| FL defense category (src) | FL defense methods | Can defend | U-P tradeoff analysis | Defender's perspective analysis |
| U-CA | EA | P-CA | DIm | DB | DV |
| Adversarial training (C) | FAT [61], RS [8], FedDynAT [39], GALP [13] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Byzantine robust aggregation techniques (S) | Krum [3], ShieldFL [30], FLTrust [5] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Data/ update analysis (S and/or C) | DeepSight [38], FL-Defender [18], SparseFed [36] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Secure multi-party computation (C) | AMPC [58], Byrd et al. [4] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| Trusted execution environments (C) | Flatee[35], Chen et al. [6], PPFL [34] | 0 | 0 | 0 | 0 | 0 | ○ | ○ |
| Differential privacy (C) | NbAFL [52], Hu et al. [16], 2DP-FL [54] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| Homomorphic encryption (C) | DCAE [62], PEFL [29], Batchcrypt[57] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| FCD (ours) src: S and C | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
",
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+ "type": "text",
+ "text": "ing both utility and privacy threats in a unified manner. This perspective is essential as it offers insights into the defense's robustness, adaptability, budget, multipurpose efficiency, visibility, and overall impact on the FL system. Further, within the domain of FL, adversaries can exhibit multifaceted intentions, spanning both utility and privacy threats. Deploying distinct defense mechanisms for each individual attack escalates the defender's budget, increases the overall computational complexity, and reduces the adaptability to practical FL trustworthy systems. Also, defenses that are easily accessible and transparent to attackers are susceptible to circumvention. While differential privacy (DP) [16, 52, 54], secure multi-party computation (MPC) [4, 58], and trusted execution environments (TEE) [6, 34, 35] have been proposed as potential solutions to strengthen privacy in FL, none of the existing solutions offers a satisfactory unification of privacy and utility defense under different attacks, coupled with reasonable efficiency [45].",
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+ "text": "In this work, we propose an approach called Federated Cryptography Defense (FCD) to counter both utility and privacy evasion attacks through the strategic implementation of fully homomorphic encryption (FHE) in the data space via row-based cryptography transformation. Unlike existing defense methods (as shown in Table 1), which typically focus on defending against either utility or privacy threats, our defense framework aligns with the defender's perspective by working in the data space rather than gradient space. It stands out as a practical, cost-effective, and multipurpose defense mechanism. One key advantage is the",
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+ "text": "preservation of data confidentiality, achieved through transferring the entire process on encrypted data using a shared secret key. Further, our approach enables server-side testing on encrypted data without the need for decryption, thus significantly reducing computational complexity. On the other hand, clients engage in training on encrypted data, where we introduce a novel distillation loss guided by the Kullback-Leibler (KL) divergence. This novel loss function minimizes the distillation loss between the pretrained teacher model and the local student model responses. In addition, the KL divergence loss, combined with the conventional cross-entropy loss, serves as a total loss for updating the local model. Furthermore, we have developed two distinct threat models, TM1 and TM2, addressing utility and privacy vulnerabilities in FL and expanding on the attacker's potential impact on the FL system. We present the first proposal for this type of unified defense in FL to the best of our knowledge. Our contributions are:",
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+ "1. Novel defense: We introduce a unified defense framework called FCD, which uses row-based transposition cipher to defend against both evasion adversarial and MIA attacks at the server aligning to the defender's perspective.",
+ "2. Loss function: To enable effective learning, we propose a novel distillation loss guided by KL divergence between teacher and student model predictions on the client's side.",
+ "3. Threat models: We introduce two realistic threat models (TM1 and TM2) concerning to utility and privacy attacks.",
+ "4. Theoretical analysis: We present a possible theoretical analysis of our method w.r.t. convergence and resilience to attacks.",
+ "5. Effective tradeoff analysis: We perform extensive experiments with five different datasets and models, namely, GTSRB (CNN), KBTS (CNN), CIFAR10 (ResNet18), CIFAR100 (VGG11), and EMNIST (LeNet), alternating between threat models. We also investigate the performance under homogeneous and heterogeneous (non-IID) data shard settings."
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+ "text": "2. Preliminaries",
+ "text_level": 1,
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+ "text": "Definition 2.1 (FL data setup.) We consider an FL system with a central server and $n$ clients. Each client $C_k$ , where $k \\in [1, n]$ , possesses its private local dataset $D_k$ , referred to as a shard. In this context, we define $D_k = \\{\\mathcal{X}_{i,k}, \\mathcal{Y}_{i,k}\\}_{i=1}^{\\mathcal{N}_k} \\subseteq \\mathbb{R}^d \\times \\mathbb{R}$ . Without loss of generality, we assume that $\\| \\mathcal{X}_{i,k} \\|_2 = 1$ holds for all $k \\in [1, n]$ , where $i \\in [1, N_k]$ and the final component of each point is fixed at $1/2$ , denoted as $\\mathcal{X}^d = 1/2$ ( $l_2$ norm data normalization). The server uses synchronous federated weighted averaging",
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+ "text": "24388",
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+ "text": "(FedAvg) [32] for aggregation, represented as:",
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+ "type": "equation",
+ "text": "\n$$\n\\theta_ {g} ^ {t + 1} = \\theta_ {g} ^ {t} + \\sum_ {k = 1} ^ {n} \\lambda_ {k} \\nabla \\theta_ {k} ^ {t} \\tag {1}\n$$\n",
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+ "text": "where $\\lambda_{k} = \\frac{\\mathcal{N}_{k}}{\\sum\\mathcal{N}_{k}}$ and $\\sum_{k}\\lambda_{k} = 1$",
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+ "text": "We investigate two FL data shard settings: (i) Homogeneous, where each client's dataset size is identical, i.e., $|\\mathcal{D}_1| = |\\mathcal{D}_2| = ..|\\mathcal{D}_n| = \\frac{|\\mathcal{D}|}{n}$ , and (ii) Heterogeneous, involving non-independent and non-identically (non-IID) distributed data achieved by partitioning the dataset using a Dirichlet distribution [33] with parameter $\\beta = 1$ among clients. Additional information on standard FL, evasion attacks in FL, homomorphic encryption, and transposition cipher is provided in the supplementary material.",
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+ "text": "Threat model. We introduce two distinct threat models, TM1 and TM2, formulated to reflect real-world FL production deployment settings, as illustrated in Figure 1. Our threat models address an honest-but-curious (HbC) adversary at the central server, as inspired by related work [41, 48]. In TM1 (evasion utility attack), the adversary conducts an indiscriminate evasion attack by manipulating test data at the central server during inference, aiming to misclassify a substantial portion of the inputs [24, 48]. In TM2 (privacy attack), the adversary's goal shifts to performing a model inversion attack (MIA) [15] with the aim of reconstructing private data. We provide our practical assumptions and more details about the threat models in the Supplementary material.",
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+ "Figure 1. Overview of two different threat models (TM) with potential vulnerabilities and attacks during inference."
+ ],
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+ "text": "3. Proposed Framework",
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+ "text": "Problem statement. In each communication round $t$ , suppose that the server receives model updates from $m$ clients, and there exists a test dataset $\\mathcal{D}_{test} = \\{\\mathcal{X}_i, \\mathcal{Y}_i\\}_{i=1}^{\\mathcal{N}_{te}}$ with a size of $\\mathcal{N}_{te}$ . The adversary introduces a $\\mu \\rho$ -bounded poisoned test dataset in TM1, denoted as $\\tilde{\\mathcal{D}}_{\\mathrm{test}} = \\{\\tilde{\\mathcal{X}}_i, \\tilde{\\mathcal{Y}}_i\\}_{i=1}^{\\tilde{\\mathcal{N}}_{te}}$ , with the dataset size $\\hat{\\mathcal{N}}_{te}$ varying based on the attack volume $\\rho$ as:",
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+ "text": "Definition 3.1 ( $\\mu \\rho$ -bounded adversary.) Let $\\mathcal{F}$ be the model's function class. An adversarial perturbation is characterized by the mapping $\\mathcal{A} := \\mathcal{F} \\times \\mathcal{X} \\times \\mathbb{R} \\to \\tilde{\\mathcal{X}}$ . For $\\mu > 0$ , we define the $l_2$ norm ball as $\\mathcal{B}_2(\\mathcal{X}, \\mu) := \\{\\tilde{\\mathcal{X}} \\in \\mathbb{R}^d : \\| \\tilde{\\mathcal{X}} - \\mathcal{X} \\| \\leq \\mu\\} \\cap \\mathcal{X}$ . We classify the adversary as $\\mu \\rho$ -bounded if it adheres to the condition $\\mathcal{A}(\\mathcal{F}, \\mathcal{X}_i, \\mathcal{Y}_i)_{i=1}^\\rho \\in \\mathcal{B}(\\mathcal{X}_i, \\mu)_{i=1}^\\rho$ . Furthermore, for a given $\\mu > 0$ , $\\rho > 0$ , we denote the worst-case adversarial perturbation as $\\mathcal{A}^+ := \\arg \\max_{\\tilde{\\mathcal{X}} \\in \\mathcal{B}(\\mathcal{X}, \\mu)} \\mathcal{L}(f_\\theta(\\tilde{\\mathcal{X}}_i, \\mathcal{Y}_i)_{i=1}^\\rho)$ , where $\\mathcal{L}$ represents the loss function of the global model.",
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+ "text": "Next, we formulate the objective of the adversary in MIA attack (TM2) using the below definition.",
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+ "text": "Definition 3.2 ( $\\epsilon$ -distorted.) Let $\\mathcal{X}^*$ represent the data reconstructed by the adversary using the MIA attack. The objective of the adversary is to minimize the reconstruction error $P$ in terms of as the mean squared error (MSE) between the reconstructed data $\\mathcal{X}^*$ and the original input data $\\mathcal{X}$ as",
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+ "text": "\n$$\nP := \\| \\mathcal {X} ^ {*} - \\mathcal {X} \\| _ {2} \\equiv \\frac {1}{\\mathcal {N} _ {t e}} \\sum_ {i = 1} ^ {\\mathcal {N} _ {t e}} \\left(\\mathcal {X} _ {i} ^ {*} - \\mathcal {X} _ {i}\\right) ^ {2} \\leq \\epsilon , f o r s o m e \\epsilon \\geq 0,\n$$\n",
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+ "text": "where $\\epsilon$ is the distortion of the reconstructed image.",
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+ "text": "Further, the global model, $\\mathcal{G}_{\\theta_g}$ attains a global test accuracy of $\\mathcal{A}_g$ before the attack and $\\mathcal{A}_g^*$ after the attack (Definition 3.1). Now, our FCD integrated clients have two primary objectives. First, they aim to minimize the impact of the attack, which is quantified by $U = (\\mathcal{A}_g - \\mathcal{A}_g^*)$ , in order to preserve the global test accuracy. Second, they seek to enhance the privacy of local data $\\mathcal{D}_k = \\{\\mathcal{X}_{i,k},\\mathcal{Y}_{i,k}\\}_{i = 1}^{\\mathcal{N}_k}$ by maximizing the reconstruction error $P$ . Here, $U$ and $P$ define the utility and privacy gains in the context of FL, where lower values of $U$ and higher values of $P$ are considered desirable.",
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+ "text": "FCD framework description. The FCD framework introduces modifications to standard FL, which can be categorized into server-side and client-side changes. Algorithm 1 and Figure 2 present our FCD integrated FL system.",
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+ "text": "Server side. To initiate the FL process, the central server initializes a key denoted as $\\mathcal{K}$ . This key consists of unique random integers in the range from 0 to $h - 1$ , where $h$ represents the height of the input image data. The key $\\mathcal{K}$ is defined as $\\mathcal{K} = [\\kappa_0,\\kappa_1,\\dots ,\\kappa_i,\\kappa_j,\\dots ,\\kappa_{h - 1}]$ , and it follows a property that if $i,j\\in 0,\\ldots ,h - 1$ with $i\\neq j$ , then $\\kappa_{i}\\neq \\kappa_{j}$ . This property guarantees that no row in the input image can be missed, ensuring the completeness of the data. During each communication round $t$ , the central server follows the standard FL procedure by sending the current version of the global model parameters $\\theta_g^t$ to all clients. Synchronous federated weighted averaging-based aggregation is employed at the server, as outlined in [32]. Further, the central server performs the FCD transformation on the evasion-attacked",
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+ "text": "test data to obtain $\\mathcal{E}(\\hat{\\mathcal{D}}_{test})$ . Subsequently, the server tests the current aggregated global model $\\mathcal{G}_{\\theta}^{t}$ on the FCD transformed test data, i.e., $\\mathcal{E}(\\hat{\\mathcal{D}}_{test})$ , as opposed to standard FL systems that test on the original poisoned test data $\\hat{\\mathcal{D}}_{test}$ . Client side. Let $\\mathcal{E}(\\mathcal{X}_k)$ represent the FCD-encrypted form of input samples from client $\\mathcal{C}_k$ based on the shared secret key $\\mathcal{K}$ . We denote $\\mathcal{Q}_k$ as the prediction probabilities when the local model $f_{\\theta,k}$ is applied to the transformed data $\\mathcal{E}(\\mathcal{X}_k)$ , which can be expressed as $\\mathcal{Q}_k = \\sigma(f_{\\theta,k}(\\mathcal{E}(\\mathcal{X}_k)))$ , where $\\sigma(.)$ denotes the softmax function. Consequently, the cross-entropy loss $\\mathcal{L}_{CE_k}$ of client $\\mathcal{C}_k$ , trained on FCD-encrypted data (in contrast to general FL systems that calculates $\\mathcal{L}_{CE_k}$ on normal data $\\mathcal{X}_k$ ), can be calculated as follows:",
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+ "text": "\n$$\n\\mathcal {L} _ {C E _ {k}} = \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\mathcal {L} _ {C E} \\left(\\mathcal {Q} _ {i, k}, \\mathcal {Y} _ {i, k}\\right) = - \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\mathcal {Y} _ {i, k} \\log \\mathcal {Q} _ {i, k}. \\tag {2}\n$$\n",
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+ "text": "To enhance the learning process and facilitate knowledge transfer, we employ the knowledge distillation (KD) approach [60], guided by the Kullback-Leibler (KL) divergence. In this approach, the student model $f_{\\theta,k}$ is trained to mimic the predictions of the teacher model. In our FCD framework, there is no distinct teacher model, instead, it is the same local model $f_{\\theta,k}$ initially trained on normal data at the beginning of the FL process. We introduce $\\mathcal{P}_k$ to denote the prediction probabilities when the local teacher model $f_{\\theta,k}$ is applied to normal data $\\mathcal{X}_k$ , represented as $\\mathcal{P}_k = \\sigma(f_{\\theta,k}(\\mathcal{X}_k))$ . Subsequently, we calculate the distillation loss guided by KL divergence $(\\mathcal{L}_{KLD_k})$ of client $k$ between the prediction probabilities of the teacher and the student local model, which has been trained on normal and FCD-encrypted data, respectively, as follows:",
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+ "text": "\n$$\n\\mathcal {L} _ {K L D _ {k}} \\left(\\mathcal {Q} _ {k} \\| \\mathcal {P} _ {k}\\right) = \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\sum_ {r = 1} ^ {\\mathcal {R}} \\mathcal {Q} _ {i, k} ^ {r} \\log \\frac {\\mathcal {Q} _ {i , k} ^ {r}}{\\mathcal {P} _ {i , k} ^ {r}}, \\tag {3}\n$$\n",
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+ "text": "where $\\mathcal{R}$ is the number of classes. Hence, the complete loss for client $\\mathcal{C}_k$ , unlike traditional FL systems that solely consider $\\mathcal{L}_{CE_k}$ , is expressed as:",
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+ "text": "\n$$\n\\mathcal {L} _ {k} = \\mathcal {L} _ {C E _ {k}} + \\alpha_ {k} \\mathcal {L} _ {K L D _ {k}}, \\tag {4}\n$$\n",
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+ "text": "where $\\alpha_{k}$ serves as the weighting factor, regulating the balance between the cross-entropy loss and the distillation loss for client $\\mathcal{C}_k$ . Finally, each client minimizes the total loss $\\mathcal{L}_k$ over its FCD-transformed data $\\mathcal{E}(\\mathcal{X}_k)$ across $E$ local iterations. The local model $f_{\\theta ,k}$ undergoes parameter updates through a backward pass, adjusting as $\\theta_k^{\\mathrm{new}} = \\theta_k^{\\mathrm{old}} - \\eta_k\\nabla_{\\theta_k^{\\mathrm{old}}}\\mathcal{L}_k$ , where $\\eta_{k}$ signifies the learning rate of client $\\mathcal{C}_k$ . Following each training phase, the client(s) transmit their local updates, denoted as $\\nabla \\theta_k^t = \\theta_k^t -\\theta_g^t$ , back to the server.",
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+ "sub_type": "algorithm",
+ "code_caption": [
+ "Algorithm 1 Standard FL with our FCD framework"
+ ],
+ "code_body": "Input: Global model $\\mathcal{G}_{\\theta ,t}$ , local data $\\mathcal{D}_k = (\\mathcal{X}_k,\\mathcal{Y}_k)$ \nOutput: Global test accuracy $\\mathcal{A}_g$ \n1: Client execution $(\\theta_g^t,\\mathcal{K})$ .. \n2: for each client $k = 1$ to n do \n3: Train teacher model $f_{\\theta ,k}$ on normal data $\\mathcal{D}_k$ \n4: $\\{\\mathcal{P}_{i,k}\\}_{i = (1,\\mathcal{N}_k)}\\gets \\{\\sigma (f_{\\theta ,k}(\\mathcal{X}_{i,k}))\\}_{i = (1,\\mathcal{N}_k)}$ \n5: $\\mathcal{E}(\\mathcal{X}_k)\\leftarrow \\mathrm{FCD}(\\mathcal{X}_k,\\mathcal{K})$ \n6: for each client $k = 1$ to n do \n7: Initialize the local model $\\theta_k^t\\gets \\theta_g^t$ \n8: for $b = 1$ to batches $\\in \\mathcal{E}(\\mathcal{X}_k)$ do \n9: $\\{\\mathcal{Q}_{b,k}\\} \\gets \\{\\sigma (f_{\\theta ,0}(\\mathcal{E}(\\mathcal{X}_k[b))))\\}$ \n10: $\\mathcal{L}_{CE_k}(\\mathcal{Q}_{b,k},\\mathcal{Y}_{b,k})\\gets$ Eq. 2, Cross-entropy loss \n11: $\\mathcal{L}_{KLD_k}(\\mathcal{Q}_{b,k}\\| \\mathcal{P}_{b,k})\\gets$ Eq. 3, Distillation loss \n12: $\\mathcal{L}_k\\gets \\mathcal{L}_{CE_k} + \\alpha_k\\mathcal{L}_{KLD_k}\\triangleright$ Total loss (Eq. 4) \n13: $\\theta_k^t\\gets \\theta_k^t -\\eta \\nabla_{\\theta_k^t}\\mathcal{L}_k$ \n14: $\\nabla \\theta_k^t\\gets \\theta_k^t -\\theta_g^t$ \n15: return $\\nabla \\theta_k^t$ \n16: Server execution $(\\nabla \\theta_k^t)$ . \n17: Share $\\theta_g^t,\\mathcal{K}$ to all the clients \n18: Receive model updates from selected clients $\\leftarrow \\nabla \\theta_k^t$ \n19: Perform model aggregation using FedAvg (Eq. 1) \n20: Update the global model parameter: $\\theta_g^{t + 1}$ \n21: $\\hat{\\chi}_{test}\\gets$ Poisoned test data ▷TM1, TM2 \n22: $\\mathcal{E}(\\hat{\\chi}_{test})\\gets$ FCD ( $\\hat{\\chi}_{test},\\mathcal{K}$ ) \n23: Compute $\\mathcal{A}_g\\gets$ Test $(\\mathcal{G}_{\\theta_g^{t + 1}},\\mathcal{E}(\\hat{\\chi}_{test}))$ \n24: return $\\mathcal{A}_g$",
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+ "text": "FCD cryptographic encryption. The missing element in the framework described above is the FCD encryption (a detailed algorithm and a visual representation are in the Supplementary material). We introduce a homomorphic encryption technique for FL utilizing a symmetric transposition cipher [43], with a secret key $\\mathcal{K}$ shared among all clients by the server. To ensure data privacy and minimize the potential for brute-force attacks, we define the key's dimension as $\\mathcal{K} \\in \\mathbb{R}^h$ , where $h$ corresponds to the image height. The FCD encryption process comprises two steps: (1) transposing the input image $x$ to obtain $x' = x^T$ , where rows are transformed into columns, and (2) reordering rows based on the sequence specified in the secret key $\\mathcal{K}$ . Essentially, the positions of pixel values in all colour channels $(R,G,B)$ are altered according to $\\mathcal{K}$ , i.e., $\\mathcal{R}_{\\mathcal{K}}(x') = x'[:$ , $\\mathcal{K},\\colon]$ , where $\\mathcal{R}_{\\mathcal{K}}(.)$ denotes the row-wise shuffling of input data based on the secret key. Further, we introduce the concept of $\\zeta$ -separability for the FCD encrypted data space. This concept relates to the distance between FCD-encrypted data space and adversarial perturbations. It specifies that the FCD data space maintains a separation of at least $\\zeta$ from attacked data, ensuring robustness against $\\mu\\rho$ -bounded adversarial perturbation (as defined in Definition 3.1). This",
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+ "Figure 2. Overview of the FCD-integrated FL system, with a focus on one client's training and server-side execution. We compute the cross-entropy loss $(\\mathcal{L}_{CE})$ for the local model on encrypted data and calculate the distillation loss $(\\mathcal{L}_{KL})$ using KL divergence between the pretrained teacher model and the local student model for normal and encrypted data, respectively. These losses collectively update the local model $f_{\\theta}$ . Similarly, test data is transformed into the encrypted space to counter potential attacks."
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+ "text": "property is crucial to ensure that the FCD-encrypted data space effectively mitigates utility evasion attacks during inference (TM1).",
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+ "text": "Definition 3.3 ( $\\zeta$ -separability.)",
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+ "text": "\n$$\n\\operatorname {L e t} \\mathcal {E} (\\mathcal {D}) := \\bigcup_ {k \\in [ 1, n ], i \\in [ 1, \\mathcal {N} _ {k} ]} \\left\\{\\left(\\mathcal {E} \\left(\\mathcal {X} _ {k, i}\\right), \\mathcal {Y} _ {k, i}\\right) \\right\\} \\subseteq \\mathbb {R} ^ {d} \\times \\mathbb {R}\n$$\n",
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+ "text": "represent the FCD encrypted local training data space. Similarly, $\\mathcal{E}(\\mathcal{D}_{test})$ denotes the FCD encrypted test data space such that for both training and test splits, $\\mathcal{E}(\\mathcal{X}) = \\mathcal{R}_{\\mathcal{K}}(\\mathcal{X}^{\\mathcal{T}})$ . We say that our FCD encrypted data space is $\\zeta$ -separable w.r.t. $\\mu \\rho$ -bounded adversarial data perturbation $\\tilde{\\mathcal{X}}$ (as given in Definition 3.1) for some $\\zeta > 0$ iff $\\| \\mathcal{E}(\\mathcal{X}_i) - \\tilde{\\mathcal{X}}_j \\| \\geq \\zeta$ holds for any $\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D})$ , $\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D}_{test})$ , and $\\tilde{\\mathcal{X}}_j \\in \\mathcal{A}_{\\mu \\rho}(\\mathcal{X}_{test})$ .",
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+ "text": "Rationale for FCD Design. The transposition of data before row-wise transformations enhances information security robustness by obscuring critical transformation details, providing defense against black-box evasion attacks. The use of a row-based transposition cipher, compatible with row-major programming languages (Python, C, and $\\mathrm{C + + }$ ) [46], ensures efficiency in memory access, parallel processing, and cryptographic operations, aligning with both security and computational considerations in our FCD integrated FL system. Please refer to the Supplementary material for a more detailed rationale.",
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+ "type": "text",
+ "text": "FCD dual property benefits aligned with defender's perspective. (i) Low visibility: The FCD transformation, incorporating both transposition and a secret key, ensures low visibility by creating unique gradients for each key, offering robust protection against various attacks. (ii) Low budget: FCD's vectorized operations and constrained key dimension facilitate efficient implementation, making it suitable for large-scale FL systems with minimal resource requirements and addressing concerns related to processing time",
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+ "type": "text",
+ "text": "and aligning w.r.t. defender's perspective. Please refer to the Supplementary material for more details about the FCD dual property.",
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+ "text": "Lemma 3.4 The expected time complexity of our FCD encryption function $\\mathcal{E}(\\mathcal{X})$ is linear, specifically $\\mathcal{O}(nh)$ , where $n$ represents the number of samples, and $h$ denotes the image height. This is a significant enhancement compared to the $\\mathcal{O}(n_w\\mathcal{N}^2\\log \\mathcal{N})$ time complexity required for encryption and decryption of model parameters, where $n_W$ represents the number of model parameters, and $\\mathcal{N}$ is the bit length of the key [19]. (Please refer Supplementary material for the proof.)",
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+ "type": "text",
+ "text": "3.1. Convergence and Feasibility Analysis of FCD",
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+ "text": "We start by establishing the convergence of our FCD integrated FL global model. Subsequently, we demonstrate the robustness of our FCD against gradient noise-based $\\mu \\rho$ -bounded data poisoning attacks (as defined in Definition 3.1). Finally, we establish the resilience of our FCD against MIAs with $\\epsilon$ -distorted characteristics (as described in Definition 3.2) at the server. We provide the respective proofs in the Supplementary material.",
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+ "text": "Theorem 3.5 FCD convergence. Under the regularity conditions of $L$ -smoothness, $\\tau$ -strong convexity, and a decaying learning rate, our FCD integrated FL with clients trained on encrypted data $\\mathcal{E}(X)$ , obtained using $FCD(\\mathcal{X},\\mathcal{K})$ , the global model converges to",
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+ "text": "\n$$\n\\mathbb {E} \\left[ \\mathcal {G} _ {\\theta_ {g}} \\right] - \\mathcal {G} ^ {*} \\leq \\frac {2 L}{\\tau (\\gamma + T)} \\left(\\frac {B + C}{\\tau} + 2 L \\| \\theta_ {g} ^ {0} - \\theta_ {g} ^ {*} \\| ^ {2}\\right) + D.\n$$\n",
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+ "text": "Here, the variables have the following meanings: $B = \\Gamma + (E - 1)^2$ , where $\\Gamma$ represents the measure of non-IID data distribution. $C$ signifies the client selection for aggregation, with $C = 0$ when all $n$ client updates are considered. $T$ denotes the number of global communication",
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+ "type": "text",
+ "text": "rounds, and $\\mathcal{G}^*$ represents the optimal global model [28]. The positive constant $D\\leq \\psi$ quantifies how distillation enhances the convergence rate. It accelerates convergence by transferring learnable knowledge from the local teacher model trained on $\\mathcal{X}$ to a student model trained on $\\mathcal{E}(\\mathcal{X})$ . The constant $\\psi$ quantifies how distillation accelerates convergence in our specific setup.",
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+ "text": "Corollary 3.5.1 (Robustness to $\\mu \\rho$ -bounded attacks.) Let $\\mathcal{A} := \\mathcal{G}_{\\theta_g} \\times \\mathcal{X}_{test} \\times \\mathbb{R} \\to \\tilde{\\mathcal{X}}_{test}$ be the induced adversarial perturbations on the test data using the black-box global model $\\mathcal{G}_{\\theta_g}$ . This perturbation adheres to the constraint $\\| \\tilde{\\mathcal{X}}_{test} - \\mathcal{X}_{test} \\| \\leq \\mu$ such that $\\mathcal{A}(\\mathcal{G}_{\\theta_g}, \\mathcal{X}_{test,i}, \\mathcal{Y}_i)_{i=1}^\\rho \\in \\mathcal{B}(\\mathcal{X}_{test,i}, \\mu)_{i=1}^\\rho$ (according to Definition 3.1). Furthermore, $\\mathcal{G}_{\\theta_g}$ is robust to $\\tilde{\\mathcal{X}}_{test}$ through the mechanism of our FCD integrated FL system. It allows $\\mathcal{G}_{\\theta_g}$ to test on encrypted data based on $\\mathcal{K}$ , i.e., $\\mathcal{E}(\\tilde{\\mathcal{X}}_{test})$ instead of $\\tilde{\\mathcal{X}}_{test}$ . This robustness is achieved because the adversary's objective is to generate perturbations in the original test data space that maximize the global loss (Definition 3.1), as denoted by $\\mathcal{A}^+ := \\arg \\max_{\\tilde{\\mathcal{X}}_{test} \\in \\mathcal{B}(\\mathcal{X}_{test}, \\mu)} \\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\tilde{\\mathcal{X}}_{test,i}, \\mathcal{Y}_i)_{i=1}^\\rho)$ . However, this adversarial strategy is effective when training and testing occur in the normal data space $\\mathcal{X}$ at the client side, matching the distribution of $\\mathcal{X}_{test}$ . In our FCD integrated FL system, training and testing occur in the encrypted data space. Notably, $\\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\tilde{\\mathcal{X}}_{test,i}, \\mathcal{Y}_i)_{i=1}^\\rho) \\neq \\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\mathcal{E}(\\tilde{\\mathcal{X}}_{test,i}), \\mathcal{Y}_i)_{i=1}^\\rho)$ as $\\mathcal{E}(\\tilde{\\mathcal{X}}_{test}) \\triangleq \\mathcal{R}_{\\mathcal{K}}(\\tilde{\\mathcal{X}}_{test}^T) \\neq \\tilde{\\mathcal{X}}_{test}$ and $\\| \\mathcal{E}(\\mathcal{X}_i) - \\tilde{\\mathcal{X}}_j \\| \\geq \\zeta$ holds for any $\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D})$ , $\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D}_{test})$ , and $\\tilde{\\mathcal{X}}_j \\in \\mathcal{A}_{\\mu\\rho}(\\mathcal{X}_{test})$ (Definition 3.3). Hence, the learning process occurs in different data spaces unknown to the adversary. Consequently, the gradients of the loss function trained on normal data are notably distinct from those trained on FCD encrypted data, effectively mitigating $\\mu\\rho$ -bounded adversarial data.",
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+ "text": "Theorem 3.6 (Resilience to $\\epsilon$ -distorted MIA attacks.) Let $\\mathcal{X}^*$ represent the data reconstructed by the adversary using MIA attack as per the conditions specified in Definition 3.2. We demonstrate that training on FCD-encrypted data space, denoted as $\\mathcal{E}(\\mathcal{X})$ , imparts resilience to $\\epsilon$ -distorted MIA attacks. Specifically, our result establishes that:",
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+ "text": "\n$$\n\\left| \\| \\mathcal {X} ^ {*} \\| - \\| \\mathcal {E} (\\mathcal {X}) \\| \\right| \\leq \\epsilon + \\delta , f o r s o m e \\epsilon \\geq 0 a n d \\delta \\geq 0.\n$$\n",
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+ "text": "4. Experiments",
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+ "text": "Datasets, implementation details, and metrics. Table 2 presents comprehensive details about the datasets, FL setup, attack percentage, and metrics. For TM1, we employed the black-box and active data poisoning technique called MSimBA [23]. Each experiment was conducted thrice, and",
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+ "text": "results were averaged with standard deviations presented. Please refer to the Supplementary material for more details. Baselines. For TM1, we have chosen FAT [61] and Randomized Smoothing (RS) [8] based on their relevance and applicability in evasion attacks within FL, as outlined in Table 1. Additionally, we explore the effectiveness under the following configurations: (a) With/without attack and defense and (b) two FCD setups under different client counts and attack percentages $(A_{p})$ . These setups include FCD-homoFL (homogeneous FL) and FCD-hetFL (heterogeneous FL). In TM2, we follow the baselines as per [27].",
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+ "text": "4.1. FCD Result Discussion",
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+ "text": "Benign setting. Table 3 showcases the performance of our FCD defense, along with other baselines and configurations, across four datasets under a no attack scenario. Notably, FCD demonstrates comparable or higher accuracy than vanilla FL, particularly excelling in datasets like GTSRB and KBTS. This improvement is attributed to FCD's ability to capture diverse transformation patterns with a secret key, facilitating reduced training loss and improved model convergence. Furthermore, with its symmetric cryptographic transformation, FCD outperforms FAT and RS for GTSRB and KBTS. However, in CIFAR10 and EMNIST, FCD's performance, while slightly lower than base FL, surpasses FAT and RS. This trade-off between utility and privacy is expected, as FCD provides both aspects with a modest decrease in base accuracy.",
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+ "text": "Attack setting. We analyse FCD effectiveness under two threat models separately.",
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+ "text": "(i) TM1: Table 4 illustrates the impact on utility $(U)$ for FCD compared to other baselines under M-SimBA evasion attack on GTSRB and KBTS datasets. The results indicate a significant utility degradation with higher attack percentages and more clients, emphasizing the impact of substantial test data modification by M-SimBA. FCD consistently demonstrates superior performance and robustness across all attack percentages. In homogeneous FL settings, FCD shows substantial improvement, maintaining a consistently lower attack impact on utility compared to other defenses. Particularly at $A_{p} = 30\\%$ , the attack impact on utility is 2-6 units lower for FCD as the number of clients increases, compared to other baselines. Similar trends are observed for $A_{p} = 50\\%$ , $100\\%$ . In heterogeneous FL settings, FCD outperforms other defenses, showcasing robustness even when clients are trained on varying amounts of non-IID data. For the KBTS dataset, FCD exhibits substantial improvement and lower attack impact across all attack percentages and FL settings compared to other defenses. The limited training data in the KBTS dataset results in the RS technique showing limited robustness, especially for $A_{p} = 30\\%$ in homogeneous and heterogeneous FL, followed by FAT. Ta",
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+ "Table 2. Comprehensive experimental details: datasets, models, FL setup, attack configuration, and metrics."
