Add Batch 410ffad4-b404-4ead-95be-522a0b5625f0

ICLR/2025/Flow Distillation Sampling_ Regularizing 3D Gaussians with Pre-trained Matching Priors/cca6b178-ee56-4965-853c-dfcbf289b10a_content_list.json

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-        "type": "text",
-        "text": "FLOW DISTILLATION SAMPLING: REGULARIZING 3D GAUSSIANS WITH PRE-TRAINED MATCHING PRIORS",
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-        "text": "Lin-Zhuo Chen *",
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-        "text": "Kangjie Liu *",
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-        "text": "Youtian Lin",
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-        "text": "Zhihao Li",
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-        "text": "Huawei Noah's Ark Lab",
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-        "text": "Siyu Zhu",
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-        "text": "Xun Cao",
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-        "text": "Yao Yao",
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-        "type": "text",
-        "text": "ABSTRACT",
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-        "type": "text",
-        "text": "3D Gaussian Splatting (3DGS) has achieved excellent rendering quality with fast training and rendering speed. However, its optimization process lacks explicit geometric constraints, leading to suboptimal geometric reconstruction in regions with sparse or no observational input views. In this work, we try to mitigate the issue by incorporating a pre-trained matching prior to the 3DGS optimization process. We introduce Flow Distillation Sampling (FDS), a technique that leverages pre-trained geometric knowledge to bolster the accuracy of the Gaussian radiance field. Our method employs a strategic sampling technique to target unobserved views adjacent to the input views, utilizing the optical flow calculated from the matching model (Prior Flow) to guide the flow analytically calculated from the 3DGS geometry (Radiance Flow). Comprehensive experiments in depth rendering, mesh reconstruction, and novel view synthesis showcase the significant advantages of FDS over state-of-the-art methods. Additionally, our interpretive experiments and analysis aim to shed light on the effects of FDS on geometric accuracy and rendering quality, potentially providing readers with insights into its performance. Project page: https://nju-3dv.github.io/projects/fds.",
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-        "type": "text",
-        "text": "1 INTRODUCTION",
-        "text_level": 1,
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-        "type": "text",
-        "text": "3D Gaussian Splatting (3DGS) (Kerbl et al., 2023) has been widely applied to the field of 3D reconstruction and rendering, including novel view synthesis of static scenes (Kerbl et al., 2023; Yu et al., 2024a), mesh surface reconstruction (Guédon and Lepetit, 2024; Yu et al., 2024b), inverse rendering (Liang et al., 2024; Gao et al., 2023), object detection (Cao et al., 2024), and dynamic 3D reconstruction (Wu et al., 2024; Lin et al., 2024). However, in scenarios with less-observed areas, such as indoor scenes and unbounded scenes, radiance field optimization often suffers from overfitting to these limited input views (Li et al., 2024), resulting in unreliable and corrupted geometry reconstruction.",
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-        "type": "text",
-        "text": "To mitigate the issue, recent research efforts (Li et al., 2024; Paliwal et al., 2024; Turkulainen et al., 2025) have focused on incorporating geometric priors from input views into the training process, thereby regulating the optimization of radiance field represented by 3D Gaussian points. For instance, DN-Splatter (Turkulainen et al., 2025) integrates sensor depth and normal cues into the reconstruction process. However, sensor depth acquisition is costly, and the depth prior information from pre-trained monocular deep models inevitably suffers from the scale ambiguity (Liu et al., 2023b). While the normal prior provides even better geometric details, the scale ambiguity still exists due to its monocular nature.",
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-        "type": "text",
-        "text": "In contrast to monocular priors, pairwise matching priors can provide absolute scale information of the scene. In this paper, we introduce Flow Distillation Sampling (FDS), an online method for distilling matching prior from a pre-trained optical flow model into the 3DGS training process. FDS aims to enhance the geometry quality of Gaussian radiance field by leveraging the matching prior",
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-        "text": "Published as a conference paper at ICLR 2025",
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-        "type": "page_footnote",
-        "text": "*Equal contribution.",
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-        "text": "† Corresponding author.",
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-        "text": "1",
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-        "image_caption": [
-            "Figure 1. Pipeline of the proposed FDS. For each input view, we apply the FDS camera sampling scheme to generate corresponding unobserved sampled view. We then compute Radiance flow base on rendered depth and the Prior flow from matching prior model. Finally the Prior Flow is used to supervise Radiance flow, which enhances the geometric quality of Gaussian Radiance Field."
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-        "type": "text",
-        "text": "into the unobserved novel view. Specifically, we observe that the flow between the input view and the unobserved view generated by the match prior model (i.e., Prior Flow), can guide and refine the flow analytically calculated from the 3DGS geometry (i.e., Radiance Flow), improving the 3DGS reconstruction quality. Moreover, better 3DGS scene will lead to more accurate Prior Flow, creating a mutually reinforcing effect between two computed flow maps. This remains effective even when the radiance field is poorly optimized and the image rendered from the unobserved viewpoint is blurry during training. In addition, a camera sampling scheme is proposed to adaptively control the overlap between input view and sampled view for better Prior Flow calculation, which allows to leverage prior geometric knowledge more profoundly and thereby better enhance the 3DGS reconstruction quality.",
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-        "text": "The proposed FDS has been extensively evaluated on MushRoom (Ren et al., 2024), ScanNet (V2) (Dai et al., 2017), and Replica (Straub et al., 2019) datasets for the task of geometry reconstruction. We apply FDS to two commonly used baseline approaches, namely 3DGS (Kerbl et al., 2023) and 2DGS (Huang et al., 2024). The results demonstrate a significant improvement in geometry reconstruction accuracy. Additionally, we provide interpretive experiments and comprehensive analysis on the effectiveness of FDS, shedding light on its influence on the quality of geometric reconstruction and novel-view rendering. Our contributions are summarized as follows:",
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-            "- FDS leverages matching prior information to recover absolute scale, significantly enhancing the geometric quality of the Gaussian radiance field.",
-            "- An adaptive camera sampling scheme is proposed to selectively choose unobserved views with controllable overlap with the input view, further improving the geometric quality of the Gaussian radiance field even in less-observed areas.",
-            "- By employing FDS and the adaptive camera sampling scheme, we bring significant improvements to state-of-the-art 3D geometry reconstruction approaches."
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-        "text": "2 RELATED WORK",
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-        "text": "Geometry Reconstruction Based on 3DGS: To enhance the geometry of 3DGS representations, some studies (Gao et al., 2024a; Waczynska et al., 2024; Lyu et al., 2024; Chen et al., 2023) focus on integrating mesh or SDF representations with 3DGS. (Gao et al., 2024a) improve the splitting of 3DGS under the guidance of mesh representation and deformation gradients. A hybrid representation that integrates 3DGS with mesh, allowing 3DGS to be modified as a mesh, is introduced in (Waczynska et al., 2024). Mani-GS (Gao et al., 2024b) binds Gaussian and shape-aware triangular",
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-        "text": "Published as a conference paper at ICLR 2025",
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-        "text": "2",
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-        "type": "text",
-        "text": "mesh to manipulate 3DGS directly. 3DGSR (Lyu et al., 2024) combines a signed distance field (SDF) network with 3DGS to enhance geometry quality, aligning the SDF network's geometry with that of 3DGS. NeuSG (Chen et al., 2023) utilizes point clouds from 3DGS to regulate NeuS, while its normals are also used to refine 3DGS. Additionally, NeuSG includes regularizers to ensure that 3DGS remains close to the surface. Some work (Guedon and Lepetit, 2024; Huang et al., 2024; Yu et al., 2024b; Chen et al., 2024) tends to extract geometry from 3DGS directly, benefiting from its fast training and rendering speed. Sugar (Guedon and Lepetit, 2024) proposes an efficient mesh extraction method from 3DGS, aligned with a regularization term during the training process and a refinement strategy. 2DGS (Huang et al., 2024) proposed a 2D surface modeling and two regularization losses which can preserve perspective-correct splattering and enhance geometry reconstruction. A ray-tracing-based volume rendering is introduced in GOF (Yu et al., 2024b), allowing the extraction of geometry from 3DGS directly. PGSR (Chen et al., 2024) proposes unbiased depth rendering, combined with single and multi-view regularization loss to preserve geometric consistency. However, in low-texture and less frequently observed areas, such as indoor scenes, 3DGS still tends to overfit to the limited input views, necessitating regulation through additional prior cues.",
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-        "text": "Prior Regulation for Rendering: Prior information is usually incorporated in ill-posed problems, including sparse view novel view synthesis, dynamic scene reconstruction, and mesh reconstruction. For sparse novel view synthesis tasks, most works utilize the prior depth information (Deng et al., 2022; Roessle et al., 2022; Song et al., 2024; Wang et al., 2023b; Zhu et al., 2023; Xiong et al., 2023; Paliwal et al., 2024), semantic information (Jain et al., 2021; Wynn and Turmukhambetov, 2023; Xiong et al., 2023), and matching information (Paliwal et al., 2024; Lao et al., 2024) to constrain the optimization process of 3DGS. Dynamic scene reconstruction is another challenging task which requires reconstructing the scene geometry and object motion at the same time. Therefore, optical flow priors are crucial as they can help distinguish between camera motion and object motion (Liu et al., 2023b) while providing motion priors between frames (Liu et al., 2023a; Gao et al., 2021; Li et al., 2023; Wang et al., 2023a; Guo et al., 2023; Tian et al., 2023). For the mesh reconstruction task, geometry quality is also enhanced by depth priors (Wei et al., 2021; Yu et al., 2022; Turkulainen et al., 2025) or normal priors (Yu et al., 2022; Turkulainen et al., 2025). However, using optical flow model priors to obtain metric depth priors to help geometric reconstruction, while leveraging unobserved regions to enhance the quality of limited view reconstruction, has not yet been explored.",
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-        "type": "text",
-        "text": "3 METHOD",
-        "text_level": 1,
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-        "type": "text",
-        "text": "Our FDS regulates the optimization of Gaussian radiance field by incorporating matching priors from the pretrained deep model. The generation of Radiance Flow and our proposed FDS loss, along with the equipped camera sampling scheme, are detailed in Sec. 3.1 and Sec. 3.2, respectively.",
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-        "text": "3.1 3D GAUSSIAN SPLATING AND RADIANCE FLOW",
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-        "type": "text",
-        "text": "We utilize 3DGS as an example to demonstrate how Radiance Flow is generated. 3DGS (Kerbl et al., 2023) employs a collection of 3D Gaussians to represent the 3D scene. The expression of  $i$ -th 3D Gaussian distribution shows below:",
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-        "type": "equation",
-        "text": "\n$$\n\\mathcal {G} _ {i} (\\boldsymbol {X}) = e ^ {- \\frac {1}{2} \\left(\\boldsymbol {X} - \\boldsymbol {\\mu} _ {i}\\right) ^ {T} \\boldsymbol {\\Sigma} _ {i} ^ {- 1} \\left(\\boldsymbol {X} - \\boldsymbol {\\mu} _ {i}\\right)}, \\tag {1}\n$$\n",
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-        "type": "text",
-        "text": "where  $\\boldsymbol{\\Sigma}_i = \\boldsymbol{R}_i\\boldsymbol{S}_i\\boldsymbol{S}_i^T\\boldsymbol{R}_i^T$ , and  $\\mu_i$  represents the position of the Gaussian, which is optimized during training. Both  $\\boldsymbol{S}_i$  and  $\\boldsymbol{R}_i$  are represented by a 3D scaling vector  $\\boldsymbol{s}_i$  and a quaternion  $q_i$ , respectively. In addition to the above parameters, each Gaussian  $\\boldsymbol{P}_i$  has extra learnable attributes, including opacity  $\\alpha_i$  and color feature  $\\boldsymbol{f}_i$ .",
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-        "type": "text",
-        "text": "To render a color for pixel  $x$ , a volume rendering based method similar to NeRF (Mildenhall et al., 2021) is adapted:",
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-        "type": "equation",
-        "text": "\n$$\n\\boldsymbol {C} (\\boldsymbol {x}) = \\sum_ {i \\in N} \\boldsymbol {c} _ {i} \\hat {\\alpha} _ {i} \\prod_ {j = 1} ^ {i - 1} (1 - \\hat {\\alpha} _ {j}), \\tag {2}\n$$\n",
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-        "text": "Published as a conference paper at ICLR 2025",
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-        "text": "3",
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-        "type": "text",
-        "text": "where  $c_{i}$  is the color of each Gaussian point, represented by spherical harmonics.  $\\hat{\\alpha}$  is the blending weight for each 2D Gaussian point which is the projection of 3D Gaussian points on the image plane.  $N$  is the number of Gaussian points. Similarly, the depth of pixel  $x$  is rendered using alpha blending.",
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-        "type": "equation",
-        "text": "\n$$\n\\boldsymbol {D} (\\boldsymbol {x}) = \\frac {\\sum_ {i \\in N} d _ {i} \\hat {\\alpha} _ {i} \\prod_ {j = 1} ^ {i - 1} (1 - \\hat {\\alpha} _ {j})}{\\sum_ {i \\in N} \\hat {\\alpha} _ {i} \\prod_ {j = 1} ^ {i - 1} (1 - \\hat {\\alpha} _ {j})}, \\tag {3}\n$$\n",
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-        "text": "where  $d_{i}$  is the distance from the target camera to the Gaussian Point  $\\pmb{P_i}$ . We can also render Radiance Flow between two views of camera  $m, n$  using their camera poses and position of Gaussian points  $\\pmb{\\mu}_{i}$ . As mentioned above, we can project pixel  $\\pmb{x} = (u_{1}, v_{1})$  in  $m$ -th view image to the  $n$ -th view by its corresponding depth and their pose transformation:",
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-        "type": "equation",
-        "text": "\n$$\n\\boldsymbol {D} ^ {n} \\left(u _ {2}, v _ {2}\\right) \\left[ \\begin{array}{l} u _ {2} \\\\ v _ {2} \\\\ 1 \\end{array} \\right] = \\boldsymbol {K} \\boldsymbol {T} _ {\\boldsymbol {m}} ^ {\\boldsymbol {n}} \\boldsymbol {K} ^ {- 1} \\boldsymbol {D} ^ {m} \\left(u _ {1}, v _ {1}\\right) \\left[ \\begin{array}{l} u _ {1} \\\\ v _ {1} \\\\ 1 \\end{array} \\right], \\tag {4}\n$$\n",
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-        "type": "text",
-        "text": "where  $T_{m}^{n}$  is the relative transformation from  $m$ -th view to  $n$ -th view,  $K$  is the intrinsic matrix, and  $D^{m}(u_{1}, v_{1})$  is the rendered depth of  $(u_{1}, v_{1})$  in  $m$ -th view.",
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-        "type": "text",
-        "text": "Next, we calculate  $\\pmb{F}^{m\\rightarrow n}(\\pmb{x})$  for pixel  $\\pmb{x} = (u_{1},v_{1})$  from view  $m$  to view  $n$ :",
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-        "type": "equation",
-        "text": "\n$$\n\\boldsymbol {F} ^ {m \\rightarrow n} (\\boldsymbol {x}) = \\left[\\begin{array}{l}u _ {2} - u _ {1}\\\\v _ {2} - v _ {1}\\end{array}\\right]. \\tag {5}\n$$\n",
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-        "type": "text",
-        "text": "For other type of Gaussian radiance field such as 2DGS (Huang et al., 2024), we only need to replace the alpha blending based depth  $D(x)$  with corresponding formulation.",
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-        "text": "3.2 FLOW DISTILLATION SAMPLING",
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-        "type": "text",
-        "text": "Given a collection of images  $\\{I^i\\}_{i = 1,2,\\dots N}$ , Gaussian Radiance Field typically employs the following loss function for rendering optimization:",
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-        "type": "equation",
-        "text": "\n$$\nL = \\frac {1}{B} \\sum_ {i = 1} ^ {B} (1 - \\lambda) L _ {1} + \\lambda L _ {D - S S I M} + \\lambda_ {\\text {n o r m a l}} L _ {n}, \\tag {6}\n$$\n",
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-        "text": "where  $B$  denotes batch size, and  $L_{n}$  represents the normal consistency loss from (Huang et al., 2024). However, when  $N$  is small, this representation suffers from overfitting (Li et al., 2024; Paliwal et al., 2024).",
-        "bbox": [
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-        "page_idx": 3
-    },
-    {
-        "type": "text",
-        "text": "We propose the Flow Distillation Sampling (FDS) method to incorporate pretrained matching priors into the rendering optimization process, thereby mitigating overfitting and improving rendering performance.",
-        "bbox": [
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-    {
-        "type": "text",
-        "text": "The camera sampling scheme and loss function design in FDS are introduced in Sec. 3.2.1 and Sec. 3.2.2.",
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-    {
-        "type": "text",
-        "text": "3.2.1 CAMERA SAMPLING SCHEME",
-        "text_level": 1,
-        "bbox": [
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-    {
-        "type": "text",
-        "text": "During the training process, FDS randomly samples unobserved camera views nearby the input view and then incorporates the matching priors",
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-    },
-    {
-        "type": "text",
-        "text": "into these views. To utilize the matching prior information between input view and sampled view more profoundly, we propose a camera sampling scheme that ensures sufficient movement to perceive the geometry of the scene while avoiding excessively abrupt motions that could make it difficult for the prior model to match.",
-        "bbox": [
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-    {
-        "type": "image",
-        "img_path": "images/4fe5fe334e79ac367b3337d8c2cc308300f4ff13e4acf4a95c0321c9f53cc599.jpg",
-        "image_caption": [
-            "Figure 2. Explanation of depth-adaptive translation radius. A fixed-radius camera sampling scheme may result in significantly different flow values (Flow 1 and Flow 2) in areas with varying depth ( $d_{1}$  and  $d_{2}$ )."
