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+# IIANet: An Intra- and Inter-Modality Attention Network for Audio-Visual Speech Separation
+
+Kai Li<sup>1</sup> Runxuan Yang<sup>1</sup> Fuchun Sun<sup>1</sup> Xiaolin Hu<sup>1,2,3</sup>
+
+# Abstract
+
+Recent research has made significant progress in designing fusion modules for audio-visual speech separation. However, they predominantly focus on multi-modal fusion at a single temporal scale of auditory and visual features without employing selective attention mechanisms, which is in sharp contrast with the brain. To address this issue, We propose a novel model called Intra- and Inter-Attention Network (IIANet), which leverages the attention mechanism for efficient audio-visual feature fusion. IIANet consists of two types of attention blocks: intra-attention (IntraA) and inter-attention (InterA) blocks, where the InterA blocks are distributed at the top, middle and bottom of IIANet. Heavily inspired by the way how human brain selectively focuses on relevant content at various temporal scales, these blocks maintain the ability to learn modality-specific features and enable the extraction of different semantics from audio-visual features. Comprehensive experiments on three standard audio-visual separation benchmarks (LRS2, LRS3, and VoxCeleb2) demonstrate the effectiveness of IIANet, outperforming previous state-of-the-art methods while maintaining comparable inference time. In particular, the fast version of IIANet (IIANet-fast) has only  $7\%$  of CTCNet's MACs and is  $40\%$  faster than CTCNet on CPUs while achieving better separation quality, showing the great potential of attention mechanism for efficient and effective multimodal fusion. The source code is released<sup>1</sup>.
+
+$^{1}$ Department of Computer Science and Technology, Institute for AI, BNRist, Tsinghua University, Beijing 100084, China  $^{2}$ Tsinghua Laboratory of Brain and Intelligence (THBI), IDG/McGovern Institute for Brain Research, Tsinghua University, Beijing 100084, China  $^{3}$ Chinese Institute for Brain Research (CIBR), Beijing 100010, China. Correspondence to: Xiaolin Hu <xlu@tsinghua.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+1https://github.com/JusperLee/IIANet
+
+# 1. Introduction
+
+In our daily lives, audio and visual signals are the primary means of information transmission, providing rich cues for humans to obtain valuable information in noisy environments (Cherry, 1953; Arons, 1992). This innate ability is known as the "cocktail party effect", also referred to as "speech separation" in computer science. Speech separation possesses extensive research significance, such as assisting individuals with hearing impairments (Summerfield, 1992), enhancing the auditory experience of wearable devices (Ryu-min et al., 2023), etc.
+
+Researchers have previously sought to address the cocktail party effect through audio streaming alone (Luo & Mesgarani, 2019; Luo et al., 2020; Hu et al., 2021). This task is called audio-only speech separation (AOSS). However, the quality of the separation system may decline dramatically when speech is corrupted by noise (Afouras et al., 2018a). To address this issue, many researchers have focused on audio-visual speech separation (AVSS), achieving remarkable progress. Existing AVSS methods can be broadly classified into two categories based on their fusion modules: CNN-based and Transformer-based. CNN-based methods (Gao & Grauman, 2021; Li et al., 2022; Wu et al., 2019) exhibit lower computational complexity and excellent local feature extraction capabilities, enabling the extraction of audio-visual information at multiple scales to capture local contextual relationships. Transformer-based methods (Lee et al., 2021; Rahimi et al., 2022; Montesinos et al., 2022) can leverage cross-attention mechanisms to learn associations across different time steps, effectively handling dynamic relationships between audio-visual information.
+
+By comparing the working mechanisms of these AVSS methods and the brain, we find that there are two major differences. First, auditory and visual information undergoes neural integration at multiple levels of the auditory and visual pathways, including the thalamus (Halverson & Freeman, 2006; Cai et al., 2019), A1 and V1 (Mesik et al., 2015; Eckert et al., 2008), and occipital cortex (Ghazanfar & Schroeder, 2006; Stein & Stanford, 2008). However, most AVSS methods integrate audio-visual information only at either coarse (Gao & Grauman, 2021; Lee et al., 2021) or fine (Li et al., 2022; Montesinos et al., 2022; Wu et al., 2019;
+
+Rahimi et al., 2022) temporal scale, thus ignoring semantic associations across different scales.
+
+Second, numerous evidences have indicated that the human brain employs selective attention to focus on relevant speech in a noisy environment (Cherry, 1953; Golumbic et al., 2013a). This also modulates neural responses in a frequency-specific manner, enabling the tracked speech to be distinguished and separated from competing speech (Golumbic et al., 2013b; Wikman et al., 2021; Mesgarani & Chang, 2012; O'sullivan et al., 2015; Kaya & Elhilali, 2017). The Transformer-based AVSS methods adopt attention mechanisms (Rahimi et al., 2022; Montesinos et al., 2022), but they commonly employ the Transformer as the backbone network to extract and fuse audio-visual features, and the attention is not explicitly designed for selecting speakers. In contrast to above complex fusion strategies, our approach aims to design a simple and efficient fusion module based on the selective attention mechanism.
+
+To achieve this goal, inspired by the structure and function of the brain, we propose a concise and efficient audio-visual fusion scheme, called Intra- and Inter-Attention Network (IIANet), which leverages different forms of attention $^2$  to extract diverse semantics from audio-visual features. IIANet consists of two hierarchical unimodal networks, mimicking the audio and visual pathways in the brain, and employs two types of attention mechanisms: intra-attention (IntraA) within single modalities and inter-attention (InterA) between modalities.
+
+Following a recent AOSS model (Li et al., 2023), the IntraA employs top-down attention to enhance the ability of the model to select relevant information in unimodal networks with the guidance of higher-level features. This mechanism is inspired by the massive top-down neural projections discovered in both auditory (Guinan Jr, 2006; Budinger et al., 2008) and visual (Angelucci et al., 2002; Felleman & Van Essen, 1991) pathways. The InterA mechanism aims to select relevant information in one modality with the help of the features in the other unimodal network. It is used at all levels of the unimodal networks. Roughly speaking, the InterA block at the top level (InterA-T) corresponds to higher associate cortical areas such as the frontal cortex and occipital cortex (Rajj et al., 2000; Keil et al., 2012; Stein & Stanford, 2008), and at the bottom level (InterA-B) corresponds to the thalamus (Halverson & Freeman, 2006; Cai et al., 2019). The InterA blocks at the middle levels (InterAM) reflect the direct neural projections between different auditory areas and visual areas, e.g., from the core and belt regions of the auditory cortex to the V1 area (Falchier et al., 2002).
+
+On three AVSS benchmarks, we found that IIANet surpassed the previous SOTA method CTCNet (Li et al., 2022) by a considerable margin, while the model inference time was nearly on par with CTCNet. We also built a fast version of IIANet, termed IIANet-fast. It achieved an inference speed on CPU that was more than twice as fast as CTCNet and still yielded better separation quality on three benchmark datasets.
+
+# 2. Related work
+
+# 2.1. Audio-visual speech separation
+
+Incorporating different modalities aligns better with the brain's processing (Calvert, 2001; Bar, 2007). Some methods (Gao & Grauman, 2021; Li et al., 2022; Wu et al., 2019; Lee et al., 2021; Rahimi et al., 2022; Montesinos et al., 2022) have attempted to add visual cues in the AOSS task to improve the clarity of separated audios, called audio-visual speech separation (AVSS). They have demonstrated impressive separation results in more complex acoustic environments, but these methods only focus on audio-visual features fusion at a single temporal scale in both modalities, e.g., only at the finest (Wu et al., 2019; Li et al., 2022) or coarsest (Gao & Grauman, 2021; Lee et al., 2021) scales, which restrains their efficient utilization of different modalities' information. While CTCNet's fusion module (Li et al., 2022) attempts to improve separation quality by utilizing visual and auditory features at different scales, it still focuses on fusion at the finest temporal scales. This unintentionally hinders the effectiveness of using visual information for guidance during the separation process. Moreover, some of the AVSS models (Lee et al., 2021; Montesinos et al., 2022; Sterpu et al., 2018) employ attention mechanisms within the process of inter-modal fusion, while the significance of intra-modal attention mechanisms is overlooked. This deficiency constrains their separation capabilities, as the global information within the auditory modality plays a crucial role in enhancing the clarity of the separated audio (Li et al., 2023). Notably, the fusion module of these methods (Lee et al., 2021; Montesinos et al., 2022; Sterpu et al., 2018) tends to use a similarity matrix to compute the attention, which dramatically increases the computational complexity. In contrast, our proposed IIANet uses the sigmoid function and element-wise product for selective attention, which can efficiently integrate audio-visual features at different temporal scales.
+
+# 2.2. Attention mechanism in speech separation
+
+Attention is a critical function of the brain (Rensink, 2000; Corbetta & Shulman, 2002; Mizokuchi et al., 2023), serving as a pivotal mechanism for humans to cope with complex internal and external environments. Our brain has the ability to focus its superior resources to selectively process
+
+task-relevant information (Schneider, 2013). It is widely accepted that numerous brain regions are involved in the transmission of attention, forming an intricate neural network of interconnections (Posner & Petersen, 1990; Fukushima, 1986). These regions play distinct roles, enabling humans to selectively focus on processing certain information at necessary times and locations while ignoring other perceivable information. Numerous evidences have shown the importance of top-down attention (Fernández et al., 2015; Mesgarani & Chang, 2012) and cross-modal attention (Golumbic et al., 2013b; Wikman et al., 2021) for the brain to select information of interest. In addition, cross-model integration happens in multiple stages, and all can be modulated by attention (Talsma et al., 2010; Ahmed et al., 2023).
+
+Recently, Kuo et al. (2022) constructed artificial neural networks to simulate the attention mechanism of auditory features, confirming that top-down attention plays a critical role in addressing the cocktail party problem. Several recent AOSS models (Shi et al., 2018; Li et al., 2023; Chen et al., 2023) have utilized top-down attention in designing speech separation models. These approaches were able to reduce computational costs while maintaining audio separation quality. They have also shown excellent performance dealing with complex mixture audio. However, the efficient integration of top-down and cross-modal attention in AVSS models remains an unclear aspect.
+
+# 3. The proposed model
+
+Let  $\mathbf{A} \in R^{1 \times T_a}$  and  $\mathbf{V} \in R^{H \times W \times T_v}$  represent the audio and video streams of an individual speaker, respectively, where  $T_a$  denotes the audio length;  $H$ ,  $W$ , and  $T_v$  denote the height, width, and the number of lip frames, respectively. We aim to separate a high-quality, clean single-speaker audio  $\mathbf{A}$  from the noisy speech  $\mathbf{S} \in R^{1 \times T_a}$  based on the video cues  $\mathbf{V}$  and filter out the remaining speech components (other interfering speakers). Specifically, our pipeline consists of four modules: audio encoder, video encoder, separation network and audio decoder (see Figure 1A).
+
+The overall pipeline of the IIANet is summarized as follows. First, we obtain a video containing two speakers from the streaming media, and the lips are extracted from each image frame. Second, we encode the lip frames and noisy speech into lip embeddings  $\mathbf{E}_V \in R^{N_v \times T_v}$  and noisy speech embeddings  $\mathbf{E}_S \in R^{N_a \times T_a'}$  using a video encoder same as CTCNet (Li et al., 2022) and an audio encoder (a 1D convolutional layer), respectively, where  $N_v$ ,  $N_a$  and  $T_a'$  are the visual and audio embedding dimensions and the number of frames for audio embedding, respectively. Third, the separation network takes  $\mathbf{E}_S$  and  $\mathbf{E}_V$  as inputs and outputs a soft mask  $\mathbf{M} \in R^{N_a \times T_a'}$  for the target speaker. We multiply  $\mathbf{E}_S$  with  $\mathbf{M}$  to estimate the target speaker's speech embedding  $(\bar{\mathbf{E}}_S = \mathbf{E}_S \odot \mathbf{M})$ , where “ $\odot$ ” denotes the element-wise prod
+
+uct. Finally, we utilize the audio decoder (a 1D transposed convolutional layer) to convert the  $\bar{\mathbf{E}}_S$  back to the waveform, generating the target speaker's speech  $\bar{\mathbf{A}}\in R^{1\times T_a}$ .
+
+# 3.1. Audio-visual separation network
+
+The core architecture of IIANet is an audio-visual separation network consisting of two types of components: (A) intraattention (IntraA) blocks, (B) inter-attention (InterA) blocks (distributed at different locations: top (InterA-T), middle (InterA-M) and bottom (InterA-B)). The architecture of the separation network is illustrated in Figure 1B, which is described as follows:
+
+1. Bottom-up pass. The audio network and video network take  $\mathbf{E}_S$  and  $\mathbf{E}_V$  as inputs and output multi-scale auditory  $\{\mathbf{S}_i\in R^{N_a\times \frac{T_a'}{2^i}}|i = 0,1\dots ,D\}$  and visual  $\{\mathbf{V}_i\in R^{N_v\times \frac{T_v}{2^i}}|i = 0,1\dots ,D\}$  features, where  $D$  denotes the total number of convolutional layers with a kernel size of 5 and a stride of 2, followed by a global normalization layer (GLN) (Luo & Mesgarani, 2019). In Figure 1B, we show an example model with  $D = 3$  
+2. AV fusion through the InterA-T block. The multiscale auditory  $\mathbf{S}_i$  and visual  $\mathbf{V}_i$  features are fused using the InterA-T block (Figure 2A) at the top of separation network to obtain the inter-modal global features  $\mathbf{S}_G \in R^{N_a \times \frac{T_a'}{2^D}}$  and  $\mathbf{V}_G \in R^{N_v \times \frac{T_v}{2^D}}$ .  
+3. AV fusion in the top-down pass through IntraA and InterA-M blocks. Firstly, the  $\mathbf{S}_G$  and  $\mathbf{V}_G$  are employed to modulate  $\mathbf{S}_i$  and  $\mathbf{V}_i$  within each modality using top-down global IntraA blocks, yielding auditory  $\bar{\mathbf{S}}_i\in R^{N_a\times \frac{T_a'}{2^i}}$  and visual  $\bar{\mathbf{V}}_i\in R^{N_v\times \frac{T_v}{2^i}}$  features. Secondly, the InterA-M block takes the auditory  $\bar{\mathbf{S}}_i$  and visual  $\bar{\mathbf{V}}_i$  features at the same temporal scale as inputs, and output the modulated auditory features  $\tilde{\mathbf{S}}_i\in R^{N_a\times \frac{T_a'}{2^i}}$ . Finally, the top-down pass generates the auditory  $\tilde{\mathbf{S}}_0\in R^{N_a\times T_a'}$  and visual  $\tilde{\mathbf{V}}_0\in R^{N_v\times T_v}$  features at the maximum temporal scale using top-down local IntraA blocks.  
+4. AV fusion through the InterA-B block. The InterA-B block (Figure 2D) takes the auditory  $\tilde{\mathbf{S}}_0$  and visual  $\tilde{\mathbf{V}}_0$  features as input and outputs audio-visual features  $\bar{\mathbf{E}}_{S,t}\in R^{N_a\times T_a'}$  and  $\bar{\mathbf{E}}_{V,t}\in R^{N_v\times T_v}$ , where  $t$  denotes the cycle index of audio-visual fusion.  
+5. Cycle of audio and video networks. The above steps complete the first cycle of the separation network. For the  $t$ -th cycle where  $t \geq 2$ , the audio-visual features  $\bar{\mathbf{E}}_{S,t-1}$  and  $\bar{\mathbf{E}}_{V,t-1}$  obtained in the  $(t-1)$ -th cycle, instead of  $\mathbf{E}_S$  and  $\mathbf{E}_V$ , serve as input to the separation network, and we repeat the above steps. After  $N_F$  cycles, the auditory features  $\bar{\mathbf{E}}_{S,t}$  are refined alone for
+
+![](images/b4ad4205aa23f6356e64ca1442d10264364bcef7bc182d18c39d9299bdfb2390.jpg)  
+A. Overall pipeline
+
+![](images/3b66e1d0df49dc8691143f85f8e0ee7d8e0c7ff3fefaa5ac352e35f295b674d9.jpg)  
+B.IIANet  
+Figure 1. The overall pipeline of IIANet. (A) IIANet consists of four main components: audio encoder, video encoder, separation network, and audio decoder. The red and blue tildes indicate that the same module is repeated several times. (B) The separation network contains two types of attention blocks: IntraA and InterA (InterA-T, InterA-M, InterA-B) blocks. The dashed lines indicate the use of global features  $\mathbf{S}_G$  and  $\mathbf{V}_G$  as top-down attention modulation for multi-scale features  $\mathbf{S}_i$  and  $\mathbf{V}_i$ . All blocks use different parameters but keep the same across different cycles.
+
+$N_{S}$  cycles in the audio network to reconstruct high-quality audio. Finally, we pass the output of the final cycle through an activation function (ReLU) to yield the separation network's output M (Figure 1A).
+
+The steps 2-4 and the blocks mentioned in these steps are described in details in what follows.
+
+Step 2: AV fusion through the InterA-T block. In the InterA-T block, we first utilize an average pooling layer with a pooling ratio  $2^{D - i}$  in the temporal dimension to down-sample the temporal dimensions of  $\mathbf{S}_i$  and  $\mathbf{V}_i$  to  $\frac{T_a'}{2^D}$  and  $\frac{T_v}{2^D}$ , respectively, and then merge them to obtain global features  $\mathbf{S}_G$  and  $\mathbf{V}_G$  using InterA-T block (Figure 2A). The detailed process of the InterA-T block is described by the
+
+equations below:
+
+$$
+\boldsymbol {f} _ {S} = \sum_ {i = 0} ^ {D - 1} p _ {i} (\mathbf {S} _ {i}) + \mathbf {S} _ {D} \text {a n d} \boldsymbol {f} _ {V} = \sum_ {i = 0} ^ {D - 1} p _ {i} (\mathbf {V} _ {i}) + \mathbf {V} _ {D},
+$$
+
+$$
+\mathbf {S} _ {G} = \operatorname {F F N} _ {S} \left(\boldsymbol {f} _ {S} \odot \sigma (\mathcal {Q} (\boldsymbol {f} _ {V}))\right),
+$$
+
+$$
+\mathbf {V} _ {G} = \operatorname {F F N} _ {V} \left(\boldsymbol {f} _ {V} \odot \sigma (\mathcal {Q} (\boldsymbol {f} _ {S}))\right), \tag {1}
+$$
+
+where  $\pmb{f}_{S} \in R^{N_{a} \times \frac{T_{a}^{\prime}}{2^{D}}}$  and  $\pmb{f}_{V} \in R^{N_{v} \times \frac{T_{v}}{2^{D}}}$  denote the cumulative multi-scale auditory and visual features,  $\mathrm{FFN}(\cdot)$  denotes a feed-forward network with three convolutional layers followed by a GLN,  $\mathcal{Q}^3$  denotes a convolutional layer followed by a GLN,  $p_{i}(\cdot)$  denotes the average pooling layers and  $\sigma$  denotes the sigmoid function. Here,  $p_{i}(\cdot)$  uses
+
+![](images/4b8528b1ac962c310406b95ce69a2b9ecb2c750501da6c9fe896eed70a6a56ac.jpg)  
+A. InterA-T
+
+![](images/2dff5f384d95d8c00105dbc6da6713b7df9f8bbe0232f93583521413f4081032.jpg)  
+B. IntraA
+
+![](images/c4786828df379da511a7edc853c94a4958d2bf64ef4606310834e65e0f426561.jpg)  
+C. InterA-M
+
+![](images/5d8b1d335875db4f6fde1f902609d087eacfc1f77f847e79092b607ea57815e7.jpg)  
+D. InterA-B  
+Figure 2. Flow diagram of InraA and InterA blocks: (A) InterA-T block, (B) IntraA block, (C) InterA-M block and (D) InterA-B block in the IIANet, where  $\odot$  denotes element-wise product and  $\sigma$  denotes the sigmoid function.
+
+Element-wise product  
+$\oplus$  Addition  $\rightarrow$  Audio flow  
+Up-sample Video flow
+
+different pooling ratios for different  $i$  such that the temporal dimensions of all  $p_i(\mathbf{S}_i)$  are  $\frac{T_a'}{2^D}$  and the temporal dimensions of all  $p_i(\mathbf{V}_i)$  are  $\frac{T_v}{2^D}$ . The hyperparameters of  $\mathcal{Q}$  are specified such that the resulting audio and video features have the same embedding dimension. The  $\sigma$  function receiving information from one modality acts as selective attention, modulating the other modality. The global features  $\mathbf{S}_G$  and  $\mathbf{V}_G$  are used to guide the separation process in the top-down pass.
+
+Step 3: AV fusion in the top-down pass through IntraA and InterA-M blocks. Given the multi-scale audio-visual features  $\{\mathbf{S}_i, \mathbf{V}_i\}$  and  $\mathbf{S}_G$  and  $\mathbf{V}_G$  as inputs, we employ the global IntraA blocks  $\phi(\mathbf{x}, \mathbf{y})$  (Figure 2B) in the top-down pass (the dotted lines in Figure 1B) to extract audio  $\bar{\mathbf{S}}_i$  and visual features  $\bar{\mathbf{V}}_i$  at each temporal scale,
+
+$$
+\bar {\mathbf {x}} = \phi (\mathbf {x}, \mathbf {y}) = \sigma \left(\mathcal {Q} (\mu (\mathbf {y}))\right) \odot \mathbf {x} + \mathcal {Q} (\mu (\mathbf {y})) \tag {2}
+$$
+
+where  $\mu (\cdot)$  denotes interpolation up-sampling. For audio signals,  $\bar{\mathbf{S}}_i = \phi (\mathbf{S}_i,\mathbf{S}_G)$ ; for video signals,  $\bar{\mathbf{V}}_i = \phi (\mathbf{V}_i,\mathbf{V}_G)$ . The hyperparameters of  $\mu$  are specified such that the resulted audio and video signals have the same temporal dimension. Since these attention blocks use the global features  $\mathbf{S}_G$  and  $\mathbf{V}_G$  to modulate the intermediate features  $\mathbf{S}_i$  and  $\mathbf{V}_i$  when reconstructing these features in the top-down process, they are called global IntraA blocks. This
+
+is inspired by the global attention mechanism used in an AOSS model (Li et al., 2023). But in that model, the global feature is simply upsampled, then goes through a sigmoid function and multiplied with the intermediate features  $\mathbf{x}$ . Mathematically, the process is written as follows
+
+$$
+\bar {\mathbf {x}} = \phi^ {\prime} (\mathbf {x}, \mathbf {y}) = \sigma (\mu (\mathbf {y})) \odot \mathbf {x}. \tag {3}
+$$
+
+We empirically found that  $\phi$  worked better than  $\phi^{\prime}$  (see Appendix A).
+
+In contrast to previous AVSS methods (Gao & Grauman, 2021; Li et al., 2022), we investigate which part of the audio network should be attended to undergo the guidance of visual features within the same temporal scale. We utilize the visual features  $\bar{\mathbf{V}}_i$  from the same temporal scale as  $\bar{\mathbf{S}}_i$  to modulate the auditory features  $\bar{\mathbf{S}}_i$  using the InterA-M blocks (Figure 2C),
+
+$$
+\tilde {\mathbf {S}} _ {i} = \sigma (\mathcal {Q} (\mu (\bar {\mathbf {V}} _ {i}))) \odot \bar {\mathbf {S}} _ {i}, \quad \forall i \in [ 0, D ], \tag {4}
+$$
+
+where  $\tilde{\mathbf{S}}_i\in R^{N\times \frac{T_a'}{2^i}}$  denotes the modulated auditory features. Same as before, the hyperparameters of  $\mathcal{Q}$  and  $\mu$  are specified such that the two terms on the two sides of  $\odot$  have the same dimension. In this way, the InterA-M blocks are expected to extract the target speaker's sound features
+
+according to the target speaker's visual features at the same scale.
+
+Then, we reconstructed auditory and visual features separately to obtain the features  $(\check{\mathbf{S}}_0,\check{\mathbf{V}}_0)$  in a top-down pass (Figure 1B). We leverage the local top-down attention mechanism (Li et al., 2023), and the corresponding blocks are called local IntraA blocks, which are able to fuse as many multi-scale features as possible while reducing information redundancy. We start from the  $(D - 1)$ -th layer. The attention signals are  $\tilde{\mathbf{S}}_D$  and  $\bar{\mathbf{V}}_D$ . Therefore,
+
+$$
+\check {\mathbf {S}} _ {D - 1} = \phi (\tilde {\mathbf {S}} _ {D - 1}, \tilde {\mathbf {S}} _ {D}), \quad \check {\mathbf {V}} _ {D - 1} = \phi (\bar {\mathbf {V}} _ {D - 1}, \bar {\mathbf {V}} _ {D}). (5)
+$$
+
+For  $i = D - 2, \dots, 0$ , the attention signals are  $\tilde{\mathbf{S}}_i$  and  $\tilde{\mathbf{V}}_i$ , therefore
+
+$$
+\check {\mathbf {S}} _ {i} = \phi (\tilde {\mathbf {S}} _ {i}, \check {\mathbf {S}} _ {i + 1}), \quad \check {\mathbf {V}} _ {i} = \phi (\bar {\mathbf {V}} _ {i}, \check {\mathbf {V}} _ {i + 1}). \qquad (6)
+$$
+
+Please note that different IntraA blocks within one cycle, as depicted in Figure 1B, use different parameters.
+
+Step 4: AV fusion through the InterA-B block. In CTC-Net (Li et al., 2022), the global audio-visual features fused all auditory and visual multi-scale features using the concatenation strategy, leading to high computational complexity. Here, we only perform inter-modal fusion on the finest-grained auditory and visual features  $(\check{\mathbf{S}}_0$  and  $\check{\mathbf{V}}_0)$  using attention strategy, greatly reducing computational complexity.
+
+Specifically, we reduce the impact of redundant features by incorporating the InterA-B block (Figure 2D) between the finest-grained auditory and visual features  $\tilde{\mathbf{S}}_0$  and  $\tilde{\mathbf{V}}_0$ . In particular, the process of global audio-visual fusion is as follows:
+
+$$
+\begin{array}{l} \bar {\mathbf {E}} _ {S, t} = \check {\mathbf {S}} _ {0} + \mathcal {Q} (\mu (\check {\mathbf {V}} _ {0}) \odot \sigma (\mathcal {Q} (\check {\mathbf {S}} _ {0}))), \\ \bar {\bar {\mathbf {S}}} = \check {\mathbf {S}} _ {1} + \mathcal {Q} ((\tilde {\mathbf {T}} _ {1}) \odot (\mathcal {Q} (\check {\mathbf {T}} _ {1}))). \end{array} \tag {7}
+$$
+
+$$
+\bar {\mathbf {E}} _ {V, t} = \bar {\mathbf {V}} _ {0} + \mathcal {Q} \left(\mu (\bar {\mathbf {S}} _ {0}) \odot \sigma \left(\mathcal {Q} (\bar {\mathbf {V}} _ {0})\right)\right).
+$$
+
+Same as in (1), the hyperparameters of  $\mathcal{Q}$  and  $\mu$  are specified such that the element-wise product and addition in (5) function correctly. The working process of this block is simple. One modality features are modulated by the other modality features through an attention function  $\sigma$ , and then the results are added to the original features. Several convolutions and up-samplings are interleaved in the process to ensure that the dimensions of results are appropriate. We keep the original features in the final results by addition because we do not want the final features to be altered too much by the other modality information.
+
+Finally, the output auditory and visual features  $\bar{\mathbf{E}}_{S,t}$  and  $\bar{\mathbf{E}}_{V,t}$  serve as inputs for the subsequent audio network and video network.
+
+# 4. Experiments
+
+# 4.1. Datasets
+
+Consistent with previous research, we experimented on three commonly used AVSS datasets: LRS2 (Afouras et al., 2018a), LRS3 (Afouras et al., 2018b), and VoxCeleb2 (Chung et al., 2018). Due to the fact that the LRS2 and VoxCeleb2 datasets were collected from YouTube, they present a more complex acoustic environment compared to the LRS3 dataset. Each audio was 2 seconds in length with a sample rate of  $16\mathrm{kHz}$ . The mixture was created by randomly selecting two different speakers from datasets and mixing their speeches with signal-to-noise ratios between  $-5\mathrm{dB}$  and  $5\mathrm{dB}$ . Lip frames were synchronized with audio, having a frame rate of 25 FPS and a size of  $88\times 88$  grayscale images. We used the same dataset split consistent with previous works (Li et al., 2022; Gao & Grauman, 2021; Lee et al., 2021). See Appendix B for details.
+
+# 4.2. Model configurations and evaluation metrics
+
+The IIANet model was implemented using PyTorch (Paszke et al., 2019), and optimized using the Adam optimizer (Kingma & Ba, 2014) with an initial learning rate of 0.001. The learning rate was cut by  $50\%$  whenever there was no improvement in the best validation loss for 15 consecutive epochs. The training was halted if the best validation loss failed to improve over 30 successive epochs. Gradient clipping was used during training to avert gradient explosion, setting the maximum  $L2$  norm to 5. We used the SI-SNR objective function to train the IIANet. See Appendix C for details. All experiments were conducted with a batch size of 6, utilizing 8 GeForce RTX 4090 GPUs. The hyperparameters are included in Appendix D.
+
+Same as previous research (Gao & Grauman, 2021; Li et al., 2022; Afouras et al., 2018a), the SDRi (Vincent et al., 2006b) and SI-SNRi (Le Roux et al., 2019) were used as a metric for speech separation. See Appendix C for details. To simulate human auditory perception and to assess and quantify the quality of speech signals, we also used PESQ (Union, 2007) as an evaluation metric, which is used to measure the clarity and intelligibility of speech signals, ensuring the reliability of the results.
+
+# 4.3. Comparisons with state-of-the-art methods
+
+To explore the efficiency of the proposed approach, we have designed a faster version, namely IIANet-fast, which runs half of  $N_{S}$  cycles compared to IIANet (see Appendix D for detailed parameters). We compared IIANet and IIANet-fast with existing AVSS methods. To conduct a fair comparison, we obtained metric values in original papers or reproduced the results using officially released models by the authors.
+
+Table 1. Separation results of different AVSS methods on LRS2, LRS3, and VoxCeleb2 datasets. These metrics represent the average values for all speakers in each test set, where larger SI-SNri, SDRi and PESQ values are better. “-” denotes results not reported in the original paper. Bolds indicate the best while wave line indicate the second best.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">LRS2</td><td colspan="3">LRS3</td><td colspan="3">VoxCeleb2</td></tr><tr><td>SI-SNRI</td><td>SDRi</td><td>PESQ</td><td>SI-SNRI</td><td>SDRi</td><td>PESQ</td><td>SI-SNRI</td><td>SDRi</td><td>PESQ</td></tr><tr><td>AVConvTasNet (Wu et al., 2019)</td><td>12.5</td><td>12.8</td><td>2.69</td><td>11.2</td><td>11.7</td><td>2.58</td><td>9.2</td><td>9.8</td><td>2.17</td></tr><tr><td>LWTNet (Afouras et al., 2020)</td><td>-</td><td>10.8</td><td>-</td><td>-</td><td>4.8</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>VisualVoice (Gao &amp; Grauman, 2021)</td><td>11.5</td><td>11.8</td><td>2.78</td><td>9.9</td><td>10.3</td><td>2.13</td><td>9.3</td><td>10.2</td><td>2.45</td></tr><tr><td>CaffNet-C (Lee et al., 2021)</td><td>-</td><td>10.0</td><td>1.15</td><td>-</td><td>9.8</td><td>-</td><td>-</td><td>7.6</td><td>-</td></tr><tr><td>AVLiT-8 (Martel et al., 2023)</td><td>12.8</td><td>13.1</td><td>2.56</td><td>13.5</td><td>13.6</td><td>2.78</td><td>9.4</td><td>9.9</td><td>2.23</td></tr><tr><td>CTCNet (Li et al., 2022)</td><td>14.3</td><td>14.6</td><td>3.08</td><td>17.4</td><td>17.5</td><td>3.24</td><td>11.9</td><td>13.1</td><td>3.00</td></tr><tr><td>IIANet (ours)</td><td>16.0</td><td>16.2</td><td>3.23</td><td>18.3</td><td>18.5</td><td>3.28</td><td>13.6</td><td>14.3</td><td>3.12</td></tr><tr><td>IIANet-fast (ours)</td><td>15.1</td><td>15.4</td><td>3.11</td><td>17.8</td><td>17.9</td><td>3.25</td><td>12.6</td><td>13.6</td><td>3.01</td></tr></table>
+
+Table 2. Parameter sizes and computational complexities of different AVSS methods. All results are measured with an input audio length of 1 second, a sample rate of  ${16}\mathrm{{kHz}}$  ,and a video frame sample rate of 25 FPS. Inference time is measured in the same testing environment without the use of any additional acceleration techniques, such as quantization or pruning.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">Computation Cost</td><td colspan="2">Inference Time</td><td rowspan="2">GPU Memory (MB)</td></tr><tr><td>MACs (G)</td><td>Params (M)</td><td>CPU (s)</td><td>GPU (ms)</td></tr><tr><td>AVConvTasNet (Wu et al., 2019)</td><td>23.8</td><td>16.5</td><td>1.06</td><td>62.51</td><td>117.05</td></tr><tr><td>VisualVoice (Gao &amp; Grauman, 2021)</td><td>9.7</td><td>77.8</td><td>2.98</td><td>110.31</td><td>313.74</td></tr><tr><td>AVLiT (Martel et al., 2023)</td><td>18.2</td><td>5.8</td><td>0.46</td><td>62.51</td><td>24.0</td></tr><tr><td>CTCNet (Li et al., 2022)</td><td>167.1</td><td>7.0</td><td>1.26</td><td>84.17</td><td>75.8</td></tr><tr><td>IIANet (ours)</td><td>18.6</td><td>3.1</td><td>1.27</td><td>110.11</td><td>12.5</td></tr><tr><td>IIANet-fast (ours)</td><td>11.9</td><td>3.1</td><td>0.52</td><td>70.94</td><td>12.5</td></tr></table>
+
+Separation quality. As demonstrated in Table 1, IIANet consistently outperformed existing methods across all datasets, with at least a  $0.9\mathrm{~dB}$  gain in SI-SN Ri. In particular, IIANet exceeded in complex environments, notably on the LRS2 and VoxCeleb2 datasets, outperforming AVSS methods by a minimum of  $1.7\mathrm{~dB}$  SI-SN Ri in separation quality. This significant improvement highlights the importance of IIANet's hierarchical audio-visual integration. In addition, IIANet-fast also achieved excellent results, which surpassed CTCNet in separation quality with an improvement of  $1.1\mathrm{~dB}$  in SI-SN Ri on the challenging LRS2 dataset.
+
+Model size and computational cost. In practical applications, model size and computational cost are often crucial considerations. We employed three hardware-agnostic metrics to circumvent disparities between different hardware, including number of parameters (Params) and Multiply-Accumulate operations  $(\mathrm{MACs})^4$ , and GPU memory usage during inference. We also selected two hardware-related metrics to measure inference speed on specific GPU and CPU hardware. The results were shown in Table 2. Please note that some of the AVSS methods (Alfouras et al., 2018; Afouras et al., 2020; Lee et al., 2021) in Table 1 are not
+
+For applications that need to run models in resource-constrained environments, such as mobile devices and edge computing, MACs is a key metric because it is directly related to power consumption, latency, and storage requirements.
+
+included in Table 2 because they are not open sourced.
+
+Compared to the previous SOTA method CTCNet, IIANet can achieve significantly higher separation quality using  $11\%$  of the MACs,  $44\%$  of the parameters and  $16\%$  of the GPU memory. IIANet's inference on CPU was as fast as CTCNet, but was slower on GPU. With better separation quality than CTCNet, IIANet-fast had much less computational cost (only  $7\%$  of CTCNet's MACs) and substantially higher inference speed ( $40\%$  of CTCNet's time on CPU and  $84\%$  of CTCNet's time on GPU). This not only showcased the efficiency of IIANet-fast but also highlighted its aptness for deployment onto environments with limited hardware resources and strict real-time requirement.
+
+In conclusion, compared to with previous AVSS methods, IIANet achieved a good trade-off between separation quality and computational cost.
+
+# 4.4. Multi-speaker performance
+
+We present results on the LRS2 dataset with 3 and 4 speakers. We created two variants of the dataset for these purposes, named LRS2-3Mix and LRS2-4Mix, both used in our model's training and testing. Correspondingly, the default LRS2 dataset has a new name LRS2-2Mix. Throughout the paper, unless otherwise specified, LRS2 dataset refers to the default LRS2-2Mix dataset. For more details on dataset con
+
+struction, training processes, and testing procedures, please refer to Appendix F.
+
+We use the following baseline methods for comparison: AVConvTasNet (Wu et al., 2019), AVLIT (Martel et al., 2023), and CTCNet (Li et al., 2022). The results are presented in Table 3. Each table column shows a different dataset, where the number of speakers in the mixed signal  $C$  varies. The models used for evaluating each dataset were specifically trained for separating a corresponding number of speakers. It is evident from the results that the proposed model significantly outperformed the previous methods across all three datasets.
+
+# 4.5. Ablation study
+
+To better understand IIANet, we compared each key component using the LRS2 dataset in a completely fair setup, i.e., using the same architecture and hyperparameters in the following experiments.
+
+Importance of IntraA and InterA. We investigated the role of IntraA and InterA in model performance. To this end, we constructed a control model (called Control 1) resulted from removing IntraA and InterA blocks from IIANet. Basically, it uses two upside down UNet (Ronneberger et al., 2015) architectures as the visual and auditory backbones and fuses visual and auditory features at the finest temporal scale (Figure 4A in Appendix). We then added IntraA blocks to Control 1 to obtain a new model, called Control 2 (Figure 4B in Appendix). The detailed description of Controls 1 and 2 can be found in Appendix G and the training and inference procedures are the same as IIANet. As shown in Table 4, Control 1 obtained poor results, and adding IntraA blocks improved the results. This improvement validated the effectiveness of the IntraA blocks in the AVSS task, though it was originally proposed for AOSS (Li et al., 2023), because the IntraA module obtained intra-modal contextual features. Adding InterA blocks to Control 2 yielded the proposed IIANet, which obtained a substantial separation quality improvement by  $2.4\mathrm{dB}$  and  $2.2\mathrm{dB}$  in SI-SNRi and SDRi metrics, respectively. This proves that the InterA module enables the visual information to exploit and extract the relevant auditory features fully. Taken together, the results show the effectiveness of the IntraA and InterA blocks.
+
+Importance of different InterA blocks. Table 5 reports the performances of different combinations of the three InterA blocks: InterA-T, InterA-M and InterA-B. Please note that excluding InterA-T block means directly using  $\sum_{i=0}^{D-1} p(\mathbf{S}_i) + \mathbf{S}_D$  and  $\sum_{i=0}^{D-1} p(\mathbf{V}_i) + \mathbf{V}_D$  as input to auditory and visual MLPs, respectively, with cross-modal attention signal removed. Excluding InterA-M block means removing visual fusion branches at each temporal scale. Excluding InterA-B block means not performing fine-grained audio-visual fusion.
+
+When only one type of attention blocks is enabled in the inter-modal settings, the InterA-M block results in the highest separation quality. When two types of attention blocks are enabled, the combination of InterA-M and InterA-B yielded the best results. The combination of all three types of blocks performed the best.
+
+Comparison different AV fusion module implementations To validate the effectiveness of IIANet, we used Control 2 shown in Figure 4B as the baseline model to investigate the performance of different AV fusion strategies (Lee et al., 2021; Montesinos et al., 2022; Sterpu et al., 2018). The reason for choosing Control 2 as the baseline is that its audio and video networks share roughly the same structure as the TDANet, an AOSS model proposed in (Li et al., 2023), which has shown excellent performance. We investigated the following three fusion strategies.
+
+(1) Concatenate the visual and auditory features along the channel dimension, utilize the self-attention mechanism from the Transformer to integrate the AV information, and return it to the visual and auditory networks. Specifically, we first downsample  $\check{\mathbf{S}}_0\in \mathbb{R}^{N_a\times T_a'}$  to the same temporal dimension as  $\tilde{\mathbf{V}}_0\in \mathbb{R}^{N_v\times T_v}$ , obtaining  $\tilde{\mathbf{S}}_0^{\prime}\in \mathbb{R}^{N_a\times T_v}$ . Similarly, we upsample  $\check{\mathbf{V}}_0$  to the same temporal dimension as  $\check{\mathbf{S}}_0$ , yielding  $\check{\mathbf{V}}_0^{\prime}\in \mathbb{R}^{N_a\times T_a'}$ . Subsequently, we concatenate  $(\check{\mathbf{S}}_0,\check{\mathbf{V}}_0^{\prime})$  and  $(\tilde{\mathbf{V}}_0,\tilde{\mathbf{S}}_0^{\prime})$  along the channel dimension, resulting in the fused features  $\check{\mathbf{F}}_{\mathbf{s}}\in \mathbb{R}^{(N_a + N_v)\times T_a^{\prime}}$  and  $\check{\mathbf{F}}_{\mathbf{v}}\in \mathbb{R}^{(N_a + N_v)\times T_v}$ , respectively. These fused features,  $\check{\mathbf{F}}_{\mathbf{s}}$  and  $\check{\mathbf{F}}_{\mathbf{v}}$ , are then fed as inputs to the self-attention mechanism implemented within the Transformer. Finally, the output features are returned to the auditory and visual networks, respectively.  
+(2) Use the visual and auditory features as the Query for two separate cross-attention layers to fuse AV features, respectively. Specifically, we first obtain  $\check{\mathbf{V}}_0^{\prime}$  and  $\check{\mathbf{S}}_0^{\prime}$  using the upsampling and downsampling operations as described in (1). Then, we assign  $\check{\mathbf{V}}_0^{\prime}$  and  $\check{\mathbf{S}}_0^{\prime}$  as the Q values for the auditory and visual attention layers, respectively. For the auditory attention layer,  $\check{\mathbf{S}}_0$  serves as both the K and V values, while for the visual attention layer,  $\check{\mathbf{V}}_0$  serves as both the K and V values. Finally, the fused features obtained from the two cross-attention layers are used as inputs to the auditory and visual networks, respectively.  
+(3) Employ two self-attention mechanisms to extract visual and auditory features separately, then integrate the audiovisual features using the second fusion strategy (see above). Specifically, we first feed  $\tilde{\mathbf{S}}_0$  and  $\tilde{\mathbf{V}}_0$  into two separate self-attention layers to extract auditory and visual features, respectively. Then, we employ the cross-attention mechanism, consistent with the imple
+
+Table 3. Performance of different models varies with the number of speakers. These results were based on training conducted using the code from published baseline methods.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">LRS2-2Mix</td><td colspan="2">LRS2-3Mix</td><td colspan="2">LRS2-4Mix</td></tr><tr><td>SI-SN Ri</td><td>SDRi</td><td>SI-SN Ri</td><td>SDRi</td><td>SI-SN Ri</td><td>SDRi</td></tr><tr><td>AVConvTasNet (Wu et al., 2019)</td><td>12.5</td><td>12.8</td><td>8.2</td><td>8.8</td><td>4.1</td><td>4.6</td></tr><tr><td>AVLIT-8 (Martel et al., 2023)</td><td>12.8</td><td>13.1</td><td>9.4</td><td>9.9</td><td>5.0</td><td>5.7</td></tr><tr><td>CTCNet (Li et al., 2022)</td><td>14.3</td><td>14.6</td><td>10.3</td><td>10.8</td><td>6.3</td><td>6.9</td></tr><tr><td>IIANet (ours)</td><td>16.0</td><td>16.2</td><td>12.6</td><td>13.1</td><td>7.8</td><td>8.3</td></tr></table>
+
+Table 4. Importance of IntraA blocks and InterA blocks on the LRS2 test set. MACs are measured with 1s input audio sampled at  ${16}\mathrm{{kHz}}$  .  
+
+<table><tr><td>Model</td><td>SI-SNRi/SDRi</td><td>Params (M)/MACs (G)</td></tr><tr><td>Control 1</td><td>13.1/13.4</td><td>2.2/16.7</td></tr><tr><td>Control 2</td><td>13.6/14.0</td><td>2.2/18.1</td></tr><tr><td>IIANet</td><td>16.0/16.2</td><td>3.1/18.6</td></tr></table>
+
+Table 5. Impact of each InterA block in IIANet on the LRS2 test set.  
+
+<table><tr><td>InterA-T</td><td>InterA-M</td><td>InterA-B</td><td>SI-SN Ri / SN Ri</td></tr><tr><td>✓</td><td>×</td><td>×</td><td>13.8 / 14.1</td></tr><tr><td>×</td><td>✓</td><td>×</td><td>14.5 / 14.7</td></tr><tr><td>×</td><td>×</td><td>✓</td><td>14.1 / 14.4</td></tr><tr><td>✓</td><td>✓</td><td>×</td><td>14.8 / 15.1</td></tr><tr><td>✓</td><td>×</td><td>✓</td><td>15.0 / 15.2</td></tr><tr><td>×</td><td>✓</td><td>✓</td><td>15.5 / 15.7</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>16.0 / 16.2</td></tr></table>
+
+mentation described in (2), to fuse the auditory and visual features. Finally, the fused features are used as inputs to the auditory and visual networks, respectively.
+
+Table 6. Results of different audio-visual fusion strategies on the LRS2 dataset.  
+
+<table><tr><td>Strategy</td><td>SI-SN Ri</td><td>SDRi</td><td>Params (M)</td><td>MACs (G)</td></tr><tr><td>(1) SA</td><td>13.3</td><td>13.7</td><td>7.8</td><td>12.7</td></tr><tr><td>(2) CA</td><td>13.7</td><td>14.0</td><td>5.2</td><td>12.3</td></tr><tr><td>(3) SA + CA</td><td>14.1</td><td>14.3</td><td>7.3</td><td>12.7</td></tr><tr><td>Ours</td><td>16.0</td><td>16.2</td><td>3.1</td><td>11.9</td></tr></table>
+
+Experimental results are shown in Table 6. Compared to IIANet, the three fusion strategies did not achieve better separation quality and had higher parameter counts and computational complexity. This demonstrates the effectiveness of IIANet's hierarchical selective attention mechanism in improving separation performance.
+
+More analyses. We also performed additional analyses of IIANet. First, we found that networks that include visual information significantly improve speech separation performance compared to networks that rely only on audio information (see Appendix E for details). Then, we improve
+
+the separation quality by gradually increasing the number of fusions (1 to 5) (Appendix H). Meanwhile, we investigated the separation quality of three different pre-trained video encoder models on the LRS2-2Mix dataset (Appendix I). In addition, we explored the relationship between the loss of visual cues (especially the speaker's side profile) and the separation quality (Appendix J). The samples from the side view angle showed a significant decrease in separation quality compared to the frontal view. Compared to other AVSS methods, IIANet had the smallest decrease of about  $4\%$ . Finally, we applied a dynamic mixing data enhancement scheme and obtained even better results (Appendix K). Unless specifically stated, experiments did not utilize dynamic mixing data enhancement.
+
+# 4.6. Qualitative evaluation
+
+We visually compare the speech separation results of AVConvTasNet, VisualVoice, CTCNet, and our IIANet, all trained on the LRS2 dataset. We use the spectrogram of the separated audios to demonstrate the quality of the separated signals, as shown in Figure 6 in Appendix L. In the left part, we observe that IIANet achieves better reconstruction results. See Appendix L for details. In addition, we have investigated cases with suboptimal separation effects (Appendix M).
+
+Besides, we evaluated different AVSS methods (VisualVoice, CTCNet) in real-world scenarios by collecting multi-speaker videos on YouTube. Some typical results are presented in website<sup>5</sup>. By listening to these results, one can confirm that IIANet generated higher-quality separated audio than other separation models.
+
+# 5. Conclusion and discussion
+
+We proposed a brain-inspired AVSS model, which is characterized by extensive use of intra-attention and inter-attention in the audio and video networks. Compared with the previous SOTA method CTCNet on three benchmark datasets, IIANet achieved significantly higher separation quality with  $11\%$  MACs,  $44\%$  parameters and  $16\%$  GPU memory. By reducing the number of cycles, IIANet ran much faster than CTCNet yet still obtained better separation results.
+
+# Impact Statement
+
+The deployment and utilization of voice separation technology, while innovative, harbors potential for significant negative impacts across various domains. This technology's capability to unauthorizedly capture and dissect individual conversations poses a stark invasion of privacy, risking the exposure and malicious analysis of personal and organizational dialogues. In the realm of security, particularly concerning systems reliant on voice recognition, such technology could be exploited by hackers to bypass protective measures by isolating and replicating specific voice commands. Additionally, it may facilitate undue surveillance in workplaces or public areas, challenging ethical and legal norms regarding individual rights. Moreover, the technology's ability to extract voices from their contexts could foster misinformation, damaging reputations and undermining trust in society. These issues underscore the critical need for careful management and ethical guidelines in deploying voice separation technology to prevent its misuse while safeguarding privacy and security.
+
+# Acknowledgements
+
+This work was supported in part by the National Key Research and Development Program of China (No. 2021ZD0200301) and the National Natural Science Foundation of China (No. U2341228).
+
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+
+# A. Evaluation on different global IntraA implementations
+
+In this experiment, we investigated the performance of two implementations of global IntraA blocks in IIANet:  $\phi (\mathbf{x},\mathbf{y})$  and  $\phi^{\prime}(\mathbf{x},\mathbf{y})$  (see eqns. (2) and (3)). The results in Table 6 indicate that global IntraA blocks with  $\phi (\mathbf{x},\mathbf{y})$  significantly improves separation quality without notably increasing computational complexity. This finding underscores the importance of global IntraA with  $\phi (\mathbf{x},\mathbf{y})$  in effectively and accurately handing intra-modal contextual features, thereby substantially improving performance.
+
+Table 7. Comparison of separation quality and efficiency among different global IntraA implementations.  
+
+<table><tr><td>Implementations</td><td>SI-SN Ri</td><td>SDRi</td><td>Params (M)</td><td>MACs (G)</td></tr><tr><td>φ(x,y)</td><td>16.0</td><td>16.2</td><td>3.1</td><td>18.64</td></tr><tr><td>φ&#x27;(x,y)</td><td>14.7</td><td>15.0</td><td>3.1</td><td>17.53</td></tr></table>
+
+# B. Dataset details
+
+Lip Reading Sentences 2 (LRS2) (Afouras et al., 2018a). The LRS2 dataset consists of thousands of BBC video clips, divided into Train, Validation, and Test folders. We used the same dataset consistent with previous works (Li et al., 2022; Gao & Grauman, 2021; Lee et al., 2021), created by randomly selecting two different speakers from LRS2 and mixing their speeches with signal-to-noise ratios between -5dB and 5dB. Since the LRS2 data contains reverberation and noise, and the overlap rate is not  $100\%$ , the dataset is closer to real-world scenarios. We use the same data split containing 11-hour training, 3-hour validation, and 1.5-hour test sets.
+
+Lip Reading Sentences 3 (LRS3) (Afouras et al., 2018b). The LRS3 dataset includes thousands of spoken sentences from YouTube but are from TED talks, generally cleaner environments. We used the same dataset consistent with previous works (Li et al., 2022; Gao & Grauman, 2021; Lee et al., 2021), which is constructed by randomly selecting the voices of two different speakers from the LRS3 data. In contrast to LRS2, the LRS3 data has relatively less noise and is closer to separation tasks in clean environments. It comprises 28-hour training, 3-hour validation, and 1.5-hour test sets.
+
+VoxCeleb2 (Chung et al., 2018). The VoxCeleb2 dataset contains over one million sentences from 6,112 individuals extracted from YouTube videos, divided into Dev and Test folders. We used the same dataset consistent with previous works (Li et al., 2022; Gao & Grauman, 2021; Lee et al., 2021), constructed by selecting  $5\%$  of the data from the Dev folder of VoxCeleb2 for creating training and validation sets. Similar to LRS2, VoxCeleb2 also contains a significant amount of noise and reverberation, making it closer to real-world scenarios, but the acoustic environment of VoxCeleb2 is more complex and challenging. It comprises 56-hour training, 3-hour validation, and 1.5-hour test sets.
+
+# C. object function and metrics
+
+We used scale-invariant source-to-noise ratio (SI-SNR) (Le Roux et al., 2019) as the objective function to be maximized for training IIANet. The SI-SNR for each speaker is defined as:
+
+$$
+\operatorname {S I - S N R} (\mathbf {A}, \bar {\mathbf {A}}) = 1 0 \log_ {1 0} \left(\frac {| | \boldsymbol {\omega} \times \mathbf {A} | | ^ {2}}{| | \bar {\mathbf {A}} - \boldsymbol {\omega} \times \mathbf {A} | | ^ {2}}\right), \text {w h e r e} \boldsymbol {\omega} = \frac {\bar {\mathbf {A}} ^ {\top} \mathbf {A}}{\mathbf {A} ^ {\top} \bar {\mathbf {A}}}, \tag {8}
+$$
+
+where  $\mathbf{A}$  and  $\bar{\mathbf{A}}$  denote the ground truth speech signal and the estimate speech signal by the model, respectively. “ $\times$ ” denotes the matrix multiplication.
+
+The speech separation quality is usually assessed using two key metrics: SI-SNR improvement (SI-SNRI) and signal distortion ratio improvement (SDRi). The use of SI-SNR and SDR improvements, which means that it considers the quality of the separated audio and the difficulty of the mixture audio. This is important because for more difficult mixture audio to separate, even if the performance of the separation system is not optimal, the improvement relative to the mixture audio may still be significant. The values of both metrics are derived from the SI-SNR (Le Roux et al., 2019) and the SDR (Vincent et al., 2006a). The higher the values of these metrics, the higher the separation quality. The formulas for SI-SNRI, SDR and SDRi are as follows:
+
+$$
+\operatorname {S I} - \operatorname {S N R i} (\mathbf {S}, \mathbf {A}, \bar {\mathbf {A}}) = \operatorname {S I} - \operatorname {S N R} (\mathbf {A}, \bar {\mathbf {A}}) - \operatorname {S I} - \operatorname {S N R} (\mathbf {A}, \mathbf {S}), \tag {9}
+$$
+
+$$
+\operatorname {S D R} (\mathbf {A}, \bar {\mathbf {A}}) = 1 0 \log_ {1 0} \left(\frac {| | \mathbf {A} | | ^ {2}}{| | \mathbf {A} - \bar {\mathbf {A}} | | ^ {2}}\right), \tag {10}
+$$
+
+$$
+\operatorname {S D R i} (\mathbf {S}, \mathbf {A}, \bar {\mathbf {A}}) = \operatorname {S D R} (\mathbf {A}, \bar {\mathbf {A}}) - \operatorname {S D R} (\mathbf {A}, \mathbf {S}),
+$$
+
+where  $\mathbf{S}$  denotes the mixture audio.
+
+# D. Model configurations
+
+IIANet's audio encoder utilizes a single convolutional layer to obtain audio embeddings from the waveform signal. In contrast, the audio decoder employs a transposed convolutional layer to convert these embeddings back into waveforms. They are widely used in time-domain audio-only speech separation methods (Luo & Mesgarani, 2019; Luo et al., 2020; Li et al., 2023). For the video encoder, we used the same pretrained model as in the CTCNet paper (Li et al., 2022), named CTCNet-Lip. It consisted of 3D convolutional layers and a standard ResNet-18 network to extract lip movement features. Given grayscale lip motion video frames  $V \in \mathbb{R}^{T_v \times H \times W}$ , we initially applied stripped 3D convolutions (with batch normalization, ReLU, and MaxPooling layers) for preprocessing and downsampling. The downscaled video features  $F_v \in \mathbb{R}^{T_v \times C_0 \times H_0 \times W_0}$  were then processed frame-by-frame through the ResNet network to obtain lip embeddings  $E_v \in \mathbb{R}^{T_{0v} \times C_v}$ .  $T_v$  and  $C_v$  represent the number of frames and feature dimensions after downsampling, respectively. We pretrained the video encoder on the lip-reading task on the LRW (Lip Reading in the Wild) dataset (Son Chung et al., 2017), where each lip-reading video has been annotated with one of 500 words. For classification predictions, we averaged the video encoder's output across the time dimension and appended a linear layer as a classifier. After pretraining, we fixed the feature extraction network's weights and reinitialized the video encoder for speech separation.