+ ],
+ "table_footnote": [],
+ "table_body": "| FL defense category (src) | FL defense methods | Can defend | U-P tradeoff analysis | Defender's perspective analysis |
| U-CA | EA | P-CA | DIm | DB | DV |
| Adversarial training (C) | FAT [61], RS [8], FedDynAT [39], GALP [13] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Byzantine robust aggregation techniques (S) | Krum [3], ShieldFL [30], FLTrust [5] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Data/ update analysis (S and/or C) | DeepSight [38], FL-Defender [18], SparseFed [36] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Secure multi-party computation (C) | AMPC [58], Byrd et al. [4] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| Trusted execution environments (C) | Flatee[35], Chen et al. [6], PPFL [34] | 0 | 0 | 0 | 0 | 0 | ○ | ○ |
| Differential privacy (C) | NbAFL [52], Hu et al. [16], 2DP-FL [54] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| Homomorphic encryption (C) | DCAE [62], PEFL [29], Batchcrypt[57] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| FCD (ours) src: S and C | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
"
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+ "angle": 0,
+ "content": "ing both utility and privacy threats in a unified manner. This perspective is essential as it offers insights into the defense's robustness, adaptability, budget, multipurpose efficiency, visibility, and overall impact on the FL system. Further, within the domain of FL, adversaries can exhibit multifaceted intentions, spanning both utility and privacy threats. Deploying distinct defense mechanisms for each individual attack escalates the defender's budget, increases the overall computational complexity, and reduces the adaptability to practical FL trustworthy systems. Also, defenses that are easily accessible and transparent to attackers are susceptible to circumvention. While differential privacy (DP) [16, 52, 54], secure multi-party computation (MPC) [4, 58], and trusted execution environments (TEE) [6, 34, 35] have been proposed as potential solutions to strengthen privacy in FL, none of the existing solutions offers a satisfactory unification of privacy and utility defense under different attacks, coupled with reasonable efficiency [45]."
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+ "content": "In this work, we propose an approach called Federated Cryptography Defense (FCD) to counter both utility and privacy evasion attacks through the strategic implementation of fully homomorphic encryption (FHE) in the data space via row-based cryptography transformation. Unlike existing defense methods (as shown in Table 1), which typically focus on defending against either utility or privacy threats, our defense framework aligns with the defender's perspective by working in the data space rather than gradient space. It stands out as a practical, cost-effective, and multipurpose defense mechanism. One key advantage is the"
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+ "content": "preservation of data confidentiality, achieved through transferring the entire process on encrypted data using a shared secret key. Further, our approach enables server-side testing on encrypted data without the need for decryption, thus significantly reducing computational complexity. On the other hand, clients engage in training on encrypted data, where we introduce a novel distillation loss guided by the Kullback-Leibler (KL) divergence. This novel loss function minimizes the distillation loss between the pretrained teacher model and the local student model responses. In addition, the KL divergence loss, combined with the conventional cross-entropy loss, serves as a total loss for updating the local model. Furthermore, we have developed two distinct threat models, TM1 and TM2, addressing utility and privacy vulnerabilities in FL and expanding on the attacker's potential impact on the FL system. We present the first proposal for this type of unified defense in FL to the best of our knowledge. Our contributions are:"
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+ "content": "1. Novel defense: We introduce a unified defense framework called FCD, which uses row-based transposition cipher to defend against both evasion adversarial and MIA attacks at the server aligning to the defender's perspective."
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+ "content": "2. Loss function: To enable effective learning, we propose a novel distillation loss guided by KL divergence between teacher and student model predictions on the client's side."
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+ "content": "3. Threat models: We introduce two realistic threat models (TM1 and TM2) concerning to utility and privacy attacks."
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+ "content": "4. Theoretical analysis: We present a possible theoretical analysis of our method w.r.t. convergence and resilience to attacks."
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+ "content": "5. Effective tradeoff analysis: We perform extensive experiments with five different datasets and models, namely, GTSRB (CNN), KBTS (CNN), CIFAR10 (ResNet18), CIFAR100 (VGG11), and EMNIST (LeNet), alternating between threat models. We also investigate the performance under homogeneous and heterogeneous (non-IID) data shard settings."
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+ "content": "Definition 2.1 (FL data setup.) We consider an FL system with a central server and \\( n \\) clients. Each client \\( C_k \\), where \\( k \\in [1, n] \\), possesses its private local dataset \\( D_k \\), referred to as a shard. In this context, we define \\( D_k = \\{\\mathcal{X}_{i,k}, \\mathcal{Y}_{i,k}\\}_{i=1}^{\\mathcal{N}_k} \\subseteq \\mathbb{R}^d \\times \\mathbb{R} \\). Without loss of generality, we assume that \\( \\| \\mathcal{X}_{i,k} \\|_2 = 1 \\) holds for all \\( k \\in [1, n] \\), where \\( i \\in [1, N_k] \\) and the final component of each point is fixed at \\( 1/2 \\), denoted as \\( \\mathcal{X}^d = 1/2 \\) (\\( l_2 \\) norm data normalization). The server uses synchronous federated weighted averaging"
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+ "content": "(FedAvg) [32] for aggregation, represented as:"
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+ "content": "\\[\n\\theta_ {g} ^ {t + 1} = \\theta_ {g} ^ {t} + \\sum_ {k = 1} ^ {n} \\lambda_ {k} \\nabla \\theta_ {k} ^ {t} \\tag {1}\n\\]"
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+ "content": "where \\(\\lambda_{k} = \\frac{\\mathcal{N}_{k}}{\\sum\\mathcal{N}_{k}}\\) and \\(\\sum_{k}\\lambda_{k} = 1\\)"
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+ "content": "We investigate two FL data shard settings: (i) Homogeneous, where each client's dataset size is identical, i.e., \\(|\\mathcal{D}_1| = |\\mathcal{D}_2| = ..|\\mathcal{D}_n| = \\frac{|\\mathcal{D}|}{n}\\), and (ii) Heterogeneous, involving non-independent and non-identically (non-IID) distributed data achieved by partitioning the dataset using a Dirichlet distribution [33] with parameter \\(\\beta = 1\\) among clients. Additional information on standard FL, evasion attacks in FL, homomorphic encryption, and transposition cipher is provided in the supplementary material."
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+ "content": "Threat model. We introduce two distinct threat models, TM1 and TM2, formulated to reflect real-world FL production deployment settings, as illustrated in Figure 1. Our threat models address an honest-but-curious (HbC) adversary at the central server, as inspired by related work [41, 48]. In TM1 (evasion utility attack), the adversary conducts an indiscriminate evasion attack by manipulating test data at the central server during inference, aiming to misclassify a substantial portion of the inputs [24, 48]. In TM2 (privacy attack), the adversary's goal shifts to performing a model inversion attack (MIA) [15] with the aim of reconstructing private data. We provide our practical assumptions and more details about the threat models in the Supplementary material."
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+ "content": "Figure 1. Overview of two different threat models (TM) with potential vulnerabilities and attacks during inference."
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+ "content": "Problem statement. In each communication round \\( t \\), suppose that the server receives model updates from \\( m \\) clients, and there exists a test dataset \\( \\mathcal{D}_{test} = \\{\\mathcal{X}_i, \\mathcal{Y}_i\\}_{i=1}^{\\mathcal{N}_{te}} \\) with a size of \\( \\mathcal{N}_{te} \\). The adversary introduces a \\( \\mu \\rho \\)-bounded poisoned test dataset in TM1, denoted as \\( \\tilde{\\mathcal{D}}_{\\mathrm{test}} = \\{\\tilde{\\mathcal{X}}_i, \\tilde{\\mathcal{Y}}_i\\}_{i=1}^{\\tilde{\\mathcal{N}}_{te}} \\), with the dataset size \\( \\hat{\\mathcal{N}}_{te} \\) varying based on the attack volume \\( \\rho \\) as:"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Definition 3.1 (\\(\\mu \\rho\\)-bounded adversary.) Let \\(\\mathcal{F}\\) be the model's function class. An adversarial perturbation is characterized by the mapping \\(\\mathcal{A} := \\mathcal{F} \\times \\mathcal{X} \\times \\mathbb{R} \\to \\tilde{\\mathcal{X}}\\). For \\(\\mu > 0\\), we define the \\(l_2\\) norm ball as \\(\\mathcal{B}_2(\\mathcal{X}, \\mu) := \\{\\tilde{\\mathcal{X}} \\in \\mathbb{R}^d : \\| \\tilde{\\mathcal{X}} - \\mathcal{X} \\| \\leq \\mu\\} \\cap \\mathcal{X}\\). We classify the adversary as \\(\\mu \\rho\\)-bounded if it adheres to the condition \\(\\mathcal{A}(\\mathcal{F}, \\mathcal{X}_i, \\mathcal{Y}_i)_{i=1}^\\rho \\in \\mathcal{B}(\\mathcal{X}_i, \\mu)_{i=1}^\\rho\\). Furthermore, for a given \\(\\mu > 0\\), \\(\\rho > 0\\), we denote the worst-case adversarial perturbation as \\(\\mathcal{A}^+ := \\arg \\max_{\\tilde{\\mathcal{X}} \\in \\mathcal{B}(\\mathcal{X}, \\mu)} \\mathcal{L}(f_\\theta(\\tilde{\\mathcal{X}}_i, \\mathcal{Y}_i)_{i=1}^\\rho)\\), where \\(\\mathcal{L}\\) represents the loss function of the global model."
+ },
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+ "type": "text",
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+ "content": "Next, we formulate the objective of the adversary in MIA attack (TM2) using the below definition."
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+ "content": "Definition 3.2 (\\(\\epsilon\\)-distorted.) Let \\(\\mathcal{X}^*\\) represent the data reconstructed by the adversary using the MIA attack. The objective of the adversary is to minimize the reconstruction error \\(P\\) in terms of as the mean squared error (MSE) between the reconstructed data \\(\\mathcal{X}^*\\) and the original input data \\(\\mathcal{X}\\) as"
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+ "type": "equation",
+ "bbox": [
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+ "content": "\\[\nP := \\| \\mathcal {X} ^ {*} - \\mathcal {X} \\| _ {2} \\equiv \\frac {1}{\\mathcal {N} _ {t e}} \\sum_ {i = 1} ^ {\\mathcal {N} _ {t e}} \\left(\\mathcal {X} _ {i} ^ {*} - \\mathcal {X} _ {i}\\right) ^ {2} \\leq \\epsilon , f o r s o m e \\epsilon \\geq 0,\n\\]"
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+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "where \\(\\epsilon\\) is the distortion of the reconstructed image."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Further, the global model, \\(\\mathcal{G}_{\\theta_g}\\) attains a global test accuracy of \\(\\mathcal{A}_g\\) before the attack and \\(\\mathcal{A}_g^*\\) after the attack (Definition 3.1). Now, our FCD integrated clients have two primary objectives. First, they aim to minimize the impact of the attack, which is quantified by \\(U = (\\mathcal{A}_g - \\mathcal{A}_g^*)\\), in order to preserve the global test accuracy. Second, they seek to enhance the privacy of local data \\(\\mathcal{D}_k = \\{\\mathcal{X}_{i,k},\\mathcal{Y}_{i,k}\\}_{i = 1}^{\\mathcal{N}_k}\\) by maximizing the reconstruction error \\(P\\). Here, \\(U\\) and \\(P\\) define the utility and privacy gains in the context of FL, where lower values of \\(U\\) and higher values of \\(P\\) are considered desirable."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "FCD framework description. The FCD framework introduces modifications to standard FL, which can be categorized into server-side and client-side changes. Algorithm 1 and Figure 2 present our FCD integrated FL system."
+ },
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+ "type": "text",
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+ "content": "Server side. To initiate the FL process, the central server initializes a key denoted as \\(\\mathcal{K}\\). This key consists of unique random integers in the range from 0 to \\(h - 1\\), where \\(h\\) represents the height of the input image data. The key \\(\\mathcal{K}\\) is defined as \\(\\mathcal{K} = [\\kappa_0,\\kappa_1,\\dots ,\\kappa_i,\\kappa_j,\\dots ,\\kappa_{h - 1}]\\), and it follows a property that if \\(i,j\\in 0,\\ldots ,h - 1\\) with \\(i\\neq j\\), then \\(\\kappa_{i}\\neq \\kappa_{j}\\). This property guarantees that no row in the input image can be missed, ensuring the completeness of the data. During each communication round \\(t\\), the central server follows the standard FL procedure by sending the current version of the global model parameters \\(\\theta_g^t\\) to all clients. Synchronous federated weighted averaging-based aggregation is employed at the server, as outlined in [32]. Further, the central server performs the FCD transformation on the evasion-attacked"
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+ "type": "text",
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+ "content": "test data to obtain \\(\\mathcal{E}(\\hat{\\mathcal{D}}_{test})\\). Subsequently, the server tests the current aggregated global model \\(\\mathcal{G}_{\\theta}^{t}\\) on the FCD transformed test data, i.e., \\(\\mathcal{E}(\\hat{\\mathcal{D}}_{test})\\), as opposed to standard FL systems that test on the original poisoned test data \\(\\hat{\\mathcal{D}}_{test}\\). Client side. Let \\(\\mathcal{E}(\\mathcal{X}_k)\\) represent the FCD-encrypted form of input samples from client \\(\\mathcal{C}_k\\) based on the shared secret key \\(\\mathcal{K}\\). We denote \\(\\mathcal{Q}_k\\) as the prediction probabilities when the local model \\(f_{\\theta,k}\\) is applied to the transformed data \\(\\mathcal{E}(\\mathcal{X}_k)\\), which can be expressed as \\(\\mathcal{Q}_k = \\sigma(f_{\\theta,k}(\\mathcal{E}(\\mathcal{X}_k)))\\), where \\(\\sigma(.)\\) denotes the softmax function. Consequently, the cross-entropy loss \\(\\mathcal{L}_{CE_k}\\) of client \\(\\mathcal{C}_k\\), trained on FCD-encrypted data (in contrast to general FL systems that calculates \\(\\mathcal{L}_{CE_k}\\) on normal data \\(\\mathcal{X}_k\\)), can be calculated as follows:"
+ },
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+ "type": "equation",
+ "bbox": [
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+ "content": "\\[\n\\mathcal {L} _ {C E _ {k}} = \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\mathcal {L} _ {C E} \\left(\\mathcal {Q} _ {i, k}, \\mathcal {Y} _ {i, k}\\right) = - \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\mathcal {Y} _ {i, k} \\log \\mathcal {Q} _ {i, k}. \\tag {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "To enhance the learning process and facilitate knowledge transfer, we employ the knowledge distillation (KD) approach [60], guided by the Kullback-Leibler (KL) divergence. In this approach, the student model \\( f_{\\theta,k} \\) is trained to mimic the predictions of the teacher model. In our FCD framework, there is no distinct teacher model, instead, it is the same local model \\( f_{\\theta,k} \\) initially trained on normal data at the beginning of the FL process. We introduce \\( \\mathcal{P}_k \\) to denote the prediction probabilities when the local teacher model \\( f_{\\theta,k} \\) is applied to normal data \\( \\mathcal{X}_k \\), represented as \\( \\mathcal{P}_k = \\sigma(f_{\\theta,k}(\\mathcal{X}_k)) \\). Subsequently, we calculate the distillation loss guided by KL divergence \\( (\\mathcal{L}_{KLD_k}) \\) of client \\( k \\) between the prediction probabilities of the teacher and the student local model, which has been trained on normal and FCD-encrypted data, respectively, as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} _ {K L D _ {k}} \\left(\\mathcal {Q} _ {k} \\| \\mathcal {P} _ {k}\\right) = \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\sum_ {r = 1} ^ {\\mathcal {R}} \\mathcal {Q} _ {i, k} ^ {r} \\log \\frac {\\mathcal {Q} _ {i , k} ^ {r}}{\\mathcal {P} _ {i , k} ^ {r}}, \\tag {3}\n\\]"
+ },
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+ "type": "text",
+ "bbox": [
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+ "content": "where \\(\\mathcal{R}\\) is the number of classes. Hence, the complete loss for client \\(\\mathcal{C}_k\\), unlike traditional FL systems that solely consider \\(\\mathcal{L}_{CE_k}\\), is expressed as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} _ {k} = \\mathcal {L} _ {C E _ {k}} + \\alpha_ {k} \\mathcal {L} _ {K L D _ {k}}, \\tag {4}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "where \\(\\alpha_{k}\\) serves as the weighting factor, regulating the balance between the cross-entropy loss and the distillation loss for client \\(\\mathcal{C}_k\\). Finally, each client minimizes the total loss \\(\\mathcal{L}_k\\) over its FCD-transformed data \\(\\mathcal{E}(\\mathcal{X}_k)\\) across \\(E\\) local iterations. The local model \\(f_{\\theta ,k}\\) undergoes parameter updates through a backward pass, adjusting as \\(\\theta_k^{\\mathrm{new}} = \\theta_k^{\\mathrm{old}} - \\eta_k\\nabla_{\\theta_k^{\\mathrm{old}}}\\mathcal{L}_k\\), where \\(\\eta_{k}\\) signifies the learning rate of client \\(\\mathcal{C}_k\\). Following each training phase, the client(s) transmit their local updates, denoted as \\(\\nabla \\theta_k^t = \\theta_k^t -\\theta_g^t\\), back to the server."
+ },
+ {
+ "type": "code_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Algorithm 1 Standard FL with our FCD framework"
+ },
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+ "type": "algorithm",
+ "bbox": [
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+ "angle": 0,
+ "content": "Input: Global model \\(\\mathcal{G}_{\\theta ,t}\\) , local data \\(\\mathcal{D}_k = (\\mathcal{X}_k,\\mathcal{Y}_k)\\) \nOutput: Global test accuracy \\(\\mathcal{A}_g\\) \n1: Client execution \\((\\theta_g^t,\\mathcal{K})\\) .. \n2: for each client \\(k = 1\\) to n do \n3: Train teacher model \\(f_{\\theta ,k}\\) on normal data \\(\\mathcal{D}_k\\) \n4: \\(\\{\\mathcal{P}_{i,k}\\}_{i = (1,\\mathcal{N}_k)}\\gets \\{\\sigma (f_{\\theta ,k}(\\mathcal{X}_{i,k}))\\}_{i = (1,\\mathcal{N}_k)}\\) \n5: \\(\\mathcal{E}(\\mathcal{X}_k)\\leftarrow \\mathrm{FCD}(\\mathcal{X}_k,\\mathcal{K})\\) \n6: for each client \\(k = 1\\) to n do \n7: Initialize the local model \\(\\theta_k^t\\gets \\theta_g^t\\) \n8: for \\(b = 1\\) to batches \\(\\in \\mathcal{E}(\\mathcal{X}_k)\\) do \n9: \\(\\{\\mathcal{Q}_{b,k}\\} \\gets \\{\\sigma (f_{\\theta ,0}(\\mathcal{E}(\\mathcal{X}_k[b))))\\}\\) \n10: \\(\\mathcal{L}_{CE_k}(\\mathcal{Q}_{b,k},\\mathcal{Y}_{b,k})\\gets\\) Eq. 2, Cross-entropy loss \n11: \\(\\mathcal{L}_{KLD_k}(\\mathcal{Q}_{b,k}\\| \\mathcal{P}_{b,k})\\gets\\) Eq. 3, Distillation loss \n12: \\(\\mathcal{L}_k\\gets \\mathcal{L}_{CE_k} + \\alpha_k\\mathcal{L}_{KLD_k}\\triangleright\\) Total loss (Eq. 4) \n13: \\(\\theta_k^t\\gets \\theta_k^t -\\eta \\nabla_{\\theta_k^t}\\mathcal{L}_k\\) \n14: \\(\\nabla \\theta_k^t\\gets \\theta_k^t -\\theta_g^t\\) \n15: return \\(\\nabla \\theta_k^t\\) \n16: Server execution \\((\\nabla \\theta_k^t)\\) . \n17: Share \\(\\theta_g^t,\\mathcal{K}\\) to all the clients \n18: Receive model updates from selected clients \\(\\leftarrow \\nabla \\theta_k^t\\) \n19: Perform model aggregation using FedAvg (Eq. 1) \n20: Update the global model parameter: \\(\\theta_g^{t + 1}\\) \n21: \\(\\hat{\\chi}_{test}\\gets\\) Poisoned test data ▷TM1, TM2 \n22: \\(\\mathcal{E}(\\hat{\\chi}_{test})\\gets\\) FCD (\\(\\hat{\\chi}_{test},\\mathcal{K}\\)) \n23: Compute \\(\\mathcal{A}_g\\gets\\) Test \\((\\mathcal{G}_{\\theta_g^{t + 1}},\\mathcal{E}(\\hat{\\chi}_{test}))\\) \n24: return \\(\\mathcal{A}_g\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.553,
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+ "angle": 0,
+ "content": "FCD cryptographic encryption. The missing element in the framework described above is the FCD encryption (a detailed algorithm and a visual representation are in the Supplementary material). We introduce a homomorphic encryption technique for FL utilizing a symmetric transposition cipher [43], with a secret key \\(\\mathcal{K}\\) shared among all clients by the server. To ensure data privacy and minimize the potential for brute-force attacks, we define the key's dimension as \\(\\mathcal{K} \\in \\mathbb{R}^h\\), where \\(h\\) corresponds to the image height. The FCD encryption process comprises two steps: (1) transposing the input image \\(x\\) to obtain \\(x' = x^T\\), where rows are transformed into columns, and (2) reordering rows based on the sequence specified in the secret key \\(\\mathcal{K}\\). Essentially, the positions of pixel values in all colour channels \\((R,G,B)\\) are altered according to \\(\\mathcal{K}\\), i.e., \\(\\mathcal{R}_{\\mathcal{K}}(x') = x'[:\\), \\(\\mathcal{K},\\colon]\\), where \\(\\mathcal{R}_{\\mathcal{K}}(.)\\) denotes the row-wise shuffling of input data based on the secret key. Further, we introduce the concept of \\(\\zeta\\)-separability for the FCD encrypted data space. This concept relates to the distance between FCD-encrypted data space and adversarial perturbations. It specifies that the FCD data space maintains a separation of at least \\(\\zeta\\) from attacked data, ensuring robustness against \\(\\mu\\rho\\)-bounded adversarial perturbation (as defined in Definition 3.1). This"
+ },
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+ "type": "page_number",
+ "bbox": [
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+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.075,
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+ ],
+ "angle": 0,
+ "content": "Figure 2. Overview of the FCD-integrated FL system, with a focus on one client's training and server-side execution. We compute the cross-entropy loss \\((\\mathcal{L}_{CE})\\) for the local model on encrypted data and calculate the distillation loss \\((\\mathcal{L}_{KL})\\) using KL divergence between the pretrained teacher model and the local student model for normal and encrypted data, respectively. These losses collectively update the local model \\(f_{\\theta}\\). Similarly, test data is transformed into the encrypted space to counter potential attacks."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "property is crucial to ensure that the FCD-encrypted data space effectively mitigates utility evasion attacks during inference (TM1)."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "Definition 3.3 (\\(\\zeta\\)-separability.)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\operatorname {L e t} \\mathcal {E} (\\mathcal {D}) := \\bigcup_ {k \\in [ 1, n ], i \\in [ 1, \\mathcal {N} _ {k} ]} \\left\\{\\left(\\mathcal {E} \\left(\\mathcal {X} _ {k, i}\\right), \\mathcal {Y} _ {k, i}\\right) \\right\\} \\subseteq \\mathbb {R} ^ {d} \\times \\mathbb {R}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "represent the FCD encrypted local training data space. Similarly, \\(\\mathcal{E}(\\mathcal{D}_{test})\\) denotes the FCD encrypted test data space such that for both training and test splits, \\(\\mathcal{E}(\\mathcal{X}) = \\mathcal{R}_{\\mathcal{K}}(\\mathcal{X}^{\\mathcal{T}})\\). We say that our FCD encrypted data space is \\(\\zeta\\)-separable w.r.t. \\(\\mu \\rho\\)-bounded adversarial data perturbation \\(\\tilde{\\mathcal{X}}\\) (as given in Definition 3.1) for some \\(\\zeta > 0\\) iff \\(\\| \\mathcal{E}(\\mathcal{X}_i) - \\tilde{\\mathcal{X}}_j \\| \\geq \\zeta\\) holds for any \\(\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D})\\), \\(\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D}_{test})\\), and \\(\\tilde{\\mathcal{X}}_j \\in \\mathcal{A}_{\\mu \\rho}(\\mathcal{X}_{test})\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Rationale for FCD Design. The transposition of data before row-wise transformations enhances information security robustness by obscuring critical transformation details, providing defense against black-box evasion attacks. The use of a row-based transposition cipher, compatible with row-major programming languages (Python, C, and \\(\\mathrm{C + + }\\)) [46], ensures efficiency in memory access, parallel processing, and cryptographic operations, aligning with both security and computational considerations in our FCD integrated FL system. Please refer to the Supplementary material for a more detailed rationale."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "FCD dual property benefits aligned with defender's perspective. (i) Low visibility: The FCD transformation, incorporating both transposition and a secret key, ensures low visibility by creating unique gradients for each key, offering robust protection against various attacks. (ii) Low budget: FCD's vectorized operations and constrained key dimension facilitate efficient implementation, making it suitable for large-scale FL systems with minimal resource requirements and addressing concerns related to processing time"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
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+ "angle": 0,
+ "content": "and aligning w.r.t. defender's perspective. Please refer to the Supplementary material for more details about the FCD dual property."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Lemma 3.4 The expected time complexity of our FCD encryption function \\(\\mathcal{E}(\\mathcal{X})\\) is linear, specifically \\(\\mathcal{O}(nh)\\), where \\(n\\) represents the number of samples, and \\(h\\) denotes the image height. This is a significant enhancement compared to the \\(\\mathcal{O}(n_w\\mathcal{N}^2\\log \\mathcal{N})\\) time complexity required for encryption and decryption of model parameters, where \\(n_W\\) represents the number of model parameters, and \\(\\mathcal{N}\\) is the bit length of the key [19]. (Please refer Supplementary material for the proof.)"
