We generate a new prompt set within one of the new superclass "fairy tales".
+
+```txt
+{ "prompt": "A photo of A sticker of a cute corgi dog, on a student's laptop, amidst an action-packed skate session, showing its teeth, at a live music gig, at an inter national airport," "concept_token": "dog", "superclass": "inanimate", "subject": "A sticker of a cute corgi dog", "settings": [ "on a student's laptop", "amidst an action-packed skate session", "adding character to urban signage", "at a live music gig", "at an international airport"] , "style": "A photo of"   
+} According to the template above, select a random number between 5 and 10, and extend the settings to that length. The extended content should describe the subject. Also, update the content in the prompt accord ingly.
+```
+
+![](images/7457ffe63cee58f91af8d5253fc3b47d60c9e290bdddc136d03f78bcf754b9a6.jpg)
+
+```txt
+Here is the updated version based on the instructions: { "prompt": "A photo of A sticker of a cute corgi de on a student's laptop, amidst an action-packed skate session, adding character to urban signage, at a live music gig, at an international airport, brightening up a park bench, stuck to a water bottle in a hiking trail," "concept_token": "dog", "superclass": "inanimate", "subject": "A sticker of a cute corgi dog", "settings": [ "on a student's laptop", "amidst an action-packed skate session", "adding character to urban signage", "at a live music gig", "at an international airport", "brightening up a park bench", "stuck to a water bottle in a hiking trail"], "style": "A photo of" } I extended the settings to a random length of 7 and updated the prompt accordingly.
+```
+
+![](images/55eeece9f30b086f503f8b339c9f34bca778df397934ba9ee99f35c7665ee2ff.jpg)
+
+```txt
+{ "prompt": "A photo of A sticker of a cute corgi dog, on a student's laptop, amidst an action-packed skate session, showing its teeth, at a live music gig, at an inter national airport," "concept_token": "dog", "superclass": "inanimate", "subject": "A sticker of a cute corgi dog", "settings": [ "on a student's laptop", "amidst an action-packed skate session", "adding character to urban signage", "at a live music gig", "at an international airport"] style": "A photo of"   
+} According to the template above, replace the superclass category with 'fair_tales' and choose a new random style, subject, and settings based on this superclass. Ensure that the concept_token represents the subject described, and the settings' length is a random number between 5 and 10, describing various actions of the sub ject.
+```
+
+![](images/f5fad6fe235131225bb2503cdd0ac30debbd4ea4a18aa077cff3e84729cc1cfc.jpg)
+
+```txt
+Here is the updated version based on the instructions:   
+{ "prompt": "An illustration of A mischievous elf, dancing in the moonlight, weaving through the forest trees, whispering to the woodland creatures, playing a flute by the riverbank, hiding behind a mushroom.", "concept_token": "elf", "superclass": "fair_tales", "subject": "A mischievous elf", "settings": [ dancing in the moonlight", "weaving through the forest trees", "whispering to the woodland creatures", "playing a flute by the riverbank", "hiding behind a mushroom"],"style": "An illustration of"   
+}   
+In this update, the superclass was changed to "fair_tale and a new subject, style, and settings were chosen to this theme. The settings' length was randomly selected to be 5.
+```
+
+- The official implementation of BLIP-Diffusion (Li et al., 2024) at https://github.com/salesforce/LAVIS/tree/main/projects/blip-diffusion.  
+- The official implementation of StoryDiffusion (Zhou et al., 2024) at https://github.com/HVision-NKU/StoryDiffusion.
+
+Since Consistory (Tewel et al., 2024) is not open-source, we reimplemented it ourselves. During the inference time, BLIP-Diffusion (Li et al., 2024), IP-Adapter (Ye et al., 2023), and PhotoMaker (Li et al., 2023b) all require a reference image as the additional input. To generate the reference image, we use their corresponding base models, providing the identity description as the input prompt. For example, if the full prompt is "a photo of a beautiful girl walking on the street", we use "a photo of a beautiful girl" to generate the reference image. The reference image is then used to generate all frames in the corresponding prompt set.