+
+We set the kernel size of the audio encoder and decoder to 16 and stride size to 8. The video encoder utilized a lip-reading pre-trained model consistent with CTCNet (Li et al., 2022) to extract visual embeddings. The number of down-sampling  $D$  for auditory and visual networks was set to 4, and the number of channels for all convolutional layers was set to 512. The auditory and visual networks used the same set of weights across different cycles, and they were cycled  $N_{F} = 4$  times; after that, the audio network was additionally cycled  $N_{S} = 12$  times. The fast version of IIANet, namely IIANet-fast, performed only  $N_{S} = 6$  cycles on the auditory network. For FFN in Inter-T blocks, we set the three convolutional layers to have channel size (512, 1024, 512), kernel size (1, 5, 1), stride size (1, 1, 1), and biases (False, True, False). To avoid overfitting, we set the dropout probability for all layers to 0.1.
+
+# E. Audio network
+
+In IIANet (audio-only), we use only audio as input and its pipeline is shown in Figure 3A. The structure of the audio network is illustrated in Figure 3B. Specifically, the audio network takes  $\mathbf{E}_{S,t + 1}$  as input and obtains multi-scale auditory features  $\mathbf{S}_i$  through a bottom-up pathway. At the coarsest temporal scale, these multi-scale features are integrated to obtain the global feature  $\mathbf{S}_G$ . Subsequently, we employ a top-down attention mechanism to generate modulated multi-scale auditory features  $\bar{\mathbf{S}}_i$ , enabling auditory focus across different temporal scales and capturing the core semantics at various scales. Finally, through top-down processing, refined auditory features  $\bar{\mathbf{E}}_{S,t} = \bar{\mathbf{S}}_0$  are produced, serving as the input for the next audio network iteration. To the end, we achieve optimization of auditory features within the audio network with shared parameters.
+
+For the training and testing of IIANet (audio-only), we used hyperparameter settings and datasets consistent with IIANet for a fair comparison with other models. We solved the permutation problem (Hershey et al., 2016) using permutation invariant training (PIT) method (Yu et al., 2017) consistent with blind source separation methods (Li et al., 2023; Luo & Mesgarani, 2019; Luo et al., 2020) to maximize the SI-SNR.
+
+As shown in Table 8, the speech separation quality with added visual information (IIANet) was improved by about 4 dB compared to IIANet (audio-only).
+
+# F. Experimental configurations for multiple speakers
+
+Dataset: We utilized the LRS2-3Mix and LRS2-4Mix datasets for training purposes. For audio data construction, we selected random audio clips from 3 to 4 random speakers mixed with a random signal-to-noise ratio (SNR) value ranging between -5 and  $5\mathrm{dB}$ . Regarding visual data construction, we employed the same experimental setup used in the LRS2-2Mix dataset. In the end, the constructed LRS2-3Mix and LRS2-4Mix datasets comprised 20,000 mixed speech samples for the training set, 5,000 for the validation set, and 300 for the testing set.
+
+![](images/65db1628bd63f286ce7cb6b612ad55f33fc65f6cdac71ae7a903b31cb93055ce.jpg)  
+A. Overall pipeline
+
+![](images/1cd69831ecbd9da7f128c2a55e03182147efd4dddf21055ee90d82b3ac2e3cd5.jpg)  
+B.IIANet (audio-only)  
+Figure 3. The overall pipeline and architecture of IIANet (audio-only).
+
+Table 8. Separation results of different AVSS methods on LRS2, LRS3, and VoxCeleb2 datasets. These metrics represent the average values for all speakers in each test set. The larger SI-SNRI, SDRi, and PESQ values are better.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">LRS2</td><td colspan="3">LRS3</td><td colspan="3">VoxCeleb2</td></tr><tr><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td></tr><tr><td>IIANet (ours)</td><td>16.0</td><td>16.2</td><td>3.23</td><td>18.3</td><td>18.5</td><td>3.28</td><td>13.6</td><td>14.3</td><td>3.12</td></tr><tr><td>IIANet (audio-only)</td><td>11.2</td><td>11.5</td><td>2.32</td><td>12.5</td><td>12.8</td><td>2.54</td><td>9.5</td><td>10.0</td><td>2.23</td></tr></table>
+
+Training Process: Consistent with the experimental configuration used for training with two speakers, we trained different baseline models and the IIANet on multi-speaker datasets. This approach allowed us to maintain uniformity in our experimental configuration while exploring the model's performance across varying numbers of speakers.
+
+Inference Process: In the case of video with multiple speakers, our inference process is as follows: Firstly, we use a face detection model (Zhang et al., 2016) to extract the facial images of the different speakers from the video. These images are then cropped to isolate the lip area, with each of these cropped images serving as inputs to the lip reading model. At the same time, the audio corresponding to the video is fed as input into the audio encoder. These processed visual and auditory inputs are subsequently fed into our visual and auditory networks in the separation network, generating visual and auditory features. Through this separation network, we obtain the audio mask of the intended speaker. Then, by employing the audio decoder, we obtain the waveform of the intended speaker. This process continues iteratively until all speakers' visual information has been processed.
+
+# G. Control models
+
+To validate the effectiveness of our proposed intra-attention and inter-attention blocks in IIANet, we constructed two control models: Control 1 and Control 2.
+
+![](images/7621a984b10f14c7a3b27bbd0b23267f2fe4a26d10014347fe15916eac3b639c.jpg)  
+A. Control 1
+
+![](images/8aabcf812862f5c98983654e73e8b25a6af0c398baa9ba9e1a07cb26af2391a5.jpg)  
+B. Control 2  
+Figure 4. The architecture of IIANet's control models. (A) Control 1. It is obtained by removing the IntraA and InterA blocks of IIANet. (B) Control 2. It is obtained by removing InterA blocks of IIANet.
+
+Control 1 was obtained by removing all IntraA and InterA blocks from IIANet, and its architecture is depicted in Figure 4A. We retained the InterA-B blocks in Control 1 to preserve essential connectivity between the audio and visual networks for audio-visual feature fusion. Control 2 was obtained by adding IntraA blocks to Control 1, and its architecture is depicted in Figure 4B.
+
+# H. Impact of different AV fusion cycles  $N_{F}$
+
+In IIANet, we gradually expanded the number of AV fusion cycles. With fixed audio-only cycle numbers of 14, the cycle numbers for AV fusion were 1, 2, 3, 4, and 5. This allowed us to gradually detect the effect of visual information through the strength of the fused information. Table 9 shows that separation quality improved as the number of fusions increased, but reached a plateau at 4. To make the model lightweight without compromising separation quality, we consider  $N_{F} = 4$  to be a good choice.
+
+# I. Comparison of different video encoders
+
+We examined the separation quality of three different pre-trained video encoder models on the LRS2-2Mix dataset: DC-TCN (Ma et al., 2021), MS-TCN (Martinez et al., 2020), and CTCNet-Lip (Li et al., 2022), where DC-TCN, MS-TCN and CTCNet-Lip are lip-reading models that are used to compute embeddings in linguistic contexts. We trained IIANet by replacing only the video encoder. The results are shown in Table 10. Interestingly, we can find that the separation quality was as good as using different video encoders (with SI-SNRi less than  $0.3\mathrm{dB}$ ). This indicates that different video encoders
+
+Table 9. Results on the LRS2 dataset for different numbers of audio-visual fusion cycles used by IIANet.  
+
+<table><tr><td>NF</td><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td><td>Params(M)</td><td>MACs(G)</td></tr><tr><td>1</td><td>14.0</td><td>14.2</td><td>3.06</td><td>3.1</td><td>17.99</td></tr><tr><td>2</td><td>14.5</td><td>14.7</td><td>3.10</td><td>3.1</td><td>18.57</td></tr><tr><td>3</td><td>15.2</td><td>15.3</td><td>3.15</td><td>3.1</td><td>18.61</td></tr><tr><td>4</td><td>16.0</td><td>16.2</td><td>3.23</td><td>3.1</td><td>18.64</td></tr><tr><td>5</td><td>16.0</td><td>16.3</td><td>3.23</td><td>3.1</td><td>18.68</td></tr></table>
+
+do not affect overall performance as much, implying the pivotal role of the audio-video fusion strategy itself.
+
+Table 10. Results of different video encoders used in IIANet on the LRS2 dataset. "Acc" denotes the accuracy of lip-reading recognition, which is an effective metric for evaluating lip-reading capabilities.  
+
+<table><tr><td>Video encoder</td><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td><td>Acc(%)</td></tr><tr><td>DC-TCN</td><td>15.7</td><td>16.0</td><td>3.19</td><td>88.0</td></tr><tr><td>MS-TCN</td><td>15.8</td><td>16.0</td><td>3.22</td><td>85.3</td></tr><tr><td>CTCNet-Lip</td><td>16.0</td><td>16.2</td><td>3.23</td><td>84.1</td></tr></table>
+
+# J. Impaired visual cues
+
+We explored the relationship between visual cue loss (speaker's facial orientation, specifically the side profile) and separation quality. We utilized the MediaPipe tool on the LRS2 dataset to assess the speaker's facial orientation and categorized 6000 audio-video pairs into two groups: frontal orientation (5331 samples) and side orientation (669 samples). A few sample visualizations are shown in Figure 5. Our findings revealed a marked decline in separation quality for samples with a side orientation compared to those facing front. Specifically, we observe that the SI-SNri value decreases by more than  $8\%$  on AVLiT-8 and CTCNet for side orientation samples compared to frontal orientation samples, whereas this decrease is only about  $4\%$  on IIANet. This result underscores the effectiveness of our proposed cross-modal fusion approach in integrating visual and auditory features capable of mitigating visual information impairment.
+
+![](images/fac147f754fe8d7d8cc3d12ed025fc29b13072c83fc7b4e9e2f75fcebcd3e455.jpg)  
+B. Side orientation  
+Figure 5. Sample visualizations of faces with different orientations in the LRS2 dataset.
+
+Table 11. Results of different models with different orientations on the LRS2 dataset.  
+
+<table><tr><td>Model</td><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td></tr><tr><td colspan="4">Frontal orientation</td></tr><tr><td>AVLiT-8</td><td>13.5</td><td>13.8</td><td>2.75</td></tr><tr><td>CTCNet</td><td>14.9</td><td>15.1</td><td>3.12</td></tr><tr><td>IIANet</td><td>16.3</td><td>16.5</td><td>3.24</td></tr><tr><td colspan="4">Side orientation</td></tr><tr><td>AVLiT-8</td><td>12.1</td><td>12.4</td><td>2.37</td></tr><tr><td>CTCNet</td><td>13.7</td><td>14.1</td><td>3.04</td></tr><tr><td>IIANet</td><td>15.7</td><td>15.9</td><td>3.22</td></tr></table>
+
+# K. Data augmentation
+
+To enhance the model's generalization capability, we apply a dynamic mixture data augmentation scheme to the audio-visual separation task. Specifically, we construct new training data during the training process by randomly sampling two speech signals and mixing them after random gain adjustments. Similar approaches have been used in speech and music separation methods (Subakan et al., 2021; Zeghidour & Grangier, 2021). As shown in Table 12, adopting the dynamic mixing training scheme can further improve the model's separation performance. We recommend using this training scheme in practice, as the cost of generating mixtures is not high (compared to fixed training data, it only adds 0.1 minutes per epoch).
+
+Table 12. Comparison of separation performance for different training schemes. "DM" stands for dynamic mixture scheme. Training time refers to the time spent training one epoch on the LRS2 training set.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">LRS2</td><td colspan="3">LRS3</td><td colspan="3">VoxCeleb2</td><td rowspan="2">Training Time (m)</td></tr><tr><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td><td>SI-SN Ri</td><td>SDRi</td><td>PESQ</td></tr><tr><td>Without DM</td><td>16.0</td><td>16.2</td><td>3.23</td><td>18.3</td><td>18.5</td><td>3.28</td><td>13.6</td><td>14.3</td><td>3.12</td><td>36.1</td></tr><tr><td>With DM</td><td>16.8</td><td>17.0</td><td>3.35</td><td>18.6</td><td>18.8</td><td>3.30</td><td>14.0</td><td>14.7</td><td>3.17</td><td>36.2</td></tr></table>
+
+# L. Visual results
+
+We performed visual analysis of the separation performance of four AVSS methods: IIANet, CTCNet, VisualVoice and AVConvTasNet. The results, based on models trained on the LRS2 dataset, are presented in Figure 6. We tested the performance of the four algorithms by randomly selecting three audio mixing samples from the LRS2 dataset.
+
+In Sample I, poor separation of high-frequency components was observed for CTCNet, VisualVoice, and AVConvTasNet. In contrast, the IIANet model exhibited higher separation quality, evidenced by its clarity and strong similarity to the ground truth.
+
+In Sample II, IIANet presented more detailed harmonic features by accurately removing low and mid-frequency components to the right of the center. In sharp contrast, the other models demonstrated significant deficiencies in recovering harmonic features, resulting in large deviations from the ground truth.
+
+In Sample III, IIANet demonstrated excellent ability to maintain the complex harmonic structure of the original audio. On the other hand, models such as CTCNet and AVConvTasNet showed blurring effects in this regard, failing to capture the fine details in the ground truth.
+
+# M. Bad cases
+
+We have incorporated additional case studies featuring suboptimal separation effects. Specifically, through sorting and analyzing the cases within the LRS2 test set based on their separation effects, we have identified instances posing challenges in separation. We have meticulously scrutinized and summarized these cases, yielding the following key observations:
+
+![](images/08d03cc7af5db56a705f1f16650a5b2b888b465439f566616ffcbbbddce7d2de.jpg)
+
+![](images/84db2c0f78eb39de12f159963b62b7af8546cce768250ecb67f2618c12a48a50.jpg)
+
+![](images/37763ad66a214decae05f41afb6dfdef6a1d78099ea207be876d75ca29b688a0.jpg)  
+Sample I  
+Sample II
+
+![](images/c5c5dcb8c963dafacef466aa5b8ed5f0d1ddb00bb6da68de5659150310dae8bf.jpg)
+
+![](images/bc90fef4c093cbd3641d3959263680f4e8093dc8261768643e0914f5f164ed95.jpg)
+
+![](images/1b0867e8b7b8c10fb33529e7bf5093fb8c8a26b48e084794e63a76c00a225385.jpg)
+
+![](images/a72b38b45dfaca6be7c252c24f99a6dacf2a47c8761c57fcf04ab37adff4c84b.jpg)
+
+![](images/ace10ae73f5e066a6a6c7776cf22994291ac73dea7b9d611ca4ee83921a6dde6.jpg)  
+Sample III
+
+![](images/2282d2c3ae26be928cf2a8bea1429ac5708e6912fe0669ba89bd968b1eac8b2d.jpg)
+
+![](images/cead1eeefdbbe9d6fed11a7cfa9853890e1007f41d47885d22ec43fa746f522b.jpg)
+
+![](images/ac9e46c036c100f7fb7f6eee05910411dde96322a1f24bb361956cf99d7245c4.jpg)  
+Figure 6. Spectrogram of separated audio from different models. Each row represents the results for the same audio mixture.
+
+![](images/f9d50f109b7dc95b18df56ff96918c68a0191e55ec855b97e7c79501196fcebb.jpg)
+
+![](images/6f1603b33df1abc6d11949eb69bb2cfd4f7a3f20b59ecf787c88754fc6e226a6.jpg)
+
+![](images/f653bd34fdc97e2735de32505ff077b864f09e67ce08dcfa4dfee2c1ad077e60.jpg)
+
+![](images/8589947e2fad4465346b00f40d6cb369dc7df9a190c2d6dd96cd805e8f8009c0.jpg)
+
+- Some audio samples exhibited pronounced noise and reverberation, rendering the audio content challenging to discern even to the human ear.  
+- The low resolution of certain video samples led to inadequate details in the lip region images, impeding the quality of separation.  
+- In scenarios where two speakers exhibited high timbre similarity, our separation model occasionally misallocated segments of speech content to non-target speakers.
+
+We hope that researchers will pay attention to these issues in the future and explore corresponding approaches to improve the performance of audio-visual separation models.

2024/IIANet_ An Intra- and Inter-Modality Attention Network for Audio-Visual Speech Separation/images.zip

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2024/ILILT_ Implicit Learning of Inverse Lithography Technologies/c0639e19-5e8f-4a49-b452-4de437af403d_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Haoyu Yang<sup>1</sup> Haoxing Ren<sup>1</sup>",
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+    {
+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Lithography, transferring chip design masks to the silicon wafer, is the most important phase in modern semiconductor manufacturing flow. Due to the limitations of lithography systems, Extensive design optimizations are required to tackle the design and silicon mismatch. Inverse lithography technology (ILT) is one of the promising solutions to perform pre-fabrication optimization, termed mask optimization. Because of mask optimization problems' constrained non-convexity, numerical ILT solvers rely heavily on good initialization to avoid getting stuck on sub-optimal solutions. Machine learning (ML) techniques are hence proposed to generate mask initialization for ILT solvers with one-shot inference, targeting faster and better convergence during ILT. This paper addresses the question of whether ML models can directly generate high-quality optimized masks without engaging ILT solvers in the loop. We propose an implicit learning ILT framework: ILILT, which leverages the implicit layer learning method and lithography-conditioned inputs to ground the model. Trained to understand the ILT optimization procedure, ILILT can outperform the state-of-the-art machine learning solutions, significantly improving efficiency and quality.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Lithography is the most important phase in semiconductor manufacturing flow, which transfers chip design patterns onto the silicon wafer. However, due to the limitations of lithography systems, there are usually fetal manufacturing gaps between the design and the patterns on the wafer. Extensive prior fabrication design optimizations, termed mask optimization, are required to compensate for the manufacturing loss (Figure 1).",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$^{1}$ Design Automation Research Group, NVIDIA, Austin, TX. Correspondence to: Haoyu Yang <haoyuy@nvidia.com>.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/59e198609b91e28e002c5b3f7612dcff7aa7d0e825b5a63d02f37b324238d21c.jpg",
+        "image_caption": [
+            "Figure 1. Limitations of lithography system requires design optimization to be correctly fabricated on the silicon wafer."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Inverse lithography technology (ILT), as a numerical solution, has shown great promise for mask optimization on advanced-node lithography (Gao et al., 2014; Yu et al., 2021; Braam et al., 2019; Pang et al., 2019). As shown in Figure 2(a), an ILT solver requires frequent calls of lithography simulator, which mathematically approximate the lithography fabrication behavior. The forward path computes the wafer image corresponding to the current mask and we measure the error between the wafer image and the original design. In the backward path, the error will be passed back through gradient method to update the mask. However, the non-convexity of mask optimization problems poses a great challenge to numerical ILT solvers, that they rely heavily on good mask initialization and can easily get stuck at sub-optimal solutions.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recent research into machine learning (ML) techniques for lithography have made ILT solvers more efficient. As shown in Figure 2(b), ML can provide better optimization starting points that significantly reduce ILT iterations and yield better convergence (Yang et al., 2018; Chen et al., 2020; Jiang et al., 2020). GAN-OPC (Yang et al., 2018) is the earliest deep learning solution for inverse lithography, where a conditional generative adversarial network is proposed for initial mask generation with litho-guided pre-training. DAMO (Chen et al., 2020) improves on this work by replacing the standard generator in GAN-OPC with nested UNet and ResNet bottleneck layers (Zhou et al., 2019; Isola et al., 2017; He et al., 2016), yielding a direct output of high-resolution masks. Jiang et al. (Jiang et al., 2020) also integrates the explicit gradient calculation of litho-guided pre-training and achieves better training convergence and mask quality. Very recently, Zhao et al. (Zhao et al., 2022) develops the AdaOPC framework that is able to learn from",
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+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/1951a342b3c97aca01378f23b1db00b107f6e758eb8a8791793ec9c13046aa7b.jpg",
+        "image_caption": [
+            "(a) ILT"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "image",
+        "img_path": "images/81bf9b7c7601f8e342f3e2b41cd8386c881a5b584f48aabb99e5ad05889a23dc.jpg",
+        "image_caption": [
+            "(b) ML-Aided"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/ff20a6cfe4d35f4f90bf656ada9291404e659ef40bb277d5ccf61dad531b10a9.jpg",
+        "image_caption": [
+            "(c) ILILT",
+            "Figure 2. Working scheme of ILT solvers. (a) Standard ILT solver that iteratively updates the mask from design till convergence. (b) Common ML approach that uses ML to produce an initial mask followed by standard ILT procedure. (c) The ILILT that uses ML to imitate the standard ILT procedure."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "layout geometry and identifies whether a pattern can be optimized by the machine learning engine. This further increase the applicability of ML for mask optimization solutions.",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, existing solutions only exploit limited ML in the ILT solver and still require significant intervention by traditional numerical solvers. For example, GAN-OPC and its variations can only learn a design-to-mask mapping, which cannot be used to fix poor-quality masks; AdaOPC (Zhao et al., 2022) has to conduct full numerical optimizations on complicated patterns. In this paper, we try to address the question of whether an ML model can directly generate high quality optimized masks without engaging ILT solvers in the loop. We believe that the key limitation of existing methods is that the trained ML models are not grounded to any actual lithography condition during inference, therefore training-inference distribution shift can not be recognized and corrected. We argue that ML models grounded on lithography estimation of intermediate masks can make an ML-based ILT framework more effective.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We propose the ILILT (Figure 2(c)) framework to formulate mask optimization as an implicit layer learning problem, trained to optimize the mask towards an equilibrium state (Bai et al., 2019). Like a standard ILT solver that iteratively updates masks from a design, ILILT iteratively modifies masks from a design but with much fewer steps and each step runs significantly faster because no gradient computing is required. This process produces a number of intermediate masks which ground the ML model with intermediate wafer images computed by lithography models.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "ILT As An Implicit Layer. Unlike common neural network layers that explicitly define specific computation graphs supporting forward and backward propagation, an",
+        "bbox": [
+            84,
+            859,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "implicit layer models a joint condition of its input and output (Jittorntrum, 1978). The nature of ILT makes it a representative example of an implicit layer, where the input is a chip design and the output is the corresponding optimized mask that results in a similar wafer pattern compared to the input after the lithography process. Inspired by the implicit layer learning approach, we design ILILT with a weight-tied sequence model, making it possible to learn an optimization trajectory from training design-mask pairs. To make the trajectory learning grounded with physics, we also integrate ILILT with fast lithography estimator that can implicitly apply lithography conditions on top of the optimization procedure. After training, ILILT is able to perform mask optimization on a given design and reaches a stable state in a few iterations. We will later show how ILILT outperforms a state-of-the-art academic ILT engine with technical details and supporting experiments on public chip design benchmarks.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The reminders of the paper are organized as follows: Section 2 covers basic terminologies and related works of mask optimization; Section 3 discusses and analyzes the technique details of the ILILT framework; Section 4 presents the experimental results supporting the effectiveness of ILILT and Section 5 concludes the paper.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2. Preliminaries",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1. ILT Solver",
+        "text_level": 1,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Mask optimization tries to find a mask design such that the remaining pattern on the silicon wafer after the lithography process is as close as possible to the desired shapes. This can be formulated as a constrained optimization problem:",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {M} l = d \\left(\\boldsymbol {Z}, \\boldsymbol {Z} ^ {*}\\right), \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            625,
+            593,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where,",
+        "bbox": [
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+            625,
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+            636
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {Z} = f _ {l} (\\boldsymbol {M}), \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            645,
+            647,
+            883,
+            662
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where  $Z$  and  $Z^{*}$  represent the wafer image and the chip design, respectively,  $M$  corresponds to the mask to be optimized,  $f_{l}$  represents the lithography process which is highly non-convex (see appendix for more details) and  $d$  is a measurement indicating the differences between  $Z$  and  $Z^{*}$ . Inverse lithography techniques (Gao et al., 2014) try to solve Problem (1) in a numerical way by repeating the following steps until convergence:",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Forward:  $Z_{t} = f_{l}(M_{t})$ .",
+            "2. Backward:  $M_{t + 1} = M_t - \\lambda \\frac{\\partial l}{\\partial Z_t}\\frac{\\partial Z_t}{\\partial M_t}$ ."
+        ],
+        "bbox": [
+            509,
+            809,
+            807,
+            863
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where  $\\lambda$  is some hyper-parameter that controls how the mask image is updated during the optimization procedure. There",
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
+            289,
+            56,
+            684,
+            70
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
+            480,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "are also variations of ILTs (Yu et al., 2021; Jiang et al., 2020; Gao et al., 2014), but they mostly share similar mask-update concepts.",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2. Implicit Layers",
+        "text_level": 1,
+        "bbox": [
+            86,
+            146,
+            227,
+            162
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Most neural network layers or computational graphs explicitly define a function  $\\pmb{y} = f(\\pmb{x})$ , which is differentiable and can be back-propagated via the chain rule. An implicit layer, instead of specifying how to compute the layer's output from the input, specifies the conditions that we want the layer's output to satisfy, separating the procedure to compute the layer from the layer definition itself. It has been used in deep equilibrium models, Neural ODEs, and differentiable optimizations.",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1 (Implicit Layer). Implicit layers try to find  $\\pmb{y}$  such that  $f(\\pmb{x},\\pmb{y}) \\in \\mathcal{C}$ , which defines joint conditions between  $\\pmb{x}$  and  $\\pmb{y}$  (Jittorntrum, 1978).",
+        "bbox": [
+            84,
+            309,
+            473,
+            354
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A fixed-point layer is an example of the implicit layer method, which directly computes the fixed point of a nonlinear transformation, i.e., the solution to the nonlinear system which models the convergent state of a recurrent sequence.",
+        "bbox": [
+            84,
+            364,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.2 (Fixed-Point Layer). A fixed-point layer can be mathematically given by",
+        "bbox": [
+            84,
+            429,
+            473,
+            460
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {y} = g (\\boldsymbol {x}, \\boldsymbol {y}). \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            468,
+            473,
+            484
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A fixed-point layer can be considered to be an infinite number of layers parameterized by the same weight  $w$  (weighted), where each layer can be represented as",
+        "bbox": [
+            84,
+            500,
+            475,
+            544
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {y} _ {t + 1} = g (\\boldsymbol {x}, \\boldsymbol {y} _ {t}, \\boldsymbol {w}), \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            553,
+            473,
+            570
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To train a fixed-point layer, (Bai et al., 2019) proposes an unrolling-free deep equilibrium (DEQ) method, which trains the fixed-point layer in two alternative steps: (1) solve for the fixed-point of the layer and (2) update network parameters. This process consists of repeated forward and backward computations as follows. The forward path is related to the finding of the fixed point of Equation (3) and can be solved with Newton's method. Let",
+        "bbox": [
+            84,
+            584,
+            475,
+            704
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh (\\boldsymbol {x}, \\boldsymbol {y} ^ {*}, \\boldsymbol {w}) = g (\\boldsymbol {x}, \\boldsymbol {y} ^ {*}, \\boldsymbol {w}) - \\boldsymbol {y} ^ {*}, \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            166,
+            713,
+            473,
+            729
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and  $\\pmb{y}^{*}$  can be found through the following step until convergence:",
+        "bbox": [
+            84,
+            737,
+            475,
+            767
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {y} _ {t + 1} = \\boldsymbol {y} _ {t} - \\lambda \\boldsymbol {J} _ {t} ^ {- 1} h (\\boldsymbol {x}, \\boldsymbol {y} _ {t}, \\boldsymbol {w}), \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            773,
+            473,
+            792
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $J_{t}$  is the Jacobian matrix of  $h$  at  $\\mathbf{y}_t$  and  $\\lambda$  is the update rate.",
+        "bbox": [
+            84,
+            799,
+            473,
+            829
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The backward process covers the update of  $\\boldsymbol{w}$ , which can be written as",
+        "bbox": [
+            84,
+            837,
+            473,
+            866
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {w} = \\boldsymbol {w} - \\lambda \\frac {\\partial l}{\\partial \\boldsymbol {w}} = \\boldsymbol {w} + \\lambda \\frac {\\partial l}{\\partial \\boldsymbol {y} ^ {*}} \\boldsymbol {J} ^ {* - 1} \\frac {\\partial g}{\\partial \\boldsymbol {w}}, \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            140,
+            869,
+            473,
+            904
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $J^{*}$  is the Jacobian matrix of  $h$  at  $\\boldsymbol{y}^{*}$ ,  $l$  is a loss function that measures whether  $f(x,y) \\in \\mathcal{C}$  satisfies the implicit layer definition, e.g. the ILT loss Equation (1). For a detailed proof of Equation (7), we refer readers to (Bai et al., 2019).",
+        "bbox": [
+            496,
+            84,
+            885,
+            160
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3. Neural Networks for ML-based Computational Lithography",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are two major groups of network architectures used in lithography applications: (1) The UNet family (Ronneberger et al., 2015; Zhou et al., 2019) and (2) The FNO family (Li et al., 2021). UNets consist of various bypass connections between convolution layer outputs. Representative works include TEMPO (Ye et al., 2020), LithoGAN (Ye et al., 2019), DAMO (Chen et al., 2020) and GAN-OPC (Yang et al., 2018). FNOs contain one or more Fourier Neural Operators along with auxiliary convolution layers. Representative works are DOINN (Yang et al., 2022b) and CFNO (Yang et al., 2022a). UNet-based models usually rely on large model capacity to learn training data distribution while FNO-based models focus on global information learning through frequency domain embedding.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. The ILILT Framework",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.1. ILT Layer",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Following the definition of implicit layer learning, we can solve the ILT equations with an implicit ILT layer defined as finding  $M$  such that",
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+    {
+        "type": "equation",
+        "text": "\n$$\nf (\\boldsymbol {M}, \\boldsymbol {Z} ^ {*}) = d \\left(f _ {l} (\\boldsymbol {M}), \\boldsymbol {Z} ^ {*}\\right) = \\epsilon , \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\epsilon$  is some minimum value achievable for Equation (1).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Because Equation (8) is a composite of a group of non-linear functions, it is difficult to solve explicitly. Therefore, we assume Equation (8) has a solution  $M^{*}$  that follows the form of",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {M} ^ {*} = g \\left(\\boldsymbol {M} ^ {*}, \\boldsymbol {Z} ^ {*}\\right), \\tag {9}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "which yields a fixed-point layer representation of the ILT solution.",
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+    },
+    {
+        "type": "text",
+        "text": "We start solving Equation (9) from the weight-tied approach, where we define",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {M} _ {t + 1} = g \\left(\\boldsymbol {M} _ {t}, \\boldsymbol {Z} ^ {*}, \\boldsymbol {w}\\right), t = 1, 2, \\dots , \\tag {10}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {t \\rightarrow \\infty} \\boldsymbol {M} _ {t} = \\lim  _ {t \\rightarrow \\infty} g \\left(\\boldsymbol {M} _ {t}, \\boldsymbol {Z} ^ {*}, \\boldsymbol {w}\\right) = g \\left(\\boldsymbol {M} ^ {*}, \\boldsymbol {Z} ^ {*}, \\boldsymbol {w}\\right) = \\boldsymbol {M} ^ {*}. \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2. Lithography Estimation",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As we discussed previously, due to the lack of prior lithography knowledge, state-of-art AI approaches have been lim",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "ited to only generate initialization for numerical ILT solvers. This motivates us to build lithography estimation inside  $g$  to ground the AI model.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We rewrite Equation (10) as:",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {M} _ {t + 1} = g \\left(\\boldsymbol {M} _ {t}, \\boldsymbol {Z} _ {t}, \\boldsymbol {Z} ^ {*}, \\boldsymbol {w}\\right), t = 1, 2, \\dots , \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $Z_{t} = f_{l}(M_{t})$  is the wafer image of  $M_{t}$  at time stamp  $t$ . Equation (12) formulates the core of the ILILT framework. A virtue of Equation (12) is the embedded condition of the lithography estimator that implicitly tells  $g$  the physical meaning of each step.",
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+    },
+    {
+        "type": "text",
+        "text": "3.3. Training",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Equation (12) defines a weight-tied fixed-point model that can be solved with the DEQ method. Although DEQ (Bai et al., 2019) reduces the computing overhead of sequence unrolling and back-propagation through time, there are several concerns when it's used for solving ILT problems:",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Solving  $M^{*}$  through Newton's method or other Root-Finding algorithms is still computationally costly considering that  $M^{*}$  usually has millions of entries.",
+            "- The solution error caused by the non-optimality in Newton's method will further propagate to the gradient calculation in backward DEQ, inducing additional errors.",
+            "- The DEQ backward pass still requires the computation of the gradient over lithography modeling in the term  $\\frac{\\partial d}{\\partial M^*}$ , showing no advantage over a traditional ILT solver."
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To address these concerns, we developed the following training architecture that solves the ILT layer efficiently and effectively. We unroll ILILT to a fixed depth  $T$  and allow the final output  $M_T$  to be trained towards a ground truth mask  $M^*$ ,",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {\\boldsymbol {w}} l = \\sum_ {t = \\left[ \\frac {T}{2} \\right]} ^ {T} \\exp \\left[ \\frac {t}{T} - 1 \\right] \\cdot \\left| \\left| \\boldsymbol {M} _ {t} - \\boldsymbol {M} ^ {*} \\right| \\right| _ {F} ^ {2}, \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "s.t.  $M_{t + 1} = g(M_t,Z_t,Z^*,\\pmb {w}), t = 0,1,\\dots,T - 1,$",
+        "bbox": [
+            104,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {Z} _ {t} = f _ {l} (\\boldsymbol {M} _ {t}), t = 0, 1,..., T,\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            133,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $t$  indicates each unrolling step.",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It should be noted that to encourage the mask to grow towards and stop at the golden solution, we did not adopt the cost function that directly minimizes the difference between  $M_T$  and  $M^*$ . Instead, we ask intermediate masks in later unrolling stages to have a trend growing toward the ground truth, which is reflected in Equation (13) having masks at",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "each time step  $M_{t}$  trained towards the ground truth  $M^{*}$ . The coefficient  $\\exp \\left[\\frac{t}{T} - 1\\right]$  applied at each loss term is to let the training process be aware of the progressive growth of masks at each time step. This trick allows the mask to grow smoothly in these steps while stopping at the fixed point. The exponential term is empirically chosen to regularize the mask-growing trends which are consistent with the behavior of the numerical ILT solver that only minor changes are observed in later optimization iterations. Solving Equation (13) is straightforward through back-propagation through time (BPTT).",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4. Discussion",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4.1. RELATIONS TO ML-AIDED SOLUTIONS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For the training procedure, we observe that ILILT resembles GAN-OPC (Yang et al., 2018) in terms of the training objectives, where we also force the model to learn a golden mask generated by some conventional methods. However, ILILT advances beyond GAN-OPC works (Yang et al., 2018; Jiang et al., 2020; Zhao et al., 2022; Chen et al., 2020) because it not only learns what a good mask should look like, but also why and how a good mask is grown thanks to the implicit lithography condition introduced by  $f_{l}(\\cdot)$ .",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4.2. RELATIONS TO IMPLICIT LAYER LEARNING",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The whole ILILT method is built upon the definition of the ILT layer and how it is solved. Representative solutions, e.g. DEQ (Bai et al., 2019), still requires costly computation of gradients of the objectives over physical models. ILILT makes it even more implicit by feeding the wafer image, target design and the mask image simultaneously into  $g(\\cdot, \\boldsymbol{w})$  at each time stamp. Finally  $g(\\cdot, \\boldsymbol{w})$  will automatically figure out the inherent relations between masks, designs and resist patterns.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4.3. RELATIONS TO LEARNING-TO-OPTIMIZE",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The ILILT system is a similar scheme to learning-to-optimize (L2O) solutions (Chen et al., 2021), particularly the algorithm unrolling approach, which unrolls a truncated algorithm into multi-stage neural networks with independent parameters. In contrast, ILILT benefits from the weight-tied structure that (1) significantly reduces model size for memory efficiency and (2) generalizes better (Bai et al., 2019).",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4.4. PRACTICAL USAGE SCENARIO",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lastly, ILILT imitates a real ILT solver during the inference runtime but with many benefits: (1) ILILT requires to query a lithography simulator in each step but updates the mask in a derivative-free manner. (2) ILILT requires significant less steps than traditional ILT solver. (3) ILILT has the opportunity to outperform numerical ILT solvers due to strong mask",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
+            289,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/3554f6abc20d2f08793e5533cb9260981d2f080ac82d61dc7b57baefaf1db8a4.jpg",
+        "image_caption": [
+            "(a) Design"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            99,
+            88,
+            207,
+            171
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/82ff16beb3f84d30778a974defd7419200f71e37e5adfb5e6446999f688b395a.jpg",
+        "image_caption": [
+            "(b) Mask"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/8a7133d90e2b4480b4974cb3e4663bd61adfd4afcc0e5ca66efb66fd413762c1.jpg",
+        "image_caption": [
+            "(c) Wafer",
+            "Figure 3. Data example from LithoBench (Zheng et al., 2023). (a) Design image contains the patterns of original circuit devices or connectivity; (b) Mask image is the optimized design that is manufacturing-friendly; (c) Wafer image is the patterns on silicon given the mask image."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "prriors learned from broad data points. (4) ILILT can serve as a real solver during chip design optimization. One representative case is when a design is not well-optimized by an existing mask optimizer. Because most AI solutions only learn a design-to-mask mapping, it is not possible to use them to fix partially optimized masks. ILILT, however, does not have this limitation, and can take over from anywhere in the optimization trajectory.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1. The Dataset and Configurations",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the effectiveness of the ILILT framework, we adopt the latest AI for computational lithography benchmark suite LithoBench (Zheng et al., 2023), where state-of-the-art mask optimization solutions are evaluated and will be used as comparison baselines. LithoBench contains over 100K design clips and each covers an area of  $2\\mu m \\times 2\\mu m$ . Each data point contains a pair of chip design patterns and the ground truth masks generated by a numerical ILT solver. These design patterns and masks are rasterized with a  $1nm^2$  pixel size, yielding  $2048 \\times 2048$  images. Example designs are displayed in Figure 7. We can observe that in real world, mask optimization does not simply finetune the design like in Figure 1. The process also produces sub-resolution assist features (SRAFs) surrounding the main design pattern to improve its robustness against process variations. This poses great challenge of ML models on optimization tasks.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To deploy ILILT, we re-implement the lithography simulator as a pytorch layer that can be integrated in the data pipeline. Detailed configurations of the generator used to reproduce our results are listed in Table 1. Here we adopt CFNO (Yang et al., 2022a) as the backbone network for  $g$  to evaluate ILILT.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2. Evaluation Metrics",
+        "text_level": 1,
+        "bbox": [
+            84,
+            851,
+            253,
+            864
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As is typical for mask optimization tasks, we evaluate the final performance through the metrics that are widely adopted",
+        "bbox": [
+            84,
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+            475,
+            906
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/968008c860383631b5fa90573a15b8d22d32b74a37ebb08a4160800aa1711188.jpg",
+        "table_caption": [
+            "Table 1. Model Configuration."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Item</td><td>g</td></tr><tr><td>Backbone</td><td>CFNO/Pix2PixHD</td></tr><tr><td>Max Epoch</td><td>5</td></tr><tr><td>Initial Learning Rate</td><td>0.004</td></tr><tr><td>Learning Rate Decay Policy</td><td>step</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Weight Decay</td><td>0.0001</td></tr><tr><td>Loss</td><td>Equation (13)</td></tr><tr><td>Sequence Unrolling Depth (T)</td><td>4-8</td></tr><tr><td>Batch Size</td><td>4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/0187f896587e62b86714b6fd6621dfb681fdf4f7e2723ddbc76d51819c23f26a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            329
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/27892f5cafc21debf47d64860f8947136fe477bacaef860584495e9203478925.jpg",
+        "image_caption": [
+            "Figure 4. Lithography edge placement error and process variations. Smaller EPE and PVB area indicates better optimization QoR."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            607,
+            347,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "in the chip manufacturing industry: edge placement error (EPE) violations and the process variation band (PVB) area. In semiconductor manufacturing processes, the lithography conditions are not always as we expected due to systematic error and mechanical perturbations, which is the root cause of process variations. Therefore, we expect the design to be robust under different manufacturing conditions. Typically, we call the expected lithography settings and configurations nominal condition, and non-nominal condition will result in larger or smaller patterns on the silicon wafer. The difference between the smallest pattern and the largest pattern will form a process variation band (PVB) as in Figure 4. A good mask optimization result should have a smaller number of EPE violations and a smaller PVB area.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 4.1 (EPE Violation(Banerjee et al., 2013)). EPE is measured as the geometric distance between the target edge and the lithographic contour printed at the nominal condition. If the EPE measured at a point is greater than a certain tolerance value, we call it an EPE violation. In our experiments, the EPE measuring points are sampled at  $40nm$  interval.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 4.2 (PVB Area(Banerjee et al., 2013)). This is evaluated by running lithography simulations at different process conditions on the final mask solution. Once run, the total area of the process variation band will be defined as PVB Area.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "page_number",
+        "text": "5",
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+        "image_caption": [
+            "(a)  $\\mathrm{T} = 1$"
+        ],
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+        "image_caption": [
+            "(b)  $\\mathrm{T} = 2$"
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+        "image_caption": [
+            "(c)  $\\mathrm{T} = 3$"
+        ],
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+        "img_path": "images/8a39a4ca39626318acd9f83e47d81d598899239bac63589223c045a652a05d87.jpg",
+        "image_caption": [
+            "(d)  $\\mathrm{T} = 4$"
+        ],
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+    {
+        "type": "image",
+        "img_path": "images/7468b471194520d05af833531063962593609a8f38c5822f92bf219233267dad.jpg",
+        "image_caption": [
+            "(e)  $\\mathrm{T} = 5$"
+        ],
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+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/0e6d886a2e50b3243b18acca8eb6423439596d307e7aa8935acf796a2ea9bc7c.jpg",
+        "image_caption": [
+            "(f)  $\\mathrm{T} = 6$"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/29a9c09ffb07faee36cbc783d3c6538067b36af23c06153800ee0f05892b9996.jpg",
+        "image_caption": [
+            "(g)  $\\mathrm{T} = 7$"
+        ],
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+        "bbox": [
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+    {
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+        "img_path": "images/950036039630f147e689c1b512cf8ba239b5fd65bdce53c6099401a0c02540ce.jpg",
+        "image_caption": [
+            "(h)  $\\mathrm{T} = 8$"
+        ],
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+    {
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+        "img_path": "images/c334c5c2d790b63c61232329a7395a007ac62f2b5ab7095774a2489229b1f07a.jpg",
+        "image_caption": [
+            "(i) EPE=12"
+        ],
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+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/e4b39924cfe6d24692fa5a59bc89539de4085ef1df0ac7be5aba6c704aaf9087.jpg",
+        "image_caption": [
+            "(j)  $\\mathrm{EPE} = 12$"
+        ],
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+        "bbox": [
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+    {
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+        "img_path": "images/1b47c37982f489b61c1f7358add0609d1a0b7952d94bf7097e8d02c700338f61.jpg",
+        "image_caption": [
+            "(k) EPE=0"
+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/6d78a2056ba015e37e96dc5196dc03dc72819fac0f4719ba57a65bae24b193d3.jpg",
+        "image_caption": [
+            "(1) EPE=0"
+        ],
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+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/6830137cb596a43106ec144e289b65d661a29ca64ef54bfb5ef4157aa67d265f.jpg",
+        "image_caption": [
+            "(m) EPE=0"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0597604f367a1130e2ff7f2b5ec8b994ed13eaf4a89fb53c933fd92c72e98d52.jpg",
+        "image_caption": [
+            "(n) EPE=0"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/b06484a4019f06d6f8b09cd6441ce4d40307d5960f7a685ea991713a8590ea22.jpg",
+        "image_caption": [
+            "(o) EPE=0"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/6adec80a2aa9d88023a969cf4e3109bc8915c3c8d15e4d346b4afcde32906fe4.jpg",
+        "image_caption": [
+            "(p) EPE=0"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/d8e9566c9a4f9e1e9f137bc5cfea4b6777e2cad6d5303ebe765382c3acf82946.jpg",
+        "image_caption": [
+            "Figure 5. Visualization of the optimization trajectory beyond the maximum unrolling depth. We use the model trained with unrolling step  $\\mathrm{T} = 4$ , and mask stays stable after the 4th step with only minor tuning. (a)-(h) correspond to the generated masks generated from time stamps 1-8. (i)-(p) are the lithography simulated wafer image with their corresponding EPE violation count.",
+            "(a) Design"
+        ],
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+        "image_caption": [
+            "(b) ILT-Mask"
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+        "img_path": "images/a5428f6c2bef5b39218ad8513954173e0a7699a5bef358ef2e18ab7dc81d064d.jpg",
+        "image_caption": [
+            "(c) ILT-Wafer"
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+    {
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+        "img_path": "images/3f32c67f0995c5fea07799a56deef692cf69a194446e1b85ec65828bee36942d.jpg",
+        "image_caption": [
+            "(d) ILILT-Mask"
+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d6c81548cbbfa6157f129ddd75fc79bea7a4943c9e914a016aabd12c49e15200.jpg",
+        "image_caption": [
+            "(e) ILILT-Wafer"
+        ],
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+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/142615c8785e3d36a90e434d7dfcd8f934d445c8708350e20e6f57adafc176f8.jpg",
+        "image_caption": [
+            "(f) Design",
+            "Figure 6. Mask prior learning from ILILT (P2PHD). Due to design pattern variations, the numerical solver sometimes fails to produce the correct solution. In this example, ILT generates rule-violating artifacts and performs feature shrinking excessively, causing the missing of design patterns as in (c). While ILILT can learn from the broader dataset and make correct predictions. For a more complicated case (f)-(j), ILILT can successfully produce optimized mask that can pass lithography printability check while numerical ILT still exhibits EPE violations in the printed wafer image."
+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d3b5aa835ed03488be68113545b1847862ece87750b2dfd30e7626fb59e6e302.jpg",
+        "image_caption": [
+            "(g) ILT-Mask"
+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0d2b99835c8cae389244066dfc8553ef04c6c2ac8d2fca488bf100633309db52.jpg",
+        "image_caption": [
+            "(h) ILT-Wafer"
+        ],
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+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a28e9ffcf3d74059fe2d714601ae43c38315dc7fa5affe38c2fcf56d9798cd5.jpg",
+        "image_caption": [
+            "(i) ILILT-Mask"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/2967971a871668b955f4468540a5c25e69909d131ae6630f317c7adfc3ec25f4.jpg",
+        "image_caption": [
+            "(j) ILILT-Wafer"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3. Comparison with State-of-the-Art",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the first experiment, we compare the performance of ILILT with state-of-the-art mask optimizers. A quantitative comparison is listed in Table 2. Columns \"EPE\", \"PVB\" and \"Throughput (s/tile)\" indicate the count of EPE violations, PVB Area in terms of  $nm^2$  and optimization throughput, respectively; \"GPU-ILT\" (Sun et al., 2023) is the latest academic numerical ILT engine with GPU acceleration, which will also be the engine used to generate the training data; \"GAN-OPC\" (Yang et al., 2018) is one of the early attempts using machine learning for mask optimization following the scheme in Figure 2(b). GAN-OPC employs UNet as a generator backbone. \"Pix2PixHD\" (Wang et al., 2018) is another representative CNN-based backbone that enjoys better gen",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "eralization capability for high-resolution image generation, which fits design-mask translation task well. \"Neural-ILT\" (Jiang et al., 2020) is a follow-up of GAN-OPC, which adopts a true lithography simulator to produce the gradient for generator training. \"CFNO\" (Yang et al., 2022a) is an FNO-based design that preserves the global perception of the Fourier neural operator while saving computation with ViT's patch embedding. CFNO and Pix2PixHD will also be the backbone frameworks of the ILILT generator, and we will show the advantage of the ILILT data pipeline over the standalone generator. ILILT- $X$  lists the results for  $X$  unrolling steps.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Overall, the ILILT with Pix2PixHD backbone achieves the smallest EPE violation count among all other ML-based",
+        "bbox": [
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+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/be5121ab9e7d3a8136a9cb4c40819da5867edf700c0a0d2eb1da671857ac8472.jpg",
+        "table_caption": [
+            "Table 2. Comparison with State-of-the-Arts."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>EPE</td><td>PVB</td><td>Throughput</td></tr><tr><td>GPU-ILT (Sun et al., 2023)</td><td>0.21</td><td>4656</td><td>4</td></tr><tr><td>GAN-OPC (Yang et al., 2018)</td><td>8.3</td><td>6686</td><td>0.008</td></tr><tr><td>Neural-ILT (Jiang et al., 2020)</td><td>6.2</td><td>8537</td><td>10</td></tr><tr><td>Pix2PixHD (Wang et al., 2018)</td><td>0.12</td><td>4685</td><td>0.1</td></tr><tr><td>CFNO (Yang et al., 2022a)</td><td>0.51</td><td>4623</td><td>0.06</td></tr><tr><td>L2O-CFNO-8-Step</td><td>1.59</td><td>4436</td><td>0.6</td></tr><tr><td>L2O-P2PHD-8-Step</td><td>0.87</td><td>4526</td><td>0.5</td></tr><tr><td>ILILT-CFNO-2-130K (ours)</td><td>0.39</td><td>4704</td><td>0.1</td></tr><tr><td>ILILT-CFNO-4-130K (ours)</td><td>0.39</td><td>4667</td><td>0.2</td></tr><tr><td>ILILT-CFNO-6-130K (ours)</td><td>0.35</td><td>4657</td><td>0.3</td></tr><tr><td>ILILT-CFNO-8-130K (ours)</td><td>0.25</td><td>4674</td><td>0.4</td></tr><tr><td>ILILT-P2PHD-2-45M (ours)</td><td>0.13</td><td>5056</td><td>0.2</td></tr><tr><td>ILILT-P2PHD-4-45M (ours)</td><td>0.13</td><td>4680</td><td>0.4</td></tr><tr><td>ILILT-P2PHD-6-45M (ours)</td><td>0.11</td><td>4670</td><td>0.6</td></tr><tr><td>ILILT-P2PHD-8-45M (ours)</td><td>0.08</td><td>4695</td><td>0.8</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "solutions with 0.08 average EPE violations (achieved by  $T = 8$ ), which outperforms the golden numerical solver. ILILT-CFNO though behaves slightly worse than ILILT-P2PHD, it can still exhibit comparable results with \"GPU-ILT\" with a  $400 \\times$  smaller model size. All methods do not show a significant advantage on PVB area, and this is because the optimization flow in these solutions do not explicitly target process variations. Lastly, the ILILT already achieves better mask quality than the other three ML-based solutions without the refinement from the legacy ILT engine. We also show a  $10 \\times$  speedup over the numerical ILT solver.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The ILILT framework also ensembles the beneath idea of learning-to-optimize (L2O), which unrolls optimization steps on multiple layers of neural networks whose weights are not shared. As a comparison, we also implement two standard L2O flows with CFNO and Pix2PixHD as backbones respectively. We choose the 8 unrolling steps that have similar computing costs to the best ILILT settings. As shown in Table 2, L2O does not preserve the performance of standalone backbone models with a significant increase in EPE violations. L2O unrolls the ILT problem into a fixed number of optimization steps, which are parameterized by different neural network layers. Such a model, with  $8 \\times$  trainable parameters in our case, requires more converging training efforts. Specifically, the error of each optimization step will propagate to the next level which causes performance degradation compared to standalone backbone models.",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4. Generalization of ILILT",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ILILT also shows better generalization capability thanks to its weight-tied structure. This can be observed in the superior performance boosting over standalone CNN and Neural Operator-based backbone models. We can observe that with the weight-tied sequence training on CFNO and Pix2PixHD, the mask quality has an improvement of  $50\\%$  and  $33\\%$  reduced EPE violation. Although the unrolled sequence induces additional runtime, the model inference time can be ignored when compared to the entire physical implementation and manufacturing flow.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5. Self-learning Capability",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Usually, ML-based solutions rely highly on the dataset quality, and hence the performance of traditional ILT solvers. However, ILT is naturally non-convex and suffers from suboptimal solutions. Under such a scenario, ILILT can take advantage of its capability to outperform numerical solvers in certain chip designs, providing faster iteration for dataset updates and serving as a powerful auxiliary candidate for traditional numerical solvers.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.6. ILILT Optimization Flow",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The learned mask optimization process is depicted in Figure 5, where we visualize the growth of a mask along the unrolled sequence for a given design. We bring the model trained with unrolling step  $\\mathrm{T} = 4$  but keep the loop running until  $\\mathrm{T} = 8$ . We show that the mask reaches the best quality at the configured unrolling depth. Interestingly, the ILILT also learns a fixed-point iteration such that if we continuously feed the mask beyond the unrolling depth, the model will no longer make significant changes to the mask except minor adjustments. At  $\\mathrm{T} = 8$ , the ILILT engine tries to remove the rule-violating artifact from the middle-bottom region. This property brings more application scenarios of the ILILT framework that include producing initial solutions and performing fine-grained fixes at the post-optimization stage. Another benefit of ILILT is its stable optimization trajectory, because quality fluctuation is common during the optimization runtime of numerical ILT solvers (Gao et al., 2014; Ma et al., 2017; Yu et al., 2021; Sun et al., 2023), causing additional overhead to identify good solutions.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.7. Investigation of Unrolling Depth",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here we discuss the effects of the maximum unrolling depth. The ILILT is trained using different max unrolling depths  $(T)$  with  $T\\in \\{2,4,6,8\\}$ . With PVB area being similar, we observe better results happen in larger unrolling steps in the configuration. When  $T$  is small, ILILT does not benefit too much from the weight-tied structure and exhibits larger",
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+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "EPE violation counts. Instead, a larger  $T$  will put stronger regularization on the model and enable temporal feature learning to increase the model generality. However, a longer unrolling sequence will linearly increase the computing cost as well as training efforts. In this paper, we experimentally show the performance boosting saturates at  $T = 8$  for a fixed number of training epochs.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.8. Mask Prior Learning",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we also observe an interesting behavior of ILILT. For some cases, ILILT can even produce better QoR than numerical solution. An example is shown in Figure 6, where the ILILT produces mask solution with zero EPE violations while numerical solver comes with four. This is because ILILT not only learns the mask optimization trajectory from the golden numerical solutions but also captures the mask solution from the broader dataset, which makes the trained network equip deep mask priors. This can also be observed in the generated mask images that ILILT avoids generating rule-violation isolated artifacts while numerical ILT does (see small white dots at the bottom of Figure 6(g)). The result can also be explained through deep image prior (Ulyanov et al., 2018), that during training neural networks capture the structural knowledge with priority. ILILT strengthens this prior through weigh-tied unrolling and lithography guidance.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5. Conclusion",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a novel solution to the mask optimization problem with inverse lithography technology. We argue that existing AI-based approaches are overly reliant on conventional numerical solvers and lack forward lithography conditions, limiting the applicability of AI-based approaches to mask optimization scenarios. To further exploit the potential of AI and machine learning, we carefully analyze the characteristics of the ILT procedure and reformulate it as an implicit layer learning problem. Instead of applying the end-to-end DEQ method, we solve the ILT layer through ILILT, a weight-tied model, where a forward lithography estimator provides lithography information in the optimization procedure. Experiments demonstrate the effectiveness of ILILT learning an ILT layer with limited intervention of a traditional numerical solver. To the best of our knowledge, this is the first time an AI solution has been proposed to imitate the working scheme of a traditional ILT system. We hope this work can further motivate research into expanding the potential of AI for solving complex design automation problems. Additional investigations on production-level designs are necessary to prototype the framework to study whether QoR meets manufacturing standards.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+    },
+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lithography is the very first topic in chip design community brought into applied deep learning. Because design layouts, masks and wafer patterns are naturally in the format of images which can benefit from many backbones and algorithms, though domain adoptions are necessary. Early efforts only fall into forward predictive tasks like hotspot detection or lithography simulation. There is limited research dealing with chip optimization problems that can directly benefit semiconductor manufacturing. This work, ILILT, we believe is the first effort to solve inverse lithography/mask optimization directly using machine learning without the intervention of traditional numerical solvers. Hopefully, this can motivate new research strategies on how is AI included in chip design and manufacturing flow, and the advancement of chip design can feedback and benefit AI in the long run.",
+        "bbox": [
+            84,
+            109,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            86,
+            354,
+            181,
+            371
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
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+            "Braam, K., Selinidis, K., Hoppe, W., Cai, H., and Xiao, G. EUV mask synthesis with rigorous ILT for process window improvement. In Proceedings of SPIE, volume 10962, 2019.",
+            "Chen, G., Chen, W., Ma, Y., Yang, H., and Yu, B. DAMO: Deep agile mask optimization for full chip scale. In IEEE/ACM International Conference on Computer-Aided Design (ICCAD), 2020.",
+            "Chen, T., Chen, X., Chen, W., Heaton, H., Liu, J., Wang, Z., and Yin, W. Learning to optimize: A primer and a benchmark. arXiv preprint arXiv:2103.12828, 2021.",
+            "Gao, J.-R., Xu, X., Yu, B., and Pan, D. Z. MOSAIC: Mask optimizing solution with process window aware inverse correction. In ACM/IEEE Design Automation Conference (DAC), pp. 52:1-52:6, 2014.",
+            "He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770-778, 2016.",
+            "Isola, P., Zhu, J.-Y., Zhou, T., and Efros, A. A. Image-to-image translation with conditional adversarial networks. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1125-1134, 2017."