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "3.1. Convergence and Feasibility Analysis of FCD"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "We start by establishing the convergence of our FCD integrated FL global model. Subsequently, we demonstrate the robustness of our FCD against gradient noise-based \\(\\mu \\rho\\)-bounded data poisoning attacks (as defined in Definition 3.1). Finally, we establish the resilience of our FCD against MIAs with \\(\\epsilon\\)-distorted characteristics (as described in Definition 3.2) at the server. We provide the respective proofs in the Supplementary material."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Theorem 3.5 FCD convergence. Under the regularity conditions of \\( L \\)-smoothness, \\( \\tau \\)-strong convexity, and a decaying learning rate, our FCD integrated FL with clients trained on encrypted data \\( \\mathcal{E}(X) \\), obtained using \\( FCD(\\mathcal{X},\\mathcal{K}) \\), the global model converges to"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} \\left[ \\mathcal {G} _ {\\theta_ {g}} \\right] - \\mathcal {G} ^ {*} \\leq \\frac {2 L}{\\tau (\\gamma + T)} \\left(\\frac {B + C}{\\tau} + 2 L \\| \\theta_ {g} ^ {0} - \\theta_ {g} ^ {*} \\| ^ {2}\\right) + D.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Here, the variables have the following meanings: \\( B = \\Gamma + (E - 1)^2 \\), where \\( \\Gamma \\) represents the measure of non-IID data distribution. \\( C \\) signifies the client selection for aggregation, with \\( C = 0 \\) when all \\( n \\) client updates are considered. \\( T \\) denotes the number of global communication"
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+ "type": "page_number",
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+ "content": "rounds, and \\(\\mathcal{G}^*\\) represents the optimal global model [28]. The positive constant \\(D\\leq \\psi\\) quantifies how distillation enhances the convergence rate. It accelerates convergence by transferring learnable knowledge from the local teacher model trained on \\(\\mathcal{X}\\) to a student model trained on \\(\\mathcal{E}(\\mathcal{X})\\). The constant \\(\\psi\\) quantifies how distillation accelerates convergence in our specific setup."
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+ "content": "Corollary 3.5.1 (Robustness to \\(\\mu \\rho\\)-bounded attacks.) Let \\(\\mathcal{A} := \\mathcal{G}_{\\theta_g} \\times \\mathcal{X}_{test} \\times \\mathbb{R} \\to \\tilde{\\mathcal{X}}_{test}\\) be the induced adversarial perturbations on the test data using the black-box global model \\(\\mathcal{G}_{\\theta_g}\\). This perturbation adheres to the constraint \\(\\| \\tilde{\\mathcal{X}}_{test} - \\mathcal{X}_{test} \\| \\leq \\mu\\) such that \\(\\mathcal{A}(\\mathcal{G}_{\\theta_g}, \\mathcal{X}_{test,i}, \\mathcal{Y}_i)_{i=1}^\\rho \\in \\mathcal{B}(\\mathcal{X}_{test,i}, \\mu)_{i=1}^\\rho\\) (according to Definition 3.1). Furthermore, \\(\\mathcal{G}_{\\theta_g}\\) is robust to \\(\\tilde{\\mathcal{X}}_{test}\\) through the mechanism of our FCD integrated FL system. It allows \\(\\mathcal{G}_{\\theta_g}\\) to test on encrypted data based on \\(\\mathcal{K}\\), i.e., \\(\\mathcal{E}(\\tilde{\\mathcal{X}}_{test})\\) instead of \\(\\tilde{\\mathcal{X}}_{test}\\). This robustness is achieved because the adversary's objective is to generate perturbations in the original test data space that maximize the global loss (Definition 3.1), as denoted by \\(\\mathcal{A}^+ := \\arg \\max_{\\tilde{\\mathcal{X}}_{test} \\in \\mathcal{B}(\\mathcal{X}_{test}, \\mu)} \\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\tilde{\\mathcal{X}}_{test,i}, \\mathcal{Y}_i)_{i=1}^\\rho)\\). However, this adversarial strategy is effective when training and testing occur in the normal data space \\(\\mathcal{X}\\) at the client side, matching the distribution of \\(\\mathcal{X}_{test}\\). In our FCD integrated FL system, training and testing occur in the encrypted data space. Notably, \\(\\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\tilde{\\mathcal{X}}_{test,i}, \\mathcal{Y}_i)_{i=1}^\\rho) \\neq \\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\mathcal{E}(\\tilde{\\mathcal{X}}_{test,i}), \\mathcal{Y}_i)_{i=1}^\\rho)\\) as \\(\\mathcal{E}(\\tilde{\\mathcal{X}}_{test}) \\triangleq \\mathcal{R}_{\\mathcal{K}}(\\tilde{\\mathcal{X}}_{test}^T) \\neq \\tilde{\\mathcal{X}}_{test}\\) and \\(\\| \\mathcal{E}(\\mathcal{X}_i) - \\tilde{\\mathcal{X}}_j \\| \\geq \\zeta\\) holds for any \\(\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D})\\), \\(\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D}_{test})\\), and \\(\\tilde{\\mathcal{X}}_j \\in \\mathcal{A}_{\\mu\\rho}(\\mathcal{X}_{test})\\) (Definition 3.3). Hence, the learning process occurs in different data spaces unknown to the adversary. Consequently, the gradients of the loss function trained on normal data are notably distinct from those trained on FCD encrypted data, effectively mitigating \\(\\mu\\rho\\)-bounded adversarial data."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Theorem 3.6 (Resilience to \\(\\epsilon\\)-distorted MIA attacks.) Let \\(\\mathcal{X}^*\\) represent the data reconstructed by the adversary using MIA attack as per the conditions specified in Definition 3.2. We demonstrate that training on FCD-encrypted data space, denoted as \\(\\mathcal{E}(\\mathcal{X})\\), imparts resilience to \\(\\epsilon\\)-distorted MIA attacks. Specifically, our result establishes that:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.091,
+ 0.732,
+ 0.457,
+ 0.765
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left| \\| \\mathcal {X} ^ {*} \\| - \\| \\mathcal {E} (\\mathcal {X}) \\| \\right| \\leq \\epsilon + \\delta , f o r s o m e \\epsilon \\geq 0 a n d \\delta \\geq 0.\n\\]"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.8,
+ 0.21,
+ 0.817
+ ],
+ "angle": 0,
+ "content": "4. Experiments"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.825,
+ 0.47,
+ 0.903
+ ],
+ "angle": 0,
+ "content": "Datasets, implementation details, and metrics. Table 2 presents comprehensive details about the datasets, FL setup, attack percentage, and metrics. For TM1, we employed the black-box and active data poisoning technique called MSimBA [23]. Each experiment was conducted thrice, and"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.092,
+ 0.892,
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+ ],
+ "angle": 0,
+ "content": "results were averaged with standard deviations presented. Please refer to the Supplementary material for more details. Baselines. For TM1, we have chosen FAT [61] and Randomized Smoothing (RS) [8] based on their relevance and applicability in evasion attacks within FL, as outlined in Table 1. Additionally, we explore the effectiveness under the following configurations: (a) With/without attack and defense and (b) two FCD setups under different client counts and attack percentages \\((A_{p})\\). These setups include FCD-homoFL (homogeneous FL) and FCD-hetFL (heterogeneous FL). In TM2, we follow the baselines as per [27]."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.499,
+ 0.286,
+ 0.714,
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+ ],
+ "angle": 0,
+ "content": "4.1. FCD Result Discussion"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Benign setting. Table 3 showcases the performance of our FCD defense, along with other baselines and configurations, across four datasets under a no attack scenario. Notably, FCD demonstrates comparable or higher accuracy than vanilla FL, particularly excelling in datasets like GTSRB and KBTS. This improvement is attributed to FCD's ability to capture diverse transformation patterns with a secret key, facilitating reduced training loss and improved model convergence. Furthermore, with its symmetric cryptographic transformation, FCD outperforms FAT and RS for GTSRB and KBTS. However, in CIFAR10 and EMNIST, FCD's performance, while slightly lower than base FL, surpasses FAT and RS. This trade-off between utility and privacy is expected, as FCD provides both aspects with a modest decrease in base accuracy."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.538,
+ 0.892,
+ 0.568
+ ],
+ "angle": 0,
+ "content": "Attack setting. We analyse FCD effectiveness under two threat models separately."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.569,
+ 0.892,
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+ ],
+ "angle": 0,
+ "content": "(i) TM1: Table 4 illustrates the impact on utility \\((U)\\) for FCD compared to other baselines under M-SimBA evasion attack on GTSRB and KBTS datasets. The results indicate a significant utility degradation with higher attack percentages and more clients, emphasizing the impact of substantial test data modification by M-SimBA. FCD consistently demonstrates superior performance and robustness across all attack percentages. In homogeneous FL settings, FCD shows substantial improvement, maintaining a consistently lower attack impact on utility compared to other defenses. Particularly at \\(A_{p} = 30\\%\\), the attack impact on utility is 2-6 units lower for FCD as the number of clients increases, compared to other baselines. Similar trends are observed for \\(A_{p} = 50\\%\\), \\(100\\%\\). In heterogeneous FL settings, FCD outperforms other defenses, showcasing robustness even when clients are trained on varying amounts of non-IID data. For the KBTS dataset, FCD exhibits substantial improvement and lower attack impact across all attack percentages and FL settings compared to other defenses. The limited training data in the KBTS dataset results in the RS technique showing limited robustness, especially for \\(A_{p} = 30\\%\\) in homogeneous and heterogeneous FL, followed by FAT. Ta"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ }
+ ],
+ [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.166,
+ 0.09,
+ 0.804,
+ 0.104
+ ],
+ "angle": 0,
+ "content": "Table 2. Comprehensive experimental details: datasets, models, FL setup, attack configuration, and metrics."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.081,
+ 0.115,
+ 0.895,
+ 0.2
+ ],
+ "angle": 0,
+ "content": "| FL defense category (src) | FL defense methods | Can defend | U-P tradeoff analysis | Defender's perspective analysis |
| U-CA | EA | P-CA | DIm | DB | DV |
| Adversarial training (C) | FAT [61], RS [8], FedDynAT [39], GALP [13] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Byzantine robust aggregation techniques (S) | Krum [3], ShieldFL [30], FLTrust [5] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Data/ update analysis (S and/or C) | DeepSight [38], FL-Defender [18], SparseFed [36] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Secure multi-party computation (C) | AMPC [58], Byrd et al. [4] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| Trusted execution environments (C) | Flatee[35], Chen et al. [6], PPFL [34] | 0 | 0 | 0 | 0 | 0 | ○ | ○ |
| Differential privacy (C) | NbAFL [52], Hu et al. [16], 2DP-FL [54] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| Homomorphic encryption (C) | DCAE [62], PEFL [29], Batchcrypt[57] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| FCD (ours) src: S and C | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
+
+ing both utility and privacy threats in a unified manner. This perspective is essential as it offers insights into the defense's robustness, adaptability, budget, multipurpose efficiency, visibility, and overall impact on the FL system. Further, within the domain of FL, adversaries can exhibit multifaceted intentions, spanning both utility and privacy threats. Deploying distinct defense mechanisms for each individual attack escalates the defender's budget, increases the overall computational complexity, and reduces the adaptability to practical FL trustworthy systems. Also, defenses that are easily accessible and transparent to attackers are susceptible to circumvention. While differential privacy (DP) [16, 52, 54], secure multi-party computation (MPC) [4, 58], and trusted execution environments (TEE) [6, 34, 35] have been proposed as potential solutions to strengthen privacy in FL, none of the existing solutions offers a satisfactory unification of privacy and utility defense under different attacks, coupled with reasonable efficiency [45].
+
+In this work, we propose an approach called Federated Cryptography Defense (FCD) to counter both utility and privacy evasion attacks through the strategic implementation of fully homomorphic encryption (FHE) in the data space via row-based cryptography transformation. Unlike existing defense methods (as shown in Table 1), which typically focus on defending against either utility or privacy threats, our defense framework aligns with the defender's perspective by working in the data space rather than gradient space. It stands out as a practical, cost-effective, and multipurpose defense mechanism. One key advantage is the
+
+preservation of data confidentiality, achieved through transferring the entire process on encrypted data using a shared secret key. Further, our approach enables server-side testing on encrypted data without the need for decryption, thus significantly reducing computational complexity. On the other hand, clients engage in training on encrypted data, where we introduce a novel distillation loss guided by the Kullback-Leibler (KL) divergence. This novel loss function minimizes the distillation loss between the pretrained teacher model and the local student model responses. In addition, the KL divergence loss, combined with the conventional cross-entropy loss, serves as a total loss for updating the local model. Furthermore, we have developed two distinct threat models, TM1 and TM2, addressing utility and privacy vulnerabilities in FL and expanding on the attacker's potential impact on the FL system. We present the first proposal for this type of unified defense in FL to the best of our knowledge. Our contributions are:
+
+1. Novel defense: We introduce a unified defense framework called FCD, which uses row-based transposition cipher to defend against both evasion adversarial and MIA attacks at the server aligning to the defender's perspective.
+2. Loss function: To enable effective learning, we propose a novel distillation loss guided by KL divergence between teacher and student model predictions on the client's side.
+3. Threat models: We introduce two realistic threat models (TM1 and TM2) concerning to utility and privacy attacks.
+4. Theoretical analysis: We present a possible theoretical analysis of our method w.r.t. convergence and resilience to attacks.
+5. Effective tradeoff analysis: We perform extensive experiments with five different datasets and models, namely, GTSRB (CNN), KBTS (CNN), CIFAR10 (ResNet18), CIFAR100 (VGG11), and EMNIST (LeNet), alternating between threat models. We also investigate the performance under homogeneous and heterogeneous (non-IID) data shard settings.
+
+# 2. Preliminaries
+
+Definition 2.1 (FL data setup.) We consider an FL system with a central server and $n$ clients. Each client $C_k$ , where $k \in [1, n]$ , possesses its private local dataset $D_k$ , referred to as a shard. In this context, we define $D_k = \{\mathcal{X}_{i,k}, \mathcal{Y}_{i,k}\}_{i=1}^{\mathcal{N}_k} \subseteq \mathbb{R}^d \times \mathbb{R}$ . Without loss of generality, we assume that $\| \mathcal{X}_{i,k} \|_2 = 1$ holds for all $k \in [1, n]$ , where $i \in [1, N_k]$ and the final component of each point is fixed at $1/2$ , denoted as $\mathcal{X}^d = 1/2$ ( $l_2$ norm data normalization). The server uses synchronous federated weighted averaging
+
+(FedAvg) [32] for aggregation, represented as:
+
+$$
+\theta_ {g} ^ {t + 1} = \theta_ {g} ^ {t} + \sum_ {k = 1} ^ {n} \lambda_ {k} \nabla \theta_ {k} ^ {t} \tag {1}
+$$
+
+where $\lambda_{k} = \frac{\mathcal{N}_{k}}{\sum\mathcal{N}_{k}}$ and $\sum_{k}\lambda_{k} = 1$
+
+We investigate two FL data shard settings: (i) Homogeneous, where each client's dataset size is identical, i.e., $|\mathcal{D}_1| = |\mathcal{D}_2| = ..|\mathcal{D}_n| = \frac{|\mathcal{D}|}{n}$ , and (ii) Heterogeneous, involving non-independent and non-identically (non-IID) distributed data achieved by partitioning the dataset using a Dirichlet distribution [33] with parameter $\beta = 1$ among clients. Additional information on standard FL, evasion attacks in FL, homomorphic encryption, and transposition cipher is provided in the supplementary material.
+
+Threat model. We introduce two distinct threat models, TM1 and TM2, formulated to reflect real-world FL production deployment settings, as illustrated in Figure 1. Our threat models address an honest-but-curious (HbC) adversary at the central server, as inspired by related work [41, 48]. In TM1 (evasion utility attack), the adversary conducts an indiscriminate evasion attack by manipulating test data at the central server during inference, aiming to misclassify a substantial portion of the inputs [24, 48]. In TM2 (privacy attack), the adversary's goal shifts to performing a model inversion attack (MIA) [15] with the aim of reconstructing private data. We provide our practical assumptions and more details about the threat models in the Supplementary material.
+
+
+Figure 1. Overview of two different threat models (TM) with potential vulnerabilities and attacks during inference.
+
+# 3. Proposed Framework
+
+Problem statement. In each communication round $t$ , suppose that the server receives model updates from $m$ clients, and there exists a test dataset $\mathcal{D}_{test} = \{\mathcal{X}_i, \mathcal{Y}_i\}_{i=1}^{\mathcal{N}_{te}}$ with a size of $\mathcal{N}_{te}$ . The adversary introduces a $\mu \rho$ -bounded poisoned test dataset in TM1, denoted as $\tilde{\mathcal{D}}_{\mathrm{test}} = \{\tilde{\mathcal{X}}_i, \tilde{\mathcal{Y}}_i\}_{i=1}^{\tilde{\mathcal{N}}_{te}}$ , with the dataset size $\hat{\mathcal{N}}_{te}$ varying based on the attack volume $\rho$ as:
+
+Definition 3.1 ( $\mu \rho$ -bounded adversary.) Let $\mathcal{F}$ be the model's function class. An adversarial perturbation is characterized by the mapping $\mathcal{A} := \mathcal{F} \times \mathcal{X} \times \mathbb{R} \to \tilde{\mathcal{X}}$ . For $\mu > 0$ , we define the $l_2$ norm ball as $\mathcal{B}_2(\mathcal{X}, \mu) := \{\tilde{\mathcal{X}} \in \mathbb{R}^d : \| \tilde{\mathcal{X}} - \mathcal{X} \| \leq \mu\} \cap \mathcal{X}$ . We classify the adversary as $\mu \rho$ -bounded if it adheres to the condition $\mathcal{A}(\mathcal{F}, \mathcal{X}_i, \mathcal{Y}_i)_{i=1}^\rho \in \mathcal{B}(\mathcal{X}_i, \mu)_{i=1}^\rho$ . Furthermore, for a given $\mu > 0$ , $\rho > 0$ , we denote the worst-case adversarial perturbation as $\mathcal{A}^+ := \arg \max_{\tilde{\mathcal{X}} \in \mathcal{B}(\mathcal{X}, \mu)} \mathcal{L}(f_\theta(\tilde{\mathcal{X}}_i, \mathcal{Y}_i)_{i=1}^\rho)$ , where $\mathcal{L}$ represents the loss function of the global model.
+
+Next, we formulate the objective of the adversary in MIA attack (TM2) using the below definition.
+
+Definition 3.2 ( $\epsilon$ -distorted.) Let $\mathcal{X}^*$ represent the data reconstructed by the adversary using the MIA attack. The objective of the adversary is to minimize the reconstruction error $P$ in terms of as the mean squared error (MSE) between the reconstructed data $\mathcal{X}^*$ and the original input data $\mathcal{X}$ as
+
+$$
+P := \| \mathcal {X} ^ {*} - \mathcal {X} \| _ {2} \equiv \frac {1}{\mathcal {N} _ {t e}} \sum_ {i = 1} ^ {\mathcal {N} _ {t e}} \left(\mathcal {X} _ {i} ^ {*} - \mathcal {X} _ {i}\right) ^ {2} \leq \epsilon , f o r s o m e \epsilon \geq 0,
+$$
+
+where $\epsilon$ is the distortion of the reconstructed image.
+
+Further, the global model, $\mathcal{G}_{\theta_g}$ attains a global test accuracy of $\mathcal{A}_g$ before the attack and $\mathcal{A}_g^*$ after the attack (Definition 3.1). Now, our FCD integrated clients have two primary objectives. First, they aim to minimize the impact of the attack, which is quantified by $U = (\mathcal{A}_g - \mathcal{A}_g^*)$ , in order to preserve the global test accuracy. Second, they seek to enhance the privacy of local data $\mathcal{D}_k = \{\mathcal{X}_{i,k},\mathcal{Y}_{i,k}\}_{i = 1}^{\mathcal{N}_k}$ by maximizing the reconstruction error $P$ . Here, $U$ and $P$ define the utility and privacy gains in the context of FL, where lower values of $U$ and higher values of $P$ are considered desirable.
+
+FCD framework description. The FCD framework introduces modifications to standard FL, which can be categorized into server-side and client-side changes. Algorithm 1 and Figure 2 present our FCD integrated FL system.
+
+Server side. To initiate the FL process, the central server initializes a key denoted as $\mathcal{K}$ . This key consists of unique random integers in the range from 0 to $h - 1$ , where $h$ represents the height of the input image data. The key $\mathcal{K}$ is defined as $\mathcal{K} = [\kappa_0,\kappa_1,\dots ,\kappa_i,\kappa_j,\dots ,\kappa_{h - 1}]$ , and it follows a property that if $i,j\in 0,\ldots ,h - 1$ with $i\neq j$ , then $\kappa_{i}\neq \kappa_{j}$ . This property guarantees that no row in the input image can be missed, ensuring the completeness of the data. During each communication round $t$ , the central server follows the standard FL procedure by sending the current version of the global model parameters $\theta_g^t$ to all clients. Synchronous federated weighted averaging-based aggregation is employed at the server, as outlined in [32]. Further, the central server performs the FCD transformation on the evasion-attacked
+
+test data to obtain $\mathcal{E}(\hat{\mathcal{D}}_{test})$ . Subsequently, the server tests the current aggregated global model $\mathcal{G}_{\theta}^{t}$ on the FCD transformed test data, i.e., $\mathcal{E}(\hat{\mathcal{D}}_{test})$ , as opposed to standard FL systems that test on the original poisoned test data $\hat{\mathcal{D}}_{test}$ . Client side. Let $\mathcal{E}(\mathcal{X}_k)$ represent the FCD-encrypted form of input samples from client $\mathcal{C}_k$ based on the shared secret key $\mathcal{K}$ . We denote $\mathcal{Q}_k$ as the prediction probabilities when the local model $f_{\theta,k}$ is applied to the transformed data $\mathcal{E}(\mathcal{X}_k)$ , which can be expressed as $\mathcal{Q}_k = \sigma(f_{\theta,k}(\mathcal{E}(\mathcal{X}_k)))$ , where $\sigma(.)$ denotes the softmax function. Consequently, the cross-entropy loss $\mathcal{L}_{CE_k}$ of client $\mathcal{C}_k$ , trained on FCD-encrypted data (in contrast to general FL systems that calculates $\mathcal{L}_{CE_k}$ on normal data $\mathcal{X}_k$ ), can be calculated as follows:
+
+$$
+\mathcal {L} _ {C E _ {k}} = \sum_ {i = 1} ^ {\mathcal {N} _ {k}} \mathcal {L} _ {C E} \left(\mathcal {Q} _ {i, k}, \mathcal {Y} _ {i, k}\right) = - \sum_ {i = 1} ^ {\mathcal {N} _ {k}} \mathcal {Y} _ {i, k} \log \mathcal {Q} _ {i, k}. \tag {2}
+$$
+
+To enhance the learning process and facilitate knowledge transfer, we employ the knowledge distillation (KD) approach [60], guided by the Kullback-Leibler (KL) divergence. In this approach, the student model $f_{\theta,k}$ is trained to mimic the predictions of the teacher model. In our FCD framework, there is no distinct teacher model, instead, it is the same local model $f_{\theta,k}$ initially trained on normal data at the beginning of the FL process. We introduce $\mathcal{P}_k$ to denote the prediction probabilities when the local teacher model $f_{\theta,k}$ is applied to normal data $\mathcal{X}_k$ , represented as $\mathcal{P}_k = \sigma(f_{\theta,k}(\mathcal{X}_k))$ . Subsequently, we calculate the distillation loss guided by KL divergence $(\mathcal{L}_{KLD_k})$ of client $k$ between the prediction probabilities of the teacher and the student local model, which has been trained on normal and FCD-encrypted data, respectively, as follows:
+
+$$
+\mathcal {L} _ {K L D _ {k}} \left(\mathcal {Q} _ {k} \| \mathcal {P} _ {k}\right) = \sum_ {i = 1} ^ {\mathcal {N} _ {k}} \sum_ {r = 1} ^ {\mathcal {R}} \mathcal {Q} _ {i, k} ^ {r} \log \frac {\mathcal {Q} _ {i , k} ^ {r}}{\mathcal {P} _ {i , k} ^ {r}}, \tag {3}
+$$
+
+where $\mathcal{R}$ is the number of classes. Hence, the complete loss for client $\mathcal{C}_k$ , unlike traditional FL systems that solely consider $\mathcal{L}_{CE_k}$ , is expressed as:
+
+$$
+\mathcal {L} _ {k} = \mathcal {L} _ {C E _ {k}} + \alpha_ {k} \mathcal {L} _ {K L D _ {k}}, \tag {4}
+$$
+
+where $\alpha_{k}$ serves as the weighting factor, regulating the balance between the cross-entropy loss and the distillation loss for client $\mathcal{C}_k$ . Finally, each client minimizes the total loss $\mathcal{L}_k$ over its FCD-transformed data $\mathcal{E}(\mathcal{X}_k)$ across $E$ local iterations. The local model $f_{\theta ,k}$ undergoes parameter updates through a backward pass, adjusting as $\theta_k^{\mathrm{new}} = \theta_k^{\mathrm{old}} - \eta_k\nabla_{\theta_k^{\mathrm{old}}}\mathcal{L}_k$ , where $\eta_{k}$ signifies the learning rate of client $\mathcal{C}_k$ . Following each training phase, the client(s) transmit their local updates, denoted as $\nabla \theta_k^t = \theta_k^t -\theta_g^t$ , back to the server.
+
+Algorithm 1 Standard FL with our FCD framework
+Input: Global model $\mathcal{G}_{\theta ,t}$ , local data $\mathcal{D}_k = (\mathcal{X}_k,\mathcal{Y}_k)$
+Output: Global test accuracy $\mathcal{A}_g$
+1: Client execution $(\theta_g^t,\mathcal{K})$ ..