-        ],
-        "image_footnote": [],
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-        "type": "header",
-        "text": "Published as a conference paper at ICLR 2025",
-        "bbox": [
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-    },
-    {
-        "type": "page_number",
-        "text": "4",
-        "bbox": [
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-    {
-        "type": "text",
-        "text": "Given the world to camera transformation matrix  $T^i$  for the  $i$ -th input view, a small translation disturbance  $t$  and rotation disturbance  $R$  are applied to get the sampled transformation matrix  $T^s$ , expressed as:",
-        "bbox": [
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-        "page_idx": 4
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\n\\boldsymbol {T} ^ {s} = \\left(\\left(\\boldsymbol {T} ^ {i}\\right) ^ {- 1} \\boldsymbol {E}\\right) ^ {- 1} \\tag {7}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
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-        ],
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-    },
-    {
-        "type": "text",
-        "text": "where  $\\pmb{E} = [\\pmb{R},\\pmb{t}]$  denotes the transformation matrix from the sampled view to the input view, and  $t$  is a translation vector satisfying  $|\\pmb{t}| = \\epsilon_t$ ,  $\\pmb{t} = (t_1,t_2,t_3)$ , indicating that unobserved view is sampled along a small spiral circle with a radius of  $\\epsilon_t$ . As noted in (Bian et al., 2021), the rotation flow in image warping is independent of depth. Therefore, we set the rotation part to identity matrix. So that  $E$  is a pure-translation transformation. For the translation radius  $\\epsilon_t$ , as illustrated in Fig. 2, using a uniform radius for all views leads to varying flows due to the differences in depths; the closer the depth, the larger the flow. To ensure flow consistency across all views while maintaining controllable overlap between the input views and sampled views, we hope to implement a depth-adaptive radius, which can help to preserve the same flow between all input views and their sampled views.",
-        "bbox": [
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-        "page_idx": 4
-    },
-    {
-        "type": "text",
-        "text": "According to (4), the pure translation transformation (Bian et al., 2021) between the input view  $i$  and its sampled view  $s$  is shown below:",
-        "bbox": [
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-            335
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-        "page_idx": 4
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\n\\boldsymbol {D} ^ {\\boldsymbol {s}} \\left(u _ {2}, v _ {2}\\right) \\left[ \\begin{array}{l} u _ {2} \\\\ v _ {2} \\\\ 1 \\end{array} \\right] = \\boldsymbol {D} ^ {\\boldsymbol {i}} \\left(u _ {1}, v _ {1}\\right) \\left[ \\begin{array}{l} u _ {1} \\\\ v _ {1} \\\\ 1 \\end{array} \\right] + \\boldsymbol {K} \\left[ \\begin{array}{l} t _ {1} \\\\ t _ {2} \\\\ t _ {3} \\end{array} \\right] \\tag {8}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
-            334,
-            356,
-            825,
-            398
-        ],
-        "page_idx": 4
-    },
-    {
-        "type": "text",
-        "text": "where  $K$  is the intrinsic matrix of the camera, after solving the above equation (Bian et al., 2021), we get:",
-        "bbox": [
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-            443
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-        "page_idx": 4
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\n\\left[ \\begin{array}{l} u _ {2} \\\\ v _ {2} \\end{array} \\right] = \\left[ \\begin{array}{c} \\frac {\\boldsymbol {D} _ {i} \\left(u _ {1} , v _ {1}\\right) u _ {1} + f _ {x} t _ {1} + c _ {x} t _ {3}}{\\boldsymbol {D} _ {i} \\left(u _ {1} , v _ {1}\\right) + t _ {3}} \\\\ \\frac {\\boldsymbol {D} _ {i} \\left(u _ {1} , v _ {1}\\right) u _ {1} + f _ {y} t _ {2} + c _ {x} t _ {3}}{\\boldsymbol {D} _ {i} \\left(u _ {1} , v _ {1}\\right) + t _ {3}} \\end{array} \\right] \\tag {9}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
-            387,
-            463,
-            825,
-            506
-        ],
-        "page_idx": 4
-    },
-    {
-        "type": "text",
-        "text": "We set  $t_3 = 0$  in our camera sampling scheme and assume camera intrinsic parameters:  $f_x \\approx f_y = f$ . The radiance flow  $F^{i \\to s}(u_1, v_1) = \\begin{bmatrix} u_2 - u_1 \\\\ v_2 - v_1 \\end{bmatrix}$  from the input view  $i$  to its sampled view  $s$  is shown below:",
-        "bbox": [
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-            826,
-            578
-        ],
-        "page_idx": 4
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\n\\boldsymbol {F} ^ {\\boldsymbol {i} \\rightarrow \\boldsymbol {s}} \\left(u _ {1}, v _ {1}\\right) = \\left[\\begin{array}{l}\\frac {f}{\\boldsymbol {D} _ {i} \\left(u _ {1} , v _ {1}\\right)} t _ {1}\\\\\\frac {f}{\\boldsymbol {D} _ {i} \\left(u _ {1} , v _ {1}\\right)} t _ {2}\\end{array}\\right] \\tag {10}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
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-            599,
-            825,
-            642
-        ],
-        "page_idx": 4
-    },
-    {
-        "type": "text",
-        "text": "We aim to keep the value of  $||F^{i\\to s}(u_1,v_1)||_2$  constant for the pixel  $x = (u_{1},v_{1})$  during each camera sampling. By setting  $||F^{i\\to s}(u_1,v_1)||_2 = \\sigma$  and incorporating this with (10), we get:",
-        "bbox": [
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-            656,
-            823,
-            686
-        ],
-        "page_idx": 4
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\n\\epsilon_ {t} = \\sqrt {t _ {1} ^ {2} + t _ {2} ^ {2}} = \\sigma \\frac {\\boldsymbol {D} _ {\\boldsymbol {i}} \\left(u _ {1} , v _ {1}\\right)}{f} \\tag {11}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
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-            738
-        ],
-        "page_idx": 4
-    },
-    {
-        "type": "text",
-        "text": "Thus, the radius of translation in our camera sampling is defined as  $\\epsilon_{t} = \\sigma \\frac{D_{i}(u_{1},v_{1})}{f}$  which helps maintain stable flow. The parameter  $\\sigma$  can be tuned as a hyperparameter, which represents the mean radiance flow between the input view and its sampled view. Given that pixel depths vary within an image, we use the mean depth  $\\bar{D}_i$  of the image and set the radius of our translation  $\\epsilon_{t} = \\sigma \\frac{\\bar{D}_{i}}{f}$ .",
-        "bbox": [
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-        "page_idx": 4
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\n\\boldsymbol {t} = \\left[ \\begin{array}{c} \\sigma \\frac {\\bar {D} _ {i}}{f} \\sin (2 \\pi \\xi) \\\\ \\sigma \\frac {\\bar {D} _ {i}}{f} \\cos (2 \\pi \\xi) \\\\ 0 \\end{array} \\right] \\tag {12}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
-            423,
-            838,
-            825,
-            895
-        ],
-        "page_idx": 4
-    },
-    {
-        "type": "text",
-        "text": "Where  $\\xi \\sim U(0,1)$  is a uniform random value between  $[0,1]$  during training.",
-        "bbox": [
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-        "page_idx": 4
-    },
-    {
-        "type": "header",
-        "text": "Published as a conference paper at ICLR 2025",
-        "bbox": [
-            171,
-            32,
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-            47
-        ],
-        "page_idx": 4
-    },
-    {
-        "type": "page_number",
-        "text": "5",
-        "bbox": [
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-        "page_idx": 4
-    },
-    {
-        "type": "text",
-        "text": "3.2.2 FLOW DISTILLATION SAMPLING LOSS",
-        "text_level": 1,
-        "bbox": [
-            171,
-            103,
-            496,
-            118
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "text",
-        "text": "Specifically, to optimize the rendered results of the  $i$ -th view, every iteration, we sample an unobserved view  $s$  near the  $i$ -th input training camera to render the color image  $C^s$ . The Radiance Flow  $F^{i \\to s}(p)$ , guided by Equ. (5), benefits from the introduced matching priors. We define the Prior Flow  $\\overline{F}^{i \\to s}$  as:",
-        "bbox": [
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-            176
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\n\\overline {{F}} ^ {i \\rightarrow s} = \\mathcal {M} _ {\\theta} \\left(I ^ {i}, C ^ {s}\\right), \\tag {13}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
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-            191,
-            825,
-            213
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "text",
-        "text": "where  $\\theta$  represents the parameters of pretrained network  $\\mathcal{M}$ ,  $I^i$  is thergb ground truth of view  $i$ , and  $C^s$  is the rendered image of sampled view  $s$ . Instead of using  $C^i$  to generate Prior Flow, we find that using  $I^i$  can help us to remove the floaters in  $C^s$  as shown in Tab. 4. However, both  $F^{i\\rightarrow s}$  and  $\\overline{F}^{i\\rightarrow s}$  face challenges:",
-        "bbox": [
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-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "list",
-        "sub_type": "text",
-        "list_items": [
-            "-  $F^{i \\to s}$ , derived from Equ. (5), is inaccurate due to the incorrect positioning of Gaussian points.",
-            "-  $\\overline{F}^{i\\rightarrow s}$ , derived from Equ. (13), suffers from the blurred rendering quality of  $C^s$ ."