+        ],
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+        "page_idx": 8
+    },
+    {
+        "type": "list",
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+        "list_items": [
+            "Jiang, B., Liu, L., Ma, Y., Zhang, H., Young, E. F. Y., and Yu, B. Neural-ILT: Migrating ILT to nerval networks for mask printability and complexity co-optimizaton\". In IEEE/ACM International Conference on Computer-Aided Design (ICCAD), 2020.",
+            "Jittorntrum, K. An implicit function theorem. Journal of Optimization Theory and Applications, 25(4):575-577, 1978.",
+            "Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A. Fourier neural operator for parametric partial differential equations. International Conference on Learning Representations (ICLR), 2021.",
+            "Ma, Y., Gao, J.-R., Kuang, J., Miao, J., and Yu, B. A unified framework for simultaneous layout decomposition and mask optimization. In IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pp. 81-88, 2017.",
+            "Pang, L., Russell, E. V., Baggenstoss, B., Lee, M., Digaum, J., Yang, M.-C., Ungar, P. J., Bouaricha, A., Wang, K., Su, B., et al. Study of mask and wafer co-design that utilizes a new extreme SIMD approach to computing in memory manufacturing: full-chip curvilinear ilt in a day. In Proceedings of SPIE, volume 11148, pp. 136-151, 2019.",
+            "Ronneberger, O., Fischer, P., and Brox, T. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), pp. 234-241, 2015.",
+            "Sun, S., Yang, F., Yu, B., Shang, L., and Zeng, X. Efficient ilt via multi-level lithography simulation. In 2023 60th ACM/IEEE Design Automation Conference (DAC), pp. 1-6. IEEE, 2023.",
+            "Ulyanov, D., Vedaldi, A., and Lempitsky, V. Deep image prior. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 9446-9454, 2018.",
+            "Wang, T.-C., Liu, M.-Y., Zhu, J.-Y., Tao, A., Kautz, J., and Catanzaro, B. High-resolution image synthesis and semantic manipulation with conditional gans. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8798-8807, 2018.",
+            "Yang, H., Li, S., Ma, Y., Yu, B., and Young, E. F. GAN-OPC: Mask optimization with lithography-guided generative adversarial nets. In ACM/IEEE Design Automation Conference (DAC), pp. 131:1-131:6, 2018."
+        ],
+        "bbox": [
+            498,
+            84,
+            887,
+            905
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
+            289,
+            56,
+            683,
+            70
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Yang, H., Li, Z., Sastry, K., Mukhopadhyay, S., Anandkumar, A., Khailany, B., Singh, V., and Ren, H. Large scale mask optimization via convolutional fourier neural operator and litho-guided self training. arXiv preprint arXiv:2207.04056, 2022a.",
+            "Yang, H., Li, Z., Sastry, K., Mukhopadhyay, S., Kilgard, M., Anandkumar, A., Khailany, B., Singh, V., and Ren, H. Generic lithography modeling with dual-band optics-inspired neural networks. In ACM/IEEE Design Automation Conference (DAC), 2022b.",
+            "Ye, W., Alawieh, M. B., Lin, Y., and Pan, D. Z. LithoGAN: End-to-end lithography modeling with generative adversarial networks. In ACM/IEEE Design Automation Conference (DAC), pp. 107:1-107:6, 2019.",
+            "Ye, W., Alawieh, M. B., Watanabe, Y., Nojima, S., Lin, Y., and Pan, D. Z. Tempo: Fast mask topography effect modeling with deep learning. In ACM International Symposium on Physical Design (ISPD), pp. 127-134, 2020.",
+            "Yu, Z., Chen, G., Ma, Y., and Yu, B. A GPU-enabled level set method for mask optimization. In IEEE/ACM Proceedings Design, Automation and Test in Eurpoe (DATE), 2021.",
+            "Zhao, W., Yao, X., Yu, Z., Chen, G., Ma, Y., Yu, B., and Wong, M. D. AdaOPC: A self-adaptive mask optimization framework for real design patterns. In IEEE/ACM International Conference on Computer-Aided Design (IC-CAD), 2022.",
+            "Zheng, S., Yang, H., Zhu, B., Yu, B., and Wong, M. D. Lithobench: Benchmarking ai computational lithography for semiconductor manufacturing. In Conference on Neural Information Processing Systems (NIPS), 2023.",
+            "Zhou, Z., Siddiquee, M. M. R., Tajbakhsh, N., and Liang, J. Unet++: Redesigning skip connections to exploit multiscale features in image segmentation. IEEE transactions on medical imaging, 39(6):1856-1867, 2019."
+        ],
+        "bbox": [
+            86,
+            84,
+            475,
+            683
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
+            289,
+            56,
+            683,
+            71
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "page_number",
+        "text": "10",
+        "bbox": [
+            477,
+            922,
+            495,
+            934
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A. Lithography Simulation Model",
+        "text_level": 1,
+        "bbox": [
+            84,
+            83,
+            375,
+            101
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Throughout the manuscript, we use  $f_{l}$  to represent the entire lithography process. In detail, this can be broken into two stages: (1) optical modeling and (2) resist modeling. The optical modeling tries to find the illumination intensity after the mask and is projected on the silicon wafer. This process can be mathematically written as,",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {I} = \\sum_ {k = 1} ^ {N} \\alpha_ {k} | | \\boldsymbol {h} _ {k} \\otimes \\boldsymbol {M} | | ^ {2}, \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where  $I$  is the light intensity projected on the silicon wafer,  $M$  denotes the mask image, and  $h_k$ 's and  $\\alpha_k$ 's are lithography system-related parameters that are calibrated in foundries.  $h_k$ 's and  $\\alpha_k$ 's are different under different process conditions.",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "During resist modeling,  $I$  will have a chemical reaction with some \"resist materials\" and etch chip design on the top of the silicon wafer, which is given by",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {Z} (i, j) = \\left\\{ \\begin{array}{l l} 1, & \\text {i f} I (i, j) > I _ {\\mathrm {t h}}, \\\\ 0, & \\text {o t h e r w i s e .} \\end{array} \\right. \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this paper,  $I_{\\mathrm{th}} = 0.225$ , which is determined by the technology profile.",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/861409920e573e250d4a297ab010ef8a65641e01de41002caa96e8085a1b2dc9.jpg",
+        "image_caption": [
+            "(a) Mask",
+            "Figure 7. Lithography process. (a) Mask image contains the patterns of optimized design; (b) Intensity image is the UV lights that are projected on the silicon wafer given by Equation (14); (c) Wafer image is the patterns on silicon given Equation (15)."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/8f0b89de105cd5ff1af2c34614270df10ee2e7fbb35ad44ad0d52a27a7857cff.jpg",
+        "image_caption": [
+            "(b) Intensity"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            429,
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/60d45a7eb0241100c79fc2b5a85d247392c986f4f9c040343fea8909b57358bc.jpg",
+        "image_caption": [
+            "(c) Wafer"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B. Numerical ILT",
+        "text_level": 1,
+        "bbox": [
+            84,
+            537,
+            238,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Because numerical solver usually requires differentiable objectives. In practical, we relax  $M \\in \\{0,1\\}$  and  $Z \\in \\{0,1\\}$  through sigmoid.",
+        "bbox": [
+            84,
+            563,
+            888,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {M} = \\frac {1}{1 + \\exp \\left(- \\beta_ {m} \\left(\\boldsymbol {M} ^ {\\prime} - 0 . 5\\right)\\right)}, \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            604,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where  $M^{\\prime}$  is some auxiliary variable that is usually initialized as the original chip design  $Z^{*}$ .",
+        "bbox": [
+            84,
+            648,
+            694,
+            666
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {Z} = \\frac {1}{1 + \\exp \\left(- \\beta_ {z} (\\boldsymbol {I} - I _ {\\mathrm {t h}}) \\right.}. \\tag {17}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            385,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "And now we can solve the following ILT problem through gradient descent.",
+        "bbox": [
+            84,
+            715,
+            586,
+            733
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {M, M ^ {\\prime}} l = \\left| \\left| Z - Z ^ {*} \\right| \\right| \\tag {18}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            743,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "s.t. Equations (14), (16) and (17).",
+        "bbox": [
+            383,
+            771,
+            609,
+            787
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "It should be noted that numerical ILT requires frequent calls of time-consuming lithography simulation.",
+        "bbox": [
+            84,
+            800,
+            766,
+            816
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C. CFNO",
+        "text_level": 1,
+        "bbox": [
+            84,
+            833,
+            171,
+            849
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "CFNO, the convolutional Fourier Neural Operator (Yang et al., 2022a), was built upon the original FNO (Li et al., 2021) design. The capability of FNO learning frequency domain features fits the lithography well. However, the original FNO design has a few challenges: (1) FNO requires computations of FFT of the entire input mask image which is computationally",
+        "bbox": [
+            84,
+            859,
+            888,
+            906
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
+            287,
+            56,
+            684,
+            71
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "page_number",
+        "text": "11",
+        "bbox": [
+            477,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/1e248c99f4ef6aba42aa769289ebc907e60171e103cd87b986c411f676648096.jpg",
+        "image_caption": [
+            "Figure 8. CFNO backbone design."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            282,
+            127,
+            681,
+            262
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/e35e201124e6c374d327b8953f862a0dcbe12971b1b1fb518ac834b60ecbebcc.jpg",
+        "image_caption": [
+            "Figure 9. The data pipeline of  $g$  used throughout experiments."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            367,
+            388,
+            609,
+            830
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
+            289,
+            56,
+            683,
+            70
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
+            477,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "costly; (2) FNO, once trained, can only be applied on the same-sized image. CFNO addresses these problems by combining the patch embedding in ViT. The core idea is to separate the input into non-overlapped patches, and each patch will go through a shared FNO layer as in Figure 8. Then we perform a token-wise convolution to learn the relations among different patches and finally form the mask embedding. The overall architecture of  $g$  is shown in Figure 9.",
+        "bbox": [
+            83,
+            85,
+            887,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D. Pix2PixHD",
+        "text_level": 1,
+        "bbox": [
+            84,
+            186,
+            209,
+            204
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Pix2PixHD (Wang et al., 2018) is an improved version of Pix2Pix which employs hierarchical local enhancers for high-resolution image generation. To make the framework fit more on the mask optimization problem we choose  $ngf = 16$ ,  $n\\_downsample\\_global = 4$ ,  $n\\_blocks\\_global = 9$  as defined in https://github.com/NVIDIA/pix2pixHD. We also make certain improvements by replacing deconvolution layers with bicubic interpolation and stride-1 convolution layers for smooth shape generation. We also apply  $4 \\times 4$  downsampling (achieved by average pooling) at the input directly to reduce computing overhead and perform interpolation to cast the image back to desired resolution. This modification yields 45M model as listed in Section 4.",
+        "bbox": [
+            83,
+            213,
+            890,
+            320
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E. Additional Visualization",
+        "text_level": 1,
+        "bbox": [
+            86,
+            338,
+            318,
+            354
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/30fc5dc866367b352e892ccca7840edcc2bb8c800534abfb79f9b2ec5d167e17.jpg",
+        "image_caption": [
+            "(a) AES-Design"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            86,
+            366,
+            251,
+            493
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/2c9dacbe2f96d31b0e974351bf3ca00375f59dfc8acf46b9b9465e27d1b0bb7d.jpg",
+        "image_caption": [
+            "(b) ILT-Mask",
+            "(c) ILILT(P2P)-Mask",
+            "(d) ILILT(CFNO)-Mask",
+            "(e) L2O(P2P)-Mask",
+            "(f) L2O(CFNO)-Mask"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            86,
+            518,
+            888,
+            647
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/42146d0977944f2953d64f3c00fec579601c0aa8b137bfa0ba6ce6ec63955cc2.jpg",
+        "image_caption": [
+            "(g) ILT-Wafer",
+            "(h) ILILT(P2P)-Wafer",
+            "(i) ILILT(CFNO)-Wafer",
+            "(j) L2O(P2P)-Wafer",
+            "(k) L2O(CFNO)-Wafer",
+            "Figure 10. Visualization sample of AES chip design for ILILT and L2O with CNN and Neural Operator backbones. L2O with Pix2PixHD backbone even produces meaningless artifacts."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            86,
+            674,
+            887,
+            801
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "ILILT: Implicit Learning of Inverse Lithography Technologies",
+        "bbox": [
+            287,
+            56,
+            684,
+            71
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            477,
+            922,
+            495,
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+        ],
+        "page_idx": 12
+    }
+]

2024/ILILT_ Implicit Learning of Inverse Lithography Technologies/c0639e19-5e8f-4a49-b452-4de437af403d_origin.pdf

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2024/ILILT_ Implicit Learning of Inverse Lithography Technologies/full.md

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+# ILILT: Implicit Learning of Inverse Lithography Technologies
+
+Haoyu Yang<sup>1</sup> Haoxing Ren<sup>1</sup>
+
+# Abstract
+
+Lithography, transferring chip design masks to the silicon wafer, is the most important phase in modern semiconductor manufacturing flow. Due to the limitations of lithography systems, Extensive design optimizations are required to tackle the design and silicon mismatch. Inverse lithography technology (ILT) is one of the promising solutions to perform pre-fabrication optimization, termed mask optimization. Because of mask optimization problems' constrained non-convexity, numerical ILT solvers rely heavily on good initialization to avoid getting stuck on sub-optimal solutions. Machine learning (ML) techniques are hence proposed to generate mask initialization for ILT solvers with one-shot inference, targeting faster and better convergence during ILT. This paper addresses the question of whether ML models can directly generate high-quality optimized masks without engaging ILT solvers in the loop. We propose an implicit learning ILT framework: ILILT, which leverages the implicit layer learning method and lithography-conditioned inputs to ground the model. Trained to understand the ILT optimization procedure, ILILT can outperform the state-of-the-art machine learning solutions, significantly improving efficiency and quality.
+
+# 1. Introduction
+
+Lithography is the most important phase in semiconductor manufacturing flow, which transfers chip design patterns onto the silicon wafer. However, due to the limitations of lithography systems, there are usually fetal manufacturing gaps between the design and the patterns on the wafer. Extensive prior fabrication design optimizations, termed mask optimization, are required to compensate for the manufacturing loss (Figure 1).
+
+$^{1}$ Design Automation Research Group, NVIDIA, Austin, TX. Correspondence to: Haoyu Yang <haoyuy@nvidia.com>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+![](images/59e198609b91e28e002c5b3f7612dcff7aa7d0e825b5a63d02f37b324238d21c.jpg)  
+Figure 1. Limitations of lithography system requires design optimization to be correctly fabricated on the silicon wafer.
+
+Inverse lithography technology (ILT), as a numerical solution, has shown great promise for mask optimization on advanced-node lithography (Gao et al., 2014; Yu et al., 2021; Braam et al., 2019; Pang et al., 2019). As shown in Figure 2(a), an ILT solver requires frequent calls of lithography simulator, which mathematically approximate the lithography fabrication behavior. The forward path computes the wafer image corresponding to the current mask and we measure the error between the wafer image and the original design. In the backward path, the error will be passed back through gradient method to update the mask. However, the non-convexity of mask optimization problems poses a great challenge to numerical ILT solvers, that they rely heavily on good mask initialization and can easily get stuck at sub-optimal solutions.
+
+Recent research into machine learning (ML) techniques for lithography have made ILT solvers more efficient. As shown in Figure 2(b), ML can provide better optimization starting points that significantly reduce ILT iterations and yield better convergence (Yang et al., 2018; Chen et al., 2020; Jiang et al., 2020). GAN-OPC (Yang et al., 2018) is the earliest deep learning solution for inverse lithography, where a conditional generative adversarial network is proposed for initial mask generation with litho-guided pre-training. DAMO (Chen et al., 2020) improves on this work by replacing the standard generator in GAN-OPC with nested UNet and ResNet bottleneck layers (Zhou et al., 2019; Isola et al., 2017; He et al., 2016), yielding a direct output of high-resolution masks. Jiang et al. (Jiang et al., 2020) also integrates the explicit gradient calculation of litho-guided pre-training and achieves better training convergence and mask quality. Very recently, Zhao et al. (Zhao et al., 2022) develops the AdaOPC framework that is able to learn from
+
+![](images/1951a342b3c97aca01378f23b1db00b107f6e758eb8a8791793ec9c13046aa7b.jpg)  
+(a) ILT
+
+![](images/81bf9b7c7601f8e342f3e2b41cd8386c881a5b584f48aabb99e5ad05889a23dc.jpg)  
+(b) ML-Aided
+
+![](images/ff20a6cfe4d35f4f90bf656ada9291404e659ef40bb277d5ccf61dad531b10a9.jpg)  
+(c) ILILT  
+Figure 2. Working scheme of ILT solvers. (a) Standard ILT solver that iteratively updates the mask from design till convergence. (b) Common ML approach that uses ML to produce an initial mask followed by standard ILT procedure. (c) The ILILT that uses ML to imitate the standard ILT procedure.
+
+layout geometry and identifies whether a pattern can be optimized by the machine learning engine. This further increase the applicability of ML for mask optimization solutions.
+
+However, existing solutions only exploit limited ML in the ILT solver and still require significant intervention by traditional numerical solvers. For example, GAN-OPC and its variations can only learn a design-to-mask mapping, which cannot be used to fix poor-quality masks; AdaOPC (Zhao et al., 2022) has to conduct full numerical optimizations on complicated patterns. In this paper, we try to address the question of whether an ML model can directly generate high quality optimized masks without engaging ILT solvers in the loop. We believe that the key limitation of existing methods is that the trained ML models are not grounded to any actual lithography condition during inference, therefore training-inference distribution shift can not be recognized and corrected. We argue that ML models grounded on lithography estimation of intermediate masks can make an ML-based ILT framework more effective.
+
+We propose the ILILT (Figure 2(c)) framework to formulate mask optimization as an implicit layer learning problem, trained to optimize the mask towards an equilibrium state (Bai et al., 2019). Like a standard ILT solver that iteratively updates masks from a design, ILILT iteratively modifies masks from a design but with much fewer steps and each step runs significantly faster because no gradient computing is required. This process produces a number of intermediate masks which ground the ML model with intermediate wafer images computed by lithography models.
+
+ILT As An Implicit Layer. Unlike common neural network layers that explicitly define specific computation graphs supporting forward and backward propagation, an
+
+implicit layer models a joint condition of its input and output (Jittorntrum, 1978). The nature of ILT makes it a representative example of an implicit layer, where the input is a chip design and the output is the corresponding optimized mask that results in a similar wafer pattern compared to the input after the lithography process. Inspired by the implicit layer learning approach, we design ILILT with a weight-tied sequence model, making it possible to learn an optimization trajectory from training design-mask pairs. To make the trajectory learning grounded with physics, we also integrate ILILT with fast lithography estimator that can implicitly apply lithography conditions on top of the optimization procedure. After training, ILILT is able to perform mask optimization on a given design and reaches a stable state in a few iterations. We will later show how ILILT outperforms a state-of-the-art academic ILT engine with technical details and supporting experiments on public chip design benchmarks.
+
+The reminders of the paper are organized as follows: Section 2 covers basic terminologies and related works of mask optimization; Section 3 discusses and analyzes the technique details of the ILILT framework; Section 4 presents the experimental results supporting the effectiveness of ILILT and Section 5 concludes the paper.
+
+# 2. Preliminaries
+
+# 2.1. ILT Solver
+
+Mask optimization tries to find a mask design such that the remaining pattern on the silicon wafer after the lithography process is as close as possible to the desired shapes. This can be formulated as a constrained optimization problem:
+
+$$
+\min  _ {M} l = d \left(\boldsymbol {Z}, \boldsymbol {Z} ^ {*}\right), \tag {1}
+$$
+
+where,
+
+$$
+\boldsymbol {Z} = f _ {l} (\boldsymbol {M}), \tag {2}
+$$
+
+where  $Z$  and  $Z^{*}$  represent the wafer image and the chip design, respectively,  $M$  corresponds to the mask to be optimized,  $f_{l}$  represents the lithography process which is highly non-convex (see appendix for more details) and  $d$  is a measurement indicating the differences between  $Z$  and  $Z^{*}$ . Inverse lithography techniques (Gao et al., 2014) try to solve Problem (1) in a numerical way by repeating the following steps until convergence:
+
+1. Forward:  $Z_{t} = f_{l}(M_{t})$ .  
+2. Backward:  $M_{t + 1} = M_t - \lambda \frac{\partial l}{\partial Z_t}\frac{\partial Z_t}{\partial M_t}$ .
+
+where  $\lambda$  is some hyper-parameter that controls how the mask image is updated during the optimization procedure. There
+
+are also variations of ILTs (Yu et al., 2021; Jiang et al., 2020; Gao et al., 2014), but they mostly share similar mask-update concepts.
+
+# 2.2. Implicit Layers
+
+Most neural network layers or computational graphs explicitly define a function  $\pmb{y} = f(\pmb{x})$ , which is differentiable and can be back-propagated via the chain rule. An implicit layer, instead of specifying how to compute the layer's output from the input, specifies the conditions that we want the layer's output to satisfy, separating the procedure to compute the layer from the layer definition itself. It has been used in deep equilibrium models, Neural ODEs, and differentiable optimizations.
+
+Definition 2.1 (Implicit Layer). Implicit layers try to find  $\pmb{y}$  such that  $f(\pmb{x},\pmb{y}) \in \mathcal{C}$ , which defines joint conditions between  $\pmb{x}$  and  $\pmb{y}$  (Jittorntrum, 1978).
+
+A fixed-point layer is an example of the implicit layer method, which directly computes the fixed point of a nonlinear transformation, i.e., the solution to the nonlinear system which models the convergent state of a recurrent sequence.
+
+Definition 2.2 (Fixed-Point Layer). A fixed-point layer can be mathematically given by
+
+$$
+\boldsymbol {y} = g (\boldsymbol {x}, \boldsymbol {y}). \tag {3}
+$$
+
+A fixed-point layer can be considered to be an infinite number of layers parameterized by the same weight  $w$  (weighted), where each layer can be represented as
+
+$$
+\boldsymbol {y} _ {t + 1} = g (\boldsymbol {x}, \boldsymbol {y} _ {t}, \boldsymbol {w}), \tag {4}
+$$
+
+To train a fixed-point layer, (Bai et al., 2019) proposes an unrolling-free deep equilibrium (DEQ) method, which trains the fixed-point layer in two alternative steps: (1) solve for the fixed-point of the layer and (2) update network parameters. This process consists of repeated forward and backward computations as follows. The forward path is related to the finding of the fixed point of Equation (3) and can be solved with Newton's method. Let
+
+$$
+h (\boldsymbol {x}, \boldsymbol {y} ^ {*}, \boldsymbol {w}) = g (\boldsymbol {x}, \boldsymbol {y} ^ {*}, \boldsymbol {w}) - \boldsymbol {y} ^ {*}, \tag {5}
+$$
+
+and  $\pmb{y}^{*}$  can be found through the following step until convergence:
+
+$$
+\boldsymbol {y} _ {t + 1} = \boldsymbol {y} _ {t} - \lambda \boldsymbol {J} _ {t} ^ {- 1} h (\boldsymbol {x}, \boldsymbol {y} _ {t}, \boldsymbol {w}), \tag {6}
+$$
+
+where  $J_{t}$  is the Jacobian matrix of  $h$  at  $\mathbf{y}_t$  and  $\lambda$  is the update rate.
+
+The backward process covers the update of  $\boldsymbol{w}$ , which can be written as
+
+$$
+\boldsymbol {w} = \boldsymbol {w} - \lambda \frac {\partial l}{\partial \boldsymbol {w}} = \boldsymbol {w} + \lambda \frac {\partial l}{\partial \boldsymbol {y} ^ {*}} \boldsymbol {J} ^ {* - 1} \frac {\partial g}{\partial \boldsymbol {w}}, \tag {7}
+$$
+
+where  $J^{*}$  is the Jacobian matrix of  $h$  at  $\boldsymbol{y}^{*}$ ,  $l$  is a loss function that measures whether  $f(x,y) \in \mathcal{C}$  satisfies the implicit layer definition, e.g. the ILT loss Equation (1). For a detailed proof of Equation (7), we refer readers to (Bai et al., 2019).
+
+# 2.3. Neural Networks for ML-based Computational Lithography
+
+There are two major groups of network architectures used in lithography applications: (1) The UNet family (Ronneberger et al., 2015; Zhou et al., 2019) and (2) The FNO family (Li et al., 2021). UNets consist of various bypass connections between convolution layer outputs. Representative works include TEMPO (Ye et al., 2020), LithoGAN (Ye et al., 2019), DAMO (Chen et al., 2020) and GAN-OPC (Yang et al., 2018). FNOs contain one or more Fourier Neural Operators along with auxiliary convolution layers. Representative works are DOINN (Yang et al., 2022b) and CFNO (Yang et al., 2022a). UNet-based models usually rely on large model capacity to learn training data distribution while FNO-based models focus on global information learning through frequency domain embedding.
+
+# 3. The ILILT Framework
+
+# 3.1. ILT Layer
+
+Following the definition of implicit layer learning, we can solve the ILT equations with an implicit ILT layer defined as finding  $M$  such that
+
+$$
+f (\boldsymbol {M}, \boldsymbol {Z} ^ {*}) = d \left(f _ {l} (\boldsymbol {M}), \boldsymbol {Z} ^ {*}\right) = \epsilon , \tag {8}
+$$
+
+where  $\epsilon$  is some minimum value achievable for Equation (1).
+
+Because Equation (8) is a composite of a group of non-linear functions, it is difficult to solve explicitly. Therefore, we assume Equation (8) has a solution  $M^{*}$  that follows the form of
+
+$$
+\boldsymbol {M} ^ {*} = g \left(\boldsymbol {M} ^ {*}, \boldsymbol {Z} ^ {*}\right), \tag {9}
+$$
+
+which yields a fixed-point layer representation of the ILT solution.
+
+We start solving Equation (9) from the weight-tied approach, where we define
+
+$$
+\boldsymbol {M} _ {t + 1} = g \left(\boldsymbol {M} _ {t}, \boldsymbol {Z} ^ {*}, \boldsymbol {w}\right), t = 1, 2, \dots , \tag {10}
+$$
+
+and
+
+$$
+\lim  _ {t \rightarrow \infty} \boldsymbol {M} _ {t} = \lim  _ {t \rightarrow \infty} g \left(\boldsymbol {M} _ {t}, \boldsymbol {Z} ^ {*}, \boldsymbol {w}\right) = g \left(\boldsymbol {M} ^ {*}, \boldsymbol {Z} ^ {*}, \boldsymbol {w}\right) = \boldsymbol {M} ^ {*}. \tag {11}
+$$
+
+# 3.2. Lithography Estimation
+
+As we discussed previously, due to the lack of prior lithography knowledge, state-of-art AI approaches have been lim
+
+ited to only generate initialization for numerical ILT solvers. This motivates us to build lithography estimation inside  $g$  to ground the AI model.
+
+We rewrite Equation (10) as:
+
+$$
+\boldsymbol {M} _ {t + 1} = g \left(\boldsymbol {M} _ {t}, \boldsymbol {Z} _ {t}, \boldsymbol {Z} ^ {*}, \boldsymbol {w}\right), t = 1, 2, \dots , \tag {12}
+$$
+
+where  $Z_{t} = f_{l}(M_{t})$  is the wafer image of  $M_{t}$  at time stamp  $t$ . Equation (12) formulates the core of the ILILT framework. A virtue of Equation (12) is the embedded condition of the lithography estimator that implicitly tells  $g$  the physical meaning of each step.
+
+# 3.3. Training
+
+Equation (12) defines a weight-tied fixed-point model that can be solved with the DEQ method. Although DEQ (Bai et al., 2019) reduces the computing overhead of sequence unrolling and back-propagation through time, there are several concerns when it's used for solving ILT problems:
+
+- Solving  $M^{*}$  through Newton's method or other Root-Finding algorithms is still computationally costly considering that  $M^{*}$  usually has millions of entries.  
+- The solution error caused by the non-optimality in Newton's method will further propagate to the gradient calculation in backward DEQ, inducing additional errors.  
+- The DEQ backward pass still requires the computation of the gradient over lithography modeling in the term  $\frac{\partial d}{\partial M^*}$ , showing no advantage over a traditional ILT solver.
+
+To address these concerns, we developed the following training architecture that solves the ILT layer efficiently and effectively. We unroll ILILT to a fixed depth  $T$  and allow the final output  $M_T$  to be trained towards a ground truth mask  $M^*$ ,
+
+$$
+\min  _ {\boldsymbol {w}} l = \sum_ {t = \left[ \frac {T}{2} \right]} ^ {T} \exp \left[ \frac {t}{T} - 1 \right] \cdot \left| \left| \boldsymbol {M} _ {t} - \boldsymbol {M} ^ {*} \right| \right| _ {F} ^ {2}, \tag {13}
+$$
+
+s.t.  $M_{t + 1} = g(M_t,Z_t,Z^*,\pmb {w}), t = 0,1,\dots,T - 1,$
+
+$$
+\boldsymbol {Z} _ {t} = f _ {l} (\boldsymbol {M} _ {t}), t = 0, 1,..., T,
+$$
+
+where  $t$  indicates each unrolling step.
+
+It should be noted that to encourage the mask to grow towards and stop at the golden solution, we did not adopt the cost function that directly minimizes the difference between  $M_T$  and  $M^*$ . Instead, we ask intermediate masks in later unrolling stages to have a trend growing toward the ground truth, which is reflected in Equation (13) having masks at
+
+each time step  $M_{t}$  trained towards the ground truth  $M^{*}$ . The coefficient  $\exp \left[\frac{t}{T} - 1\right]$  applied at each loss term is to let the training process be aware of the progressive growth of masks at each time step. This trick allows the mask to grow smoothly in these steps while stopping at the fixed point. The exponential term is empirically chosen to regularize the mask-growing trends which are consistent with the behavior of the numerical ILT solver that only minor changes are observed in later optimization iterations. Solving Equation (13) is straightforward through back-propagation through time (BPTT).
+
+# 3.4. Discussion
+
+# 3.4.1. RELATIONS TO ML-AIDED SOLUTIONS
+
+For the training procedure, we observe that ILILT resembles GAN-OPC (Yang et al., 2018) in terms of the training objectives, where we also force the model to learn a golden mask generated by some conventional methods. However, ILILT advances beyond GAN-OPC works (Yang et al., 2018; Jiang et al., 2020; Zhao et al., 2022; Chen et al., 2020) because it not only learns what a good mask should look like, but also why and how a good mask is grown thanks to the implicit lithography condition introduced by  $f_{l}(\cdot)$ .
+
+# 3.4.2. RELATIONS TO IMPLICIT LAYER LEARNING
+
+The whole ILILT method is built upon the definition of the ILT layer and how it is solved. Representative solutions, e.g. DEQ (Bai et al., 2019), still requires costly computation of gradients of the objectives over physical models. ILILT makes it even more implicit by feeding the wafer image, target design and the mask image simultaneously into  $g(\cdot, \boldsymbol{w})$  at each time stamp. Finally  $g(\cdot, \boldsymbol{w})$  will automatically figure out the inherent relations between masks, designs and resist patterns.
+
+# 3.4.3. RELATIONS TO LEARNING-TO-OPTIMIZE
+
+The ILILT system is a similar scheme to learning-to-optimize (L2O) solutions (Chen et al., 2021), particularly the algorithm unrolling approach, which unrolls a truncated algorithm into multi-stage neural networks with independent parameters. In contrast, ILILT benefits from the weight-tied structure that (1) significantly reduces model size for memory efficiency and (2) generalizes better (Bai et al., 2019).
+
+# 3.4.4. PRACTICAL USAGE SCENARIO
+
+Lastly, ILILT imitates a real ILT solver during the inference runtime but with many benefits: (1) ILILT requires to query a lithography simulator in each step but updates the mask in a derivative-free manner. (2) ILILT requires significant less steps than traditional ILT solver. (3) ILILT has the opportunity to outperform numerical ILT solvers due to strong mask
+
+![](images/3554f6abc20d2f08793e5533cb9260981d2f080ac82d61dc7b57baefaf1db8a4.jpg)  
+(a) Design
+
+![](images/82ff16beb3f84d30778a974defd7419200f71e37e5adfb5e6446999f688b395a.jpg)  
+(b) Mask
+
+![](images/8a7133d90e2b4480b4974cb3e4663bd61adfd4afcc0e5ca66efb66fd413762c1.jpg)  
+(c) Wafer  
+Figure 3. Data example from LithoBench (Zheng et al., 2023). (a) Design image contains the patterns of original circuit devices or connectivity; (b) Mask image is the optimized design that is manufacturing-friendly; (c) Wafer image is the patterns on silicon given the mask image.
+
+prriors learned from broad data points. (4) ILILT can serve as a real solver during chip design optimization. One representative case is when a design is not well-optimized by an existing mask optimizer. Because most AI solutions only learn a design-to-mask mapping, it is not possible to use them to fix partially optimized masks. ILILT, however, does not have this limitation, and can take over from anywhere in the optimization trajectory.
+
+# 4. Experiments
+
+# 4.1. The Dataset and Configurations
+
+To demonstrate the effectiveness of the ILILT framework, we adopt the latest AI for computational lithography benchmark suite LithoBench (Zheng et al., 2023), where state-of-the-art mask optimization solutions are evaluated and will be used as comparison baselines. LithoBench contains over 100K design clips and each covers an area of  $2\mu m \times 2\mu m$ . Each data point contains a pair of chip design patterns and the ground truth masks generated by a numerical ILT solver. These design patterns and masks are rasterized with a  $1nm^2$  pixel size, yielding  $2048 \times 2048$  images. Example designs are displayed in Figure 7. We can observe that in real world, mask optimization does not simply finetune the design like in Figure 1. The process also produces sub-resolution assist features (SRAFs) surrounding the main design pattern to improve its robustness against process variations. This poses great challenge of ML models on optimization tasks.
+
+To deploy ILILT, we re-implement the lithography simulator as a pytorch layer that can be integrated in the data pipeline. Detailed configurations of the generator used to reproduce our results are listed in Table 1. Here we adopt CFNO (Yang et al., 2022a) as the backbone network for  $g$  to evaluate ILILT.
+
+# 4.2. Evaluation Metrics
+
+As is typical for mask optimization tasks, we evaluate the final performance through the metrics that are widely adopted
+
+Table 1. Model Configuration.  
+
+<table><tr><td>Item</td><td>g</td></tr><tr><td>Backbone</td><td>CFNO/Pix2PixHD</td></tr><tr><td>Max Epoch</td><td>5</td></tr><tr><td>Initial Learning Rate</td><td>0.004</td></tr><tr><td>Learning Rate Decay Policy</td><td>step</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Weight Decay</td><td>0.0001</td></tr><tr><td>Loss</td><td>Equation (13)</td></tr><tr><td>Sequence Unrolling Depth (T)</td><td>4-8</td></tr><tr><td>Batch Size</td><td>4</td></tr></table>
+
+![](images/0187f896587e62b86714b6fd6621dfb681fdf4f7e2723ddbc76d51819c23f26a.jpg)
+
+![](images/27892f5cafc21debf47d64860f8947136fe477bacaef860584495e9203478925.jpg)  
+Figure 4. Lithography edge placement error and process variations. Smaller EPE and PVB area indicates better optimization QoR.
+
+in the chip manufacturing industry: edge placement error (EPE) violations and the process variation band (PVB) area. In semiconductor manufacturing processes, the lithography conditions are not always as we expected due to systematic error and mechanical perturbations, which is the root cause of process variations. Therefore, we expect the design to be robust under different manufacturing conditions. Typically, we call the expected lithography settings and configurations nominal condition, and non-nominal condition will result in larger or smaller patterns on the silicon wafer. The difference between the smallest pattern and the largest pattern will form a process variation band (PVB) as in Figure 4. A good mask optimization result should have a smaller number of EPE violations and a smaller PVB area.
+
+Definition 4.1 (EPE Violation(Banerjee et al., 2013)). EPE is measured as the geometric distance between the target edge and the lithographic contour printed at the nominal condition. If the EPE measured at a point is greater than a certain tolerance value, we call it an EPE violation. In our experiments, the EPE measuring points are sampled at  $40nm$  interval.
+
+Definition 4.2 (PVB Area(Banerjee et al., 2013)). This is evaluated by running lithography simulations at different process conditions on the final mask solution. Once run, the total area of the process variation band will be defined as PVB Area.
+
+![](images/94a95e8e4176e8bd5f391f33b7bcb1ae800fc33eac78093377fc3ea9c777dde0.jpg)  
+(a)  $\mathrm{T} = 1$
+
+![](images/c936be93886ce61bedc0ab36f9f81c71e9179f22412ac1a1c0016f0d747da542.jpg)  
+(b)  $\mathrm{T} = 2$
+
+![](images/c9f506c1328ee7e8f73000bc8a6e630a7a82306f22e3da7d13ce713ad5e41a5e.jpg)  
+(c)  $\mathrm{T} = 3$
+
+![](images/8a39a4ca39626318acd9f83e47d81d598899239bac63589223c045a652a05d87.jpg)  
+(d)  $\mathrm{T} = 4$
+
+![](images/7468b471194520d05af833531063962593609a8f38c5822f92bf219233267dad.jpg)  
+(e)  $\mathrm{T} = 5$
+
+![](images/0e6d886a2e50b3243b18acca8eb6423439596d307e7aa8935acf796a2ea9bc7c.jpg)  
+(f)  $\mathrm{T} = 6$
+
+![](images/29a9c09ffb07faee36cbc783d3c6538067b36af23c06153800ee0f05892b9996.jpg)  
+(g)  $\mathrm{T} = 7$
+
+![](images/950036039630f147e689c1b512cf8ba239b5fd65bdce53c6099401a0c02540ce.jpg)  
+(h)  $\mathrm{T} = 8$
+
+![](images/c334c5c2d790b63c61232329a7395a007ac62f2b5ab7095774a2489229b1f07a.jpg)  
+(i) EPE=12
+
+![](images/e4b39924cfe6d24692fa5a59bc89539de4085ef1df0ac7be5aba6c704aaf9087.jpg)  
+(j)  $\mathrm{EPE} = 12$
+
+![](images/1b47c37982f489b61c1f7358add0609d1a0b7952d94bf7097e8d02c700338f61.jpg)  
+(k) EPE=0
+
+![](images/6d78a2056ba015e37e96dc5196dc03dc72819fac0f4719ba57a65bae24b193d3.jpg)  
+(1) EPE=0
+
+![](images/6830137cb596a43106ec144e289b65d661a29ca64ef54bfb5ef4157aa67d265f.jpg)  
+(m) EPE=0
+
+![](images/0597604f367a1130e2ff7f2b5ec8b994ed13eaf4a89fb53c933fd92c72e98d52.jpg)  
+(n) EPE=0
+
+![](images/b06484a4019f06d6f8b09cd6441ce4d40307d5960f7a685ea991713a8590ea22.jpg)  
+(o) EPE=0
+
+![](images/6adec80a2aa9d88023a969cf4e3109bc8915c3c8d15e4d346b4afcde32906fe4.jpg)  
+(p) EPE=0
+
+![](images/d8e9566c9a4f9e1e9f137bc5cfea4b6777e2cad6d5303ebe765382c3acf82946.jpg)  
+Figure 5. Visualization of the optimization trajectory beyond the maximum unrolling depth. We use the model trained with unrolling step  $\mathrm{T} = 4$ , and mask stays stable after the 4th step with only minor tuning. (a)-(h) correspond to the generated masks generated from time stamps 1-8. (i)-(p) are the lithography simulated wafer image with their corresponding EPE violation count.  
+(a) Design
+
+![](images/a54474d3502d0a79c432c1669d4e032067fbfc67c8fd5a4991d908f0fd6e05c2.jpg)  
+(b) ILT-Mask
+
+![](images/a5428f6c2bef5b39218ad8513954173e0a7699a5bef358ef2e18ab7dc81d064d.jpg)  
+(c) ILT-Wafer
+
+![](images/3f32c67f0995c5fea07799a56deef692cf69a194446e1b85ec65828bee36942d.jpg)  
+(d) ILILT-Mask
+
+![](images/d6c81548cbbfa6157f129ddd75fc79bea7a4943c9e914a016aabd12c49e15200.jpg)  
+(e) ILILT-Wafer
+
+![](images/142615c8785e3d36a90e434d7dfcd8f934d445c8708350e20e6f57adafc176f8.jpg)  
+(f) Design  
+Figure 6. Mask prior learning from ILILT (P2PHD). Due to design pattern variations, the numerical solver sometimes fails to produce the correct solution. In this example, ILT generates rule-violating artifacts and performs feature shrinking excessively, causing the missing of design patterns as in (c). While ILILT can learn from the broader dataset and make correct predictions. For a more complicated case (f)-(j), ILILT can successfully produce optimized mask that can pass lithography printability check while numerical ILT still exhibits EPE violations in the printed wafer image.
+
+![](images/d3b5aa835ed03488be68113545b1847862ece87750b2dfd30e7626fb59e6e302.jpg)  
+(g) ILT-Mask
+
+![](images/0d2b99835c8cae389244066dfc8553ef04c6c2ac8d2fca488bf100633309db52.jpg)  
+(h) ILT-Wafer
+
+![](images/1a28e9ffcf3d74059fe2d714601ae43c38315dc7fa5affe38c2fcf56d9798cd5.jpg)  
+(i) ILILT-Mask
+
+![](images/2967971a871668b955f4468540a5c25e69909d131ae6630f317c7adfc3ec25f4.jpg)  
+(j) ILILT-Wafer
+
+# 4.3. Comparison with State-of-the-Art
+
+In the first experiment, we compare the performance of ILILT with state-of-the-art mask optimizers. A quantitative comparison is listed in Table 2. Columns "EPE", "PVB" and "Throughput (s/tile)" indicate the count of EPE violations, PVB Area in terms of  $nm^2$  and optimization throughput, respectively; "GPU-ILT" (Sun et al., 2023) is the latest academic numerical ILT engine with GPU acceleration, which will also be the engine used to generate the training data; "GAN-OPC" (Yang et al., 2018) is one of the early attempts using machine learning for mask optimization following the scheme in Figure 2(b). GAN-OPC employs UNet as a generator backbone. "Pix2PixHD" (Wang et al., 2018) is another representative CNN-based backbone that enjoys better gen
+
+eralization capability for high-resolution image generation, which fits design-mask translation task well. "Neural-ILT" (Jiang et al., 2020) is a follow-up of GAN-OPC, which adopts a true lithography simulator to produce the gradient for generator training. "CFNO" (Yang et al., 2022a) is an FNO-based design that preserves the global perception of the Fourier neural operator while saving computation with ViT's patch embedding. CFNO and Pix2PixHD will also be the backbone frameworks of the ILILT generator, and we will show the advantage of the ILILT data pipeline over the standalone generator. ILILT- $X$  lists the results for  $X$  unrolling steps.
+
+Overall, the ILILT with Pix2PixHD backbone achieves the smallest EPE violation count among all other ML-based
+
+Table 2. Comparison with State-of-the-Arts.  
+
+<table><tr><td>Method</td><td>EPE</td><td>PVB</td><td>Throughput</td></tr><tr><td>GPU-ILT (Sun et al., 2023)</td><td>0.21</td><td>4656</td><td>4</td></tr><tr><td>GAN-OPC (Yang et al., 2018)</td><td>8.3</td><td>6686</td><td>0.008</td></tr><tr><td>Neural-ILT (Jiang et al., 2020)</td><td>6.2</td><td>8537</td><td>10</td></tr><tr><td>Pix2PixHD (Wang et al., 2018)</td><td>0.12</td><td>4685</td><td>0.1</td></tr><tr><td>CFNO (Yang et al., 2022a)</td><td>0.51</td><td>4623</td><td>0.06</td></tr><tr><td>L2O-CFNO-8-Step</td><td>1.59</td><td>4436</td><td>0.6</td></tr><tr><td>L2O-P2PHD-8-Step</td><td>0.87</td><td>4526</td><td>0.5</td></tr><tr><td>ILILT-CFNO-2-130K (ours)</td><td>0.39</td><td>4704</td><td>0.1</td></tr><tr><td>ILILT-CFNO-4-130K (ours)</td><td>0.39</td><td>4667</td><td>0.2</td></tr><tr><td>ILILT-CFNO-6-130K (ours)</td><td>0.35</td><td>4657</td><td>0.3</td></tr><tr><td>ILILT-CFNO-8-130K (ours)</td><td>0.25</td><td>4674</td><td>0.4</td></tr><tr><td>ILILT-P2PHD-2-45M (ours)</td><td>0.13</td><td>5056</td><td>0.2</td></tr><tr><td>ILILT-P2PHD-4-45M (ours)</td><td>0.13</td><td>4680</td><td>0.4</td></tr><tr><td>ILILT-P2PHD-6-45M (ours)</td><td>0.11</td><td>4670</td><td>0.6</td></tr><tr><td>ILILT-P2PHD-8-45M (ours)</td><td>0.08</td><td>4695</td><td>0.8</td></tr></table>
+
+solutions with 0.08 average EPE violations (achieved by  $T = 8$ ), which outperforms the golden numerical solver. ILILT-CFNO though behaves slightly worse than ILILT-P2PHD, it can still exhibit comparable results with "GPU-ILT" with a  $400 \times$  smaller model size. All methods do not show a significant advantage on PVB area, and this is because the optimization flow in these solutions do not explicitly target process variations. Lastly, the ILILT already achieves better mask quality than the other three ML-based solutions without the refinement from the legacy ILT engine. We also show a  $10 \times$  speedup over the numerical ILT solver.
+
+The ILILT framework also ensembles the beneath idea of learning-to-optimize (L2O), which unrolls optimization steps on multiple layers of neural networks whose weights are not shared. As a comparison, we also implement two standard L2O flows with CFNO and Pix2PixHD as backbones respectively. We choose the 8 unrolling steps that have similar computing costs to the best ILILT settings. As shown in Table 2, L2O does not preserve the performance of standalone backbone models with a significant increase in EPE violations. L2O unrolls the ILT problem into a fixed number of optimization steps, which are parameterized by different neural network layers. Such a model, with  $8 \times$  trainable parameters in our case, requires more converging training efforts. Specifically, the error of each optimization step will propagate to the next level which causes performance degradation compared to standalone backbone models.
+
+# 4.4. Generalization of ILILT
+
+ILILT also shows better generalization capability thanks to its weight-tied structure. This can be observed in the superior performance boosting over standalone CNN and Neural Operator-based backbone models. We can observe that with the weight-tied sequence training on CFNO and Pix2PixHD, the mask quality has an improvement of  $50\%$  and  $33\%$  reduced EPE violation. Although the unrolled sequence induces additional runtime, the model inference time can be ignored when compared to the entire physical implementation and manufacturing flow.
+
+# 4.5. Self-learning Capability
+
+Usually, ML-based solutions rely highly on the dataset quality, and hence the performance of traditional ILT solvers. However, ILT is naturally non-convex and suffers from suboptimal solutions. Under such a scenario, ILILT can take advantage of its capability to outperform numerical solvers in certain chip designs, providing faster iteration for dataset updates and serving as a powerful auxiliary candidate for traditional numerical solvers.
+
+# 4.6. ILILT Optimization Flow
+
+The learned mask optimization process is depicted in Figure 5, where we visualize the growth of a mask along the unrolled sequence for a given design. We bring the model trained with unrolling step  $\mathrm{T} = 4$  but keep the loop running until  $\mathrm{T} = 8$ . We show that the mask reaches the best quality at the configured unrolling depth. Interestingly, the ILILT also learns a fixed-point iteration such that if we continuously feed the mask beyond the unrolling depth, the model will no longer make significant changes to the mask except minor adjustments. At  $\mathrm{T} = 8$ , the ILILT engine tries to remove the rule-violating artifact from the middle-bottom region. This property brings more application scenarios of the ILILT framework that include producing initial solutions and performing fine-grained fixes at the post-optimization stage. Another benefit of ILILT is its stable optimization trajectory, because quality fluctuation is common during the optimization runtime of numerical ILT solvers (Gao et al., 2014; Ma et al., 2017; Yu et al., 2021; Sun et al., 2023), causing additional overhead to identify good solutions.
+
+# 4.7. Investigation of Unrolling Depth
+
+Here we discuss the effects of the maximum unrolling depth. The ILILT is trained using different max unrolling depths  $(T)$  with  $T\in \{2,4,6,8\}$ . With PVB area being similar, we observe better results happen in larger unrolling steps in the configuration. When  $T$  is small, ILILT does not benefit too much from the weight-tied structure and exhibits larger
+
+EPE violation counts. Instead, a larger  $T$  will put stronger regularization on the model and enable temporal feature learning to increase the model generality. However, a longer unrolling sequence will linearly increase the computing cost as well as training efforts. In this paper, we experimentally show the performance boosting saturates at  $T = 8$  for a fixed number of training epochs.
+
+# 4.8. Mask Prior Learning
+
+In our experiments, we also observe an interesting behavior of ILILT. For some cases, ILILT can even produce better QoR than numerical solution. An example is shown in Figure 6, where the ILILT produces mask solution with zero EPE violations while numerical solver comes with four. This is because ILILT not only learns the mask optimization trajectory from the golden numerical solutions but also captures the mask solution from the broader dataset, which makes the trained network equip deep mask priors. This can also be observed in the generated mask images that ILILT avoids generating rule-violation isolated artifacts while numerical ILT does (see small white dots at the bottom of Figure 6(g)). The result can also be explained through deep image prior (Ulyanov et al., 2018), that during training neural networks capture the structural knowledge with priority. ILILT strengthens this prior through weigh-tied unrolling and lithography guidance.
+
+# 5. Conclusion
+
+In this paper, we propose a novel solution to the mask optimization problem with inverse lithography technology. We argue that existing AI-based approaches are overly reliant on conventional numerical solvers and lack forward lithography conditions, limiting the applicability of AI-based approaches to mask optimization scenarios. To further exploit the potential of AI and machine learning, we carefully analyze the characteristics of the ILT procedure and reformulate it as an implicit layer learning problem. Instead of applying the end-to-end DEQ method, we solve the ILT layer through ILILT, a weight-tied model, where a forward lithography estimator provides lithography information in the optimization procedure. Experiments demonstrate the effectiveness of ILILT learning an ILT layer with limited intervention of a traditional numerical solver. To the best of our knowledge, this is the first time an AI solution has been proposed to imitate the working scheme of a traditional ILT system. We hope this work can further motivate research into expanding the potential of AI for solving complex design automation problems. Additional investigations on production-level designs are necessary to prototype the framework to study whether QoR meets manufacturing standards.
+
+# Impact Statement
+
+Lithography is the very first topic in chip design community brought into applied deep learning. Because design layouts, masks and wafer patterns are naturally in the format of images which can benefit from many backbones and algorithms, though domain adoptions are necessary. Early efforts only fall into forward predictive tasks like hotspot detection or lithography simulation. There is limited research dealing with chip optimization problems that can directly benefit semiconductor manufacturing. This work, ILILT, we believe is the first effort to solve inverse lithography/mask optimization directly using machine learning without the intervention of traditional numerical solvers. Hopefully, this can motivate new research strategies on how is AI included in chip design and manufacturing flow, and the advancement of chip design can feedback and benefit AI in the long run.
+
+# References
+
+Bai, S., Kolter, J. Z., and Koltun, V. Deep equilibrium models. In Conference on Neural Information Processing Systems (NIPS), 2019.  
+Banerjee, S., Li, Z., and Nassif, S. R. ICCAD-2013 CAD contest in mask optimization and benchmark suite. In IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pp. 271-274, 2013.  
+Braam, K., Selinidis, K., Hoppe, W., Cai, H., and Xiao, G. EUV mask synthesis with rigorous ILT for process window improvement. In Proceedings of SPIE, volume 10962, 2019.  
+Chen, G., Chen, W., Ma, Y., Yang, H., and Yu, B. DAMO: Deep agile mask optimization for full chip scale. In IEEE/ACM International Conference on Computer-Aided Design (ICCAD), 2020.  
+Chen, T., Chen, X., Chen, W., Heaton, H., Liu, J., Wang, Z., and Yin, W. Learning to optimize: A primer and a benchmark. arXiv preprint arXiv:2103.12828, 2021.  
+Gao, J.-R., Xu, X., Yu, B., and Pan, D. Z. MOSAIC: Mask optimizing solution with process window aware inverse correction. In ACM/IEEE Design Automation Conference (DAC), pp. 52:1-52:6, 2014.  
+He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770-778, 2016.  
+Isola, P., Zhu, J.-Y., Zhou, T., and Efros, A. A. Image-to-image translation with conditional adversarial networks. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1125-1134, 2017.
+
+Jiang, B., Liu, L., Ma, Y., Zhang, H., Young, E. F. Y., and Yu, B. Neural-ILT: Migrating ILT to nerval networks for mask printability and complexity co-optimizaton". In IEEE/ACM International Conference on Computer-Aided Design (ICCAD), 2020.  
+Jittorntrum, K. An implicit function theorem. Journal of Optimization Theory and Applications, 25(4):575-577, 1978.  
+Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A. Fourier neural operator for parametric partial differential equations. International Conference on Learning Representations (ICLR), 2021.  
+Ma, Y., Gao, J.-R., Kuang, J., Miao, J., and Yu, B. A unified framework for simultaneous layout decomposition and mask optimization. In IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pp. 81-88, 2017.  
+Pang, L., Russell, E. V., Baggenstoss, B., Lee, M., Digaum, J., Yang, M.-C., Ungar, P. J., Bouaricha, A., Wang, K., Su, B., et al. Study of mask and wafer co-design that utilizes a new extreme SIMD approach to computing in memory manufacturing: full-chip curvilinear ilt in a day. In Proceedings of SPIE, volume 11148, pp. 136-151, 2019.  
+Ronneberger, O., Fischer, P., and Brox, T. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), pp. 234-241, 2015.  
+Sun, S., Yang, F., Yu, B., Shang, L., and Zeng, X. Efficient ilt via multi-level lithography simulation. In 2023 60th ACM/IEEE Design Automation Conference (DAC), pp. 1-6. IEEE, 2023.  
+Ulyanov, D., Vedaldi, A., and Lempitsky, V. Deep image prior. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 9446-9454, 2018.  
+Wang, T.-C., Liu, M.-Y., Zhu, J.-Y., Tao, A., Kautz, J., and Catanzaro, B. High-resolution image synthesis and semantic manipulation with conditional gans. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8798-8807, 2018.  
+Yang, H., Li, S., Ma, Y., Yu, B., and Young, E. F. GAN-OPC: Mask optimization with lithography-guided generative adversarial nets. In ACM/IEEE Design Automation Conference (DAC), pp. 131:1-131:6, 2018.
+
+Yang, H., Li, Z., Sastry, K., Mukhopadhyay, S., Anandkumar, A., Khailany, B., Singh, V., and Ren, H. Large scale mask optimization via convolutional fourier neural operator and litho-guided self training. arXiv preprint arXiv:2207.04056, 2022a.  
+Yang, H., Li, Z., Sastry, K., Mukhopadhyay, S., Kilgard, M., Anandkumar, A., Khailany, B., Singh, V., and Ren, H. Generic lithography modeling with dual-band optics-inspired neural networks. In ACM/IEEE Design Automation Conference (DAC), 2022b.  
+Ye, W., Alawieh, M. B., Lin, Y., and Pan, D. Z. LithoGAN: End-to-end lithography modeling with generative adversarial networks. In ACM/IEEE Design Automation Conference (DAC), pp. 107:1-107:6, 2019.  
+Ye, W., Alawieh, M. B., Watanabe, Y., Nojima, S., Lin, Y., and Pan, D. Z. Tempo: Fast mask topography effect modeling with deep learning. In ACM International Symposium on Physical Design (ISPD), pp. 127-134, 2020.  
+Yu, Z., Chen, G., Ma, Y., and Yu, B. A GPU-enabled level set method for mask optimization. In IEEE/ACM Proceedings Design, Automation and Test in Eurpoe (DATE), 2021.  
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+Zhou, Z., Siddiquee, M. M. R., Tajbakhsh, N., and Liang, J. Unet++: Redesigning skip connections to exploit multiscale features in image segmentation. IEEE transactions on medical imaging, 39(6):1856-1867, 2019.