+2: for each client $k = 1$ to n do
+3: Train teacher model $f_{\theta ,k}$ on normal data $\mathcal{D}_k$
+4: $\{\mathcal{P}_{i,k}\}_{i = (1,\mathcal{N}_k)}\gets \{\sigma (f_{\theta ,k}(\mathcal{X}_{i,k}))\}_{i = (1,\mathcal{N}_k)}$
+5: $\mathcal{E}(\mathcal{X}_k)\leftarrow \mathrm{FCD}(\mathcal{X}_k,\mathcal{K})$
+6: for each client $k = 1$ to n do
+7: Initialize the local model $\theta_k^t\gets \theta_g^t$
+8: for $b = 1$ to batches $\in \mathcal{E}(\mathcal{X}_k)$ do
+9: $\{\mathcal{Q}_{b,k}\} \gets \{\sigma (f_{\theta ,0}(\mathcal{E}(\mathcal{X}_k[b))))\}$
+10: $\mathcal{L}_{CE_k}(\mathcal{Q}_{b,k},\mathcal{Y}_{b,k})\gets$ Eq. 2, Cross-entropy loss
+11: $\mathcal{L}_{KLD_k}(\mathcal{Q}_{b,k}\| \mathcal{P}_{b,k})\gets$ Eq. 3, Distillation loss
+12: $\mathcal{L}_k\gets \mathcal{L}_{CE_k} + \alpha_k\mathcal{L}_{KLD_k}\triangleright$ Total loss (Eq. 4)
+13: $\theta_k^t\gets \theta_k^t -\eta \nabla_{\theta_k^t}\mathcal{L}_k$
+14: $\nabla \theta_k^t\gets \theta_k^t -\theta_g^t$
+15: return $\nabla \theta_k^t$
+16: Server execution $(\nabla \theta_k^t)$ .
+17: Share $\theta_g^t,\mathcal{K}$ to all the clients
+18: Receive model updates from selected clients $\leftarrow \nabla \theta_k^t$
+19: Perform model aggregation using FedAvg (Eq. 1)
+20: Update the global model parameter: $\theta_g^{t + 1}$
+21: $\hat{\chi}_{test}\gets$ Poisoned test data ▷TM1, TM2
+22: $\mathcal{E}(\hat{\chi}_{test})\gets$ FCD ( $\hat{\chi}_{test},\mathcal{K}$ )
+23: Compute $\mathcal{A}_g\gets$ Test $(\mathcal{G}_{\theta_g^{t + 1}},\mathcal{E}(\hat{\chi}_{test}))$
+24: return $\mathcal{A}_g$
+
+FCD cryptographic encryption. The missing element in the framework described above is the FCD encryption (a detailed algorithm and a visual representation are in the Supplementary material). We introduce a homomorphic encryption technique for FL utilizing a symmetric transposition cipher [43], with a secret key $\mathcal{K}$ shared among all clients by the server. To ensure data privacy and minimize the potential for brute-force attacks, we define the key's dimension as $\mathcal{K} \in \mathbb{R}^h$ , where $h$ corresponds to the image height. The FCD encryption process comprises two steps: (1) transposing the input image $x$ to obtain $x' = x^T$ , where rows are transformed into columns, and (2) reordering rows based on the sequence specified in the secret key $\mathcal{K}$ . Essentially, the positions of pixel values in all colour channels $(R,G,B)$ are altered according to $\mathcal{K}$ , i.e., $\mathcal{R}_{\mathcal{K}}(x') = x'[:$ , $\mathcal{K},\colon]$ , where $\mathcal{R}_{\mathcal{K}}(.)$ denotes the row-wise shuffling of input data based on the secret key. Further, we introduce the concept of $\zeta$ -separability for the FCD encrypted data space. This concept relates to the distance between FCD-encrypted data space and adversarial perturbations. It specifies that the FCD data space maintains a separation of at least $\zeta$ from attacked data, ensuring robustness against $\mu\rho$ -bounded adversarial perturbation (as defined in Definition 3.1). This
+
+
+Figure 2. Overview of the FCD-integrated FL system, with a focus on one client's training and server-side execution. We compute the cross-entropy loss $(\mathcal{L}_{CE})$ for the local model on encrypted data and calculate the distillation loss $(\mathcal{L}_{KL})$ using KL divergence between the pretrained teacher model and the local student model for normal and encrypted data, respectively. These losses collectively update the local model $f_{\theta}$ . Similarly, test data is transformed into the encrypted space to counter potential attacks.
+
+property is crucial to ensure that the FCD-encrypted data space effectively mitigates utility evasion attacks during inference (TM1).
+
+# Definition 3.3 ( $\zeta$ -separability.)
+
+$$
+\operatorname {L e t} \mathcal {E} (\mathcal {D}) := \bigcup_ {k \in [ 1, n ], i \in [ 1, \mathcal {N} _ {k} ]} \left\{\left(\mathcal {E} \left(\mathcal {X} _ {k, i}\right), \mathcal {Y} _ {k, i}\right) \right\} \subseteq \mathbb {R} ^ {d} \times \mathbb {R}
+$$
+
+represent the FCD encrypted local training data space. Similarly, $\mathcal{E}(\mathcal{D}_{test})$ denotes the FCD encrypted test data space such that for both training and test splits, $\mathcal{E}(\mathcal{X}) = \mathcal{R}_{\mathcal{K}}(\mathcal{X}^{\mathcal{T}})$ . We say that our FCD encrypted data space is $\zeta$ -separable w.r.t. $\mu \rho$ -bounded adversarial data perturbation $\tilde{\mathcal{X}}$ (as given in Definition 3.1) for some $\zeta > 0$ iff $\| \mathcal{E}(\mathcal{X}_i) - \tilde{\mathcal{X}}_j \| \geq \zeta$ holds for any $\mathcal{E}(\mathcal{X}_i) \in \mathcal{E}(\mathcal{D})$ , $\mathcal{E}(\mathcal{X}_i) \in \mathcal{E}(\mathcal{D}_{test})$ , and $\tilde{\mathcal{X}}_j \in \mathcal{A}_{\mu \rho}(\mathcal{X}_{test})$ .
+
+Rationale for FCD Design. The transposition of data before row-wise transformations enhances information security robustness by obscuring critical transformation details, providing defense against black-box evasion attacks. The use of a row-based transposition cipher, compatible with row-major programming languages (Python, C, and $\mathrm{C + + }$ ) [46], ensures efficiency in memory access, parallel processing, and cryptographic operations, aligning with both security and computational considerations in our FCD integrated FL system. Please refer to the Supplementary material for a more detailed rationale.
+
+FCD dual property benefits aligned with defender's perspective. (i) Low visibility: The FCD transformation, incorporating both transposition and a secret key, ensures low visibility by creating unique gradients for each key, offering robust protection against various attacks. (ii) Low budget: FCD's vectorized operations and constrained key dimension facilitate efficient implementation, making it suitable for large-scale FL systems with minimal resource requirements and addressing concerns related to processing time
+
+and aligning w.r.t. defender's perspective. Please refer to the Supplementary material for more details about the FCD dual property.
+
+Lemma 3.4 The expected time complexity of our FCD encryption function $\mathcal{E}(\mathcal{X})$ is linear, specifically $\mathcal{O}(nh)$ , where $n$ represents the number of samples, and $h$ denotes the image height. This is a significant enhancement compared to the $\mathcal{O}(n_w\mathcal{N}^2\log \mathcal{N})$ time complexity required for encryption and decryption of model parameters, where $n_W$ represents the number of model parameters, and $\mathcal{N}$ is the bit length of the key [19]. (Please refer Supplementary material for the proof.)
+
+# 3.1. Convergence and Feasibility Analysis of FCD
+
+We start by establishing the convergence of our FCD integrated FL global model. Subsequently, we demonstrate the robustness of our FCD against gradient noise-based $\mu \rho$ -bounded data poisoning attacks (as defined in Definition 3.1). Finally, we establish the resilience of our FCD against MIAs with $\epsilon$ -distorted characteristics (as described in Definition 3.2) at the server. We provide the respective proofs in the Supplementary material.
+
+Theorem 3.5 FCD convergence. Under the regularity conditions of $L$ -smoothness, $\tau$ -strong convexity, and a decaying learning rate, our FCD integrated FL with clients trained on encrypted data $\mathcal{E}(X)$ , obtained using $FCD(\mathcal{X},\mathcal{K})$ , the global model converges to
+
+$$
+\mathbb {E} \left[ \mathcal {G} _ {\theta_ {g}} \right] - \mathcal {G} ^ {*} \leq \frac {2 L}{\tau (\gamma + T)} \left(\frac {B + C}{\tau} + 2 L \| \theta_ {g} ^ {0} - \theta_ {g} ^ {*} \| ^ {2}\right) + D.
+$$
+
+Here, the variables have the following meanings: $B = \Gamma + (E - 1)^2$ , where $\Gamma$ represents the measure of non-IID data distribution. $C$ signifies the client selection for aggregation, with $C = 0$ when all $n$ client updates are considered. $T$ denotes the number of global communication
+
+rounds, and $\mathcal{G}^*$ represents the optimal global model [28]. The positive constant $D\leq \psi$ quantifies how distillation enhances the convergence rate. It accelerates convergence by transferring learnable knowledge from the local teacher model trained on $\mathcal{X}$ to a student model trained on $\mathcal{E}(\mathcal{X})$ . The constant $\psi$ quantifies how distillation accelerates convergence in our specific setup.
+
+Corollary 3.5.1 (Robustness to $\mu \rho$ -bounded attacks.) Let $\mathcal{A} := \mathcal{G}_{\theta_g} \times \mathcal{X}_{test} \times \mathbb{R} \to \tilde{\mathcal{X}}_{test}$ be the induced adversarial perturbations on the test data using the black-box global model $\mathcal{G}_{\theta_g}$ . This perturbation adheres to the constraint $\| \tilde{\mathcal{X}}_{test} - \mathcal{X}_{test} \| \leq \mu$ such that $\mathcal{A}(\mathcal{G}_{\theta_g}, \mathcal{X}_{test,i}, \mathcal{Y}_i)_{i=1}^\rho \in \mathcal{B}(\mathcal{X}_{test,i}, \mu)_{i=1}^\rho$ (according to Definition 3.1). Furthermore, $\mathcal{G}_{\theta_g}$ is robust to $\tilde{\mathcal{X}}_{test}$ through the mechanism of our FCD integrated FL system. It allows $\mathcal{G}_{\theta_g}$ to test on encrypted data based on $\mathcal{K}$ , i.e., $\mathcal{E}(\tilde{\mathcal{X}}_{test})$ instead of $\tilde{\mathcal{X}}_{test}$ . This robustness is achieved because the adversary's objective is to generate perturbations in the original test data space that maximize the global loss (Definition 3.1), as denoted by $\mathcal{A}^+ := \arg \max_{\tilde{\mathcal{X}}_{test} \in \mathcal{B}(\mathcal{X}_{test}, \mu)} \mathcal{L}(\mathcal{G}_{\theta_g}(\tilde{\mathcal{X}}_{test,i}, \mathcal{Y}_i)_{i=1}^\rho)$ . However, this adversarial strategy is effective when training and testing occur in the normal data space $\mathcal{X}$ at the client side, matching the distribution of $\mathcal{X}_{test}$ . In our FCD integrated FL system, training and testing occur in the encrypted data space. Notably, $\mathcal{L}(\mathcal{G}_{\theta_g}(\tilde{\mathcal{X}}_{test,i}, \mathcal{Y}_i)_{i=1}^\rho) \neq \mathcal{L}(\mathcal{G}_{\theta_g}(\mathcal{E}(\tilde{\mathcal{X}}_{test,i}), \mathcal{Y}_i)_{i=1}^\rho)$ as $\mathcal{E}(\tilde{\mathcal{X}}_{test}) \triangleq \mathcal{R}_{\mathcal{K}}(\tilde{\mathcal{X}}_{test}^T) \neq \tilde{\mathcal{X}}_{test}$ and $\| \mathcal{E}(\mathcal{X}_i) - \tilde{\mathcal{X}}_j \| \geq \zeta$ holds for any $\mathcal{E}(\mathcal{X}_i) \in \mathcal{E}(\mathcal{D})$ , $\mathcal{E}(\mathcal{X}_i) \in \mathcal{E}(\mathcal{D}_{test})$ , and $\tilde{\mathcal{X}}_j \in \mathcal{A}_{\mu\rho}(\mathcal{X}_{test})$ (Definition 3.3). Hence, the learning process occurs in different data spaces unknown to the adversary. Consequently, the gradients of the loss function trained on normal data are notably distinct from those trained on FCD encrypted data, effectively mitigating $\mu\rho$ -bounded adversarial data.
+
+Theorem 3.6 (Resilience to $\epsilon$ -distorted MIA attacks.) Let $\mathcal{X}^*$ represent the data reconstructed by the adversary using MIA attack as per the conditions specified in Definition 3.2. We demonstrate that training on FCD-encrypted data space, denoted as $\mathcal{E}(\mathcal{X})$ , imparts resilience to $\epsilon$ -distorted MIA attacks. Specifically, our result establishes that:
+
+$$
+\left| \| \mathcal {X} ^ {*} \| - \| \mathcal {E} (\mathcal {X}) \| \right| \leq \epsilon + \delta , f o r s o m e \epsilon \geq 0 a n d \delta \geq 0.
+$$
+
+# 4. Experiments
+
+Datasets, implementation details, and metrics. Table 2 presents comprehensive details about the datasets, FL setup, attack percentage, and metrics. For TM1, we employed the black-box and active data poisoning technique called MSimBA [23]. Each experiment was conducted thrice, and
+
+results were averaged with standard deviations presented. Please refer to the Supplementary material for more details. Baselines. For TM1, we have chosen FAT [61] and Randomized Smoothing (RS) [8] based on their relevance and applicability in evasion attacks within FL, as outlined in Table 1. Additionally, we explore the effectiveness under the following configurations: (a) With/without attack and defense and (b) two FCD setups under different client counts and attack percentages $(A_{p})$ . These setups include FCD-homoFL (homogeneous FL) and FCD-hetFL (heterogeneous FL). In TM2, we follow the baselines as per [27].
+
+# 4.1. FCD Result Discussion
+
+Benign setting. Table 3 showcases the performance of our FCD defense, along with other baselines and configurations, across four datasets under a no attack scenario. Notably, FCD demonstrates comparable or higher accuracy than vanilla FL, particularly excelling in datasets like GTSRB and KBTS. This improvement is attributed to FCD's ability to capture diverse transformation patterns with a secret key, facilitating reduced training loss and improved model convergence. Furthermore, with its symmetric cryptographic transformation, FCD outperforms FAT and RS for GTSRB and KBTS. However, in CIFAR10 and EMNIST, FCD's performance, while slightly lower than base FL, surpasses FAT and RS. This trade-off between utility and privacy is expected, as FCD provides both aspects with a modest decrease in base accuracy.
+
+Attack setting. We analyse FCD effectiveness under two threat models separately.
+
+(i) TM1: Table 4 illustrates the impact on utility $(U)$ for FCD compared to other baselines under M-SimBA evasion attack on GTSRB and KBTS datasets. The results indicate a significant utility degradation with higher attack percentages and more clients, emphasizing the impact of substantial test data modification by M-SimBA. FCD consistently demonstrates superior performance and robustness across all attack percentages. In homogeneous FL settings, FCD shows substantial improvement, maintaining a consistently lower attack impact on utility compared to other defenses. Particularly at $A_{p} = 30\%$ , the attack impact on utility is 2-6 units lower for FCD as the number of clients increases, compared to other baselines. Similar trends are observed for $A_{p} = 50\%$ , $100\%$ . In heterogeneous FL settings, FCD outperforms other defenses, showcasing robustness even when clients are trained on varying amounts of non-IID data. For the KBTS dataset, FCD exhibits substantial improvement and lower attack impact across all attack percentages and FL settings compared to other defenses. The limited training data in the KBTS dataset results in the RS technique showing limited robustness, especially for $A_{p} = 30\%$ in homogeneous and heterogeneous FL, followed by FAT. Ta
+
+Table 2. Comprehensive experimental details: datasets, models, FL setup, attack configuration, and metrics.
+
+| FL defense category (src) | FL defense methods | Can defend | U-P tradeoff analysis | Defender's perspective analysis |
| U-CA | EA | P-CA | DIm | DB | DV |
| Adversarial training (C) | FAT [61], RS [8], FedDynAT [39], GALP [13] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Byzantine robust aggregation techniques (S) | Krum [3], ShieldFL [30], FLTrust [5] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Data/ update analysis (S and/or C) | DeepSight [38], FL-Defender [18], SparseFed [36] | 0 | 0 | ○ | ○ | 0 | ○ | ○ |
| Secure multi-party computation (C) | AMPC [58], Byrd et al. [4] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| Trusted execution environments (C) | Flatee[35], Chen et al. [6], PPFL [34] | 0 | 0 | 0 | 0 | 0 | ○ | ○ |
| Differential privacy (C) | NbAFL [52], Hu et al. [16], 2DP-FL [54] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| Homomorphic encryption (C) | DCAE [62], PEFL [29], Batchcrypt[57] | ○ | 0 | 0 | 0 | 0 | ○ | ○ |
| FCD (ours) src: S and C | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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+ "content": "We investigate two FL data shard settings: (i) Homogeneous, where each client's dataset size is identical, i.e., "
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+ "content": "|\\mathcal{D}_1| = |\\mathcal{D}_2| = ..|\\mathcal{D}_n| = \\frac{|\\mathcal{D}|}{n}"
+ },
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+ "type": "text",
+ "content": ", and (ii) Heterogeneous, involving non-independent and non-identically (non-IID) distributed data achieved by partitioning the dataset using a Dirichlet distribution [33] with parameter "
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+ "content": "\\beta = 1"
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+ "type": "text",
+ "content": " among clients. Additional information on standard FL, evasion attacks in FL, homomorphic encryption, and transposition cipher is provided in the supplementary material."
+ }
+ ]
+ }
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+ "type": "text",
+ "content": "Threat model. We introduce two distinct threat models, TM1 and TM2, formulated to reflect real-world FL production deployment settings, as illustrated in Figure 1. Our threat models address an honest-but-curious (HbC) adversary at the central server, as inspired by related work [41, 48]. In TM1 (evasion utility attack), the adversary conducts an indiscriminate evasion attack by manipulating test data at the central server during inference, aiming to misclassify a substantial portion of the inputs [24, 48]. In TM2 (privacy attack), the adversary's goal shifts to performing a model inversion attack (MIA) [15] with the aim of reconstructing private data. We provide our practical assumptions and more details about the threat models in the Supplementary material."
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+ "content": "Figure 1. Overview of two different threat models (TM) with potential vulnerabilities and attacks during inference."
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+ "content": "3. Proposed Framework"
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+ "content": "Problem statement. In each communication round "
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+ "type": "text",
+ "content": ", suppose that the server receives model updates from "
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+ "type": "text",
+ "content": " clients, and there exists a test dataset "
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+ "type": "inline_equation",
+ "content": "\\mathcal{D}_{test} = \\{\\mathcal{X}_i, \\mathcal{Y}_i\\}_{i=1}^{\\mathcal{N}_{te}}"
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+ "content": " with a size of "
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+ "content": "\\mathcal{N}_{te}"
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+ "type": "text",
+ "content": ". The adversary introduces a "
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+ "type": "text",
+ "content": "-bounded poisoned test dataset in TM1, denoted as "
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+ "content": "\\tilde{\\mathcal{D}}_{\\mathrm{test}} = \\{\\tilde{\\mathcal{X}}_i, \\tilde{\\mathcal{Y}}_i\\}_{i=1}^{\\tilde{\\mathcal{N}}_{te}}"
+ },
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+ "content": ", with the dataset size "
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+ "type": "text",
+ "content": " varying based on the attack volume "
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+ "content": "\\rho"
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+ "type": "text",
+ "content": " as:"
+ }
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+ "type": "text",
+ "content": "Definition 3.1 ("
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+ "type": "inline_equation",
+ "content": "\\mu \\rho"
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+ "type": "text",
+ "content": "-bounded adversary.) Let "
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+ {
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+ "type": "inline_equation",
+ "content": "\\mathcal{F}"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " be the model's function class. An adversarial perturbation is characterized by the mapping "
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+ "type": "inline_equation",
+ "content": "\\mathcal{A} := \\mathcal{F} \\times \\mathcal{X} \\times \\mathbb{R} \\to \\tilde{\\mathcal{X}}"
+ },
+ {
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+ "type": "text",
+ "content": ". For "
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+ "type": "inline_equation",
+ "content": "\\mu > 0"
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+ {
+ "bbox": [
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+ "type": "text",
+ "content": ", we define the "
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+ "type": "inline_equation",
+ "content": "l_2"
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+ "type": "text",
+ "content": " norm ball as "
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+ "type": "inline_equation",
+ "content": "\\mathcal{B}_2(\\mathcal{X}, \\mu) := \\{\\tilde{\\mathcal{X}} \\in \\mathbb{R}^d : \\| \\tilde{\\mathcal{X}} - \\mathcal{X} \\| \\leq \\mu\\} \\cap \\mathcal{X}"
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+ "type": "text",
+ "content": ". We classify the adversary as "
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+ "type": "inline_equation",
+ "content": "\\mu \\rho"
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+ "type": "text",
+ "content": "-bounded if it adheres to the condition "
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+ "type": "inline_equation",
+ "content": "\\mathcal{A}(\\mathcal{F}, \\mathcal{X}_i, \\mathcal{Y}_i)_{i=1}^\\rho \\in \\mathcal{B}(\\mathcal{X}_i, \\mu)_{i=1}^\\rho"
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+ "type": "text",
+ "content": ". Furthermore, for a given "
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+ "type": "text",
+ "content": ", "
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+ "type": "inline_equation",
+ "content": "\\rho > 0"
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+ "type": "text",
+ "content": ", we denote the worst-case adversarial perturbation as "
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+ "type": "inline_equation",
+ "content": "\\mathcal{A}^+ := \\arg \\max_{\\tilde{\\mathcal{X}} \\in \\mathcal{B}(\\mathcal{X}, \\mu)} \\mathcal{L}(f_\\theta(\\tilde{\\mathcal{X}}_i, \\mathcal{Y}_i)_{i=1}^\\rho)"
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+ "type": "text",
+ "content": ", where "
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+ "type": "inline_equation",
+ "content": "\\mathcal{L}"
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+ "type": "text",
+ "content": " represents the loss function of the global model."
+ }
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+ }
+ ],
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+ "type": "text",
+ "content": "Next, we formulate the objective of the adversary in MIA attack (TM2) using the below definition."
+ }
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+ }
+ ],
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+ "type": "text",
+ "content": "Definition 3.2 ("
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+ "type": "inline_equation",
+ "content": "\\epsilon"
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+ "type": "text",
+ "content": "-distorted.) Let "
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+ "content": "\\mathcal{X}^*"
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+ {
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+ "type": "text",
+ "content": " represent the data reconstructed by the adversary using the MIA attack. The objective of the adversary is to minimize the reconstruction error "
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": " in terms of as the mean squared error (MSE) between the reconstructed data "
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+ "content": "\\mathcal{X}^*"
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+ "type": "text",
+ "content": " and the original input data "
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+ "type": "inline_equation",
+ "content": "\\mathcal{X}"
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+ {
+ "bbox": [
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+ "type": "text",
+ "content": " as"
+ }
+ ]
+ }
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+ "type": "interline_equation",
+ "content": "P := \\| \\mathcal {X} ^ {*} - \\mathcal {X} \\| _ {2} \\equiv \\frac {1}{\\mathcal {N} _ {t e}} \\sum_ {i = 1} ^ {\\mathcal {N} _ {t e}} \\left(\\mathcal {X} _ {i} ^ {*} - \\mathcal {X} _ {i}\\right) ^ {2} \\leq \\epsilon , f o r s o m e \\epsilon \\geq 0,",
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+ "type": "text",
+ "content": "where "
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+ "content": "\\epsilon"
+ },
+ {
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+ "type": "text",
+ "content": " is the distortion of the reconstructed image."
+ }
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+ }
+ ],
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+ "type": "text",
+ "content": "Further, the global model, "
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+ "type": "inline_equation",
+ "content": "\\mathcal{G}_{\\theta_g}"
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+ {
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+ "type": "text",
+ "content": " attains a global test accuracy of "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "\\mathcal{A}_g"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " before the attack and "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{A}_g^*"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": " after the attack (Definition 3.1). Now, our FCD integrated clients have two primary objectives. First, they aim to minimize the impact of the attack, which is quantified by "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "U = (\\mathcal{A}_g - \\mathcal{A}_g^*)"
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+ "type": "text",
+ "content": ", in order to preserve the global test accuracy. Second, they seek to enhance the privacy of local data "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{D}_k = \\{\\mathcal{X}_{i,k},\\mathcal{Y}_{i,k}\\}_{i = 1}^{\\mathcal{N}_k}"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " by maximizing the reconstruction error "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "P"
+ },
+ {
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+ "type": "text",
+ "content": ". Here, "
+ },
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+ "content": "U"
+ },
+ {
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+ "type": "text",
+ "content": " and "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "P"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " define the utility and privacy gains in the context of FL, where lower values of "
+ },
+ {
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+ "content": "\\mathcal {L} _ {C E _ {k}} = \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\mathcal {L} _ {C E} \\left(\\mathcal {Q} _ {i, k}, \\mathcal {Y} _ {i, k}\\right) = - \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\mathcal {Y} _ {i, k} \\log \\mathcal {Q} _ {i, k}. \\tag {2}",
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+ "type": "text",
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": ". Subsequently, we calculate the distillation loss guided by KL divergence "
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+ "type": "inline_equation",
+ "content": "(\\mathcal{L}_{KLD_k})"
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": " between the prediction probabilities of the teacher and the student local model, which has been trained on normal and FCD-encrypted data, respectively, as follows:"
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+ "content": "\\mathcal {L} _ {K L D _ {k}} \\left(\\mathcal {Q} _ {k} \\| \\mathcal {P} _ {k}\\right) = \\sum_ {i = 1} ^ {\\mathcal {N} _ {k}} \\sum_ {r = 1} ^ {\\mathcal {R}} \\mathcal {Q} _ {i, k} ^ {r} \\log \\frac {\\mathcal {Q} _ {i , k} ^ {r}}{\\mathcal {P} _ {i , k} ^ {r}}, \\tag {3}",
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+ },
+ {
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+ "type": "text",
+ "content": " is the number of classes. Hence, the complete loss for client "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "\\mathcal{C}_k"
+ },
+ {
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+ "type": "text",
+ "content": ", unlike traditional FL systems that solely consider "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "\\mathcal{L}_{CE_k}"
+ },
+ {
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+ "type": "text",
+ "content": ", is expressed as:"
+ }
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+ }
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+ "type": "interline_equation",
+ "content": "\\mathcal {L} _ {k} = \\mathcal {L} _ {C E _ {k}} + \\alpha_ {k} \\mathcal {L} _ {K L D _ {k}}, \\tag {4}",
+ "image_path": "22f8f05ae80f433de6aca0d587ac3712e216dd57319445b469720dfc8fc8bcf1.jpg"
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+ }
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+ "type": "text",
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+ "type": "text",
+ "content": " serves as the weighting factor, regulating the balance between the cross-entropy loss and the distillation loss for client "
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": ". Finally, each client minimizes the total loss "
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+ {
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+ "type": "inline_equation",
+ "content": "\\mathcal{L}_k"
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+ {
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+ "type": "text",
+ "content": " over its FCD-transformed data "
+ },
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+ "type": "inline_equation",
+ "content": "\\mathcal{E}(\\mathcal{X}_k)"
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+ {
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+ "type": "text",
+ "content": " across "
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+ "type": "inline_equation",
+ "content": "E"
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+ "type": "text",
+ "content": " local iterations. The local model "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "f_{\\theta ,k}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " undergoes parameter updates through a backward pass, adjusting as "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\theta_k^{\\mathrm{new}} = \\theta_k^{\\mathrm{old}} - \\eta_k\\nabla_{\\theta_k^{\\mathrm{old}}}\\mathcal{L}_k"
+ },
+ {
+ "bbox": [
+ 46,
+ 594,
+ 289,
+ 713
+ ],
+ "type": "text",
+ "content": ", where "
+ },
+ {
+ "bbox": [
+ 46,
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+ 713
+ ],
+ "type": "inline_equation",
+ "content": "\\eta_{k}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " signifies the learning rate of client "
+ },
+ {
+ "bbox": [
+ 46,
+ 594,
+ 289,
+ 713
+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{C}_k"
+ },
+ {
+ "bbox": [
+ 46,
+ 594,
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+ 713
+ ],
+ "type": "text",
+ "content": ". Following each training phase, the client(s) transmit their local updates, denoted as "
+ },
+ {
+ "bbox": [
+ 46,
+ 594,
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+ ],
+ "type": "inline_equation",
+ "content": "\\nabla \\theta_k^t = \\theta_k^t -\\theta_g^t"
+ },
+ {
+ "bbox": [
+ 46,
+ 594,
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+ ],
+ "type": "text",
+ "content": ", back to the server."