+
+Identity Prompt: "A photo of a fox."
+
+Frame Prompt: 1 "wearing a scarf in a meadow" 2 "playing in the snow" 3 "at the edge of a river"
+
+![](images/d02e01db08c18d651c54ba5c6f9e5e57597599d172836b56add1167eeca8b437.jpg)  
+Order: 123
+
+![](images/38e7e9554f9282aa6e3dd93d0f56cb2024d965f0a2ce5ac566c4b89e7476e572.jpg)
+
+![](images/aec2494bc503005dd33b183a847315082f1a790a2053407f9d301247cd491f01.jpg)  
+Order: 132
+
+![](images/550056a2a15f331689d81d6f4396c47ee1c47b8437c4356226a5e2193c7bcd88.jpg)
+
+![](images/22899a87fad48226d3039a2adfa170eb5d906753aa53dbac35923cef0870a3eb.jpg)
+
+![](images/40168a9550a54273858f9184658dbdacf1365ddfc4f1343f9642db48389ff837.jpg)
+
+![](images/b2fda4b7b6f3b47227053cfa5a4ca832808d158e4b9f25a3e4bb00fedfc1f6c0.jpg)  
+Order: 213
+
+![](images/4be69941acc05da5967f6bee64468e355de0af4da6ca8cabb2c665c3417fef13.jpg)
+
+![](images/63beae799a101d20fc8f7080b3b8eda28a76546405f66e863729436cb9ae5484.jpg)
+
+![](images/118fdf57ea3886f07764ef3961d3884dd4ebf1985b9a2acee8b46be67c2f2d7b.jpg)  
+Order: 231
+
+![](images/3537a3e5adb76d76cd50a0b63c340172fa4a1698693de26c4d9f79576680f613.jpg)
+
+![](images/829a01210caaf0ddf5230261d59d7e43e6b31637d8085e31e85e198b7aceef25.jpg)
+
+![](images/382ce4b84ffc21d7767a4a9de200feca1d9de450b207af1cddc09e77f87db3a5.jpg)  
+Order: 312
+
+![](images/e53b0cdbab22c098def90897f1df72a457493b9c425cc516dc871c8ae65c9ecb.jpg)  
+Figure 10: Robustness to frame prompts order. With the same set of frame prompts but in different orders, our method  $IPrompt1Story$  consistently generates images with a unified identity.
+
+![](images/15ee252b2491425c3f5a2a9af1c18853c57d36a38ed189c38dbf33fdfcd2b4f0.jpg)
+
+![](images/28da9cc4f1c465cfda753333bf570e3efb3173bf252d7bf26cc131d2514e9fa7.jpg)  
+Order: 3 2 1
+
+![](images/e4a761b740c44fdc8e93180152e2fff2bfc902295dbbaab95d7a0e580242cad5.jpg)
+
+![](images/ad726ed19c4b9261a894c502b2f6b7e3870736346e041f4223a22ed2dfbb47b0.jpg)
+
+# C ADDITIONAL ABLATION STUDY
+
+# C.1 ROBUSTNESS TO DIVERSE DESCRIPTION ORDERS
+
+To validate the robustness of our method regarding the order of frame prompts, we used the same three frame prompts: "wearing a scarf in a meadow", "playing in the snow", and "at the edge of a river" to create six different sequences for images generation. The identity prompt was consistently set to "a photo of a fox" and each sequence used the same seed for a generation. As shown in Fig. 10, our method  $1Prompt1Story$  generates images with identity consistency across different orders. Furthermore, the content of the images generated from varying sequences is closely aligned with the text descriptions, further demonstrating our method Singular-Value Reweighting effectiveness in suppressing content of unrelated frame prompts.