+
+# A. Lithography Simulation Model
+
+Throughout the manuscript, we use  $f_{l}$  to represent the entire lithography process. In detail, this can be broken into two stages: (1) optical modeling and (2) resist modeling. The optical modeling tries to find the illumination intensity after the mask and is projected on the silicon wafer. This process can be mathematically written as,
+
+$$
+\boldsymbol {I} = \sum_ {k = 1} ^ {N} \alpha_ {k} | | \boldsymbol {h} _ {k} \otimes \boldsymbol {M} | | ^ {2}, \tag {14}
+$$
+
+where  $I$  is the light intensity projected on the silicon wafer,  $M$  denotes the mask image, and  $h_k$ 's and  $\alpha_k$ 's are lithography system-related parameters that are calibrated in foundries.  $h_k$ 's and  $\alpha_k$ 's are different under different process conditions.
+
+During resist modeling,  $I$  will have a chemical reaction with some "resist materials" and etch chip design on the top of the silicon wafer, which is given by
+
+$$
+\boldsymbol {Z} (i, j) = \left\{ \begin{array}{l l} 1, & \text {i f} I (i, j) > I _ {\mathrm {t h}}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {15}
+$$
+
+In this paper,  $I_{\mathrm{th}} = 0.225$ , which is determined by the technology profile.
+
+![](images/861409920e573e250d4a297ab010ef8a65641e01de41002caa96e8085a1b2dc9.jpg)  
+(a) Mask  
+Figure 7. Lithography process. (a) Mask image contains the patterns of optimized design; (b) Intensity image is the UV lights that are projected on the silicon wafer given by Equation (14); (c) Wafer image is the patterns on silicon given Equation (15).
+
+![](images/8f0b89de105cd5ff1af2c34614270df10ee2e7fbb35ad44ad0d52a27a7857cff.jpg)  
+(b) Intensity
+
+![](images/60d45a7eb0241100c79fc2b5a85d247392c986f4f9c040343fea8909b57358bc.jpg)  
+(c) Wafer
+
+# B. Numerical ILT
+
+Because numerical solver usually requires differentiable objectives. In practical, we relax  $M \in \{0,1\}$  and  $Z \in \{0,1\}$  through sigmoid.
+
+$$
+\boldsymbol {M} = \frac {1}{1 + \exp \left(- \beta_ {m} \left(\boldsymbol {M} ^ {\prime} - 0 . 5\right)\right)}, \tag {16}
+$$
+
+where  $M^{\prime}$  is some auxiliary variable that is usually initialized as the original chip design  $Z^{*}$ .
+
+$$
+\boldsymbol {Z} = \frac {1}{1 + \exp \left(- \beta_ {z} (\boldsymbol {I} - I _ {\mathrm {t h}}) \right.}. \tag {17}
+$$
+
+And now we can solve the following ILT problem through gradient descent.
+
+$$
+\min  _ {M, M ^ {\prime}} l = \left| \left| Z - Z ^ {*} \right| \right| \tag {18}
+$$
+
+s.t. Equations (14), (16) and (17).
+
+It should be noted that numerical ILT requires frequent calls of time-consuming lithography simulation.
+
+# C. CFNO
+
+CFNO, the convolutional Fourier Neural Operator (Yang et al., 2022a), was built upon the original FNO (Li et al., 2021) design. The capability of FNO learning frequency domain features fits the lithography well. However, the original FNO design has a few challenges: (1) FNO requires computations of FFT of the entire input mask image which is computationally
+
+![](images/1e248c99f4ef6aba42aa769289ebc907e60171e103cd87b986c411f676648096.jpg)  
+Figure 8. CFNO backbone design.
+
+![](images/e35e201124e6c374d327b8953f862a0dcbe12971b1b1fb518ac834b60ecbebcc.jpg)  
+Figure 9. The data pipeline of  $g$  used throughout experiments.
+
+costly; (2) FNO, once trained, can only be applied on the same-sized image. CFNO addresses these problems by combining the patch embedding in ViT. The core idea is to separate the input into non-overlapped patches, and each patch will go through a shared FNO layer as in Figure 8. Then we perform a token-wise convolution to learn the relations among different patches and finally form the mask embedding. The overall architecture of  $g$  is shown in Figure 9.
+
+# D. Pix2PixHD
+
+Pix2PixHD (Wang et al., 2018) is an improved version of Pix2Pix which employs hierarchical local enhancers for high-resolution image generation. To make the framework fit more on the mask optimization problem we choose  $ngf = 16$ ,  $n\_downsample\_global = 4$ ,  $n\_blocks\_global = 9$  as defined in https://github.com/NVIDIA/pix2pixHD. We also make certain improvements by replacing deconvolution layers with bicubic interpolation and stride-1 convolution layers for smooth shape generation. We also apply  $4 \times 4$  downsampling (achieved by average pooling) at the input directly to reduce computing overhead and perform interpolation to cast the image back to desired resolution. This modification yields 45M model as listed in Section 4.
+
+# E. Additional Visualization
+
+![](images/30fc5dc866367b352e892ccca7840edcc2bb8c800534abfb79f9b2ec5d167e17.jpg)  
+(a) AES-Design
+
+![](images/2c9dacbe2f96d31b0e974351bf3ca00375f59dfc8acf46b9b9465e27d1b0bb7d.jpg)  
+(b) ILT-Mask  
+(c) ILILT(P2P)-Mask  
+(d) ILILT(CFNO)-Mask  
+(e) L2O(P2P)-Mask  
+(f) L2O(CFNO)-Mask
+
+![](images/42146d0977944f2953d64f3c00fec579601c0aa8b137bfa0ba6ce6ec63955cc2.jpg)  
+(g) ILT-Wafer  
+(h) ILILT(P2P)-Wafer  
+(i) ILILT(CFNO)-Wafer  
+(j) L2O(P2P)-Wafer  
+(k) L2O(CFNO)-Wafer  
+Figure 10. Visualization sample of AES chip design for ILILT and L2O with CNN and Neural Operator backbones. L2O with Pix2PixHD backbone even produces meaningless artifacts.

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+        "image_caption": [
+            "Figure 1: IM-3D generates high-quality and faithful 3D assets from text/image pair. It eschews Score Distillation Sampling (SDS) for robust 3D reconstruction of the output of a video diffusion model, tuned to generate a  $360^{\\circ}$  video of the object."
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+    {
+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
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+        "text": "Most text-to-3D generators build upon off-the-shelf text-to-image models trained on billions of images. They use variants of Score Distillation Sampling (SDS), which is slow,",
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+        "text": "somewhat unstable, and prone to artifacts. A mitigation is to fine-tune the 2D generator to be multi-view aware, which can help distillation or can be combined with reconstruction networks to output 3D objects directly. In this paper, we further explore the design space of text-to-3D models. We significantly improve multi-view generation by considering video instead of image generators. Combined with a 3D reconstruction algorithm which, by using Gaussian splatting, can optimize a robust image-based loss, we directly produce high-quality 3D outputs from the generated views. Our new method, IM-3D, reduces the number of evalua",
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+        "type": "page_footnote",
+        "text": "*Equal contribution  ${}^{1}$  Meta  ${}^{2}$  University of Oxford,Oxford,UK. Correspondence to: Filippos Kokkinos <fkokkinos@meta.com>, Luke Melas-Kyriazi <lukemk@robots.ox.ac.uk>.",
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+    {
+        "type": "page_footnote",
+        "text": "Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).",
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+    {
+        "type": "page_number",
+        "text": "1",
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+    },
+    {
+        "type": "text",
+        "text": "tions of the 2D generator network  $10 - 100\\times$ , resulting in a much more efficient pipeline, better quality, fewer geometric inconsistencies, and a high yield of usable 3D assets.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "All state-of-the-art open-world text-to-3D generators are built on top of off-the-shelf 2D generators trained on billions of images. This is necessary because there is not enough 3D training data to directly learn generators that can understand language and operate in an open-ended manner. However, the best way of building such models is still debated.",
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+        "type": "text",
+        "text": "One approach is to perform 3D distillation by adopting Score Distillation Sampling (SDS) (Poole et al., 2023) or one of its variants. These models can work on top of nearly any modern 2D generator, but they require tens of thousands of evaluations of the 2D generator and can take hours to generate a single asset. They are also prone to artifacts and may fail to converge. Mitigating these shortcomings inspired a significant body of research (Wang et al., 2023b).",
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+        "type": "text",
+        "text": "The fundamental reason for these limitations is that the underlying 2D generator is not 3D aware. SDS slowly makes the different views of the 3D object agree with the 2D model, which characterizes them independently of each other. Several authors (Shi et al., 2023b; Wang & Shi, 2023; Shi et al., 2023a) have shown that fine-tuning the 2D generator to understand the correlation between different views of the object significantly facilitates distillation. More recently, approaches such as (Li et al., 2023) avoid distillation entirely and instead just reconstruct the 3D object from the generated views. However, in order to compensate for defects in multi-view generation, they must incorporate very large 3D reconstruction networks. Ultimately, these approaches are many times faster than distillation, but quality is limited.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we explore the benefits of further increasing the quality of multi-view generation and how this might affect the design space of future text-to-3D models. We are inspired by the fact that, in the limit, a 2D generator could output enough consistent views of the object to afford simple multi-view reconstruction, sidestepping distillation and reconstruction networks entirely.",
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+        "text": "To this end, we introduce IM-3D, a text-to-3D generation approach that leverages Iterative Multiview diffusion and reconstruction (Figures 1 and 2). IM-3D is based on significantly boosting the quality of the multi-view generation network by switching from a text-to-image to a text-to-video generator network. Specifically, we pick Emu Video (Girdhar et al., 2023), a video generator conditioned both on a reference image and a textual prompt. Our first contribution is to show that Emu Video can be fine-tuned, using a relatively small number of synthetic 3D assets, to generate directly up to 16 high-resolution consistent views ( $512 \\times 512$ ) of",
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+    {
+        "type": "text",
+        "text": "the object. While Emu Video is in itself an iterative model based on diffusion, by adopting a fast sampling algorithm, the views can be generated in a few seconds and in a small number of iterations.",
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+    {
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+        "text": "Our second contribution is to show that we can extract a high-quality 3D object by directly fitting a 3D model to the resulting views—without distillation or reconstruction networks—quickly and robustly. To do so, we rely on a 3D reconstruction algorithm based on Gaussian splatting (GS) (Kerbl et al., 2023). The importance of GS is that it affords fast differentiable rendering of the 3D object, which allows the use of image-based losses like LPIPS. The latter is key to bridging the small inconsistencies left by the 2D generator without requiring ad-hoc reconstruction models.",
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+    {
+        "type": "text",
+        "text": "Third, we notice that, while this process results in mostly very good 3D models, some inconsistencies may still remain. We thus propose to close the loop and feed the 3D reconstruction back to the 2D generator. In order to do so, we simply render noised images of the 3D object and restart the video diffusion process from those. This approach is closer in spirit to SDS as it builds consensus progressively, but the feedback loop is closed two or three times per generated asset instead of tens of thousands of times.",
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+        "text": "There are many advantages to our approach. Compared to SDS, it reduces dramatically the number of evaluations of the 2D generator network. Using a fast sampler, generating the first version of the multi-view images requires only around 40 evaluations. Iterated generations are much shorter (as they start from a partially denoised result), at most doubling the total number of evaluations. This is a  $10-100 \\times$  reduction compared to SDS. The 3D reconstruction is also very fast, taking only a minute for the first version of the asset and a few seconds for the second or third. It also sidesteps typical SDS issues such as artifacts (e.g., saturated colors, Janus problem), lack of diversity (by avoiding mode seeking), and low yield (failure to converge). Compared to methods like (Li et al., 2023), IM-3D is slower but achieves much higher quality and does not require learning large reconstruction networks, offloading most of the work to 2D generation instead.",
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+        "text": "In a nutshell, our contribution is to show how video generator networks can improve consistent multi-view generators to a point where it is possible to obtain state-of-the-art and efficient text/image-to-3D results without distillation and without training reconstruction networks.",
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+    {
+        "type": "text",
+        "text": "2. Related work",
+        "text_level": 1,
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+        "text": "3D Distillation. 3D distillation is the process of extracting a 3D object from a 2D neural network trained to generate images from text or otherwise match them to text. For example, methods like DreamFields (Jain et al., 2022) do so starting",
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+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "2",
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+    {
+        "type": "text",
+        "text": "from the CLIP image similarity score. However, most recent methods build on diffusion-based image generators that utilize variants of the Score Distillation Sampling (SDS) loss introduced with DreamFusion (Poole et al., 2023). Fantasia 3D (Chen et al., 2023a) disentangles illumination from materials. Magic3D (Lin et al., 2022) reconstructs high-resolution texture meshes. RealFusion (Melas-Kyriazi et al., 2023) starts from a reference image and fine-tunes the prompt of a 2D generator to match it, distilling a 3D object afterwards. Make-it-3D (Tang et al., 2023b) also starts from a 2D image, combining SDS with a CLIP loss with respect to the reference image and a depth prior. HiFi-123 (Yu et al., 2023) uses DDIM inversion to obtain the code for the reference image. ATT3D (Lorraine et al., 2023) develops an amortized version of SDS, where several variants of the same object are distilled in parallel. HiFA (Zhu & Zhuang, 2023) reformulates the SDS loss and anneals the diffusion noise. DreamTime (Huang et al., 2023) also proposes to optimize the noise schedule. ProlificDreamer (Wang et al., 2023b), SteinDreamer (Wang et al., 2023a), Collaborative SDS (Kim et al., 2023) and Noise-free SDS (Katzir et al., 2023) improve the variance of the SDS gradient estimate. DreamGaussian (Tang et al., 2023a), GaussianDreamer (Yi et al., 2023) and (Chen et al., 2023c) apply Gaussian splatting to the SDS loss.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Methods using multi-view generation. Many methods have proposed to use multi-view generation to improve 3D generation. For multi-view generation, the most common approach is Zero-1-to-3 (Liu et al., 2023b), which fine-tunes the Stable Diffusion (SD) model to generate novel views of an object. Zero123++ (Shi et al., 2023a) further improves on this base model in various ways, including generating directly a grid of several multi-view images. CascadeZero123 (Chen et al., 2023b) proposes to apply two such models in sequence: the first to obtain approximate multiple views of the object, and the second to achieve better quality views conditioned on the approximate ones.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Magic123 (Qian et al., 2023) and DreamCraft3D (Sun et al., 2023) combine Zero-1-to-3 and SD. They start from a generated 2D image, extract depth and normals, and apply the RealFusion / DreamBooth technique to fine-tune the 2D diffusion model to generate different views of the object.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "MVDream (Shi et al., 2023b) directly generates four fixed viewpoints of an object from a text prompt. Consistent123 (Weng et al., 2023) uses a different form of cross-view attention and generates several views sufficient for direct reconstruction. ConsistNet (Yang et al., 2023) introduces an explicit 3D pooling mechanism to exchange information between views. ImageDream (Wang & Shi, 2023) extends MVDream to start from a given input image and proposes a new variant of image conditioning compared to that of Zero-1-to-3. RichDreamer (Qiu et al., 2023)",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "further learns to generate normals and separation between material and lighting.",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Viewset Diffusion (Szymanowicz et al., 2023), Forward Diffusion (Tewari et al., 2023), SyncDreamer (Liu et al., 2023c) and DMV3D (Xu et al., 2023) denoise multiple views of the 3D object simultaneously to improve consistency. 3DGen (Gupta et al., 2023) learns a latent space to encode 3D objects using a VAE-like technique followed by a diffusion model. However, this approach is not very scalable as it requires training the model from scratch using 3D data. HexaGen3D (Mercier et al., 2024) extends 3DGen to use features from an SD model instead, thus increasing the scalability of the approach.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Two concurrent works are Stable Video Diffusion (Blattmann et al., 2023) and ViVid-1-to-3 (Kwak et al., 2023). Both use video models as multiview generators, but they only produce novel views, not 3D assets.",
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+    {
+        "type": "text",
+        "text": "Non-SDS methods. Some text-to-3D methods perform \"direct\" 3D reconstruction on top of generated views without using SDS. One-2-3-45 (Liu et al., 2023a) compensates for the shortcomings of the multi-view generator by training a reconstruction network. Instant3D (Li et al., 2023) is similar but based on a much larger reconstruction model (Hong et al., 2023). Wonder3D (Long et al., 2023) further learns to generate multiple views of a given input image together with the corresponding normal maps, which are then used to reconstruct the 3D object. AGG (Xu et al., 2024) builds a single-image reconstruction network on top of Gaussian splatting. CAD (Wan et al., 2023) learns a 3D generator network from image samples using a 2D diffusion model, replacing the SDS loss with adversarial training.",
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+    {
+        "type": "text",
+        "text": "Our approach also eschews the SDS loss but shows that it is possible to offload most of the modelling burden to the 2D generator network, utilizing a straightforward and efficient 3D reconstruction algorithm.",
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+    {
+        "type": "text",
+        "text": "3. Method",
+        "text_level": 1,
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+        "text": "We first describe our video-based multi-view generator network in Section 3.1 and its training data in Section 3.2, followed by a description of the robust 3D reconstruction module in Section 3.3 and of iterative refinement in Section 3.4. An overview of our method is shown in Figure 2.",
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+    {
+        "type": "text",
+        "text": "3.1. Multi-view as video generation",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our multi-view generation model is based on finetuning an existing text-to-video (T2V) generator network Emu Video (Girdhar et al., 2023). First, it utilizes a text-to-image (T2I) model (Emu (Dai et al., 2023)) to generate an initial image  $\\mathbf{I}$  corresponding to the given textual prompt  $\\pmb{p}$ . Second, the image  $\\mathbf{I} \\in \\mathbb{R}^{3 \\times H \\times W}$  and the text prompt  $\\pmb{p}$  are",
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+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
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+        "type": "page_number",
+        "text": "3",
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+    {
+        "type": "image",
+        "img_path": "images/33351a0d8b088693e5d616e1234196d8bf99c14c997a6da41a219d0844ff653d.jpg",
+        "image_caption": [
+            "Figure 2: Overview of IM-3D. Our model starts from an input image (e.g., generated from a T2I model). It feeds the latter into an image-to-video diffusion model to generate a turn-table like video. The latter is plugged into 3D Gaussian Splating to directly reconstruct the 3D object using image-based losses for robustness. Optionally, renders of the objects are generated and fed back to the video diffusion model, repeating the process for refinement."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "fed into a second generator, which produces up to  $K = 16$  frames of video  $\\mathbf{J} \\in \\mathbb{R}^{K \\times 3 \\times H \\times W}$ , utilizing  $\\mathbf{I}$  as guidance for the first frame. Notice that, while the model is trained such that  $\\mathbf{I} \\approx \\mathbf{J}_1$ , this is not an exact equality. Instead, the model draws a sample  $\\mathbf{J}$  from a learned conditional distribution  $p(\\mathbf{J}|\\mathbf{I},\\mathbf{p})$ , which allows it to slightly deviate from the input image to better fit in the generated video. An advantage of Emu Video compared to other video generators is that the video frames  $\\mathbf{J}$  are already high-quality and high-resolution without requiring sophisticated coarse-to-fine sampling schemes. It is archetected as a fine-tuned version of the original T2I Emu network with some modifications to account for the temporal dependencies between frames.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Starting from the pre-trained Emu Video model, we then fine-tune it to generate a particular kind of video, where the camera moves around a given 3D object, effectively generating simultaneously several views of it, in a turntable-like fashion. In order to do so, we consider an internal dataset of synthetic 3D objects, further described in Section 3.2. This dataset provides us with training videos  $\\mathcal{J} = \\{(\\mathbf{J}_n,\\mathbf{I}_n,\\pmb {p}_n)\\}_{n = 1}^N$ , each containing  $K = 16$  views of the object taken at fixed angular interval and a random but fixed elevation, the initial image  $\\mathbf{I}_n = [\\mathbf{J}_n]_1$ , and the textual prompt  $\\pmb{p}_n$ . The camera distance is fixed across all renders.",
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+    {
+        "type": "text",
+        "text": "Differently from many prior multi-view generation networks, we do not pass the camera parameters to the model; instead, we use a fixed camera distance and orientation, randomizing only the elevation. The model simply learns to produce a set of views that follow this distribution.",
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+    {
+        "type": "text",
+        "text": "Like most image and video generators, Emu Video is based on diffusion and implements a denoising neural",
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+    },
+    {
+        "type": "text",
+        "text": "network  $\\hat{\\epsilon} (\\mathbf{J}_t,t,\\mathbf{I},\\pmb {p})$  that takes as input a noised video  $\\mathbf{J}_t = \\sqrt{1 - \\sigma_t^2}\\mathbf{J} + \\sigma_t\\epsilon$ , where  $\\epsilon \\sim N(0,I)$  is Gaussian noise and  $\\sigma_t\\in [0,1]$  is the noise level, and tries to estimate the noise  $\\epsilon$  from it. The training uses the standard diffusion loss  $\\mathcal{L}_{\\mathrm{diff}}(\\hat{\\epsilon} |\\mathbf{J},\\mathbf{I},\\pmb {p},t,\\epsilon) = w_t^{\\mathrm{diff}}\\cdot \\| \\hat{\\epsilon} (\\mathbf{J}_t,t,\\mathbf{I},\\pmb {p}) - \\epsilon \\| ^2$  where  $(\\mathbf{J},\\mathbf{I},\\pmb {p})\\in \\mathcal{I}$  is a training video,  $\\epsilon$  is a Gaussian sample,  $t$  is a time step, also randomly sampled, and  $w_{t}$  is a corresponding weighing factor. To finetune Emu Video, we use  $\\mathcal{L}_{\\mathrm{diff}}$ , but freeze all parameters except for the temporal convolutional and attention layers.",
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+    {
+        "type": "text",
+        "text": "3.2.Data",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The dataset  $\\mathcal{I}$  used to train our model consists of turn-table-like videos of synthetic 3D objects. Several related papers in multi-view generation also use synthetic data, taking Objaverse (Deitke et al., 2022) or Objaverse-XL (Deitke et al., 2023) as a source. Here, we utilize an in-house collection of 3D assets of comparable quality, for which we generate textual descriptions using an image captioning network.",
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+    {
+        "type": "text",
+        "text": "Similar to prior works (Li et al., 2023), we use a subset of  $100\\mathrm{k}$  assets selected for quality, as determined by the CLIP (Radford et al., 2021) alignment between rendered images and textual descriptions. Each video  $\\mathbf{J} \\in \\mathcal{J}$  is obtained by sampling one of the  $100\\mathrm{k}$  assets, choosing a random elevation in  $[0, \\pi/4]$ , and then placing the camera around the object at uniform  $(2\\pi/K$  degree) intervals.",
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+    {
+        "type": "text",
+        "text": "3.3. Fast and robust reconstruction",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To generate a 3D asset from a prompt  $\\pmb{p}$ , we first sample an image  $\\mathbf{I} \\sim p(\\mathbf{I}|\\pmb{p})$  from the Emu image model, followed by sampling a multi-view video  $\\mathbf{J} \\sim p(\\mathbf{J}|\\mathbf{I},\\pmb{p})$  from the fine",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "page_number",
+        "text": "4",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "tuned Emu Video model. Given the video  $\\mathbf{J}$ , we then directly fit a 3D model  $G$ . While there are many possible choices for this model, here we use Gaussian splatting (Kerbl et al., 2023), a radiance field that uses a large number of 3D Gaussians to approximate the 3D opacity and color functions.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given the 3D model  $G$  and a camera viewpoint  $\\Pi$ , the differentiable Gaussian splattering renderer produces an image  $\\hat{\\mathbf{I}} = \\mathcal{R}(G,\\Pi)$ . Compared to other methods such as NeRF (Mildenhall et al., 2020), or even faster versions such as DVGO (Sun et al., 2022) or TensoRF (Chen et al., 2022), the key advantage of Gaussian splattering is the efficiency of the differentiable renderer, both in time and space, which allows rendering a full high-resolution image  $\\mathbf{I}$  at each training iteration instead of just selected pixels as in most prior works. Because of this fact, we can utilize image-level losses such as LPIPS (Zhang et al., 2018), i.e.,  $\\mathcal{L}_{\\mathrm{LPIPS}}(\\hat{\\mathbf{I}},\\mathbf{I}) = \\sum_{q=1}^{Q}\\|w_q\\odot(\\Phi_q(\\hat{\\mathbf{I}})-\\Phi_q(\\mathbf{I}))\\|^2$  where  $\\Phi_q:\\mathbb{R}^{3\\times H\\times W}\\to \\mathbb{R}^C$  is a family of  $Q$  patch-wise feature extractors implemented by the VGG-VD neural network (Simonyan & Zisserman, 2015). We also utilize a second image-based loss  $\\mathcal{L}_{\\mathrm{MS - SSIM}}$ , the multi-scale structural similarity index measure (MS-SSIM) (Wang et al., 2003). Finally, we use a mask loss  $\\mathcal{L}_{\\mathrm{Mask}}$  with masks obtained using the method introduced in (Qin et al., 2022). In our ablation studies, we show the significant benefits of using these image-based losses rather than the standard pixel-wise RGB loss  $\\mathcal{L}_{\\mathrm{RGB}}(\\hat{\\mathbf{I}},\\mathbf{I}) = \\|\\hat{\\mathbf{I}} -\\mathbf{I}\\|^2$ . Our final loss is the weighted loss combination  $\\mathcal{L} = w_{\\mathrm{LPIPS}}\\mathcal{L}_{\\mathrm{LPIPS}} + w_{\\mathrm{SSIM}}\\mathcal{L}_{\\mathrm{SSIM}} + w_{\\mathrm{Mask}}\\mathcal{L}_{\\mathrm{Mask}}$ . The object  $G$  is thus reconstructed via direct optimization, i.e.,  $G^{*} = \\operatorname*{argmin}_{G}\\sum_{k=1}^{K}\\mathcal{L}(\\mathcal{R}(G,\\Pi_k),[\\mathbf{J}]_k)$  where  $[\\mathbf{J}]_k$  denotes the  $k$ -th image in the video.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.4. Fast sampling and iterative generation",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The SDS loss can be seen as a way to bridge the gap between image generators that are unaware of 3D objects and their 3D reconstructions, absorbing multi-view consistency defects in the generation. Because our model is rather view-consistent from the outset, and because we can use robust reconstruction losses, the SDS loss is unnecessary. Instead, given a prompt  $\\pmb{p}$ , we simply generate an image  $\\mathbf{I}$ , followed by video  $\\mathbf{J}$ , and then fit a 3D object  $G$  to the latter.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One main advantage is that this dramatically reduces the number of model evaluations compared to using the SDS loss. Optimizing the SDS loss is (approximately) the same as ascending the score, i.e., the gradient  $\\nabla \\log p(\\mathbf{J}_t|\\mathbf{I},\\pmb {p})$  of the log distribution over noised videos (or images), so optimizing the asset  $G$  can be seen as a form of multi-view mode seeking. The score is obtained from the same network  $\\hat{\\epsilon}$ . However, despite the conditioning on a specific textual prompt  $\\pmb{p}$  and input view  $\\mathbf{I}$ , the sampled distribution is rather wide, requiring a very large number (thousands) of iterations to converge to a mode; furthermore, regressing to a mode",
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+    },
+    {
+        "type": "text",
+        "text": "reduces the diversity and quality of the output.",
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+    },
+    {
+        "type": "text",
+        "text": "In our case, the network  $\\hat{\\epsilon}$  is used to generate directly a single video  $\\mathbf{J}$ , which is then reconstructed without further invocations to the model. Because the video  $\\mathbf{J}$  is already sufficiently view-consistent, the 3D reconstruction converges quickly to a good solution. Furthermore, we can adopt fast stochastic ODE solvers such as DPM++ (Lu et al., 2022) to further reduce the number of model evaluations to obtain the video in the first place. Overall, compared to using the SDS loss, the number of model evaluations is reduced by a factor  $10 - 100\\times$  (see the Appendix for additional analysis).",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Despite the overall consistency of generated videos  $\\mathbf{J}$ , they are still not perfect. We thus additionally compensate for such inconsistencies during model fitting, but still without resorting to the SDS loss. Instead, we alternate 3D reconstruction and video generation. To do so, once the first video  $\\mathbf{J}$  and corresponding 3D model  $G^{*}$  are obtained, we use the latter to generate a video  $\\hat{\\mathbf{J}} = \\mathcal{R}(G^{*},\\Pi)$  using the 3D renderer, sample an intermediated noised video  $\\hat{\\mathbf{J}}_t$  by adding noise to it as shown above and then invoking the video generator again to obtain a denoised and updated video  $\\mathbf{J}'$ .",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We iterate this process two times. This is vastly faster than using the SDS loss while still being highly robust.",
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+    {
+        "type": "text",
+        "text": "4. Experiments",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our method generates 3D objects from a textual description  $\\pmb{p}$  and a reference image  $\\mathbf{I}$ . In order to compare to prior work, we consider, in particular, the set of textual prompts from (Shi et al., 2023b), which are often used for evaluation.",
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+        "type": "text",
+        "text": "Given an input image and prompt  $(\\mathbf{I},\\pmb {p})$ , previous methods either synthesize a multi-view image sequence  $\\mathbf{J}$  (usually by means of a generator network) or output a 3D model, or both. We compare the quality of the produced artifacts visually, utilizing the image sequence  $\\mathbf{J}$  directly, or corresponding renders  $\\hat{\\mathbf{J}}$  of the 3D model. In general, we can expect the quality and faithfulness of  $\\mathbf{J}$  to be better than that of  $\\hat{\\mathbf{J}}$  because the generated image sequence needs not to be perfectly view-consistent. On the other hand, the renders  $\\hat{\\mathbf{J}}$  from the 3D model are consistent by construction, but may be blurrier than  $\\mathbf{J}$ , or contain other defects.",
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+    {
+        "type": "text",
+        "text": "4.1. Comparison to the state-of-the-art",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we compare IM-3D to relevant state-of-the-art methods in the literature, including MVDream (Shi et al., 2023b), Zero123XL (Deitke et al., 2023), Magic123 (Qian et al., 2023), SyncDreamer (Liu et al., 2023c), ImageDream (Wang & Shi, 2023), LRM (Hong et al., 2023) and One2345++ (Liu et al., 2023a). For LRM, since no public models are available, we utilize the open-source OpenLRM (He & Wang, 2023). For One2345++, which is only available via a web interface, we manually upload each",
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+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
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+    {
+        "type": "page_number",
+        "text": "5",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/b01db652a709717797760c6a795a2fa6f849ec38f054f9df3f8f59de052e9192.jpg",
+        "image_caption": [
+            "Figure 3: Qualitative Comparisons. Our method IM-3D (last row) and others for the same text/image prompt pairs. For IM-3D, we show the final GS reconstruction (which guarantees multi-view consistency). We match the input image faithfully and obtain high-quality, detailed reconstructions in just 3 minutes. Faster methods such as OpenLRM are also much worse."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Iteration 1 (Initial Reconstruction - Above) vs Iteration 2 (After Generation - Below)",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Iteration 1 (Initial Reconstruction - Above) vs Iteration 2 (After Generation - Below)",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/d7f6ec80cdc1b31b7a6671aea7cf48e3098a2355b9f6048edb7ee5d9f10be721.jpg",
+        "image_caption": [
+            "Figure 4: A visualization of reconstruction quality over multiple iterations of multiview diffusion and reconstruction. We compare the initial reconstructions obtained by our model (i.e. the result of training on our initial generated videos) to the result after one iteration of reconstruction and refinement. We see that although the initial reconstructions have reasonable shapes, they lack fine-grained details due to small inconsistencies in the generated multiview images. After one iteration of noising, denoising, and reconstruction, our method resolves these inconsistencies and produces 3D assets with significantly higher levels of detail (as highlighted by the red circles above)."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
+            200,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1481d78e7576406d8280d49808560848721499e0f7f3e7e617c30842a9cabc9e.jpg",
+        "table_caption": [
+            "Table 1: Faithfulness to the textual and visual prompts of image sequences synthesised or rendered by various methods. Assessed on the prompt list from (Wang & Shi, 2023; Shi et al., 2023b)."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"2\">synthesized view</td><td colspan=\"2\">re-rendered view</td></tr><tr><td>model</td><td>Time (min)</td><td>CLIP (Text)</td><td>CLIP (Image)</td><td>CLIP (Text)</td><td>CLIP (Image)</td></tr><tr><td>SDXL (Podell et al., 2023) [upper bound]</td><td>0.03</td><td>33.33</td><td>100</td><td>—</td><td>—</td></tr><tr><td>MVDream (Shi et al., 2023b)</td><td>72</td><td>31.26 ±2.9</td><td>76.44 ±6.5</td><td>30.63 ±2.7</td><td>76.94 ±5.2</td></tr><tr><td>Zero123XL (Deitke et al., 2023)</td><td>10</td><td>19.58 ±1.3</td><td>60.29 ±5.8</td><td>29.06 ±3.3</td><td>81.33 ±6.9</td></tr><tr><td>Magic123 (Qian et al., 2023)</td><td>15</td><td>—</td><td>—</td><td>29.51 ±4.7</td><td>84.14 ±10.2</td></tr><tr><td>SyncDreamer (Liu et al., 2023c)</td><td>13</td><td>27.76 ±3.0</td><td>77.26 ±7.2</td><td>26.22 ±3.4</td><td>74.95 ±6.6</td></tr><tr><td>ImageDream (Wang &amp; Shi, 2023)</td><td>120</td><td>31.08 ±3.4</td><td>85.39 ±5.8</td><td>30.73 ±2.3</td><td>83.77 ±5.2</td></tr><tr><td>OpenLRM (Hong et al., 2023)</td><td>0.17</td><td>—</td><td>—</td><td>29.75 ±3.2</td><td>83.08 ±9.5</td></tr><tr><td>One2345++ (Liu et al., 2023a)</td><td>0.75</td><td>—</td><td>—</td><td>29.71 ±2.3</td><td>83.78 ±6.4</td></tr><tr><td>IM-3D (ours)</td><td>3</td><td>31.92 ±1.6</td><td>92.38 ±5.1</td><td>31.66 ±1.7</td><td>91.40 ±5.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/acbfe2f46160663235e54046399a339b51038fe77af14d759f8310c33276938e.jpg",
+        "image_caption": [
+            "Figure 5: Human evaluation. We perform human evaluation of IM-3D vs state-of-the-art in Image-to-3D and Text-to-3D. Human raters preferred IM-3D to all competitors with regard to both generation quality and faithfulness, often by a large margin."
+        ],
+        "image_footnote": [],
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "image in the evaluation set. We carry out both quantitative and qualitative comparisons using the set of prompts and images from (Shi et al., 2023b; Wang & Shi, 2023).",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Quantitative comparison. Table 1 provides a quantitative comparison of our method to others. We adopt the same metrics as (Shi et al., 2023b; Wang & Shi, 2023), which are based on the CLIP (Radford et al., 2021) similarity scores. Specifically, we utilize the ability of CLIP to embed text and images in the same space. We then use the embeddings to compare the textual prompt  $\\pmb{p}$  and the image prompt  $\\mathbf{I}$  to the images  $\\mathbf{J}$  of the object (either synthesized or rendered). A high CLIP similarity score means high faithfulness to the prompt. As an upper bound, we also report the CLIP scores of the prompt images  $\\mathbf{I}$ , which were generated using the SDXL (Podell et al., 2023) model.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The key takeaway from Table 1 is that IM-3D outperforms all others in terms of both textual and visual faithfulness. This is true for both the image sequences  $\\mathbf{J}$  output by the video generator as well as the renders  $\\hat{\\mathbf{J}}$  from the fitted 3D",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0f1f83991624ef144ed18e29be1dfc4119cbdce9492ccfa833dae26b4bd9decf.jpg",
+        "table_caption": [
+            "Table 2: Ablation on the importance of loss terms and 3D representation during the fitting stage."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Loss / Representation</td><td>CLIP (Text)</td><td>CLIP (Image)</td></tr><tr><td>IM-3D (ours)</td><td>31.66 ±1.7</td><td>91.40 ±5.5</td></tr><tr><td>- LLPiPS</td><td>29.38 ±2.1</td><td>84.71 ±6.4</td></tr><tr><td>- RGB instead of LLPiPS</td><td>29.67 ±2.0</td><td>84.99 ±5.9</td></tr><tr><td>- LSSIM</td><td>31.53 ±1.8</td><td>90.64 ±5.7</td></tr><tr><td>- Mask</td><td>31.43 ±1.9</td><td>90.14 ±6.0</td></tr><tr><td>w/ NeRF instead of GS</td><td>30.42 ±2.1</td><td>87.37 ±5.4</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "GS models  $G$ . IM-3D is particularly strong when it comes to visual faithfulness, which also means that the images we generate are of a quality comparable to the input image  $\\mathbf{I}$ . Additionally, our method requires significantly less time than most (3 minutes vs hours for some models).",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Human evaluation. Automated metrics for the evaluation of generative models are not fully representative of the value of the output in applications. Thus, we also conduct a human study. We ask annotators to evaluate our model against a competitor based on (1) Image Alignment and (2) 3D quality. We present to annotators the outputs of two different methods, rendered as  $360^{\\circ}$  videos, and ask them to indicate a preference based on these two criteria. Table 1 shows the win rate when comparing our methods against others. Our method surpasses in performance all other baselines in both studies indicating that the proposed method produces high-quality 3D results that closely align with the image prompt. Further details are provided in the Appendix.",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.2. Ablations",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effect of iterative refinements. In Figure 4, we demonstrate the efficacy of our proposed iterative refinement process. Our model's initial reconstructions (derived from training on our initially generated videos) are compared to the outcome following a single iteration of multiview diffusion",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/b5ec89fee5e3cb4590277c35c810cbd9e582507af5f56be2b909ff5d9392d17c.jpg",
+        "image_caption": [
+            "Figure 6: Reconstruction Quality with Pixel-Level and Image-Level Losses. We find that image-level losses are crucial to the success of our method. With pixel-level losses such as the L2 loss, small inconsistencies in the generated images are effectively averaged together, resulting in unnatural and blurry-looking reconstructions."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            98,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "and reconstruction. While the initial reconstructions exhibit satisfactory shapes, they miss out on intricate details due to minor inconsistencies in the initial multiview images. In a few instances, some parts of these initial reconstructions look as if two copies of a shape have been superimposed upon one another as the reconstruction process tries to satisfy two inconsistent views. However, our technique rectifies these discrepancies with one iteration of denoising and reconstruction, significantly enhancing the level of detail.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Image-Level Losses. In Table 2 and Figure 6, we compare the results of optimization with pixel-level and image-level loss functions. We find that image-level losses are central to our method's ability to generate high-quality 3D assets. The use of pixel-level losses such as L2 loss is detrimental, as minor inconsistencies in the multiview images are emphasized by the optimization process and effectively averaged together. This averaging results in a low CLIP score (29.67 vs 31.66 for LPIPS) as well as blurry and unnatural generations.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparing 3D Representations The last line of Table 2 provides a comparison of 3D representations, showing the effect of using NeRF as an underlying 3D representation rather than Gaussian splatting (GS). We find that the visual quality of models generated using NeRF is slightly worse than GS. The true benefit of GS is that it is much faster and much more memory-efficient; training with GS takes 3 minutes, whereas training with NeRF takes 40 minutes. Additionally, the memory-efficient nature of Gaussian splatting makes it easy to render at our diffusion model's native resolution of  $512\\mathrm{px}$ , whereas for NeRF one has to use ray microbatching or optimize at a lower resolution.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Ablation on Using Fewer Frames. We show quantitative performance when performing our reconstruction and generation using fewer frames.",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td># Frames</td><td>CLIP (Text)</td><td>CLIP (Image)</td></tr><tr><td>16</td><td>31.66 ±1.7</td><td>91.40 ±5.5</td></tr><tr><td>8</td><td>31.38 ±1.8</td><td>90.06 ±6.3</td></tr><tr><td>4</td><td>30.06 ±2.6</td><td>86.96 ±8.6</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Using Fewer Frames Differently from the vast majority of other diffusion-based text-to-3D and image-to-3D approaches, which generate only 1-4 frames, IM-3D generates 16 frames simultaneously. We demonstrate the significance of this in Table 3, finding that our quantitative performance improves as we increase the number of generated frames.",
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+    {
+        "type": "text",
+        "text": "4.3. Limitations",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The fine-tuned video generator is generally very view-consistent, but it still has limitations. One interesting failure case is that for highly dynamic subjects (e.g., horses, which are often captured running), the model sometimes renders spurious animations (e.g., walking or galloping) despite our fine-tuning, which is problematic for 3D reconstruction. This occurs more often when the prompt contains verbs describing motion; see the Appendix for an example.",
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+    {
+        "type": "text",
+        "text": "5. Conclusions",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work, we have shown that starting from a video generator network instead of an image generator can result in better multi-view generation to a point where it can impact the design of future text-to-3D models. In fact, we have shown that the quality is sufficient to eschew distillation losses like SDS as well as large reconstruction networks. Instead, one can simply fit the 3D object to the generated",
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+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+    },
+    {
+        "type": "text",
+        "text": "views using a robust image-based loss. Reconstruction can be further alternated with refining the target video, quickly converging to a better 3D object with minimal impact on efficiency. Compared to works that rely on SDS, our approach significantly reduces the number of evaluations of the 2D generator network, resulting in a faster and more memory-efficient pipeline without compromising on quality.",
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+    },
+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our work uses Generative AI, whose potential impacts are and have been extensively discussed in the academic, business and public spheres. Our work does not change these issues qualitatively. The Emu models (Dai et al., 2023) were explicitly designed with fairness and safety in mind, and fine-tuning them on curated 3D models is likely to further reduce the potential for harm.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
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+        ],
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
+            200,
+            56,
+            767,
+            70
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "page_number",
+        "text": "11",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A. Appendix",
+        "text_level": 1,
+        "bbox": [
+            84,
+            83,
+            197,
+            99
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1. Training details",
+        "text_level": 1,
+        "bbox": [
+            84,
+            109,
+            233,
+            125
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In line with (Girdhar et al., 2023), we maintain the spatial convolutional and attention layers of Emu Video, fine-tuning only the temporal layers. We minimize the standard diffusion loss over a span of 5 days, employing 80 A100 GPUs with a total batch size of 240 and a learning rate of  $Ie - 5$ . Our findings indicate that prolonged training effectively counters the network's inclination to generate 360 videos of deforming objects, given that the initialization is a video generation model. Contrary to MVDream (Shi et al., 2023b) and Instant3D (Li et al., 2023), we observe no degradation in texture quality with extended training. This can be ascribed to the fact that the spatial layers remain static and the network is image-conditioned, necessitating that the generated 360 video retain the high-frequency texture elements of the input.",
+        "bbox": [
+            84,
+            132,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For Gaussian fitting, we initialize 5000 points at the center of the 3D space, and densify and prune the Gaussians every 50 iterations. We conduct optimization for 1200 iterations and execute Emu Video twice for 10 iterations each using the DPM solver (Lu et al., 2022) during this process, repeating this every 500 iterations. Empirically, we found that setting the weights to  $w_{\\mathrm{LPIPS}} = 10$ ,  $w_{\\mathrm{SSIM}} = 0.2$  and  $w_{\\mathrm{Mask}} = 1$  yields the best results during the fitting stage.",
+        "bbox": [
+            84,
+            246,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Prior to fitting, we need to estimate the elevation of the very first generated video. To that end, we trained an elevation estimator on top of DINO (Caron et al., 2021) features using the 100k 3D renderings. The network averages the features of 4 uniformly distributed frames and uses a 2-layer MLP to regress the elevation in radians.",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2. Human evaluation",
+        "text_level": 1,
+        "bbox": [
+            84,
+            376,
+            251,
+            390
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In our study, we employed the prompt set delineated in (Shi et al., 2023b) to conduct a human annotation evaluation via Amazon Mechanical Turk (AMT). The task assigned to the annotators involved choosing between two 3D assets, both rendered as  $360^{\\circ}$  videos, with one of the assets being the output of our proposed method. To ensure a robust and unbiased evaluation, we randomized the presentation order of the methods for each question. Each question was assessed by five annotators, and we reported the consensus opinion. Since each question corresponds to a triplet of (competing method, 3D reconstruction, and quality/faithfulness), this is a total of 5 annotators  $\\times$  (5 methods  $\\times$  39 reconstructions  $\\times$  2 question types) = 1950 annotations. We instructed the annotators to overlook any disparities in background colors, as normalizing all methods to yield the same scale is a non-trivial task.",
+        "bbox": [
+            84,
+            398,
+            885,
+            521
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3. Further information on network efficiency",
+        "text_level": 1,
+        "bbox": [
+            84,
+            537,
+            418,
+            551
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/94bdbcc01e6ffe027df1d4df9893db23369547d6aa3847b83859450aa09ddd02.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>ProlificDreamer</td><td>MVDream</td><td>ImageDream</td><td>Zero123XL</td><td>SyncDreamer</td><td>IM-3D</td></tr><tr><td>Number of network calls</td><td>320000</td><td>20000</td><td>25000</td><td>1200</td><td>200</td><td>80</td></tr></table>",
+        "bbox": [
+            135,
+            580,
+            836,
+            645
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Number of diffusion network calls to generate one 3D asset. The proposed method, IM-3D, requires only a fraction of the model evaluations to compute a 3D asset.",
+        "bbox": [
+            84,
+            655,
+            885,
+            686
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Table 4, we present the number of diffusion model forward passes used to reconstruct a single 3D object for various 3D generation methods. Whereas some other methods require thousands or tens of thousands of iterations, our method requires less than one hundred.",
+        "bbox": [
+            84,
+            719,
+            885,
+            765
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.4. Further comparisons with recent methods",
+        "text_level": 1,
+        "bbox": [
+            84,
+            781,
+            416,
+            796
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Appendix A.4, we present additional comparisons with recent and concurrently-developed methods (Long et al., 2023; Tochilkin et al., 2024; Tang et al., 2024; Kwak et al., 2023).",
+        "bbox": [
+            84,
+            805,
+            885,
+            835
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.5. Results on human portraits",
+        "text_level": 1,
+        "bbox": [
+            84,
+            852,
+            313,
+            867
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Our training data does not contain humans; however, in Figure 7, we show the results of our model on human portrait reconstruction. We find that our model performs well despite the inputs being outside of the training distribution.",
+        "bbox": [
+            84,
+            875,
+            885,
+            906
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
+            200,
+            56,
+            767,
+            70
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
+            477,
+            922,
+            495,
+            934
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/77914f4825798d1274ed1c236e3417d252c5164d3e3ac81ba95037a616cae587.jpg",
+        "image_caption": [
+            "Figure 7: Qualitative examples on generated human portraits. The input images for the figure are generated by a 2D image generation model."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            89,
+            169,
+            869,
+            378
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b761fef9543e69524016fe325d78c78d3e1cfad53a3499ecd26ae16946663f7f.jpg",
+        "image_caption": [
+            "Failure Case",
+            "Figure 8: Visualisation of a failure case. In this case, our finetuned video network generated an animated video of a horse walking rather than a static video. As a result, the 3D reconstruction process produces erroneous geometry (e.g. the head of the horse is barely visible from some views)."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            261,
+            625,
+            707,
+            757
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
+            200,
+            56,
+            769,
+            70
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            477,
+            922,
+            495,
+            934
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/e030a90b7a6b557227647d09723a6831992c8bd4d6a21cf71812871c80b6349f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CLIP-Text</td><td>CLIP-Image</td></tr><tr><td>Wonder3D (Long et al., 2023)</td><td>29.0</td><td>79.8</td></tr><tr><td>TripoSR (Tochilkin et al., 2024)</td><td>29.0</td><td>81.5</td></tr><tr><td>LGM (Tang et al., 2024)</td><td>30.1</td><td>87.9</td></tr><tr><td>Vivid-1-to-3 (Kwak et al., 2023)</td><td>29.5</td><td>87.1</td></tr><tr><td>Ours</td><td>31.3</td><td>90.5</td></tr></table>",
+        "bbox": [
+            271,
+            80,
+            697,
+            189
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Further comparison results. We outperform recent and concurrent methods.",
+        "bbox": [
+            199,
+            200,
+            769,
+            215
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b783d34fc7e3c3d22fafc36bcc90fab862dced72a7eb4543bcd686c2816bcb0.jpg",
+        "image_caption": [
+            "Figure 9: Visualisation of Meshes. We convert our Gaussian Splitting representation to DMTet (Shen et al., 2021) using marching cubes and optimize the resulting meshes using our iterative multiview diffusion and reconstruction process."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            102,
+            229,
+            864,
+            467
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.6. Failure Cases",
+        "text_level": 1,
+        "bbox": [
+            84,
+            535,
+            217,
+            547
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As described in the limitations section of the main paper, the video generator does not produce perfect results in all cases. A notable instance of failure is observed with subjects that exhibit high dynamism, such as horses often depicted in motion. In such cases, the model occasionally produces unwarranted animations like walking or galloping, which disrupts the 3D reconstruction process. We show an example in Figure 8.",
+        "bbox": [
+            84,
+            559,
+            888,
+            619
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.7. Conversion to Meshes",
+        "text_level": 1,
+        "bbox": [
+            86,
+            635,
+            276,
+            648
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Although Gaussian Splatting is being rapidly adopted by the computer vision and graphics communities, some production applications require that objects be converted to meshes. We show that in these cases, it is straightforward to extract high-quality meshes from our Gaussian Splatting representation: one can run marching cubes (Cline, 1988) and then optionally continue to optimize the resulting mesh using DMTet (Shen et al., 2021). We show visual results of converting our models to meshes in Figure 9.",
+        "bbox": [
+            84,
+            659,
+            887,
+            736
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation",
+        "bbox": [
+            200,
+            56,
+            769,
+            70
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
+            477,
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+        ],
+        "page_idx": 13
+    }
+]

2024/IM-3D_ Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation/e28e34fa-0ac3-4859-ad51-209cd75a78bd_origin.pdf

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2024/IM-3D_ Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation/full.md

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+# IM-3D: Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation
+
+Luke Melas-Kyriazi  $^{*12}$  Iro Laina  $^{2}$  Christian Rupprecht  $^{2}$  Natalia Neverova  $^{1}$  Andrea Vedaldi  $^{1}$  Oran Gafni  $^{1}$  Filippos Kokkinos  $^{*1}$
+
+![](images/7a3991692b5c0b03a2a5854ddf9100667c6c105dbfbca6a25afd3271d126554b.jpg)  
+Figure 1: IM-3D generates high-quality and faithful 3D assets from text/image pair. It eschews Score Distillation Sampling (SDS) for robust 3D reconstruction of the output of a video diffusion model, tuned to generate a  $360^{\circ}$  video of the object.
+
+# Abstract
+
+Most text-to-3D generators build upon off-the-shelf text-to-image models trained on billions of images. They use variants of Score Distillation Sampling (SDS), which is slow,
+
+somewhat unstable, and prone to artifacts. A mitigation is to fine-tune the 2D generator to be multi-view aware, which can help distillation or can be combined with reconstruction networks to output 3D objects directly. In this paper, we further explore the design space of text-to-3D models. We significantly improve multi-view generation by considering video instead of image generators. Combined with a 3D reconstruction algorithm which, by using Gaussian splatting, can optimize a robust image-based loss, we directly produce high-quality 3D outputs from the generated views. Our new method, IM-3D, reduces the number of evalua
+
+tions of the 2D generator network  $10 - 100\times$ , resulting in a much more efficient pipeline, better quality, fewer geometric inconsistencies, and a high yield of usable 3D assets.
+
+# 1. Introduction
+
+All state-of-the-art open-world text-to-3D generators are built on top of off-the-shelf 2D generators trained on billions of images. This is necessary because there is not enough 3D training data to directly learn generators that can understand language and operate in an open-ended manner. However, the best way of building such models is still debated.
+
+One approach is to perform 3D distillation by adopting Score Distillation Sampling (SDS) (Poole et al., 2023) or one of its variants. These models can work on top of nearly any modern 2D generator, but they require tens of thousands of evaluations of the 2D generator and can take hours to generate a single asset. They are also prone to artifacts and may fail to converge. Mitigating these shortcomings inspired a significant body of research (Wang et al., 2023b).
+
+The fundamental reason for these limitations is that the underlying 2D generator is not 3D aware. SDS slowly makes the different views of the 3D object agree with the 2D model, which characterizes them independently of each other. Several authors (Shi et al., 2023b; Wang & Shi, 2023; Shi et al., 2023a) have shown that fine-tuning the 2D generator to understand the correlation between different views of the object significantly facilitates distillation. More recently, approaches such as (Li et al., 2023) avoid distillation entirely and instead just reconstruct the 3D object from the generated views. However, in order to compensate for defects in multi-view generation, they must incorporate very large 3D reconstruction networks. Ultimately, these approaches are many times faster than distillation, but quality is limited.
+
+In this paper, we explore the benefits of further increasing the quality of multi-view generation and how this might affect the design space of future text-to-3D models. We are inspired by the fact that, in the limit, a 2D generator could output enough consistent views of the object to afford simple multi-view reconstruction, sidestepping distillation and reconstruction networks entirely.
+
+To this end, we introduce IM-3D, a text-to-3D generation approach that leverages Iterative Multiview diffusion and reconstruction (Figures 1 and 2). IM-3D is based on significantly boosting the quality of the multi-view generation network by switching from a text-to-image to a text-to-video generator network. Specifically, we pick Emu Video (Girdhar et al., 2023), a video generator conditioned both on a reference image and a textual prompt. Our first contribution is to show that Emu Video can be fine-tuned, using a relatively small number of synthetic 3D assets, to generate directly up to 16 high-resolution consistent views ( $512 \times 512$ ) of
+
+the object. While Emu Video is in itself an iterative model based on diffusion, by adopting a fast sampling algorithm, the views can be generated in a few seconds and in a small number of iterations.
+
+Our second contribution is to show that we can extract a high-quality 3D object by directly fitting a 3D model to the resulting views—without distillation or reconstruction networks—quickly and robustly. To do so, we rely on a 3D reconstruction algorithm based on Gaussian splatting (GS) (Kerbl et al., 2023). The importance of GS is that it affords fast differentiable rendering of the 3D object, which allows the use of image-based losses like LPIPS. The latter is key to bridging the small inconsistencies left by the 2D generator without requiring ad-hoc reconstruction models.
+
+Third, we notice that, while this process results in mostly very good 3D models, some inconsistencies may still remain. We thus propose to close the loop and feed the 3D reconstruction back to the 2D generator. In order to do so, we simply render noised images of the 3D object and restart the video diffusion process from those. This approach is closer in spirit to SDS as it builds consensus progressively, but the feedback loop is closed two or three times per generated asset instead of tens of thousands of times.
+
+There are many advantages to our approach. Compared to SDS, it reduces dramatically the number of evaluations of the 2D generator network. Using a fast sampler, generating the first version of the multi-view images requires only around 40 evaluations. Iterated generations are much shorter (as they start from a partially denoised result), at most doubling the total number of evaluations. This is a  $10-100 \times$  reduction compared to SDS. The 3D reconstruction is also very fast, taking only a minute for the first version of the asset and a few seconds for the second or third. It also sidesteps typical SDS issues such as artifacts (e.g., saturated colors, Janus problem), lack of diversity (by avoiding mode seeking), and low yield (failure to converge). Compared to methods like (Li et al., 2023), IM-3D is slower but achieves much higher quality and does not require learning large reconstruction networks, offloading most of the work to 2D generation instead.
+
+In a nutshell, our contribution is to show how video generator networks can improve consistent multi-view generators to a point where it is possible to obtain state-of-the-art and efficient text/image-to-3D results without distillation and without training reconstruction networks.
+
+# 2. Related work
+
+3D Distillation. 3D distillation is the process of extracting a 3D object from a 2D neural network trained to generate images from text or otherwise match them to text. For example, methods like DreamFields (Jain et al., 2022) do so starting
+
+from the CLIP image similarity score. However, most recent methods build on diffusion-based image generators that utilize variants of the Score Distillation Sampling (SDS) loss introduced with DreamFusion (Poole et al., 2023). Fantasia 3D (Chen et al., 2023a) disentangles illumination from materials. Magic3D (Lin et al., 2022) reconstructs high-resolution texture meshes. RealFusion (Melas-Kyriazi et al., 2023) starts from a reference image and fine-tunes the prompt of a 2D generator to match it, distilling a 3D object afterwards. Make-it-3D (Tang et al., 2023b) also starts from a 2D image, combining SDS with a CLIP loss with respect to the reference image and a depth prior. HiFi-123 (Yu et al., 2023) uses DDIM inversion to obtain the code for the reference image. ATT3D (Lorraine et al., 2023) develops an amortized version of SDS, where several variants of the same object are distilled in parallel. HiFA (Zhu & Zhuang, 2023) reformulates the SDS loss and anneals the diffusion noise. DreamTime (Huang et al., 2023) also proposes to optimize the noise schedule. ProlificDreamer (Wang et al., 2023b), SteinDreamer (Wang et al., 2023a), Collaborative SDS (Kim et al., 2023) and Noise-free SDS (Katzir et al., 2023) improve the variance of the SDS gradient estimate. DreamGaussian (Tang et al., 2023a), GaussianDreamer (Yi et al., 2023) and (Chen et al., 2023c) apply Gaussian splatting to the SDS loss.