+ }
+ ]
+ }
+ ],
+ "index": 6
+ },
+ {
+ "type": "code",
+ "bbox": [
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+ "blocks": [
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+ "bbox": [
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+ "type": "text",
+ "content": "Algorithm 1 Standard FL with our FCD framework"
+ }
+ ]
+ }
+ ],
+ "index": 7,
+ "angle": 0,
+ "type": "code_caption"
+ },
+ {
+ "bbox": [
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+ "spans": [
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+ "type": "text",
+ "content": "Input: Global model "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "\\mathcal{G}_{\\theta ,t}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " , local data "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{D}_k = (\\mathcal{X}_k,\\mathcal{Y}_k)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \nOutput: Global test accuracy "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{A}_g"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n1: Client execution "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "(\\theta_g^t,\\mathcal{K})"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " .. \n2: for each client "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "k = 1"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " to n do \n3: Train teacher model "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "f_{\\theta ,k}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " on normal data "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{D}_k"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n4: "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\{\\mathcal{P}_{i,k}\\}_{i = (1,\\mathcal{N}_k)}\\gets \\{\\sigma (f_{\\theta ,k}(\\mathcal{X}_{i,k}))\\}_{i = (1,\\mathcal{N}_k)}"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " \n5: "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{E}(\\mathcal{X}_k)\\leftarrow \\mathrm{FCD}(\\mathcal{X}_k,\\mathcal{K})"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n6: for each client "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "k = 1"
+ },
+ {
+ "bbox": [
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+ 545,
+ 415
+ ],
+ "type": "text",
+ "content": " to n do \n7: Initialize the local model "
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "\\theta_k^t\\gets \\theta_g^t"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n8: for "
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "b = 1"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " to batches "
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "\\in \\mathcal{E}(\\mathcal{X}_k)"
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "text",
+ "content": " do \n9: "
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "\\{\\mathcal{Q}_{b,k}\\} \\gets \\{\\sigma (f_{\\theta ,0}(\\mathcal{E}(\\mathcal{X}_k[b))))\\}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n10: "
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{L}_{CE_k}(\\mathcal{Q}_{b,k},\\mathcal{Y}_{b,k})\\gets"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " Eq. 2, Cross-entropy loss \n11: "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{L}_{KLD_k}(\\mathcal{Q}_{b,k}\\| \\mathcal{P}_{b,k})\\gets"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " Eq. 3, Distillation loss \n12: "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{L}_k\\gets \\mathcal{L}_{CE_k} + \\alpha_k\\mathcal{L}_{KLD_k}\\triangleright"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " Total loss (Eq. 4) \n13: "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\theta_k^t\\gets \\theta_k^t -\\eta \\nabla_{\\theta_k^t}\\mathcal{L}_k"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " \n14: "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\nabla \\theta_k^t\\gets \\theta_k^t -\\theta_g^t"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n15: return "
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "\\nabla \\theta_k^t"
+ },
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+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n16: Server execution "
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "(\\nabla \\theta_k^t)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " . \n17: Share "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\theta_g^t,\\mathcal{K}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " to all the clients \n18: Receive model updates from selected clients "
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "\\leftarrow \\nabla \\theta_k^t"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n19: Perform model aggregation using FedAvg (Eq. 1) \n20: Update the global model parameter: "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\theta_g^{t + 1}"
+ },
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+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " \n21: "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "\\hat{\\chi}_{test}\\gets"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " Poisoned test data ▷TM1, TM2 \n22: "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{E}(\\hat{\\chi}_{test})\\gets"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " FCD ("
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "inline_equation",
+ "content": "\\hat{\\chi}_{test},\\mathcal{K}"
+ },
+ {
+ "bbox": [
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+ 415
+ ],
+ "type": "text",
+ "content": ") \n23: Compute "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{A}_g\\gets"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " Test "
+ },
+ {
+ "bbox": [
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+ 545,
+ 415
+ ],
+ "type": "inline_equation",
+ "content": "(\\mathcal{G}_{\\theta_g^{t + 1}},\\mathcal{E}(\\hat{\\chi}_{test}))"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " \n24: return "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{A}_g"
+ }
+ ]
+ }
+ ],
+ "index": 8,
+ "angle": 0,
+ "type": "code_body"
+ }
+ ],
+ "index": 8,
+ "sub_type": "algorithm"
+ },
+ {
+ "bbox": [
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+ "spans": [
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+ "type": "text",
+ "content": "FCD cryptographic encryption. The missing element in the framework described above is the FCD encryption (a detailed algorithm and a visual representation are in the Supplementary material). We introduce a homomorphic encryption technique for FL utilizing a symmetric transposition cipher [43], with a secret key "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{K}"
+ },
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+ "type": "text",
+ "content": " shared among all clients by the server. To ensure data privacy and minimize the potential for brute-force attacks, we define the key's dimension as "
+ },
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+ "type": "inline_equation",
+ "content": "\\mathcal{K} \\in \\mathbb{R}^h"
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+ "type": "text",
+ "content": ", where "
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+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "h"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " corresponds to the image height. The FCD encryption process comprises two steps: (1) transposing the input image "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "x"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " to obtain "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "x' = x^T"
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+ "type": "text",
+ "content": ", where rows are transformed into columns, and (2) reordering rows based on the sequence specified in the secret key "
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+ "type": "inline_equation",
+ "content": "\\mathcal{K}"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": ". Essentially, the positions of pixel values in all colour channels "
+ },
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+ "type": "inline_equation",
+ "content": "(R,G,B)"
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+ "type": "text",
+ "content": " are altered according to "
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+ "type": "text",
+ "content": ", i.e., "
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+ "type": "inline_equation",
+ "content": "\\mathcal{R}_{\\mathcal{K}}(x') = x'[:"
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+ "type": "text",
+ "content": ", "
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+ "type": "inline_equation",
+ "content": "\\mathcal{K},\\colon]"
+ },
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+ "type": "text",
+ "content": ", where "
+ },
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+ "type": "inline_equation",
+ "content": "\\mathcal{R}_{\\mathcal{K}}(.)"
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+ "type": "text",
+ "content": " denotes the row-wise shuffling of input data based on the secret key. Further, we introduce the concept of "
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+ "type": "inline_equation",
+ "content": "\\zeta"
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+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "-separability for the FCD encrypted data space. This concept relates to the distance between FCD-encrypted data space and adversarial perturbations. It specifies that the FCD data space maintains a separation of at least "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "\\zeta"
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+ "type": "text",
+ "content": " from attacked data, ensuring robustness against "
+ },
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+ "type": "inline_equation",
+ "content": "\\mu\\rho"
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+ {
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+ ],
+ "type": "text",
+ "content": "-bounded adversarial perturbation (as defined in Definition 3.1). This"
+ }
+ ]
+ }
+ ],
+ "index": 9
+ }
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+ "type": "text",
+ "content": "Figure 2. Overview of the FCD-integrated FL system, with a focus on one client's training and server-side execution. We compute the cross-entropy loss "
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+ "type": "inline_equation",
+ "content": "(\\mathcal{L}_{CE})"
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+ "content": " denotes the FCD encrypted test data space such that for both training and test splits, "
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+ "content": "\\mathcal{E}(\\mathcal{X}) = \\mathcal{R}_{\\mathcal{K}}(\\mathcal{X}^{\\mathcal{T}})"
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+ "content": "\\| \\mathcal{E}(\\mathcal{X}_i) - \\tilde{\\mathcal{X}}_j \\| \\geq \\zeta"
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+ "content": "Rationale for FCD Design. The transposition of data before row-wise transformations enhances information security robustness by obscuring critical transformation details, providing defense against black-box evasion attacks. The use of a row-based transposition cipher, compatible with row-major programming languages (Python, C, and "
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+ "type": "text",
+ "content": ") [46], ensures efficiency in memory access, parallel processing, and cryptographic operations, aligning with both security and computational considerations in our FCD integrated FL system. Please refer to the Supplementary material for a more detailed rationale."
+ }
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+ "content": "FCD dual property benefits aligned with defender's perspective. (i) Low visibility: The FCD transformation, incorporating both transposition and a secret key, ensures low visibility by creating unique gradients for each key, offering robust protection against various attacks. (ii) Low budget: FCD's vectorized operations and constrained key dimension facilitate efficient implementation, making it suitable for large-scale FL systems with minimal resource requirements and addressing concerns related to processing time"
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+ "content": "and aligning w.r.t. defender's perspective. Please refer to the Supplementary material for more details about the FCD dual property."
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+ "content": " is linear, specifically "
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+ "content": " denotes the image height. This is a significant enhancement compared to the "
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+ "type": "inline_equation",
+ "content": "\\mathcal{O}(n_w\\mathcal{N}^2\\log \\mathcal{N})"
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+ "type": "text",
+ "content": " time complexity required for encryption and decryption of model parameters, where "
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+ "content": " represents the number of model parameters, and "
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+ "content": " is the bit length of the key [19]. (Please refer Supplementary material for the proof.)"
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+ "content": "-distorted characteristics (as described in Definition 3.2) at the server. We provide the respective proofs in the Supplementary material."
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+ "content": ", the global model converges to"
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+ "content": " quantifies how distillation accelerates convergence in our specific setup."
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+ "content": "-bounded attacks.) Let "
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+ "content": "\\mathcal{A} := \\mathcal{G}_{\\theta_g} \\times \\mathcal{X}_{test} \\times \\mathbb{R} \\to \\tilde{\\mathcal{X}}_{test}"
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+ "content": " be the induced adversarial perturbations on the test data using the black-box global model "
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+ "content": " through the mechanism of our FCD integrated FL system. It allows "
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+ "content": ". This robustness is achieved because the adversary's objective is to generate perturbations in the original test data space that maximize the global loss (Definition 3.1), as denoted by "
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+ "content": "\\mathcal{A}^+ := \\arg \\max_{\\tilde{\\mathcal{X}}_{test} \\in \\mathcal{B}(\\mathcal{X}_{test}, \\mu)} \\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\tilde{\\mathcal{X}}_{test,i}, \\mathcal{Y}_i)_{i=1}^\\rho)"
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+ "content": ". However, this adversarial strategy is effective when training and testing occur in the normal data space "
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+ "content": ". In our FCD integrated FL system, training and testing occur in the encrypted data space. Notably, "
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+ "type": "inline_equation",
+ "content": "\\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\tilde{\\mathcal{X}}_{test,i}, \\mathcal{Y}_i)_{i=1}^\\rho) \\neq \\mathcal{L}(\\mathcal{G}_{\\theta_g}(\\mathcal{E}(\\tilde{\\mathcal{X}}_{test,i}), \\mathcal{Y}_i)_{i=1}^\\rho)"
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+ "content": "\\mathcal{E}(\\tilde{\\mathcal{X}}_{test}) \\triangleq \\mathcal{R}_{\\mathcal{K}}(\\tilde{\\mathcal{X}}_{test}^T) \\neq \\tilde{\\mathcal{X}}_{test}"
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+ "content": "\\| \\mathcal{E}(\\mathcal{X}_i) - \\tilde{\\mathcal{X}}_j \\| \\geq \\zeta"
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+ "content": " holds for any "
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+ "content": "\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D})"
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+ "content": "\\mathcal{E}(\\mathcal{X}_i) \\in \\mathcal{E}(\\mathcal{D}_{test})"
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+ "content": "\\tilde{\\mathcal{X}}_j \\in \\mathcal{A}_{\\mu\\rho}(\\mathcal{X}_{test})"
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+ "type": "text",
+ "content": " (Definition 3.3). Hence, the learning process occurs in different data spaces unknown to the adversary. Consequently, the gradients of the loss function trained on normal data are notably distinct from those trained on FCD encrypted data, effectively mitigating "
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+ "type": "text",
+ "content": "-bounded adversarial data."
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+ "content": "\\epsilon"
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+ "content": " represent the data reconstructed by the adversary using MIA attack as per the conditions specified in Definition 3.2. We demonstrate that training on FCD-encrypted data space, denoted as "
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+ "content": "-distorted MIA attacks. Specifically, our result establishes that:"
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+ "content": "\\left| \\| \\mathcal {X} ^ {*} \\| - \\| \\mathcal {E} (\\mathcal {X}) \\| \\right| \\leq \\epsilon + \\delta , f o r s o m e \\epsilon \\geq 0 a n d \\delta \\geq 0.",
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+ "content": "Datasets, implementation details, and metrics. Table 2 presents comprehensive details about the datasets, FL setup, attack percentage, and metrics. For TM1, we employed the black-box and active data poisoning technique called MSimBA [23]. Each experiment was conducted thrice, and"
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+ "content": "results were averaged with standard deviations presented. Please refer to the Supplementary material for more details. Baselines. For TM1, we have chosen FAT [61] and Randomized Smoothing (RS) [8] based on their relevance and applicability in evasion attacks within FL, as outlined in Table 1. Additionally, we explore the effectiveness under the following configurations: (a) With/without attack and defense and (b) two FCD setups under different client counts and attack percentages "
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+ "content": ". These setups include FCD-homoFL (homogeneous FL) and FCD-hetFL (heterogeneous FL). In TM2, we follow the baselines as per [27]."
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+ "content": "Benign setting. Table 3 showcases the performance of our FCD defense, along with other baselines and configurations, across four datasets under a no attack scenario. Notably, FCD demonstrates comparable or higher accuracy than vanilla FL, particularly excelling in datasets like GTSRB and KBTS. This improvement is attributed to FCD's ability to capture diverse transformation patterns with a secret key, facilitating reduced training loss and improved model convergence. Furthermore, with its symmetric cryptographic transformation, FCD outperforms FAT and RS for GTSRB and KBTS. However, in CIFAR10 and EMNIST, FCD's performance, while slightly lower than base FL, surpasses FAT and RS. This trade-off between utility and privacy is expected, as FCD provides both aspects with a modest decrease in base accuracy."
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+ "content": "Attack setting. We analyse FCD effectiveness under two threat models separately."
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+ "type": "text",
+ "content": " for FCD compared to other baselines under M-SimBA evasion attack on GTSRB and KBTS datasets. The results indicate a significant utility degradation with higher attack percentages and more clients, emphasizing the impact of substantial test data modification by M-SimBA. FCD consistently demonstrates superior performance and robustness across all attack percentages. In homogeneous FL settings, FCD shows substantial improvement, maintaining a consistently lower attack impact on utility compared to other defenses. Particularly at "
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+ "content": ", the attack impact on utility is 2-6 units lower for FCD as the number of clients increases, compared to other baselines. Similar trends are observed for "
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+ "content": ". In heterogeneous FL settings, FCD outperforms other defenses, showcasing robustness even when clients are trained on varying amounts of non-IID data. For the KBTS dataset, FCD exhibits substantial improvement and lower attack impact across all attack percentages and FL settings compared to other defenses. The limited training data in the KBTS dataset results in the RS technique showing limited robustness, especially for "
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+ "content": " in homogeneous and heterogeneous FL, followed by FAT. Ta"
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| ② Content | (P4) Watermark | Text overlay with αw=[51, 153, 255] and Rz=[2, 4, 6] | 09.05 |
| (P5) Background | Random image inserting in background with Ni=[1, 3, 5] | 26.70 |
| ③ Inconsistency | (P6) Illumination | Region illumination change with Vl=[51, 102, 153] or Vs=[0.5, 0.25, 0.17] | 10.67 |
| (P7) Ink-bleeding | Erosion simulation with kernel size Ke=[3, 7, 11] | 08.91 |
| (P8) Ink-holdout | Dilation simulation with kernel size Kd=[3, 7, 11] | 15.64 |
| ④ Blur | (P9) Defocus | Gaussian blur simulation with kernel size Kg=[1, 3, 5] | 08.10 |
| (P10) Vibration | Motion blur simulation with kernel size Km=[3, 9, 15] | 16.29 |
| ⑤ Noise | (P11) Speckle | Blotches noise simulation with Db=[1 × 10-4,3 × 10-4,5 × 10-4] | 24.31 |
| (P12) Texture | Fibrous noise simulation with Nf=[300, 900, 1500] | 40.57 |
| Overall | The average of all perturbations | 22.90 |
",
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+ {
+ "type": "text",
+ "text": "$R_{z}$ and alpha channel opacity $\\alpha_{w}$ . (P5) Complex Background, which is often leveraged in printed media to enrich the document's visual content. Images from ILSVRC dataset [49], are randomly embedded within the document images for the simulation of background. The severity level is determined by the number of embedded images $N_{i}$ .",
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+ },
+ {
+ "type": "text",
+ "text": "3 Inconsistency Distortion, which addresses distortions, e.g., (P6) Non-uniform Illumination, simulated through a $50\\%$ probability of glare or shadow, with levels determined by the brightness $V_{l}$ or shadow $V_{s}$ , (P7) Ink-bleeding, achieved by using erosion process within the document image, and (P8) Ink-holdout, which are accomplished by using morphological dilation operations with elliptical kernel $K_{e}$ and $K_{d}$ , respectively.",
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+ {
+ "type": "text",
+ "text": "$\\Theta$ Blur. Blur effects arise from (P9) Defocus, which is approximated with Gaussian kernel function, i.e., the point spread function (PSF) with severity levels controlled by kernel size $K_{g}$ , and (P10) Vibration, which is achieved by using a convolution kernel rotated by a random angle to replicate directional motion in size $K_{m}$ . This perturbation can be applied via a Gaussian filtering operation.",
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+ },
+ {
+ "type": "text",
+ "text": "⑤ Noise. Two unique noise types in document images are included. (P11) Speckle caused by ink clumping, can be achieved by using Gaussian noise modulated by blob density $D_{b}$ . It reproduces both foreground and background noise, reflecting the complex stochastic nature of noise artifacts. (P12) Texture, resembling the fibers in paper, is added to simulate the natural fiber structures of archival documents. In each fiber segment, curvature and length are independently simulated with trigonometric functions and Cauchy distribution, forming the complete fiber through segment assembly. The number of fibers $N_{f}$ varies across noise levels to represent different paper qualities.",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "The principles of design and division of these perturbations are three-fold: (1) All perturbations are realistic and",
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "inspired by the real-world disturbance or document layout analysis. (2) The leveraged perturbations are comprehensive and occur in document from top to bottom, from global to local, from dense to sparse, and from content-wise to pixel-wise. (3) All perturbation levels are reasonably determined by Image Quality Assessment (IQA) metrics, e.g., MS-SSIM [62] and CW-SSIM [51], Degradation and our proposed metrics which are detailed as the following.",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "3.3. Perturbation Evaluation Metrics",
+ "text_level": 1,
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Quantifying the document perturbations is a crucial task, which lacks straightforward metrics. Previous methods [25, 28, 33, 66] are constrained by using a pre-selected baseline model performance as a reference to measure the perturbation effect, which will conflate the perturbation effect with the model robustness due to the inherent variability.",
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+ "type": "text",
+ "text": "To design an effective perturbation metric, our considerations are two-fold: (1) Evaluation should be relatively independent to any specific model performance. (2) Evaluation should be inclusive and sensitive to all perturbations. Inspired by the IQA methods [12], we analyse all 12 perturbations through 4 metrics, as presented in Fig. 3.",
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+ "type": "text",
+ "text": "MS-SSIM (Multi-Scale Structural Similarity Index) [62] evaluates image quality by considering difference in structural information across different resolutions, as:",
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+ {
+ "type": "equation",
+ "text": "\n$$\nf ^ {\\mathrm {M S - S S I M}} (x, y) = \\left[ l _ {M} (x, y) \\right] ^ {\\alpha_ {M}} \\cdot \\prod_ {j = 1} ^ {M} \\left[ c _ {j} (x, y) \\right] ^ {\\beta_ {j}} \\left[ s _ {j} (x, y) \\right] ^ {\\gamma_ {j}}, \\tag {1}\n$$\n",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "where $l_{M}(x,y)$ represents the luminance difference at the $M$ -th scale, and $c_{j}(x,y)$ , $s_{j}(x,y)$ are the contrast and structure comparison at the $j$ -th scale. It measures the impact of most perturbations effectively but lacks sensitivity to watermark and warping (Fig. 3b), and is overly sensitive to rotation and keystoning, as shown in Fig. 3a.",
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+ "type": "page_number",
+ "text": "15559",
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+ "img_path": "images/91a89b0cdc051aa3edd62939b651085ef4374fbf5e10c02875f236ef10373304.jpg",
+ "image_caption": [
+ "(a) Comparison of four evaluation metrics on 12 perturbations"
+ ],
+ "image_footnote": [],
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+ "type": "image",
+ "img_path": "images/da8190bc8f8a4852f4ae9e411e7ab221317a48c8c798d7f50d6d7c82d208f689.jpg",
+ "image_caption": [
+ "(b) Comparison of the sensitivity of metrics to different perturbation levels.",
+ "Figure 3. Analysis of perturbation evaluation metrics. (a) Comparison of perturbation metrics, including MS-SSIM, CW-SSIM, Degradation w.r.t a baseline, and the proposed mean Perturbation Effect (mPE). mPE is more balanced and inclusive to different perturbations. (b) Six documents perturbed by warping and defocus and their scores indicate that mPE is more sensitive to measure different levels."
+ ],
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+ {
+ "type": "text",
+ "text": "CW-SSIM (Complex Wavelet Structural Similarity Index) [51] assesses the similarity of local patterns of pixel intensities transformed into the wavelet domain, which is robust to spatial distortions, e.g., keystoneing and rotation, formulated by the following equation:",
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+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf ^ {\\mathrm {C W - S S I M}} (x, y) = \\frac {2 \\left| \\sum_ {l = 1} ^ {L} w _ {l} x _ {l} y _ {l} ^ {*} \\right| + K}{\\sum_ {l = 1} ^ {L} \\left| x _ {l} \\right| ^ {2} + \\sum_ {l = 1} ^ {L} \\left| y _ {l} \\right| ^ {2} + K}, \\tag {2}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "where $x_{l}$ and $y_{l}$ denote the wavelet coefficients, $w_{l}$ is the $l$ -th coefficient weight, $L$ indicates the local region number, $K$ is tiny value, and * denotes the complex conjugation. As shown in Fig. 3a, it reflects the perturbations impact but fails for quantitative evaluation at different levels for some perturbations, e.g., defocus, demonstrated in Fig. 3b.",
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+ },
+ {
+ "type": "text",
+ "text": "Degradation across severity levels, as indicated by ImageNet-C [25], is a common practice for various robustness benchmark [25, 28, 33, 66]. Degradation $D$ , where $D = 1 - mAP$ similar with ImageNet-C, illustrates the performance impact of perturbation on a baseline model. From Fig. 3a, we notice that the Degradation metric is highly dependent on the choice of the baseline and tends to be overly sensitive to perturbations, e.g., background and texture.",
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+ },
+ {
+ "type": "text",
+ "text": "Mean Perturbation Effect (mPE) is a new metric proposed to assess the compound effects of document perturbations. For specific perturbation $p$ at each severity level $s$ , degradation of model $g$ is written as $D_{s,p}^{g}$ , and IQA metrics is written as $f_{s,p}^{i}$ , i.e., from Eq. (1) and Eq. (2). The mPE to measure the effect of perturbation $p$ is calculated as:",
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbf {m P E} _ {p} = \\frac {1}{N M K} \\sum_ {s = 1} ^ {N} \\left(\\sum_ {i = 1} ^ {M} f _ {s, p} ^ {i} + \\sum_ {g = 1} ^ {K} D _ {s, p} ^ {g}\\right). \\tag {3}\n$$\n",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "mPE aims to isolate the model's robustness from the image alterations impact. We employ MS-SSIM and CW-SSIM as IQA metrics, and we select Faster R-CNN [48] as a baseline in Degradation metric. As evident from Fig. 3b, our mPE metric effectively quantifies the impact of each perturbation and severity level, while other metrics are less sensitive.",
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+ },
+ {
+ "type": "text",
+ "text": "3.4. Perturbation Robustness Benchmarks",
+ "text_level": 1,
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "To systematically benchmark the robustness of DLA models, we apply the mentioned 12 perturbations to 3 datasets.",
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "PubLayNet-P [72] is a commonly used large-scale DLA dataset. It contains over 360,000 document images sourced from PubMed Central, and focuses on 5 principal types of layout elements, i.e., text, title, list, table, and figure.",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "DocLayNet-P [46] includes academic papers, brochures, business letters, technical papers, and more with 80,863 document pages. It contains annotations for 11 layout components, aiming at enabling generalizable DLA models.",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "$\\mathbf{M}^6\\mathbf{Doc - P}$ [8] provides more nuanced annotations (74 classes), capturing finer aspects of document layouts with 9,080 document images. This dataset is tailored to advance research in sophisticated document understanding tasks.",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "While PubLayNet provides vast array of annotated scientific documents, DocLayNet broadens the variety of document types, and M6Doc delves into more detailed and complex annotations, catering to advanced document analysis.",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "4. Robust Document Layout Analyzer",
+ "text_level": 1,
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "4.1. Framework Overview",
+ "text_level": 1,
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "To boost the robustness performance on perturbed documents, we propose a Robust Document Layout Analyzer (RoDLA) model. As shown in Fig. 4, our proposed model is inspired by the architecture of DINO [69] but introduces channel attention blocks and average pooling blocks in the Encoder, allowing for more robust feature extraction. Additionally, we incorporate InternImage[61] as a backbone, which is pre-trained on ImageNet22K [10] dataset. This pretraining setting enhances multi-scale feature extraction for more stable performance. The overall architecture and robustness enhancement design is depicted in Fig. 4.",
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+ {
+ "type": "text",
+ "text": "4.2. Robustness Enhancement Design",
+ "text_level": 1,
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "As identified in [19], self-attention tends to overemphasize focus on irrelevant tokens, that leads to attention shifting in",
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+ {
+ "type": "page_number",
+ "text": "15560",
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+ {
+ "type": "image",
+ "img_path": "images/79db0dabeb1fa0901d071046f4604fed14190c028ea98273dc6a09f187e4af94.jpg",
+ "image_caption": [
+ "Figure 4. The architecture of RoDLA model. RoDLA is comprised of Encoder, Query Selection, Decoder, and Matching components. It optimizes the attention mechanism in Encoder, heightening focus on crucial tokens and reinforcing key token connections in multi-scale features to extract stable features."