-        ],
-        "bbox": [
-            215,
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-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "text",
-        "text": "Despite these inaccuracies, we observe that  $\\overline{F}^{i\\rightarrow s}$  is more robust and precise during training compared with  $F^{i\\rightarrow s}$ . Motivated by this observation, we aim to utilize  $\\overline{F}^{i\\rightarrow s}$  to refine  $F^{i\\rightarrow s}$ , which can make  $F^{i\\rightarrow s}$  more accurate, enhancing the positioning of Gaussian points and subsequently improving the rendering quality of  $C^s$ . This process also leads to a more accurate  $\\overline{F}^{i\\rightarrow s}$ . Based on these observations, we propose the FDS loss, which distills matching priors from a pretrained deep model through mutual refinement of the two flows:",
-        "bbox": [
-            169,
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-            455
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\nL _ {f d s} = \\left\\| \\overline {{\\boldsymbol {F}}} ^ {i \\rightarrow s} - \\boldsymbol {F} ^ {i \\rightarrow s} \\right\\| _ {2} \\tag {14}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
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-    },
-    {
-        "type": "text",
-        "text": "Algorithm 1 Flow Distillation Sampling",
-        "text_level": 1,
-        "bbox": [
-            171,
-            511,
-            442,
-            526
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "text",
-        "text": "Input: A batch of input training image:  $\\{I_i\\}_{i=1}^N$ , Transformation Matrix:  $\\{T_i\\}_{i=1}^N$ , Prior Matching Network  $\\mathcal{M}_{\\theta}$ , Gaussian Points  $\\{P_i\\}_{i=1}^M$  with  $\\{r_i, t_i, f_i, \\mu_i, \\alpha_i\\}$ .",
-        "bbox": [
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-            531,
-            823,
-            563
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "text",
-        "text": "Output:  $L_{fds}$",
-        "bbox": [
-            173,
-            563,
-            272,
-            575
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "list",
-        "sub_type": "text",
-        "list_items": [
-            "1: for  $i$  in  $\\{1,2,\\dots,B\\}$  do",
-            "2:  $\\xi \\sim U(0,1), R \\gets I$",
-            "3:  $t_1 \\gets \\sigma \\frac{\\pmb{D}_i}{f} \\sin(2\\pi \\xi), t_2 \\gets \\sigma \\frac{\\pmb{D}_i}{f} \\cos(2\\pi \\xi), t_3 \\gets 0$",
-            "4:  $\\pmb{E} \\gets [\\pmb{R}, \\pmb{t}], \\pmb{T}^s \\gets ((\\pmb{T}^i)^{-1}\\pmb{E})^{-1}$",
-            "5:  $C^s, D^s \\gets \\text{Render}(\\pmb{T}^s, \\pmb{P})$",
-            "6:  $C^i, D^i \\gets \\text{Render}(\\pmb{T}^i, \\pmb{P})$",
-            "7:  $X^{s} \\gets K T_{i}^{s} K^{-1} D^{i}(X^{i}) X^{i} / D(X^{s})$",
-            "8:  $F^{i\\to s}\\gets X^s -X^i$",
-            "9:  $\\overline{F}^{i\\to s}\\gets \\mathcal{M}_{\\theta}(I^i,C^s)$",
-            "10:  $L_{fds} \\gets L_{fds} + 1 / B||\\overline{F}^{i \\to s} - F^{i \\to s}||_2$",
-            "11: return  $L_{fds}$"
-        ],
-        "bbox": [
-            176,
-            577,
-            552,
-            750
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "text",
-        "text": "To maintain training stability and reduce computational complexity, we detach  $\\overline{F}^{i\\rightarrow s}$  from  $P_{i}$  when we calculate loss. This prevents the propagation of gradients, as computing them is resource-intensive and time-consuming, as noted in (Poole et al., 2022). Additionally, this helps to ensure that  $\\overline{F}^{i\\rightarrow s}$  is not directly influenced by the less reliable  $F^{i\\rightarrow s}$ .",
-        "bbox": [
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-        "page_idx": 5
-    },
-    {
-        "type": "text",
-        "text": "In summary, the procedures of our proposed Flow Distillation Sampling are presented in Algorithm 1. The overall training loss is:",
-        "bbox": [
-            169,
-            839,
-            826,
-            869
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "equation",
-        "text": "\n$$\nL = \\frac {1}{B} \\sum_ {i = 1} ^ {B} (1 - \\lambda) L _ {1} + \\lambda L _ {D - S S I M} + \\lambda_ {\\text {n o r m a l}} L _ {n} + \\lambda_ {f d s} L _ {f d s}. \\tag {15}\n$$\n",
-        "text_format": "latex",
-        "bbox": [
-            281,
-            886,
-            825,
-            926
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "header",
-        "text": "Published as a conference paper at ICLR 2025",
-        "bbox": [
-            171,
-            32,
-            478,
-            47
-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "page_number",
-        "text": "6",
-        "bbox": [
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-        ],
-        "page_idx": 5
-    },
-    {
-        "type": "image",
-        "img_path": "images/3afe1218b31c40d771666c443cc31158e1c0dc7d681413879e63d0e04852d393.jpg",
-        "image_caption": [
-            "Figure 3. Comparison of depth reconstruction on Mushroom and ScanNet datasets. The original 3DGS or 2DGS model equipped with FDS can remove unwanted floaters and reconstruct geometry more precisely."
-        ],
-        "image_footnote": [],
-        "bbox": [
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-        ],
-        "page_idx": 6
-    },
-    {
-        "type": "text",
-        "text": "4 EXPERIMENTS",
-        "text_level": 1,
-        "bbox": [
-            171,
-            508,
-            328,
-            523
-        ],
-        "page_idx": 6
-    },
-    {
-        "type": "text",
-        "text": "4.1 SETUPS",
-        "text_level": 1,
-        "bbox": [
-            171,
-            540,
-            269,
-            553
-        ],
-        "page_idx": 6
-    },
-    {
-        "type": "text",
-        "text": "4.1.1 IMPLEMENTATION DETAILS",
-        "text_level": 1,
-        "bbox": [
-            171,
-            566,
-            419,
-            579
-        ],
-        "page_idx": 6
-    },
-    {
-        "type": "text",
-        "text": "We apply our FDS method to two types of 3DGS: the original 3DGS, and 2DGS (Huang et al., 2024). The number of iterations in our optimization process is 35,000. We follow the default training configuration and apply our FDS method after 15,000 iterations, then we add normal consistency loss for both 3DGS and 2DGS after 25000 iterations. The weight for FDS,  $\\lambda_{fds}$ , is set to 0.015, the  $\\sigma$  is set to 23, and the weight for normal consistency is set to 0.15 for all experiments. We removed the depth distortion loss in 2DGS because we found that it degrades its results in indoor scenes. The Gaussian point cloud is initialized using Colmap for all datasets. We tested the impact of using Sea Raft (Wang et al., 2024) and Raft (Teed and Deng, 2020) on FDS performance. Due to the blurriness of the ScanNet dataset, additional prior constraints are required. Thus, we incorporate normal prior supervision on the rendered normals in ScanNet (V2) dataset by default. The normal prior is predicted by the Stable Normal model (Ye et al., 2024) across all types of 3DGS. The entire framework is implemented in PyTorch (Paszke et al., 2019), and all experiments are conducted on a single NVIDIA 4090D GPU.",
-        "bbox": [
-            169,
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-        "page_idx": 6
-    },
-    {
-        "type": "text",
-        "text": "4.1.2 DATASETS AND METRICS",
-        "text_level": 1,
-        "bbox": [
-            171,
-            787,
-            405,
-            801
-        ],
-        "page_idx": 6
-    },
-    {
-        "type": "text",
-        "text": "We evaluate our method for 3D reconstruction and novel view synthesis tasks on Mushroom (Ren et al., 2024), ScanNet (v2) (Dai et al., 2017), and Replica (Straub et al., 2019) datasets, which feature challenging indoor scenes with both sparse and dense image sampling. The Mushroom dataset is an indoor dataset with sparse image sampling and two distinct camera trajectories. We train our model on the training split of the long capture sequence and evaluate novel view synthesis on the test split of the long capture sequences. Five scenes are selected to evaluate our FDS, including \"coffee room\", \"honka\", \"kokko\", \"sauna\", and \"vr room\". ScanNet(V2) (Dai et al., 2017) consists of 1,613 indoor scenes with annotated camera poses and depth maps. We select 5 scenes from the ScanNet (V2)",
-        "bbox": [
-            169,
-            811,
-            826,
-            925
-        ],
-        "page_idx": 6
-    },
-    {
-        "type": "header",
-        "text": "Published as a conference paper at ICLR 2025",
-        "bbox": [
-            173,
-            32,
-            478,
-            47
-        ],
-        "page_idx": 6
-    },
-    {
-        "type": "page_number",
-        "text": "7",
-        "bbox": [
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-        "page_idx": 6
-    },
-    {
-        "type": "table",
-        "img_path": "images/170aa9ab285eb9ab83655936765c4eb30d8267f03050b405a98ee47ade945889.jpg",
-        "table_caption": [
-            "Table 1. 3D Reconstruction and novel view synthesis results on Mushroom dataset. * Represents that FDS uses the Raft model."
-        ],
-        "table_footnote": [],
-        "table_body": "<table><tr><td>Method</td><td>Acc↓</td><td>Comp ↓</td><td>C-L1 ↓</td><td>NC ↑</td><td>F-Score ↑</td><td>Abs Rel ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>Time ↓</td></tr><tr><td>GOF</td><td>0.1812</td><td>0.1093</td><td>0.1453</td><td>0.6292</td><td>0.3665</td><td>0.2380</td><td>21.37</td><td>0.7762</td><td>0.3132</td><td>≈1.4h</td></tr><tr><td>PGSR</td><td>0.0971</td><td>0.1420</td><td>0.1196</td><td>0.7193</td><td>0.5105</td><td>0.1723</td><td>22.13</td><td>0.7773</td><td>0.2918</td><td>≈1.2h</td></tr><tr><td>3DGS</td><td>0.1167</td><td>0.1033</td><td>0.1100</td><td>0.7954</td><td>0.3739</td><td>0.1214</td><td>24.18</td><td>0.8392</td><td>0.2511</td><td>≈0.8h</td></tr><tr><td>3DGS + FDS*</td><td>0.0569</td><td>0.0676</td><td>0.0623</td><td>0.8105</td><td>0.6573</td><td>0.0603</td><td>24.72</td><td>0.8489</td><td>0.2379</td><td>≈1.3h</td></tr><tr><td>3DGS + FDS</td><td>0.0527</td><td>0.0565</td><td>0.0546</td><td>0.8178</td><td>0.6958</td><td>0.0568</td><td>24.76</td><td>0.8486</td><td>0.2381</td><td>≈1.3h</td></tr><tr><td>2DGS</td><td>0.1078</td><td>0.0850</td><td>0.0964</td><td>0.7835</td><td>0.5170</td><td>0.1002</td><td>23.56</td><td>0.8166</td><td>0.2730</td><td>≈0.8h</td></tr><tr><td>2DGS + FDS*</td><td>0.0689</td><td>0.0646</td><td>0.0667</td><td>0.8042</td><td>0.6582</td><td>0.0589</td><td>23.98</td><td>0.8255</td><td>0.2621</td><td>≈1.3h</td></tr><tr><td>2DGS + FDS</td><td>0.0615</td><td>0.0534</td><td>0.0574</td><td>0.8151</td><td>0.6974</td><td>0.0561</td><td>24.06</td><td>0.8271</td><td>0.2610</td><td>≈1.3h</td></tr></table>",
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-            "Table 2. 3D Reconstruction and novel view synthesis results on ScanNet dataset."
-        ],
-        "table_footnote": [],
-        "table_body": "<table><tr><td>Method</td><td>Acc ↓</td><td>Comp ↓</td><td>C-L1 ↓</td><td>NC ↑</td><td>F-Score ↑</td><td>Abs Rel ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td></tr><tr><td>GOF</td><td>1.8671</td><td>0.0805</td><td>0.9738</td><td>0.5622</td><td>0.2526</td><td>0.1597</td><td>21.55</td><td>0.7575</td><td>0.3881</td></tr><tr><td>PGSR</td><td>0.2928</td><td>0.5103</td><td>0.4015</td><td>0.5567</td><td>0.1926</td><td>0.1661</td><td>21.71</td><td>0.7699</td><td>0.3899</td></tr><tr><td>3DGS</td><td>0.4867</td><td>0.1211</td><td>0.3039</td><td>0.7342</td><td>0.3059</td><td>0.1227</td><td>22.19</td><td>0.7837</td><td>0.3907</td></tr><tr><td>3DGS + FDS</td><td>0.2458</td><td>0.0787</td><td>0.1622</td><td>0.7831</td><td>0.4482</td><td>0.0573</td><td>22.83</td><td>0.7911</td><td>0.3826</td></tr><tr><td>2DGS</td><td>0.2658</td><td>0.0845</td><td>0.1752</td><td>0.7504</td><td>0.4464</td><td>0.0831</td><td>22.59</td><td>0.7881</td><td>0.3854</td></tr><tr><td>2DGS + FDS</td><td>0.1457</td><td>0.0679</td><td>0.1068</td><td>0.7883</td><td>0.5459</td><td>0.0432</td><td>22.91</td><td>0.7928</td><td>0.3800</td></tr></table>",
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-        "type": "text",
-        "text": "dataset, uniformly sampling one-tenth of the views, following the approach in (Guo et al., 2022). To further improve the geometry rendering quality of 3DGS, Replica (Straub et al., 2019) contains small-scale real-world indoor scans. We evaluate our FDS on five scenes from Replica: office0, office1, office2, office3 and office4, selecting one-tenth of the views for training. The results for Replica are provided in the supplementary materials. To evaluate the rendering quality and geometry of 3DGS, we report PSNR, SSIM, and LPIPS for rendering quality, along with Absolute Relative Distance (Abs Rel) as a depth quality metrics. Additionally, for mesh evaluation, we use metrics including Accuracy, Completion, Chamfer-L1 distance, Normal Consistency, and F-scores.",
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-        "text": "4.2 RESULTS",
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-        "text": "4.2.1 DEPTH RENDERING AND NOVEL VIEW SYNTHESIS",
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-        "text": "The comparison results on Mushroom and ScanNet are presented in Tab. 1 and Tab. 2, respectively. Due to the sparsity of sampling in the Mushroom dataset, challenges are posed for both GOF (Yu et al., 2024b) and PGSR (Chen et al., 2024), leading to their relative poor performance on the Mushroom dataset. Our approach achieves the best performance with the FDS method applied during the training process. The FDS significantly enhances the geometric quality of 3DGS on the Mushroom dataset, improving the \"abs rel\" metric by more than  $50\\%$ . We found that Sea Raft (Wang et al., 2024) outperforms Raft (Teed and Deng, 2020) on FDS, indicating that a better optical flow model can lead to more significant improvements. Additionally, the render quality of RGB images shows a slight improvement, by 0.58 in 3DGS and 0.50 in 2DGS, benefiting from the incorporation of cross-view consistency in FDS. On the Mushroom dataset, adding the FDS loss increases the training time by half an hour, which maintains the same level as baseline. Similarly, our method shows a notable improvement on the ScanNet dataset as well using Sea Raft (Wang et al., 2024) Model. The \"abs rel\" metric in 2DGS is improved nearly  $50\\%$ . This demonstrates the robustness and effectiveness of the FDS method across different datasets.",
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-        "type": "text",
-        "text": "The qualitative comparisons on the Mushroom and ScanNet dataset are illustrated in Fig. 3. As seen in the first row of Fig. 3, both the original 3DGS and 2DGS suffer from overfitting, leading to corrupted geometry generation. Our FDS effectively mitigates this issue by supervising the matching relationship between the input and sampled views, helping to recover the geometry. FDS also improves the refinement of geometric details, as shown in other rows. By incorporating the matching prior through FDS, the quality of the rendered depth is significantly improved.",
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-        "text": "4.2.2 MESH EXTRACTION",
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-        "type": "text",
-        "text": "To further demonstrate the improvement in geometry quality, we applied methods used in (Turku-lainen et al., 2025) to extract meshes from the input views of optimized 3DGS and 2DGS. The",
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-        "text": "Published as a conference paper at ICLR 2025",
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-        "text": "8",
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-            "Table 3. Ablation study on geometry priors."