+
+Methods using multi-view generation. Many methods have proposed to use multi-view generation to improve 3D generation. For multi-view generation, the most common approach is Zero-1-to-3 (Liu et al., 2023b), which fine-tunes the Stable Diffusion (SD) model to generate novel views of an object. Zero123++ (Shi et al., 2023a) further improves on this base model in various ways, including generating directly a grid of several multi-view images. CascadeZero123 (Chen et al., 2023b) proposes to apply two such models in sequence: the first to obtain approximate multiple views of the object, and the second to achieve better quality views conditioned on the approximate ones.
+
+Magic123 (Qian et al., 2023) and DreamCraft3D (Sun et al., 2023) combine Zero-1-to-3 and SD. They start from a generated 2D image, extract depth and normals, and apply the RealFusion / DreamBooth technique to fine-tune the 2D diffusion model to generate different views of the object.
+
+MVDream (Shi et al., 2023b) directly generates four fixed viewpoints of an object from a text prompt. Consistent123 (Weng et al., 2023) uses a different form of cross-view attention and generates several views sufficient for direct reconstruction. ConsistNet (Yang et al., 2023) introduces an explicit 3D pooling mechanism to exchange information between views. ImageDream (Wang & Shi, 2023) extends MVDream to start from a given input image and proposes a new variant of image conditioning compared to that of Zero-1-to-3. RichDreamer (Qiu et al., 2023)
+
+further learns to generate normals and separation between material and lighting.
+
+Viewset Diffusion (Szymanowicz et al., 2023), Forward Diffusion (Tewari et al., 2023), SyncDreamer (Liu et al., 2023c) and DMV3D (Xu et al., 2023) denoise multiple views of the 3D object simultaneously to improve consistency. 3DGen (Gupta et al., 2023) learns a latent space to encode 3D objects using a VAE-like technique followed by a diffusion model. However, this approach is not very scalable as it requires training the model from scratch using 3D data. HexaGen3D (Mercier et al., 2024) extends 3DGen to use features from an SD model instead, thus increasing the scalability of the approach.
+
+Two concurrent works are Stable Video Diffusion (Blattmann et al., 2023) and ViVid-1-to-3 (Kwak et al., 2023). Both use video models as multiview generators, but they only produce novel views, not 3D assets.
+
+Non-SDS methods. Some text-to-3D methods perform "direct" 3D reconstruction on top of generated views without using SDS. One-2-3-45 (Liu et al., 2023a) compensates for the shortcomings of the multi-view generator by training a reconstruction network. Instant3D (Li et al., 2023) is similar but based on a much larger reconstruction model (Hong et al., 2023). Wonder3D (Long et al., 2023) further learns to generate multiple views of a given input image together with the corresponding normal maps, which are then used to reconstruct the 3D object. AGG (Xu et al., 2024) builds a single-image reconstruction network on top of Gaussian splatting. CAD (Wan et al., 2023) learns a 3D generator network from image samples using a 2D diffusion model, replacing the SDS loss with adversarial training.
+
+Our approach also eschews the SDS loss but shows that it is possible to offload most of the modelling burden to the 2D generator network, utilizing a straightforward and efficient 3D reconstruction algorithm.
+
+# 3. Method
+
+We first describe our video-based multi-view generator network in Section 3.1 and its training data in Section 3.2, followed by a description of the robust 3D reconstruction module in Section 3.3 and of iterative refinement in Section 3.4. An overview of our method is shown in Figure 2.
+
+# 3.1. Multi-view as video generation
+
+Our multi-view generation model is based on finetuning an existing text-to-video (T2V) generator network Emu Video (Girdhar et al., 2023). First, it utilizes a text-to-image (T2I) model (Emu (Dai et al., 2023)) to generate an initial image  $\mathbf{I}$  corresponding to the given textual prompt  $\pmb{p}$ . Second, the image  $\mathbf{I} \in \mathbb{R}^{3 \times H \times W}$  and the text prompt  $\pmb{p}$  are
+
+![](images/33351a0d8b088693e5d616e1234196d8bf99c14c997a6da41a219d0844ff653d.jpg)  
+Figure 2: Overview of IM-3D. Our model starts from an input image (e.g., generated from a T2I model). It feeds the latter into an image-to-video diffusion model to generate a turn-table like video. The latter is plugged into 3D Gaussian Splating to directly reconstruct the 3D object using image-based losses for robustness. Optionally, renders of the objects are generated and fed back to the video diffusion model, repeating the process for refinement.
+
+fed into a second generator, which produces up to  $K = 16$  frames of video  $\mathbf{J} \in \mathbb{R}^{K \times 3 \times H \times W}$ , utilizing  $\mathbf{I}$  as guidance for the first frame. Notice that, while the model is trained such that  $\mathbf{I} \approx \mathbf{J}_1$ , this is not an exact equality. Instead, the model draws a sample  $\mathbf{J}$  from a learned conditional distribution  $p(\mathbf{J}|\mathbf{I},\mathbf{p})$ , which allows it to slightly deviate from the input image to better fit in the generated video. An advantage of Emu Video compared to other video generators is that the video frames  $\mathbf{J}$  are already high-quality and high-resolution without requiring sophisticated coarse-to-fine sampling schemes. It is archetected as a fine-tuned version of the original T2I Emu network with some modifications to account for the temporal dependencies between frames.
+
+Starting from the pre-trained Emu Video model, we then fine-tune it to generate a particular kind of video, where the camera moves around a given 3D object, effectively generating simultaneously several views of it, in a turntable-like fashion. In order to do so, we consider an internal dataset of synthetic 3D objects, further described in Section 3.2. This dataset provides us with training videos  $\mathcal{J} = \{(\mathbf{J}_n,\mathbf{I}_n,\pmb {p}_n)\}_{n = 1}^N$ , each containing  $K = 16$  views of the object taken at fixed angular interval and a random but fixed elevation, the initial image  $\mathbf{I}_n = [\mathbf{J}_n]_1$ , and the textual prompt  $\pmb{p}_n$ . The camera distance is fixed across all renders.
+
+Differently from many prior multi-view generation networks, we do not pass the camera parameters to the model; instead, we use a fixed camera distance and orientation, randomizing only the elevation. The model simply learns to produce a set of views that follow this distribution.
+
+Like most image and video generators, Emu Video is based on diffusion and implements a denoising neural
+
+network  $\hat{\epsilon} (\mathbf{J}_t,t,\mathbf{I},\pmb {p})$  that takes as input a noised video  $\mathbf{J}_t = \sqrt{1 - \sigma_t^2}\mathbf{J} + \sigma_t\epsilon$ , where  $\epsilon \sim N(0,I)$  is Gaussian noise and  $\sigma_t\in [0,1]$  is the noise level, and tries to estimate the noise  $\epsilon$  from it. The training uses the standard diffusion loss  $\mathcal{L}_{\mathrm{diff}}(\hat{\epsilon} |\mathbf{J},\mathbf{I},\pmb {p},t,\epsilon) = w_t^{\mathrm{diff}}\cdot \| \hat{\epsilon} (\mathbf{J}_t,t,\mathbf{I},\pmb {p}) - \epsilon \| ^2$  where  $(\mathbf{J},\mathbf{I},\pmb {p})\in \mathcal{I}$  is a training video,  $\epsilon$  is a Gaussian sample,  $t$  is a time step, also randomly sampled, and  $w_{t}$  is a corresponding weighing factor. To finetune Emu Video, we use  $\mathcal{L}_{\mathrm{diff}}$ , but freeze all parameters except for the temporal convolutional and attention layers.
+
+# 3.2.Data
+
+The dataset  $\mathcal{I}$  used to train our model consists of turn-table-like videos of synthetic 3D objects. Several related papers in multi-view generation also use synthetic data, taking Objaverse (Deitke et al., 2022) or Objaverse-XL (Deitke et al., 2023) as a source. Here, we utilize an in-house collection of 3D assets of comparable quality, for which we generate textual descriptions using an image captioning network.
+
+Similar to prior works (Li et al., 2023), we use a subset of  $100\mathrm{k}$  assets selected for quality, as determined by the CLIP (Radford et al., 2021) alignment between rendered images and textual descriptions. Each video  $\mathbf{J} \in \mathcal{J}$  is obtained by sampling one of the  $100\mathrm{k}$  assets, choosing a random elevation in  $[0, \pi/4]$ , and then placing the camera around the object at uniform  $(2\pi/K$  degree) intervals.
+
+# 3.3. Fast and robust reconstruction
+
+To generate a 3D asset from a prompt  $\pmb{p}$ , we first sample an image  $\mathbf{I} \sim p(\mathbf{I}|\pmb{p})$  from the Emu image model, followed by sampling a multi-view video  $\mathbf{J} \sim p(\mathbf{J}|\mathbf{I},\pmb{p})$  from the fine
+
+tuned Emu Video model. Given the video  $\mathbf{J}$ , we then directly fit a 3D model  $G$ . While there are many possible choices for this model, here we use Gaussian splatting (Kerbl et al., 2023), a radiance field that uses a large number of 3D Gaussians to approximate the 3D opacity and color functions.
+
+Given the 3D model  $G$  and a camera viewpoint  $\Pi$ , the differentiable Gaussian splattering renderer produces an image  $\hat{\mathbf{I}} = \mathcal{R}(G,\Pi)$ . Compared to other methods such as NeRF (Mildenhall et al., 2020), or even faster versions such as DVGO (Sun et al., 2022) or TensoRF (Chen et al., 2022), the key advantage of Gaussian splattering is the efficiency of the differentiable renderer, both in time and space, which allows rendering a full high-resolution image  $\mathbf{I}$  at each training iteration instead of just selected pixels as in most prior works. Because of this fact, we can utilize image-level losses such as LPIPS (Zhang et al., 2018), i.e.,  $\mathcal{L}_{\mathrm{LPIPS}}(\hat{\mathbf{I}},\mathbf{I}) = \sum_{q=1}^{Q}\|w_q\odot(\Phi_q(\hat{\mathbf{I}})-\Phi_q(\mathbf{I}))\|^2$  where  $\Phi_q:\mathbb{R}^{3\times H\times W}\to \mathbb{R}^C$  is a family of  $Q$  patch-wise feature extractors implemented by the VGG-VD neural network (Simonyan & Zisserman, 2015). We also utilize a second image-based loss  $\mathcal{L}_{\mathrm{MS - SSIM}}$ , the multi-scale structural similarity index measure (MS-SSIM) (Wang et al., 2003). Finally, we use a mask loss  $\mathcal{L}_{\mathrm{Mask}}$  with masks obtained using the method introduced in (Qin et al., 2022). In our ablation studies, we show the significant benefits of using these image-based losses rather than the standard pixel-wise RGB loss  $\mathcal{L}_{\mathrm{RGB}}(\hat{\mathbf{I}},\mathbf{I}) = \|\hat{\mathbf{I}} -\mathbf{I}\|^2$ . Our final loss is the weighted loss combination  $\mathcal{L} = w_{\mathrm{LPIPS}}\mathcal{L}_{\mathrm{LPIPS}} + w_{\mathrm{SSIM}}\mathcal{L}_{\mathrm{SSIM}} + w_{\mathrm{Mask}}\mathcal{L}_{\mathrm{Mask}}$ . The object  $G$  is thus reconstructed via direct optimization, i.e.,  $G^{*} = \operatorname*{argmin}_{G}\sum_{k=1}^{K}\mathcal{L}(\mathcal{R}(G,\Pi_k),[\mathbf{J}]_k)$  where  $[\mathbf{J}]_k$  denotes the  $k$ -th image in the video.
+
+# 3.4. Fast sampling and iterative generation
+
+The SDS loss can be seen as a way to bridge the gap between image generators that are unaware of 3D objects and their 3D reconstructions, absorbing multi-view consistency defects in the generation. Because our model is rather view-consistent from the outset, and because we can use robust reconstruction losses, the SDS loss is unnecessary. Instead, given a prompt  $\pmb{p}$ , we simply generate an image  $\mathbf{I}$ , followed by video  $\mathbf{J}$ , and then fit a 3D object  $G$  to the latter.
+
+One main advantage is that this dramatically reduces the number of model evaluations compared to using the SDS loss. Optimizing the SDS loss is (approximately) the same as ascending the score, i.e., the gradient  $\nabla \log p(\mathbf{J}_t|\mathbf{I},\pmb {p})$  of the log distribution over noised videos (or images), so optimizing the asset  $G$  can be seen as a form of multi-view mode seeking. The score is obtained from the same network  $\hat{\epsilon}$ . However, despite the conditioning on a specific textual prompt  $\pmb{p}$  and input view  $\mathbf{I}$ , the sampled distribution is rather wide, requiring a very large number (thousands) of iterations to converge to a mode; furthermore, regressing to a mode
+
+reduces the diversity and quality of the output.
+
+In our case, the network  $\hat{\epsilon}$  is used to generate directly a single video  $\mathbf{J}$ , which is then reconstructed without further invocations to the model. Because the video  $\mathbf{J}$  is already sufficiently view-consistent, the 3D reconstruction converges quickly to a good solution. Furthermore, we can adopt fast stochastic ODE solvers such as DPM++ (Lu et al., 2022) to further reduce the number of model evaluations to obtain the video in the first place. Overall, compared to using the SDS loss, the number of model evaluations is reduced by a factor  $10 - 100\times$  (see the Appendix for additional analysis).
+
+Despite the overall consistency of generated videos  $\mathbf{J}$ , they are still not perfect. We thus additionally compensate for such inconsistencies during model fitting, but still without resorting to the SDS loss. Instead, we alternate 3D reconstruction and video generation. To do so, once the first video  $\mathbf{J}$  and corresponding 3D model  $G^{*}$  are obtained, we use the latter to generate a video  $\hat{\mathbf{J}} = \mathcal{R}(G^{*},\Pi)$  using the 3D renderer, sample an intermediated noised video  $\hat{\mathbf{J}}_t$  by adding noise to it as shown above and then invoking the video generator again to obtain a denoised and updated video  $\mathbf{J}'$ .
+
+We iterate this process two times. This is vastly faster than using the SDS loss while still being highly robust.
+
+# 4. Experiments
+
+Our method generates 3D objects from a textual description  $\pmb{p}$  and a reference image  $\mathbf{I}$ . In order to compare to prior work, we consider, in particular, the set of textual prompts from (Shi et al., 2023b), which are often used for evaluation.
+
+Given an input image and prompt  $(\mathbf{I},\pmb {p})$ , previous methods either synthesize a multi-view image sequence  $\mathbf{J}$  (usually by means of a generator network) or output a 3D model, or both. We compare the quality of the produced artifacts visually, utilizing the image sequence  $\mathbf{J}$  directly, or corresponding renders  $\hat{\mathbf{J}}$  of the 3D model. In general, we can expect the quality and faithfulness of  $\mathbf{J}$  to be better than that of  $\hat{\mathbf{J}}$  because the generated image sequence needs not to be perfectly view-consistent. On the other hand, the renders  $\hat{\mathbf{J}}$  from the 3D model are consistent by construction, but may be blurrier than  $\mathbf{J}$ , or contain other defects.
+
+# 4.1. Comparison to the state-of-the-art
+
+In this section, we compare IM-3D to relevant state-of-the-art methods in the literature, including MVDream (Shi et al., 2023b), Zero123XL (Deitke et al., 2023), Magic123 (Qian et al., 2023), SyncDreamer (Liu et al., 2023c), ImageDream (Wang & Shi, 2023), LRM (Hong et al., 2023) and One2345++ (Liu et al., 2023a). For LRM, since no public models are available, we utilize the open-source OpenLRM (He & Wang, 2023). For One2345++, which is only available via a web interface, we manually upload each
+
+![](images/b01db652a709717797760c6a795a2fa6f849ec38f054f9df3f8f59de052e9192.jpg)  
+Figure 3: Qualitative Comparisons. Our method IM-3D (last row) and others for the same text/image prompt pairs. For IM-3D, we show the final GS reconstruction (which guarantees multi-view consistency). We match the input image faithfully and obtain high-quality, detailed reconstructions in just 3 minutes. Faster methods such as OpenLRM are also much worse.
+
+Iteration 1 (Initial Reconstruction - Above) vs Iteration 2 (After Generation - Below)
+
+Iteration 1 (Initial Reconstruction - Above) vs Iteration 2 (After Generation - Below)
+
+![](images/d7f6ec80cdc1b31b7a6671aea7cf48e3098a2355b9f6048edb7ee5d9f10be721.jpg)  
+Figure 4: A visualization of reconstruction quality over multiple iterations of multiview diffusion and reconstruction. We compare the initial reconstructions obtained by our model (i.e. the result of training on our initial generated videos) to the result after one iteration of reconstruction and refinement. We see that although the initial reconstructions have reasonable shapes, they lack fine-grained details due to small inconsistencies in the generated multiview images. After one iteration of noising, denoising, and reconstruction, our method resolves these inconsistencies and produces 3D assets with significantly higher levels of detail (as highlighted by the red circles above).
+
+Table 1: Faithfulness to the textual and visual prompts of image sequences synthesised or rendered by various methods. Assessed on the prompt list from (Wang & Shi, 2023; Shi et al., 2023b).  
+
+<table><tr><td colspan="2"></td><td colspan="2">synthesized view</td><td colspan="2">re-rendered view</td></tr><tr><td>model</td><td>Time (min)</td><td>CLIP (Text)</td><td>CLIP (Image)</td><td>CLIP (Text)</td><td>CLIP (Image)</td></tr><tr><td>SDXL (Podell et al., 2023) [upper bound]</td><td>0.03</td><td>33.33</td><td>100</td><td>—</td><td>—</td></tr><tr><td>MVDream (Shi et al., 2023b)</td><td>72</td><td>31.26 ±2.9</td><td>76.44 ±6.5</td><td>30.63 ±2.7</td><td>76.94 ±5.2</td></tr><tr><td>Zero123XL (Deitke et al., 2023)</td><td>10</td><td>19.58 ±1.3</td><td>60.29 ±5.8</td><td>29.06 ±3.3</td><td>81.33 ±6.9</td></tr><tr><td>Magic123 (Qian et al., 2023)</td><td>15</td><td>—</td><td>—</td><td>29.51 ±4.7</td><td>84.14 ±10.2</td></tr><tr><td>SyncDreamer (Liu et al., 2023c)</td><td>13</td><td>27.76 ±3.0</td><td>77.26 ±7.2</td><td>26.22 ±3.4</td><td>74.95 ±6.6</td></tr><tr><td>ImageDream (Wang &amp; Shi, 2023)</td><td>120</td><td>31.08 ±3.4</td><td>85.39 ±5.8</td><td>30.73 ±2.3</td><td>83.77 ±5.2</td></tr><tr><td>OpenLRM (Hong et al., 2023)</td><td>0.17</td><td>—</td><td>—</td><td>29.75 ±3.2</td><td>83.08 ±9.5</td></tr><tr><td>One2345++ (Liu et al., 2023a)</td><td>0.75</td><td>—</td><td>—</td><td>29.71 ±2.3</td><td>83.78 ±6.4</td></tr><tr><td>IM-3D (ours)</td><td>3</td><td>31.92 ±1.6</td><td>92.38 ±5.1</td><td>31.66 ±1.7</td><td>91.40 ±5.5</td></tr></table>
+
+![](images/acbfe2f46160663235e54046399a339b51038fe77af14d759f8310c33276938e.jpg)  
+Figure 5: Human evaluation. We perform human evaluation of IM-3D vs state-of-the-art in Image-to-3D and Text-to-3D. Human raters preferred IM-3D to all competitors with regard to both generation quality and faithfulness, often by a large margin.
+
+image in the evaluation set. We carry out both quantitative and qualitative comparisons using the set of prompts and images from (Shi et al., 2023b; Wang & Shi, 2023).
+
+Quantitative comparison. Table 1 provides a quantitative comparison of our method to others. We adopt the same metrics as (Shi et al., 2023b; Wang & Shi, 2023), which are based on the CLIP (Radford et al., 2021) similarity scores. Specifically, we utilize the ability of CLIP to embed text and images in the same space. We then use the embeddings to compare the textual prompt  $\pmb{p}$  and the image prompt  $\mathbf{I}$  to the images  $\mathbf{J}$  of the object (either synthesized or rendered). A high CLIP similarity score means high faithfulness to the prompt. As an upper bound, we also report the CLIP scores of the prompt images  $\mathbf{I}$ , which were generated using the SDXL (Podell et al., 2023) model.
+
+The key takeaway from Table 1 is that IM-3D outperforms all others in terms of both textual and visual faithfulness. This is true for both the image sequences  $\mathbf{J}$  output by the video generator as well as the renders  $\hat{\mathbf{J}}$  from the fitted 3D
+
+Table 2: Ablation on the importance of loss terms and 3D representation during the fitting stage.  
+
+<table><tr><td>Loss / Representation</td><td>CLIP (Text)</td><td>CLIP (Image)</td></tr><tr><td>IM-3D (ours)</td><td>31.66 ±1.7</td><td>91.40 ±5.5</td></tr><tr><td>- LLPiPS</td><td>29.38 ±2.1</td><td>84.71 ±6.4</td></tr><tr><td>- RGB instead of LLPiPS</td><td>29.67 ±2.0</td><td>84.99 ±5.9</td></tr><tr><td>- LSSIM</td><td>31.53 ±1.8</td><td>90.64 ±5.7</td></tr><tr><td>- Mask</td><td>31.43 ±1.9</td><td>90.14 ±6.0</td></tr><tr><td>w/ NeRF instead of GS</td><td>30.42 ±2.1</td><td>87.37 ±5.4</td></tr></table>
+
+GS models  $G$ . IM-3D is particularly strong when it comes to visual faithfulness, which also means that the images we generate are of a quality comparable to the input image  $\mathbf{I}$ . Additionally, our method requires significantly less time than most (3 minutes vs hours for some models).
+
+Human evaluation. Automated metrics for the evaluation of generative models are not fully representative of the value of the output in applications. Thus, we also conduct a human study. We ask annotators to evaluate our model against a competitor based on (1) Image Alignment and (2) 3D quality. We present to annotators the outputs of two different methods, rendered as  $360^{\circ}$  videos, and ask them to indicate a preference based on these two criteria. Table 1 shows the win rate when comparing our methods against others. Our method surpasses in performance all other baselines in both studies indicating that the proposed method produces high-quality 3D results that closely align with the image prompt. Further details are provided in the Appendix.
+
+# 4.2. Ablations
+
+Effect of iterative refinements. In Figure 4, we demonstrate the efficacy of our proposed iterative refinement process. Our model's initial reconstructions (derived from training on our initially generated videos) are compared to the outcome following a single iteration of multiview diffusion
+
+![](images/b5ec89fee5e3cb4590277c35c810cbd9e582507af5f56be2b909ff5d9392d17c.jpg)  
+Figure 6: Reconstruction Quality with Pixel-Level and Image-Level Losses. We find that image-level losses are crucial to the success of our method. With pixel-level losses such as the L2 loss, small inconsistencies in the generated images are effectively averaged together, resulting in unnatural and blurry-looking reconstructions.
+
+and reconstruction. While the initial reconstructions exhibit satisfactory shapes, they miss out on intricate details due to minor inconsistencies in the initial multiview images. In a few instances, some parts of these initial reconstructions look as if two copies of a shape have been superimposed upon one another as the reconstruction process tries to satisfy two inconsistent views. However, our technique rectifies these discrepancies with one iteration of denoising and reconstruction, significantly enhancing the level of detail.
+
+Image-Level Losses. In Table 2 and Figure 6, we compare the results of optimization with pixel-level and image-level loss functions. We find that image-level losses are central to our method's ability to generate high-quality 3D assets. The use of pixel-level losses such as L2 loss is detrimental, as minor inconsistencies in the multiview images are emphasized by the optimization process and effectively averaged together. This averaging results in a low CLIP score (29.67 vs 31.66 for LPIPS) as well as blurry and unnatural generations.
+
+Comparing 3D Representations The last line of Table 2 provides a comparison of 3D representations, showing the effect of using NeRF as an underlying 3D representation rather than Gaussian splatting (GS). We find that the visual quality of models generated using NeRF is slightly worse than GS. The true benefit of GS is that it is much faster and much more memory-efficient; training with GS takes 3 minutes, whereas training with NeRF takes 40 minutes. Additionally, the memory-efficient nature of Gaussian splatting makes it easy to render at our diffusion model's native resolution of  $512\mathrm{px}$ , whereas for NeRF one has to use ray microbatching or optimize at a lower resolution.
+
+Table 3: Ablation on Using Fewer Frames. We show quantitative performance when performing our reconstruction and generation using fewer frames.
+
+<table><tr><td># Frames</td><td>CLIP (Text)</td><td>CLIP (Image)</td></tr><tr><td>16</td><td>31.66 ±1.7</td><td>91.40 ±5.5</td></tr><tr><td>8</td><td>31.38 ±1.8</td><td>90.06 ±6.3</td></tr><tr><td>4</td><td>30.06 ±2.6</td><td>86.96 ±8.6</td></tr></table>
+
+Using Fewer Frames Differently from the vast majority of other diffusion-based text-to-3D and image-to-3D approaches, which generate only 1-4 frames, IM-3D generates 16 frames simultaneously. We demonstrate the significance of this in Table 3, finding that our quantitative performance improves as we increase the number of generated frames.
+
+# 4.3. Limitations
+
+The fine-tuned video generator is generally very view-consistent, but it still has limitations. One interesting failure case is that for highly dynamic subjects (e.g., horses, which are often captured running), the model sometimes renders spurious animations (e.g., walking or galloping) despite our fine-tuning, which is problematic for 3D reconstruction. This occurs more often when the prompt contains verbs describing motion; see the Appendix for an example.
+
+# 5. Conclusions
+
+In this work, we have shown that starting from a video generator network instead of an image generator can result in better multi-view generation to a point where it can impact the design of future text-to-3D models. In fact, we have shown that the quality is sufficient to eschew distillation losses like SDS as well as large reconstruction networks. Instead, one can simply fit the 3D object to the generated
+
+views using a robust image-based loss. Reconstruction can be further alternated with refining the target video, quickly converging to a better 3D object with minimal impact on efficiency. Compared to works that rely on SDS, our approach significantly reduces the number of evaluations of the 2D generator network, resulting in a faster and more memory-efficient pipeline without compromising on quality.
+
+# Impact Statement
+
+Our work uses Generative AI, whose potential impacts are and have been extensively discussed in the academic, business and public spheres. Our work does not change these issues qualitatively. The Emu models (Dai et al., 2023) were explicitly designed with fairness and safety in mind, and fine-tuning them on curated 3D models is likely to further reduce the potential for harm.
+
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+Simonyan, K. and Zisserman, A. Very deep convolutional networks for large-scale image recognition. In Proc. ICLR, 2015.  
+Sun, C., Sun, M., and Chen, H. Direct voxel grid optimization: Super-fast convergence for radiance fields reconstruction. In Proc. CVPR, 2022.  
+Sun, J., Zhang, B., Shao, R., Wang, L., Liu, W., Xie, Z., and Liu, Y. DreamCraft3D: Hierarchical 3D generation with bootstrapped diffusion prior. arXiv.cs, abs/2310.16818, 2023.
+
+Szymanowicz, S., Rupprecht, C., and Vedaldi, A. Viewset diffusion: (0-)image-conditioned 3D generative models from 2D data. In Proceedings of the International Conference on Computer Vision (ICCV), 2023. URL https://szymanowicz.github.io/viewset-diffusion.  
+Tang, J., Ren, J., Zhou, H., Liu, Z., and Zeng, G. Dream-Gaussian: Generative gaussian splatting for efficient 3D content creation. arXiv, (2309.16653), 2023a.  
+Tang, J., Wang, T., Zhang, B., Zhang, T., Yi, R., Ma, L., and Chen, D. Make-it-3d: High-fidelity 3d creation from A single image with diffusion prior. arXiv.cs, abs/2303.14184, 2023b.  
+Tang, J., Chen, Z., Chen, X., Wang, T., Zeng, G., and Liu, Z. Lgm: Large multi-view gaussian model for high-resolution 3d content creation. arXiv preprint arXiv:2402.05054, 2024.  
+Tewari, A., Yin, T., Cazenavette, G., Rezchikov, S., Tenenbaum, J. B., Durand, F., Freeman, W. T., and Sitzmann, V. Diffusion with forward models: Solving stochastic inverse problems without direct supervision. arXiv.cs, abs/2306.11719, 2023.  
+Tochilkin, D., Pankratz, D., Liu, Z., Huang, Z., Letts, A., Li, Y., Liang, D., Laforte, C., Jampani, V., and Cao, Y.-P. Triposr: Fast 3d object reconstruction from a single image. arXiv preprint arXiv:2403.02151, 2024.  
+Wan, Z., Paschalidou, D., Huang, I., Liu, H., Shen, B., Xiang, X., Liao, J., and Guibas, L. Cad: Photorealistic 3d generation via adversarial distillation. arXiv, (2312.06663), 2023.  
+Wang, P. and Shi, Y. ImageDream: Image-prompt multi-view diffusion for 3D generation. arXiv.cs, abs/2312.02201, 2023.  
+Wang, P., Fan, Z., Xu, D., Wang, D., Mohan, S., Iandola, F., Ranjan, R., Li, Y., Liu, Q., Wang, Z., and Chandra, V. SteinDreamer: Variance reduction for text-to-3d score distillation via stein identity. arXiv, 2023a.  
+Wang, Z., Simoncelli, E., and Bovik, A. Multiscale structural similarity for image quality assessment. In The Thirty-Seventh Asilomar Conference on Signals, Systems & Computers, 2003, volume 2, 2003. doi: 10.1109/ACSSC.2003.1292216.  
+Wang, Z., Lu, C., Wang, Y., Bao, F., Li, C., Su, H., and Zhu, J. ProlificDreamer: High-fidelity and diverse text-to-3D generation with variational score distillation. arXiv.cs, abs/2305.16213, 2023b.
+
+Weng, H., Yang, T., Wang, J., Li, Y., Zhang, T., Chen, C. L. P., and Zhang, L. Consistent123: Improve consistency for one image to 3d object synthesis. arXiv, 2023.  
+Xu, D., Yuan, Y., Mardani, M., Liu, S., Song, J., Wang, Z., and Vahda, A. AGG: Amortized generative 3d gaussians for single image to 3d. arXiv.cs, 2024.  
+Xu, Y., Tan, H., Luan, F., Bi, S., Wang, P., Li, J., Shi, Z., Sunkavalli, K., Wetzstein, G., Xu, Z., and Zhang, K. Dmv3d: Denoising multi-view diffusion using 3d large reconstruction model, 2023.  
+Yang, J., Cheng, Z., Duan, Y., Ji, P., and Li, H. Consistnet: Enforcing 3d consistency for multi-view images diffusion. arXiv.cs, abs/2310.10343, 2023.  
+Yi, T., Fang, J., Wu, G., Xie, L., Zhang, X., Liu, W., Tian, Q., and Wang, X. Gaussian dreamer: Fast generation from text to 3d gaussian splattering with point cloud priors. arXiv.cs, abs/2310.08529, 2023.  
+Yu, W., Yuan, L., Cao, Y., Gao, X., Li, X., Quan, L., Shan, Y., and Tian, Y. HiFi-123: Towards high-fidelity one image to 3d content generation. arXiv.cs, abs/2310.06744, 2023.  
+Zhang, R., Isola, P., Efros, A. A., Shechtman, E., and Wang, O. The unreasonable effectiveness of deep features as a perceptual metric. In Proc. CVPR, pp. 586-595, 2018.  
+Zhu, J. and Zhuang, P. HiFA: High-fidelity text-to-3D with advanced diffusion guidance. CoRR, abs/2305.18766, 2023.
+
+# A. Appendix
+
+# A.1. Training details
+
+In line with (Girdhar et al., 2023), we maintain the spatial convolutional and attention layers of Emu Video, fine-tuning only the temporal layers. We minimize the standard diffusion loss over a span of 5 days, employing 80 A100 GPUs with a total batch size of 240 and a learning rate of  $Ie - 5$ . Our findings indicate that prolonged training effectively counters the network's inclination to generate 360 videos of deforming objects, given that the initialization is a video generation model. Contrary to MVDream (Shi et al., 2023b) and Instant3D (Li et al., 2023), we observe no degradation in texture quality with extended training. This can be ascribed to the fact that the spatial layers remain static and the network is image-conditioned, necessitating that the generated 360 video retain the high-frequency texture elements of the input.
+
+For Gaussian fitting, we initialize 5000 points at the center of the 3D space, and densify and prune the Gaussians every 50 iterations. We conduct optimization for 1200 iterations and execute Emu Video twice for 10 iterations each using the DPM solver (Lu et al., 2022) during this process, repeating this every 500 iterations. Empirically, we found that setting the weights to  $w_{\mathrm{LPIPS}} = 10$ ,  $w_{\mathrm{SSIM}} = 0.2$  and  $w_{\mathrm{Mask}} = 1$  yields the best results during the fitting stage.
+
+Prior to fitting, we need to estimate the elevation of the very first generated video. To that end, we trained an elevation estimator on top of DINO (Caron et al., 2021) features using the 100k 3D renderings. The network averages the features of 4 uniformly distributed frames and uses a 2-layer MLP to regress the elevation in radians.
+
+# A.2. Human evaluation
+
+In our study, we employed the prompt set delineated in (Shi et al., 2023b) to conduct a human annotation evaluation via Amazon Mechanical Turk (AMT). The task assigned to the annotators involved choosing between two 3D assets, both rendered as  $360^{\circ}$  videos, with one of the assets being the output of our proposed method. To ensure a robust and unbiased evaluation, we randomized the presentation order of the methods for each question. Each question was assessed by five annotators, and we reported the consensus opinion. Since each question corresponds to a triplet of (competing method, 3D reconstruction, and quality/faithfulness), this is a total of 5 annotators  $\times$  (5 methods  $\times$  39 reconstructions  $\times$  2 question types) = 1950 annotations. We instructed the annotators to overlook any disparities in background colors, as normalizing all methods to yield the same scale is a non-trivial task.
+
+# A.3. Further information on network efficiency
+
+<table><tr><td></td><td>ProlificDreamer</td><td>MVDream</td><td>ImageDream</td><td>Zero123XL</td><td>SyncDreamer</td><td>IM-3D</td></tr><tr><td>Number of network calls</td><td>320000</td><td>20000</td><td>25000</td><td>1200</td><td>200</td><td>80</td></tr></table>
+
+Table 4: Number of diffusion network calls to generate one 3D asset. The proposed method, IM-3D, requires only a fraction of the model evaluations to compute a 3D asset.
+
+In Table 4, we present the number of diffusion model forward passes used to reconstruct a single 3D object for various 3D generation methods. Whereas some other methods require thousands or tens of thousands of iterations, our method requires less than one hundred.
+
+# A.4. Further comparisons with recent methods
+
+In Appendix A.4, we present additional comparisons with recent and concurrently-developed methods (Long et al., 2023; Tochilkin et al., 2024; Tang et al., 2024; Kwak et al., 2023).
+
+# A.5. Results on human portraits
+
+Our training data does not contain humans; however, in Figure 7, we show the results of our model on human portrait reconstruction. We find that our model performs well despite the inputs being outside of the training distribution.
+
+![](images/77914f4825798d1274ed1c236e3417d252c5164d3e3ac81ba95037a616cae587.jpg)  
+Figure 7: Qualitative examples on generated human portraits. The input images for the figure are generated by a 2D image generation model.
+
+![](images/b761fef9543e69524016fe325d78c78d3e1cfad53a3499ecd26ae16946663f7f.jpg)  
+Failure Case  
+Figure 8: Visualisation of a failure case. In this case, our finetuned video network generated an animated video of a horse walking rather than a static video. As a result, the 3D reconstruction process produces erroneous geometry (e.g. the head of the horse is barely visible from some views).
+
+<table><tr><td></td><td>CLIP-Text</td><td>CLIP-Image</td></tr><tr><td>Wonder3D (Long et al., 2023)</td><td>29.0</td><td>79.8</td></tr><tr><td>TripoSR (Tochilkin et al., 2024)</td><td>29.0</td><td>81.5</td></tr><tr><td>LGM (Tang et al., 2024)</td><td>30.1</td><td>87.9</td></tr><tr><td>Vivid-1-to-3 (Kwak et al., 2023)</td><td>29.5</td><td>87.1</td></tr><tr><td>Ours</td><td>31.3</td><td>90.5</td></tr></table>
+
+Table 5: Further comparison results. We outperform recent and concurrent methods.
+
+![](images/6b783d34fc7e3c3d22fafc36bcc90fab862dced72a7eb4543bcd686c2816bcb0.jpg)  
+Figure 9: Visualisation of Meshes. We convert our Gaussian Splitting representation to DMTet (Shen et al., 2021) using marching cubes and optimize the resulting meshes using our iterative multiview diffusion and reconstruction process.
+
+# A.6. Failure Cases
+
+As described in the limitations section of the main paper, the video generator does not produce perfect results in all cases. A notable instance of failure is observed with subjects that exhibit high dynamism, such as horses often depicted in motion. In such cases, the model occasionally produces unwarranted animations like walking or galloping, which disrupts the 3D reconstruction process. We show an example in Figure 8.
+
+# A.7. Conversion to Meshes
+
+Although Gaussian Splatting is being rapidly adopted by the computer vision and graphics communities, some production applications require that objects be converted to meshes. We show that in these cases, it is straightforward to extract high-quality meshes from our Gaussian Splatting representation: one can run marching cubes (Cline, 1988) and then optionally continue to optimize the resulting mesh using DMTet (Shen et al., 2021). We show visual results of converting our models to meshes in Figure 9.

2024/IM-3D_ Iterative Multiview Diffusion and Reconstruction for High-Quality 3D Generation/images.zip

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2024/IM-Unpack_ Training and Inference with Arbitrarily Low Precision Integers/full.md

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+# IM-Unpack: Training and Inference with Arbitrarily Low Precision Integers
+
+Zhanpeng Zeng $^{1}$  Karthikeyan Sankaralingam $^{1,2}$  Vikas Singh $^{1}$
+
+# Abstract
+
+GEneral Matrix Multiply (GEMM) is a central operation in deep learning and corresponds to a large chunk of the compute footprint. Therefore, improving its efficiency is an active topic of research. A popular strategy is the use of low bit-width integers to approximate the original matrix entries. This allows efficiency gains, but often requires sophisticated techniques to control the rounding error. In this work, we first verify that when the low bit-width restriction is removed, for a variety of Transformer-based models, integers are, in fact, sufficient for all GEMMs need – for both training and inference stages, and achieve parity (with floating point). No sophisticated techniques are needed. We find that while a large majority of entries in matrices (encountered in such models) can be easily represented by low bit-width integers, the existence of a few heavy hitter entries make it difficult to achieve efficiency gains via the exclusive use of low bit-width GEMMs alone. To address this issue, we develop a simple algorithm, Integer Matrix Unpacking (IM-Unpack), to unpack a matrix with large integer entries into a larger matrix whose entries all lie within the representable range of arbitrarily low bit-width integers. This allows equivalence with the original GEMM, i.e., the exact result can be obtained using purely low bit-width integer GEMMs. This comes at the cost of additional operations – we show that for many popular models, this overhead is quite small. Code is available at https://github.com/vsingh-group/im-unpack.
+
+# 1. Introduction
+
+Calculating the product of two matrices using GEneral Matrix Multiply (GEMM) is one of the most widely used opera
+
+<sup>1</sup>University of Wisconsin-Madison <sup>2</sup>NVIDIA Research. Correspondence to: Zhanpeng Zeng <zeng38@wisc.edu>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+tions in modern machine learning. Given matrices  $\mathbf{A}$  and  $\mathbf{B}$  of size  $n\times d$  and  $h\times d$  respectively, the output of a GEMM is calculated as
+
+$$
+\mathbf {C} = \mathbf {A B} ^ {\top}
+$$
+
+Choosing the appropriate numerical precision or data type (FP32, FP16, or BF16) for GEMM is often important, and hinges on several factors including the specific application, characteristics of the data, model architecture, as well as numerical behavior such as convergence. This choice affects compute and memory efficiency most directly, since a disproportionately large chunk of the compute footprint of a model involves the GEMM operator. A good example is the large improvement in latency and memory achieved via low bit-width GEMM, and made possible due to extensive ongoing work on quantization (to low bit-width data types) and low-precision training (Banner et al., 2019; Nagel et al., 2019; Kim et al., 2021; Dettmers et al., 2022; Li & Gu, 2023; Xiao et al., 2023; Dettmers et al., 2022; Liu et al., 2023b;a; Lin et al., 2022; Li & Gu, 2023; Yuan et al., 2022; Ding et al., 2022; Li et al., 2023; Wang et al., 2018; Wu et al., 2018; Zhu et al., 2020; Wortsman et al., 2023). Integer quantization is being actively pursued for inference efficiency, and the use of low bit-width integers is universal to deliver the efficiency gains. However, this strategy often incurs large rounding errors when representing all matrix entries as low bit-width integers, and explains the drop in performance and thereby, a need for error correction techniques (Frantar et al., 2023; Xiao et al., 2023; Chee et al., 2023; Adepu et al., 2024). So how much of the performance degradation is due to (a) rounding to integers versus (b) restricting to low bit-width integers? To answer this question, it appears worthwhile to check whether integer GEMMs will achieve parity without sophisticated techniques (for the inference stage, and more aspirationally, for training) for popular models if we do not restrict to low bit-width integers.
+
+Overview. The starting point of our work is to first experimentally verify that the aforementioned hypothesis – that integer GEMM may work – is true (see §2). But by itself, this finding offers no value proposition for efficiency. Nonetheless, this experiment is useful for the following reason. For a particular class of models (e.g., Transformers), we can easily contrast the corresponding input matrices A and B between (a) integer GEMM and (b) low bit-width integer GEMM and probe if any meaningful structure can
+
+![](images/f7765128bd882f475396edaf3ea012d1579a6c3c6bd7e335fd1f2a7c8a9464fb.jpg)  
+Figure 1. Overall Illustration. We verify the Efficacy of Integers (Contribution 1) in §2, but note that the integer matrices contain heavy hitters (§3). Then, we describe our proposed algorithm, IM-Unpack (Contribution 2), to resolve these heavy hitters in §4.
+
+be exploited. While there is a clear difference in the outputs of (a) integer GEMM versus (b) low bit-width integer GEMM, we find that a large majority of entries of A and B can be represented using low bit-width integers – and the difference in the outputs can be attributed to a few heavy hitter entries in A and B, that cannot be represented using low bit-width integers. Other works have also run into this issue of “outliers” and use high precision (Dettmers et al., 2022) or a separate quantization for these entries (Yuan et al., 2022; Xiao et al., 2023). Based on the observation that we can represent a large integer by a series of smaller integers, our algorithm, Integer Matrix Unpack (IM-Unpack), enables unpacking any integer into a series of low-bit integers. The benefit is that the calculation can be carried out entirely using low bit-width integer arithmetic and thus unifies calculations needed for heavy hitters and the other entries (already amenable to low-bit integer arithmetic). Specifically, IM-Unpack unpacks an integer matrix such that all values of the unpacked matrices always stay within the representable range of low bit-width integers (bit-width can be chosen arbitrarily, as low as two). We obtain the exact result of the original integer GEMM using purely low bit-width integer GEMMs. Since the bit-width of integer arithmetic is independent of the actual range of the original matrices, the construction will simplify the hardware/compiler support by only needing support for one specific bit-width. The structure/contributions of this paper is shown in Fig. 1.
+
+Notations. To simplify the presentation, we will narrow the scope of our discussion exclusively to Transformers. We first define notations for all relevant GEMMs. For the linear layer, let the input activation and parameter matrix be  $\mathbf{X}$  and  $\mathbf{W}$ . Let the query, key, value matrices needed in self-attention be  $\mathbf{Q},\mathbf{K},\mathbf{V}$ . Below, we itemize all GEMMs:
+
+$$
+\mathbf {Y} = \mathbf {X} \mathbf {W} ^ {\top} \quad \mathbf {P} = \mathbf {Q} \mathbf {K} ^ {\top} \quad \mathbf {O} = \mathbf {M} \mathbf {V}
+$$
+
+where  $\mathbf{M}$  is the attention score between  $\mathbf{Q}$  and  $\mathbf{K}$  defined as  $\mathbf{M} = \mathrm{softmax}(\mathbf{P})$  (omitting scaling factors). Now, given the gradient for  $\mathbf{Y},\mathbf{P},\mathbf{O}$  denoted as  $\nabla_{\mathbf{Y}},\nabla_{\mathbf{P}},\nabla_{\mathbf{O}}$ , the other gradients are calculated via GEMMs as well:
+
+$$
+\nabla_ {\mathbf {X}} = \nabla_ {\mathbf {Y}} \mathbf {W} \quad \nabla_ {\mathbf {Q}} = \nabla_ {\mathbf {P}} \mathbf {K} \quad \nabla_ {\mathbf {M}} = \nabla_ {\mathbf {O}} \mathbf {V} ^ {\top}
+$$
+
+$$
+\nabla_ {\mathbf {W}} = \nabla_ {\mathbf {Y}} ^ {\top} \mathbf {X} \quad \nabla_ {\mathbf {K}} = \nabla_ {\mathbf {P}} ^ {\top} \mathbf {Q} \quad \nabla_ {\mathbf {V}} = \mathbf {M} ^ {\top} \nabla_ {\mathbf {O}}
+$$
+
+These notations will help refer to each type of GEMM later.
+
+# 2. Round to Nearest: What do we lose?
+
+Let us start by using the simplest Rounding To Nearest (RTN) to map FP to integers, and check the extent to which integer GEMMs work satisfactorily for both training and inference, if we do not restrict to low bit-width integers. Specifically, for matrix  $\mathbf{A}$ , all entries of  $\mathbf{A}$  are quantized via
+
+$$
+\mathbf {A} _ {q} = \operatorname {r o u n d} \left(0. 5 \beta / \alpha_ {p} (\mathbf {A}) \mathbf {A}\right) \tag {1}
+$$
+
+where  $\alpha_{p}(\mathbf{A})$  gives the  $p$ -th percentile (see §A.1 for reasons behind using percentile) based on the magnitude of entries in  $\mathbf{A}$ , i.e.,  $p\%$  of entries in  $\mathbf{A}$  fall in the interval  $[- \alpha_{p}(\mathbf{A}), \alpha_{p}(\mathbf{A})]$ . We only need  $\alpha_{p}(\mathbf{A})$  as a meaningful estimate of the approximate range of values, and so we set  $p = 95\%$  for all experiments except a few cases noted explicitly. The hyperparameter  $\beta$  is the number of distinct integers that we want to use to encode values that are within  $[- \alpha_{p}(\mathbf{A}), \alpha_{p}(\mathbf{A})]$ . Then, after quantization, the GEMM for the original matrices can be approximated (because we incur a rounding error) in the quantized domain using integer GEMMs. The approximated GEMM is computed using the quantized  $\mathbf{A}$  and  $\mathbf{B}$ :
+
+$$
+\mathbf {C} \approx \frac {\alpha_ {p} (\mathbf {A}) \alpha_ {p} (\mathbf {B})}{(0 . 5 \beta) ^ {2}} \mathbf {A} _ {q} \mathbf {B} _ {q} ^ {\top} \tag {2}
+$$
+
+The scaling factor in (2) is used to undo the scaling in (1). Here,  $\mathbf{A}_q\mathbf{B}_q^\top$  is an integer GEMM, as desired. For notational simplicity, if clear from context, we will drop the  $q$  subscript from  $\mathbf{A}$  and  $\mathbf{B}$ .
+
+In §2.1–§2.2, we evaluate the efficacy of integer GEMMs as a replacement to FP GEMMs by evaluating how well simple RTN works for inference and training (error analysis of RTN is discussed in §A.2). We do not strictly constrain the quantized integers to be within the representable range of certain bit-widths (the maximal value of quantized integers can be very large, up to maximum of INT32), so we use INT without specifying bit-width to denote the data type used in RTN in these subsections. The use of integers with specific bit-widths will be discussed later in §3–§4.
+
+# 2.1. Efficacy of Integers: Inference
+
+A majority of the literature on quantized low precision calculations focuses on inference efficiency (Frantar et al., 2023;
+
+Table 1. Inference: Comparison on LLaMA-7B zero-shot performance and ViT ImageNet classification when quantize parameter only.  $\overline{\mathrm{Bits}}$  is the average bit required to store a weight parameter. HS: HellaSwag, WG: WinoGrande.  
+
+<table><tr><td rowspan="10">LLaMA-7B</td><td>Method</td><td>β</td><td>Bits</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HS</td><td>PIQA</td><td>WG</td></tr><tr><td>Full-Precision</td><td>-</td><td>16</td><td>43.1</td><td>76.3</td><td>77.8</td><td>57.2</td><td>78.0</td><td>68.8</td></tr><tr><td>GPTQ</td><td>-</td><td>4</td><td>37.4</td><td>72.7</td><td>73.3</td><td>54.9</td><td>77.9</td><td>67.9</td></tr><tr><td>LLM-FP4</td><td>-</td><td>4</td><td>40.4</td><td>74.9</td><td>74.2</td><td>55.8</td><td>77.8</td><td>69.9</td></tr><tr><td>QuIP</td><td>-</td><td>2</td><td>22.3</td><td>42.8</td><td>50.3</td><td>34.0</td><td>61.8</td><td>52.6</td></tr><tr><td>RTN+HE</td><td>5</td><td>2.5</td><td>39.3</td><td>72.8</td><td>69.9</td><td>53.4</td><td>74.9</td><td>66.4</td></tr><tr><td></td><td>7</td><td>2.9</td><td>42.6</td><td>73.9</td><td>72.3</td><td>55.9</td><td>77.0</td><td>67.4</td></tr><tr><td></td><td>11</td><td>3.5</td><td>43.9</td><td>76.1</td><td>77.3</td><td>56.3</td><td>77.3</td><td>69.3</td></tr><tr><td></td><td>15</td><td>4.0</td><td>43.0</td><td>75.7</td><td>77.5</td><td>57.0</td><td>78.0</td><td>69.2</td></tr><tr><td></td><td>31</td><td>5.0</td><td>42.7</td><td>76.1</td><td>76.1</td><td>57.3</td><td>77.3</td><td>69.3</td></tr><tr><td rowspan="7">ViT</td><td>Method</td><td>β</td><td>Bits</td><td>Tiny</td><td>Small</td><td>Base</td><td>Large</td><td colspan="2">Huge</td></tr><tr><td>Full-Precision</td><td>-</td><td>32</td><td>75.5</td><td>81.4</td><td>85.1</td><td>85.8</td><td colspan="2">87.6</td></tr><tr><td>PTQ4ViT</td><td>-</td><td>3</td><td>18.3</td><td>36.2</td><td>21.4</td><td>81.3</td><td colspan="2">78.9</td></tr><tr><td>RTN+HE</td><td>3</td><td>1.8</td><td>0.5</td><td>8.3</td><td>63.6</td><td>81.9</td><td colspan="2">83.3</td></tr><tr><td></td><td>5</td><td>2.4</td><td>38.2</td><td>69.0</td><td>81.1</td><td>84.9</td><td colspan="2">86.7</td></tr><tr><td></td><td>7</td><td>2.9</td><td>63.6</td><td>76.7</td><td>83.6</td><td>85.4</td><td colspan="2">87.2</td></tr><tr><td></td><td>15</td><td>4.0</td><td>73.4</td><td>80.5</td><td>84.8</td><td>85.7</td><td colspan="2">87.6</td></tr></table>
+
+Chee et al., 2023; Liu et al., 2023a; Yuan et al., 2022; Frantar et al., 2023; Chee et al., 2023; Liu et al., 2023a; Yuan et al., 2022; Lin et al., 2022; Li & Gu, 2023; Yuan et al., 2022; Ding et al., 2022; Li et al., 2023). Here, given a trained model, quantization seeks to reduce the precision of parameters and input activations to low precision. This allows faster low precision arithmetic for compute efficiency while maintaining model performance. So, we first evaluate how well RTN preserves model performance compared to baselines in this inference regime using downstream tasks: ARC (Clark et al., 2018), BoolQ (Clark et al., 2019), HellaSwag (Zellers et al., 2019), PIQA (Bisk et al., 2020), and WinoGrande (Sakaguchi et al., 2021). ( $\S$ A.3 includes more experiments). Most quantization schemes for LLMs focus on quantizing GEMMs in the Linear layers, while quantization methods for Vision Transformers are more ambitious and quantize all GEMMs in a Transformer. We follow this convention for baselines, but present all variants for RTN.
+
+Quantize Parameters Only. One direction of quantization research focuses on quantizing the parameters for better storage and memory usage (Frantar et al., 2023; Chee et al., 2023). We also evaluate how well RTN works for storage and memory efficiency. After quantization, the quantized  $\mathbf{W}_q$  usually contains a few hundreds of distinct integers. Simply representing  $\mathbf{W}_q$  in plain integer format would not be efficient and usually requires larger than 8 bits per value for memory. By inspecting the value distribution of  $\mathbf{W}_q$ , we find that a few values occur much more frequently than others, which create a clear opportunity for compression. We simply apply Huffman Encoding (HE), which was also used in (Han et al., 2016) to compress models for memory efficiency, to use shorter encoding for more frequent values. As shown in Table 1, with RTN and HE, we are able to significantly reduce the average bits per value with small
+
+Table 2. Inference: Comparison on LLaMA-7B zero-shot performance and ViT ImageNet classification when quantize computation in all linear layers. The super-script  $\ddagger$  indicates that LLM.int8() uses mixed-precision (INT8+FP16) to process outliers using FP16.
+
+<table><tr><td rowspan="11">LLaMA-7B</td><td>Method</td><td>β</td><td>Type</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HS</td><td>PIQA</td><td>WG</td></tr><tr><td>Full-Precision</td><td>-</td><td>BF16</td><td>43.1</td><td>76.3</td><td>77.8</td><td>57.2</td><td>78.0</td><td>68.8</td></tr><tr><td>LLM.int8()</td><td>-</td><td>INT8‡</td><td>43.8</td><td>75.5</td><td>77.8</td><td>57.4</td><td>77.6</td><td>68.7</td></tr><tr><td>SmoothQuant</td><td>-</td><td>INT8</td><td>37.4</td><td>74.4</td><td>74.0</td><td>55.0</td><td>77.5</td><td>69.6</td></tr><tr><td>LLM-QAT</td><td>-</td><td>INT4</td><td>30.2</td><td>50.3</td><td>63.5</td><td>55.6</td><td>64.3</td><td>52.9</td></tr><tr><td>LLM-FP4</td><td>-</td><td>FP4</td><td>33.6</td><td>65.9</td><td>64.2</td><td>47.8</td><td>73.5</td><td>63.7</td></tr><tr><td rowspan="5">RTN</td><td>5</td><td>INT</td><td>39.3</td><td>72.8</td><td>69.9</td><td>53.4</td><td>74.9</td><td>66.4</td></tr><tr><td>7</td><td>INT</td><td>42.6</td><td>73.9</td><td>72.3</td><td>55.9</td><td>77.0</td><td>67.4</td></tr><tr><td>11</td><td>INT</td><td>43.9</td><td>76.1</td><td>77.3</td><td>56.3</td><td>77.3</td><td>69.3</td></tr><tr><td>15</td><td>INT</td><td>43.0</td><td>75.7</td><td>77.5</td><td>57.0</td><td>78.0</td><td>69.2</td></tr><tr><td>31</td><td>INT</td><td>42.7</td><td>76.1</td><td>76.1</td><td>57.3</td><td>77.3</td><td>69.3</td></tr><tr><td rowspan="5">ViT</td><td>Method</td><td>β</td><td>Type</td><td>Tiny</td><td>Small</td><td>Base</td><td>Large</td><td colspan="2">Huge</td></tr><tr><td>Full-Precision</td><td>-</td><td>FP32</td><td>75.5</td><td>81.4</td><td>85.1</td><td>85.8</td><td colspan="2">87.6</td></tr><tr><td rowspan="3">RTN</td><td>5</td><td>INT</td><td>3.9</td><td>36.9</td><td>78.7</td><td>83.6</td><td colspan="2">85.3</td></tr><tr><td>7</td><td>INT</td><td>41.0</td><td>70.9</td><td>82.8</td><td>84.9</td><td colspan="2">86.7</td></tr><tr><td>15</td><td>INT</td><td>71.4</td><td>79.8</td><td>84.6</td><td>85.6</td><td colspan="2">87.5</td></tr></table>
+
+Table 3. Inference: Comparison on LLaMA-7B and ViT when quantize computation in all GEMMs. \*: PTQ4ViT uses a twin uniform quantization so GEMMs cannot be performed on INT6 directly and requires some modifications.  