+ ],
+ "image_footnote": [],
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+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "visual document analysis, where globally accumulated perturbations may interfere with the model performance. To address this, we integrate spatial-wise average pooling layers with varying dilation into the Robust Encoder, accompanied by a Dilation Predictor employing two $3 \\times 3$ convolutional layers to predict the utilization of dilated average pooling layers. This approach leverages the attention of neighboring tokens to mitigate the overemphasis on tokens, and dynamically constrains long-distance perception based on context. By diminishing attention on irrelevant tokens, this approach effectively captures robust feature.",
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+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "As outlined in [73], self-attention inherently facilitates visual grouping, filtering out irrelevant information. Thus, we have incorporated a Channel-wise Attention (CA) module within the Encoder. This module implements a self-attention operation in channel dimension $d$ , as: $\\mathbf{Y} = \\sigma (\\frac{\\text{Softmax}(\\mathbf{Q})\\cdot\\text{Softmax}(\\mathbf{K}^T)}{\\sqrt{n}}\\cdot_{\\text{MLP}}(\\mathbf{V}))$ , where $\\mathbf{Q},\\mathbf{K},\\mathbf{V}\\in \\mathbb{R}^{d\\times n}$ . This mechanism not only simplifies computational complexity but also aggregates local features in channel-wise. The encoder module synergies channel-wise and spatial-wise self-attention, aggregating strongly correlated feature channels while directing attention towards spatially relevant key tokens. This enhancement of global attention minimizes the shifting of attention tokens, leading to robust feature extraction for complex analysis task.",
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+ {
+ "type": "text",
+ "text": "5. Benchmarking and Analysis",
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+ },
+ {
+ "type": "text",
+ "text": "5.1. Experiment Settings",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "Selected Baselines. To establish a comprehensive robustness benchmark for DLA, we not only include three distinct datasets but also reproduce a variety of state-of-the-art methods and backbones, including CNN-based backbones",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "(ResNet [22], LayoutParser [54], InternImage [61]), Transformer-based structures (SwinDocSegmenter [4], DiT [31], and LayoutLMv3 [26]), and methods from CNN-based structures (Faster R-CNN [48], Mask R-CNN [23], Cascade R-CNN [6]), to Transformer-based frameworks (DocSegTr [5], DINO [69], Co-DINO [75], and our RoDLA). Note that all the methods are pre-trained on ImageNet dataset, while DiT and LayoutLMv3 are pre-trained on the IIT-CDIP Test Collection 1.0 document dataset [30], which gives a comparison of the pre-training data.",
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+ {
+ "type": "text",
+ "text": "Implementation Details. For fair comparison, we reproduce all methods in the MMDetection [7] framework. Hyperparameters are from their original settings or papers. All models are trained and validated exclusively on clean data. Only during the testing phase, the impact of perturbations is evaluated on all methods. This approach ensures that our robustness benchmark is conducted under controlled and unbiased conditions. More details are in supplementary.",
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+ {
+ "type": "text",
+ "text": "5.2. Robustness Evaluation Metrics",
+ "text_level": 1,
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+ {
+ "type": "text",
+ "text": "From our analysis in Fig. 2 and Table 1, it is evident that various types of perturbations impact the same document images differently, leading to a range of effects on the models' performance. Previous robustness benchmarks [25, 33, 39, 66] measure the perturbations impact solely relying on a pre-selected baseline model. This approach tends to overlook the varying degrees of robustness different models exhibit towards the perturbations impact, overly relying on the benchmark performance of a particular model. To address this limitation, we propose Mean Robustness Degradation (mRD) for evaluating model robustness. mRD is a comprehensive measure of the average performance of models under different levels and types of perturbations. Additionally, inspired by ImageNet-C, we opt to incorporate Degradation $D$ into mRD because we found that different perturbations pose varying levels of difficulty. Thus, the RD of a perturbation $p$ is calculated as:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbf {R D} _ {p} = \\frac {1}{N} \\sum_ {s = 1} ^ {N} \\frac {D _ {s , p} ^ {g}}{\\mathrm {m P E} _ {s , p}}, \\tag {4}\n$$\n",
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+ {
+ "type": "text",
+ "text": "where mPE is from Eq. (3). The overall robustness score $\\mathbf{mRD}$ is the average score of $\\mathbf{RD}_p$ across perturbations $p\\in [1,12]$ . Exceeding 100 indicates the model's performance degrades more than expected due to perturbations, while falling below 100 suggests performance improvements despite these impacts. The lower the better. This metric offers a comprehensive assessment of model robustness, enabling evaluation under diverse and realistic conditions. To ensure a more comprehensive evaluation, we test DLA models on clean datasets with mAP scores, as well as the average mAP performance on 12 perturbations (i.e., P-Avg).",
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+ "text": "15561",
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+ "type": "table",
+ "img_path": "images/02cec31ab1b3e4bd834103996f7a2c787ae62ceab1fba634f46c66730b27ac04.jpg",
+ "table_caption": [
+ "Table 2. The robustness benchmark on PubLayNet-P dataset. V, L, and T stand for Visual, Layout, and Textual modality. 'Ext.' means using extra pre-training data. mAP scores are evaluated on the clean data, the 12 perturbation types (P1-P12), and the perturbation average (P-Avg). mRD is the proposed mean Robustness Degradation, the smaller the better."
+ ],
+ "table_footnote": [],
+ "table_body": "| Document Perturbation (P) | Description | mPE↓ |
| None | The original clean data | 0 |
| ① Spatial | (P1) Rotation | Rotation simulation with θ=[5°, 10°, 15°] | 58.30 |
| (P2) Warping | Elastic transformation simulation with Rσ=[0.2, 0.06, 0.04] and Rα=[2, 0.6, 0.4] | 22.00 |
| (P3) Keystoning | Perspective transformation simulation with Rk=[0.02, 0.06, 0.1] | 34.49 |
| ② Content | (P4) Watermark | Text overlay with αw=[51, 153, 255] and Rz=[2, 4, 6] | 09.05 |
| (P5) Background | Random image inserting in background with Ni=[1, 3, 5] | 26.70 |
| ③ Inconsistency | (P6) Illumination | Region illumination change with Vl=[51, 102, 153] or Vs=[0.5, 0.25, 0.17] | 10.67 |
| (P7) Ink-bleeding | Erosion simulation with kernel size Ke=[3, 7, 11] | 08.91 |
| (P8) Ink-holdout | Dilation simulation with kernel size Kd=[3, 7, 11] | 15.64 |
| ④ Blur | (P9) Defocus | Gaussian blur simulation with kernel size Kg=[1, 3, 5] | 08.10 |
| (P10) Vibration | Motion blur simulation with kernel size Km=[3, 9, 15] | 16.29 |
| ⑤ Noise | (P11) Speckle | Blotches noise simulation with Db=[1 × 10-4,3 × 10-4,5 × 10-4] | 24.31 |
| (P12) Texture | Fibrous noise simulation with Nf=[300, 900, 1500] | 40.57 |
| Overall | The average of all perturbations | 22.90 |
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+ "content": "\\(R_{z}\\) and alpha channel opacity \\(\\alpha_{w}\\). (P5) Complex Background, which is often leveraged in printed media to enrich the document's visual content. Images from ILSVRC dataset [49], are randomly embedded within the document images for the simulation of background. The severity level is determined by the number of embedded images \\(N_{i}\\)."
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+ "content": "3 Inconsistency Distortion, which addresses distortions, e.g., (P6) Non-uniform Illumination, simulated through a \\(50\\%\\) probability of glare or shadow, with levels determined by the brightness \\(V_{l}\\) or shadow \\(V_{s}\\), (P7) Ink-bleeding, achieved by using erosion process within the document image, and (P8) Ink-holdout, which are accomplished by using morphological dilation operations with elliptical kernel \\(K_{e}\\) and \\(K_{d}\\), respectively."
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+ "content": "\\(\\Theta\\) Blur. Blur effects arise from (P9) Defocus, which is approximated with Gaussian kernel function, i.e., the point spread function (PSF) with severity levels controlled by kernel size \\(K_{g}\\), and (P10) Vibration, which is achieved by using a convolution kernel rotated by a random angle to replicate directional motion in size \\(K_{m}\\). This perturbation can be applied via a Gaussian filtering operation."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.686,
+ 0.47,
+ 0.867
+ ],
+ "angle": 0,
+ "content": "⑤ Noise. Two unique noise types in document images are included. (P11) Speckle caused by ink clumping, can be achieved by using Gaussian noise modulated by blob density \\( D_{b} \\). It reproduces both foreground and background noise, reflecting the complex stochastic nature of noise artifacts. (P12) Texture, resembling the fibers in paper, is added to simulate the natural fiber structures of archival documents. In each fiber segment, curvature and length are independently simulated with trigonometric functions and Cauchy distribution, forming the complete fiber through segment assembly. The number of fibers \\( N_{f} \\) varies across noise levels to represent different paper qualities."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.871,
+ 0.47,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "The principles of design and division of these perturbations are three-fold: (1) All perturbations are realistic and"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.358,
+ 0.892,
+ 0.48
+ ],
+ "angle": 0,
+ "content": "inspired by the real-world disturbance or document layout analysis. (2) The leveraged perturbations are comprehensive and occur in document from top to bottom, from global to local, from dense to sparse, and from content-wise to pixel-wise. (3) All perturbation levels are reasonably determined by Image Quality Assessment (IQA) metrics, e.g., MS-SSIM [62] and CW-SSIM [51], Degradation and our proposed metrics which are detailed as the following."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.49,
+ 0.787,
+ 0.505
+ ],
+ "angle": 0,
+ "content": "3.3. Perturbation Evaluation Metrics"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.514,
+ 0.892,
+ 0.605
+ ],
+ "angle": 0,
+ "content": "Quantifying the document perturbations is a crucial task, which lacks straightforward metrics. Previous methods [25, 28, 33, 66] are constrained by using a pre-selected baseline model performance as a reference to measure the perturbation effect, which will conflate the perturbation effect with the model robustness due to the inherent variability."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.606,
+ 0.892,
+ 0.697
+ ],
+ "angle": 0,
+ "content": "To design an effective perturbation metric, our considerations are two-fold: (1) Evaluation should be relatively independent to any specific model performance. (2) Evaluation should be inclusive and sensitive to all perturbations. Inspired by the IQA methods [12], we analyse all 12 perturbations through 4 metrics, as presented in Fig. 3."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.697,
+ 0.892,
+ 0.742
+ ],
+ "angle": 0,
+ "content": "MS-SSIM (Multi-Scale Structural Similarity Index) [62] evaluates image quality by considering difference in structural information across different resolutions, as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.506,
+ 0.755,
+ 0.892,
+ 0.81
+ ],
+ "angle": 0,
+ "content": "\\[\nf ^ {\\mathrm {M S - S S I M}} (x, y) = \\left[ l _ {M} (x, y) \\right] ^ {\\alpha_ {M}} \\cdot \\prod_ {j = 1} ^ {M} \\left[ c _ {j} (x, y) \\right] ^ {\\beta_ {j}} \\left[ s _ {j} (x, y) \\right] ^ {\\gamma_ {j}}, \\tag {1}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.811,
+ 0.892,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "where \\( l_{M}(x,y) \\) represents the luminance difference at the \\( M \\)-th scale, and \\( c_{j}(x,y) \\), \\( s_{j}(x,y) \\) are the contrast and structure comparison at the \\( j \\)-th scale. It measures the impact of most perturbations effectively but lacks sensitivity to watermark and warping (Fig. 3b), and is overly sensitive to rotation and keystoning, as shown in Fig. 3a."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.481,
+ 0.945,
+ 0.519,
+ 0.957
+ ],
+ "angle": 0,
+ "content": "15559"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.078,
+ 0.093,
+ 0.457,
+ 0.228
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.103,
+ 0.23,
+ 0.431,
+ 0.243
+ ],
+ "angle": 0,
+ "content": "(a) Comparison of four evaluation metrics on 12 perturbations"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.462,
+ 0.096,
+ 0.888,
+ 0.228
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.477,
+ 0.23,
+ 0.873,
+ 0.243
+ ],
+ "angle": 0,
+ "content": "(b) Comparison of the sensitivity of metrics to different perturbation levels."
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.075,
+ 0.255,
+ 0.893,
+ 0.298
+ ],
+ "angle": 0,
+ "content": "Figure 3. Analysis of perturbation evaluation metrics. (a) Comparison of perturbation metrics, including MS-SSIM, CW-SSIM, Degradation w.r.t a baseline, and the proposed mean Perturbation Effect (mPE). mPE is more balanced and inclusive to different perturbations. (b) Six documents perturbed by warping and defocus and their scores indicate that mPE is more sensitive to measure different levels."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.312,
+ 0.47,
+ 0.388
+ ],
+ "angle": 0,
+ "content": "CW-SSIM (Complex Wavelet Structural Similarity Index) [51] assesses the similarity of local patterns of pixel intensities transformed into the wavelet domain, which is robust to spatial distortions, e.g., keystoneing and rotation, formulated by the following equation:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.1,
+ 0.398,
+ 0.47,
+ 0.437
+ ],
+ "angle": 0,
+ "content": "\\[\nf ^ {\\mathrm {C W - S S I M}} (x, y) = \\frac {2 \\left| \\sum_ {l = 1} ^ {L} w _ {l} x _ {l} y _ {l} ^ {*} \\right| + K}{\\sum_ {l = 1} ^ {L} \\left| x _ {l} \\right| ^ {2} + \\sum_ {l = 1} ^ {L} \\left| y _ {l} \\right| ^ {2} + K}, \\tag {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.445,
+ 0.47,
+ 0.536
+ ],
+ "angle": 0,
+ "content": "where \\( x_{l} \\) and \\( y_{l} \\) denote the wavelet coefficients, \\( w_{l} \\) is the \\( l \\)-th coefficient weight, \\( L \\) indicates the local region number, \\( K \\) is tiny value, and * denotes the complex conjugation. As shown in Fig. 3a, it reflects the perturbations impact but fails for quantitative evaluation at different levels for some perturbations, e.g., defocus, demonstrated in Fig. 3b."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.536,
+ 0.471,
+ 0.656
+ ],
+ "angle": 0,
+ "content": "Degradation across severity levels, as indicated by ImageNet-C [25], is a common practice for various robustness benchmark [25, 28, 33, 66]. Degradation \\(D\\), where \\(D = 1 - mAP\\) similar with ImageNet-C, illustrates the performance impact of perturbation on a baseline model. From Fig. 3a, we notice that the Degradation metric is highly dependent on the choice of the baseline and tends to be overly sensitive to perturbations, e.g., background and texture."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.657,
+ 0.47,
+ 0.749
+ ],
+ "angle": 0,
+ "content": "Mean Perturbation Effect (mPE) is a new metric proposed to assess the compound effects of document perturbations. For specific perturbation \\( p \\) at each severity level \\( s \\), degradation of model \\( g \\) is written as \\( D_{s,p}^{g} \\), and IQA metrics is written as \\( f_{s,p}^{i} \\), i.e., from Eq. (1) and Eq. (2). The mPE to measure the effect of perturbation \\( p \\) is calculated as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.125,
+ 0.759,
+ 0.469,
+ 0.802
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {m P E} _ {p} = \\frac {1}{N M K} \\sum_ {s = 1} ^ {N} \\left(\\sum_ {i = 1} ^ {M} f _ {s, p} ^ {i} + \\sum_ {g = 1} ^ {K} D _ {s, p} ^ {g}\\right). \\tag {3}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.81,
+ 0.47,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "mPE aims to isolate the model's robustness from the image alterations impact. We employ MS-SSIM and CW-SSIM as IQA metrics, and we select Faster R-CNN [48] as a baseline in Degradation metric. As evident from Fig. 3b, our mPE metric effectively quantifies the impact of each perturbation and severity level, while other metrics are less sensitive."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.499,
+ 0.312,
+ 0.829,
+ 0.327
+ ],
+ "angle": 0,
+ "content": "3.4. Perturbation Robustness Benchmarks"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.335,
+ 0.892,
+ 0.365
+ ],
+ "angle": 0,
+ "content": "To systematically benchmark the robustness of DLA models, we apply the mentioned 12 perturbations to 3 datasets."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.366,
+ 0.892,
+ 0.426
+ ],
+ "angle": 0,
+ "content": "PubLayNet-P [72] is a commonly used large-scale DLA dataset. It contains over 360,000 document images sourced from PubMed Central, and focuses on 5 principal types of layout elements, i.e., text, title, list, table, and figure."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.427,
+ 0.892,
+ 0.487
+ ],
+ "angle": 0,
+ "content": "DocLayNet-P [46] includes academic papers, brochures, business letters, technical papers, and more with 80,863 document pages. It contains annotations for 11 layout components, aiming at enabling generalizable DLA models."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.487,
+ 0.892,
+ 0.547
+ ],
+ "angle": 0,
+ "content": "\\(\\mathbf{M}^6\\mathbf{Doc - P}\\) [8] provides more nuanced annotations (74 classes), capturing finer aspects of document layouts with 9,080 document images. This dataset is tailored to advance research in sophisticated document understanding tasks."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.548,
+ 0.892,
+ 0.608
+ ],
+ "angle": 0,
+ "content": "While PubLayNet provides vast array of annotated scientific documents, DocLayNet broadens the variety of document types, and M6Doc delves into more detailed and complex annotations, catering to advanced document analysis."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.499,
+ 0.623,
+ 0.82,
+ 0.64
+ ],
+ "angle": 0,
+ "content": "4. Robust Document Layout Analyzer"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.648,
+ 0.706,
+ 0.662
+ ],
+ "angle": 0,
+ "content": "4.1. Framework Overview"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.671,
+ 0.892,
+ 0.838
+ ],
+ "angle": 0,
+ "content": "To boost the robustness performance on perturbed documents, we propose a Robust Document Layout Analyzer (RoDLA) model. As shown in Fig. 4, our proposed model is inspired by the architecture of DINO [69] but introduces channel attention blocks and average pooling blocks in the Encoder, allowing for more robust feature extraction. Additionally, we incorporate InternImage[61] as a backbone, which is pre-trained on ImageNet22K [10] dataset. This pretraining setting enhances multi-scale feature extraction for more stable performance. The overall architecture and robustness enhancement design is depicted in Fig. 4."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.847,
+ 0.791,
+ 0.864
+ ],
+ "angle": 0,
+ "content": "4.2. Robustness Enhancement Design"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.5,
+ 0.871,
+ 0.892,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "As identified in [19], self-attention tends to overemphasize focus on irrelevant tokens, that leads to attention shifting in"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.481,
+ 0.945,
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+ ],
+ "angle": 0,
+ "content": "15560"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.087,
+ 0.092,
+ 0.462,
+ 0.312
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.076,
+ 0.316,
+ 0.471,
+ 0.386
+ ],
+ "angle": 0,
+ "content": "Figure 4. The architecture of RoDLA model. RoDLA is comprised of Encoder, Query Selection, Decoder, and Matching components. It optimizes the attention mechanism in Encoder, heightening focus on crucial tokens and reinforcing key token connections in multi-scale features to extract stable features."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.401,
+ 0.47,
+ 0.567
+ ],
+ "angle": 0,
+ "content": "visual document analysis, where globally accumulated perturbations may interfere with the model performance. To address this, we integrate spatial-wise average pooling layers with varying dilation into the Robust Encoder, accompanied by a Dilation Predictor employing two \\(3 \\times 3\\) convolutional layers to predict the utilization of dilated average pooling layers. This approach leverages the attention of neighboring tokens to mitigate the overemphasis on tokens, and dynamically constrains long-distance perception based on context. By diminishing attention on irrelevant tokens, this approach effectively captures robust feature."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.568,
+ 0.471,
+ 0.78
+ ],
+ "angle": 0,
+ "content": "As outlined in [73], self-attention inherently facilitates visual grouping, filtering out irrelevant information. Thus, we have incorporated a Channel-wise Attention (CA) module within the Encoder. This module implements a self-attention operation in channel dimension \\(d\\), as: \\(\\mathbf{Y} = \\sigma (\\frac{\\text{Softmax}(\\mathbf{Q})\\cdot\\text{Softmax}(\\mathbf{K}^T)}{\\sqrt{n}}\\cdot_{\\text{MLP}}(\\mathbf{V}))\\), where \\(\\mathbf{Q},\\mathbf{K},\\mathbf{V}\\in \\mathbb{R}^{d\\times n}\\). This mechanism not only simplifies computational complexity but also aggregates local features in channel-wise. The encoder module synergies channel-wise and spatial-wise self-attention, aggregating strongly correlated feature channels while directing attention towards spatially relevant key tokens. This enhancement of global attention minimizes the shifting of attention tokens, leading to robust feature extraction for complex analysis task."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.793,
+ 0.338,
+ 0.81
+ ],
+ "angle": 0,
+ "content": "5. Benchmarking and Analysis"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.817,
+ 0.271,
+ 0.833
+ ],
+ "angle": 0,
+ "content": "5.1. Experiment Settings"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.84,
+ 0.471,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "Selected Baselines. To establish a comprehensive robustness benchmark for DLA, we not only include three distinct datasets but also reproduce a variety of state-of-the-art methods and backbones, including CNN-based backbones"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.092,
+ 0.892,
+ 0.243
+ ],
+ "angle": 0,
+ "content": "(ResNet [22], LayoutParser [54], InternImage [61]), Transformer-based structures (SwinDocSegmenter [4], DiT [31], and LayoutLMv3 [26]), and methods from CNN-based structures (Faster R-CNN [48], Mask R-CNN [23], Cascade R-CNN [6]), to Transformer-based frameworks (DocSegTr [5], DINO [69], Co-DINO [75], and our RoDLA). Note that all the methods are pre-trained on ImageNet dataset, while DiT and LayoutLMv3 are pre-trained on the IIT-CDIP Test Collection 1.0 document dataset [30], which gives a comparison of the pre-training data."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.245,
+ 0.892,
+ 0.366
+ ],
+ "angle": 0,
+ "content": "Implementation Details. For fair comparison, we reproduce all methods in the MMDetection [7] framework. Hyperparameters are from their original settings or papers. All models are trained and validated exclusively on clean data. Only during the testing phase, the impact of perturbations is evaluated on all methods. This approach ensures that our robustness benchmark is conducted under controlled and unbiased conditions. More details are in supplementary."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.499,
+ 0.381,
+ 0.776,
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+ ],
+ "angle": 0,
+ "content": "5.2. Robustness Evaluation Metrics"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.407,
+ 0.892,
+ 0.664
+ ],
+ "angle": 0,
+ "content": "From our analysis in Fig. 2 and Table 1, it is evident that various types of perturbations impact the same document images differently, leading to a range of effects on the models' performance. Previous robustness benchmarks [25, 33, 39, 66] measure the perturbations impact solely relying on a pre-selected baseline model. This approach tends to overlook the varying degrees of robustness different models exhibit towards the perturbations impact, overly relying on the benchmark performance of a particular model. To address this limitation, we propose Mean Robustness Degradation (mRD) for evaluating model robustness. mRD is a comprehensive measure of the average performance of models under different levels and types of perturbations. Additionally, inspired by ImageNet-C, we opt to incorporate Degradation \\(D\\) into mRD because we found that different perturbations pose varying levels of difficulty. Thus, the RD of a perturbation \\(p\\) is calculated as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.615,
+ 0.679,
+ 0.892,
+ 0.72
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {R D} _ {p} = \\frac {1}{N} \\sum_ {s = 1} ^ {N} \\frac {D _ {s , p} ^ {g}}{\\mathrm {m P E} _ {s , p}}, \\tag {4}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.735,
+ 0.892,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "where mPE is from Eq. (3). The overall robustness score \\(\\mathbf{mRD}\\) is the average score of \\(\\mathbf{RD}_p\\) across perturbations \\(p\\in [1,12]\\). Exceeding 100 indicates the model's performance degrades more than expected due to perturbations, while falling below 100 suggests performance improvements despite these impacts. The lower the better. This metric offers a comprehensive assessment of model robustness, enabling evaluation under diverse and realistic conditions. To ensure a more comprehensive evaluation, we test DLA models on clean datasets with mAP scores, as well as the average mAP performance on 12 perturbations (i.e., P-Avg)."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.481,
+ 0.945,
+ 0.518,
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+ ],
+ "angle": 0,
+ "content": "15561"
+ }
+ ],
+ [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.077,
+ 0.089,
+ 0.892,
+ 0.131
+ ],
+ "angle": 0,
+ "content": "Table 2. The robustness benchmark on PubLayNet-P dataset. V, L, and T stand for Visual, Layout, and Textual modality. 'Ext.' means using extra pre-training data. mAP scores are evaluated on the clean data, the 12 perturbation types (P1-P12), and the perturbation average (P-Avg). mRD is the proposed mean Robustness Degradation, the smaller the better."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.081,
+ 0.132,
+ 0.89,
+ 0.317
+ ],
+ "angle": 0,
+ "content": "| Document Perturbation (P) | Description | mPE↓ |
| None | The original clean data | 0 |
| ① Spatial | (P1) Rotation | Rotation simulation with θ=[5°, 10°, 15°] | 58.30 |
| (P2) Warping | Elastic transformation simulation with Rσ=[0.2, 0.06, 0.04] and Rα=[2, 0.6, 0.4] | 22.00 |
| (P3) Keystoning | Perspective transformation simulation with Rk=[0.02, 0.06, 0.1] | 34.49 |
| ② Content | (P4) Watermark | Text overlay with αw=[51, 153, 255] and Rz=[2, 4, 6] | 09.05 |
| (P5) Background | Random image inserting in background with Ni=[1, 3, 5] | 26.70 |
| ③ Inconsistency | (P6) Illumination | Region illumination change with Vl=[51, 102, 153] or Vs=[0.5, 0.25, 0.17] | 10.67 |
| (P7) Ink-bleeding | Erosion simulation with kernel size Ke=[3, 7, 11] | 08.91 |
| (P8) Ink-holdout | Dilation simulation with kernel size Kd=[3, 7, 11] | 15.64 |
| ④ Blur | (P9) Defocus | Gaussian blur simulation with kernel size Kg=[1, 3, 5] | 08.10 |
| (P10) Vibration | Motion blur simulation with kernel size Km=[3, 9, 15] | 16.29 |
| ⑤ Noise | (P11) Speckle | Blotches noise simulation with Db=[1 × 10-4,3 × 10-4,5 × 10-4] | 24.31 |
| (P12) Texture | Fibrous noise simulation with Nf=[300, 900, 1500] | 40.57 |
| Overall | The average of all perturbations | 22.90 |
+
+$R_{z}$ and alpha channel opacity $\alpha_{w}$ . (P5) Complex Background, which is often leveraged in printed media to enrich the document's visual content. Images from ILSVRC dataset [49], are randomly embedded within the document images for the simulation of background. The severity level is determined by the number of embedded images $N_{i}$ .
+
+3 Inconsistency Distortion, which addresses distortions, e.g., (P6) Non-uniform Illumination, simulated through a $50\%$ probability of glare or shadow, with levels determined by the brightness $V_{l}$ or shadow $V_{s}$ , (P7) Ink-bleeding, achieved by using erosion process within the document image, and (P8) Ink-holdout, which are accomplished by using morphological dilation operations with elliptical kernel $K_{e}$ and $K_{d}$ , respectively.
+
+$\Theta$ Blur. Blur effects arise from (P9) Defocus, which is approximated with Gaussian kernel function, i.e., the point spread function (PSF) with severity levels controlled by kernel size $K_{g}$ , and (P10) Vibration, which is achieved by using a convolution kernel rotated by a random angle to replicate directional motion in size $K_{m}$ . This perturbation can be applied via a Gaussian filtering operation.
+
+⑤ Noise. Two unique noise types in document images are included. (P11) Speckle caused by ink clumping, can be achieved by using Gaussian noise modulated by blob density $D_{b}$ . It reproduces both foreground and background noise, reflecting the complex stochastic nature of noise artifacts. (P12) Texture, resembling the fibers in paper, is added to simulate the natural fiber structures of archival documents. In each fiber segment, curvature and length are independently simulated with trigonometric functions and Cauchy distribution, forming the complete fiber through segment assembly. The number of fibers $N_{f}$ varies across noise levels to represent different paper qualities.
+
+The principles of design and division of these perturbations are three-fold: (1) All perturbations are realistic and
+
+inspired by the real-world disturbance or document layout analysis. (2) The leveraged perturbations are comprehensive and occur in document from top to bottom, from global to local, from dense to sparse, and from content-wise to pixel-wise. (3) All perturbation levels are reasonably determined by Image Quality Assessment (IQA) metrics, e.g., MS-SSIM [62] and CW-SSIM [51], Degradation and our proposed metrics which are detailed as the following.
+
+# 3.3. Perturbation Evaluation Metrics
+
+Quantifying the document perturbations is a crucial task, which lacks straightforward metrics. Previous methods [25, 28, 33, 66] are constrained by using a pre-selected baseline model performance as a reference to measure the perturbation effect, which will conflate the perturbation effect with the model robustness due to the inherent variability.
+
+To design an effective perturbation metric, our considerations are two-fold: (1) Evaluation should be relatively independent to any specific model performance. (2) Evaluation should be inclusive and sensitive to all perturbations. Inspired by the IQA methods [12], we analyse all 12 perturbations through 4 metrics, as presented in Fig. 3.
+
+MS-SSIM (Multi-Scale Structural Similarity Index) [62] evaluates image quality by considering difference in structural information across different resolutions, as:
+
+$$
+f ^ {\mathrm {M S - S S I M}} (x, y) = \left[ l _ {M} (x, y) \right] ^ {\alpha_ {M}} \cdot \prod_ {j = 1} ^ {M} \left[ c _ {j} (x, y) \right] ^ {\beta_ {j}} \left[ s _ {j} (x, y) \right] ^ {\gamma_ {j}}, \tag {1}
+$$
+
+where $l_{M}(x,y)$ represents the luminance difference at the $M$ -th scale, and $c_{j}(x,y)$ , $s_{j}(x,y)$ are the contrast and structure comparison at the $j$ -th scale. It measures the impact of most perturbations effectively but lacks sensitivity to watermark and warping (Fig. 3b), and is overly sensitive to rotation and keystoning, as shown in Fig. 3a.
+
+
+(a) Comparison of four evaluation metrics on 12 perturbations
+
+
+(b) Comparison of the sensitivity of metrics to different perturbation levels.
+Figure 3. Analysis of perturbation evaluation metrics. (a) Comparison of perturbation metrics, including MS-SSIM, CW-SSIM, Degradation w.r.t a baseline, and the proposed mean Perturbation Effect (mPE). mPE is more balanced and inclusive to different perturbations. (b) Six documents perturbed by warping and defocus and their scores indicate that mPE is more sensitive to measure different levels.