+
+# C.2 Singular-Value Reweighting ANALYSIS
+
+Our Singular-Value Reweighting algorithm comprises two successive components: SVR+ enhances the frame prompts we wish to express, while SVR- iteratively weakens the frame prompts we aim to suppress. In our experiments, we first apply SVR+, followed by SVR-. In particular, we found that performing SVR- before SVR+ also yields similar results (see Fig. 11-left).
+
+In the process of applying SVR-, we employed a strategy of iteratively suppressing each frame prompt. In fact, we could also concatenate the text embeddings corresponding to all frame prompts for suppression. To explore this, we conducted further ablation study specifically on the SVR-component. Assuming that we have  $n$  frames to generate, we discovered that merging the text embeddings corresponding to the  $n - 1$  frames we wish to suppress with  $c^{\mathrm{EOT}}$  and subsequently performing the SVD decomposition does not effectively extract the main components of all frame prompts included in  $c^{\mathrm{EOT}}$ . Consequently, applying Eq. 3 to weaken the eigenvalues based on their magnitude fails to adequately eliminate the descriptions of all suppressed frames. we refer to this as "joint suppress", as illustrated in Fig. 11 (right, the first row). In contrast, if we handle each frame prompt to be suppressed individually and iteratively perform SVD and the operations from Eq. 3, which we term "iterative suppress", we can more effectively suppress all irrelevant frame prompts, as shown in Fig. 11 (right, the second row).
+
+![](images/0bd22d6d40dac9c8822565fedf1c76b298849128e7c8a68860f14457c3aa0a5a.jpg)  
+Figure 11: (Left): "SVR+ First" indicates that SVR+ is applied before SVR- in the Singular-Value Reweighting process, while "SVR- First" means the opposite order. We found that both sequences yield similar results (same seed). (Right): Compared to "Joint Suppress", "Iterative Suppress" is more effective at minimizing the influence of other frame prompts when generating images for the current frame. "Joint Suppress" produces images with similar backgrounds (the first row, first and third columns).
+
+![](images/5e5c4585d373784180b0991f00b0b1296d41cb7e256b11aa93b5e825c639dd86.jpg)
+
+"A watercolor illustration of A puppy<sup>0</sup>, wearing a small sweater<sup>1</sup>, digging a hole<sup>2</sup>, wearing a training harness<sup>3</sup>, sticking head out of the window<sup>4</sup>, swimming<sup>5</sup>, playing in the yard<sup>6</sup)."
+
+![](images/7de949fe9a439ea7dbae81e7bac86333aa18dbf76e87fdfc97b733c1f4efb555.jpg)  
+Figure 12: SVR with identity enhancement. The first row represents the original SVR with enhancements applied only to the frame prompt. The second row builds upon the original by further enhancing the identity prompt in the  $\mathrm{SVR + }$  module. The results indicate that while the second method improves identity consistency, it also leads to more similar object poses and backgrounds.
+
+In our SVR, we enhance only the current frame prompt that needs to be expressed. An alternative option is to enhance the identity prompt simultaneously. We found that doing so can make the object's identity more consistent; however, it also introduces some negative effects, the background and subject's pose appearing more similar across images, as shown in Fig. 12. Furthermore, to demonstrate the role of the  $c^{\mathrm{EOT}}$  in SVR, we conducted an ablation study on the  $c^{\mathrm{EOT}}$  component. Specifically, we kept the  $c^{\mathrm{EOT}}$  part of the text embedding unchanged during the SVR process and used this text embedding to generate images. As shown in Fig. 13, the results indicate that without performing SVR on the  $c^{\mathrm{EOT}}$ , the backgrounds of different frame prompts tend to blend together.
+
+# C.3 Naive Prompt Reweighting ABLATION STUDY
+
+Similar to the Singular-Value Reweighting (SVR) experiment, we conducted an ablation study to verify the effectiveness of Naive Prompt Reweighting (NPR) in terms of identity preservation and prompt alignment compared to our method 1Prompt1Story. We denote  $\mathrm{NPR + }$  as applying a scaling factor of 2 to the text embedding corresponding to the current frame prompt that needs to be expressed. Conversely, NPR- denotes applying a scaling factor of 0.5 to the text embeddings of all other frame prompts that need to be suppressed. NPR represents the combination of both  $\mathrm{NPR + }$  and NPR- operations.