+
+<table><tr><td rowspan="7">LLaMA-7B</td><td>Method</td><td>β</td><td>Type</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HS</td><td>PIQA</td><td>WG</td></tr><tr><td>Full-Precision</td><td>-</td><td>BF16</td><td>43.1</td><td>76.3</td><td>77.8</td><td>57.2</td><td>78.0</td><td>68.8</td></tr><tr><td rowspan="5">RTN</td><td>5</td><td>INT</td><td>23.5</td><td>34.3</td><td>54.8</td><td>32.5</td><td>57.6</td><td>49.7</td></tr><tr><td>7</td><td>INT</td><td>34.2</td><td>64.0</td><td>64.6</td><td>50.1</td><td>70.3</td><td>61.2</td></tr><tr><td>11</td><td>INT</td><td>41.6</td><td>72.4</td><td>68.7</td><td>55.1</td><td>75.4</td><td>65.1</td></tr><tr><td>15</td><td>INT</td><td>44.0</td><td>75.0</td><td>74.6</td><td>56.4</td><td>77.0</td><td>66.3</td></tr><tr><td>31</td><td>INT</td><td>43.4</td><td>75.8</td><td>76.8</td><td>57.5</td><td>77.4</td><td>68.4</td></tr><tr><td rowspan="10">ViT</td><td>Method</td><td>β</td><td>Type</td><td>Tiny</td><td>Small</td><td>Base</td><td>Large</td><td colspan="2">Huge</td></tr><tr><td>Full-Precision</td><td>-</td><td>FP32</td><td>75.5</td><td>81.4</td><td>85.1</td><td>85.8</td><td colspan="2">87.6</td></tr><tr><td>FQ-ViT</td><td>-</td><td>INT8</td><td>-</td><td>-</td><td>83.3</td><td>85.0</td><td colspan="2">-</td></tr><tr><td>I-ViT</td><td>-</td><td>INT8</td><td>-</td><td>81.3</td><td>84.8</td><td>-</td><td colspan="2">-</td></tr><tr><td>PTQ4ViT</td><td>-</td><td>INT6*</td><td>66.7</td><td>78.3</td><td>82.9</td><td>84.9</td><td colspan="2">86.6</td></tr><tr><td>APQ-ViT</td><td>-</td><td>INT4</td><td>17.6</td><td>48.0</td><td>41.4</td><td>-</td><td colspan="2">-</td></tr><tr><td>RepQ-ViT</td><td>-</td><td>INT4</td><td>-</td><td>65.1</td><td>68.5</td><td>-</td><td colspan="2">-</td></tr><tr><td rowspan="3">RTN</td><td>5</td><td>INT</td><td>3.5</td><td>28.5</td><td>76.9</td><td>83.2</td><td colspan="2">84.9</td></tr><tr><td>7</td><td>INT</td><td>39.0</td><td>69.9</td><td>82.1</td><td>84.7</td><td colspan="2">86.5</td></tr><tr><td>15</td><td>INT</td><td>71.1</td><td>79.8</td><td>84.5</td><td>85.6</td><td colspan="2">87.5</td></tr></table>
+
+or no performance degradation and result in significantly better efficiency compared to baselines (Frantar et al., 2023; Chee et al., 2023; Liu et al., 2023a; Yuan et al., 2022) for both Transformer based LLMs and Vision Transformers.
+
+Quantize GEMMs in Linear layers. It is common (Xiao et al., 2023; Liu et al., 2023a) to try and quantize the weight and input activation of linear layers to low precision for compute efficiency. We summarize our comparisons in Tab. 2. Here, we compare RTN to (Xiao et al., 2023; Dettmers et al., 2022; Liu et al., 2023b;a). As shown in Tab. 2, a simple RTN works remarkably well compared to other baselines. We use INT as a data type for RTN here; in §4, we show that we can compute integer GEMMs of any bit-widths using arbitrarily low bit-width GEMMs.
+
+Quantize all GEMMs. A more ambitious goal is to quan-
+
+![](images/c78cad3afbeb031ca690ada73d8f46d284220585dda363a0b18403acbd6bae84.jpg)
+
+![](images/6acc4d4638a73e9ab8d5ff29a8d5f7cd4d1f530da63da57226427b7942edc6ac.jpg)  
+Figure 2. Training: Comparison of RoBERTa loss curves.
+
+![](images/6a172c3f0ab3d4dc73be463af6f6e432a77d4a85e48e954a9fff4379031f8031.jpg)
+
+![](images/f2d98d9ae23eb2c187d414c7dcd3250b49066c98ad7fda7315bbc4b310c857ce.jpg)
+
+![](images/160db895bbc8c13e0ebd693a80a07ed04584ff05d3506edc536cf6405268d936.jpg)  
+Figure 3. Training: Comparison of ViT-Small.  $\dagger$  and  $*$ : we set  $\beta = 16383$  and  $\beta = 1023$ , respectively, for the set  $\{\nabla_{\mathbf{Y}}, \nabla_{\mathbf{P}}, \nabla_{\mathbf{O}}\}$ .
+
+![](images/5990ead8bc88b7fbc8054b8f0c9e7c2c1e3a23a3e14d6224e916ef668c2a6e3e.jpg)
+
+Table 4. Training: Validation log perplexity of RoBERTa.  
+
+<table><tr><td>Size</td><td>FP32</td><td>BF16</td><td>β = 255</td><td>β = 31</td><td>β = 15</td></tr><tr><td>Small</td><td>1.869</td><td>1.868</td><td>1.823</td><td>1.840</td><td>1.891</td></tr><tr><td>Base</td><td>1.611</td><td>-</td><td>-</td><td>1.601</td><td>-</td></tr></table>
+
+Table 5. Training: Validation top-1 accuracy of ViT-Small.  
+
+<table><tr><td>FP32</td><td>FP16</td><td>β = 63†</td><td>β = 31†</td><td>β = 31*</td></tr><tr><td>78.91</td><td>79.16</td><td>78.94</td><td>79.33</td><td>79.17</td></tr></table>
+
+tize every GEMM in a Transformer model for higher efficiency. The comparison results with (Lin et al., 2022; Li & Gu, 2023; Yuan et al., 2022; Ding et al., 2022; Li et al., 2023) are summarized in Tab. 3. We can draw a similar conclusion that a simple RTN offers strong performance.
+
+# 2.2. Efficacy of Integers: Training
+
+The transition from FP32 to FP16 and BF16 for GEMMs has doubled the compute efficiency of modern deep learning models. However, far fewer efforts have focused on low precision training (relative to inference) and this usually requires more sophisticated modifications (Wang et al., 2018; Wu et al., 2018; Zhu et al., 2020; Wortman et al., 2023). In this subsection, we evaluate how well quantizing all GEMMs (both forward and backward pass) using RTN works for training Transformer models. To ensure that the updates can be properly accumulated for the parameters, we use FP32 for storing the parameters and use the quantized version for GEMMs. To limit the amount of compute but still gather useful feedback, we evaluate RTN on RoBERTa (Liu et al., 2019) pretraining using masked language model
+
+ing (Devlin et al., 2019) on the English Wikipedia corpus (Foundation) and ImageNet classification (Deng et al., 2009) using ViT (Dosovitskiy et al., 2021) (and see T5-Large (Raffel et al., 2020) finetuning in §A.3). All hyperparameters (including random seed) are the same for full-precision and RTN quantized training. See §A.3 for more details of training configurations.
+
+RoBERTa. As shown in Fig. 2, when  $p = 95\%$ , for both Small and Base models, the RTN quantized training gives an almost identical log perplexity (loss) curves as FP32 training for  $\beta \in \{15, 31, 255\}$ . For larger  $\beta$ , the curve is even closer to the FP32 training curve. We see that  $\beta = 31$  already gives a remarkably good result. Surprisingly, despite a marginally higher training log perplexity when using RTN, the validation log perplexity of RTN ( $\beta = 31$  and  $\beta = 255$ ) is marginally lower than FP32 and BF16, see Tab. 4.
+
+ViT. For ViT, compared to RoBERTa pretraining, we found that it may be necessary to allow the gradients  $\nabla_{\mathbf{Y}}, \nabla_{\mathbf{P}}, \nabla_{\mathbf{O}}$  of the model to have higher bit-widths. As shown in Fig. 3, when  $\beta$  is the same ( $\beta = 31$  and  $\beta = 127$  for the set  $\{\mathbf{X}, \mathbf{W}, \mathbf{Q}, \mathbf{K}, \mathbf{M}, \mathbf{V}\}$  and  $\{\nabla_{\mathbf{Y}}, \nabla_{\mathbf{P}}, \nabla_{\mathbf{O}}\}$ , we see divergence in the middle of training. Alternatively, when using a larger  $\beta$  for only the set  $\{\nabla_{\mathbf{Y}}, \nabla_{\mathbf{P}}, \nabla_{\mathbf{O}}\}$ , the loss curve of RTN quantized training is almost identical to FP32 training. Surprisingly, we observed similar results as RoBERTa training: despite marginally higher training loss when using RTN, the validation top-1 accuracy of RTN is higher than FP32 as shown in Fig. 3 and Tab. 5.
+
+# 3. What happens with Low Bit-Width?
+
+Converting floating point to integers alone will not provide efficiency benefits. Rather, we want to use a representation that can be efficiently computed (and this is why low bit-width integers are common in integer quantization). Notice that as a direct consequence of RTN, by (1),  $95\%$  of values can be represented using  $\beta$  distinct numbers, which requires only  $\log_2(\beta + 1)$  bits. For example, if  $\beta = 15$ , then we can represent these  $95\%$  of values with 4-bit signed integers, which is already low bit-width. So, is there still a problem?
+
+It turns out that the challenge involves dealing with the remaining  $5\%$  of entries. To get a sense of how large these values are, we calculate the ratio  $\alpha_{100}(\cdot) / \alpha_{95}(\cdot)$  between the maximum and  $95^{\mathrm{th}}$ -percentile of the magnitude of each matrix in GEMMs when performing (a) inference (forward pass) of LLaMA-7B and ViT-Large and (b) training (forward pass and backward pass) of RoBERTa-Small at different training phases. We can check the ratios in Tab. 6 and Tab. 7, respectively. We see extremely large values across both training and inference and across the entire duration of training, so simply increasing the representation bit width of low precision integers by a few more bits will not be
+
+Table 6. Maximal ratios between the maximum and 95-percentile of magnitudes of each matrix involved in GEMMs.  
+
+<table><tr><td>Model</td><td>X</td><td>W</td><td>Q</td><td>K</td><td>M</td><td>V</td></tr><tr><td>LLaMA-7B</td><td>141312.0</td><td>47.8</td><td>8.4</td><td>8.1</td><td>4448.0</td><td>36.2</td></tr><tr><td>ViT-Large</td><td>284402.4</td><td>34.8</td><td>4.3</td><td>4.3</td><td>120.0</td><td>8.9</td></tr></table>
+
+Table 7. Maximal ratios between the maximum and 95-percentile of magnitudes of each matrix involved in GEMMs during the training of RoBERTa-Small.  
+
+<table><tr><td>Progress</td><td>X</td><td>W</td><td>∇Y</td><td>Q</td><td>K</td><td>∇P</td><td>M</td><td>V</td><td>∇O</td></tr><tr><td>1/3</td><td>28.7</td><td>7.1</td><td>292.5</td><td>3.7</td><td>3.0</td><td>309365.2</td><td>3924.6</td><td>3.1</td><td>25.8</td></tr><tr><td>2/3</td><td>25.7</td><td>13.8</td><td>235.4</td><td>4.2</td><td>2.7</td><td>283742.8</td><td>2283.3</td><td>3.3</td><td>32.4</td></tr><tr><td>3/3</td><td>22.0</td><td>16.0</td><td>290.3</td><td>4.0</td><td>3.0</td><td>218376.0</td><td>2018.6</td><td>3.4</td><td>28.9</td></tr></table>
+
+Table 8. Catastrophic performance degradation when restricting outliers to a representable range of quantized domain or clipping the outliers on zero-shot inference of LLaMA-7B and ImageNet classification of quantized ViT models.  $p = 100$  means we keep outliers within representable range of  $\beta$  distinct integers.  $\beta = \infty$  means that we do not quantize the values. Clip means we clip the values that are larger than  $p$ -percentile.  
+
+<table><tr><td rowspan="5">LLaMA-7B</td><td>p</td><td>β</td><td>Clip</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HS</td><td>PIQA</td><td>WG</td></tr><tr><td colspan="3">Full-Precision</td><td>43.1</td><td>76.3</td><td>77.8</td><td>57.2</td><td>78.0</td><td>68.8</td></tr><tr><td>100</td><td>255</td><td>No</td><td>35.8</td><td>66.2</td><td>57.8</td><td>47.4</td><td>71.3</td><td>63.9</td></tr><tr><td>99.5</td><td>∞</td><td>Yes</td><td>21.4</td><td>25.5</td><td>60.2</td><td>25.8</td><td>53.5</td><td>49.9</td></tr><tr><td>95</td><td>31</td><td>No</td><td>43.4</td><td>75.8</td><td>76.8</td><td>57.5</td><td>77.4</td><td>68.4</td></tr><tr><td rowspan="5">ViT</td><td>p</td><td>β</td><td>Clip</td><td>Tiny</td><td>Small</td><td>Base</td><td>Large</td><td colspan="2">Huge</td></tr><tr><td colspan="3">Full-Precision</td><td>75.5</td><td>81.4</td><td>85.1</td><td>85.8</td><td colspan="2">87.6</td></tr><tr><td>100</td><td>127</td><td>No</td><td>53.9</td><td>69.1</td><td>72.0</td><td>81.6</td><td colspan="2">83.6</td></tr><tr><td>99.5</td><td>∞</td><td>Yes</td><td>11.3</td><td>24.1</td><td>9.0</td><td>15.8</td><td colspan="2">0.6</td></tr><tr><td>95</td><td>15</td><td>No</td><td>71.1</td><td>79.8</td><td>84.5</td><td>85.6</td><td colspan="2">87.5</td></tr></table>
+
+sufficient to represent these heavy hitters.
+
+We performed experiments studying different ways of handling these heavy hitters when quantizing all GEMMs (linear layers and self-attention computation) in Transformer models. Unless  $\beta$  is inordinately large (based on Tab. 6 and Tab. 7, more than  $10^{5}$  times larger than our choice of  $\beta$  for  $p = 95\%$ ), simply ensuring that the heavy hitters lie within the representable range of  $\beta$  for  $\beta = 255$  or  $\beta = 127$  results in a huge performance drop as shown in Tab. 8. On the other hand, clipping the extreme heavy hitters (at the 99.5-percentile) also fails as shown in Tab. 8. Our observations for training are similar – we can see the loss curves for  $p = 100\%$ ,  $\beta = 255$  and  $p = 95\%$ ,  $\beta = 31$  in Fig. 2.
+
+As briefly mentioned earlier, some ideas have been proposed to process these so-called outliers. The approach in Dettmers et al. (2022) exploits the location structure of where these outliers occur and moves the columns or rows of each matrix (with these outliers) into a different matrix, then GEMM is performed using FP16. The authors in Xiao et al. (2023) propose to smooth the outliers in activation and mitigate the quantization difficulty to parameters via a transformation. These strategy requires GEMM hardware
+
+![](images/c047b5b93c8432f2e0f862296014a6d43d7f6f6c37ee103be21982b6c2ef425d.jpg)  
+Figure 4. Illustration of unpacking row vectors. The solid, dashed, and dotted arrows correspond to lines 5, 4, and 6 in Algo. 1
+
+support for different precisions or may lower the performance as shown in our baseline comparisons in  $\S 2.1$ .
+
+Goals. We desire an approach that does not alter the results of integer GEMMs; in other words, all results in §2.1 and §2.2 must remain exactly the same, yet we should not need calculations using different precisions. This may appear unrealistic but our simple procedure, IM-Unpack, allows representing heavy hitters using low bit-width integers. Calculations are carried out using low bit-width integer arithmetic. Specifically, IM-Unpack unpacks a matrix containing heavy hitters into a larger unpacked matrix (we study how large the expansion will be in §4.2) whose values are all representable by low bit-width integers. IM-Unpack obtains the exact output of the original GEMM using purely low bit-width integer GEMMs on these unpacked matrices.
+
+# 4. IM-Unpack: Integer Matrix Unpacking
+
+Our approach starts with a simple observation that, for example, a 32-bit integer  $v$  can be represented as
+
+$$
+v = v _ {0} + 1 2 8 v _ {1} + 1 2 8 ^ {2} v _ {2} + 1 2 8 ^ {3} v _ {3} + 1 2 8 ^ {4} v _ {4}
+$$
+
+where  $v_{i}$  are 8-bit integers. Multiplication/addition of two 32-bit integers can be performed on these decomposed 8-bit integers followed by some post-processing steps (scaling via bit shifting and accumulation). Note that while this unpacking enables performing high bit-width arithmetic using lower bit-width, it does need more operations. For example, one 32-bit addition now becomes five 8-bit additions with some follow up processing, and one 32-bit multiplication becomes twenty five 8-bit multiplications (distributive law).
+
+Remark 4.1. The reason why this unpacking is still useful is because the additional work depends on the number and spatial distribution of the heavy hitters/outliers. We harvest gains because outliers account for a very small portion of the matrices that appear in practice in training/inference stages of Transformer models. Exploitation of the sparsity of outliers in quantization can also be found in Dettmers et al. (2022); Xiao et al. (2023).
+
+Let  $b$  be the target bit-width of low bit-width integers and  $s = 2^{b - 1}$  be the representable range of bit-width  $b$ :  $b$ -bit integers can represent a set  $\{-s + 1, \dots, 0, \dots, s - 1\}$ . We refer to any integers inside of this set as In-Bound (IB) values and
+
+Algorithm 1 UnpackRow(A, b)  
+1: Let  $\Pi \gets \mathbf{I}$  and  $s\gets 2^{b - 1}$  and  $i\gets 0$    
+2: while  $\mathbf{A}[i,:]$  exists do   
+3: if  $\mathbf{A}[i,:]$  contains OB entries then   
+4: Append floor(A[i,:]/s) as a new row to A   
+5:  $\mathbf{A}[i,:]\leftarrow \mathbf{A}[i,:]\mathrm{mod}s$    
+6: Append s[;, i] as a new column to I   
+7: end if   
+8:  $i\gets i + 1$    
+9: end while   
+10: return A,II
+
+any integers outside of this set as Out-of-Bound (OB) values, which will be used in later discussion to refer to the values that need to be unpacked. We will first show how to unpack a vector to multiple low bit-width vectors. Then, we will discuss how to unpack a matrix using different strategies to achieve better results in different cases in §4.1. Lastly, we will evaluate how well IM-Unpack works in §4.2, and provide an end to end quantization baseline comparison and discuss model speedup in §A.3 when employing our IM-Unpack as supplement to §2.
+
+Unpacking an integer vector. Let  $\mathbf{v}$  be an integer vector and define a function:
+
+$$
+m (\mathbf {v}, s, i) = \operatorname {f l o o r} (\mathbf {v} / s ^ {i}) \bmod s \tag {3}
+$$
+
+such that for all  $i$ , all entries of  $m(\mathbf{v}, s, i)$  are bounded (IB), i.e., lie in the interval  $[-s + 1, s - 1]$ . When  $s$  is clear from the context, we shorten the LHS of (3) to just  $m(\mathbf{v}, i)$ . Then,
+
+$$
+\mathbf {v} = \sum_ {i = 0} ^ {\infty} s ^ {i} m (\mathbf {v}, i) \tag {4}
+$$
+
+Note that  $\mathbf{v} / s^i$  decreases to 0 exponentially fast, so we are able to unpack a vector with just a few low bit-width vectors.
+
+# 4.1. Variants of Matrix Unpacking
+
+In this subsection, we discuss different strategies of matrix unpacking for different structure-types of matrices. First, we discuss the case where  $\mathbf{A}$  is the matrix containing OB values to be unpacked and  $\mathbf{B}$  is a matrix whose values are all IB. Next, we discuss how unpacking works when both  $\mathbf{A}$  and  $\mathbf{B}$  contains OB values.
+
+Unpacking row vectors. We start with the simplest means of unpacking a matrix: unpacking the row vectors. Given a matrix  $\mathbf{A}$ , if one row of  $\mathbf{A}$  contains OB values, we can unpack the row to multiple rows whose entries are all bounded. The exact procedure is described in Alg. 1 and illustrated in Fig. 4. In Fig. 4, when the second row in  $\mathbf{A}$  contains OB values, we can unpack it to two row vectors (the second and fifth row) and the post-processing step takes the form of applying  $\Pi_A$  to the unpacked matrix  $\mathbf{A}_u$ .
+
+Reconstructing A. A can be reconstructed using the unpacked matrix  $\mathbf{A}_u$  whose entries are IB and a sparse matrix
+
+![](images/c92f14f1ba5e07af219a806610baf8d9c46e82fd986061e850ccf8d48695a8ca.jpg)  
+Figure 5. Left: Failure case for unpacking rows. Right: Failure case for unpacking rows or columns alone.
+
+$\Pi$  whose column contains only one non-zero:
+
+$$
+\mathbf {A} _ {u}, \Pi_ {A} = \operatorname {U n p a c k R o w} (\mathbf {A}, b)
+$$
+
+$$
+\mathbf {A} = \boldsymbol {\Pi} _ {A} \mathbf {A} _ {u}
+$$
+
+Here, applying  $\mathbf{\Pi}_A$  to  $\mathbf{A}_u$  can be efficiently computed easily (for example, via torch.index_add).
+
+Are we done? If we do not care about maximizing efficiency, then the above scheme already provides a way to perform high bit-width GEMM using low bit-width GEMM. However, this might not be the optimal unpacking strategy for some matrices. For example, consider the left matrix shown in Fig. 5. Since every row of this matrix contains OB values, every row needs to be unpacked, resulting in a much larger matrix. In this case, it might be better to try and unpack the column vectors. Let us apply a similar idea of unpacking row vectors to unpack column vectors of A:
+
+$$
+\mathbf {A} = \mathbf {A} _ {u} ^ {\prime} \boldsymbol {\Pi} _ {A} ^ {\prime} \tag {5}
+$$
+
+$$
+\mathbf {A} \mathbf {B} ^ {\top} = \mathbf {A} _ {u} ^ {\prime} \boldsymbol {\Pi} _ {A} ^ {\prime} \mathbf {B} ^ {\top} \tag {5}
+$$
+
+While unpacking column vectors is reasonable, the sparse matrix  $\Pi_A^\prime$  creates an issue when performing a GEMM of two lower bit-width matrices:  $\Pi_A^\prime$  has to be applied to  $\mathbf{A}_u^\prime$  or  $\mathbf{B}^{\top}$  before GEMM, but the result/output may contain OB entries after application, disabling low bit-width integer GEMM. This problem is similar to the per-channel quantization. It is not simple to handle and becomes more involved when  $\mathbf{B}$  also need to be unpacked.
+
+Unpacking column vectors. Alternatively, let us look at how  $\mathbf{AB}^{\top}$  is computed via outer product of column vectors:
+
+$$
+\mathbf {C} = \mathbf {A} \mathbf {B} ^ {\top} = \sum_ {i = 1} ^ {d} \mathbf {A} [:, i ] \mathbf {B} [:, i ] ^ {\top}
+$$
+
+Let us look at the  $i$ -th outer product. Let us try unpacking  $\mathbf{A}(:, i]$  using (4), then we have
+
+$$
+\mathbf {A} [:, i ] \mathbf {B} [:, i ] ^ {\intercal} = \sum_ {j = 0} ^ {\infty} s ^ {j} m (\mathbf {A} [:, i ], j) \mathbf {B} [:, i ] ^ {\intercal}
+$$
+
+Suppose that  $m(\mathbf{A}(:,i],j) = 0$  for  $j\geq k$ , then we can unpack one outer product to  $k$  outer products. This is equivalent to appending  $m(\mathbf{A}(:,i],j)$  for  $0\leq j < k$  to the columns of  $\mathbf{A}$ , appending  $k$  identical  $\mathbf{B}(:,i]$  to the columns of  $\mathbf{B}$ , and maintaining a diagonal matrix to keep track of the scaling factor  $s^j$ . The exact procedure is described in Alg. 2, and
+
+![](images/ade135df7aa9fe26ed104235dc17e3ae173307e163f679d4a9c14b05a0577333.jpg)  
+Figure 6. Illustration of unpacking column vectors. The blue solid, dashed, and dotted arrows correspond to lines 5, 4, and 7 in Algo. 1, and the gray dashed arrow corresponds to line 6 in Algo. 1.  
+Fig. 6 shows a visualization of unpacking columns. Using column unpacking, we have
+
+Algorithm 2 UnpackColumn(A,B,S,b)  
+1: Let  $s \gets 2^{b-1}$  and  $i \gets 0$   
+2: while  $\mathbf{A}(:, i]$  exists do  
+3: if  $\mathbf{A}(:, i]$  contains OB entries then  
+4: Append floor  $(\mathbf{A}(:, i)/s)$  as a new column to  $\mathbf{A}$   
+5:  $\mathbf{A}(:, i] \gets \mathbf{A}(:, i] \mod s$   
+6: Append  $\mathbf{B}(:, i]$  as a new column to  $\mathbf{B}$   
+7: Append  $s\mathbf{S}[i, i]$  as a new diagonal entry to  $\mathbf{S}$   
+8: end if  
+9:  $i \gets i + 1$   
+10: end while  
+11: return  $\mathbf{A}, \mathbf{B}, \mathbf{S}$
+
+Algorithm 3 ScaledMatMul(A,B,S)  
+1: Let  $\mathbf{C} \gets 0$   
+2: for all distinct diagonal entry  $s^i$  in  $\mathbf{S}$  do  
+3: Let  $\mathcal{I}$  be the index set where  $\mathbf{S}[j,j] = s^i$  for  $j \in \mathcal{I}$   
+4:  $\mathbf{C} \gets \mathbf{C} + s^i\mathbf{A}(:,\mathcal{I})\mathbf{B}(:,\mathcal{I}]^\top$   
+5: end for  
+6: return  $\mathbf{C}$
+
+$$
+\mathbf {A} _ {u}, \mathbf {B} _ {e}, \mathbf {S} _ {u} = \text {U n p a c k C o l u m n} (\mathbf {A}, \mathbf {B}, \mathbf {I}, b)
+$$
+
+$$
+\mathbf {A} \mathbf {B} ^ {\top} = \mathbf {A} _ {u} \mathbf {S} _ {u} \mathbf {B} _ {e} ^ {\top}
+$$
+
+Naively, this still suffers from the same problem as discussed in (5) in that there is a diagonal scaling matrix between two low bit-width matrices making low bit-width GEMMs difficult. However, since  $\mathbf{S}_u$  is a diagonal matrix whose diagonal entries consist of a few distinct factors in  $\{1,s,s^2,\ldots\}$ , we can easily compute one GEMM for each distinct diagonal entry as shown in Alg. 3.
+
+$$
+\mathbf {A} \mathbf {B} ^ {\top} = \operatorname {S c a l e d M a t M u l} \left(\mathbf {A} _ {u}, \mathbf {B} _ {e}, \mathbf {S} _ {u}\right)
+$$
+
+Further, since  $s$  is a power of 2, the scaling can be efficiently implemented via bit shifting.
+
+Are we done yet? Unpacking columns is efficient for the left matrix shown in Fig. 5. However, neither unpacking rows nor unpacking columns will be efficient for unpacking the right matrix shown in Fig. 5. All rows and columns contains OB values. Unpacking rows or columns alone will not be ideal. For the right matrix in Fig. 5, a better strategy is to unpack the second row and the second column simultaneously.
+
+Unpacking both rows and columns simultaneously. Our
+
+Algorithm 4 UnpackBoth(A,B,S,b)  
+1: Let  $s \gets 2^{b-1}$  and  
+2: while True do  
+3: Let  $(c_0, i), (c_1, j)$  be the tuples of top OB count in row/column vectors and corresponding index  
+4: if  $c_0 = 0$  and  $c_1 = 0$  then  
+5: break  
+6: else if  $c_0 \geq c_1$  then  
+7: Append floor( $\mathbf{A}[i, :]/s$ ) as a new row to  $\mathbf{A}$   
+8:  $\mathbf{A}[i, :] \gets \mathbf{A}[i, :]\mod s$   
+9: Append  $s\Pi[:, i]$  as a new column to  $\Pi$   
+10: else  
+11: Append floor( $\mathbf{A}[:, j]/s$ ) as a new column to  $\mathbf{A}$   
+12:  $\mathbf{A}[:, j] \gets \mathbf{A}[:, j]\mod s$   
+13: Append  $\mathbf{B}[:, j]$  as a new column to  $\mathbf{B}$   
+14: Append  $s\mathbf{S}[j, j]$  as a new diagonal entry to  $\mathbf{S}$   
+15: end if  
+16: end while  
+17: return  $\mathbf{A}, \mathbf{B}, \mathbf{S}, \Pi$
+
+final strategy combines row and column unpacking together and selectively performs row unpack or column unpack based on the number of OB values that can be eliminated. The procedure is described in Alg. 4, and we provide an illustration of unpacking both dimensions in Fig. 7. With this procedure, we can obtain the output of high bit-width GEMM using low bit-width as:
+
+$$
+\mathbf {A} _ {u}, \mathbf {B} _ {e}, \mathbf {S} _ {u}, \boldsymbol {\Pi} _ {A} = \text {U n p a c k B o t h} (\mathbf {A}, \mathbf {B}, \mathbf {I}, b) \tag {6}
+$$
+
+$$
+\mathbf {A} \mathbf {B} ^ {\top} = \boldsymbol {\Pi} _ {A} \mathbf {A} _ {u} \mathbf {S} _ {u} \mathbf {B} _ {e} ^ {\top}
+$$
+
+Here,  $\mathbf{A}_u\mathbf{S}_u\mathbf{B}^\top$  can be calculated via Alg. 3, and applying  $\Pi_A$  can be carried out efficiently as discussed.
+
+Combining everything. Since we have different strategies for unpacking, let us first define a unified interface in Alg. 5. One can verify that for any strategies  $s_A$ :
+
+$$
+\mathbf {A} _ {u}, \mathbf {B} _ {e}, \mathbf {S} _ {u}, \boldsymbol {\Pi} _ {A} = \text {U n p a c k} (\mathbf {A}, \mathbf {B}, \mathbf {I}, b, s _ {A}) \tag {7}
+$$
+
+$$
+\mathbf {A} \mathbf {B} ^ {\top} = \boldsymbol {\Pi} _ {A} \mathbf {A} _ {u} \mathbf {S} _ {u} \mathbf {B} _ {e} ^ {\top}
+$$
+
+In the previous discussion,  $\mathbf{B}$  was assumed to have all values IB. When  $\mathbf{B}$  contains OB values, we note that  $\mathbf{B}$  can be unpacked in a similar manner, and the choice of unpacking strategies for  $\mathbf{B}$  is independent of the unpacking strategy for  $\mathbf{A}$ . For example,  $\mathbf{A}$  can be unpacked row-wise, while  $\mathbf{B}$  is unpacked column-wise. By taking the unpacked  $\mathbf{A}_u, \mathbf{B}_e, \mathbf{S}_u, \Pi_A$  from (7), we can further unpack  $\mathbf{B}$  using
+
+![](images/6bcf0f7a55b1abe239a9843bbd4e8c649c30743112a344222b69c5f53a33333c.jpg)  
+Figure 7. Illustration of unpacking both rows and columns based on the OOB counts. The red solid, dashed, and dotted arrows correspond to lines 8, 7, and 9 in Algo. 4. The blue solid, dashed, and dotted arrows correspond to lines 12, 11, and 14 in Algo. 4, and the gray dashed arrow corresponds to line 13 in Algo. 4.
+
+Algorithm 5 Unpack(A, B, S, b, strategy)
+
+1: if strategy is UnpackRow then  
+2:  $\mathbf{A}_u, \Pi_A \gets \mathrm{UnpackRow}(\mathbf{A}, b)$  
+3:  $\mathbf{S}_u, \mathbf{B}_e \gets \mathbf{S}, \mathbf{B}$  
+4: else if strategy is UnpackColumn then  
+5:  $\mathbf{A}_u,\mathbf{B}_e,\mathbf{S}_u\gets$  UnpackColumn(A,B,S,b)  
+6:  $\Pi_A\gets \mathbf{I}$  
+7: else  
+8:  $\mathbf{A}_u, \mathbf{B}_e, \mathbf{S}_u, \Pi_A \gets \text{UnpackBoth}(\mathbf{A}, \mathbf{B}, \mathbf{S}, b)$  
+9: end if  
+10: return  $\mathbf{A}_u, \mathbf{B}_e, \mathbf{S}_u, \Pi_A$
+
+strategy  $s_B$ :
+
+$$
+\begin{array}{l} \mathbf {B} _ {e u}, \mathbf {A} _ {u e}, \mathbf {S} _ {u u}, \boldsymbol {\Pi} _ {B} = \operatorname {U n p a c k} \left(\mathbf {B} _ {e}, \mathbf {A} _ {u}, \mathbf {S} _ {u}, b, s _ {B}\right) \\ \mathbf {A} \mathbf {B} ^ {\top} = \boldsymbol {\Pi} _ {A} \mathbf {A} _ {u e} \mathbf {S} _ {u u} \mathbf {B} _ {e u} ^ {\top} \boldsymbol {\Pi} _ {B} ^ {\top} \\ \end{array}
+$$
+
+Here, values in both  $\mathbf{A}_{ue}$  and  $\mathbf{B}_{eu}$  are IB, and the result can be obtained similar to discussion in Eq. (6).
+
+As we noted during the discussion,  $\Pi_A,\Pi_B,\mathbf{S}_{uu}$  are special sparse matrices.  $\Pi_{A}$  and  $\Pi_B$  are sparse matrices whose column contains only one non-zero, and  $\mathbf{S}_{uu}$  are diagonal matrices whose diagonal entries consist of a few distinct factors. As a result,  $\mathbf{A}_{ue}\mathbf{S}_{uu}\mathbf{B}_{eu}^{\top}$  is computed via Alg. 3 using GEMMs on  $\mathbf{A}_{ue}$  and  $\mathbf{B}_{eu}$ , and applying  $\Pi_A$  and  $\Pi_B$  is possible simply via torch.index_add.
+
+Summary. We introduced three strategies to unpack a matrix to low bit-width integer matrices for different structures of OB values in a matrix. While these strategies work for arbitrary matrices, we can clearly see that these unpacking strategies are most efficient when the OB values concentrate in a few columns and rows. Luckily, the matrices of interest in Transformer models indeed have this property, which is studied and exploited in several works (Dettmers et al., 2022; Xiao et al., 2023).
+
+# 4.2. Evaluating Unpacking Overhead
+
+The idea of IM-Unpack is to use more low bit-width arithmetic operations to compute a high bit-width operation. As we see in the description of IM-Unpack algorithm, the number of row and column vectors will increase, so the unpacked matrices  $\mathbf{A}_{ue}$  and  $\mathbf{B}_{eu}$  is larger in terms of size compared to  $\mathbf{A}$  and  $\mathbf{B}$ , which obviously increases the computational cost of low bit-width GEMMs. In this subsection, we evaluate how much this cost will increase. For two ma
+
+Table 9. Averaged unpack ratios of each type of GEMMs in LLaMA-7B: linear layers (computing  $\mathbf{Y}$ ), attention score (computing  $\mathbf{P}$ ), and attention output (computing  $\mathbf{O}$ ) when using different unpack strategies and integer bit-width  $b$  under quantization  $\beta$  settings. AS: Attention Score, AO: Attention Output.
+
+<table><tr><td colspan="3">β</td><td colspan="3">5</td><td colspan="3">15</td><td colspan="3">31</td></tr><tr><td colspan="3">Integer Bits b</td><td>3</td><td>4</td><td>5</td><td>4</td><td>5</td><td>6</td><td>5</td><td>6</td><td>7</td></tr><tr><td rowspan="7">Linear (Y)</td><td rowspan="6">X</td><td>Row</td><td>Row</td><td>2.67</td><td>1.93</td><td>1.57</td><td>2.47</td><td>2.02</td><td>1.73</td><td>2.12</td><td>2.00</td></tr><tr><td>Row</td><td>Col</td><td>10.76</td><td>2.35</td><td>1.61</td><td>9.91</td><td>5.36</td><td>1.84</td><td>8.44</td><td>5.62</td></tr><tr><td>W</td><td>Both</td><td>5.46</td><td>1.95</td><td>1.57</td><td>5.15</td><td>2.20</td><td>1.73</td><td>4.71</td><td>2.24</td></tr><tr><td>Col</td><td>Row</td><td>3.80</td><td>1.32</td><td>1.06</td><td>3.98</td><td>1.64</td><td>1.16</td><td>3.93</td><td>1.68</td></tr><tr><td>Col</td><td>Col</td><td>15.40</td><td>1.62</td><td>1.09</td><td>16.00</td><td>4.01</td><td>1.25</td><td>15.69</td><td>4.33</td></tr><tr><td>Col</td><td>Both</td><td>5.21</td><td>1.34</td><td>1.06</td><td>6.04</td><td>1.76</td><td>1.16</td><td>5.98</td><td>1.82</td></tr><tr><td colspan="2">Mix</td><td>2.6</td><td>1.27</td><td>1.06</td><td>2.44</td><td>1.4</td><td>1.15</td><td>2.1</td><td>1.42</td><td>1.16</td></tr><tr><td rowspan="5">AS (P)</td><td rowspan="4">Q</td><td>Row</td><td>Row</td><td>1.97</td><td>1.60</td><td>1.0</td><td>2.00</td><td>1.87</td><td>1.15</td><td>2.00</td><td>1.87</td></tr><tr><td>Row</td><td>K</td><td>3.22</td><td>1.64</td><td>1.0</td><td>5.35</td><td>2.07</td><td>1.17</td><td>5.36</td><td>2.09</td></tr><tr><td>Col</td><td>Row</td><td>1.81</td><td>1.04</td><td>1.0</td><td>2.91</td><td>1.14</td><td>1.01</td><td>2.91</td><td>1.15</td></tr><tr><td>Col</td><td>Col</td><td>3.36</td><td>1.08</td><td>1.0</td><td>8.66</td><td>1.32</td><td>1.03</td><td>8.67</td><td>1.35</td></tr><tr><td colspan="2">Mix</td><td>1.72</td><td>1.03</td><td>1.0</td><td>1.95</td><td>1.13</td><td>1.01</td><td>1.95</td><td>1.14</td><td>1.01</td></tr><tr><td rowspan="5">AO (O)</td><td rowspan="4">M</td><td>Row</td><td>Row</td><td>6.02</td><td>4.18</td><td>3.27</td><td>4.72</td><td>3.65</td><td>3.02</td><td>3.93</td><td>3.24</td></tr><tr><td>Row</td><td>V</td><td>15.10</td><td>4.53</td><td>3.35</td><td>18.21</td><td>4.64</td><td>3.16</td><td>15.07</td><td>4.21</td></tr><tr><td>Col</td><td>Row</td><td>16.29</td><td>8.14</td><td>5.12</td><td>11.28</td><td>6.98</td><td>4.91</td><td>8.41</td><td>5.84</td></tr><tr><td>Col</td><td>Col</td><td>42.21</td><td>8.76</td><td>5.21</td><td>43.57</td><td>9.11</td><td>5.09</td><td>32.31</td><td>7.74</td></tr><tr><td colspan="2">Mix</td><td>5.98</td><td>4.11</td><td>3.16</td><td>4.7</td><td>3.62</td><td>2.97</td><td>3.92</td><td>3.22</td><td>2.77</td></tr></table>
+
+Table 10. Averaged unpack ratios of each type of quantized GEMMs in both forward and backward of a RoBERTa-Small when using different integer bit length  $b$  at different training phrases of the  $\beta = {31}$  experiment in Fig. 2. The optimal strategies (Mix as in Tab. 9) for each GEMM is used.
+
+<table><tr><td colspan="2">Progress</td><td colspan="3">1/3</td><td colspan="3">2/3</td><td colspan="3">3/3</td></tr><tr><td colspan="2">Integer Bits b</td><td>5</td><td>6</td><td>7</td><td>5</td><td>6</td><td>7</td><td>5</td><td>6</td><td>7</td></tr><tr><td rowspan="3">Linear</td><td>Y</td><td>2.00</td><td>1.31</td><td>1.08</td><td>2.00</td><td>1.32</td><td>1.07</td><td>2.00</td><td>1.32</td><td>1.05</td></tr><tr><td>∇x</td><td>1.50</td><td>1.31</td><td>1.15</td><td>1.50</td><td>1.30</td><td>1.16</td><td>1.50</td><td>1.30</td><td>1.15</td></tr><tr><td>∇w</td><td>1.98</td><td>1.25</td><td>1.04</td><td>1.98</td><td>1.25</td><td>1.03</td><td>1.98</td><td>1.25</td><td>1.03</td></tr><tr><td rowspan="3">AS</td><td>P</td><td>1.66</td><td>1.04</td><td>1.00</td><td>1.42</td><td>1.05</td><td>1.0</td><td>1.40</td><td>1.04</td><td>1.00</td></tr><tr><td>∇Q</td><td>2.22</td><td>1.90</td><td>1.71</td><td>2.22</td><td>1.91</td><td>1.7</td><td>2.24</td><td>1.92</td><td>1.71</td></tr><tr><td>∇K</td><td>1.79</td><td>1.06</td><td>1.00</td><td>1.49</td><td>1.07</td><td>1.0</td><td>1.45</td><td>1.07</td><td>1.00</td></tr><tr><td rowspan="3">AO</td><td>O</td><td>3.11</td><td>2.71</td><td>2.30</td><td>3.10</td><td>2.68</td><td>2.24</td><td>3.10</td><td>2.62</td><td>2.22</td></tr><tr><td>∇M</td><td>1.21</td><td>1.10</td><td>1.04</td><td>1.21</td><td>1.10</td><td>1.04</td><td>1.21</td><td>1.10</td><td>1.04</td></tr><tr><td>∇v</td><td>2.88</td><td>2.52</td><td>2.12</td><td>2.87</td><td>2.48</td><td>2.10</td><td>2.86</td><td>2.41</td><td>2.09</td></tr></table>
+
+trices  $\mathbf{A}$  and  $\mathbf{B}$ , the complexity of a GEMM is  $\mathcal{O}(ndh)$ . Similarly, let  $n'$ ,  $d'$  be the size of  $\mathbf{A}_{ue}$  and  $h'$  be the number of rows of  $\mathbf{B}_{eu}$ . The cost of  $\mathbf{A}_{ue}\mathbf{S}_{uu}\mathbf{B}_{eu}^{\top}$  is  $\mathcal{O}(n'd'h')$ , we can directly measure the unpack ratio
+
+$$
+r = \frac {n ^ {\prime} d ^ {\prime} h ^ {\prime}}{n d h}
+$$
+
+![](images/86cec552095eeb0e2880d636b7e53e3e9d9ac3bd7df1da4ba09d776cd426001d.jpg)  
+Figure 8. Illustration of the bit representation of a matrix. Heavy-hitters have higher order non-zero bits. When a matrix contains heavy-hitters, its bit representation has a sparsity structure in the higher order bits as illustrated.
+
+Table 11. Averaged ratios of quantized GEMMs  $(\beta = 15)$  in linear layers on ViT-Large when using different strategies and a range of integer bit-widths  $b$  to the lowest bit-width possible.  
+
+<table><tr><td></td><td colspan="2">Integer Bits b</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td></tr><tr><td rowspan="9">X</td><td>Row</td><td>Row</td><td>7.24</td><td>3.80</td><td>2.63</td><td>2.22</td><td>1.54</td><td>1.43</td></tr><tr><td>Row</td><td>Col</td><td>194.89</td><td>27.52</td><td>10.46</td><td>4.31</td><td>1.62</td><td>1.43</td></tr><tr><td>Row</td><td>Both</td><td>85.92</td><td>13.80</td><td>6.22</td><td>2.76</td><td>1.56</td><td>1.43</td></tr><tr><td>Col</td><td>Row</td><td>19.27</td><td>4.85</td><td>3.06</td><td>1.46</td><td>1.25</td><td>1.12</td></tr><tr><td>Col W</td><td>Col</td><td>526.31</td><td>35.86</td><td>12.22</td><td>2.81</td><td>1.32</td><td>1.13</td></tr><tr><td>Col</td><td>Both</td><td>27.06</td><td>13.31</td><td>7.59</td><td>1.78</td><td>1.26</td><td>1.13</td></tr><tr><td>Both</td><td>Row</td><td>7.62</td><td>3.39</td><td>2.58</td><td>1.64</td><td>1.42</td><td>1.33</td></tr><tr><td>Both</td><td>Col</td><td>206.32</td><td>24.45</td><td>10.27</td><td>3.15</td><td>1.49</td><td>1.34</td></tr><tr><td>Both</td><td>Both</td><td>79.19</td><td>11.16</td><td>6.09</td><td>2.01</td><td>1.43</td><td>1.34</td></tr><tr><td></td><td>Mix</td><td></td><td>6.29</td><td>2.98</td><td>2.24</td><td>1.40</td><td>1.23</td><td>1.11</td></tr></table>
+
+to understand by how much the cost for low bit-width GEMMs increases. We uses LLaMA-7B to study the unpack ratio  $r$  when using different unpacking strategies (Tab. 9). Note that since unpacking both requires keeping track of the OB count in each row and column vector which is not as fast as the other two strategies, we only use it for unpacking parameters  $\mathbf{W}$  for inference since it can be performed once when loading the model. The Mix in Tab. 9 means that for each GEMM, we compare different strategies and choose the optimal strategy that results in the smallest unpack ratio. We note that the unpack ratios of computing  $\mathbf{Y}$  and  $\mathbf{P}$  are quite reasonable, but the ratios of computing  $\mathbf{O}$  is larger. This is expected since the large outliers of the self-attention matrix  $\mathbf{M}$  mainly concentrate in the diagonal (Beltagy et al., 2020). We also study the unpack ratios of each type of quantized GEMMs at different training phases, and show the results of Mix strategy in Tab. 10. The ratios stay relatively unchanged as training progresses. Also, we can observe similar high unpack ratio when computing  $\mathbf{O}$  and  $\nabla_{\mathbf{V}}$  since these GEMMs involve the self-attention matrix  $\mathbf{M}$ . Lastly, we verify that we can unpack matrices to arbitrarily low integer matrices (Tab. 11). The 2-bit setting is the lowest bit width that can be used for symmetric signed integers  $\{-1,0,1\}$ . In §A.1, we also describe a small adjustment to reduce this 2-bit setting to 1-bit arithmetic.
+
+# 4.3. Limitations
+
+To simplify the presentation, we used the simplest RTN quantization, which might not deliver the optimal performance. More sophisticated techniques are likely to further improve the results. For example, we may be able to remove the demands of large  $\beta$  for the set  $\{\nabla_{\mathbf{Y}},\nabla_{\mathbf{P}},\nabla_{\mathbf{O}}\}$  for ViT
+
+training. The current unpacking strategies cannot handle the self-attention matrix  $\mathbf{M}$  efficiently since the outliers mainly concentrate on the diagonal region rather than rows or columns; this needs further investigation.
+
+# 5. Conclusions
+
+In this paper, we verify the efficacy of integer GEMMs in both training and inference for Transformer-based models in language modeling and vision. A simple RTN quantization strategy works well compared to baselines. But the presence of large outliers/heavy-hitters makes it difficult to make use of efficient low bit-width integer GEMMs since these outliers are much larger than the representable range of low bit-width integers. We take a "multi-resolution" view (Zeng et al., 2022) in how we extract a spectrum of bit-width tradeoffs. This is loosely similar to sparsity but here, instead of making a zero versus non-zero distinction between the entries, our heavy-hitters (which need higher bit-width representations) are analogous to "non-sparse" entries (as illustrated in Fig. 8). To handle high bit-width heavy-hitters, we develop an algorithm to unpack integer matrices that contains arbitrarily large values to slightly larger matrices with the property that all values lie within the representable range of low bit-width integers and a procedure to obtain the GEMM output of original matrices using only low bit-width integer GEMMs on the unpacked matrices followed by some scaling (using bit shifting) and accumulation. Our algorithm can greatly simplify the design of hardware and improve the power efficiency by only supporting low bit-width integer GEMMs for both training and inference.
+
+# Acknowledgments
+
+The authors are grateful to the doctors, nurses, and staff at the UW Neuro ICU for their exceptional care and support in aiding one of the author's recovery during the preparation of this paper. This work was supported in part by funding from the Vilas Board of Trustees and UW-Madison Office of the Vice Chancellor for Research and Graduate Education. The authors thank Michael Davies for many helpful discussions.
+
+# Impact Statement
+
+This paper presents an approach to use low precision computing for training and inference. The goal is to accelerate computing speed and reduce energy consumption and thus promote more sustainable computational practices. We do not identify any specific societal implications or ethical considerations that must be highlighted.
+
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+
+Table 12. Standard deviation vs percentile when removing largest outliers.  $\mathbf{X}$  has  ${2.25} \times  {10}^{7}$  entries and  $\mathbf{W}$  has  ${1.68} \times  {10}^{7}$  entries.  
+
+<table><tr><td colspan="2">Number of Largest Outliers Removed</td><td>0</td><td>10</td><td>102</td><td>103</td></tr><tr><td rowspan="2">W</td><td>Standard Deviation</td><td>0.0082</td><td>0.0082</td><td>0.0082</td><td>0.0082</td></tr><tr><td>95-Percentile</td><td>0.0177</td><td>0.0177</td><td>0.0177</td><td>0.0177</td></tr><tr><td rowspan="2">X</td><td>Standard Deviation</td><td>0.0330</td><td>0.0327</td><td>0.0320</td><td>0.0214</td></tr><tr><td>95-Percentile</td><td>0.0280</td><td>0.0280</td><td>0.0280</td><td>0.0278</td></tr></table>
+
+# A. Appendix
+
+We provide more details about design choices and experimental setup as well as additional experiments that were omitted from the main paper due to space.
+
+# A.1. Rationale behind certain design choices
+
+Why Use Percentiles? We need a way of mapping the actual range of values in a floating point matrix to an integer range. In this process, we should ensure that most values fall within the desired range and fill up the representable range as much as possible, so we need a statistic to estimate the range of values in a FP matrix. We compared percentile and standard deviation and inspected different parameter matrices  $\mathbf{W}$  and the corresponding inputs  $\mathbf{X}$  in the LLaMA-7B model (Touvron et al., 2023). The outlier problem in  $\mathbf{W}$  is moderate: we can see in Tab. 12, that both standard deviation and percentile estimates are stable across columns. On the other hand, the outliers in  $\mathbf{X}$  is a bit more problematic and includes a few entries that are much larger than the non-outliers. The estimation of standard deviation varies much more as shown in Tab. 12: even removing an extremely small subset of the largest outliers can sizable alter the estimates. In contrast, the percentile is more stable. As a result, we choose percentile as the estimation of value range.
+
+Can We Use as Low as 1-bit Encoding? As discussed in §4.2, we use 2 bits to encode  $\{-1,0, + 1\}$  for symmetric encoding, so the 2 bit setting is the lowest bit width that our method supports. However, with small adjustments, it is possible in principle to derive a 1-bit encoding scheme for this 2-bit encoding. Given A and B matrices whose values are  $\{-1,0, + 1\}$ , we can easily decompose
+
+$$
+\mathbf {A} = \mathbf {A} _ {p} - \mathbf {A} _ {n}
+$$
+
+$$
+\mathbf {B} = \mathbf {B} _ {p} - \mathbf {B} _ {n}
+$$
+
+where  $\mathbf{A}_p, \mathbf{A}_n, \mathbf{B}_p, \mathbf{B}_n$  consist of values  $\{0,1\}$ . Then,
+
+$$
+\mathbf {A} \mathbf {B} ^ {\intercal} = \mathbf {A} _ {p} \mathbf {B} _ {p} ^ {\intercal} - \mathbf {A} _ {n} \mathbf {B} _ {p} ^ {\intercal} - \mathbf {A} _ {n} \mathbf {B} _ {p} ^ {\intercal} + \mathbf {A} _ {n} \mathbf {B} _ {n} ^ {\intercal}
+$$
+
+becomes four 1-bit GEMMs, so 1-bit encoding is feasible in this sense.
+
+One benefit of 1 bit quantization is that multiplication in GEMM becomes logic AND and accumulation becomes counting the number of 1's. Notice that similar benefits can be exploited in the 2-bit encoding for  $-1,0, + 1$ . The mapping between 2-bit binary representation and decimal numbers is  $00_{b} = 0,01_{b} = 1,10_{b} = 0,11_{b} = -1$ . We call the left bit a sign bit and right bit a value bit. We note that the multiplication becomes logical XOR in the sign bit and logical AND in the value bit, and accumulation becomes counting the number of 1's in value bit and then subtracting the number of  $11_{b}$ . We have not performed experiments evaluating this approach yet.
+
+# A.2. Analyzing the Error of Rounding to Nearest
+
+Our method consists of two main steps: 1. Rounding to Nearest (RTN) integer, which maps floating-point entries in the matrices to integer entries; and 2. Our main algorithm IM-Unpack, which unpacks integer matrices to the desired bit-width. As described at the end of §3, Step 2 does not incur any loss and so, can be ignored in the discussion below.
+
+We can perform a simple technical analysis for Step 1. Since the scaling in (1) and (2) will not affect the analysis, we can assume, without loss of generality, that the scaling is simply set to 1. We are interested in the error
+
+$$
+\mathbf {E} = \mathbf {A B} ^ {\intercal} - \mathbf {A} _ {q} \mathbf {B} _ {q} ^ {\mathbf {T}}
+$$
+
+We can examine a specific entry in the product C. Let  $\mathbf{u}$  and  $\mathbf{v}$  be the rows of A and B, respectively, whose inner product  $\sum_{i}^{d}\mathbf{u}[i]\mathbf{v}[i]$  is a specific entry in C. In Step 1, each  $\mathbf{u}[i]$  and  $\mathbf{v}[i]$  is rounded to the nearest integer,  $\hat{\mathbf{u}}[i]$  and  $\hat{\mathbf{v}}[i]$ , resulting in a deviation of at most  $\pm 0.5$  from the original entries.
+
+Let the error incurred in this  $i$ -th term due to rounding be denoted as
+
+$$
+\epsilon [ i ] = \hat {\mathbf {u}} [ i ] \hat {\mathbf {v}} [ i ] - \mathbf {u} [ i ] \mathbf {v} [ i ]
+$$
+
+After some calculations, we see that the product  $\hat{\mathbf{u}}[i]\hat{\mathbf{v}}[i]$  of the rounded values will deviate by at most  $\pm 0.5(|\mathbf{u}[i]| + |\mathbf{v}[i]| + 0.5)$  from the product  $\mathbf{u}[i]\mathbf{v}[i]$  of the original values. Let  $\mathbf{c}[i] = 0.5(|\mathbf{u}[i]| + |\mathbf{v}[i]| + 0.5)$ , and let  $c$  be a vector whose  $i$ -th entry is  $\mathbf{c}[i]$ . Then, it follows that  $-\mathbf{c}[i] \leq \epsilon[i] \leq \mathbf{c}[i]$ . We can now examine the total error in the inner product  $\sum_{i} \mathbf{u}[i]\mathbf{v}[i]$ .
+
+When  $\epsilon [i] > 0$ , this term over-contributes, and when  $\epsilon [i] < 0$ , it under-contributes. Assuming that these two scenarios are equally likely, the cumulative error in  $\sum_{i}\mathbf{u}[i]\mathbf{v}[i]$  can be represented as a sum involving a Rademacher-distributed variable  $X$  (half-half chance of being  $+1$  or  $-1$ ), modulated by  $\epsilon [i]$  as coefficients. Our interest is in the error  $\sum_{i}\mathbf{x}[i]\epsilon [i]$ , where  $\mathbf{x}[i]$  is a set of random variables following a Rademacher distribution. We want to check whether the probability of a bad event, where this sum (error) exceeds a suitably high threshold, decays quickly as the threshold increases. This is related to Tomaszewski's conjecture, and in particular, we now know that
+
+$$
+\mathcal {P} (\sum_ {i} \mathbf {x} [ i ] \epsilon [ i ] > t \| \epsilon \| _ {2}) <   \exp (- t ^ {2} / 2) \quad \mathrm {a n d} \quad \mathcal {P} (| \sum_ {i} \mathbf {x} [ i ] \epsilon [ i ] | > t \| \epsilon \| _ {2}) <   2 \exp (- t ^ {2} / 2)
+$$
+
+implying an exponential decay in the probability of the error exceeding a threshold as desired. By incorporating the maximal possible error, denoted as  $\mathbf{c}[i]$ , we obtain:
+
+$$
+\mathcal {P} (\mid \sum_ {i} \mathbf {x} [ i ] \epsilon [ i ] \mid > t \left\| \mathbf {c} \right\| _ {2}) <   2 \exp (- t ^ {2} / 2)
+$$
+
+Since  $-\mathbf{c}[i] \leq \epsilon [i] \leq \mathbf{c}[i]$ , we can also use Hoeffding's inequality, which is less tight but has a similar form of dependency.