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-        "table_body": "<table><tr><td>Method</td><td>Acc↓</td><td>Comp ↓</td><td>C-L1 ↓</td><td>NC ↑</td><td>F-Score ↑</td><td>Abs Rel ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td></tr><tr><td>2DGS</td><td>0.1078</td><td>0.0850</td><td>0.0964</td><td>0.7835</td><td>0.5170</td><td>0.1002</td><td>23.56</td><td>0.8166</td><td>0.2730</td></tr><tr><td>2DGS+Depth</td><td>0.0862</td><td>0.0702</td><td>0.0782</td><td>0.8153</td><td>0.5965</td><td>0.0672</td><td>23.92</td><td>0.8227</td><td>0.2619</td></tr><tr><td>2DGS+MVDepth</td><td>0.2065</td><td>0.0917</td><td>0.1491</td><td>0.7832</td><td>0.3178</td><td>0.0792</td><td>23.74</td><td>0.8193</td><td>0.2692</td></tr><tr><td>2DGS+Normal</td><td>0.0939</td><td>0.0637</td><td>0.0788</td><td>0.8359</td><td>0.5782</td><td>0.0768</td><td>23.78</td><td>0.8197</td><td>0.2676</td></tr><tr><td>2DGS+FDS</td><td>0.0615</td><td>0.0534</td><td>0.0574</td><td>0.8151</td><td>0.6974</td><td>0.0561</td><td>24.06</td><td>0.8271</td><td>0.2610</td></tr><tr><td>2DGS+Depth+FDS</td><td>0.0561</td><td>0.0519</td><td>0.0540</td><td>0.8295</td><td>0.7282</td><td>0.0454</td><td>24.22</td><td>0.8291</td><td>0.2570</td></tr><tr><td>2DGS+Normal+FDS</td><td>0.0529</td><td>0.0450</td><td>0.0490</td><td>0.8477</td><td>0.7430</td><td>0.0443</td><td>24.10</td><td>0.8283</td><td>0.2590</td></tr><tr><td>2DGS+Depth+Normal</td><td>0.0695</td><td>0.0513</td><td>0.0604</td><td>0.8540</td><td>0.6723</td><td>0.0523</td><td>24.09</td><td>0.8264</td><td>0.2575</td></tr><tr><td>2DGS+Depth+Normal+FDS</td><td>0.0506</td><td>0.0423</td><td>0.0464</td><td>0.8598</td><td>0.7613</td><td>0.0403</td><td>24.22</td><td>0.8300</td><td>0.0403</td></tr></table>",
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-            "Figure 4. The error map of Radiance Flow and Prior Flow. RF: Radiance Flow, PF: Prior Flow, * means that there is no FDS loss supervision during optimization."
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-        "type": "text",
-        "text": "comparison results are presented in Tab. 1. With the integration of FDS, the mesh quality is significantly enhanced compared to the baseline, featuring fewer floaters and more well-defined shapes.",
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-        "text": "4.3 ABLATION STUDY",
-        "text_level": 1,
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-        "text": "Ablation study on geometry priors: To highlight the advantage of incorporating matching priors, we incorporated various types of priors generated by different models into 2DGS. These include a monocular depth estimation model (Depth Anything v2) (Yang et al., 2024), a two-view depth estimation model (Unimatch) (Xu et al., 2023), and a monocular normal estimation model (StableNormal) (Ye et al., 2024). We adapt the scale and shift-invariant loss in Midas (Birkl et al., 2023) for monocular depth supervision and L1 loss for two-view depth supervision. We use Sea Raft (Wang et al., 2024) as our default optical flow model. The comparison results on Mushroom dataset are shown in Tab. 3. We observe that the normal prior provides accurate shape information, enhancing the geometric quality of the radiance field. The multi-view depth prior, hindered by the limited feature overlap between input views, fails to offer reliable geometric information. We test average \"Abs Rel\" of multi-view depth prior, and the result is 0.19, which performs worse than the \"Abs Rel\" results rendered by original 2DGS. From the results, it can be seen that depth order information provided by monocular depth improves reconstruction accuracy. Meanwhile, our FDS achieves the best performance among all the priors, and by integrating all three components, we obtained the optimal results.",
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-        "text": "Ablation study on FDS: In this section, we present the design of our FDS method through an ablation study on the Mushroom dataset to validate its effectiveness. The optional configurations of FDS are outlined in Tab. 4. Our base model is the 2DGS equipped with FDS, and its results are shown in the first row. The goal of this analysis is to evaluate the impact of various strategies on FDS sampling and loss design. We observe that when we replace  $I_{i}$  in (13) with  $C_i$ , as shown in the second row, the geometric quality of 2DGS deteriorates. Using  $I_{i}$  instead of  $C_i$  help us to remove the",
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-        "text": "Published as a conference paper at ICLR 2025",
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-        "text": "9",
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-        "table_caption": [
-            "Table 4. Ablation study on FDS strategies."
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-        "table_footnote": [],
-        "table_body": "<table><tr><td colspan=\"2\">Mθ(X, Cs)</td><td colspan=\"3\">Loss</td><td colspan=\"3\">Metric</td></tr><tr><td>X = Ci</td><td>X = Ii</td><td>Next Input view</td><td>Sampled view</td><td>Fixed Sampled view</td><td>Abs Rel ↓</td><td>F-score ↑</td><td>NC ↑</td></tr><tr><td></td><td>✓</td><td></td><td>✓</td><td></td><td>0.0561</td><td>0.6974</td><td>0.8151</td></tr><tr><td>✓</td><td></td><td></td><td>✓</td><td></td><td>0.0839</td><td>0.6242</td><td>0.8030</td></tr><tr><td></td><td>✓</td><td>✓</td><td></td><td></td><td>0.0877</td><td>0.6091</td><td>0.7614</td></tr><tr><td></td><td>✓</td><td></td><td></td><td>✓</td><td>0.0724</td><td>0.6312</td><td>0.8015</td></tr></table>",
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-        "text": "floaters in  $C^s$ , which are also remained in  $C^i$ . We also experiment with modifying the FDS loss. For example, in the third row, we use the next training input view as the sampling view, and replace the render result of next training view with ground truth image of its input view. Due to the significant movement between images, the Prior Flow fails to accurately match the pixel between them, leading to a further degradation in geometric quality. Finally, we attempt to fix the sampling view and found that this severely damaged the geometric quality, indicating that random sampling is essential for the stability of the mean error in the Prior Flow.",
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-            "Figure 5. Limitation of FDS."
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-        "type": "text",
-        "text": "Interpretive Experiments: To demonstrate the mutual refinement of two flows in our FDS, for each view, we sample the unobserved views multiple times to compute the mean error of both Radiance Flow and Prior Flow. We use Raft (Teed and Deng, 2020) as our default optical flow model for visualization. The ground truth flow is calculated based on Eq. (4) and Eq. (5) utilizing ground truth depth in dataset. We introduce our FDS loss after 16000 iterations during optimization of 2DGS. The error maps are shown in Fig. 4. Our analysis reveals that Radiance Flow tends to exhibit significant geometric errors, whereas Prior Flow can more accurately estimate the geometry, effectively disregarding errors introduced by floating Gaussian points.",
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-        "text": "4.4 LIMITATION AND FURTHER WORK",
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-        "type": "text",
-        "text": "Firstly, our FDS faces challenges in scenes with significant lighting variations between different views, as shown in the lamp of first row in Fig. 5. Incorporating exposure compensation into FDS could help address this issue. Additionally, our method struggles with reflective surfaces and motion blur, leading to incorrect matching. In the future, we plan to explore the potential of FDS in monocular video reconstruction tasks, using only a single input image at each time step.",
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-        "text": "5 CONCLUSIONS",
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-        "text": "In this paper, we propose Flow Distillation Sampling (FDS), which leverages the matching prior between input views and sampled unobserved views from the pretrained optical flow model, to improve the geometry quality of Gaussian radiance field. Our method can be applied to different approaches (3DGS and 2DGS) to enhance the geometric rendering quality of the corresponding neural radiance fields. We apply our method to the 3DGS-based framework, and the geometry is enhanced on the Mushroom, ScanNet, and Replica datasets.",
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-        "text": "This work was supported by National Key R&D Program of China (2023YFB3209702), the National Natural Science Foundation of China (62441204, 62472213), and Gusu Innovation & Entrepreneurship Leading Talents Program (ZXL2024361)",
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-        "text": "Lijie Fan $^{1,*}$  Tianhong Li $^{2,\\dagger}$  Siyang Qin $^{1,\\dagger}$  Yuanzhen Li $^{1}$  Chen Sun $^{1}$  Michael Rubinstein $^{1}$  Deqing Sun $^{1}$  Kaiming He $^{2}$  Yonglong Tian $^{1,*}$",
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-        "text": "Scaling up autoregressive models in vision has not proven as beneficial as in large language models. In this work, we investigate this scaling problem in the context of text-to-image generation, focusing on two critical factors: whether models use discrete or continuous tokens, and whether tokens are generated in a random or fixed raster order using BERT- or GPT-like transformer architectures. Our empirical results show that, while all models scale effectively in terms of validation loss, their evaluation performance—measured by FID, GenEval score, and visual quality—follows different trends. Models based on continuous tokens achieve significantly better visual quality than those using discrete tokens. Furthermore, the generation order and attention mechanisms significantly affect the GenEval score: random-order models achieve notably better GenEval scores compared to raster-order models. Inspired by these findings, we train Fluid, a random-order autoregressive model on continuous tokens. Fluid 10.5B model achieves a new state-of-the-art zero-shot FID of 6.16 on MS-COCO 30K, and 0.69 overall score on the GenEval benchmark. We hope our findings and results will encourage future efforts to further bridge the scaling gap between vision and language models.",
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-        "type": "text",
-        "text": "Scaling laws underpin the unprecedented success of large language models (LLMs). Empirically, increasing the number of parameters in autoregressive models consistently leads to significant performance improvements and the emergence of new capabilities in natural language processing (NLP) tasks (Devlin et al., 2018; Radford et al., 2018; Brown et al., 2020; Kaplan et al., 2020; Wei et al., 2022). This empirical relationship has inspired numerous efforts to scale up language models, resulting in the development of many highly capable models (Bubeck et al., 2023; Team et al., 2023; Achiam et al., 2023).",
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-        "text": "Encouraged by this success, many attempts have been made to adopt and scale up autoregressive models in computer vision, particularly for generative tasks like text-to-image generation (Yu et al., 2021; 2023; Bai et al., 2024; El-Nouby et al., 2024). However, the performance and visual quality of content generated by these models often fall short compared to other generative models, such as diffusion models (Ho et al., 2020; Sahara et al., 2022; Rombach et al., 2022a; Esser et al., 2024), leaving it unclear whether similar scaling laws apply to the vision domain.",
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-        "text": "We propose several hypotheses for the performance gap. First, the vector quantization (VQ) (van den Oord et al., 2017) step, which is required for most visual autoregressive models, may introduce significant information loss, ultimately limiting model performance. Second, unlike the inherently sequential nature of language, generating visual content might benefit more from a different autoregressive prediction order. Third, there is often a confusion between two levels of generalizability when evaluating scaling laws in vision models: (a) generalization to new data using the same metric as the training loss (commonly referred to as validation loss), and (b) generalization to a new metric or problem different from the training objective, such as FID (Heusel et al., 2017), the GenEval benchmark (Ghosh et al., 2024), or visual quality. We hypothesize that power-law scaling (Kaplan et al., 2020) applies to (a) for autoregressive models on vision data, but not necessarily to (b).",
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-        "type": "text",
-        "text": "To investigate these hypotheses, we conduct a comprehensive empirical study on the scaling behavior of autoregressive models in the context of text-to-image generation. Specifically, we explore two key factors: whether the model operates on continuous or discrete tokens, and whether tokens are generated in a random or fixed raster order. To this end, we utilize the Diffusion Loss (Li et al., 2024) to make autoregressive models compatible with continuous tokens. We generalize BERT-like vision model MaskGIT (Chang et al., 2022) as random-order autoregression, as it conceptually predicts output tokens in a randomized order while retaining the autoregressive nature of \"predicting next tokens based on known ones\". We analyze the behavior of four autoregressive variants, each employing different combinations of these two factors. We scale their parameters from 150M to 3B and evaluate their performance using three metrics: validation loss, FID (Heusel et al., 2017), and GenEval score (Ghosh et al., 2024). We also inspect the visual quality of the generated images.",
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-        "type": "text",
-        "text": "Our experiments indicate that VQ-based models, regardless of whether they use a random or fixed raster order, exhibit a slower improvement in FID scores when scaling up model size compared to models operating on continuous tokens. VQ models also produce images of lower visual quality, likely due to information loss introduced by vector quantization.",
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-        "type": "text",
-        "text": "Furthermore, the token generation order and the associated attention mechanism primarily influence the global structure of the generated image. In the GenEval benchmark, random-order models with bidirectional attention significantly outperform raster-order models with causal attention, particularly when generating multiple objects. Random-order models can readjust the global structure at every prediction step, whereas raster-order models cannot. This suggests that the token generation order plays a crucial role in achieving better text-to-image alignment.",
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-        "type": "text",
-        "text": "Our experiments also demonstrate that validation loss scales as a power-law with model size, no matter whether the model operates with continuous or discrete tokens. This implies scalable behavior at the level of (a) generalization—generalizing to new data using the same metric as the training loss—which aligns with observations in language models (Kaplan et al., 2020). However, as for generalizing to a different metric, such as FID or GenEval score, although performance consistently improves with better validation loss, the trend may not follow a strict power-law.",
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-        "type": "text",
-        "text": "Building on these findings, we scale the Fluid model, i.e., random-order model with continuous tokens, up to 10.5B parameters and train it using the WebLI dataset (Chen et al., 2022). The resulting Fluid 10.5B model achieves a zero-shot FID of 6.16 on MS-COCO and a GenEval overall score of",
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-        "text": "Published as a conference paper at ICLR 2025",
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-        "type": "page_number",
-        "text": "2",
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-        "type": "text",
-        "text": "0.69, comparing favorably with leading text-to-image generative models such as DALL-E 3 (Betker et al., 2023) and Stable Diffusion 3 (Esser et al., 2024). We hope that our empirical findings and positive results could shed light on the scaling behavior for text-to-image generation models and further innovation in this frontier.",
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-        "type": "text",
-        "text": "2 RELATED WORK",
-        "text_level": 1,
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-        "type": "text",
-        "text": "Text-to-image diffusion models. The dominant approaches are based on diffusion models. Dalle 2 (Ramesh et al., 2022), Imagen (Ho et al., 2022), and Stable Diffusion (Rombach et al., 2022a) revolutionized text-to-image generation. Most recently, SD v3 (Esser et al., 2024) and Imagen3 (Baldridge et al., 2024) can generate realistic images that are hard to tell from real images. However, generating samples is usually computationally expensive due to the multiple forward passes.",
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-        "type": "text",
-        "text": "Autoregressive (AR) models. While being the defacto model for language modeling, AR models are lagging behind diffusion models for text-to-image generation. AR models (Yu et al., 2022; Chang et al., 2023; Li et al., 2023; Yu et al., 2023) are often used together with discrete tokenizers (van den Oord et al., 2017; Esser et al., 2021), which often limit the modeling capability. For example, Parti (Yu et al., 2022) scales up the model to 20B, which obtains a slightly lower/better FID score of 7.23 on MS-COCO than 7.27 by the 3.4B diffusion-based Imagen. Here we find that, with a continuous tokenizer, our small 369M model can already achieve the same FID score as Parti with 20B model.",
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-        "type": "text",
-        "text": "Recently Li et al. (2024) challenges the conventional wisdom and replaces the discrete tokenizer with a continuous tokenizer via a diffusion loss. They introduced masked autoregressive (MAR) model obtains strong results for class-conditioning generation on ImageNet. However, scaling MAR models for text-to-image generation is unexplored. Here we empirically study the scaling behavior for MAR and report several findings important for the research community.",
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-        "type": "text",
-        "text": "Scaling language models. Kaplan et al. (2020) empirically observed that, for language model, the validation loss scales as a power-law with model size, dataset size, and the amount of compute used for training. Hoffmann et al. (2022) discovered that contemporary LLMs are under-trained, and that for compute-optimal training, the model size and the number of training tokens should be scaled equally. Under the same compute budget while using 4x more more data, their 70B Chinchilla outperforms much larger models, such as the 530B Megatron-Turing NLG (Smith et al., 2022). Wei et al. (2022) found that larger models have emergent abilities that are not present in smaller models. These observations have inspired significant efforts to scale up language models to trillions of parameters (Achiam et al., 2023; Team et al., 2023; Dubey et al., 2024).",
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-    {
-        "type": "text",
-        "text": "Scaling vision models. Similar scaling law has been obscure for computer vision. For recognition models (Tan, 2019; Zhai et al., 2022; He et al., 2022; Dehghani et al., 2023), scaling often comes with diminishing returns. For example, Dehghani et al. (2023) scale ViT up from 3B to 22B but only observed  $0.25\\%$  accuracy increase in ImageNet linear probing. The scaling of generative models is more promising - DiT (Peebles & Xie, 2023) shows consistent improvement in generation quality when scaling up compute and model size (despite only up to 600M). The follow-up Sora (OpenAI, 2024) further shows the potential to scale up for video generation. In this paper, we perform a comprehensive empirical study of AR models for text-to-image generation.",
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-        "type": "text",
-        "text": "3 PRELIMINARY: AUTOREGRESSIVE IMAGE GENERATION",
-        "text_level": 1,
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-        "type": "text",
-        "text": "Given a sequence of tokens  $\\{x^1, x^2, \\ldots, x^n\\}$  where the superscript  $1 \\leq i \\leq n$  specifies an order, autoregressive models (Gregor et al., 2014; van den Oord et al., 2016b;a; Parmar et al., 2018; Chen et al., 2018; 2020) formulate the generation problem as \"next token prediction\":",
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-        "type": "equation",
-        "text": "\n$$\np \\left(x ^ {1}, \\dots , x ^ {n}\\right) = \\prod_ {i = 1} ^ {n} p \\left(x ^ {i} \\mid x ^ {1}, \\dots , x ^ {i - 1}\\right). \\tag {1}\n$$\n",
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-        "type": "text",
-        "text": "Following the chain rule, the network is trained to model  $p(x^i \\mid x^1, \\dots, x^{i-1})$  and generate tokens iteratively. While all autoregressive models share this fundamental approach, differences in their",
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-        "type": "header",
-        "text": "Published as a conference paper at ICLR 2025",
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-        "type": "page_number",
-        "text": "3",
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-    {
-        "type": "image",
-        "img_path": "images/14fab35773c8d9aac66790afb7e218e2cbccc92979651577e9ee6140f6fc462a.jpg",
-        "image_caption": [
-            "Figure 2: Autoregressive models with different orders. (a) A raster-order autoregressive model predicts one next token based on the known ones, implemented using a GPT-like transformer with causal attention. (b) A random-order autoregressive model predicts one or multiple tokens simultaneously given a random order, implemented using a BERT-like transformer with bidirectional attention."