+
+As shown in Fig. 14, images generated using the NPR+, NPR-, and NPR methods all exhibit varying degrees of interference from other frame prompts. In contrast, our method effectively removes
+
+"A digital portrait of a 14-year-old boy, in a rose garden, making a sandcastle, playing in snow."
+
+![](images/68f01708354627a3d05bf6d888b052f9a050c04499409b2808bf35787f81f18b.jpg)  
+Figure 13: Ablation study for  $c^{\mathrm{EOT}}$ . The left three images demonstrate the SVR process with a fixed  $c^{\mathrm{EOT}}$  while the right illustrates the SVR procedure described in the main text. The results indicate that keeping  $c^{\mathrm{EOT}}$  unchanged leads to background blending across images generated for different frame prompts, highlighting the importance of updating  $c^{\mathrm{EOT}}$  dynamically.
+
+"A watercolor illustration of a male child<sup>0</sup>, in a toy store<sup>1</sup>, exploring an exhibit<sup>2</sup>, in a backyard, playing with a puppy<sup>3</sup>, enjoying a carousel ride<sup>4</sup>, building a fort<sup>5</sup)."
+
+![](images/a526480cff9a8db774cc62038a3dd56af7727e15633124f7712b506cacc1c22c.jpg)  
+Figure 14: Naive Prompt Reweighting ablation study. NPR+, NPR-, and NPR are ineffective at suppressing the influence of other frame prompts. For example, the "puppy", which appears only in the frame prompt of the third frame, also shows up in the first and second frames using the aforementioned methods. In contrast, our method (the last row) effectively suppresses unwanted semantic information from other frame prompts.
+
+irrelevant semantic information from other frame subject descriptions in the single-prompt setting, resulting in images that are more aligned with their corresponding frame prompts.
+
+# C.4 SEED VARIETY
+
+Since our method 1Prompt1Story does not modify the original parameters of the diffusion model, it preserves the inherent ability of the model to generate images with diverse identities and backgrounds using different seeds. By varying the initial noise while keeping the input prompt set constant, our method can produce a range of characters and backgrounds, all while maintaining strong identity consistency and prompt alignment, as shown in Fig. 15.
+
+![](images/7c39299bac43a5de006ae93eaae057e3f1c4727855e495925d2f0380940c4929.jpg)  
+Figure 15: Seed variation. By using different seeds, our method 1Prompt1Story can generate images with diverse backgrounds while maintaining a consistent identity.
+
+# D ADDITIONAL RESULTS OF OUR METHOD 1Prompt1Story
+
+# D.1 CONSISTENT STORY GENERATION WITH MULTIPLE SUBJECTS.
+
+Our method is capable of generating stories involving multiple subjects. By specifying several subjects in the identity prompt and appending corresponding frame prompts, we can directly produce a series of images that maintain consistent identities across these subjects, as demonstrated in Fig. 16. However, this approach has a limitation: all generated images will include every character referenced in the identity prompt, which poses a constraint on the flexibility of our method.
+
+# D.2 STORY GENERATION OF ANY LENGTH.
+
+To generate stories of any length, we designed a "sliding window" technique to overcome the input text length limitations of diffusion models like SDXL. Suppose we aim to generate a story with  $n$  images, each corresponding to  $n$  frame prompts, using a window size  $t$ , where  $t < n$ . Similarly, we represent the identity prompt as  $\mathcal{P}_0$  and the frame prompts as  $\mathcal{P}_i$ , where  $i \in [1, n]$ . For generating the image corresponding to the  $i$ -th frame, if  $i \leq t$ , we use  $\mathcal{P} = [\mathcal{P}_0; \mathcal{P}_1; \ldots; \mathcal{P}_t]$  as input prompt and apply our method 1Prompt1Story to generate the images. If  $i > t$ , we use  $\mathcal{P} = [\mathcal{P}_0; \mathcal{P}_{i - t + 1}; \ldots; \mathcal{P}_i]$  to generate the images. As shown in Fig. 19, we applied an ultra-long prompt to generate 42 images with consistent identities, using a window size of 10.