+
+However, the concentration bounds do not adequately explain the strong empirical behavior. To assess impact on the impact on training, if desired, we can use results such as the one in (De Sa et al., 2018) for convergence analysis but with some modifications to the optimization. For example, if we use LP-SVRG in (De Sa et al., 2018) and just use stochastic rounding instead of deterministic RTN (fixed point arithmetic in (De Sa et al., 2018) is the same as the integer arithmetic with scaling that we use in (2)), then under the assumption that the objective function is  $\mu$ -strongly convex with respect to parameters (and an additional L-Lipschitz requirement), with an additional cost of variance reduction, we can get a linear convergence rate.
+
+# A.3. Experimental Details and Additional Results
+
+Additional details of training experiments. We run all of our experiments on NVIDIA RTX 3090. Below we provide the training hyperparameters. RoBERTa-Small is a 4-layer Transformer encoder whose model dimension is 512, hidden dimension is 2048, and number of heads is 8. For RoBERTa-Small models, we train each model for 200K steps with batches of 256 512-length sequences. We use an AdamW optimizer with 1e-4 learning rate, 10,000 warm-up steps, 0.01 weight decay, and linear decay. For RoBERTa-Base models, we train each model for 300K steps with batches of 128 512-length sequences. We use an AdamW optimizer with 5e-5 learning rate, 10,000 warm-up steps, 0.01 weight decay, and linear decay. We use timm to train our ViT-Small models. The hyperparameters of all experiments are the same: batch size 1024, optimizer AdamW, learning rate 0.001, weight decay 0.05, augmentation rand-m9-mstd0.5-inc1, mixup 0.8, cutmix 1.0.
+
+Unpack Ratios of ViT-Large. Similar to Tab. 9 in the main paper, we also evaluate the unpack ratios of ViT-Large, which are shown in Tab. 13. The overall results are similar to what was observed in unpack ratios of LLaMA-7B (Tab. 9).
+
+More Empirical Results on LLM Quantization. To evaluate how well RTN works for inference of different models and different model sizes, beside the experiments shown in the main text, we also run experiments on LLaMA-13B (Touvron et al., 2023), Mistral-7B (Jiang et al., 2023), and Phi-2 (Javaheripi et al., 2023). The results are summarized in Tab. 14, Tab. 15, and Tab. 16. To minimize code change, we only evaluate the quantization of linear layers, consistent with quantization-focused papers for Mistral-7B and and Phi-2.
+
+More Empirical Results on Training. We wanted to understand how well RTN works for training of larger models but wanted to avoid allocating a significant compute budget to perform such an experiment. So, we resorted to finetuning a T5-Large model on the first 50K instance of the XSum summarization dataset (Narayan et al., 2018) using BF16 and RTN, and show the results in Fig. 9. The validation metrics are shown in Tab. 17. We can draw a similar conclusion as in the main paper that RTN quantized training gives similar results to BF16 training.
+
+Table 13. Averaged unpack ratios of each type of GEMMs in ViT-Large: linear layers (computing  $\mathbf{Y}$ ), attention score (computing  $\mathbf{P}$ ), and attention output (computing  $\mathbf{O}$ ) when using different unpack strategies and integer bit length  $b$  under quantization  $\beta$  settings. AS: Attention Score, AO: Attention Output.  
+
+<table><tr><td></td><td colspan="3">β</td><td colspan="3">5</td><td colspan="3">7</td><td colspan="3">15</td></tr><tr><td></td><td colspan="3">Integer Bits b</td><td>3</td><td>4</td><td>5</td><td>3</td><td>4</td><td>5</td><td>4</td><td>5</td><td>6</td></tr><tr><td rowspan="6">Linear(Y)</td><td rowspan="6">X</td><td>Row</td><td>Row</td><td>2.90</td><td>2.00</td><td>1.55</td><td>3.01</td><td>2.38</td><td>1.59</td><td>2.63</td><td>2.22</td><td>1.54</td></tr><tr><td>Row</td><td>Col</td><td>10.97</td><td>2.32</td><td>1.56</td><td>12.34</td><td>4.12</td><td>1.65</td><td>10.46</td><td>4.31</td><td>1.62</td></tr><tr><td>Row</td><td>Both</td><td>6.24</td><td>2.08</td><td>1.55</td><td>6.84</td><td>2.82</td><td>1.60</td><td>6.22</td><td>2.76</td><td>1.56</td></tr><tr><td>Col</td><td>Row</td><td>2.33</td><td>1.39</td><td>1.20</td><td>3.38</td><td>1.51</td><td>1.26</td><td>3.06</td><td>1.46</td><td>1.25</td></tr><tr><td>Col</td><td>Col</td><td>8.89</td><td>1.64</td><td>1.22</td><td>13.97</td><td>2.63</td><td>1.32</td><td>12.22</td><td>2.81</td><td>1.32</td></tr><tr><td>Col</td><td>Both</td><td>4.99</td><td>1.44</td><td>1.20</td><td>7.99</td><td>1.76</td><td>1.27</td><td>7.59</td><td>1.78</td><td>1.26</td></tr><tr><td></td><td></td><td>Mix</td><td></td><td>2.60</td><td>1.27</td><td>1.06</td><td>2.44</td><td>1.40</td><td>1.15</td><td>2.10</td><td>1.42</td><td>1.16</td></tr><tr><td rowspan="4">AS(P)</td><td rowspan="4">Q</td><td>Row</td><td>Row</td><td>1.84</td><td>1.07</td><td>1.00</td><td>2.01</td><td>1.35</td><td>1.00</td><td>1.99</td><td>1.40</td><td>1.00</td></tr><tr><td>Row</td><td>Col</td><td>3.06</td><td>1.07</td><td>1.00</td><td>6.38</td><td>1.39</td><td>1.00</td><td>6.34</td><td>1.46</td><td>1.00</td></tr><tr><td>Col</td><td>Row</td><td>1.34</td><td>1.01</td><td>1.00</td><td>2.50</td><td>1.04</td><td>1.00</td><td>2.49</td><td>1.05</td><td>1.00</td></tr><tr><td>Col</td><td>Col</td><td>2.39</td><td>1.01</td><td>1.00</td><td>8.25</td><td>1.08</td><td>1.00</td><td>8.24</td><td>1.10</td><td>1.00</td></tr><tr><td></td><td></td><td>Mix</td><td></td><td>1.33</td><td>1.01</td><td>1.00</td><td>1.91</td><td>1.04</td><td>1.00</td><td>1.90</td><td>1.04</td><td>1.00</td></tr><tr><td rowspan="4">AO(O)</td><td rowspan="4">M</td><td>Row</td><td>Row</td><td>2.84</td><td>2.07</td><td>1.65</td><td>3.07</td><td>2.24</td><td>1.80</td><td>2.56</td><td>2.11</td><td>1.78</td></tr><tr><td>Row</td><td>Col</td><td>5.78</td><td>2.12</td><td>1.65</td><td>11.12</td><td>2.47</td><td>1.81</td><td>9.22</td><td>2.35</td><td>1.79</td></tr><tr><td>Col</td><td>Row</td><td>3.98</td><td>2.26</td><td>1.64</td><td>4.69</td><td>2.57</td><td>1.81</td><td>3.58</td><td>2.33</td><td>1.77</td></tr><tr><td>Col</td><td>Col</td><td>8.42</td><td>2.29</td><td>1.64</td><td>16.97</td><td>2.83</td><td>1.81</td><td>12.92</td><td>2.60</td><td>1.77</td></tr><tr><td></td><td></td><td>Mix</td><td></td><td>2.25</td><td>1.61</td><td>1.32</td><td>2.55</td><td>1.77</td><td>1.42</td><td>2.22</td><td>1.70</td><td>1.42</td></tr></table>
+
+Table 14. Baseline comparison on LLaMA-13B when quantize computation in all linear layers.  
+
+<table><tr><td>Method</td><td>β</td><td>Type</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HellaSwag</td><td>PIQA</td><td>WinoGrande</td></tr><tr><td>Full-Precision</td><td>-</td><td>BF16</td><td>48.0</td><td>79.5</td><td>80.6</td><td>60.0</td><td>79.2</td><td>72.1</td></tr><tr><td rowspan="2">SmoothQuant</td><td>-</td><td>INT8</td><td>45.5</td><td>76.3</td><td>76.5</td><td>58.0</td><td>78.0</td><td>72.1</td></tr><tr><td>-</td><td>INT4</td><td>25.1</td><td>49.9</td><td>57.6</td><td>56.0</td><td>61.3</td><td>52.6</td></tr><tr><td>LLM-FP4</td><td>-</td><td>FP4</td><td>39.9</td><td>71.7</td><td>71.9</td><td>53.3</td><td>74.8</td><td>66.7</td></tr><tr><td rowspan="4">RTN</td><td>5</td><td>INT</td><td>37.6</td><td>70.0</td><td>69.1</td><td>51.9</td><td>72.4</td><td>64.6</td></tr><tr><td>7</td><td>INT</td><td>44.1</td><td>76.1</td><td>73.5</td><td>57.3</td><td>76.7</td><td>67.6</td></tr><tr><td>15</td><td>INT</td><td>46.9</td><td>78.8</td><td>79.4</td><td>59.0</td><td>78.2</td><td>72.5</td></tr><tr><td>31</td><td>INT</td><td>48.0</td><td>79.7</td><td>80.2</td><td>59.9</td><td>78.0</td><td>71.3</td></tr></table>
+
+Table 15. Baseline comparison on LLaMA-13B when quantize all GEMMs in a Transformer.  
+
+<table><tr><td>Method</td><td>β</td><td>Type</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HellaSwag</td><td>PIQA</td><td>WinoGrande</td></tr><tr><td>Full-Precision</td><td>-</td><td>BF16</td><td>48.0</td><td>79.5</td><td>80.6</td><td>60.0</td><td>79.2</td><td>72.1</td></tr><tr><td rowspan="4">RTN</td><td>5</td><td>INT</td><td>25.1</td><td>44.4</td><td>54.8</td><td>37.7</td><td>57.9</td><td>52.0</td></tr><tr><td>7</td><td>INT</td><td>38.0</td><td>66.9</td><td>70.1</td><td>53.3</td><td>72.5</td><td>64.2</td></tr><tr><td>15</td><td>INT</td><td>45.9</td><td>77.6</td><td>80.0</td><td>59.5</td><td>77.5</td><td>71.5</td></tr><tr><td>31</td><td>INT</td><td>47.9</td><td>79.3</td><td>80.0</td><td>60.5</td><td>78.6</td><td>70.9</td></tr></table>
+
+Table 16. RTN performance on Mistral-7B and Phi-2 when quantize computation in all linear layers.  
+
+<table><tr><td></td><td>Method</td><td>β</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HellaSwag</td><td>PIQA</td><td>WinoGrande</td></tr><tr><td rowspan="5">Mistral7B</td><td>Full-Precision</td><td>-</td><td>50.3</td><td>80.9</td><td>83.6</td><td>61.3</td><td>80.7</td><td>73.8</td></tr><tr><td rowspan="4">RTN</td><td>5</td><td>38.1</td><td>70.5</td><td>69.9</td><td>53.9</td><td>73.3</td><td>61.4</td></tr><tr><td>7</td><td>44.9</td><td>75.0</td><td>76.0</td><td>58.7</td><td>77.8</td><td>68.6</td></tr><tr><td>15</td><td>48.8</td><td>79.7</td><td>80.3</td><td>60.8</td><td>79.6</td><td>73.2</td></tr><tr><td>31</td><td>50.3</td><td>80.1</td><td>83.5</td><td>61.5</td><td>80.7</td><td>74.4</td></tr><tr><td rowspan="5">Phi-2</td><td>Full-Precision</td><td>-</td><td>20.6</td><td>26.1</td><td>41.3</td><td>25.8</td><td>54.3</td><td>49.3</td></tr><tr><td rowspan="4">RTN</td><td>5</td><td>22.1</td><td>26.7</td><td>41.5</td><td>25.6</td><td>52.3</td><td>48.1</td></tr><tr><td>7</td><td>21.3</td><td>25.8</td><td>40.9</td><td>25.8</td><td>53.9</td><td>49.5</td></tr><tr><td>15</td><td>21.3</td><td>27.3</td><td>45.4</td><td>25.8</td><td>53.4</td><td>48.8</td></tr><tr><td>31</td><td>21.0</td><td>25.8</td><td>40.8</td><td>25.7</td><td>53.0</td><td>50.7</td></tr></table>
+
+![](images/a51f2200fc92967fbff6a6ba3dbde1003b3f61dfbe56ac9e54b43a6ef5f2cb9e.jpg)  
+Figure 9. Loss curves of T5-Large finetuning on  $1/4$  of XSum dataset for 1 epoch.
+
+Table 17. Validation metrics of T5-Large finetuning on 1/4 of XSum dataset for 1 epoch.  
+
+<table><tr><td>Method</td><td>β</td><td>Type</td><td>Loss</td><td>Rouge1</td><td>Rouge2</td><td>Rougel</td><td>Rougelsum</td></tr><tr><td>Full-Precision</td><td>-</td><td>BF16</td><td>1.65</td><td>36.12</td><td>13.00</td><td>29.21</td><td>29.20</td></tr><tr><td>RTN</td><td>31</td><td>INT</td><td>1.66</td><td>36.03</td><td>13.83</td><td>29.03</td><td>29.04</td></tr></table>
+
+Table 18. Inference: end to end baseline comparison combining information from Tab. 2 and Tab. 9 for LLaMA-7B.  
+
+<table><tr><td>Method</td><td>β</td><td>Type</td><td>r</td><td>bits × r/16</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HS</td><td>PIQA</td><td>WG</td></tr><tr><td>Full-Precision</td><td>-</td><td>BF16</td><td>1</td><td>1</td><td>43.1</td><td>76.3</td><td>77.8</td><td>57.2</td><td>78.0</td><td>68.8</td></tr><tr><td>LLM.int8()</td><td>-</td><td>INT8+FP16</td><td>1</td><td>&gt;0.5</td><td>43.8</td><td>75.5</td><td>77.8</td><td>57.4</td><td>77.6</td><td>68.7</td></tr><tr><td>SmoothQuant</td><td>-</td><td>INT8</td><td>1</td><td>0.5</td><td>37.4</td><td>74.4</td><td>74.0</td><td>55.0</td><td>77.5</td><td>69.6</td></tr><tr><td>LLM-QAT</td><td>-</td><td>INT4</td><td>1</td><td>0.25</td><td>30.2</td><td>50.3</td><td>63.5</td><td>55.6</td><td>64.3</td><td>52.9</td></tr><tr><td>LLM-FP4</td><td>-</td><td>FP4</td><td>1</td><td>0.25</td><td>33.6</td><td>65.9</td><td>64.2</td><td>47.8</td><td>73.5</td><td>63.7</td></tr><tr><td rowspan="10">RTN+IMUnpack</td><td>5</td><td>INT4</td><td>1.27</td><td>0.318</td><td>39.3</td><td>72.8</td><td>69.9</td><td>53.4</td><td>74.9</td><td>66.4</td></tr><tr><td>5</td><td>INT5</td><td>1.06</td><td>0.331</td><td>39.3</td><td>72.8</td><td>69.9</td><td>53.4</td><td>74.9</td><td>66.4</td></tr><tr><td>7</td><td>INT4</td><td>1.41</td><td>0.352</td><td>42.6</td><td>73.9</td><td>72.3</td><td>55.9</td><td>77.0</td><td>67.4</td></tr><tr><td>7</td><td>INT5</td><td>1.14</td><td>0.356</td><td>42.6</td><td>73.9</td><td>72.3</td><td>55.9</td><td>77.0</td><td>67.4</td></tr><tr><td>11</td><td>INT5</td><td>1.27</td><td>0.397</td><td>43.9</td><td>76.1</td><td>77.3</td><td>56.3</td><td>77.3</td><td>69.3</td></tr><tr><td>11</td><td>INT6</td><td>1.07</td><td>0.401</td><td>43.9</td><td>76.1</td><td>77.3</td><td>56.3</td><td>77.3</td><td>69.3</td></tr><tr><td>15</td><td>INT5</td><td>1.40</td><td>0.438</td><td>43.0</td><td>75.7</td><td>77.5</td><td>57.0</td><td>78.0</td><td>69.2</td></tr><tr><td>15</td><td>INT6</td><td>1.15</td><td>0.431</td><td>43.0</td><td>75.7</td><td>77.5</td><td>57.0</td><td>78.0</td><td>69.2</td></tr><tr><td>31</td><td>INT6</td><td>1.42</td><td>0.532</td><td>42.7</td><td>76.1</td><td>76.1</td><td>57.3</td><td>77.3</td><td>69.3</td></tr><tr><td>31</td><td>INT7</td><td>1.16</td><td>0.507</td><td>42.7</td><td>76.1</td><td>76.1</td><td>57.3</td><td>77.3</td><td>69.3</td></tr></table>
+
+End-to-End Quantization Baseline Comparison. An end-to-end baseline comparison requires combining information from Tab. 1, Tab. 2, and Tab. 3 with Tab. 9 and Tab. 13. As we change the bit-width of integers that we will use, the unpack ratio  $r$  can be determined from Tab. 9 and 13. Since after unpacking, the calculation is similar or the same as standard GEMMs (except that the input sizes are different), we could first estimate the runtime of GEMMs, and discuss the runtime overhead of unpacking later.
+
+Note that INT2, INT3, INT5, INT6, INT7 GEMM implementations are not yet publicly available. Although INT4 and INT8 are now available in NVIDIA's CUTLASS, their integration (e.g., in PyTorch) is ongoing (more discussion below). As a result, we can only estimate the runtime of these GEMMs based on publicly available information. According to NVIDIA, INT4 offers a  $4 \times$  performance bump compared to FP16, and INT8 is  $2 \times$  performance compared to FP16. We measured the runtime of INT4 and INT8 GEMMs in CUTLASS standalone, and the performance is indeed  $4 \times$  and  $2 \times$ , respectively, compared to FP16. We can reasonably assume that the performance of  $x$ -bit INT GEMMs can approach  $\frac{16}{x}$  performance compared to FP16, and the runtime approaches  $\frac{x}{16}$  of the runtime of FP16. Further, the runtime of GEMMs is linearly proportional to  $n \times d \times h$  for matrices of size  $n \times d$  and  $h \times d$ . When comparing runtime of GEMMs for different sizes, the ratio  $r = \frac{n \times d \times h}{n' \times d' \times h'}$  can give an accurate estimate of the runtime comparison when these GEMMs finally become available (we verified this linear relationship in FP16 GEMMs for a contiguous range of matrix sizes, not just a power of 2). As a result, we can use  $\frac{x r}{16}$  as a proxy for the runtime of our GEMMs. In Tab. 18, we provided  $\frac{x r}{16}$  values (bits  $\times r / 16$  column) for comparing across different baselines and the corresponding model accuracy. We see that our method has better model accuracy with faster runtimes under the proposed proxy when not accounting for the runtime overhead of quantization.
+
+The only remaining task is to check the runtime overhead of quantization and unpacking. Since the goal is to speed up
+
+Table 19. Runtime overhead of UnpackColumn and UnpackRow compared to GEMM and Clone for different bit-widths, unpack ratios  $r$  , and input matrix shapes.  
+
+<table><tr><td>Shape n × d × h</td><td>GEMM FP32</td><td>FP16</td><td>Clone FP32</td><td colspan="8">UnpackCol FP32</td><td colspan="5">UnpackRow FP32</td></tr><tr><td>Bit Width</td><td>-</td><td>-</td><td>-</td><td colspan="2">2</td><td colspan="2">4</td><td colspan="2">8</td><td colspan="2">2</td><td colspan="2">4</td><td colspan="2">8</td><td></td></tr><tr><td>Unpack Ratio</td><td>-</td><td>-</td><td>-</td><td>1.21</td><td>1.44</td><td>1.21</td><td>1.44</td><td>1.21</td><td>1.44</td><td>1.21</td><td>1.44</td><td>1.21</td><td>1.44</td><td>1.21</td><td>1.44</td><td></td></tr><tr><td>213×213×213</td><td>63.4</td><td>5.0</td><td>0.8</td><td>1.0</td><td>0.9</td><td>1.0</td><td>1.0</td><td>1.1</td><td>1.1</td><td>1.1</td><td>1.1</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td></td></tr><tr><td>213×213×214</td><td>116.9</td><td>10.4</td><td>1.2</td><td>1.4</td><td>1.3</td><td>1.4</td><td>1.4</td><td>1.5</td><td>1.5</td><td>1.5</td><td>1.4</td><td>1.4</td><td>1.4</td><td>1.5</td><td>1.5</td><td></td></tr><tr><td>213×213×215</td><td>252.7</td><td>25.1</td><td>2.0</td><td>2.2</td><td>2.2</td><td>2.2</td><td>2.2</td><td>2.4</td><td>2.5</td><td>2.5</td><td>2.2</td><td>2.3</td><td>2.3</td><td>2.4</td><td>2.4</td><td></td></tr><tr><td>214×213×213</td><td>126.3</td><td>10.5</td><td>1.2</td><td>1.4</td><td>1.4</td><td>1.4</td><td>1.4</td><td>1.5</td><td>1.5</td><td>1.5</td><td>1.4</td><td>1.4</td><td>1.4</td><td>1.5</td><td>1.5</td><td></td></tr><tr><td>214×213×214</td><td>253.8</td><td>20.9</td><td>1.6</td><td>1.8</td><td>1.8</td><td>1.8</td><td>1.8</td><td>2.0</td><td>2.0</td><td>2.0</td><td>1.8</td><td>1.8</td><td>1.9</td><td>1.9</td><td>2.0</td><td></td></tr><tr><td>214×213×215</td><td>510.8</td><td>60.1</td><td>2.4</td><td>2.6</td><td>2.6</td><td>2.6</td><td>2.7</td><td>2.9</td><td>3.0</td><td>3.0</td><td>2.6</td><td>2.7</td><td>2.7</td><td>2.9</td><td>2.9</td><td></td></tr><tr><td>215×213×213</td><td>255.5</td><td>20.9</td><td>2.0</td><td>2.2</td><td>2.2</td><td>2.2</td><td>2.2</td><td>2.4</td><td>2.5</td><td>2.4</td><td>2.2</td><td>2.3</td><td>2.3</td><td>2.4</td><td>2.5</td><td></td></tr><tr><td>215×213×214</td><td>514.3</td><td>60.0</td><td>2.4</td><td>2.6</td><td>2.6</td><td>2.6</td><td>2.7</td><td>2.9</td><td>2.9</td><td>3.0</td><td>2.6</td><td>2.7</td><td>2.7</td><td>2.9</td><td>2.9</td><td></td></tr><tr><td>215×213×215</td><td>957.7</td><td>121.5</td><td>3.2</td><td>3.4</td><td>3.4</td><td>3.5</td><td>3.5</td><td>3.8</td><td>3.9</td><td>3.9</td><td>3.5</td><td>3.6</td><td>3.6</td><td>3.8</td><td>3.9</td><td></td></tr></table>
+
+Table 20. Inference: estimated speedup of the entire model for LLaMA-7B.  
+
+<table><tr><td>Method</td><td>β</td><td>Type</td><td>r</td><td>bits × r/16</td><td>Speedup</td><td>ARC-c</td><td>ARC-e</td><td>BoolQ</td><td>HS</td><td>PIQA</td><td>WG</td></tr><tr><td>Full-Precision</td><td>-</td><td>BF16</td><td>1</td><td>1</td><td>-</td><td>43.1</td><td>76.3</td><td>77.8</td><td>57.2</td><td>78.0</td><td>68.8</td></tr><tr><td rowspan="5">RTN+IMUnpack</td><td>5</td><td>INT4</td><td>1.27</td><td>0.318</td><td>82%</td><td>39.3</td><td>72.8</td><td>69.9</td><td>53.4</td><td>74.9</td><td>66.4</td></tr><tr><td>7</td><td>INT4</td><td>1.41</td><td>0.352</td><td>74%</td><td>42.6</td><td>73.9</td><td>72.3</td><td>55.9</td><td>77.0</td><td>67.4</td></tr><tr><td>11</td><td>INT5</td><td>1.27</td><td>0.397</td><td>64%</td><td>43.9</td><td>76.1</td><td>77.3</td><td>56.3</td><td>77.3</td><td>69.3</td></tr><tr><td>15</td><td>INT6</td><td>1.15</td><td>0.431</td><td>57%</td><td>43.0</td><td>75.7</td><td>77.5</td><td>57.0</td><td>78.0</td><td>69.2</td></tr><tr><td>31</td><td>INT7</td><td>1.16</td><td>0.507</td><td>44%</td><td>42.7</td><td>76.1</td><td>76.1</td><td>57.3</td><td>77.3</td><td>69.3</td></tr></table>
+
+GEMMs, the quantization overhead must be small or else, the overhead will eliminate the gain of faster GEMMs. As a result, the code needs to be implemented as custom CUDA kernels to minimize the overhead. We implemented UnpackRow and UnpackCol CUDA kernels for FP32 matrices A and B. The supported unpack bit widths are 2, 4, 8 by packing the bit representation of 16 INT2 or 8 INT4 or 4 INT8 into an INT32. For other bit-widths, an efficient implementation would need additional hardware support. We expect the runtime of these kernels would be halved if the input date type is FP16.
+
+We compare these kernels to the tensor.clone() operation. The clone operation is the simplest pytorch operation that copies the values of a memory chunk (tensor) to another memory chunk (new tensor). The scaling and rounding operations in common quantization methods share a similar runtime as clone operation. We profiled UnpackRow and UnpackCol kernels (scaling and rounding are also computed within the kernel) for matrices of different sizes, bit-widths, and unpack ratio  $r$ . The runtime overhead of quantization and unpacking together is only between  $1 \times 1.5 \times 1$  of the runtime of clone operation applied to the matrices of the same size. The runtime ratio approaches 1 as the matrix size increases! We note that the kernels are still not fully optimized, and therefore, further code optimization might further lower the runtime. The overhead is very small relative to clone operation, so any other quantization scheme is unlikely to be much faster than our method in terms of overhead. For estimating the overhead compared to GEMMs, we also provide runtime for FP32 GEMM and FP16 GEMM for these matrix sizes alongside the runtimes for clone, UnpackRow, and UnpackCol as shown in Tab. 19.
+
+Estimated Overall Speedup of Entire Models. We have carefully calculated the speedup of the entire model runtime. We measure the breakdowns of different operators in LLaMA-7B and LLaMA-70B for an input of shape [16, 512] where 16 is the batch size and 512 is the sequence length. The overall model runtime of LLaMA-7B and LLaMA-70B is 1.84s and 18.92s, respectively. The overall runtime of GEMMs for Linear layers accounts for  $77.32\%$  and  $88.96\%$  of the overall model runtime, respectively. We exclude GEMMs in attention computation for simplicity which are accounted for in the much smaller  $22.68\%$  and  $11.04\%$  of the model's runtime (this percentage includes GEMMs in attention computation, activation functions, layer norm, residual connection, etc.). Let  $p$  be the percentage spent of GEMMs for Linear layers and  $h$  be the runtime ratio of the runtime overhead of IM-Unpack compared to the runtime of GEMMs, if we use  $\frac{x}{16}$  as runtime of INT- $x$  GEMM compared to FP16 GEMMs, we estimate the efficiency gain of using INT- $x$  for model computation. The speedup of the overall model runtime can be estimated via  $\frac{1}{\left(\frac{x r}{16} + h\right)p + (1 - p)}$  using Amdahl's law. We know from Tab. 19 that  $h < 0.1$ , so we can populate the speedup of the overall model runtime in Tab. 20 using the worst possible overhead  $h = 0.1$ . Since hardware and software support for different bit-widths is not publicly available, this estimate is based on calculations using actual measurements and profiling.

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+# IOI: Invisible One-Iteration Adversarial Attack on No-Reference Image- and Video-Quality Metrics
+
+Ekaterina Shumitskaya  $^{123}$  Anastasia Antsiferova  $^{21}$  Dmitriy Vatolin  $^{123}$
+
+# Abstract
+
+No-reference image- and video-quality metrics are widely used in video processing benchmarks. The robustness of learning-based metrics under video attacks has not been widely studied. In addition to having success, attacks on metrics that can be employed in video processing benchmarks must be fast and imperceptible. This paper introduces an Invisible One-Iteration (IOI) adversarial attack on no-reference image and video quality metrics. The proposed method uses two modules to ensure high visual quality and temporal stability of adversarial videos and runs for one iteration, which makes it fast. We compared our method alongside eight prior approaches using image and video datasets via objective and subjective tests. Our method exhibited superior visual quality across various attacked metric architectures while maintaining comparable attack success and speed. We made the code available on GitHub: https://github.com/katiashh/loi-attack.
+
+# 1. Introduction
+
+No-reference (NR) image- and video-quality assessment poses a significant challenge in computer vision. In contrast to full-reference (FR) quality metrics, NR metrics do not estimate the similarity of a distorted image or video to the original one but evaluate its visual appeal. The rapid integration of deep-learning-based NR image-and video-quality assessment metrics (Ying et al., 2020; Talebi & Milanfar, 2018; Su et al., 2020; Golestaneh et al., 2022) led to the importance of investigating their vulnerabilities to transformations of input. One of the most common types of input
+
+$^{1}$ ISP RAS Research Center for Trusted Artificial Intelligence, Moscow, Russia  $^{2}$ MSU Institute for Artificial Intelligence Moscow, Russia  $^{3}$ Lomonosov Moscow State University, Moscow, Russia. Correspondence to: Ekaterina Shumitskaya <ekaterina.shumitskaya@graphics.cs.msu.ru>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+transformations is adversarial attacks. In the case of image- and video-quality metrics, adversarial attacks are modifications of input images or videos that change the predicted quality score without significant influence on perceptual quality. Several studies (Zhang et al., 2022a; Korhonen & You, 2022a; Sang et al., 2022; Shumitskaya et al., 2022; 2023b; Meftah et al., 2023; Yang et al., 2024) unveiled vulnerabilities in NR image quality metrics when exposed to adversarial attacks.
+
+NR metrics are employed in various image- and video-processing benchmarks, such as super-resolution (PIRM-SR, 2019; Khrulkov & Babenko, 2021; Ma et al., 2017), video generation (VideoGeneration, 2024), video compression (Ghadiyaram et al., 2017). For some tasks, NR metrics show even better performance than FR metrics; for example, video super-resolution (SR-MSU, 2023) or video compression by new encoding standards such as H.266/VVC (Antsiferova et al., 2022). The developers of video-processing algorithms can integrate adversarial attacks on quality metrics into their methods to achieve higher positions in public benchmarks. Today, such cheating can be detected in benchmarks that publish subjective comparisons along with objective ones. For example, in MSU Codec Comparison 2021 (CC-MSU, 2021), the leaderboard by a learning-based metric VMAF, which was shown to be vulnerable to attacks (Siniukov et al., 2022; Zvezdakova et al., 2019), differs from a subjective one. Algorithms in these benchmarks compete in both visual quality and speed, and a high speed is essential for some real-life applications like universal encoding for video compression (from one to ten frames per second). Thus, to cheat in video processing benchmarks, it is profitable for an attacker to inject an imperceptible perturbation into the video without significantly decreasing the method speed.
+
+In the literature, most of the existing approaches evaluate NR metrics robustness in the image domain. However, to evaluate the robustness of NR metrics for videos, an adversary has to satisfy several essential conditions, making such a task more challenging:
+
+1. Quantitative success of an attack. For NR metrics, the success of an attack is measured by the amount of the metric's score change. Both decreasing and increasing
+
+a metric's score can be considered an attack; however, making a score higher has more practical applications.
+
+2. High speed of an attack. The attack must operate at high speed for practical viability. A slow attack holds limited practical significance, as its integration into video processing algorithms will greatly slow down the method and lower its position in benchmarks.  
+3. Temporal consistency of a transformed video. The per-frame implementation of adversarial attacks designed for images in videos leads to noticeable flickering effects that look suspicious in subjective comparisons.
+
+Our research aims to investigate the potential of injecting fast, invisible and temporally consistent adversarial attacks on NR quality metrics. This paper introduces the Invisible One-Iteration (IOI) adversarial attack for images and videos. To achieve high attack speed, our method yields perturbation by calculating the gradient of an attacked model using one access to the model. We further show that a one-iteration attack for each frame is more efficient than a many-iteration attack applied to only some frames. To keep the temporal stability of a perturbed video, we make our attack invisible to the human eye by using weighting and frequency modules. The proposed attack operates in a white-box scenario, which does not limit its applicability for benchmarks: usually, a comparison methodology is known, and quality metrics are published for reproducibility. The primary contributions of this work can be summarized as follows:
+
+- We propose an Invisible One-Iteration (IOI) adversarial attack that increases NR image- and video-quality metrics scores. It produces perturbations that are imperceptible and temporally stable in videos. The attack is fast: it does not require convergence and works efficiently with one iteration.  
+- We propose a methodology for comparing adversarial attacks at NR metrics. It is based on aligning attack speed and relative metrics increase after the attack, yielding to comparing only objective and subjective quality of adversarial videos.  
+- We conducted comprehensive experiments using two datasets and three NR models. Four quality metrics were used to demonstrate that the proposed attack generates adversarial images and videos of superior quality compared to prior methods.  
+- We conducted a subjective study on the proposed method's perceptual quality and temporal stability. A crowd-sourced subjective comparison with 685 subjects showed that the proposed attack produces adversarial videos of better visual quality than previous methods.
+
+Our code is publicly available at https://github. com/katiashh/ioi-attack.
+
+# 2. Related Work
+
+Adversarial attacks usually have constraints on perturbations and minimize the  $L_{\infty}$  norm between adversarial examples and their original inputs. Other  $L_{p}$  norms (Su et al., 2019; Szegedy et al., 2013) are less common due to a higher computational complexity. We further consider methods that use only  $L_{\infty}$  constraint to work faster on high-dimensional data like videos. Among such kinds of attacks, Goodfellow et al. proposed the FGSM method (Goodfellow et al., 2015) that generates adversarial examples by leveraging gradients from the targeted model. Recently, UAP (Shumitskaya et al., 2022) and FACPA (Shumitskaya et al., 2023b) attacks on NR image quality metrics have been proposed. These methods also used  $L_{\infty}$  norm constraints at the training stage.
+
+Numerous studies showed that  $L_{p}$  norms are not suitable as a distance metric to evaluate perceptual image quality (Sharif et al., 2018; Fezza et al., 2019; Wang et al., 2004; Johnson et al., 2016; Isola et al., 2017). Perturbations in images generated under  $L_{p}$  norm constraints often result in noisy pixels within smooth areas of the original image, which is easily perceptible to the human eye. Several adversarial attacks prioritize the visual quality of the generated adversarial images and use more sophisticated restrictions on perturbations than the bare implementation of  $L_{p}$  constraints.
+
+Zhang et al. (2020) introduces a novel approach AdvJND by incorporating just noticeable difference (JND) (Yang et al., 2005) coefficients into the  $L_{\infty}$  norm constraint during adversarial example generation. These coefficients account for the human eye's ability to perceive the threshold of changes in an image. The authors employed I-FGSM and FGSM algorithms as baselines. To enhance the visual fidelity of adversarial images, they amplified original perturbations obtained from the FGSM or I-FGSM methods by scaled JND coefficients.
+
+SSAH (Luo et al., 2022) adversarial attack targets two objectives: semantic similarity of images and low-frequency constraint. The first component is crafted with a focus on image classification tasks. The second component adds perturbations within high-frequency regions by minimizing the difference between low-frequency information of adversarial and original images. Usually, it requires many iterations to converge and produce good visual quality.
+
+Korhonen et al. (2022b) introduced an iterative attack method targeting NR quality metrics, employing a Sobel filter to hide distortions within textured regions. In each iteration, the model's gradient under attack concerning the input is multiplied by a spatial activity map derived from
+
+the original image via the Sobel filter. This map highlights areas with substantial texture, enhancing the visual fidelity of the resulting adversarial images.
+
+Zhang et al. (2022a) introduced an iterative adversarial image crafting approach leveraging various FR metrics like Chebyshev distance, SSIM, LPIPS, and DISTS to manage visual distortions. Their method involved the iterative minimization of a loss function consisting of two components: the attacked loss and a loss based on some differential FR image quality metric.
+
+Karli et al. (2021) introduced the Normalized Variance Weighting (NVW) method aimed at amplifying perturbations within high-variance regions of images. This technique can be utilized alongside gradient attacks like FGSM or I-FGSM. Additionally, the authors proposed the LPIPS-minimization method, aiming to enhance perceptual quality by minimizing the LPIPS distance between the original and adversarial images while ensuring the classifier remains deceived. However, this minimization method is exclusively effective for discrete tasks, such as classification or detection, while applying it to quality metrics will lead to eliminating relative gain.
+
+Table 1 summarizes existing attacks on image- and video-quality metrics. The primary drawback of the prior methods is their requirement to run the attack via many iterations with small steps. This leads to low attack speed, particularly on high-resolution video data. They can be used efficiently to attack videos only when applied to each  $k$ -th frame, which, as we further show, reduces relative gain and temporal consistency.
+
+Table 1. Comparison of existing adversarial attacks on image- and video-quality metrics regarding visual quality regulation features and the requirement to converge for an attack to succeed.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Visual quality</td><td rowspan="2">Speed Needn’t converg.</td></tr><tr><td>Weights</td><td>Freq. reg.</td></tr><tr><td>FGSM (2015)</td><td>×</td><td>×</td><td>✓</td></tr><tr><td>UAP (2022)</td><td>×</td><td>×</td><td>✓</td></tr><tr><td>FACPA (2023b)</td><td>×</td><td>×</td><td>✓</td></tr><tr><td>AdvJND (2020)</td><td>JND map</td><td>×</td><td>✓</td></tr><tr><td>SSAH (2022)</td><td>×</td><td>DWT</td><td>×</td></tr><tr><td>Korhonen et al. (2022b)</td><td>Sobel map</td><td>×</td><td>✓</td></tr><tr><td>Zhang et al. (2022a)</td><td>×</td><td>×</td><td>×</td></tr><tr><td>NVW (2021)</td><td>Local STD</td><td>×</td><td>✓</td></tr><tr><td>IOI (proposed)</td><td>Local STD-based</td><td>FFT</td><td>✓</td></tr></table>
+
+# 3. Proposed method
+
+# 3.1. Problem formulation
+
+The adversarial attack on the NR quality metric  $M$  is usually modelled as follows:
+
+$$
+\underset {I ^ {a}} {\operatorname {a r g m a x}} \left\{M \left(I ^ {a}\right) - M (I) \right\}, \| I ^ {a} - I \| _ {p} \leq \epsilon , \tag {1}
+$$
+
+where  $I$  is a clear video frame,  $I^a$  is an attacked frame,  $\epsilon$  is a small constant.
+
+To ensure a high visual quality of the perturbed image, instead of using  $l_{p}$  norms that are inefficient for this task, we formulate the problem as follows:
+
+$$
+\operatorname {a r g m a x} \left\{M \left(I ^ {a}\right) - M (I) \right\},
+$$
+
+$$
+\frac {I ^ {a}}{| | L _ {f} (I ^ {a}) | | _ {2} | | L _ {f} (I) | | _ {2}} \geq 1 - \epsilon^ {*}, \tag {2}
+$$
+
+where  $L_{f}$  is a low-frequency filter,  $\epsilon^{*}$  is a small constant. When low-frequency components of the clear and adversarial images are closely aligned, the perturbations mainly affect high-frequency areas. As a result, distortions in adversarial images are nearly invisible to the human eye, according to the contrast masking theory of human vision (Legge & Foley, 1980). For measuring the difference between attacked and original images/frames during a perturbation construction, we use adversarial mean absolute error in the frequency domain  $(MAE^{*})$ , which is calculated as follows:
+
+$$
+M A E ^ {*} (I ^ {a}, I) = \frac {1}{H W} \sum_ {i = 0} ^ {(H - 1)} \sum_ {j = 0} ^ {(W - 1)} \left| I _ {i j} ^ {a *} - I _ {i j} ^ {*} \right|, \tag {3}
+$$
+
+where  $I_{ij}^{a*}$  and  $I_{ij}^*$  are Fast Fourier Transform (FFT) coefficients of attacked and original images correspondingly,  $H$  and  $W$  - image dimensions.
+
+As discussed in the introduction, we focus on the scenario where metric scores increase after an attack. In some other studies, NR metrics are modified in two directions, and the attack success is measured by a decrease in the metric's correlation with subjective quality (Zhang et al., 2022b; 2023). While this approach holds theoretical significance in evaluating the stability of metric scores, it carries certain limitations. Only increasing the target metric score will keep the same correlation with subjective quality; thus, an attack remains undetected. Instead of using correlation as a measure of NR metrics' adversarial robustness, we evaluate attack success using relative gain (RG):
+
+$$
+R G = \frac {M (I ^ {a}) - M (I)}{M _ {\text {r a n g e}}} \tag {4}
+$$
+
+where  $M_{range}$  is the range of scores produced by  $M$ .
+
+# 3.2. One-iteration attack
+
+Figure 1 provides an overview of our method. Initially, the proposed attack perturbs the image using a baseline gradient attack. Subsequently, it processes the perturbed image using two modules to enhance the visual quality of an adversarial image/video: the frequency module and the weighting module. For the baseline gradient attack, we employ FGSM (Goodfellow et al., 2015):
+
+$$
+I ^ {p} = I + \epsilon * \operatorname {s i g n} \left(\nabla_ {I} M (I)\right) \tag {5}
+$$
+
+# 3.3. Frequency module
+
+The frequency module extracts features from the original  $I$  and perturbed  $I^p$  images for further processing. It decomposes them into low-frequency (LF) and high-frequency (HF) components using the Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT). The LF component keeps fundamental content information, while the HF component contains noise and texture details. A threshold  $tr^{*}$  for dividing frequencies in  $\mathrm{f\%}$  highest and  $(1 - \mathrm{f})\%$  the lowest coefficients is used to calculate indexes  $d_{f}(I)$  of the highest FFT coefficient of the original image  $I$  (Equation 6).
+
+$$
+\begin{array}{l} t r ^ {*} = \underset {t r} {\operatorname {a r g m i n}} | f - \frac {\sum_ {u = 0} ^ {H - 1} \sum_ {v = 0} ^ {W - 1} \mathbb {1} \left[ I _ {u , v} ^ {*} > t r \right]}{H W} | \tag {6} \\ d _ {f} (I) = \left\{\left(u, v\right): I _ {u, v} ^ {*} > t r ^ {*} \right\} \\ \end{array}
+$$
+
+We use  $d_{f}(I)$  to extract LF and HF components from the original image, as shown in Equation 7. These components are further utilized in the weighting module to enhance the visual quality of adversarial video frames. HF components of the perturbed image  $I^{p}$  are extracted using indexes  $d_{f}(I)$  from the original image. Thus, the perturbations will be transferred into HF areas of the original image (Equation 8). The parameter  $f$  allows fine-tuning the visibility of perturbations and relative attack gain at different levels. Lower values of  $f$  result in more visible perturbations within the frequency module and higher relative gain.
+
+$$
+\begin{array}{l} L _ {u, v} ^ {d _ {f} (I)} (I) = \left\{ \begin{array}{l l} I _ {u, v} ^ {*} & \text {i f} (u, v) \in d _ {f} (I) \\ 0 & \text {o t h e r w i s e} \end{array} \right. \\ H _ {u, v} ^ {d _ {f} (I)} (I) = \left\{ \begin{array}{l l} I _ {u, v} ^ {*} & \text {i f} (u, v) \notin d _ {f} (I) \\ 0 & \text {o t h e r w i s e} \end{array} \right. (7) \\ H _ {u, v} ^ {d _ {f} (I)} \left(I ^ {p}\right) = \left\{ \begin{array}{l l} {\left(I ^ {p}\right) _ {u, v} ^ {*}} & \text {i f} (u, v) \notin d _ {f} (I) \\ 0 & \text {o t h e r w i s e} \end{array} \right. (8) \\ \end{array}
+$$
+
+# 3.4. Weighting module
+
+Various image processing applications (Lin et al., 2005; Liu et al., 2010) operate under the assumption that distortions
+
+in low-variance regions are more visible to the human eye than in high-variance areas. The weighting module generates a weighting map for the input image based on the variance map. A similar approach was used in previous studies (Croce & Hein, 2019; Karli et al., 2021) for generating adversarial examples. We introduce additional features to enhance the weights obtained from the variance map method. The proposed weights map generation method is described in Equation 9. Initially, we compute local variance and local mean maps for an image  $x$ , determining standard deviations and means for both axes using a window size of 3 for each colour channel. Next, we derive the relative local variance of the image by dividing the local variance map by the local mean map and normalizing the weights to the range [0, 1]. In the next step, we zero out  $1\%$  and compute the square root of the weights.
+
+$$
+\begin{array}{l} \sigma_ {i, j} = \sqrt {\frac {\sum x _ {i , j} ^ {2}}{n} - (\frac {\sum x _ {i , j}}{n}) ^ {2}}, m _ {i, j} = \frac {\sum x _ {i , j}}{n} \\ \gamma = \frac {\sigma}{m}, \gamma_ {m a x} = \max  _ {i, j} \left(\gamma_ {i, j}\right), \gamma_ {n o r m} = \frac {\gamma}{\gamma_ {m a x}} \tag {9} \\ w _ {i, j} = \left\{ \begin{array}{l} \sqrt {(\gamma_ {\text {n o r m}}) _ {i , j}}  , \text {i f}   (\gamma_ {\text {n o r m}}) _ {i, j} \geq 0. 0 1 \\ 0  , \text {i f}   (\gamma_ {\text {n o r m}}) _ {i, j} <   0. 0 1 \end{array} \right. \\ \end{array}
+$$
+
+Figure 2 illustrates the weights map derived by the proposed method and three prior methods (Korhonen et al. (2022a), AdvJND (Zhang et al., 2020), and NVW (Karli et al., 2021)). NVW and AdvJND methods assign non-zero weights for a noisy background, often resulting in visible distortions within smooth regions. The image area covered by non-zero weights in Korhonen et al.'s map is small, potentially limiting the strength of the attack it can generate.
+
+We use the proposed weights map to guide the weighting of adversarial image HF components. As a result, the ultimate perturbation remains absent in smooth areas of an image. The construction of the final adversarial image involves the composition of three elements: the original LF component, the perturbed HF component multiplied by the weights map, and the original HF component multiplied by inverse weights:
+
+$$
+I ^ {a} = L _ {c} ^ {d _ {f} (I)} (I) + w H _ {c} ^ {d _ {f} (I)} \left(I ^ {p}\right) + (1 - w) H _ {c} ^ {d _ {f} (I)} (I) \tag {10}
+$$
+
+# 3.5. Mathematical properties
+
+This section provides theoretical restrictions of the generated adversarial images or video frames by the proposed method.
+
+Theorem 1. Let  $I$  and  $I^p$  be original and perturbed image correspondingly,  $I^a$  - adversarial image after IOI attack that is based on  $I^p$  with truncating parameter  $f$ . Then inequality 11 is correct, where  $MAE^{*}(\cdot ,\cdot)$  is given by Equation 3.
+
+$$
+\left| \left| I ^ {a} - I \right| \right| _ {\infty} \leq (1 - f) M A E ^ {*} \left(I ^ {p}, I\right) \tag {11}
+$$
+
+![](images/ab3e58aa4641c3cedde64c00e66850a84219b39fedd939acef21c98aca9edb03.jpg)  
+Figure 1. An overview of the proposed IOI adversarial attack.  $I$  is stands for input image,  $I^p$  - FGSM attacked image and  $I^a$  - the final IOI attacked image. Weights map is calculated using formula 9.
+
+![](images/82caa753d4a5c5bf9d0ef42cd82b75bd57709178db5476236c491b81a2a0115c.jpg)  
+image with noised
+
+![](images/1dc13f94ee5400e3387fd051d62511dede5b089d757d18770e59ff9d5d6f6a0a.jpg)
+
+![](images/8b47667359d2e98bc8e6341c19a26e84bf93e0ec0d1c62d71ddd15b00ddc8b4b.jpg)  
+Korhonen weights
+
+![](images/6a8735b129c23d92c1da136291f6428ecd637a61bb7eb6ff275e2c44b26bc9d2.jpg)  
+AdvJND weights
+
+![](images/2faebaaeaa7f0a9a3c031af307144197228bae2654b6b03d2cc0201f6b9fde1a.jpg)  
+NVW weights  
+Figure 2. Comparison of weights used in prior and proposed methods. NVW and AdvJND assign non-zero weights for a background. Korhonen et al.'s weight total area is relatively small.
+
+![](images/9d11f4b1fa3d003a8f6f9933e1aafc797251d612cc47f48378fd6f93ef721c04.jpg)  
+Proposed weights
+
+Proof. Each element of the  $(I^a - I)$  is estimated using Equation 10 and representation of high-frequency FFT components as re-transformed 2D discrete Fourier transform without  $f\%$  of highest frequencies. The complete proof is presented in the Appendix A.
+
+The statement above demonstrates that the proposed method guarantees theoretical restriction of  $l_{\infty}$  norm of the adversarial image, which depends on initial attack strength expressed in  $MAE^{*}$  and parameter  $f$  for truncating frequencies. Higher  $f$  leads to saving more frequencies and providing better visual quality of the generated IOI adversarial image/video.
+
+# 4. Experiments
+
+We compared our method to the previous approaches targeting three learning-based NR image- and video-quality metrics on two datasets of images and videos.
+
+Datasets. NIPS2017 image dataset (2017) was used to evaluate attacks on three NR metrics. It includes 1,000 images of a  $299 \times 299$  resolution. For evaluating methods on videos, we used 12 videos with  $1280 \times 720$  resolution from the DERF dataset (2001). Descriptions of these videos are available in the Appendix J. We extracted 75 frames from each original video and saved an attacked video with a frame rate of 25 frames per second, resulting in 3-second videos. The datasets licenses allow usage for research purposes.
+
+Attacked models. For experiments on images, we selected PaQ-2-PiQ (Ying et al., 2020), Hyper-IQA (Su et al., 2020), and TReS (Golestaneh et al., 2022) NR models. For videos, we attacked the PaQ-2-PiQ (Ying et al., 2020) metric. These metrics were chosen to cover different architectures. PaQ-2-PiQ (Ying et al., 2020) employs RoIPool layers, which allows the flexibility to aggregate at different scales. Hyper-IQA (Su et al., 2020) utilizes ResNet50 for semantic feature extraction with further processing in the proposed Content Understanding Hyper Network. TReS (Golestaneh et al., 2022) is based on transformer architecture.
+
+Methodology. To evaluate the efficiency of adversarial attacks, we considered three factors: relative gain (Equation 4), speed and the objective or subjective quality of perturbed images/videos. To compare these three factors, we fixed two of them by aligning the speed and relative gain of all methods and compared the objective and subjective visual quality of adversarial images/videos. Since most of the attack time is spent on backpropagation, we executed each attack for one iteration to standardize the speed of all attacks. We also conducted additional experiments with multiple
+
+iterations, results presented in the Appendix C. In a one-iteration comparison, we combined the results of Zhang et al. and SSAH attacks with FGSM attacks in all tables because one iteration of these attacks is equivalent to one iteration of FGSM. Zhang et al. use an FR quality metric to preserve image fidelity, and SSAH uses the distance between low-frequency information of two images. Since the distorted image used in both methods appears only during the first iteration, these fidelity-preserving components yield zero gradients.
+
+We employed an automatic process described in Algorithm 1 to ensure equal relative gain. Each attack has a parameter to regulate its strength. We denote this parameter as  $lr$ . Initially, we ran the proposed method with fixed parameters ( $lr = 0.1$  and  $f = 0.07$  for image data and  $lr = 0.1$  and  $f = 0.05$  for video data). Then, for each other attack and each image/video, we searched the minimal  $lr$  parameter to achieve the same relative gain. The search process also halted if the reached relative gain did not improve for  $n = 5$  search iterations. For videos, the quality score was calculated as the mean of quality scores on each frame (PaQ-2-PiQ (Ying et al., 2020), Hyper-IQA (Su et al., 2020), and TReS (Golestaneh et al., 2022) are metrics that run per-frame on videos).
+
+Algorithm 1 Relative gain aligning  
+Inputs: data element  $\mathbf{X}$  target relative gain  $RG_{t}$  adversarial attack Adv, attacked model M, search step d, stop parameter n, range of M  $M_{range}$  Output: attacked data element  $\mathbf{X}_{\mathrm{adv}}$ $lr = 0$  counter  $= 0$ $RG_{prev} = 0$  ,flag  $=$  False while not flag do  $\mathbf{X}_{\mathrm{adv}} = \mathbf{Adv}(\mathbf{X},\mathbf{M},lr)$ $RG = \frac{\mathbf{M}(\mathbf{X}_{\mathrm{adv}}) - \mathbf{M}(\mathbf{X})}{M_{range}}$  if  $RG\geq RG_{t}$  do flag  $=$  True if  $RG\leq RG_{prev}$  do counter  $=$  counter +1 if counter  $= = n$  do flag  $=$  True  $lr = lr + d$ $RG_{prev} = RG$    
+end while
+
+Quality metrics and subjective study. We compared the objective quality of adversarial images and videos using four FR image- and video-quality metrics: PSNR, SSIM (Wang et al., 2004), VIF (Sheikh & Bovik, 2006), and LPIPS (Zhang et al., 2018).
+
+We conducted a crowd-sourced subjective comparison using Subjectify.us (sub, Accessed: Jan 2024) to get subjective scores for adversarial videos. Original and adversarial videos were compressed using the x264 video codec with a CRF value 16 (preset "Medium"). Each participant was asked to choose the video of the superior visual quality from a random pair of videos shown sequentially. An option "Can't choose" was also available for them. Videos
+
+were pre-download in the browser to prevent delays in playback, and participants had the flexibility to replay the videos multiple times. Each participant compared 12 video pairs; two of the 12 pairs were special verification pairs with apparent differences in visual quality. Answers from 200 participants who failed the verification were excluded. We collected 8220 responses from 685 participants who passed verification and calculated subjective scores using the Bradley-Terry model (Bradley & Terry, 1952). More details about the subjective experiment setup are presented in the Appendix K.
+
+# 5. Results
+
+Table 11 compares the proposed IOI adversarial attack and eight prior attacks at one iteration on the NIPS2017 image dataset (2017). The comparison involves three attacked image quality models, and the relative gain is aligned using the proposed Algorithm 1. On average, the relative gain achieved by all attacks was  $7.7\%$  for all models.
+
+The proposed IOI method showed higher SSIM, VIF, and LPIPS scores for all attacked NR metrics. The PSNR score of our attack method is lower than that of other methods, which means that IOI changes more information in images; however, the perturbations are hidden in the images' texture/contrast regions. AdvJND method showed promising results for attacking Hyper-IQA and TReS models but failed on PaQ-2-PiQ. NVW performed well on PaQ-2-PiQ and TReS.
+
+Table 3 contains the results of objective comparison on 12 videos from the DERF dataset (2001) and attacking the PaQ-2-PiQ (Ying et al., 2020) metric. We applied attacks with greater intensity on videos to enhance distinguishability for further subjective comparison, so the average relative gain was  $15.3\%$ . The objective results are similar to the comparison on the images: IOI outperformed prior methods on SSIM, VIF, and LPIPS metrics and showed comparable PSNR scores. The per-video results are in the Appendix L.
+
+Subjective comparison. The subjective scores obtained from pairwise comparisons showed that the IOI attack generates adversarial videos of better visual quality: it holds a quality of 2.97, while other methods' scores are below 2.16. Confidence intervals for IOI and other methods do not intersect. The intervals intersect for some prior methods, since their adversarial videos are noisy and flickering, making their subjective quality difficult to rank. The video with the highest distinguishability was "Blue Sky" with tree branches swaying against the smooth sky. The video with the lowest distinguishability was "Rush Hour" with a highly noisy background. As shown in Figure 3, FGSM, NVW, and Korhonen et al. methods generate imperceptible distortion on the stone region but fail to suppress the
+
+Table 2. The objective quality of adversarial images generated by existing and proposed methods on the NIPS2017 image dataset (2017) for three attacked models: PaQ-2-PiQ (Ying et al., 2020), Hyper-IQA (Su et al., 2020), and TReS (Golestaneh et al., 2022). The table presents FR metrics scores for adversarial images averaged across the dataset, with aligned relative gain and  $95\%$  confidence intervals. Each attack run for one iteration.  