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-        "type": "text",
-        "text": "design can affect the performance. Two key design choices are the representation of  $x$ , i.e., discrete or continuous, and the generation order, which we elaborate below.",
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-        "type": "text",
-        "text": "Discrete vs. continuous tokens. The goal of an autoregressive model is to estimate  $p(x^{i} \\mid x^{1}, \\dots, x^{i-1})$ . Traditionally, this is done by transforming the image into a set of discrete tokens with a finite vocabulary and then estimating a categorical distribution over the vocabulary. The training objective is to minimize the cross-entropy loss, and sampling can be efficiently performed using categorical sampling. Most autoregressive image generation models rely on this form of token discretization (Esser et al., 2021; Chang et al., 2022; Tian et al., 2024; Yu et al., 2022).",
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-        "type": "text",
-        "text": "However, such discretization often leads to a significant loss of information from the image (Figure 4). Recent work (Li et al., 2024) has shown the possibility of applying a small diffusion model to approximate the distribution of each image token in a continuous fashion. This approach eliminates the need for vector quantization, and allows modeling images with continuous tokenizers which yield much better reconstruction visual quality. In this paper, we explore the scaling behavior of autoregressive image models on both discrete and continuous tokens.",
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-        "type": "text",
-        "text": "Raster Order + GPT vs. Random Order + BERT. In autoregressive image generation, there are two primary generation orders: raster and random. As illustrated in Figure 2, raster order generates tokens sequentially from left to right, top to bottom. This fixed-order generation is well-suited for a GPT-like transformer architecture, which predicts the next token in a causal manner. In contrast, random order allows multiple tokens to be generated in each step. The selection of these tokens can either be completely random or based on a sampling mechanism that prioritizes tokens with higher predicted confidence scores (Chang et al. (2023); Li et al. (2024)).",
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-        "type": "text",
-        "text": "Each generation order has its pros and cons. Raster order models with GPT-like transformer support fast inference via key-value (kv) caching. However, this causal structure can also introduce performance degradation. On the other hand, random order generation is usually achieved with a BERT-like bidirectional attention mechanism. While this approach prevents the usage of kv-caching, it enables the model to decode multiple tokens at each autoregressive step, allowing global editing. Despite their individual strengths, it remains unclear in the literature which generation order scales better for text-to-image generation tasks. In this work, we compare the performance and scaling behaviors of raster-order and random-order autoregressive models.",
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-        "type": "text",
-        "text": "4 IMPLEMENTATION",
-        "text_level": 1,
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-        "text": "The overall framework of our text-to-image model training is straightforward. An image tokenizer first converts the original image into tokens. These tokens are then partially masked, and a transformer is trained to reconstruct the masked tokens conditioned on the text. Below, we provide a detailed description of each component of our framework (shown in Figure 3).",
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-        "type": "header",
-        "text": "Published as a conference paper at ICLR 2025",
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-        "text": "4",
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-        "img_path": "images/cab25326fb05093119f9baaf67aa3757a4811a1f019cee5e1188bb0c8fc98710.jpg",
-        "image_caption": [
-            "Figure 3: Our text-to-image generation framework. A pre-trained image tokenizer converts the image into either discrete or continuous tokens. The text is embedded using a pre-trained T5 encoder, followed by a trainable text aligner. The transformer then takes cross-attention from the text embeddings to predict the missing tokens (only random order model is shown here)."
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-        "type": "text",
-        "text": "Image Tokenizer. We use a pre-trained image tokenizer to encode  $256 \\times 256$  images into a token space. Such a tokenizer can be either discrete or continuous, facilitating different training objectives of the autoregressive model. In our experiments, the discrete tokenizer is an VQGAN model (Esser et al., 2021) pretrained on the WebLI dataset (Chen et al., 2022). We follow Muse (Chang et al., 2023) to encode each image into  $16 \\times 16$  discrete tokens with a vocabulary size of 8192. For the continuous tokenizer, we",
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-        "type": "text",
-        "text": "adopt a widely-used one from Stable Diffusion (Rombach et al., 2022b), which encodes the image into  $32 \\times 32$  continuous tokens, each containing 4 channels. To be consistent in sequence length with the discrete tokenizer, we further group each  $2 \\times 2$  patch of continuous tokens into a single token, resulting in a final sequence length of 256, with each token containing 16 channels. As shown in Figure 4, the continuous tokenizer can achieve notably higher reconstruction quality than the discrete one.",
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-        "type": "text",
-        "text": "Text Encoder. The raw text (maximum length of 128) is tokenized by SentencePiece (Kudo, 2018), and embedded through a pre-trained T5-XXL encoder (Raffel et al., 2020), which has 4.7B parameters and is frozen during training. To further align the text embeddings for image generation, we add a small text aligner consisting of six trainable transformer blocks on top of the T5 embeddings, to extract the final text representation.",
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-        "type": "text",
-        "text": "Transformer. After encoding the original image into a sequence of tokens, we use a standard decoder-only transformer model (Vaswani et al., 2017) for autoregressive generation. Each block consists of three consecutive layers - self-attention, cross-attention, and MLP. The self-attention and MLP layers are only applied to visual tokens, while the cross attention layer takes visual and textual tokens as queries and keys, respectively. As shown in Figure 2, for raster-order models, the transformer predicts the next token based on previous tokens using causal attention for the self-attention block, similar to GPT. In random-order models, unknown tokens are masked by a learnable token, and the transformer predicts these masked tokens using bidirectional attention, similar to BERT.",
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-        "type": "text",
-        "text": "Output head. For discrete tokens, we follow the common practice with autoregressive models. The outputs are transformed into categorical distributions by softmax following a linear layer, whose weights are reused from the input embedding layer. For continuous tokens, we apply a six layer light-weight MLP as the diffusion head (Li et al., 2024) to model the per-token distribution. The embedding dimension of this head is the same as the backbone transformer. The per-token diffusion process follows (Nichol & Dhariwal, 2021; Li et al., 2024). The noise schedule has a cosine shape, with 1000 steps at training time; at inference time, it is resampled to 100 steps.",
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-        "image_caption": [
-            "Figure 4: Reconstruction quality of the tokenizers. Image resolution is  $256 \\times 256$ . The discrete tokenizer is significantly worse than the continuous tokenizer."
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-        "image_caption": [
-            "Discrete",
-            "PSNR: 26.6"
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-            "Continuous",
-            "PSNR: 31.5"
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-        "type": "header",
-        "text": "Published as a conference paper at ICLR 2025",
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-        "type": "page_number",
-        "text": "5",
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-        "type": "text",
-        "text": "5 EXPERIMENTS",
-        "text_level": 1,
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-        "type": "text",
-        "text": "Dataset. We use a subset of the WebLI (Web Language Image) dataset (Chen et al., 2022) as our training set, which consists of image-text pairs from the web with high scores for both image quality and alt-text relevance. By default, the images are center-cropped and resized to  $256 \\times 256$ .",
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-        "type": "text",
-        "text": "**Training.** Unless otherwise specified, we use the AdamW optimizer ( $\\beta_{1} = 0.9$ ,  $\\beta_{2} = 0.95$ ) (Loshchilov & Hutter, 2019) with a weight decay of 0.02 to train each model for 1M steps with a batch size of 2048. This is equivalent to approximately 3 epochs on our dataset. For continuous tokens, we employ a constant learning rate schedule with a 65K-step linear warmup and a maximum learning rate of  $1 \\times 10^{-4}$ ; for discrete tokens, we use a cosine learning rate schedule as we find it to be better. For training the random-order models, we randomly sample the masking ratio from [0, 1] following a cosine schedule, similar to MaskGIT (Chang et al., 2022), to mask each image. For all models, exponential moving average of the weights are gathered by a decay rate of 0.9999 and then used for evaluation.",
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-        "type": "text",
-        "text": "Inference. We follow the practices established by Imagen (Saharia et al., 2022), Muse (Chang et al., 2023), and Parti (Yu et al., 2022) to generate images from text prompts without rejection sampling. For random-order models, we use 64 steps for generation with a cosine schedule (Chang et al., 2022). To further enhance generation performance, we apply temperature and classifier-free guidance, as is commonly practiced.",
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-        "type": "text",
-        "text": "Evaluation. We evaluate the scaling behavior of different autoregressive model variants both quantitatively and qualitatively. Quantitatively, we evaluate the validation loss on 30K images from the MS-COCO 2014 training set, as well as two widely-adopted metrics: zero-shot Frechet Inception Distance (FID) on MS-COCO, and the GenEval score (Ghosh et al., 2024). Inference hyper-parameters, such as temperature and classifier-free guidance, are optimized for each evaluation metric. FID is computed over 30K randomly selected image-text pairs from the MS-COCO 2014 training set, providing a metric that evaluates both the fidelity and diversity of generated images. The GenEval benchmark, on the other hand, measures the model's ability to generate images that accurately reflect the given prompt. For qualitative evaluation, we generate images from several prompts using each model and compare the visual quality of the generated images.",
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-        "text": "5.1 SCALING BEHAVIORS",
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-        "text": "In this section, we explore how two key design choices in autoregressive image generative models—token representation and generation order—affect performance and scaling behavior. We construct models with different combinations of these two design choices, resulting in four distinct variants of autoregressive image generation models. We also explore the generalizability of these models across different data and evaluation metrics. Our experiments reveal several intriguing properties.",
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-        "text": "Validation losses consistently scale with model size. In Figure 5, we examine the scaling behavior of the four autoregressive variants in terms of validation loss. We observe a linear relationship between validation loss and model size in the log space, as we increases model size from 150 million to 3 billion parameters. This aligns with the power-law finding in Henighan et al. (2020). This demonstrates that the improvements in training loss resulting from increased model size generalize well to validation loss on data different from the training data.",
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-        "text": "Random-order models with continuous tokens scale the best in evaluation scores. In Figure 6, we analyze the scaling behavior of the four autoregressive variants in terms of FID and GenEval overall scores. We find that the improvements observed in validation loss do not always translate linearly to better evaluation metrics, implying that there is no strict power-law relationship between these metrics and model size. For example, raster-order models with discrete tokens (blue line) reach a plateau in both FID and GenEval scores around 1B parameters. Among the four variants, random-order models with continuous tokens (i.e., Fluid) show consistent improvements in evaluation metrics up to 3B parameters, achieving the best overall performance. Therefore, we focus on further investigating the scaling behavior of this model.",
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-        "text": "Published as a conference paper at ICLR 2025",
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-            "Figure 5: Validation loss scales as a power-law with model size. The validation loss is evaluated on 30K images randomly sampled from the MS-COCO 2014 training set. The x and y axes are in log-scale. The change in y is relatively small for each plot, making the log-scale alike linear-scale."
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-            "Figure 6: Random-order models using continuous tokens (orange) achieve the best performance on evaluation metrics. FID (lower is better) is evaluated on 30K images randomly sampled from the MS-COCO 2014 training set, while the GenEval overall score (higher is better) is assessed using the 553 prompts provided by the official benchmark, with four images generated for each prompt. Among all models, random-order models on continuous tokens consistently show an improvement in evaluation metrics as model size increases and achieve the best FID and GenEval scores."