+
+"A photo of a happy hedgehog with its cheese<sup>0</sup>, amid blooming spring flowers<sup>1</sup>, beside a sparkling stream<sup>2</sup>, peeking from a cozy burrow<sup>3</sup>, in an autumn forest<sup>4</sup>, next to a tiny cheese wheel<sup>5</sup>, sitting on a mushroom<sup>6</sup)."
+
+![](images/f75a45d22644f2102fb84db48e9d8e46ca1dd804370c631850112e7bb4781ef8.jpg)
+
+![](images/4595a053cd28576929bdb979626fa0fa924db9441e7e51d1052c73216e3d39cb.jpg)
+
+![](images/e7d4c36eeed619ab220e8324aab7e0d12af0e78964d1c308bfa091a60d47222c.jpg)
+
+![](images/2e5728e5bb486eb3ed97c68b46c95d508f169154d8c91ad77edfe6721dc23247.jpg)
+
+![](images/27d10fd0cf362084b697575385e6e5cc56f63b8390ebc39ee642224bdbcfd9da.jpg)
+
+![](images/3d7cfb437704c9613deafa91e324abaa578fcb2cadd9a19607fe9684764b7ec3.jpg)
+
+"A hyper-realistic digital painting of a young ginger boy with his ball, by an old brick wall covered in colorful graffiti, in the middle of a street filled with cars, near a bustling playground, next to a lake reflecting the early morning light, set against the backdrop of sunset, standing in a quiet meadow, under a cloudy sky."
+
+![](images/0bb78009bc33e686be2cc223c5b9adb76cdc849efc20bea38caec093062a7943.jpg)
+
+![](images/461b002417c36af320a6c77980d1a78f65f0ebae83e13f0f4b684d75264e204f.jpg)
+
+![](images/22d7a40219e1d41a5fc581664552bcbf0ba2466b31e92db3a26e3a53a13beb2c.jpg)
+
+![](images/2f429cfea622b8a43f935ecec49f40ae78e7dbe829de155cb723bb47932ecd9d.jpg)
+
+![](images/8b49c842366d7f1ff39e12aea764a65270a42ed0afaaf4f3c7061b7e7b60e59a.jpg)
+
+![](images/8dd4f04ead246c7bdd1b5684bcb3420b1204a621b3426f7134f301e784a713d6.jpg)
+
+"A cinematic portrait of a man and a woman<sup>0</sup>, in a cozy coffee shop with large windows<sup>1</sup>, walking along a sandy beach at sunset<sup>2</sup>, on a bustling city street at night<sup>3</sup>, on a quiet park bench amidst falling leaves<sup>4</sup>, under an umbrella during a soft rain<sup>5</sup>, in a vibrant art gallery surrounded by paintings<sup>6</sup>."
+
+![](images/01ff02a4ddc9c0a5241bab1fec8c6f81ef8ba1e9a9eca54c970d680e78a5b09b.jpg)  
+Figure 16: Multi-subject story generation. By defining multiple subjects in the identity prompt, our method generates images featuring multiple characters, each maintaining good identity consistency.