+
+<table><tr><td>Attacked model</td><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td></tr><tr><td rowspan="7">PaQ-2-PiQ (2020)</td><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.884±0.007</td><td>33.6±0.3</td><td>0.635±0.010</td><td>0.134±0.009</td></tr><tr><td>NVW (2021)</td><td>0.897±0.007</td><td>34.7±0.5</td><td>0.648±0.011</td><td>0.120±0.008</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.872±0.008</td><td>33.1±0.3</td><td>0.617±0.011</td><td>0.151±0.011</td></tr><tr><td>AdvJND (2020)</td><td>0.740±0.008</td><td>29.5±0.2</td><td>0.384±0.008</td><td>0.208±0.007</td></tr><tr><td>UAP (2022)</td><td>0.737±0.004</td><td>26.3±0.2</td><td>0.371±0.004</td><td>0.314±0.005</td></tr><tr><td>FACPA (2023b)</td><td>0.863±0.003</td><td>30.5±0.2</td><td>0.539±0.005</td><td>0.182±0.004</td></tr><tr><td>IOI (ours)</td><td>0.950±0.002</td><td>33.4±0.2</td><td>0.695±0.005</td><td>0.059±0.003</td></tr><tr><td rowspan="7">Hyper-IQA (2020)</td><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.746±0.017</td><td>30.6±0.6</td><td>0.542±0.019</td><td>0.326±0.023</td></tr><tr><td>NVW (2021)</td><td>0.801±0.015</td><td>33.4±0.7</td><td>0.610±0.019</td><td>0.255±0.021</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.765±0.016</td><td>31.1±0.6</td><td>0.562±0.019</td><td>0.303±0.022</td></tr><tr><td>AdvJND (2020)</td><td>0.909±0.004</td><td>37.1±0.3</td><td>0.660±0.011</td><td>0.073±0.005</td></tr><tr><td>UAP (2022)</td><td>0.545±0.010</td><td>21.4±0.3</td><td>0.192±0.007</td><td>0.447±0.008</td></tr><tr><td>FACPA (2023b)</td><td>0.627±0.008</td><td>24.8±0.2</td><td>0.270±0.007</td><td>0.299±0.007</td></tr><tr><td>IOI (ours)</td><td>0.952±0.002</td><td>33.5±0.2</td><td>0.722±0.005</td><td>0.058±0.003</td></tr><tr><td rowspan="7">TReS (2022)</td><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.876±0.011</td><td>35.9±0.4</td><td>0.719±0.015</td><td>0.134±0.013</td></tr><tr><td>NVW (2021)</td><td>0.902±0.010</td><td>37.7±0.5</td><td>0.754±0.014</td><td>0.107±0.011</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.888±0.011</td><td>36.3±0.4</td><td>0.734±0.015</td><td>0.123±0.013</td></tr><tr><td>AdvJND (2020)</td><td>0.915±0.006</td><td>39.1±0.4</td><td>0.736±0.013</td><td>0.064±0.006</td></tr><tr><td>UAP (2022)</td><td>0.445±0.008</td><td>17.5±0.1</td><td>0.120±0.003</td><td>0.715±0.008</td></tr><tr><td>FACPA (2023b)</td><td>0.611±0.007</td><td>23.4±0.2</td><td>0.221±0.007</td><td>0.530±0.011</td></tr><tr><td>IOI (ours)</td><td>0.945±0.002</td><td>33.4±0.2</td><td>0.756±0.005</td><td>0.059±0.003</td></tr></table>
+
+perturbation on the sky background. AdvJND, UAP and FACPA cause visible distortions on the whole image. To produce the same relative gain as IOI using one iteration,  $lr$  for other methods, was high, yielding visible perturbations. As shown in the Appendix C, all methods (except FGSM, UAP and FACPA) produce almost equivalent results at 20 iterations. But at one iteration, there is a crucial difference. The videos used for the comparison are available at https://github.com/katiashh/ioi-attack.
+
+# 6. Discussion
+
+Different frame frequency and attack success. We conducted additional experiments to show the importance of a one-iteration setup when attacking NR quality metrics for videos. This section demonstrates that employing a single iteration for each frame produces superior results compared to the sporadic application of multiple iterations, such as ten iterations for every tenth frame. We selected the PaQ-2-PiQ (2020) NR metric, "Controlled Burn" video from the DERF dataset (2001), and applied the I-FGSM attack (Kurakin et al., 2018). I-FGSM is an extension of FGSM, involving multiple iterations.
+
+Let  $n$  represent the number of iterations,  $\epsilon$  a small constant,
+
+$I$  the original video frame, and  $M$  the target NR quality metric. The  $k$ -th iteration of I-FGSM is formulated as shown in Equation 12 ( $k \in [0, n]$ ,  $I_0^p = I$ ).
+
+$$
+I _ {k + 1} ^ {p} = I _ {k} ^ {p} + \frac {\epsilon}{n} * \operatorname {s i g n} \left(\nabla_ {I _ {k} ^ {p}} M \left(I _ {k} ^ {p}\right)\right) \tag {12}
+$$
+
+We executed I-FGSM for different iteration counts:  $n = 1$ ,  $n = 2$ ,  $n = 4$ ,  $n = 6$ ,  $n = 8$ , and  $n = 10$ . Only some frames underwent attack in each experiment, specifically  $\frac{1}{n}$ . The selection of frames for the attack was done uniformly. The results of these experiments are illustrated in Figure 4. As the number of iterations in the attack increases, there is a corresponding rise in the attack's relative gain on each particular frame. However, evaluating the overall relative gain involves averaging relative gains across all frames. The optimal averaged relative gain occurs when each frame is attacked with just one iteration, and this gain decreases monotonically with the increase in the parameter  $n$ . We also measured the computation time for attacks with different values of  $n$ , which remained nearly constant at 3 seconds for all attacks. This consistency arises from the fact that each experiment's total number of adversarial iterations was the same.
+
+Compared with other values of  $n$ , we can see that a one-
+
+Table 3. Subjective comparison results on 12 videos from the DERF dataset (2001). Adversarial videos generated for PaQ-2-PiQ model (Ying et al., 2020) at equal speed and relative gain of all attacks. The table presents averaged quality metrics and subjective scores with  $95\%$  confidence intervals. Each attack runs for one iteration on each video frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.859±0.005</td><td>33.1±0.2</td><td>0.555±0.007</td><td>0.195±0.006</td><td>1.95±0.16</td></tr><tr><td>NVW (2021)</td><td>0.871±0.005</td><td>33.4±0.2</td><td>0.570±0.007</td><td>0.178±0.006</td><td>2.16±0.16</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.855±0.005</td><td>33.0±0.2</td><td>0.550±0.007</td><td>0.204±0.007</td><td>2.06±0.16</td></tr><tr><td>AdvJND (2020)</td><td>0.848±0.005</td><td>34.5±0.2</td><td>0.516±0.008</td><td>0.153±0.006</td><td>1.76±0.16</td></tr><tr><td>UAP (2022)</td><td>0.809±0.003</td><td>29.8±0.2</td><td>0.450±0.003</td><td>0.301±0.004</td><td>0.19±0.19</td></tr><tr><td>FACPA (2023b)</td><td>0.887±0.002</td><td>32.9±0.2</td><td>0.578±0.004</td><td>0.207±0.003</td><td>0.87±0.17</td></tr><tr><td>IOI (ours)</td><td>0.941±0.016</td><td>34.3±1.7</td><td>0.669±0.046</td><td>0.098±0.030</td><td>2.97±0.16</td></tr></table>
+
+![](images/0890b8bbda21825d0d78f258c83765459be0e68140a3161efcf0adcf7b89fc29.jpg)  
+Figure 3. Comparison of adversarial images generated using FGSM (2015), SSAH (2022), Zhang et al. (2022b), NVW (2021), Korhonen et al. (2022b), AdvJND (2020), UAP (2022), FACPA (2023b) and IOI (ours) attack methods when attacking PaQ-2-PiQ (2020) NR quality metric at one iteration with relative gain aligned by Algorithm 1.
+
+iteration attack yields superior averaged relative gain within the same attack time. This highlights that attack on video quality metrics differ from the classification task, where an attacker can affect only several frames to fool the classifier. From the results of this experiment, we can conclude that the effectiveness of adversarial attacks for video quality metrics is defined by their effectiveness at a one-iteration setup.
+
+Speed of the proposed method. The PyTorch realization of the IOI attack allows reaching 8 fps on the NVIDIA Tesla T4 GPU. Details presented in Appendix B.
+
+IOI performance under defences. We did additional experiments (Table 4) to check the robustness of the proposed method to three adversarial defences: video compression
+
+(Shaham et al., 2018), random crop and resize used in (Shumitskaya et al., 2023a) for NR metrics. Defences were evaluated for videos from the DERF dataset. Although the proposed method affects only high-frequency information, video compression reduced relative gain only by  $2.4\%$ . Random cropping confuses the attack and reduces relative gain approximately two times. Frames resizing almost completely mitigates the relative gain from  $14.6\%$  to  $1\%$ ; however, an NR metric increase by  $1\%$  is still significant for benchmarks; sometimes, teams compete to achieve a  $0.1\%$  metric increase to win the competition. More details are presented in the Appendix E.
+
+Limitations. Our method works in a white-box scenario that implies that an attacker knows and has access to the target model. The white-box scenario is less universal than
+
+![](images/331534bcb29e74ce2dd3ff48fcfd1115f27fa8781c870d765e75ba10a2ead151.jpg)
+
+![](images/172380eeab7ba9c30d8bc8ce755a55efd03bf90cb86eea546065eb374493cc54.jpg)
+
+![](images/2e133254d516cd1b7b84cc6c9b5cd2cbaca124777fc74ee004a584c3a4eb8868.jpg)
+
+![](images/2985d485780f692774308c47d7843c15ddecb2104ae81a07265e26cb3267c3a5.jpg)  
+Figure 4. Results of experiments when attacking PaQ-2-PiQ (2020) NR metric on the "Controlled Burn" video through I-FGSM attack (2018) with different numbers of iterations and altered frames.
+
+![](images/41861699112c125fcaf0f7f722753081dd077984a2570a0d6f1632ac0ab9e46c.jpg)
+
+![](images/1d0b4d6fd29d096b2ea72056825fffba87f66a2a3590b545a383eaa0b3fb040f.jpg)
+
+<table><tr><td></td><td>IOI</td><td>IOI + compress.</td><td>IOI + random crop</td><td>IOI + resize</td></tr><tr><td>RG</td><td>14.6%</td><td>12.2%</td><td>6.30%</td><td>0.98%</td></tr></table>
+
+Table 4. IOI performance under adversarial purification defences (video compression, random cropping, and resizing). Relative gain averaged for 12 videos.
+
+a black-box; however, as described in the introduction, NR quality metrics are usually published as part of the benchmark methodology. We made additional experiments of analysing black-box transferability (Appendix F) and IOI performance in black-box settings (Appendix G). We found out low transferability across different models and significant difference in operation speed of white-box and black-box attacks. Black-box attacks are unlikely to be injected into video processing algorithms that compete in quality and speed, which is the scenario we target in our work. We considered only NR metrics, as FR metrics are much more difficult to attack in real-life scenarios. The robustness of FR metrics has been studied in (Ghildyal & Liu, 2023).
+
+Additional experiments. We made the following additional experiments: experiment to compare the proposed method with prior methods when applying different parameters (Appendix D), metric score decreasing experiment (Appendix H), IOI attack on segment-level video quality model (Ap-
+
+pendix I). We also measured time spending for one-iteration for all methods tested in this paper (Appendix B).
+
+# 7. Conclusion
+
+This paper introduces the IOI adversarial attack on NR image- and video-quality metrics. Its primary objective is to generate imperceptible perturbations for images or videos using only one iteration. Through extensive experiments, we showed that existing methods fail to produce high-fidelity adversarial videos in near real-time scenarios (1 - 10 fps). In contrast, our proposed method demonstrates better effectiveness at high speed. Subjective and objective comparisons showed that the proposed method produces adversarial images and videos of superior visual quality, achieving the same attack success and speed as prior methods. The proposed attack is a potent tool for experimentally assessing the vulnerability of NR quality metrics. By publishing our method, we provide a tool for verification of NR metrics robustness for benchmark organizers and contribute to the future development of robust image- and video-quality metrics. The proposed method can be used as a part of an adversarial training technique to improve the robustness of image- and video-quality metrics. Our code is openly accessible at https://github.com/katiashh/ioi-attack.
+
+# Acknowledgements
+
+The authors would like to thank the CMC Faculty of MSU, especially A.V. Gulyaev for providing the necessary computing resources, which enabled us to undertake some of the calculations for this paper. We also would like to express our gratitude to Mikhail Pautov for discussing the results of this research and assistance in theoretical statements.
+
+The work was supported by a grant for research centers in the field of artificial intelligence, provided by the Analytical Center in accordance with the subsidy agreement (agreement identifier 000000D730321P5Q0002) and the agreement with the Ivannikov Institute for System Programming of dated November 2, 2021 No. 70-2021-00142.
+
+# Impact Statement
+
+In this paper, we propose a new method that can be used to exploit vulnerabilities in video quality assessment metrics. NR quality assessment is widely used for public competitions. However, little research has been published in this area. Using NR metrics vulnerabilities is profitable for benchmark participants, so they are unlikely to publish their findings. Also, no robust NR metrics have been proposed so far due to the complexity of the task: adversarial training and purification methods reduce the performance of defended methods, significantly reducing the usability of NR metrics as a substitution for subjective tests.
+
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+
+# A. Proof of Theorem 1
+
+Theorem 1. Let  $I$  and  $I^p$  be original and perturbed image correspondingly,  $I^a - IOI$  adversarial image based on  $I^p$  with truncating parameter  $f$ . Then inequality 13 is correct, where  $MAE^{*}(\cdot, \cdot)$  is given by Equation 3.
+
+$$
+\left| \left| I ^ {a} - I \right| \right| _ {\infty} \leq (1 - f) M A E ^ {*} \left(I ^ {p}, I\right) \tag {13}
+$$
+
+Proof. We can estimate the difference between  $I^a$  and  $I$ . Since  $||w||_{\infty} \leq 1$ , we can write the following:
+
+$$
+\begin{array}{l} I ^ {a} = L _ {c} ^ {d _ {f} (I)} (I) + w H _ {c} ^ {d _ {f} (I)} \left(I ^ {p}\right) + (1 - w) H _ {c} ^ {d _ {f} (I)} (I) \\ I = L _ {c} ^ {d _ {f} (I)} (I) + H _ {c} ^ {d _ {f} (I)} (I)) \\ \left. \left| I ^ {a} - I \right| \right| _ {\infty} = \left| \left| w \left(H _ {c} ^ {d _ {f} (I)} \left(I ^ {p}\right) - H _ {c} ^ {d _ {f} (I)} (I)\right) \right| \right| _ {\infty} \leq \left| \left| w \right| \right| _ {\infty} \left| \left| \left(H _ {c} ^ {d _ {f} (I)} \left(I ^ {p}\right) - H _ {c} ^ {d _ {f} (I)} (I)\right) \right| \right| _ {\infty} \leq \tag {14} \\ \leq \left| \left| \left(H _ {c} ^ {d _ {f} (I)} \left(I ^ {p}\right) - H _ {c} ^ {d _ {f} (I)} (I)\right) \right| \right| _ {\infty} \\ \end{array}
+$$
+
+Since FFT and IFFT are linear transformations and indexes  $d_{f}(I)$  for truncating  $I^p$  and  $I$  are the same:
+
+$$
+H _ {c} ^ {d _ {f} (I)} \left(I ^ {p}\right) - H _ {c} ^ {d _ {f} (I)} (I) = H _ {c} ^ {d _ {f} (I)} \left(I ^ {p} - I\right) \tag {15}
+$$
+
+We can write high-frequency component as re-transformed two-dimensional discrete Fourier transform without  $\mathrm{f\%}$  of highest frequencies and estimate the module of each element of  $H_{c}^{d_{f}(I)}(I^{p} - I)$ , where  $k_{r}, l_{s}$  – indexes of the sorter FFT coefficients, such that  $|(I^{p} - I)_{k_{n},l_{n}}^{*}| > = |(I^{p} - I)_{k_{n + 1},l_{n + 1}}^{*}| \forall n$ :
+
+$$
+\begin{array}{l} \left| H _ {c} ^ {d _ {f} (I)} \left(I ^ {p} - I\right) _ {u, v} \right| = \frac {1}{H W} \left| \sum_ {s = f (H - 1) (W - 1)} ^ {(H - 1) (W - 1)} \left(I ^ {p} - I\right) _ {k _ {s}, l _ {s}} ^ {*} e ^ {i 2 \pi \left(\frac {k _ {r} u}{H} + \frac {l _ {s} v}{W}\right)} \right| \leq \\ \leq \frac {1}{H W} \sum_ {s = f (H - 1) (W - 1)} ^ {(H - 1) (W - 1)} | (I ^ {p} - I) _ {k _ {s}, l _ {s}} ^ {*} | = \frac {1}{H W} \sum_ {s = f (H - 1) (W - 1)} ^ {(H - 1) (W - 1)} | (I ^ {p}) _ {k _ {s}, l _ {s}} ^ {*} - (I) _ {k _ {s}, l _ {s}} ^ {*} | = \beta - \alpha \tag {16} \\ \end{array}
+$$
+
+where  $\alpha$  and  $\beta$  are given by Equation 17.
+
+$$
+\beta = \frac {1}{H W} \sum_ {\substack {s = 0 \\ f (H - 1) (W - 1)}} ^ {(H - 1) (W - 1)} | (I ^ {p}) _ {k _ {s}, l _ {s}} ^ {*} - (I) _ {k _ {s}, l _ {s}} ^ {*} | \tag{17}
+$$
+
+$$
+\alpha = \frac {1}{H W} \quad \sum_ {s = 0} \quad | (I ^ {p}) _ {k _ {s}, l _ {s}} ^ {*} - (I) _ {k _ {s}, l _ {s}} ^ {*} |
+$$
+
+Considering the facts that  $\beta = MAE^{*}(I^p, I)$  (by definition) and  $\alpha >= fMAE^{*}(I^p, I)$  (since  $\alpha$  is the sum of modules of  $f\%$  highest coefficients and  $MAE^{*}(I^p, I)$  is the sum of all coefficients), we get the resulting estimate:
+
+$$
+\left| \left| I ^ {a} - I \right| \right| _ {\infty} \leq M A E ^ {*} \left(I ^ {p}, I\right) - f M A E ^ {*} \left(I ^ {p}, I\right) = (1 - f) M A E ^ {*} \left(I ^ {p}, I\right) \tag {18}
+$$
+
+The equation above demonstrates that the proposed method guarantees theoretical restriction of  $l_{\infty}$  norm of the adversarial image, which depends on initial attack strength and  $f$  parameter for truncating frequencies. It is worth noting that there was a rough estimate of  $||w||_{\infty} \leq 1$  in Equation 14. In practice, multiplication on weights highly improves the  $l_{2}$  norm of an adversarial image.
+
+# B. Speed for one iteration
+
+Table 5 presents the calculation times of attacks at one iteration that were used for comparison in this paper when targeting the PaQ-2-PiQ (Ying et al., 2020) NR metric on one image from the NIPS2017 dataset (2017) and one video from the DERF
+
+dataset (2001). We measured the calculation time on a server with an NVIDIA Tesla T4 GPU and averaged the results over 20 runs. The AdvJND is notably slower than others due to the computational complexity of calculating JND coefficients.
+
+Table 5. GPU calculation times of attacks at one iteration when attacking PaQ-2-PiQ (2020) NR metric on images and videos.  
+
+<table><tr><td>METHOD</td><td>ONE ITERATION TIME ON IMAGE</td><td>ONE ITERATION FPS ON VIDEO</td></tr><tr><td>FGSM (2015), SSAH (2022), ZHANG ET AL. (2022B)</td><td>0.025 SEC</td><td>8.92 FPS</td></tr><tr><td>NVW (2021)</td><td>0.059 SEC</td><td>2.78 FPS</td></tr><tr><td>KORHONEN ET AL. (2022B)</td><td>0.037 SEC</td><td>7.05 FPS</td></tr><tr><td>ADVJND (2020)</td><td>10.38 SEC</td><td>0.01 FPS</td></tr><tr><td>IOI (OURS) PyTorch</td><td>0.028 SEC</td><td>7.81 FPS</td></tr></table>
+
+# C. Experiment with multiple iterations
+
+We conducted an additional experiment to evaluate the performance of visual-oriented methods employed in this paper in the context of multiple iterations. For  $n = 10$  and  $n = 20$  iterations, we run the proposed method with  $\epsilon = 0.1$  and step size of  $\frac{2\epsilon}{n}$ . These experiments used the PaQ-2-PiQ NR model (Ying et al., 2020); the relative gain was  $13\%$  for both 10 and 20 iterations. Utilizing Algorithm 1, we searched for the minimal  $lr$  parameter in other attacks to achieve the same relative gain. Subsequently, we evaluated the visual quality of the resulting adversarial images using four FR metrics: SSIM, PSNR, VIF, and LPIPS. The results for  $n = 10$  and  $n = 20$  iterations are presented in Tables 6 and 7 respectively. At 20 iterations, all methods produce nearly identical results, highlighting the primary strength of the proposed IOI method in its effectiveness in one-iteration settings.
+
+From the results of this experiment, we can conclude that in the setup of multiple iteration attack, there are no significant differences in which method from Table 7 to use (except SSAH and AdvJND - they need more than 20 iterations for convergence). The primary strength of the proposed IOI method is its effectiveness and superiority in one-iteration settings, but in multi-iteration setup, it also shows competitive results.
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td></tr><tr><td>SSAH (2022)</td><td>0.890±0.005</td><td>33.6±0.4</td><td>0.680±0.009</td><td>0.106±0.005</td></tr><tr><td>Zhang et al. (2022b)</td><td>0.938±0.003</td><td>35.7±0.2</td><td>0.700±0.007</td><td>0.073±0.004</td></tr><tr><td>NVW (2021)</td><td>0.957±0.001</td><td>37.1±0.4</td><td>0.725±0.006</td><td>0.055±0.002</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.974±0.001</td><td>37.5±0.2</td><td>0.757±0.005</td><td>0.035±0.002</td></tr><tr><td>AdvJND (2020)</td><td>0.889±0.003</td><td>34.2±0.2</td><td>0.546±0.007</td><td>0.097±0.004</td></tr><tr><td>IOI (ours)</td><td>0.965±0.001</td><td>34.6±0.2</td><td>0.758±0.004</td><td>0.043±0.002</td></tr></table>
+
+Table 6. Comparison results for 10 iterations with relative gain aligning.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td></tr><tr><td>SSAH (2022)</td><td>0.949±0.002</td><td>36.6±0.2</td><td>0.790±0.005</td><td>0.048±0.002</td></tr><tr><td>Zhang et al. (2022b)</td><td>0.971±0.001</td><td>38.1±0.2</td><td>0.786±0.005</td><td>0.034±0.002</td></tr><tr><td>NVW (2021)</td><td>0.970±0.001</td><td>38.7±0.4</td><td>0.778±0.005</td><td>0.038±0.002</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.978±0.001</td><td>38.2±0.2</td><td>0.781±0.004</td><td>0.029±0.001</td></tr><tr><td>AdvJND (2020)</td><td>0.916±0.003</td><td>35.9±0.2</td><td>0.613±0.007</td><td>0.076±0.004</td></tr><tr><td>IOI (ours)</td><td>0.972±0.001</td><td>35.5±0.2</td><td>0.779±0.004</td><td>0.035±0.002</td></tr></table>
+
+Table 7. Comparison results for 20 iterations with relative gain aligning.
+
+# D. Different parameter's comparison
+
+We made an additional experiment to compare the proposed method with prior methods when applying different parameters. This allows us to compare methods in slightly different attack strengths. Results in the Table 8 showed that the proposed
+
+method generates images with better visual quality when achieving the same increase in metric score for three different increase levels.
+
+Table 8. Experiment of methods comparison under different relative gains, corresponding to different  $\epsilon$  levels in the proposed IOI method.  
+
+<table><tr><td>ε</td><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td></tr><tr><td rowspan="7">0.08</td><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.934±0.006</td><td>36.4±0.2</td><td>0.733±0.009</td><td>0.082±0.008</td></tr><tr><td>NVW (2021)</td><td>0.940±0.006</td><td>37.4±0.4</td><td>0.745±0.009</td><td>0.072±0.007</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.932±0.005</td><td>36.2±0.2</td><td>0.727±0.009</td><td>0.083±0.007</td></tr><tr><td>AdvJND (2020)</td><td>0.812±0.005</td><td>31.9±0.1</td><td>0.466±0.007</td><td>0.151±0.006</td></tr><tr><td>UAP (2022)</td><td>0.792±0.004</td><td>27.9±0.2</td><td>0.432±0.005</td><td>0.251±0.004</td></tr><tr><td>FACPA (2023b)</td><td>0.888±0.003</td><td>31.7±0.2</td><td>0.586±0.005</td><td>0.151±0.003</td></tr><tr><td>IOI (ours)</td><td>0.966±0.002</td><td>35.5±0.1</td><td>0.786±0.004</td><td>0.037±0.002</td></tr><tr><td rowspan="7">0.1</td><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.884±0.007</td><td>33.6±0.3</td><td>0.635±0.010</td><td>0.134±0.009</td></tr><tr><td>NVW (2021)</td><td>0.897±0.007</td><td>34.7±0.5</td><td>0.648±0.011</td><td>0.120±0.008</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.872±0.008</td><td>33.1±0.3</td><td>0.617±0.011</td><td>0.151±0.011</td></tr><tr><td>AdvJND (2020)</td><td>0.740±0.008</td><td>29.5±0.2</td><td>0.384±0.008</td><td>0.208±0.007</td></tr><tr><td>UAP (2022)</td><td>0.737±0.004</td><td>26.3±0.2</td><td>0.371±0.004</td><td>0.314±0.005</td></tr><tr><td>FACPA (2023b)</td><td>0.863±0.003</td><td>30.5±0.2</td><td>0.539±0.005</td><td>0.182±0.004</td></tr><tr><td>IOI (ours)</td><td>0.950±0.002</td><td>33.4±0.2</td><td>0.695±0.005</td><td>0.059±0.003</td></tr><tr><td rowspan="7">0.12</td><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.789±0.016</td><td>30.7±0.5</td><td>0.548±0.015</td><td>0.274±0.022</td></tr><tr><td>NVW (2021)</td><td>0.795±0.016</td><td>31.5±0.6</td><td>0.555±0.016</td><td>0.264±0.022</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.774±0.016</td><td>30.0±0.5</td><td>0.524±0.016</td><td>0.295±0.022</td></tr><tr><td>AdvJND (2020)</td><td>0.696±0.007</td><td>28.4±0.1</td><td>0.340±0.006</td><td>0.239±0.008</td></tr><tr><td>UAP (2022)</td><td>0.705±0.004</td><td>25.4±0.1</td><td>0.342±0.004</td><td>0.348±0.007</td></tr><tr><td>FACPA (2023b)</td><td>0.758±0.003</td><td>26.9±0.2</td><td>0.392±0.004</td><td>0.291±0.005</td></tr><tr><td>IOI (ours)</td><td>0.936±0.003</td><td>32.2±0.2</td><td>0.681±0.005</td><td>0.077±0.004</td></tr></table>
+
+# E. IOI performance under defences
+
+We conducted additional experiments to evaluate the robustness of the proposed IOI attack to three defences: compression (Shaham et al., 2018), random crop and resize (Shumitskaya et al., 2023a). Random crop and resize defences were previously studied in (Shumitskaya et al., 2023a) for evaluation of UAP (Shumitskaya et al., 2022) against NR quality metrics. Authors showed that random crop and resize to  $80\%$  of the original image/video size help to improve NR quality metric robustness to adversarial attacks without significant loss in correlations with subjective scores. Because of that, we chose parameter  $80\%$  for our experiments. Defended transformation in the case of random crop defence was selecting a random crop of the frame with  $80\%$  of the original frame size. An image was resized to  $80\%$  of the original frame size for resize-based defence. Shaham et al. showed that compression can be used as defence (Shaham et al., 2018). In our experiment, defended transformation for compression defence was compression with a CRF value 16 using x264 video codec (preset "Medium").
+
+For the experiment, we used 12 adversarial videos generated against PaQ-2-PiQ (source videos from the DERF dataset (der, 2001)) and original ones. We measured relative gain for the IOI attack with and without defences. For relative gain measurement under defence, we calculated PaQ-2-PiQ scores for original and adversarial videos after defended transformation and then, based on these scores, calculated relative gain. Results are presented in the Table 9. Compression defence reduced relative gain only by  $2.4\%$ . Random crop confuse attack and reduce relative gain approximately two times. Resize defence almost completely mitigate the relative gain, however gain in  $1\%$  still can be significant in benchmarks. From the results of these experiments, we can conclude that the proposed IOI attack is robust to compression and random crop defences but vulnerable to the resize defence.
+
+# F. Black-box transferability
+
+We conducted an additional experiment to analyse the applicability of IOI in transferable black-box setting. The experiment was organised as follows: for all generated adversarial images from NIPS2017 dataset, we measured PaQ-2-PiQ, Hyper-IQA
+
+<table><tr><td rowspan="2">Video</td><td colspan="4">Relative gain</td></tr><tr><td>IOI</td><td>IOI + compression</td><td>IOI + random crop</td><td>IOI + resize</td></tr><tr><td>Blue Sky</td><td>13.1%</td><td>11.2%</td><td>6.17%</td><td>0.88%</td></tr><tr><td>Rush Hour</td><td>23.9%</td><td>18.0%</td><td>11.3%</td><td>0.77%</td></tr><tr><td>Old Town Cross</td><td>15.6%</td><td>13.9%</td><td>5.33%</td><td>1.29%</td></tr><tr><td>Crowd Run</td><td>15.2%</td><td>12.4%</td><td>4.10%</td><td>0.64%</td></tr><tr><td>Aspen</td><td>9.50%</td><td>6.70%</td><td>4.11%</td><td>0.51%</td></tr><tr><td>Sunflower</td><td>18.9%</td><td>15.8%</td><td>13.3%</td><td>1.33%</td></tr><tr><td>Life</td><td>10.0%</td><td>7.64%</td><td>1.76%</td><td>-0.09%</td></tr><tr><td>Controlled Burn</td><td>16.2%</td><td>15.4%</td><td>7.12%</td><td>1.85%</td></tr><tr><td>Red Kayak</td><td>16.3%</td><td>14.3%</td><td>7.08%</td><td>2.40%</td></tr><tr><td>Ducks Take Off</td><td>7.94%</td><td>6.24%</td><td>2.80%</td><td>0.30%</td></tr><tr><td>Tractor</td><td>11.6%</td><td>9.57%</td><td>6.00%</td><td>0.74%</td></tr><tr><td>Park Joy</td><td>16.8%</td><td>14.8%</td><td>6.47%</td><td>1.10%</td></tr><tr><td>Mean</td><td>14.6%</td><td>12.2%</td><td>6.30%</td><td>0.98%</td></tr></table>
+
+Table 9. Results of performance proposed IOI attack under compression, random crop and resize defences.
+
+and TReS, i.e. for adversarial images created to attack PaQ-2-PiQ we also measured quality scores by Hyper-IQA and TReS. Based on these metrics scores, we calculated relative gains. Results are presented in the Table 10. All eight methods tested in the paper showed low transferability to unseen models. Low transferability can be explained by these metrics having completely different architectures: PaQ-2-PiQ employs RoIPool layers, HyperIQA utilizes ResNet50 and TReS is based on transformer architecture. Also, it's important to note that improving transferability for image quality models is more challenging than for classifiers or detectors. Transferability can occur in classification and detection tasks because different classifiers "look" at the same regions of images where classified objects are located (Meng et al., 2019). In contrast, different image quality metrics can look at different regions to estimate the score, which inherently complicates the achievement of transferability. Thus, we can infer that developing imperceptible and, at the same time, transferable one-iteration attacks on video-quality models is a challenging problem that we will consider for further research.
+
+Table 10. Results of experiment on transferability of methods used in the paper in one-iteration setting.  
+
+<table><tr><td>Attack Test</td><td>PaQ-2-PiQ</td><td>PaQ-2-PiQ Hyper-IQA</td><td>TReS</td><td>PaQ-2-PiQ</td><td>Hyper-IQA Hyper-IQA</td><td>TReS</td><td>PaQ-2-PiQ</td><td>TReS Hyper-IQA</td><td>TReS</td></tr><tr><td>FGSM, SSAH, Zhang</td><td>7.40%</td><td>-8.76%</td><td>-14.93%</td><td>0.16%</td><td>1.03%</td><td>-18.99%</td><td>0.01%</td><td>-0.39%</td><td>4.07%</td></tr><tr><td>NVW</td><td>7.17%</td><td>-8.27%</td><td>-14.19%</td><td>0.13%</td><td>0.70%</td><td>-15.32%</td><td>0.08%</td><td>0.09%</td><td>4.50%</td></tr><tr><td>Korhonen</td><td>7.36%</td><td>-8.85%</td><td>-14.81%</td><td>0.23%</td><td>0.42%</td><td>-15.97%</td><td>0.07%</td><td>0.29%</td><td>5.04%</td></tr><tr><td>AdvJND</td><td>6.61%</td><td>-0.57%</td><td>-19.84%</td><td>-0.18%</td><td>4.09%</td><td>-3.96%</td><td>0.04%</td><td>-0.43%</td><td>7.02%</td></tr><tr><td>UAP</td><td>11.62%</td><td>-6.03%</td><td>-15.55%</td><td>3.13%</td><td>4.83%</td><td>6.65%</td><td>2.44%</td><td>-0.52%</td><td>0.85%</td></tr><tr><td>FACPA</td><td>9.68%</td><td>-5.01%</td><td>-9.14%</td><td>-0.30%</td><td>1.48%</td><td>9.52%</td><td>-6.02%</td><td>-0.51%</td><td>1.14%</td></tr><tr><td>IOI (ours)</td><td>7.33%</td><td>-3.56%</td><td>-6.25%</td><td>0.26%</td><td>7.86%</td><td>-1.73%</td><td>0.39%</td><td>2.34%</td><td>7.40%</td></tr></table>
+
+# G. IOI performance in black-box setting
+
+We conducted an additional experiment aimed to show that the proposed method can be adapted for use in a black-box setting. To do this, we replaced an FGSM-generated perturbation with a black-box-generated perturbation, e.g. Square Attack (Andriushchenko et al., 2020). We will call this modification BB-IOI. It's important to note that such modification transforms the method into a multi-iteration. To verify the efficiency of BB-IOI, we conducted an additional experiment involving the implementation of the BB-IOI attack that consists of two stages: 1) generating the adversarial perturbation in a black-box manner using the Square Attack (Andriushchenko et al., 2020) method and 2) processing this perturbation using frequency and weighting modules. As a result, BB-IOI provides high-quality adversarial images by objective metrics. Although the property of imperceptibility remains, it's important to note that the computation complexity of the method has increased, which is common for black-box methods. For attacking PaQ-2-PiQ on the NIPS2017 dataset, BB-IOI achieved
+
+an average  $1.21\%$  gain operating at 40 seconds per image. Results are presented in the Table 11. This performance is  $6.12\%$  lower and 1400 times slower than IOI. Furthermore, the proposed IOI method can be adapted for use in transfer-based settings by combining it with transfer-based perturbation generation techniques (Long et al., 2022), (Li et al., 2023).
+
+Table 11. Comparison of the proposed method used in white-box setting (IOI) and black-box setting (BB-IOI).  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Relative gain ↑</td><td>Time on one image ↓</td></tr><tr><td>IOI</td><td>0.945±0.002</td><td>33.4±0.2</td><td>0.756±0.005</td><td>0.059±0.003</td><td>7.33%</td><td>0.028 sec</td></tr><tr><td>BB-IOI</td><td>0.988±0.001</td><td>36.9±0.1</td><td>0.815±0.003</td><td>0.024±0.001</td><td>1.21%</td><td>40 sec</td></tr></table>
+
+# H. Metric score decreasing experiment
+
+In this section, we show the possibility of metrics score decreasing. Given that the proposed method consists of two parts (generating a perturbed image using FGSM and subsequent processing in frequency and weighting modules), changing the optimization direction in FGSM leads to guiding an attack in the opposite direction. We conducted additional experiments to show that the proposed method can decrease PaQ-2-PiQ metric scores. We applied the proposed method to attack it on the same NIPS2017 dataset we used to increase this metric. The results revealed that increasing metric scores yielded a  $7.32\%$  score increase, while decreasing metric scores resulted in a  $7.61\%$  score decrease (almost the same). However, it's worth noting that we focused on increasing metrics' scores because decreasing quality metrics' scores holds less practical significance. An attacker can decrease the metrics for quality camouflage (2020), and it is the only real-life scenario known to the authors.
+
+# I. IOI attack on complex VQA metric
+
+The proposed method applies to segment-level video quality models. We conducted an additional experiment targeting the segment-level VQA metric MDTVSFA (Li et al., 2021) to show this. To apply the proposed attack, we first slightly modified MDTVSFA to get access to its gradient (we removed torch.no_grad() context-manager from the feature extraction module and modified the forward process in the inference model to process batches rather than dictionaries). Then, we applied IOI to attack three videos on a per-frame basis, processing one frame at a time. Subsequently, we constructed adversarial videos from these frames and calculated quality scores using the original segment-level MDTVSFA on these videos. Results are presented in the Table 12. Remarkably, this approach yielded a significant relative gain, with a  $15\%$  increase in scores. This experiment showed that attacking only spatial features of a VQA metric without accounting for temporal and other features is enough to achieve high attack success.
+
+Table 12. Experiments of IOI attack targeting segment-level video-quality metric MDTVSFA. The attack was performed per-frame. Resulting gain was calculated using original segment-level MDTVSFA.  
+
+<table><tr><td>Video</td><td>MDTVSFA clean</td><td>MDTVSFA IOI attacked</td></tr><tr><td>Blue Sky</td><td>0.544</td><td>0.659 (↑11.5%)</td></tr><tr><td>Crowd Run</td><td>0.555</td><td>0.759 (↑20.4%)</td></tr><tr><td>Pedestrian Area</td><td>0.584</td><td>0.737 (↑15.3%)</td></tr></table>
+
+# J. Video sequences
+
+We used the following 12 videos with a resolution of  $1280 \times 720$  from the DERF dataset (2001):
+
+1. "Blue Sky": https://media.xiph.org/video/derf/y4m/blue skies_1080p25.y4m  
+2. "Aspen": https://media.xiph.org/video/derf/y4m/aspen_1080p.y4m  
+3. "Sunflower": https://media.xiph.org/video/derf/y4m/sunflower_1080p25.y4m  
+4. "Crowd Run": https://media.xiph.org/video/derf/y4m/crowd_run_1080p50.y4m
+
+5. "Old Town Cross": https://media.xiph.org/video/derf/y4m/old_town CROSS_1080p50.y4m  
+6. "Life": https://media.xiph.org/video/derf/y4m/life_1080p30.y4m  
+7. "Controlled Burn": https://media.xiph.org/video/derf/y4m/controlled_burn_1080p.y4m  
+8. "Rush Hour": https://media.xiph.org/video/derf/y4m/rush_hour_1080p25.y4m  
+9. "Red Kayak": https://media.xiph.org/video/derf/y4m/red_kayak_1080p.y4m  
+10. "Ducks Take Off": https://media.xiph.org/video/derf/y4m/ducks.take_off_1080p50.y4m  
+11. "Tractor": https://media.xiph.org/video/derf/y4m/tractor_1080p25.y4m  
+12. "Park Joy": https://media.xiph.org/video/derf/y4m/park_joy_1080p50.y4m
+
+We extracted 75 frames from each original video and saved an attacked video with a frame rate of 25 frames per second, resulting in videos with a duration of 3 seconds. Figure 5 contains spatial and temporal information for these videos. Figure 6 contains the first frames of videos.
+
+![](images/820b317836d5d1f813af83da4807857740fcb0c27a1bb221cfff6ea03661d7bb.jpg)  
+Figure 5. Spatial and temporal information for videos.
+
+![](images/75d13c5d9bac1abd07577bef02f19a12c52ed534ca6b9507be4fbfefef2354e9.jpg)  
+Figure 6. First frames of videos.
+
+# K. Subjective experiment setup
+
+To derive subjective scores for adversarial videos, we conducted a crowd-sourced subjective comparison on the Subjectify.us service (sub, Accessed: Jan 2024).
+
+For comparison, we compressed all videos, including the original ones, using the x264 video codec with a CRF value of 16 (preset "Medium"). Each pair shown to participants consisted of two samples of the same source video attacked by various attack methods. Each participant was presented with a random pair of videos sequentially and was asked to choose the video with the superior visual quality. An option "Can't choose" was also provided. Videos were pre-loaded in the browser to prevent delays in playback, and participants had the flexibility to replay the videos multiple times. Each participant compared 12 video pairs, of which two were for verification. Answers from 200 participants who failed the verification were excluded.
+
+We collected 8220 responses from 685 successful participants and calculated subjective scores using the Bradley-Terry model (Bradley & Terry, 1952). The average payment to crowdworkers per a pair of sequences was  $0.05. We estimate the overall cost of the subjective tests was$ 410. Figure 7 presents the subjective experiment's general process.
+
+Details about the crowdsourced study:
+
+1. Screen resolutions were from  $360 \times 800$  to  $3440 \times 1440$ . Table 13 shows the most popular.  
+2. Participants were from 31 countries.  
+3. Participant ages ranged from 18 to 93, with an average of 39. Figure 8 shows the distribution.
+
+Command line for encoding. Given the directory of video frames in PNG format (set of images 000.png, 001.png, ..., 074.png) we run the following FFmpeg command line:
+
+ffmpeg -pattern_type glob -i *.png -c:v libx264 -crf 16 -pix fmt yuv420p res.mp4
+
+![](images/14c99327ad10eaa8ff46dae50521c6a8ef007fc35bdcb67b9f61bf29e84d8a38.jpg)  
+Figure 7. Subjective-assessment scheme.
+
+# L. Per-video results
+
+Tables 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 contain results of subjective comparison of proposed IOI adversarial attack with eight prior attacks for each video.
+
+# M. Attack examples on images and videos
+
+IOI adversarial images and videos are available in the zip archive: https://drive.google.com/file/d/1nrvV70Q4W0vh-2FdWrXHUhMDzYcI6zYl/view?usp=sharing.
+
+<table><tr><td>Resolution</td><td>Number of users</td></tr><tr><td>1920×1080</td><td>194</td></tr><tr><td>1366×768</td><td>167</td></tr><tr><td>1536×864</td><td>100</td></tr><tr><td>1280×1024</td><td>39</td></tr><tr><td>1600×900</td><td>35</td></tr><tr><td>1280×720</td><td>22</td></tr><tr><td>1440×900</td><td>19</td></tr><tr><td>1024×768</td><td>11</td></tr><tr><td>2560×1440</td><td>10</td></tr><tr><td>1680×1050</td><td>10</td></tr><tr><td>1360×768</td><td>10</td></tr></table>
+
+![](images/ce37917a3dc1b42faa247069bab29548a52acd24c9ab5fc291b929802abe2aa8.jpg)  
+Table 13. Most popular screen resolutions among crowdworkers.  
+Figure 8. Age distribution of crowdworkers.
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.739</td><td>30.8</td><td>0.390</td><td>0.352</td><td>1.44±0.66</td></tr><tr><td>NVW (2021)</td><td>0.719</td><td>30.1</td><td>0.370</td><td>0.364</td><td>1.91±0.64</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.699</td><td>29.8</td><td>0.361</td><td>0.380</td><td>1.56±0.65</td></tr><tr><td>AdvJND (2020)</td><td>0.795</td><td>33.9</td><td>0.422</td><td>0.247</td><td>2.20±0.63</td></tr><tr><td>UAP (2022)</td><td>0.809</td><td>31.6</td><td>0.453</td><td>0.338</td><td>0.70±0.70</td></tr><tr><td>FACPA (2023b)</td><td>0.890</td><td>34.4</td><td>0.557</td><td>0.253</td><td>1.11±0.67</td></tr><tr><td>IOI (ours)</td><td>0.956</td><td>33.9</td><td>0.649</td><td>0.048</td><td>4.05±0.62</td></tr></table>
+
+Table 14. Comparison results on the "Blue Sky" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.956</td><td>40.0</td><td>0.720</td><td>0.064</td><td>2.88±0.47</td></tr><tr><td>NVW (2021)</td><td>0.959</td><td>40.3</td><td>0.733</td><td>0.057</td><td>3.13±0.48</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.957</td><td>40.2</td><td>0.725</td><td>0.062</td><td>2.91±0.48</td></tr><tr><td>AdvJND (2020)</td><td>0.950</td><td>41.1</td><td>0.687</td><td>0.052</td><td>2.57±0.48</td></tr><tr><td>UAP (2022)</td><td>0.842</td><td>32.8</td><td>0.465</td><td>0.264</td><td>0.65±0.65</td></tr><tr><td>FACPA (2023b)</td><td>0.915</td><td>36.0</td><td>0.605</td><td>0.174</td><td>1.18±0.58</td></tr><tr><td>IOI (ours)</td><td>0.981</td><td>38.3</td><td>0.823</td><td>0.044</td><td>3.33±0.48</td></tr></table>
+
+Table 15. Comparison results on the "Aspen" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.919</td><td>37.2</td><td>0.604</td><td>0.206</td><td>2.71±0.52</td></tr><tr><td>NVW (2021)</td><td>0.924</td><td>37.4</td><td>0.619</td><td>0.195</td><td>2.96±0.52</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.844</td><td>34.0</td><td>0.465</td><td>0.326</td><td>2.28±0.53</td></tr><tr><td>AdvJND (2020)</td><td>0.741</td><td>32.7</td><td>0.328</td><td>0.365</td><td>1.57±0.56</td></tr><tr><td>UAP (2022)</td><td>0.846</td><td>32.8</td><td>0.463</td><td>0.345</td><td>0.65±0.65</td></tr><tr><td>FACPA (2023b)</td><td>0.922</td><td>35.9</td><td>0.618</td><td>0.209</td><td>1.20±0.60</td></tr><tr><td>IOI (ours)</td><td>0.946</td><td>36.9</td><td>0.690</td><td>0.181</td><td>3.39±0.52</td></tr></table>
+
+Table 16. Comparison results on the "Sunflower" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.921</td><td>33.3</td><td>0.689</td><td>0.071</td><td>3.61±0.51</td></tr><tr><td>NVW (2021)</td><td>0.924</td><td>33.4</td><td>0.694</td><td>0.068</td><td>3.16±0.51</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.922</td><td>33.4</td><td>0.693</td><td>0.070</td><td>3.51±0.51</td></tr><tr><td>AdvJND (2020)</td><td>0.844</td><td>31.7</td><td>0.525</td><td>0.089</td><td>2.12±0.56</td></tr><tr><td>UAP (2022)</td><td>0.798</td><td>27.2</td><td>0.465</td><td>0.204</td><td>0.73±0.73</td></tr><tr><td>FACPA (2023b)</td><td>0.885</td><td>30.5</td><td>0.604</td><td>0.140</td><td>1.44±0.64</td></tr><tr><td>IOI (ours)</td><td>0.951</td><td>32.4</td><td>0.695</td><td>0.039</td><td>4.00±0.51</td></tr></table>
+
+Table 17. Comparison results on the "Crowd Run" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.854</td><td>31.8</td><td>0.512</td><td>0.151</td><td>2.77±1.16</td></tr><tr><td>NVW (2021)</td><td>0.856</td><td>31.9</td><td>0.517</td><td>0.147</td><td>2.92±1.16</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.854</td><td>31.9</td><td>0.513</td><td>0.150</td><td>3.05±1.16</td></tr><tr><td>AdvJND (2020)</td><td>0.844</td><td>33.5</td><td>0.471</td><td>0.120</td><td>1.90±1.18</td></tr><tr><td>UAP (2022)</td><td>0.827</td><td>30.4</td><td>0.473</td><td>0.237</td><td>1.21±1.21</td></tr><tr><td>FACPA (2023b)</td><td>0.913</td><td>34.1</td><td>0.623</td><td>0.168</td><td>2.59±1.16</td></tr><tr><td>IOI (ours)</td><td>0.908</td><td>31.1</td><td>0.539</td><td>0.118</td><td>3.61±1.15</td></tr></table>
+
+Table 18. Comparison results on the "Old Town Cross" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.737</td><td>29.7</td><td>0.419</td><td>0.216</td><td>2.44±0.66</td></tr><tr><td>NVW (2021)</td><td>0.818</td><td>31.4</td><td>0.497</td><td>0.153</td><td>2.83±0.64</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.873</td><td>33.8</td><td>0.588</td><td>0.109</td><td>4.14±0.59</td></tr><tr><td>AdvJND (2020)</td><td>0.832</td><td>34.1</td><td>0.512</td><td>0.092</td><td>3.57±0.61</td></tr><tr><td>UAP (2022)</td><td>0.717</td><td>28.1</td><td>0.396</td><td>0.264</td><td>0.79±0.79</td></tr><tr><td>FACPA (2023b)</td><td>0.814</td><td>30.7</td><td>0.502</td><td>0.188</td><td>1.50±0.72</td></tr><tr><td>IOI (ours)</td><td>0.936</td><td>34.8</td><td>0.668</td><td>0.063</td><td>4.89±0.59</td></tr></table>
+
+Table 19. Comparison results on the "Life" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.833</td><td>30.7</td><td>0.532</td><td>0.256</td><td>1.64±0.52</td></tr><tr><td>NVW (2021)</td><td>0.853</td><td>31.1</td><td>0.553</td><td>0.225</td><td>1.93±0.52</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.777</td><td>28.8</td><td>0.458</td><td>0.328</td><td>1.18±0.54</td></tr><tr><td>AdvJND (2020)</td><td>0.833</td><td>32.7</td><td>0.508</td><td>0.176</td><td>1.54±0.52</td></tr><tr><td>UAP (2022)</td><td>0.798</td><td>28.8</td><td>0.476</td><td>0.314</td><td>0.57±0.57</td></tr><tr><td>FACPA (2023b)</td><td>0.903</td><td>32.9</td><td>0.648</td><td>0.195</td><td>1.01±0.55</td></tr><tr><td>IOI (ours)</td><td>0.932</td><td>32.5</td><td>0.652</td><td>0.117</td><td>2.90±0.53</td></tr></table>
+
+Table 20. Comparison results on the "Controlled Burn" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.956</td><td>40.0</td><td>0.685</td><td>0.093</td><td>3.04±0.49</td></tr><tr><td>NVW (2021)</td><td>0.960</td><td>40.4</td><td>0.704</td><td>0.082</td><td>3.08±0.49</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.959</td><td>40.4</td><td>0.701</td><td>0.084</td><td>3.39±0.49</td></tr><tr><td>AdvJND (2020)</td><td>0.974</td><td>44.1</td><td>0.767</td><td>0.035</td><td>3.51±0.49</td></tr><tr><td>UAP (2022)</td><td>0.838</td><td>32.8</td><td>0.421</td><td>0.357</td><td>0.71±0.71</td></tr><tr><td>FACPA (2023b)</td><td>0.913</td><td>36.0</td><td>0.557</td><td>0.247</td><td>1.58±0.59</td></tr><tr><td>IOI (ours)</td><td>0.952</td><td>38.5</td><td>0.695</td><td>0.118</td><td>3.14±0.49</td></tr></table>
+
+Table 21. Comparison results on the "Rush Hour" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.790</td><td>30.7</td><td>0.466</td><td>0.333</td><td>1.36±0.56</td></tr><tr><td>NVW (2021)</td><td>0.843</td><td>32.3</td><td>0.535</td><td>0.249</td><td>2.20±0.54</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.792</td><td>30.7</td><td>0.469</td><td>0.329</td><td>1.84±0.54</td></tr><tr><td>AdvJND (2020)</td><td>0.759</td><td>31.7</td><td>0.393</td><td>0.272</td><td>1.42±0.56</td></tr><tr><td>UAP (2022)</td><td>0.840</td><td>31.5</td><td>0.517</td><td>0.280</td><td>0.61±0.61</td></tr><tr><td>FACPA (2023b)</td><td>0.900</td><td>34.0</td><td>0.638</td><td>0.219</td><td>1.20±0.57</td></tr><tr><td>IOI (ours)</td><td>0.942</td><td>34.3</td><td>0.675</td><td>0.116</td><td>3.38±0.55</td></tr></table>
+
+Table 22. Comparison results on the "Red Kayak" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.904</td><td>30.6</td><td>0.549</td><td>0.172</td><td>3.42±0.56</td></tr><tr><td>NVW (2021)</td><td>0.906</td><td>30.7</td><td>0.551</td><td>0.170</td><td>3.62±0.56</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.905</td><td>30.7</td><td>0.553</td><td>0.168</td><td>3.42±0.56</td></tr><tr><td>AdvJND (2020)</td><td>0.895</td><td>32.0</td><td>0.515</td><td>0.103</td><td>3.04±0.57</td></tr><tr><td>UAP (2022)</td><td>0.826</td><td>26.0</td><td>0.400</td><td>0.346</td><td>0.77±0.77</td></tr><tr><td>FACPA (2023b)</td><td>0.880</td><td>28.6</td><td>0.491</td><td>0.194</td><td>1.48±0.68</td></tr><tr><td>IOI (ours)</td><td>0.963</td><td>34.1</td><td>0.693</td><td>0.055</td><td>4.09±0.55</td></tr></table>
+
+Table 23. Comparison results on the "Ducks Take Off" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.893</td><td>33.3</td><td>0.598</td><td>0.182</td><td>3.48±0.56</td></tr><tr><td>NVW (2021)</td><td>0.902</td><td>33.5</td><td>0.612</td><td>0.168</td><td>4.29±0.54</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.899</td><td>33.7</td><td>0.615</td><td>0.168</td><td>3.56±0.56</td></tr><tr><td>AdvJND (2020)</td><td>0.892</td><td>35.0</td><td>0.584</td><td>0.111</td><td>3.66±0.55</td></tr><tr><td>UAP (2022)</td><td>0.800</td><td>28.8</td><td>0.444</td><td>0.333</td><td>0.86±0.86</td></tr><tr><td>FACPA (2023b)</td><td>0.877</td><td>31.5</td><td>0.561</td><td>0.221</td><td>1.75±0.73</td></tr><tr><td>IOI (ours)</td><td>0.944</td><td>34.7</td><td>0.696</td><td>0.119</td><td>4.72±0.54</td></tr></table>
+
+Table 24. Comparison results on the "Tractor" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.  
+
+<table><tr><td>Method</td><td>SSIM ↑</td><td>PSNR ↑</td><td>VIF ↑</td><td>LPIPS ↓</td><td>Subjective score ↑</td></tr><tr><td>FGSM (2015), SSAH (2022), Zhang et al. (2022b)</td><td>0.807</td><td>29.7</td><td>0.491</td><td>0.240</td><td>2.55±0.65</td></tr><tr><td>NVW (2021)</td><td>0.784</td><td>28.8</td><td>0.458</td><td>0.263</td><td>2.07±0.66</td></tr><tr><td>Korhonen et al. (2022b)</td><td>0.780</td><td>28.8</td><td>0.456</td><td>0.269</td><td>2.23±0.65</td></tr><tr><td>AdvJND (2020)</td><td>0.812</td><td>31.8</td><td>0.485</td><td>0.175</td><td>1.94±0.66</td></tr><tr><td>UAP (2022)</td><td>0.764</td><td>27.3</td><td>0.424</td><td>0.325</td><td>0.74±0.74</td></tr><tr><td>FACPA (2023b)</td><td>0.834</td><td>29.7</td><td>0.525</td><td>0.273</td><td>1.41±0.69</td></tr><tr><td>IOI (ours)</td><td>0.881</td><td>30.2</td><td>0.549</td><td>0.161</td><td>3.29±0.63</td></tr></table>
+
+Table 25. Comparison results on the "Park Joy" video and PaQ-2-PiQ attacked model (Ying et al., 2020) with relative gain aligning. FR quality metric score for video is calculated as a mean of quality scores on each frame.

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