+
+CW-SSIM (Complex Wavelet Structural Similarity Index) [51] assesses the similarity of local patterns of pixel intensities transformed into the wavelet domain, which is robust to spatial distortions, e.g., keystoneing and rotation, formulated by the following equation:
+
+$$
+f ^ {\mathrm {C W - S S I M}} (x, y) = \frac {2 \left| \sum_ {l = 1} ^ {L} w _ {l} x _ {l} y _ {l} ^ {*} \right| + K}{\sum_ {l = 1} ^ {L} \left| x _ {l} \right| ^ {2} + \sum_ {l = 1} ^ {L} \left| y _ {l} \right| ^ {2} + K}, \tag {2}
+$$
+
+where $x_{l}$ and $y_{l}$ denote the wavelet coefficients, $w_{l}$ is the $l$ -th coefficient weight, $L$ indicates the local region number, $K$ is tiny value, and * denotes the complex conjugation. As shown in Fig. 3a, it reflects the perturbations impact but fails for quantitative evaluation at different levels for some perturbations, e.g., defocus, demonstrated in Fig. 3b.
+
+Degradation across severity levels, as indicated by ImageNet-C [25], is a common practice for various robustness benchmark [25, 28, 33, 66]. Degradation $D$ , where $D = 1 - mAP$ similar with ImageNet-C, illustrates the performance impact of perturbation on a baseline model. From Fig. 3a, we notice that the Degradation metric is highly dependent on the choice of the baseline and tends to be overly sensitive to perturbations, e.g., background and texture.
+
+Mean Perturbation Effect (mPE) is a new metric proposed to assess the compound effects of document perturbations. For specific perturbation $p$ at each severity level $s$ , degradation of model $g$ is written as $D_{s,p}^{g}$ , and IQA metrics is written as $f_{s,p}^{i}$ , i.e., from Eq. (1) and Eq. (2). The mPE to measure the effect of perturbation $p$ is calculated as:
+
+$$
+\mathbf {m P E} _ {p} = \frac {1}{N M K} \sum_ {s = 1} ^ {N} \left(\sum_ {i = 1} ^ {M} f _ {s, p} ^ {i} + \sum_ {g = 1} ^ {K} D _ {s, p} ^ {g}\right). \tag {3}
+$$
+
+mPE aims to isolate the model's robustness from the image alterations impact. We employ MS-SSIM and CW-SSIM as IQA metrics, and we select Faster R-CNN [48] as a baseline in Degradation metric. As evident from Fig. 3b, our mPE metric effectively quantifies the impact of each perturbation and severity level, while other metrics are less sensitive.
+
+# 3.4. Perturbation Robustness Benchmarks
+
+To systematically benchmark the robustness of DLA models, we apply the mentioned 12 perturbations to 3 datasets.
+
+PubLayNet-P [72] is a commonly used large-scale DLA dataset. It contains over 360,000 document images sourced from PubMed Central, and focuses on 5 principal types of layout elements, i.e., text, title, list, table, and figure.
+
+DocLayNet-P [46] includes academic papers, brochures, business letters, technical papers, and more with 80,863 document pages. It contains annotations for 11 layout components, aiming at enabling generalizable DLA models.
+
+$\mathbf{M}^6\mathbf{Doc - P}$ [8] provides more nuanced annotations (74 classes), capturing finer aspects of document layouts with 9,080 document images. This dataset is tailored to advance research in sophisticated document understanding tasks.
+
+While PubLayNet provides vast array of annotated scientific documents, DocLayNet broadens the variety of document types, and M6Doc delves into more detailed and complex annotations, catering to advanced document analysis.
+
+# 4. Robust Document Layout Analyzer
+
+# 4.1. Framework Overview
+
+To boost the robustness performance on perturbed documents, we propose a Robust Document Layout Analyzer (RoDLA) model. As shown in Fig. 4, our proposed model is inspired by the architecture of DINO [69] but introduces channel attention blocks and average pooling blocks in the Encoder, allowing for more robust feature extraction. Additionally, we incorporate InternImage[61] as a backbone, which is pre-trained on ImageNet22K [10] dataset. This pretraining setting enhances multi-scale feature extraction for more stable performance. The overall architecture and robustness enhancement design is depicted in Fig. 4.
+
+# 4.2. Robustness Enhancement Design
+
+As identified in [19], self-attention tends to overemphasize focus on irrelevant tokens, that leads to attention shifting in
+
+
+Figure 4. The architecture of RoDLA model. RoDLA is comprised of Encoder, Query Selection, Decoder, and Matching components. It optimizes the attention mechanism in Encoder, heightening focus on crucial tokens and reinforcing key token connections in multi-scale features to extract stable features.
+
+visual document analysis, where globally accumulated perturbations may interfere with the model performance. To address this, we integrate spatial-wise average pooling layers with varying dilation into the Robust Encoder, accompanied by a Dilation Predictor employing two $3 \times 3$ convolutional layers to predict the utilization of dilated average pooling layers. This approach leverages the attention of neighboring tokens to mitigate the overemphasis on tokens, and dynamically constrains long-distance perception based on context. By diminishing attention on irrelevant tokens, this approach effectively captures robust feature.
+
+As outlined in [73], self-attention inherently facilitates visual grouping, filtering out irrelevant information. Thus, we have incorporated a Channel-wise Attention (CA) module within the Encoder. This module implements a self-attention operation in channel dimension $d$ , as: $\mathbf{Y} = \sigma (\frac{\text{Softmax}(\mathbf{Q})\cdot\text{Softmax}(\mathbf{K}^T)}{\sqrt{n}}\cdot_{\text{MLP}}(\mathbf{V}))$ , where $\mathbf{Q},\mathbf{K},\mathbf{V}\in \mathbb{R}^{d\times n}$ . This mechanism not only simplifies computational complexity but also aggregates local features in channel-wise. The encoder module synergies channel-wise and spatial-wise self-attention, aggregating strongly correlated feature channels while directing attention towards spatially relevant key tokens. This enhancement of global attention minimizes the shifting of attention tokens, leading to robust feature extraction for complex analysis task.
+
+# 5. Benchmarking and Analysis
+
+# 5.1. Experiment Settings
+
+Selected Baselines. To establish a comprehensive robustness benchmark for DLA, we not only include three distinct datasets but also reproduce a variety of state-of-the-art methods and backbones, including CNN-based backbones
+
+(ResNet [22], LayoutParser [54], InternImage [61]), Transformer-based structures (SwinDocSegmenter [4], DiT [31], and LayoutLMv3 [26]), and methods from CNN-based structures (Faster R-CNN [48], Mask R-CNN [23], Cascade R-CNN [6]), to Transformer-based frameworks (DocSegTr [5], DINO [69], Co-DINO [75], and our RoDLA). Note that all the methods are pre-trained on ImageNet dataset, while DiT and LayoutLMv3 are pre-trained on the IIT-CDIP Test Collection 1.0 document dataset [30], which gives a comparison of the pre-training data.
+
+Implementation Details. For fair comparison, we reproduce all methods in the MMDetection [7] framework. Hyperparameters are from their original settings or papers. All models are trained and validated exclusively on clean data. Only during the testing phase, the impact of perturbations is evaluated on all methods. This approach ensures that our robustness benchmark is conducted under controlled and unbiased conditions. More details are in supplementary.
+
+# 5.2. Robustness Evaluation Metrics
+
+From our analysis in Fig. 2 and Table 1, it is evident that various types of perturbations impact the same document images differently, leading to a range of effects on the models' performance. Previous robustness benchmarks [25, 33, 39, 66] measure the perturbations impact solely relying on a pre-selected baseline model. This approach tends to overlook the varying degrees of robustness different models exhibit towards the perturbations impact, overly relying on the benchmark performance of a particular model. To address this limitation, we propose Mean Robustness Degradation (mRD) for evaluating model robustness. mRD is a comprehensive measure of the average performance of models under different levels and types of perturbations. Additionally, inspired by ImageNet-C, we opt to incorporate Degradation $D$ into mRD because we found that different perturbations pose varying levels of difficulty. Thus, the RD of a perturbation $p$ is calculated as:
+
+$$
+\mathbf {R D} _ {p} = \frac {1}{N} \sum_ {s = 1} ^ {N} \frac {D _ {s , p} ^ {g}}{\mathrm {m P E} _ {s , p}}, \tag {4}
+$$
+
+where mPE is from Eq. (3). The overall robustness score $\mathbf{mRD}$ is the average score of $\mathbf{RD}_p$ across perturbations $p\in [1,12]$ . Exceeding 100 indicates the model's performance degrades more than expected due to perturbations, while falling below 100 suggests performance improvements despite these impacts. The lower the better. This metric offers a comprehensive assessment of model robustness, enabling evaluation under diverse and realistic conditions. To ensure a more comprehensive evaluation, we test DLA models on clean datasets with mAP scores, as well as the average mAP performance on 12 perturbations (i.e., P-Avg).
+
+Table 2. The robustness benchmark on PubLayNet-P dataset. V, L, and T stand for Visual, Layout, and Textual modality. 'Ext.' means using extra pre-training data. mAP scores are evaluated on the clean data, the 12 perturbation types (P1-P12), and the perturbation average (P-Avg). mRD is the proposed mean Robustness Degradation, the smaller the better.
+
+| Document Perturbation (P) | Description | mPE↓ |
| None | The original clean data | 0 |
| ① Spatial | (P1) Rotation | Rotation simulation with θ=[5°, 10°, 15°] | 58.30 |
| (P2) Warping | Elastic transformation simulation with Rσ=[0.2, 0.06, 0.04] and Rα=[2, 0.6, 0.4] | 22.00 |
| (P3) Keystoning | Perspective transformation simulation with Rk=[0.02, 0.06, 0.1] | 34.49 |
| ② Content | (P4) Watermark | Text overlay with αw=[51, 153, 255] and Rz=[2, 4, 6] | 09.05 |
| (P5) Background | Random image inserting in background with Ni=[1, 3, 5] | 26.70 |
| ③ Inconsistency | (P6) Illumination | Region illumination change with Vl=[51, 102, 153] or Vs=[0.5, 0.25, 0.17] | 10.67 |
| (P7) Ink-bleeding | Erosion simulation with kernel size Ke=[3, 7, 11] | 08.91 |
| (P8) Ink-holdout | Dilation simulation with kernel size Kd=[3, 7, 11] | 15.64 |
| ④ Blur | (P9) Defocus | Gaussian blur simulation with kernel size Kg=[1, 3, 5] | 08.10 |
| (P10) Vibration | Motion blur simulation with kernel size Km=[3, 9, 15] | 16.29 |
| ⑤ Noise | (P11) Speckle | Blotches noise simulation with Db=[1 × 10-4,3 × 10-4,5 × 10-4] | 24.31 |
| (P12) Texture | Fibrous noise simulation with Nf=[300, 900, 1500] | 40.57 |
| Overall | The average of all perturbations | 22.90 |
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+ "content": "\\mathbf {R D} _ {p} = \\frac {1}{N} \\sum_ {s = 1} ^ {N} \\frac {D _ {s , p} ^ {g}}{\\mathrm {m P E} _ {s , p}}, \\tag {4}",
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| I (Baseline): DKM [17] | 17.0 | 7.3 | 5.8 |
| II: I, Fcoarse,θ = RN50, Ffine,θ = RN50 | 16.0 | 6.1 | 4.5 |
| III: II, Ffine,θ = VGG19 | 14.5 | 5.4 | 4.5 |
| IV: III, Dθ = Transformer | 14.4 | 5.4 | 4.1 |
| V: IV, Fcoarse,θ = DINOv2 | 14.3 | 4.6 | 3.2 |
| VI: V, Lcoarse = reg.-by-class. | 13.6 | 4.1 | 2.8 |
| VII (Ours): VI, Lrefine = robust | 13.1 | 4.0 | 2.7 |
| VIII: VII, Dθ = ConvNet | 14.0 | 4.9 | 3.5 |
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+ "text": "\n$$\n\\begin{array}{l} D _ {\\mathrm {K L}} \\left(q \\left(x _ {i} ^ {\\mathcal {B}}, x _ {i} ^ {\\mathcal {A}}; s = 2 ^ {i} c\\right) \\| p _ {i, \\theta} \\left(x _ {i} ^ {\\mathcal {B}}, x _ {i} ^ {\\mathcal {A}} \\mid \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right)\\right) = (18) \\\\ \\mathbb {E} _ {x _ {i} ^ {\\mathcal {A}}, x _ {i} ^ {\\mathcal {B}} \\sim q} \\left[\\left(\\left|\\left| \\mu_ {\\theta} \\left(x _ {i} ^ {\\mathcal {A}}, \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right) - x _ {i} ^ {\\mathcal {B}} \\right|\\right| ^ {2} + s ^ {2}\\right) ^ {1 / 4} \\right] + \\\\ \\mathbb {E} _ {x _ {i} ^ {\\mathcal {A}}, x _ {i} ^ {\\mathcal {B}} \\sim q} \\left[ - \\log p _ {i, \\theta} \\left(x _ {i} ^ {\\mathcal {A}} | \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right)\\right]. (19) \\\\ \\end{array}\n$$\n",
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+ "text": "and use a binary cross-entropy loss on $p_{i,\\theta}(x_i^{\\mathcal{A}}|\\hat{W}_{i + 1}^{\\mathcal{A}\\rightarrow \\mathcal{B}})$ , we additionally utilize clipping as in DKM [17], further details are found in the supplementary.",
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+ "table_body": "| Setup ↓ 100-PCK@ → | 1px | 3px | 5px |
| I (Baseline): DKM [17] | 17.0 | 7.3 | 5.8 |
| II: I, Fcoarse,θ = RN50, Ffine,θ = RN50 | 16.0 | 6.1 | 4.5 |
| III: II, Ffine,θ = VGG19 | 14.5 | 5.4 | 4.5 |
| IV: III, Dθ = Transformer | 14.4 | 5.4 | 4.1 |
| V: IV, Fcoarse,θ = DINOv2 | 14.3 | 4.6 | 3.2 |
| VI: V, Lcoarse = reg.-by-class. | 13.6 | 4.1 | 2.8 |
| VII (Ours): VI, Lrefine = robust | 13.1 | 4.0 | 2.7 |
| VIII: VII, Dθ = ConvNet | 14.0 | 4.9 | 3.5 |
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+ "content": "\\[\n\\log p _ {\\theta} \\left(x _ {i} ^ {\\mathcal {B}} \\mid x _ {i} ^ {\\mathcal {A}}, \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right) = \\tag {16}\n\\]"
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+ "content": "\\[\n- \\left(\\left\\| \\mu_ {\\theta} \\left(x _ {i} ^ {\\mathcal {A}}, \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right) - x _ {i} ^ {\\mathcal {B}} \\right\\| ^ {2} + s ^ {2}\\right) ^ {1 / 4}, \\tag {17}\n\\]"
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+ "content": "where \\(\\mu_{\\theta}(x_i^{\\mathcal{A}},\\hat{W}_{i + 1}^{\\mathcal{A}\\rightarrow \\mathcal{B}})\\) is the estimated mean of the distribution, and \\(s = 2^{i}c\\) . In practice, we choose \\(c = 0.03\\) The Kullback-Leibler divergence for each fine scale \\(i\\in\\) \\(\\{0,1,2,3\\}\\) (where terms that are constant with respect to \\(\\theta\\) are ignored) reads"
+ },
+ {
+ "type": "equation",
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+ "content": "\\[\n\\begin{array}{l} D _ {\\mathrm {K L}} \\left(q \\left(x _ {i} ^ {\\mathcal {B}}, x _ {i} ^ {\\mathcal {A}}; s = 2 ^ {i} c\\right) \\| p _ {i, \\theta} \\left(x _ {i} ^ {\\mathcal {B}}, x _ {i} ^ {\\mathcal {A}} \\mid \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right)\\right) = (18) \\\\ \\mathbb {E} _ {x _ {i} ^ {\\mathcal {A}}, x _ {i} ^ {\\mathcal {B}} \\sim q} \\left[\\left(\\left|\\left| \\mu_ {\\theta} \\left(x _ {i} ^ {\\mathcal {A}}, \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right) - x _ {i} ^ {\\mathcal {B}} \\right|\\right| ^ {2} + s ^ {2}\\right) ^ {1 / 4} \\right] + \\\\ \\mathbb {E} _ {x _ {i} ^ {\\mathcal {A}}, x _ {i} ^ {\\mathcal {B}} \\sim q} \\left[ - \\log p _ {i, \\theta} \\left(x _ {i} ^ {\\mathcal {A}} | \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right)\\right]. (19) \\\\ \\end{array}\n\\]"
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+ "content": "In practice, we approximate \\(q\\) with a discrete set of known correspondences \\(\\mathcal{C}\\)"
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+ "content": "\\[\n\\begin{array}{l} \\mathcal {L} _ {\\text {f i n e}} = \\sum_ {i} \\sum_ {x ^ {\\mathcal {A}}, x ^ {\\mathcal {B}} \\in \\mathcal {C}} \\left(\\left|\\left| \\mu_ {\\theta} \\left(x _ {i} ^ {\\mathcal {A}}, \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right) - x _ {i} ^ {\\mathcal {B}} \\right|\\right| ^ {2} + s ^ {2}\\right) ^ {1 / 4} \\\\ - \\lambda \\log p _ {i, \\theta} \\left(x _ {i} ^ {\\mathcal {A}} \\mid \\hat {W} _ {i + 1} ^ {\\mathcal {A} \\rightarrow \\mathcal {B}}\\right) \\tag {20} \\\\ \\end{array}\n\\]"
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+ "content": "and use a binary cross-entropy loss on \\( p_{i,\\theta}(x_i^{\\mathcal{A}}|\\hat{W}_{i + 1}^{\\mathcal{A}\\rightarrow \\mathcal{B}}) \\), we additionally utilize clipping as in DKM [17], further details are found in the supplementary."
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+ "content": "Our combined loss yields:"
+ },
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+ "content": "\\[\n\\mathcal {L} = \\mathcal {L} _ {\\text {c o a r s e}} + \\mathcal {L} _ {\\text {f i n e}}. \\tag {21}\n\\]"
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+ "content": "Table 3. SotA comparison on IMC2022 [26]. Measured in mAA (higher is better)."
+ },
+ {
+ "type": "table",
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+ "angle": 0,
+ "content": "| Setup ↓ 100-PCK@ → | 1px | 3px | 5px |
| I (Baseline): DKM [17] | 17.0 | 7.3 | 5.8 |
| II: I, Fcoarse,θ = RN50, Ffine,θ = RN50 | 16.0 | 6.1 | 4.5 |
| III: II, Ffine,θ = VGG19 | 14.5 | 5.4 | 4.5 |
| IV: III, Dθ = Transformer | 14.4 | 5.4 | 4.1 |
| V: IV, Fcoarse,θ = DINOv2 | 14.3 | 4.6 | 3.2 |
| VI: V, Lcoarse = reg.-by-class. | 13.6 | 4.1 | 2.8 |
| VII (Ours): VI, Lrefine = robust | 13.1 | 4.0 | 2.7 |
| VIII: VII, Dθ = ConvNet | 14.0 | 4.9 | 3.5 |
+
+We model the output of the refinement at scale $i$ as a generalized Charbonnier [3] (with $\alpha = 0.5$ ) distribution, for which the refiners estimate the mean $\mu$ . The generalized Charbonnier distribution behaves locally like a Normal distribution, but has a flatter tail. When used as a loss, the gradients behave locally like L2, but decay towards 0, see Figure 4. Its logarithm, (ignoring terms that do not contribute to the gradient, and up-to-scale) reads
+
+$$
+\log p _ {\theta} \left(x _ {i} ^ {\mathcal {B}} \mid x _ {i} ^ {\mathcal {A}}, \hat {W} _ {i + 1} ^ {\mathcal {A} \rightarrow \mathcal {B}}\right) = \tag {16}
+$$
+
+$$
+- \left(\left\| \mu_ {\theta} \left(x _ {i} ^ {\mathcal {A}}, \hat {W} _ {i + 1} ^ {\mathcal {A} \rightarrow \mathcal {B}}\right) - x _ {i} ^ {\mathcal {B}} \right\| ^ {2} + s ^ {2}\right) ^ {1 / 4}, \tag {17}
+$$
+
+where $\mu_{\theta}(x_i^{\mathcal{A}},\hat{W}_{i + 1}^{\mathcal{A}\rightarrow \mathcal{B}})$ is the estimated mean of the distribution, and $s = 2^{i}c$ . In practice, we choose $c = 0.03$ The Kullback-Leibler divergence for each fine scale $i\in$ $\{0,1,2,3\}$ (where terms that are constant with respect to $\theta$ are ignored) reads
+
+$$
+\begin{array}{l} D _ {\mathrm {K L}} \left(q \left(x _ {i} ^ {\mathcal {B}}, x _ {i} ^ {\mathcal {A}}; s = 2 ^ {i} c\right) \| p _ {i, \theta} \left(x _ {i} ^ {\mathcal {B}}, x _ {i} ^ {\mathcal {A}} \mid \hat {W} _ {i + 1} ^ {\mathcal {A} \rightarrow \mathcal {B}}\right)\right) = (18) \\ \mathbb {E} _ {x _ {i} ^ {\mathcal {A}}, x _ {i} ^ {\mathcal {B}} \sim q} \left[\left(\left|\left| \mu_ {\theta} \left(x _ {i} ^ {\mathcal {A}}, \hat {W} _ {i + 1} ^ {\mathcal {A} \rightarrow \mathcal {B}}\right) - x _ {i} ^ {\mathcal {B}} \right|\right| ^ {2} + s ^ {2}\right) ^ {1 / 4} \right] + \\ \mathbb {E} _ {x _ {i} ^ {\mathcal {A}}, x _ {i} ^ {\mathcal {B}} \sim q} \left[ - \log p _ {i, \theta} \left(x _ {i} ^ {\mathcal {A}} | \hat {W} _ {i + 1} ^ {\mathcal {A} \rightarrow \mathcal {B}}\right)\right]. (19) \\ \end{array}
+$$
+
+In practice, we approximate $q$ with a discrete set of known correspondences $\mathcal{C}$
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {f i n e}} = \sum_ {i} \sum_ {x ^ {\mathcal {A}}, x ^ {\mathcal {B}} \in \mathcal {C}} \left(\left|\left| \mu_ {\theta} \left(x _ {i} ^ {\mathcal {A}}, \hat {W} _ {i + 1} ^ {\mathcal {A} \rightarrow \mathcal {B}}\right) - x _ {i} ^ {\mathcal {B}} \right|\right| ^ {2} + s ^ {2}\right) ^ {1 / 4} \\ - \lambda \log p _ {i, \theta} \left(x _ {i} ^ {\mathcal {A}} \mid \hat {W} _ {i + 1} ^ {\mathcal {A} \rightarrow \mathcal {B}}\right) \tag {20} \\ \end{array}
+$$
+
+and use a binary cross-entropy loss on $p_{i,\theta}(x_i^{\mathcal{A}}|\hat{W}_{i + 1}^{\mathcal{A}\rightarrow \mathcal{B}})$ , we additionally utilize clipping as in DKM [17], further details are found in the supplementary.
+
+Our combined loss yields:
+
+$$
+\mathcal {L} = \mathcal {L} _ {\text {c o a r s e}} + \mathcal {L} _ {\text {f i n e}}. \tag {21}
+$$
+
+Table 3. SotA comparison on IMC2022 [26]. Measured in mAA (higher is better).
+
+| Setup ↓ 100-PCK@ → | 1px | 3px | 5px |
| I (Baseline): DKM [17] | 17.0 | 7.3 | 5.8 |
| II: I, Fcoarse,θ = RN50, Ffine,θ = RN50 | 16.0 | 6.1 | 4.5 |
| III: II, Ffine,θ = VGG19 | 14.5 | 5.4 | 4.5 |
| IV: III, Dθ = Transformer | 14.4 | 5.4 | 4.1 |
| V: IV, Fcoarse,θ = DINOv2 | 14.3 | 4.6 | 3.2 |
| VI: V, Lcoarse = reg.-by-class. | 13.6 | 4.1 | 2.8 |
| VII (Ours): VI, Lrefine = robust | 13.1 | 4.0 | 2.7 |
| VIII: VII, Dθ = ConvNet | 14.0 | 4.9 | 3.5 |
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| \\( {10}^{ \\circ } \\) | \\( {20}^{ \\circ } \\) | 0.1 | 0.2 |
| Ours | \\( {16.51}^{ \\circ } \\) | 16.70 | 32.24 | 0.299 | 12.96 | 23.10 |
| Hartley [15] | \\( {21.52}^{ \\circ } \\) | 12.38 | 24.26 | 1.090 | 0.45 | 0.99 |
| Fetzer [10] | \\( {70.40}^{ \\circ } \\) | 2.31 | 5.20 | 0.999 | 6.86 | 11.64 |
| Sturm [41] | \\( {17.98}^{ \\circ } \\) | 20.64 | 33.30 | 0.582 | 10.39 | 17.55 |
| Minimal [40] | \\( {36.18}^{ \\circ } \\) | 16.42 | 25.74 | 0.291 | 14.43 | 24.29 |
| Prior | \\( {24.60}^{ \\circ } \\) | 11.24 | 20.92 | 1.119 | 0.00 | 0.00 |
| GT intrinsics | \\( {6.40}^{ \\circ } \\) | 41.73 | 54.16 | - | - | - |
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+ "text": "Table 3. Mean average accuracy scores for poses $(\\mathrm{mAA}_p)$ and estimated focal lengths $(\\mathrm{mAA}_f)$ on 12 scenes from the ETH3D High Resolution Multiview dataset [37]. The focal lengths of both cameras are assumed to be equal. The last row includes reference results for pose accuracy when ground truth intrinsics are used.",
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+ "text": "Results: The median errors and mean average accuracy scores for both pose and focal lengths are reported in Table 1. On both datasets, our method significantly outperforms others in terms of the accuracy of the estimated focal lengths. The superiority of our method in terms of focal length accuracy can also be seen in Fig. 4a and 4b. In terms of pose accuracy, our method also provides more accurate results than all the compared methods.",
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+ "text": "Computational Speed: Table 2 shows the mean computational times for the various methods. Our method is significantly faster than other iterative methods. Note that the implementation of our method is not optimized and there is still some space for speed-up.",
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+ "text": "Equal Focal Lengths: We also evaluate our method for the case when both focal lengths can be assumed to be equal. In this case, we use the modified version of our iterative algorithm from Section 3.1. To perform the evalua",
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+ "text": "tion, we use the ETH3D Multiview dataset [37]. We use all image pairs satisfying the co-visibility criterion for each of the 12 scenes from the test set, yielding 3830 total pairs. The dataset combines laser scans and photometric errors to obtain the ground truth. For comparison, we modified the method by Fetzer et al. to optimize for a single focal length. We use Hartley and Silpa-Anan's method as usual, but on output we average the two focal lengths produced. Instead of the Bougnoux formula, we use the formula proposed by Sturm [41] which is specific to the case of equal focal lengths. We also evaluate the minimal 6-point algorithm [40] implemented within LO-RANSAC [8, 23]. Further details and additional experiments are available in SM.",
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+ "text": "The results are shown in Table 3 and Fig. 4c. Our method provides superior results to other methods based on the decomposition of the fundamental matrix, and performs comparably to the minimal solver [40] in terms of estimated focal lengths and shows minor improvements in terms of pose accuracy. Our method thus provides a good alternative to the 6-point minimal solver in the case when the focal lengths of the two cameras are known to be equal.",
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+ "text": "We address the important problem of robust self-calibration of the focal lengths of two cameras from a given fundamental matrix. Our new efficient iterative method shows substantial improvements over existing techniques. Synthetic experiments show that our method performs reliably even when the cameras are in or close to degenerate configurations. In real-world evaluations on two large-scale datasets, our approach demonstrates superior accuracy in terms of estimated poses and focal lengths, even when we use inaccurate initial priors, while also being faster than competing iterative approaches.",
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+ "text": "We have additionally proposed to perform a computationally simple check within the standard 7-point algorithm to remove models that lead to imaginary focal lengths. Real-world experiments show that this approach not only decreases computational time of the whole RANSAC pipeline, but, in general, also leads to improved pose and focal length accuracy across multiple RANSAC variants and implementations.",
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+ "text": "Acknowledgments: V. K. was supported by the Slovak Grant Agency for Science (VEGA), project no. 1/0373/23. and the TERAIS project, a Horizon-Widera-2021 program of the European Union under the Grant agreement number 101079338. The results were obtained using the computational resources procured in the project National competence centre for high performance computing (project code: 311070AKF2) funded by European Regional Development Fund, EU Structural Funds Informatization of society, Operational Program Integrated Infrastructure. Z. K. was supported by the Czech Science Foundation (GACR) JUNIOR STAR Grant No. 22-23183M.",
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+ "content": "| Method | Median \\( {p}_{err} \\) | \\( {\\mathrm{{mAA}}}_{p} \\) | Median \\( {f}_{err} \\) | \\( {\\mathrm{{mAA}}}_{f} \\) |
| \\( {10}^{ \\circ } \\) | \\( {20}^{ \\circ } \\) | 0.1 | 0.2 |
| Ours | \\( {16.51}^{ \\circ } \\) | 16.70 | 32.24 | 0.299 | 12.96 | 23.10 |
| Hartley [15] | \\( {21.52}^{ \\circ } \\) | 12.38 | 24.26 | 1.090 | 0.45 | 0.99 |
| Fetzer [10] | \\( {70.40}^{ \\circ } \\) | 2.31 | 5.20 | 0.999 | 6.86 | 11.64 |
| Sturm [41] | \\( {17.98}^{ \\circ } \\) | 20.64 | 33.30 | 0.582 | 10.39 | 17.55 |
| Minimal [40] | \\( {36.18}^{ \\circ } \\) | 16.42 | 25.74 | 0.291 | 14.43 | 24.29 |
| Prior | \\( {24.60}^{ \\circ } \\) | 11.24 | 20.92 | 1.119 | 0.00 | 0.00 |
| GT intrinsics | \\( {6.40}^{ \\circ } \\) | 41.73 | 54.16 | - | - | - |
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+ "content": "Computational Speed: Table 2 shows the mean computational times for the various methods. Our method is significantly faster than other iterative methods. Note that the implementation of our method is not optimized and there is still some space for speed-up."