+
+![](images/ae0cfff52363b0e91b15fc85b008a14104dfea140ffb9944e88ffc66ace2da0c.jpg)
+
+![](images/5b69d683dc8fb0ea901eb947bcc98034698b8803a0a6730982b7396689316d37.jpg)
+
+![](images/50378a3b5af1c1bb5d1ab46fa1e845486269ceb1b19ad7e594ddf7202e372b71.jpg)
+
+![](images/307eeff4972aa065762e6f23136372bd328429e14a2d06781145b401ce58eafd.jpg)
+
+![](images/6553d5a0aa2f5a4e9037937a960a4d4734df8db9aa1495f5aeb5c0b1211894f5.jpg)
+
+Figure 17: Additional result with PhotoMaker. We compared additional results of our method combined with PhotoMaker, where a lower DreamSim score indicates better ID consistency between the generated images. The results demonstrate that our method has the potential to enhance the performance of PhotoMaker.  
+![](images/9961fddc22afd62e72bd7ce796c6342dfdee5363c1a5f3d59f8b9d1ae77a7020.jpg)  
+"a pixar style illustration of a man, riding a bicycle through the city1, walking her dog along the beach2, writing in her journal at a library3."
+
+![](images/a389290dd6ac72c38c33ef7badab8610041f98832727830a40fdea782dc88232.jpg)  
+"a watarcolor illustration of a kid, flying a kite in a wide open field1, riding tricycle down the driveway2, painting a colorful picture at the kitchen table3."
+
+# D.3 COMBINE WITH DIFFERENT DIFFUSION MODELS.
+
+Since our method exclusively modifies the text-embedding and cross-attention modules of the diffusion model, it can be directly adapted to other diffusion models. In this study, we implemented our approach within the SDXL framework. Other models utilizing the SDXL framework, such as playground-v2.5 $^6$ , RealVisXL_V4.0 $^7$  and Juggernaut-X-v10 $^8$ , can apply our method without any additional modifications or fine-tuning. Our experimental results (see Fig. 20) indicate that these models can also achieve image generation with enhanced identity consistency when employing our method 1Prompt1Story.
+
+<table><tr><td>Metric</td><td>SD1.5</td><td>SDXL</td><td>BLIP-Diffusion</td><td>Textual Inversion</td><td>The Chosen One</td><td>PhotoMaker</td><td>IP-Adapter</td><td>ConsiStory</td><td>Story Diffusion</td><td>NPR</td><td>Ours</td></tr><tr><td>VQAScore↑</td><td>0.7157</td><td>0.8473</td><td>0.5735</td><td>0.6655</td><td>0.6990</td><td>0.8178</td><td>0.7834</td><td>0.8184</td><td>0.8335</td><td>0.8044</td><td>0.8275</td></tr><tr><td>DSG w/ dependency↑</td><td>0.7354</td><td>0.8524</td><td>0.6128</td><td>0.7219</td><td>0.6667</td><td>0.8108</td><td>0.7564</td><td>0.8196</td><td>0.8400</td><td>0.8407</td><td>0.8520</td></tr><tr><td>DSG w/o dependency↑</td><td>0.8095</td><td>0.8961</td><td>0.6909</td><td>0.8051</td><td>0.7495</td><td>0.8700</td><td>0.8122</td><td>0.8696</td><td>0.8853</td><td>0.8863</td><td>0.8945</td></tr><tr><td>FID↓</td><td>-</td><td>-</td><td>65.32</td><td>48.94</td><td>83.74</td><td>55.27</td><td>66.76</td><td>45.20</td><td>51.63</td><td>44.02</td><td>44.16</td></tr></table>
+
+Table 4: Additional metrics comparison. SD1.5 and SDXL are shown as references and excluded from this comparison. The bold and underlined are the best and second best results respectively.
+
+# E ADDITIONAL EXPERIMENTS
+
+# E.1 ADDITIONAL PROMPT ALIGNMENT METRICS
+
+In addition to the primary evaluation metrics, we conduct an experiment using the recent prompt alignment metrics DSG(Cho et al., 2023) and VQAScore(Lin et al., 2025). Both DSG and VQA are metrics that measure the consistency between images and text by evaluating questions and their corresponding answers. These metrics have been shown to provide more reliable strengths in fine-grained diagnosis and align closely with human judgment. We present our comparison with all other methods in Table 4, results show that our method  $IPrompt1Story$  outperforms other training-based methods and achieves the highest value on the DSG metric.