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-        "type": "text",
-        "text": "Random-order models with continuous tokens scale with training computes. In Figure 7, we plot validation loss, FID, and GenEval scores as functions of total training steps and compute for different model sizes of Fluid. We observe consistent improvements in both validation loss and evaluation performance with increased training steps and compute. However, the benefits from additional training steps saturate around 1M steps, indicating that training smaller models for more steps is less compute-efficient compared to training larger models for fewer steps. This behavior aligns with observations in language models, highlighting the potential for scaling up model sizes with sufficient training.",
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-        "type": "text",
-        "text": "Strong correlation between validation loss and evaluation metrics. In Figure 8, we plot FID and GenEval scores against validation loss for different model sizes of Fluid and observe a strong correlation. To quantify this, we fit the data points using linear regression. The Pearson correlation coefficients for FID and GenEval scores are 0.917 and -0.931, respectively, indicating a nearly linear relationship between validation loss and these evaluation metrics across model sizes ranging from  $150\\mathrm{M}$  to  $3\\mathrm{B}^{1}$ . Encouraged by this positive trend, we trained a model with 10.5B parameters and a batch size of 4096 for 1M steps, achieving state-of-the-art text-to-image generation performance, as discussed in the next section.",
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-        "text": "Published as a conference paper at ICLR 2025",
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-        "type": "page_footnote",
-        "text": "<sup>1</sup>Since both FID and GenEval scores have lower/upper bounds, this linear relationship cannot hold indefinitely. Future work should explore the limits of this correlation.",
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-        "text": "7",
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-        "image_caption": [
-            "Figure 7: Validation losses and evaluation performance scale with increasing training steps and computes. We use random-order models with continuous tokens. Results for other autoregressive variants are included in the appendix. The training compute is computed as model GFLOPs  $\\times$  batch size  $\\times$  training steps  $\\times 3$ , where the factor of 3 accounts for the backward pass being approximately twice as compute-intensive as the forward pass."
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-        "image_caption": [
-            "Figure 8: Validation loss and evaluation metrics are highly correlated. We use random-order models with continuous tokens. The Pearson correlation coefficients for FID and GenEval scores are 0.917 and -0.931, respectively. We also observe that the linear correlation slightly weakens and becomes less pronounced for the 3.1B model."
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-        "text": "Continuous tokens and large models are crucial for visual quality. In Figure 9, we compare the visual quality of images generated by the four autoregressive variants. The visual quality of models using discrete tokens is significantly worse than that of models using continuous tokens, e.g., the eyes of the corgi is asymmetric for discrete token based models and scaling up can not solve this problem. This limitation is largely because of the discrete tokenizer, which introduces substantial information loss. For instance, even with 3B parameters, the discrete token models cannot generate",
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-        "text": "Published as a conference paper at ICLR 2025",
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-        "text": "8",
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-        "image_caption": [
-            "Figure 9: Visual quality and image-text alignment improves with increasing model size. Best viewed zoomed-in. Fluid  $\\bullet$  achieves the highest visual quality and best image-text alignment."
-        ],
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-        "type": "text",
-        "text": "an accurate Mona Lisa due to poor reconstruction quality of the tokenizer (Figure 4). In contrast, models with continuous tokens produce much higher-quality images.",
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-        "type": "text",
-        "text": "Additionally, larger models show consistent improvements in both visual quality and image-text alignment. For example, a random-order model with 0.2B parameters struggles to generate \"an angry duck doing heavy weightlifting at the gym\", while the same model with 3B parameters can generate the corresponding images successfully. This demonstrates that modeling continuous tokens and increasing model size are crucial for achieving high visual quality in autoregressive image generation models.",
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-        "type": "header",
-        "text": "Published as a conference paper at ICLR 2025",
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-        "text": "9",
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-    {
-        "type": "table",
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-        "table_caption": [
-            "Table 1: System-level comparison. Fluid achieves leading results on both MS-COCO zero-shot FID-30K and GenEval benchmark (Ghosh et al., 2024). †: CM3Leon result is reported without retrieval."
-        ],
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-        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">#params</td><td rowspan=\"2\">MS-COCOFID-30K↓</td><td colspan=\"7\">GenEval</td></tr><tr><td>Single Obj.</td><td>Two Obj.</td><td>Counting</td><td>Colors</td><td>Position</td><td>Color Attri.</td><td>Overall</td></tr><tr><td colspan=\"10\">diffusion model</td></tr><tr><td>LDM</td><td>1.4B</td><td>12.64</td><td>0.92</td><td>0.29</td><td>0.23</td><td>0.70</td><td>0.02</td><td>0.05</td><td>0.37</td></tr><tr><td>DALL-E 2</td><td>4.2B</td><td>10.39</td><td>0.94</td><td>0.66</td><td>0.49</td><td>0.77</td><td>0.10</td><td>0.19</td><td>0.52</td></tr><tr><td>DALL-E 3</td><td>-</td><td>-</td><td>0.96</td><td>0.87</td><td>0.47</td><td>0.83</td><td>0.43</td><td>0.45</td><td>0.67</td></tr><tr><td>Imagen</td><td>3B</td><td>7.27</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>SD3</td><td>8B</td><td>-</td><td>0.98</td><td>0.84</td><td>0.66</td><td>0.74</td><td>0.40</td><td>0.43</td><td>0.68</td></tr><tr><td>Transfusion</td><td>7.3B</td><td>6.78</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.63</td></tr><tr><td>RAPHAEL</td><td>3B</td><td>6.61</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td colspan=\"10\">autoregressive model</td></tr><tr><td>CM3Leon†</td><td>7B</td><td>10.82</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Show-o</td><td>1.3B</td><td>9.24</td><td>0.95</td><td>0.52</td><td>0.49</td><td>0.82</td><td>0.11</td><td>0.28</td><td>0.53</td></tr><tr><td>Muse</td><td>3B</td><td>7.88</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Parti</td><td>20B</td><td>7.23</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Fluid (our work)</td><td>369M</td><td>7.23</td><td>0.96</td><td>0.64</td><td>0.53</td><td>0.78</td><td>0.33</td><td>0.46</td><td>0.62</td></tr><tr><td></td><td>665M</td><td>6.84</td><td>0.96</td><td>0.73</td><td>0.51</td><td>0.77</td><td>0.42</td><td>0.51</td><td>0.65</td></tr><tr><td></td><td>1.1B</td><td>6.59</td><td>0.96</td><td>0.77</td><td>0.61</td><td>0.78</td><td>0.34</td><td>0.53</td><td>0.67</td></tr><tr><td></td><td>3.1B</td><td>6.41</td><td>0.98</td><td>0.83</td><td>0.60</td><td>0.82</td><td>0.41</td><td>0.53</td><td>0.70</td></tr><tr><td></td><td>10.5B</td><td>6.16</td><td>0.96</td><td>0.83</td><td>0.63</td><td>0.80</td><td>0.39</td><td>0.51</td><td>0.69</td></tr></table>",
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-        "type": "text",
-        "text": "5.2 BENCHMARKING WITH PREVIOUS SYSTEMS",
-        "text_level": 1,
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-        "text": "In this section, we compare our Fluid, i.e., continuous random-order autoregressive model, with leading text-to-image generation systems in Table 1 (Rombach et al., 2022b; Ramesh et al., 2022; Betker et al., 2023; Sahara et al., 2022; Esser et al., 2024; Zhou et al., 2024; Xue et al., 2024; Yu et al., 2023; Xie et al., 2024; Chang et al., 2023; Yu et al., 2022). The Fluid smallest model, with 369M parameters, achieves a zero-shot FID of 7.23 on MS-COCO and a GenEval overall score of 0.62, matching the performance of many state-of-the-art models with several billion parameters (e.g., Parti with 20B parameters only achieves 7.23). The Fluid largest model, with 10.5B parameters, further improves the zero-shot FID on MS-COCO to 6.16 and increases the GenEval overall score to  $0.69^2$ , with a speed of 1.571 seconds per image per TPU (evaluated on 32 TPU v5 with a batch size of 2048). Detailed model configurations and generation speed results are included in the appendix. We hope these strong results and promising scaling behavior provide valuable insights and support for the scalability of autoregressive models in visual generative modeling.",
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-        "type": "text",
-        "text": "6 DISCUSSION",
-        "text_level": 1,
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-        "type": "text",
-        "text": "In this paper, we present an empirical study on the scaling behavior of autoregressive models for text-to-image generation. We investigate two critical design factors: random order versus raster order, and discrete tokens versus continuous tokens. Our results show that random-order models with continuous tokens achieve the best performance and scaling behavior across various evaluation metrics and in terms of visual quality. Building on these findings, we scale up the random-order model with continuous tokens, namely Fluid, to 10.5B parameters, and achieves state-of-the-art text-to-image generation performance. We hope that our findings and promising results could provide valuable insights into the scaling behavior of autoregressive models for image generation and help bridge the gap between the scaling performance of vision models and language models.",
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-        "type": "text",
-        "text": "Reproducibility Statement. To aid reproducibility, we have provided the implementation details of our framework in Section 4, training hyper-parameters in Section 5, and model configurations in the appendix. For the diffusion loss used for continuous tokens, we have strictly followed the open-sourced code of Li et al. (2024).",
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-        "type": "text",
-        "text": "Acknowledgements. We would like to express our gratitude to Amy Shen and Alex Rizkowsky for their assistance in securing computational resources, and to Yash Katariya, Ivy Zheng, and Roy Frostig for their valuable insights on JAX-related questions. We also thank Matan Cohen for his support with precomputed T5 embeddings. Our appreciation extends to Han Zhang, Jason Baldridge, Dilip Krishnan, David Salesin and VisCam team for their helpful discussions and valuable feedback.",
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-        "text": "Published as a conference paper at ICLR 2025",
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-    {
-        "type": "page_footnote",
-        "text": "2We observe that the GenEval scores plateau for the 10.5B Fluid compared to the 3.1B Fluid; however, it continues to show consistent improvements in visual quality and FID.",
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-        "text": "10",
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-        "type": "text",
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-            "content": "Figure 1: Samples from our Fluid 10.5B autoregressive model with continuous tokens."
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-            "content": "Scaling up autoregressive models in vision has not proven as beneficial as in large language models. In this work, we investigate this scaling problem in the context of text-to-image generation, focusing on two critical factors: whether models use discrete or continuous tokens, and whether tokens are generated in a random or fixed raster order using BERT- or GPT-like transformer architectures. Our empirical results show that, while all models scale effectively in terms of validation loss, their evaluation performance—measured by FID, GenEval score, and visual quality—follows different trends. Models based on continuous tokens achieve significantly better visual quality than those using discrete tokens. Furthermore, the generation order and attention mechanisms significantly affect the GenEval score: random-order models achieve notably better GenEval scores compared to raster-order models. Inspired by these findings, we train Fluid, a random-order autoregressive model on continuous tokens. Fluid 10.5B model achieves a new state-of-the-art zero-shot FID of 6.16 on MS-COCO 30K, and 0.69 overall score on the GenEval benchmark. We hope our findings and results will encourage future efforts to further bridge the scaling gap between vision and language models."
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-            "content": "1 INTRODUCTION"
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-            "content": "Scaling laws underpin the unprecedented success of large language models (LLMs). Empirically, increasing the number of parameters in autoregressive models consistently leads to significant performance improvements and the emergence of new capabilities in natural language processing (NLP) tasks (Devlin et al., 2018; Radford et al., 2018; Brown et al., 2020; Kaplan et al., 2020; Wei et al., 2022). This empirical relationship has inspired numerous efforts to scale up language models, resulting in the development of many highly capable models (Bubeck et al., 2023; Team et al., 2023; Achiam et al., 2023)."
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-            "content": "Encouraged by this success, many attempts have been made to adopt and scale up autoregressive models in computer vision, particularly for generative tasks like text-to-image generation (Yu et al., 2021; 2023; Bai et al., 2024; El-Nouby et al., 2024). However, the performance and visual quality of content generated by these models often fall short compared to other generative models, such as diffusion models (Ho et al., 2020; Sahara et al., 2022; Rombach et al., 2022a; Esser et al., 2024), leaving it unclear whether similar scaling laws apply to the vision domain."
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-            "content": "We propose several hypotheses for the performance gap. First, the vector quantization (VQ) (van den Oord et al., 2017) step, which is required for most visual autoregressive models, may introduce significant information loss, ultimately limiting model performance. Second, unlike the inherently sequential nature of language, generating visual content might benefit more from a different autoregressive prediction order. Third, there is often a confusion between two levels of generalizability when evaluating scaling laws in vision models: (a) generalization to new data using the same metric as the training loss (commonly referred to as validation loss), and (b) generalization to a new metric or problem different from the training objective, such as FID (Heusel et al., 2017), the GenEval benchmark (Ghosh et al., 2024), or visual quality. We hypothesize that power-law scaling (Kaplan et al., 2020) applies to (a) for autoregressive models on vision data, but not necessarily to (b)."
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-            "content": "To investigate these hypotheses, we conduct a comprehensive empirical study on the scaling behavior of autoregressive models in the context of text-to-image generation. Specifically, we explore two key factors: whether the model operates on continuous or discrete tokens, and whether tokens are generated in a random or fixed raster order. To this end, we utilize the Diffusion Loss (Li et al., 2024) to make autoregressive models compatible with continuous tokens. We generalize BERT-like vision model MaskGIT (Chang et al., 2022) as random-order autoregression, as it conceptually predicts output tokens in a randomized order while retaining the autoregressive nature of \"predicting next tokens based on known ones\". We analyze the behavior of four autoregressive variants, each employing different combinations of these two factors. We scale their parameters from 150M to 3B and evaluate their performance using three metrics: validation loss, FID (Heusel et al., 2017), and GenEval score (Ghosh et al., 2024). We also inspect the visual quality of the generated images."
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-            "content": "Our experiments indicate that VQ-based models, regardless of whether they use a random or fixed raster order, exhibit a slower improvement in FID scores when scaling up model size compared to models operating on continuous tokens. VQ models also produce images of lower visual quality, likely due to information loss introduced by vector quantization."
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-            "content": "Furthermore, the token generation order and the associated attention mechanism primarily influence the global structure of the generated image. In the GenEval benchmark, random-order models with bidirectional attention significantly outperform raster-order models with causal attention, particularly when generating multiple objects. Random-order models can readjust the global structure at every prediction step, whereas raster-order models cannot. This suggests that the token generation order plays a crucial role in achieving better text-to-image alignment."
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-            "content": "Our experiments also demonstrate that validation loss scales as a power-law with model size, no matter whether the model operates with continuous or discrete tokens. This implies scalable behavior at the level of (a) generalization—generalizing to new data using the same metric as the training loss—which aligns with observations in language models (Kaplan et al., 2020). However, as for generalizing to a different metric, such as FID or GenEval score, although performance consistently improves with better validation loss, the trend may not follow a strict power-law."