+
+# E.2 VISUAL QUALITY COMPARSION
+
+To evaluate the impact of different methods on image quality under ID consistency generation, we use images generated by the base model as the real dataset and images generated by each method itself as the fake dataset. Then, we calculate the FID(Heusel et al., 2017). As shown in Table 4 (the last row), Naive Prompt Reweighting (NPR) and our method 1Prompt1Story achieved the best and second-best results in terms of FID. This indicates that our method has a smaller impact on the image generation quality of the base model compared to other methods.
+
+# E.3 CONTEXT CONSISTENCY IN TEXT EMBEDDINGS
+
+Besides the separate t-SNE dimensionality reduction conducted for multi-prompt and single-prompt setups in sec. 3.1.1, we extended our analysis by performing a joint t-SNE reduction on the combined text embeddings from both setups. This unified approach allows for a direct visual comparison of the embeddings' spatial arrangements within the text representation space. As illustrated in Fig. 18 (left), the text embeddings originating from the multi-prompt setup remain widely dispersed (red dots), indicative of their diverse semantic properties. Conversely, embeddings from the single-prompt setup (blue dots) exhibit noticeably tighter clustering. To substantiate these observations, we also perform statistical analysis on our benchmark dataset, as shown in Fig. 18 (right).
+
+# F USER STUDY DETAILS
+
+In the user study, we compared our method with three state-of-the-art approaches: IP-Adapter, Consistory, and Story Diffusion. We selected 30 prompt sets from our ConsiStory+ benchmark to generate test images, with each prompt set producing four frames.
+
+In the questionnaire, participants were first provided with guidance on selecting images. They were instructed to choose the set that exhibited the most balanced performance across three criteria: identity consistency, prompt alignment, and image diversity, according to their personal preferences. As illustrated in Fig. 21, we de
+
+tailed these criteria at the beginning of the questionnaire. Additionally, we provided an example to demonstrate our recommended best choice, including justifications for both selecting and not selecting each set, thereby aiding participants in making informed decisions.
+
+![](images/0a7800a193d75d4feb68823e9055abdf36e4cd99cc360add3a5d15de1665701f.jpg)  
+Figure 18: Additional t-SNE visualization of text embeddings (Left) and statistical results (Right).
+
+![](images/68b8ba76e2872c5253379af5b46c20690ea995648a8fb139ec8d7e23761781e8.jpg)  
+Identity Prompt: "A teenage boy with black hair."  
+Figure 19: Long story generation. By using the "sliding window" technique, our method 1Prompt1Story can generate stories of any length with consistent identity throughout.
+
+SDXL: "A vintage-style poster of a vase with flowers<sup>0</sup>, adding charm to homely setting<sup>1</sup>, holding a vibrant arrangement of sunflowers<sup>2</sup>, displaying exotic orchids<sup>3</sup>, containing cherry blossoms<sup>4</sup>, filled with lavender and wild daisies<sup>5</sup>, holding bouquet of flowers<sup>6</sup)."
+
+![](images/6f0562997c1732f7078e4ff5a37972b56e7f6bf20f8165c7357fdfdf78fac14a.jpg)  
+Figure 20: Evaluation with different models. We test our method on various T2I diffusion models, and without requiring fine-tuning, our approach could directly generate images with a consistent identity.
+
+# Selection Guidance
+
+In this survey, you will evaluate four sets of images based on three criteria: "Identity Consistency" "Prompt Alignment" and "Image Diversity". Your task is to select the set that performs best across all three aspects. Identity Consistency: Refers to the visual coherence of the subject's appearance across the set, indicating that the same subject is depicted in all images. Prompt Alignment: Indicates how well each image in the set matches its corresponding text description. Image Diversity: Refers to the variety of poses, object arrangements, and overall composition within the set of images.
+
+# Example
+
+Each row represents one of the four image sets: A, B, C, and D. Each column corresponds to the same frame descriptions: ['wearing a superhero cape', 'at the beach', 'wearing a headscarf', 'wearing a birthday hat'].
+
+![](images/7c95112b2765c2f4d1e8479e2c7f5ecd66646005ec883ac7d564a85980acfc53.jpg)
+
+![](images/c1747e97066b67b200c0d69e4b8cf9494aabff59b46761f12f1122960b62ceae.jpg)
+
+![](images/c0e460f97199f9e9f0a7823dd8dd42aba3de3082ff1127ca8120a5fea539b224.jpg)
+
+![](images/79b5d318af09d492dd031c48e1e7a7dac9d258e5eaa4171638abeb998a2f32c8.jpg)
+
+![](images/6d97e6330aa021d7ca2aa4dc0db70e6d2a972fc4f1f4429dd729d04e6b9c90aa.jpg)
+
+![](images/db0746b1ca39ba9aa2f8c4ac66a2254017f6543773498f7e8d94959bec896b09.jpg)
+
+![](images/850e9411d5c63477d72cb65fdc3302870094b1b886be7b1f5798a940bade07b8.jpg)
+
+![](images/c7a0f4b08e959e1897656270f882242f72c64a1c58d766d497eef2188148962a.jpg)
+
+![](images/5ded00729764d3f08f85086238bf68e55393d3257465ddd4326ca96ab32be797.jpg)  
+Figure 21: User study questionnaire. Before filling out the questionnaire, participants were provided with selection guidelines, including detailed explanations of the three evaluation criteria: identity consistency, prompt alignment, and image diversity. Additionally, an example was provided, along with our recommended best choice and the reasoning behind the selection.
+
+![](images/28b8bda30d0bbe79457b99d5cbe95a3158415f88c0d82df0f873fa69ea0cc296.jpg)
+
+![](images/be86d929027030c13d7119e4dbf9412f9012e7b41c8902e1e7f3324bd197d114.jpg)
+
+![](images/c308a40098c30ebd7c9973a35b18fa9ebeb5dee73eb4d8cbace1f6fb9f74266b.jpg)
+
+![](images/9d3bbf8c69f20adcae9dc4045d41489c0393f485be9e1073bbe29824103d6143.jpg)  
+In this example, the best choice is set A (the first row).
+
+![](images/b5f0555367c4efe4368989d1c815c3069d208117467e586b7f49626d3b9a8e74.jpg)
+
+![](images/842ccf39fd8df76b99df26c0e7313ee3f14aecbdbb480818bd5f6f4d2727c832.jpg)
+
+![](images/d35c8ad4f75d718c560b3426b885431acf6e918c4b4c21cf1bdfad842db6f93c.jpg)
+
+# Reason for Selection
+
+Set A (the first row) performs well in terms of "Identity Consistency," "Text Alignment," and "Image Diversity".
+
+Set B (the second row) is not chosen because its identity consistency is poor.
+
+Set C (the third row) is not selected despite its high identity consistency because its text alignment and image diversity are lacking.
+
+Set D (the fourth row) is also not chosen due to its poor identity consistency.
+
+![](images/4ac1d6cc1439a5d526d001121068a400743ff964e0361233993f9146be721e44.jpg)  
+Figure 22: Additional qualitative comparison. We also compared our method with other existing approaches. The characters generated by vanilla SD1.5 and vanilla SDXL exhibit significant variations in both form and appearance. In contrast, some training-based methods, such as Textual Inversion and The Chosen One, generate characters with consistent forms, but their appearance lacks similarity. While NPR can produce characters with consistent identities, the backgrounds often blend across images. In contrast, our method not only ensures identity consistency but also generates backgrounds that closely align with the corresponding text descriptions.

2025/One-Prompt-One-Story_ Free-Lunch Consistent Text-to-Image Generation Using a Single Prompt/images.zip

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