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-            "content": "Building on these findings, we scale the Fluid model, i.e., random-order model with continuous tokens, up to 10.5B parameters and train it using the WebLI dataset (Chen et al., 2022). The resulting Fluid 10.5B model achieves a zero-shot FID of 6.16 on MS-COCO and a GenEval overall score of"
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-            "content": "0.69, comparing favorably with leading text-to-image generative models such as DALL-E 3 (Betker et al., 2023) and Stable Diffusion 3 (Esser et al., 2024). We hope that our empirical findings and positive results could shed light on the scaling behavior for text-to-image generation models and further innovation in this frontier."
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-            "content": "2 RELATED WORK"
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-            "type": "text",
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-            "angle": 0,
-            "content": "Text-to-image diffusion models. The dominant approaches are based on diffusion models. Dalle 2 (Ramesh et al., 2022), Imagen (Ho et al., 2022), and Stable Diffusion (Rombach et al., 2022a) revolutionized text-to-image generation. Most recently, SD v3 (Esser et al., 2024) and Imagen3 (Baldridge et al., 2024) can generate realistic images that are hard to tell from real images. However, generating samples is usually computationally expensive due to the multiple forward passes."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-                0.308,
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-                0.42
-            ],
-            "angle": 0,
-            "content": "Autoregressive (AR) models. While being the defacto model for language modeling, AR models are lagging behind diffusion models for text-to-image generation. AR models (Yu et al., 2022; Chang et al., 2023; Li et al., 2023; Yu et al., 2023) are often used together with discrete tokenizers (van den Oord et al., 2017; Esser et al., 2021), which often limit the modeling capability. For example, Parti (Yu et al., 2022) scales up the model to 20B, which obtains a slightly lower/better FID score of 7.23 on MS-COCO than 7.27 by the 3.4B diffusion-based Imagen. Here we find that, with a continuous tokenizer, our small 369M model can already achieve the same FID score as Parti with 20B model."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.17,
-                0.426,
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-                0.498
-            ],
-            "angle": 0,
-            "content": "Recently Li et al. (2024) challenges the conventional wisdom and replaces the discrete tokenizer with a continuous tokenizer via a diffusion loss. They introduced masked autoregressive (MAR) model obtains strong results for class-conditioning generation on ImageNet. However, scaling MAR models for text-to-image generation is unexplored. Here we empirically study the scaling behavior for MAR and report several findings important for the research community."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.17,
-                0.51,
-                0.827,
-                0.637
-            ],
-            "angle": 0,
-            "content": "Scaling language models. Kaplan et al. (2020) empirically observed that, for language model, the validation loss scales as a power-law with model size, dataset size, and the amount of compute used for training. Hoffmann et al. (2022) discovered that contemporary LLMs are under-trained, and that for compute-optimal training, the model size and the number of training tokens should be scaled equally. Under the same compute budget while using 4x more more data, their 70B Chinchilla outperforms much larger models, such as the 530B Megatron-Turing NLG (Smith et al., 2022). Wei et al. (2022) found that larger models have emergent abilities that are not present in smaller models. These observations have inspired significant efforts to scale up language models to trillions of parameters (Achiam et al., 2023; Team et al., 2023; Dubey et al., 2024)."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.17,
-                0.65,
-                0.827,
-                0.763
-            ],
-            "angle": 0,
-            "content": "Scaling vision models. Similar scaling law has been obscure for computer vision. For recognition models (Tan, 2019; Zhai et al., 2022; He et al., 2022; Dehghani et al., 2023), scaling often comes with diminishing returns. For example, Dehghani et al. (2023) scale ViT up from 3B to 22B but only observed \\(0.25\\%\\) accuracy increase in ImageNet linear probing. The scaling of generative models is more promising - DiT (Peebles & Xie, 2023) shows consistent improvement in generation quality when scaling up compute and model size (despite only up to 600M). The follow-up Sora (OpenAI, 2024) further shows the potential to scale up for video generation. In this paper, we perform a comprehensive empirical study of AR models for text-to-image generation."
-        },
-        {
-            "type": "title",
-            "bbox": [
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-            "angle": 0,
-            "content": "3 PRELIMINARY: AUTOREGRESSIVE IMAGE GENERATION"
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "Given a sequence of tokens \\(\\{x^1, x^2, \\ldots, x^n\\}\\) where the superscript \\(1 \\leq i \\leq n\\) specifies an order, autoregressive models (Gregor et al., 2014; van den Oord et al., 2016b;a; Parmar et al., 2018; Chen et al., 2018; 2020) formulate the generation problem as \"next token prediction\":"
-        },
-        {
-            "type": "equation",
-            "bbox": [
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-            "angle": 0,
-            "content": "\\[\np \\left(x ^ {1}, \\dots , x ^ {n}\\right) = \\prod_ {i = 1} ^ {n} p \\left(x ^ {i} \\mid x ^ {1}, \\dots , x ^ {i - 1}\\right). \\tag {1}\n\\]"
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            ],
-            "angle": 0,
-            "content": "Following the chain rule, the network is trained to model \\( p(x^i \\mid x^1, \\dots, x^{i-1}) \\) and generate tokens iteratively. While all autoregressive models share this fundamental approach, differences in their"
-        },
-        {
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-            "angle": 0,
-            "content": "3"
-        }
-    ],
-    [
-        {
-            "type": "header",
-            "bbox": [
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-                0.049
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-            "angle": 0,
-            "content": "Published as a conference paper at ICLR 2025"
-        },
-        {
-            "type": "image",
-            "bbox": [
-                0.179,
-                0.104,
-                0.825,
-                0.257
-            ],
-            "angle": 0,
-            "content": null
-        },
-        {
-            "type": "image_caption",
-            "bbox": [
-                0.17,
-                0.268,
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-                0.34
-            ],
-            "angle": 0,
-            "content": "Figure 2: Autoregressive models with different orders. (a) A raster-order autoregressive model predicts one next token based on the known ones, implemented using a GPT-like transformer with causal attention. (b) A random-order autoregressive model predicts one or multiple tokens simultaneously given a random order, implemented using a BERT-like transformer with bidirectional attention."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.171,
-                0.355,
-                0.825,
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-            ],
-            "angle": 0,
-            "content": "design can affect the performance. Two key design choices are the representation of \\( x \\), i.e., discrete or continuous, and the generation order, which we elaborate below."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.17,
-                0.402,
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-            ],
-            "angle": 0,
-            "content": "Discrete vs. continuous tokens. The goal of an autoregressive model is to estimate \\( p(x^{i} \\mid x^{1}, \\dots, x^{i-1}) \\). Traditionally, this is done by transforming the image into a set of discrete tokens with a finite vocabulary and then estimating a categorical distribution over the vocabulary. The training objective is to minimize the cross-entropy loss, and sampling can be efficiently performed using categorical sampling. Most autoregressive image generation models rely on this form of token discretization (Esser et al., 2021; Chang et al., 2022; Tian et al., 2024; Yu et al., 2022)."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "However, such discretization often leads to a significant loss of information from the image (Figure 4). Recent work (Li et al., 2024) has shown the possibility of applying a small diffusion model to approximate the distribution of each image token in a continuous fashion. This approach eliminates the need for vector quantization, and allows modeling images with continuous tokenizers which yield much better reconstruction visual quality. In this paper, we explore the scaling behavior of autoregressive image models on both discrete and continuous tokens."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.17,
-                0.594,
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-            ],
-            "angle": 0,
-            "content": "Raster Order + GPT vs. Random Order + BERT. In autoregressive image generation, there are two primary generation orders: raster and random. As illustrated in Figure 2, raster order generates tokens sequentially from left to right, top to bottom. This fixed-order generation is well-suited for a GPT-like transformer architecture, which predicts the next token in a causal manner. In contrast, random order allows multiple tokens to be generated in each step. The selection of these tokens can either be completely random or based on a sampling mechanism that prioritizes tokens with higher predicted confidence scores (Chang et al. (2023); Li et al. (2024))."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.17,
-                0.699,
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-            ],
-            "angle": 0,
-            "content": "Each generation order has its pros and cons. Raster order models with GPT-like transformer support fast inference via key-value (kv) caching. However, this causal structure can also introduce performance degradation. On the other hand, random order generation is usually achieved with a BERT-like bidirectional attention mechanism. While this approach prevents the usage of kv-caching, it enables the model to decode multiple tokens at each autoregressive step, allowing global editing. Despite their individual strengths, it remains unclear in the literature which generation order scales better for text-to-image generation tasks. In this work, we compare the performance and scaling behaviors of raster-order and random-order autoregressive models."
-        },
-        {
-            "type": "title",
-            "bbox": [
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-            "angle": 0,
-            "content": "4 IMPLEMENTATION"
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "The overall framework of our text-to-image model training is straightforward. An image tokenizer first converts the original image into tokens. These tokens are then partially masked, and a transformer is trained to reconstruct the masked tokens conditioned on the text. Below, we provide a detailed description of each component of our framework (shown in Figure 3)."
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-            "angle": 0,
-            "content": "4"
-        }
-    ],
-    [
-        {
-            "type": "header",
-            "bbox": [
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-                0.049
-            ],
-            "angle": 0,
-            "content": "Published as a conference paper at ICLR 2025"
-        },
-        {
-            "type": "image",
-            "bbox": [
-                0.239,
-                0.101,
-                0.761,
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-            ],
-            "angle": 0,
-            "content": null
-        },
-        {
-            "type": "image_caption",
-            "bbox": [
-                0.171,
-                0.273,
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-            ],
-            "angle": 0,
-            "content": "Figure 3: Our text-to-image generation framework. A pre-trained image tokenizer converts the image into either discrete or continuous tokens. The text is embedded using a pre-trained T5 encoder, followed by a trainable text aligner. The transformer then takes cross-attention from the text embeddings to predict the missing tokens (only random order model is shown here)."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.171,
-                0.348,
-                0.452,
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-            ],
-            "angle": 0,
-            "content": "Image Tokenizer. We use a pre-trained image tokenizer to encode \\(256 \\times 256\\) images into a token space. Such a tokenizer can be either discrete or continuous, facilitating different training objectives of the autoregressive model. In our experiments, the discrete tokenizer is an VQGAN model (Esser et al., 2021) pretrained on the WebLI dataset (Chen et al., 2022). We follow Muse (Chang et al., 2023) to encode each image into \\(16 \\times 16\\) discrete tokens with a vocabulary size of 8192. For the continuous tokenizer, we"
-        },
-        {
-            "type": "text",
-            "bbox": [
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-                0.528,
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-            ],
-            "angle": 0,
-            "content": "adopt a widely-used one from Stable Diffusion (Rombach et al., 2022b), which encodes the image into \\(32 \\times 32\\) continuous tokens, each containing 4 channels. To be consistent in sequence length with the discrete tokenizer, we further group each \\(2 \\times 2\\) patch of continuous tokens into a single token, resulting in a final sequence length of 256, with each token containing 16 channels. As shown in Figure 4, the continuous tokenizer can achieve notably higher reconstruction quality than the discrete one."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            ],
-            "angle": 0,
-            "content": "Text Encoder. The raw text (maximum length of 128) is tokenized by SentencePiece (Kudo, 2018), and embedded through a pre-trained T5-XXL encoder (Raffel et al., 2020), which has 4.7B parameters and is frozen during training. To further align the text embeddings for image generation, we add a small text aligner consisting of six trainable transformer blocks on top of the T5 embeddings, to extract the final text representation."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            ],
-            "angle": 0,
-            "content": "Transformer. After encoding the original image into a sequence of tokens, we use a standard decoder-only transformer model (Vaswani et al., 2017) for autoregressive generation. Each block consists of three consecutive layers - self-attention, cross-attention, and MLP. The self-attention and MLP layers are only applied to visual tokens, while the cross attention layer takes visual and textual tokens as queries and keys, respectively. As shown in Figure 2, for raster-order models, the transformer predicts the next token based on previous tokens using causal attention for the self-attention block, similar to GPT. In random-order models, unknown tokens are masked by a learnable token, and the transformer predicts these masked tokens using bidirectional attention, similar to BERT."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "Output head. For discrete tokens, we follow the common practice with autoregressive models. The outputs are transformed into categorical distributions by softmax following a linear layer, whose weights are reused from the input embedding layer. For continuous tokens, we apply a six layer light-weight MLP as the diffusion head (Li et al., 2024) to model the per-token distribution. The embedding dimension of this head is the same as the backbone transformer. The per-token diffusion process follows (Nichol & Dhariwal, 2021; Li et al., 2024). The noise schedule has a cosine shape, with 1000 steps at training time; at inference time, it is resampled to 100 steps."
-        },
-        {
-            "type": "image",
-            "bbox": [
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-            ],
-            "angle": 0,
-            "content": null
-        },
-        {
-            "type": "image_caption",
-            "bbox": [
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-            ],
-            "angle": 0,
-            "content": "Discrete"
-        },
-        {
-            "type": "image",
-            "bbox": [
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-                0.364,
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-            "angle": 0,
-            "content": null
-        },
-        {
-            "type": "image_caption",
-            "bbox": [
-                0.607,
-                0.454,
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-                0.467
-            ],
-            "angle": 0,
-            "content": "PSNR: 26.6"
-        },
-        {
-            "type": "image_caption",
-            "bbox": [
-                0.721,
-                0.349,
-                0.805,
-                0.363
-            ],
-            "angle": 0,
-            "content": "Continuous"
-        },
-        {
-            "type": "image",
-            "bbox": [
-                0.706,
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-            ],
-            "angle": 0,
-            "content": null
-        },
-        {
-            "type": "image_caption",
-            "bbox": [
-                0.722,
-                0.454,
-                0.804,
-                0.467
-            ],
-            "angle": 0,
-            "content": "PSNR: 31.5"
-        },
-        {
-            "type": "image_caption",
-            "bbox": [
-                0.463,
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-            ],
-            "angle": 0,
-            "content": "Figure 4: Reconstruction quality of the tokenizers. Image resolution is \\(256 \\times 256\\). The discrete tokenizer is significantly worse than the continuous tokenizer."
-        },
-        {
-            "type": "page_number",
-            "bbox": [
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-            "angle": 0,
-            "content": "5"
-        }
-    ],
-    [
-        {
-            "type": "header",
-            "bbox": [
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-                0.049
-            ],
-            "angle": 0,
-            "content": "Published as a conference paper at ICLR 2025"
-        },
-        {
-            "type": "title",
-            "bbox": [
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-            "angle": 0,
-            "content": "5 EXPERIMENTS"
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "Dataset. We use a subset of the WebLI (Web Language Image) dataset (Chen et al., 2022) as our training set, which consists of image-text pairs from the web with high scores for both image quality and alt-text relevance. By default, the images are center-cropped and resized to \\(256 \\times 256\\)."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "**Training.** Unless otherwise specified, we use the AdamW optimizer (\\(\\beta_{1} = 0.9\\), \\(\\beta_{2} = 0.95\\)) (Loshchilov & Hutter, 2019) with a weight decay of 0.02 to train each model for 1M steps with a batch size of 2048. This is equivalent to approximately 3 epochs on our dataset. For continuous tokens, we employ a constant learning rate schedule with a 65K-step linear warmup and a maximum learning rate of \\(1 \\times 10^{-4}\\); for discrete tokens, we use a cosine learning rate schedule as we find it to be better. For training the random-order models, we randomly sample the masking ratio from [0, 1] following a cosine schedule, similar to MaskGIT (Chang et al., 2022), to mask each image. For all models, exponential moving average of the weights are gathered by a decay rate of 0.9999 and then used for evaluation."
-        },
-        {
-            "type": "text",
-            "bbox": [
-                0.17,
-                0.338,
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-            ],
-            "angle": 0,
-            "content": "Inference. We follow the practices established by Imagen (Saharia et al., 2022), Muse (Chang et al., 2023), and Parti (Yu et al., 2022) to generate images from text prompts without rejection sampling. For random-order models, we use 64 steps for generation with a cosine schedule (Chang et al., 2022). To further enhance generation performance, we apply temperature and classifier-free guidance, as is commonly practiced."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "Evaluation. We evaluate the scaling behavior of different autoregressive model variants both quantitatively and qualitatively. Quantitatively, we evaluate the validation loss on 30K images from the MS-COCO 2014 training set, as well as two widely-adopted metrics: zero-shot Frechet Inception Distance (FID) on MS-COCO, and the GenEval score (Ghosh et al., 2024). Inference hyper-parameters, such as temperature and classifier-free guidance, are optimized for each evaluation metric. FID is computed over 30K randomly selected image-text pairs from the MS-COCO 2014 training set, providing a metric that evaluates both the fidelity and diversity of generated images. The GenEval benchmark, on the other hand, measures the model's ability to generate images that accurately reflect the given prompt. For qualitative evaluation, we generate images from several prompts using each model and compare the visual quality of the generated images."
-        },
-        {
-            "type": "title",
-            "bbox": [
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-            "angle": 0,
-            "content": "5.1 SCALING BEHAVIORS"
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "In this section, we explore how two key design choices in autoregressive image generative models—token representation and generation order—affect performance and scaling behavior. We construct models with different combinations of these two design choices, resulting in four distinct variants of autoregressive image generation models. We also explore the generalizability of these models across different data and evaluation metrics. Our experiments reveal several intriguing properties."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "Validation losses consistently scale with model size. In Figure 5, we examine the scaling behavior of the four autoregressive variants in terms of validation loss. We observe a linear relationship between validation loss and model size in the log space, as we increases model size from 150 million to 3 billion parameters. This aligns with the power-law finding in Henighan et al. (2020). This demonstrates that the improvements in training loss resulting from increased model size generalize well to validation loss on data different from the training data."
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "Random-order models with continuous tokens scale the best in evaluation scores. In Figure 6, we analyze the scaling behavior of the four autoregressive variants in terms of FID and GenEval overall scores. We find that the improvements observed in validation loss do not always translate linearly to better evaluation metrics, implying that there is no strict power-law relationship between these metrics and model size. For example, raster-order models with discrete tokens (blue line) reach a plateau in both FID and GenEval scores around 1B parameters. Among the four variants, random-order models with continuous tokens (i.e., Fluid) show consistent improvements in evaluation metrics up to 3B parameters, achieving the best overall performance. Therefore, we focus on further investigating the scaling behavior of this model."
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-            "type": "header",
-            "bbox": [
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-            "angle": 0,
-            "content": "Published as a conference paper at ICLR 2025"
-        },
-        {
-            "type": "image",
-            "bbox": [
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-            "bbox": [
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-            "angle": 0,
-            "content": "Figure 5: Validation loss scales as a power-law with model size. The validation loss is evaluated on 30K images randomly sampled from the MS-COCO 2014 training set. The x and y axes are in log-scale. The change in y is relatively small for each plot, making the log-scale alike linear-scale."
-        },
-        {
-            "type": "image",
-            "bbox": [
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-        {
-            "type": "image_caption",
-            "bbox": [
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-            "angle": 0,
-            "content": "Figure 6: Random-order models using continuous tokens (orange) achieve the best performance on evaluation metrics. FID (lower is better) is evaluated on 30K images randomly sampled from the MS-COCO 2014 training set, while the GenEval overall score (higher is better) is assessed using the 553 prompts provided by the official benchmark, with four images generated for each prompt. Among all models, random-order models on continuous tokens consistently show an improvement in evaluation metrics as model size increases and achieve the best FID and GenEval scores."
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-            "content": "Random-order models with continuous tokens scale with training computes. In Figure 7, we plot validation loss, FID, and GenEval scores as functions of total training steps and compute for different model sizes of Fluid. We observe consistent improvements in both validation loss and evaluation performance with increased training steps and compute. However, the benefits from additional training steps saturate around 1M steps, indicating that training smaller models for more steps is less compute-efficient compared to training larger models for fewer steps. This behavior aligns with observations in language models, highlighting the potential for scaling up model sizes with sufficient training."
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-            "bbox": [
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-            "angle": 0,
-            "content": "Strong correlation between validation loss and evaluation metrics. In Figure 8, we plot FID and GenEval scores against validation loss for different model sizes of Fluid and observe a strong correlation. To quantify this, we fit the data points using linear regression. The Pearson correlation coefficients for FID and GenEval scores are 0.917 and -0.931, respectively, indicating a nearly linear relationship between validation loss and these evaluation metrics across model sizes ranging from \\(150\\mathrm{M}\\) to \\(3\\mathrm{B}^{1}\\). Encouraged by this positive trend, we trained a model with 10.5B parameters and a batch size of 4096 for 1M steps, achieving state-of-the-art text-to-image generation performance, as discussed in the next section."
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-            "content": "<sup>1</sup>Since both FID and GenEval scores have lower/upper bounds, this linear relationship cannot hold indefinitely. Future work should explore the limits of this correlation."
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-            "content": "Published as a conference paper at ICLR 2025"
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-        {
-            "type": "image",
-            "bbox": [
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-        {
-            "type": "image_caption",
-            "bbox": [
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-            "content": "Figure 7: Validation losses and evaluation performance scale with increasing training steps and computes. We use random-order models with continuous tokens. Results for other autoregressive variants are included in the appendix. The training compute is computed as model GFLOPs \\(\\times\\) batch size \\(\\times\\) training steps \\(\\times 3\\), where the factor of 3 accounts for the backward pass being approximately twice as compute-intensive as the forward pass."
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-            "type": "image",
-            "bbox": [
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-            "angle": 0,
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-        {
-            "type": "image_caption",
-            "bbox": [
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-            "angle": 0,
-            "content": "Figure 8: Validation loss and evaluation metrics are highly correlated. We use random-order models with continuous tokens. The Pearson correlation coefficients for FID and GenEval scores are 0.917 and -0.931, respectively. We also observe that the linear correlation slightly weakens and becomes less pronounced for the 3.1B model."
-        },
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-            "content": "Continuous tokens and large models are crucial for visual quality. In Figure 9, we compare the visual quality of images generated by the four autoregressive variants. The visual quality of models using discrete tokens is significantly worse than that of models using continuous tokens, e.g., the eyes of the corgi is asymmetric for discrete token based models and scaling up can not solve this problem. This limitation is largely because of the discrete tokenizer, which introduces substantial information loss. For instance, even with 3B parameters, the discrete token models cannot generate"
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-            "angle": 0,
-            "content": "Published as a conference paper at ICLR 2025"
-        },
-        {
-            "type": "image",
-            "bbox": [
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-            "type": "image_caption",
-            "bbox": [
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-            "angle": 0,
-            "content": "Figure 9: Visual quality and image-text alignment improves with increasing model size. Best viewed zoomed-in. Fluid \\(\\bullet\\) achieves the highest visual quality and best image-text alignment."
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-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "an accurate Mona Lisa due to poor reconstruction quality of the tokenizer (Figure 4). In contrast, models with continuous tokens produce much higher-quality images."
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-            "type": "text",
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-            "angle": 0,
-            "content": "Additionally, larger models show consistent improvements in both visual quality and image-text alignment. For example, a random-order model with 0.2B parameters struggles to generate \"an angry duck doing heavy weightlifting at the gym\", while the same model with 3B parameters can generate the corresponding images successfully. This demonstrates that modeling continuous tokens and increasing model size are crucial for achieving high visual quality in autoregressive image generation models."
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-            "content": "9"
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-            "content": "Published as a conference paper at ICLR 2025"
-        },
-        {
-            "type": "table_caption",
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-            "content": "Table 1: System-level comparison. Fluid achieves leading results on both MS-COCO zero-shot FID-30K and GenEval benchmark (Ghosh et al., 2024). †: CM3Leon result is reported without retrieval."
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-        {
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-            "content": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">#params</td><td rowspan=\"2\">MS-COCOFID-30K↓</td><td colspan=\"7\">GenEval</td></tr><tr><td>Single Obj.</td><td>Two Obj.</td><td>Counting</td><td>Colors</td><td>Position</td><td>Color Attri.</td><td>Overall</td></tr><tr><td colspan=\"10\">diffusion model</td></tr><tr><td>LDM</td><td>1.4B</td><td>12.64</td><td>0.92</td><td>0.29</td><td>0.23</td><td>0.70</td><td>0.02</td><td>0.05</td><td>0.37</td></tr><tr><td>DALL-E 2</td><td>4.2B</td><td>10.39</td><td>0.94</td><td>0.66</td><td>0.49</td><td>0.77</td><td>0.10</td><td>0.19</td><td>0.52</td></tr><tr><td>DALL-E 3</td><td>-</td><td>-</td><td>0.96</td><td>0.87</td><td>0.47</td><td>0.83</td><td>0.43</td><td>0.45</td><td>0.67</td></tr><tr><td>Imagen</td><td>3B</td><td>7.27</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>SD3</td><td>8B</td><td>-</td><td>0.98</td><td>0.84</td><td>0.66</td><td>0.74</td><td>0.40</td><td>0.43</td><td>0.68</td></tr><tr><td>Transfusion</td><td>7.3B</td><td>6.78</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.63</td></tr><tr><td>RAPHAEL</td><td>3B</td><td>6.61</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td colspan=\"10\">autoregressive model</td></tr><tr><td>CM3Leon†</td><td>7B</td><td>10.82</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Show-o</td><td>1.3B</td><td>9.24</td><td>0.95</td><td>0.52</td><td>0.49</td><td>0.82</td><td>0.11</td><td>0.28</td><td>0.53</td></tr><tr><td>Muse</td><td>3B</td><td>7.88</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Parti</td><td>20B</td><td>7.23</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Fluid (our work)</td><td>369M</td><td>7.23</td><td>0.96</td><td>0.64</td><td>0.53</td><td>0.78</td><td>0.33</td><td>0.46</td><td>0.62</td></tr><tr><td></td><td>665M</td><td>6.84</td><td>0.96</td><td>0.73</td><td>0.51</td><td>0.77</td><td>0.42</td><td>0.51</td><td>0.65</td></tr><tr><td></td><td>1.1B</td><td>6.59</td><td>0.96</td><td>0.77</td><td>0.61</td><td>0.78</td><td>0.34</td><td>0.53</td><td>0.67</td></tr><tr><td></td><td>3.1B</td><td>6.41</td><td>0.98</td><td>0.83</td><td>0.60</td><td>0.82</td><td>0.41</td><td>0.53</td><td>0.70</td></tr><tr><td></td><td>10.5B</td><td>6.16</td><td>0.96</td><td>0.83</td><td>0.63</td><td>0.80</td><td>0.39</td><td>0.51</td><td>0.69</td></tr></table>"
-        },
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-            "type": "title",
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-            "angle": 0,
-            "content": "5.2 BENCHMARKING WITH PREVIOUS SYSTEMS"
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "In this section, we compare our Fluid, i.e., continuous random-order autoregressive model, with leading text-to-image generation systems in Table 1 (Rombach et al., 2022b; Ramesh et al., 2022; Betker et al., 2023; Sahara et al., 2022; Esser et al., 2024; Zhou et al., 2024; Xue et al., 2024; Yu et al., 2023; Xie et al., 2024; Chang et al., 2023; Yu et al., 2022). The Fluid smallest model, with 369M parameters, achieves a zero-shot FID of 7.23 on MS-COCO and a GenEval overall score of 0.62, matching the performance of many state-of-the-art models with several billion parameters (e.g., Parti with 20B parameters only achieves 7.23). The Fluid largest model, with 10.5B parameters, further improves the zero-shot FID on MS-COCO to 6.16 and increases the GenEval overall score to \\(0.69^2\\), with a speed of 1.571 seconds per image per TPU (evaluated on 32 TPU v5 with a batch size of 2048). Detailed model configurations and generation speed results are included in the appendix. We hope these strong results and promising scaling behavior provide valuable insights and support for the scalability of autoregressive models in visual generative modeling."
-        },
-        {
-            "type": "title",
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-            "content": "6 DISCUSSION"
-        },
-        {
-            "type": "text",
-            "bbox": [
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-            "angle": 0,
-            "content": "In this paper, we present an empirical study on the scaling behavior of autoregressive models for text-to-image generation. We investigate two critical design factors: random order versus raster order, and discrete tokens versus continuous tokens. Our results show that random-order models with continuous tokens achieve the best performance and scaling behavior across various evaluation metrics and in terms of visual quality. Building on these findings, we scale up the random-order model with continuous tokens, namely Fluid, to 10.5B parameters, and achieves state-of-the-art text-to-image generation performance. We hope that our findings and promising results could provide valuable insights into the scaling behavior of autoregressive models for image generation and help bridge the gap between the scaling performance of vision models and language models."
-        },
-        {
-            "type": "text",
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-            "angle": 0,
-            "content": "Reproducibility Statement. To aid reproducibility, we have provided the implementation details of our framework in Section 4, training hyper-parameters in Section 5, and model configurations in the appendix. For the diffusion loss used for continuous tokens, we have strictly followed the open-sourced code of Li et al. (2024)."
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-            "content": "Acknowledgements. We would like to express our gratitude to Amy Shen and Alex Rizkowsky for their assistance in securing computational resources, and to Yash Katariya, Ivy Zheng, and Roy Frostig for their valuable insights on JAX-related questions. We also thank Matan Cohen for his support with precomputed T5 embeddings. Our appreciation extends to Han Zhang, Jason Baldridge, Dilip Krishnan, David Salesin and VisCam team for their helpful discussions and valuable feedback."
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-            "content": "2We observe that the GenEval scores plateau for the 10.5B Fluid compared to the 3.1B Fluid; however, it continues to show consistent improvements in visual quality and FID."
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-            "content": "Published as a conference paper at ICLR 2025"
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-            "type": "title",
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-            "content": "REFERENCES"
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