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+ "content": "Equal Focal Lengths: We also evaluate our method for the case when both focal lengths can be assumed to be equal. In this case, we use the modified version of our iterative algorithm from Section 3.1. To perform the evalua"
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+ "content": "tion, we use the ETH3D Multiview dataset [37]. We use all image pairs satisfying the co-visibility criterion for each of the 12 scenes from the test set, yielding 3830 total pairs. The dataset combines laser scans and photometric errors to obtain the ground truth. For comparison, we modified the method by Fetzer et al. to optimize for a single focal length. We use Hartley and Silpa-Anan's method as usual, but on output we average the two focal lengths produced. Instead of the Bougnoux formula, we use the formula proposed by Sturm [41] which is specific to the case of equal focal lengths. We also evaluate the minimal 6-point algorithm [40] implemented within LO-RANSAC [8, 23]. Further details and additional experiments are available in SM."
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+ "content": "The results are shown in Table 3 and Fig. 4c. Our method provides superior results to other methods based on the decomposition of the fundamental matrix, and performs comparably to the minimal solver [40] in terms of estimated focal lengths and shows minor improvements in terms of pose accuracy. Our method thus provides a good alternative to the 6-point minimal solver in the case when the focal lengths of the two cameras are known to be equal."
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+ "content": "6. Conclusion"
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+ "angle": 0,
+ "content": "We address the important problem of robust self-calibration of the focal lengths of two cameras from a given fundamental matrix. Our new efficient iterative method shows substantial improvements over existing techniques. Synthetic experiments show that our method performs reliably even when the cameras are in or close to degenerate configurations. In real-world evaluations on two large-scale datasets, our approach demonstrates superior accuracy in terms of estimated poses and focal lengths, even when we use inaccurate initial priors, while also being faster than competing iterative approaches."
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+ "type": "text",
+ "bbox": [
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+ "content": "We have additionally proposed to perform a computationally simple check within the standard 7-point algorithm to remove models that lead to imaginary focal lengths. Real-world experiments show that this approach not only decreases computational time of the whole RANSAC pipeline, but, in general, also leads to improved pose and focal length accuracy across multiple RANSAC variants and implementations."
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+ "content": "Acknowledgments: V. K. was supported by the Slovak Grant Agency for Science (VEGA), project no. 1/0373/23. and the TERAIS project, a Horizon-Widera-2021 program of the European Union under the Grant agreement number 101079338. The results were obtained using the computational resources procured in the project National competence centre for high performance computing (project code: 311070AKF2) funded by European Regional Development Fund, EU Structural Funds Informatization of society, Operational Program Integrated Infrastructure. Z. K. was supported by the Czech Science Foundation (GACR) JUNIOR STAR Grant No. 22-23183M."
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+# Robust Self-calibration of Focal Lengths from the Fundamental Matrix
+
+Viktor Kocur $^{1,2}$ Daniel Kyselica $^{1}$ Zuzana Kukelova $^{2}$
+
+| Method | Median \\( {p}_{err} \\) | \\( {\\mathrm{{mAA}}}_{p} \\) | Median \\( {f}_{err} \\) | \\( {\\mathrm{{mAA}}}_{f} \\) |
| \\( {10}^{ \\circ } \\) | \\( {20}^{ \\circ } \\) | 0.1 | 0.2 |
| Ours | \\( {16.51}^{ \\circ } \\) | 16.70 | 32.24 | 0.299 | 12.96 | 23.10 |
| Hartley [15] | \\( {21.52}^{ \\circ } \\) | 12.38 | 24.26 | 1.090 | 0.45 | 0.99 |
| Fetzer [10] | \\( {70.40}^{ \\circ } \\) | 2.31 | 5.20 | 0.999 | 6.86 | 11.64 |
| Sturm [41] | \\( {17.98}^{ \\circ } \\) | 20.64 | 33.30 | 0.582 | 10.39 | 17.55 |
| Minimal [40] | \\( {36.18}^{ \\circ } \\) | 16.42 | 25.74 | 0.291 | 14.43 | 24.29 |
| Prior | \\( {24.60}^{ \\circ } \\) | 11.24 | 20.92 | 1.119 | 0.00 | 0.00 |
| GT intrinsics | \\( {6.40}^{ \\circ } \\) | 41.73 | 54.16 | - | - | - |
",
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+ "content": "Table 3. Mean average accuracy scores for poses "
+ },
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+ "type": "inline_equation",
+ "content": "(\\mathrm{mAA}_p)"
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+ "type": "text",
+ "content": " and estimated focal lengths "
+ },
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+ "content": "(\\mathrm{mAA}_f)"
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+ "type": "text",
+ "content": " on 12 scenes from the ETH3D High Resolution Multiview dataset [37]. The focal lengths of both cameras are assumed to be equal. The last row includes reference results for pose accuracy when ground truth intrinsics are used."
+ }
+ ]
+ }
+ ],
+ "index": 3,
+ "angle": 0,
+ "type": "text"
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+ "type": "text",
+ "content": "tialize the priors for the focal lengths at 1.2 times the maximum dimension of the image This initial setting implies an approximate 50-degree field-of-view for the cameras."
+ }
+ ]
+ }
+ ],
+ "index": 4
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+ "content": "Results: The median errors and mean average accuracy scores for both pose and focal lengths are reported in Table 1. On both datasets, our method significantly outperforms others in terms of the accuracy of the estimated focal lengths. The superiority of our method in terms of focal length accuracy can also be seen in Fig. 4a and 4b. In terms of pose accuracy, our method also provides more accurate results than all the compared methods."
+ }
+ ]
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+ "content": "The results also show that performing RFC for real focal lengths within RANSAC as proposed in subsection 3.2 generally leads to improvements in terms of both the pose and focal length accuracy across the compared methods."
+ }
+ ]
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+ "type": "text",
+ "content": "Computational Speed: Table 2 shows the mean computational times for the various methods. Our method is significantly faster than other iterative methods. Note that the implementation of our method is not optimized and there is still some space for speed-up."
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+ "content": "Equal Focal Lengths: We also evaluate our method for the case when both focal lengths can be assumed to be equal. In this case, we use the modified version of our iterative algorithm from Section 3.1. To perform the evalua"
+ }
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+ "type": "text",
+ "content": "tion, we use the ETH3D Multiview dataset [37]. We use all image pairs satisfying the co-visibility criterion for each of the 12 scenes from the test set, yielding 3830 total pairs. The dataset combines laser scans and photometric errors to obtain the ground truth. For comparison, we modified the method by Fetzer et al. to optimize for a single focal length. We use Hartley and Silpa-Anan's method as usual, but on output we average the two focal lengths produced. Instead of the Bougnoux formula, we use the formula proposed by Sturm [41] which is specific to the case of equal focal lengths. We also evaluate the minimal 6-point algorithm [40] implemented within LO-RANSAC [8, 23]. Further details and additional experiments are available in SM."
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+ "content": "The results are shown in Table 3 and Fig. 4c. Our method provides superior results to other methods based on the decomposition of the fundamental matrix, and performs comparably to the minimal solver [40] in terms of estimated focal lengths and shows minor improvements in terms of pose accuracy. Our method thus provides a good alternative to the 6-point minimal solver in the case when the focal lengths of the two cameras are known to be equal."
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+ "content": "Acknowledgments: V. K. was supported by the Slovak Grant Agency for Science (VEGA), project no. 1/0373/23. and the TERAIS project, a Horizon-Widera-2021 program of the European Union under the Grant agreement number 101079338. The results were obtained using the computational resources procured in the project National competence centre for high performance computing (project code: 311070AKF2) funded by European Regional Development Fund, EU Structural Funds Informatization of society, Operational Program Integrated Infrastructure. Z. K. was supported by the Czech Science Foundation (GACR) JUNIOR STAR Grant No. 22-23183M."
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+ {
+ "type": "text",
+ "text": "Robust Synthetic-to-Real Transfer for Stereo Matching",
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+ "type": "text",
+ "text": "Jiawei Zhang $^{1}$ , Jiahe Li $^{1}$ , Lei Huang $^{2}$ , Xiaohan Yu $^{3}$ , Lin Gu $^{4,5}$ , Jin Zheng $^{1}$ , Xiao Bai $^{1*}$",
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+ "text": "$^{1}$ School of Computer Science and Engineering, State Key Laboratory of",
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Complex & Critical Software Environment, Jiangxi Research Institute, Beihang University",
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "$^{2}$ SKLCCSE, Institute of Artificial Intelligence, Beihang University",
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "$^{3}$ School of Computing, Macquarie University, Australia $^{4}$ RIKEN AIP $^{5}$ The University of Tokyo",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Abstract",
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+ {
+ "type": "text",
+ "text": "With advancements in domain generalized stereo matching networks, models pre-trained on synthetic data demonstrate strong robustness to unseen domains. However, few studies have investigated the robustness after fine-tuning them in real-world scenarios, during which the domain generalization ability can be seriously degraded. In this paper, we explore fine-tuning stereo matching networks without compromising their robustness to unseen domains. Our motivation stems from comparing Ground Truth (GT) versus Pseudo Label (PL) for fine-tuning: GT degrades, but PL preserves the domain generalization ability. Empirically, we find the difference between GT and PL implies valuable information that can regularize networks during fine-tuning. We also propose a framework to utilize this difference for fine-tuning, consisting of a frozen Teacher, an exponential moving average (EMA) Teacher, and a Student network. The core idea is to utilize the EMA Teacher to measure what the Student has learned and dynamically improve GT and PL for fine-tuning. We integrate our framework with state-of-the-art networks and evaluate its effectiveness on several real-world datasets. Extensive experiments show that our method effectively preserves the domain generalization ability during fine-tuning. Code is available at: https://github.com/jiaw-z/DKT-Stereo.",
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+ "type": "text",
+ "text": "1. Introduction",
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+ {
+ "type": "text",
+ "text": "Estimating 3D geometry from 2D images is a fundamental problem in computer vision. Stereo matching is a solution that identifies matching correspondences and recovers depth information through triangulation. With the development of deep learning, stereo matching networks have shown impressive performance on various benchmarks.",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "A major obstacle to their applications is the high cost of collecting real-world annotations. To make up for the lack",
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+ "page_idx": 0
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+ "image_caption": [
+ "Figure 1. Target-domain and cross-domain performance of networks pre-trained on synthetic data, fine-tuned with GT, and with our method (best visualized in colors). IGEV-Stereo [44] is used as the baseline model. D1 error (the lower the better) is used for evaluation. (a) and (b) fine-tune networks on the KITTI and Booster datasets, respectively. Our method achieves great performance in target and unseen domains simultaneously. We also evaluate the robustness of KITTI fine-tuned networks on DrivingStereo in (a.2), where our method is more robust to challenging weather."
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+ "page_idx": 0
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+ {
+ "type": "text",
+ "text": "of real data, existing networks are commonly pre-trained on synthetic data. With recent advancements in building domain generalized networks [7, 17, 21, 55], they have demonstrated strong robustness to unseen domains. However, as shown in Figure 1, fine-tuning pre-trained networks in real-world scenarios degrades the domain generalization ability. We provide visualization results in Figure 2. The degradation of robustness to unseen scenarios can render networks unreliable for real-world applications. To solve it, we explore what degrades the domain generalization ability of stereo networks during fine-tuning and provide a solution by combining Ground Truth (GT) with Pseudo Label (PL) predicted by pre-trained stereo matching networks.",
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+ "text": "Our exploration stems from the observation that using PL for fine-tuning stereo matching networks can preserve the domain generalization ability (cf. Section 3.3.1). Since there is always a gap between predicted PL and GT, we consider whether the difference between them implies certain knowledge. We are also motivated by previous studies [9, 14, 57] of knowledge distillation utilizing wrong response to non-target classes, which they refer to as \"dark",
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+ "text": "This CVPR paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
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+ "text": "*Corresponding author: Xiao Bai (baixiao@buaa.edu.cn).",
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+ "text": "20247",
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+ "img_path": "images/90c59ccb7ed2450e21660e6b07343534d8588acdda41491d84dc51eab625cd85.jpg",
+ "image_caption": [
+ "Figure 2. Visualization results on both target and unseen domains. IGEV-Stereo [44] is used as the baseline model. The network pretrained on synthetic data shows robustness to unseen domains but can still fail to handle unseen challenges such as transparent or mirror (ToM) surfaces. Fine-tuning the network with GT improves the target-domain performance, however, it seriously degrades the domain generalization ability. Our DKT fine-tuning framework performs well on target and unseen domains simultaneously."
+ ],
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+ "text": "knowledge\" of a transfer set by introducing PL with GT.",
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+ "type": "text",
+ "text": "Starting from the observation, we aim to identify how GT and the predicted PL behave differently and further answer why fine-tuning with GT degrades the domain generalization ability of stereo matching networks. Specifically, we divide pixels into consistent and inconsistent regions based on the difference between GT and PL (cf. Section 3.1). GT and PL of the consistent region are similar, otherwise, it is defined as the inconsistent region. We empirically find two primary causes that degrade the domain generalization ability: (i) In the inconsistent region, networks encounter new knowledge not learned during pre-training on synthetic data. Learning new knowledge without sufficient consistent region for regularization seriously degrades the domain generalization ability. (ii) In the consistent region, stereo matching networks can still overfit the details of GT.",
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+ "type": "text",
+ "text": "Based on the exploration, we propose the framework utilizing Dark Knowledge to Transfer (DKT) stereo matching networks. DKT consists of a frozen Teacher, an exponential moving average (EMA) Teacher, and a Student network, all initialized with the same pre-trained weights. We use the frozen Teacher to predict PL. The EMA Teacher is updated by the Student's weights, serving as a dynamic measure of consistent and inconsistent regions. With the EMA Teacher's prediction, we propose the Filter and Ensemble (F&E) module to improve disparity maps. For GT, F&E filters out the inconsistent region with a probability, resulting in the remaining GT that has a reduced inconsistent region, avoiding insufficient consistent region for regularization. It also performs an ensemble between GT and the",
+ "bbox": [
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+ "page_idx": 1
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+ "text": "EMA Teacher's prediction in the consistent region, adding fine-grained permutations to prevent networks from overfitting GT details. For PL, F&E filters out the inconsistent region between PL and the EMA Teacher's prediction and ensembles them in the consistent region, enhancing the accuracy of PL predicted by the frozen Teacher. We train the Student with improved GT and PL jointly. After finetuning, we keep the Student for inference.",
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+ "type": "text",
+ "text": "Our main contributions are as follows:",
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+ "- To the best of our knowledge, we make the first attempt to address the domain generalization ability degradation for fine-tuning stereo matching networks. We divide pixels into consistent and inconsistent regions based on the difference between ground truth and pseudo label and demonstrate their varied roles during fine-tuning. Our further analysis of their roles identifies two primary causes of the degradation: learning new knowledge without sufficient regularization and overfitting ground truth details.",
+ "- We propose the F&E module to address these two causes, filtering out the inconsistent region to avoid insufficient regularization and ensembling disparities in the consistent region to prevent overfitting ground truth details.",
+ "- We introduce a dynamic adjustment for different regions by incorporating the exponential moving average Teacher, achieving the balance of preserving domain generalization ability and learning target domain knowledge.",
+ "- We develop the DKT fine-tuning framework, which can be easily applied to existing networks, significantly improving their robustness to unseen domains and achieving competitive target-domain performance simultaneously."
+ ],
+ "bbox": [
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+ "text": "20248",
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+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "We believe this exploration will stimulate further consideration of stereo matching networks' robustness and domain generalization ability when fine-tuning them in real-world scenarios, crucial for their practical applications.",
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+ "page_idx": 2
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+ "type": "text",
+ "text": "2. Related Work",
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Stereo Matching Networks. Estimating 3D geometry from 2D images is an important topic in computer vision [37, 39-41, 59], wherein stereo matching computes disparity between images. MC-CNN [51] first introduced a convolutional neural network to compute matching cost and predict disparity maps using SGM [15]. Since then, numerous deep-learning-based methods have been developed for stereo matching [3, 6, 16, 21, 42]. According to their strategy to perform cost construction, they can be divided into two categories. One type builds 3D cost volume and aggregates the cost with 2D convolutions. DispNetC [22] introduces end-to-end regression and builds a correlation-based 3D cost volume. Many works [19, 24, 35, 46, 49] adopt the correlation-based strategy and achieve impressive performance. Another type concatenates features to construct 4D cost volume and perform aggregation with 3D convolutions [3, 5, 16, 43, 53, 56]. Methods in this category can leverage more comprehensive information from the original images. Most recently, stereo matching networks [21, 38, 44, 52, 58] based on iterative optimization [34] are proposed and show state-of-the-art accuracy and strong robustness.",
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+ "text": "Robust Stereo Matching. Recently, building stereo matching networks that are robust to scenario changes has received increased interest. Existing studies can be categorized into joint generalization and cross-domain generalization types. Joint generalization involves training networks jointly on multiple datasets to perform well with a shared set of parameters [20, 30, 32]. Cross-domain generalization aims to improve generalization performance on unseen scenarios, with a current focus on synthetic-to-real generalization [2, 4, 7, 26, 36, 54, 55]. These methods enhance performance in unseen domains by avoiding networks overfitting synthetic data during pre-training. However, they have not investigated the domain generalization ability after fine-tuning. This paper is unique as it attempts to address the domain generalization degradation during fine-tuning.",
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+ "text": "Dark Knowledge. Knowledge Distillation (KD) is explored for transferring knowledge from one teacher model to another student model [11, 14, 48]. In the original form for classification, it constructs a transfer set by using the soft target distribution produced by teachers. Dark knowledge refers to the extra information beyond one-hot GT contained in the modified training set [27], associated with the distribution of responses to non-target classes. Building on this concept, a series of studies [9, 50, 57] explore the effects of incorrect predictions in KD. BAN [9] decomposes the KD into a dark knowledge term and ground truth component",
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+ "Image",
+ "Consistent GT",
+ "Figure 3. Visualization of each region resulting from our division. We divide GT and PL based on their consistency."
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+ "text": "and develops a born-again procedure. DKD [57] reformulates KD into target class and non-target class distillation and decouples the two parts to transfer dark knowledge. We adopt this concept to decouple different regions of disparity maps, analyzing what degrades the domain generalization ability of stereo matching networks during fine-tuning.",
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+ "type": "text",
+ "text": "3. Fine-tuning Stereo Matching Networks with Ground Truth and Pseudo Label",
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+ "text": "3.1. Preliminary and Definition",
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+ "text": "We explore whether stereo matching PL contains valuable knowledge that can preserve the domain generalization ability during fine-tuning and investigate the factors that make GT and PL behave differently. Stereo matching networks based on 2D or 3D cost aggregation are formed to obtain predictions by conducting Soft-Argmin on probability volume [16], while the recently developed iterative optimization methods directly predict continuous values [21, 44, 58] and achieve superior robustness. The difference between GT and PL lies at the level of continuous values, making our study applicable to any form of stereo matching network. Given a rectified stereo pair $(I^l, I^r)$ with GT map $D^*$ , a pre-trained stereo matching network $\\theta$ predicts a dense disparity map $\\hat{D} = \\theta(I^l, I^r)$ as PL. We adopt the pixel distance $\\hat{D} - D^*$ to measure responses to non-target disparities. To identify what makes GT and PL behave differently, we divide pixels into different regions based on the consistency between PL and GT. The visualization of each region is shown in Figure 3. We define different regions as follows:",
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+ "text": "Definition 1 Consistent region $X_{c}(\\tau)$ . $X_{c}(\\tau)$ contains pixels where the differences between GT and PL are less than the threshold $\\tau$ . $X_{c}(\\tau) = \\{x||\\hat{D}(x_{i}) - D^{*}(x_{i})| < \\tau \\}$ . This region represents areas where GT and PL align closely.",
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+ "text": "Definition 2 Inconsistent region $X_{inc}(\\tau)$ . $X_{inc}(\\tau)$ contains pixels where the differences between GT and PL are not less than the threshold $\\tau$ . $X_{inc}(\\tau) = \\{x||\\hat{D}(x_i) - D^*(x_i)| \\geq \\tau \\}$ . Stereo matching networks can encounter unseen challenges in $X_{inc}(\\tau)$ .",
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+ "text": "Definition 3 Invalid region $X_{\\text{invalid}}$ . It contains pixels without available annotations due to the sparsity of GT.",
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+ "text": "3.2. Experiment Setup",
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+ "type": "text",
+ "text": "Dataset. Our experiments are conducted on KITTI [10, 23], DrivingStereo [47], Booster [25], Middlebury [28], and ETH3D [29]. KITTI and DrivingStereo datasets collect outdoor driving scenes, Booster and Middlebury provide indoor scenes, and ETH3D includes indoor and outdoor scenes in grayscale. We use the half resolution of Middlebury and DrivingStereo, and the quarter resolution of Booster. Except for online submissions, we split datasets for local evaluation due to the submission policy. More details are shown in supplementary materials. In our tables, we use blue for target domains and red for unseen domains.",
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+ },
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+ "type": "text",
+ "text": "Metric. We calculate the percentage of pixels with an absolute error larger than a certain distance for evaluation. 3 pixel is used for KITTI and DrivingStereo datasets. 2 pixel is used for Middlebury and Booster datasets. 1 pixel is used for ETH3D dataset.",
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+ "text": "Implementation. We adopt IGEV-Stereo [44] as the basic network architecture here, and results of other networks are shown in supplementary materials. We initialize networks with SceneFlow [22] pre-trained weights. For KITTI datasets, we fine-tune networks for 50k steps with a batch size of 8. For the Booster dataset, we fine-tune networks for 25k steps with a batch size of 2. We set the learning rate to 2e-4 for KITTI and 1e-5 for Booster. We use the data augmentation strategy in [21] during fine-tuning.",
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+ {
+ "type": "text",
+ "text": "3.3. Ground Truth versus Pseudo Label",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "3.3.1 PL Preserves Domain Generalization Ability",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "We compare the fine-tuning of pre-trained networks using GT and PL. Contrast observations lead us to believe that the difference between GT and PL contains valuable information. Analyzing the validation curves presented in Figure 4, we observe a significant degradation of domain generalization ability during GT fine-tuning, occurring at an early stage. In contrast, there is no substantial degradation in domain generalization ability when using PL. This finding demonstrates that PL, as naive knowledge produced by pre-trained stereo matching networks, can preserve the original domain generalization ability.",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "3.3.2 Investigating the Difference between GT and PL",
+ "text_level": 1,
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+ "text": "To understand the role of each region in GT and PL and to delve into the reasons behind the degradation of domain generalization ability, we conduct fine-tuning experiments by isolating different regions. The comparative results across various settings are presented in Table 1.",
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+ "text": "Learning new knowledge without sufficient regularization can degrade domain generalization ability. We start by comparing the baseline, which uses all valid regions of GT, with a setting that uses only the consistent region $X_{c}(3)$ . By removing the inconsistent region $X_{inc}(3)$ of GT,",
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+ "img_path": "images/a39bff85df5ef209b40679ccae991e91c0ef605bb0d66c89e1137623a30ba2b7.jpg",
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+ "type": "image",
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+ "img_path": "images/1e9c7c363209e2ebc3b3fd6b5085c12fb12a23d97179c1ccc7d4f03a9ba05546.jpg",
+ "image_caption": [
+ "(a) Ground Truth",
+ "(b) Pseudo Label",
+ "Figure 4. Evaluation curves of the target and cross domain performance during fine-tuning with GT or PL."
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ {
+ "type": "image",
+ "img_path": "images/f60528153d4b2a8e3f3b06a7fc1a8def4f305e3f86fee1a992e60eb277f063ca.jpg",
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+ "type": "text",
+ "text": "we observe an effective alleviation of generalization degradation, particularly notable for the Booster dataset. This observation highlights the significant impact of GT in the inconsistent region on degrading generalization ability. Next, we isolate only the inconsistent region $X_{inc}(3)$ for finetuning, where we observe a catastrophic degradation in generalization ability. Networks trained with only $\\mathrm{GT}(X_{inc}(3))$ even suffer difficulties generalizing to the target-domain validation data. This comparison highlights the crucial regularization role played by GT in the consistent region for networks learning to address unseen challenges in the inconsistent region. We conclude that while GT supervision in the inconsistent region is valuable for networks adapting to unseen challenges, learning new knowledge in this region without sufficient regularization can seriously degrade domain generalization ability.",
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+ "type": "text",
+ "text": "Fine-grained details in GT can cause overfitting, degrading domain generalization ability. Unlike the inconsistent region, where handling is more challenging, networks find handling the consistent region less difficult since most necessary knowledge has been involved in synthetic data. Subsequently, we investigate whether learning in $X_{c}(\\tau)$ can also result in domain generalization degradation. In our observations, a clear degradation in domain generalization ability occurs when using GT compared to PL in the same consistent region $X_{c}(3)$ . Further tightening the threshold for the consistent region to $X_{c}(1)$ provides some relief, however, we find that even inaccurate components within a one-pixel range can lead to varying domain generalization abilities. We attribute this to stereo matching",
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+ {
+ "type": "page_number",
+ "text": "20250",
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+ {
+ "type": "table",
+ "img_path": "images/b723924912c426b1eed3e56162afadf4814cbcebbcac95d6338a196f4a826e39.jpg",
+ "table_caption": [],
+ "table_footnote": [],
+ "table_body": "| Symbol | Definition |
| x, x, X, and X | A scalar, a vector, a matrix, and a tensor |
| v = 1,···, V | Indices range from 1 to their capital version |
| R, C | Fields of real numbers and complex numbers |
| conj(X) | the complex conjugate of X |
| [x] | the one greater than or equal to x |
| X ∈ R1×I2×I3 | A 3-rd order tensor |
| X(:, :, i3) | the i3-th frontal slice of X |
| N, V, C, M | Number of samples, views, clusters, anchors |
| φ(Xv) ∈ RM×N | Anchor graph in the v-th view |
| Uv ∈ RM×K | Projection matrix in the v-th view |
| Bv ∈ RK×N | embedding feature in the v-th view |
| B ∈ RK×V×N | The rotated embedding feature tensor |
",
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+ },
+ {
+ "type": "text",
+ "text": "2.2. Preliminaries on t-SVD",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Definition 1 (T-product [27]) The T-product of two third-order tensors $\\mathcal{X} \\in \\mathbb{R}^{I_1 \\times I_2 \\times I_3}$ and $\\mathcal{Y} \\in \\mathbb{R}^{I_2 \\times J \\times I_3}$ is defined as",
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathcal {Z} = \\mathcal {X} * \\mathcal {Y} \\in \\mathbb {R} ^ {I _ {1} \\times J \\times I _ {3}}, \\tag {1}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "which can be computed by multiple matrix multiplication operations in the Fourier transform domain.",
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+ },
+ {
+ "type": "text",
+ "text": "Definition 2 (Tensor transpose [15]) Under the Fourier transform, the conjugate transpose of a tensor $\\mathcal{X} \\in \\mathbb{C}^{I_1 \\times I_2 \\times I_3}$ is denoted as $\\mathcal{X}^{\\mathrm{T}} \\in \\mathbb{C}^{I_2 \\times I_1 \\times I_3}$ . It satisfies",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {X} ^ {\\mathrm {T}} (:,: 1) = \\left(\\mathcal {X} (:,: 1)\\right) ^ {\\mathrm {T}}, \\tag {2}\n$$\n",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "and",
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathcal {X} ^ {\\mathrm {T}} (:,: i _ {3}) = \\left(\\mathcal {X} (:,: I _ {3} + 2 - i _ {3})\\right) ^ {\\top}, i _ {3} = 2, \\dots , I _ {3}, \\tag {3}\n$$\n",
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+ {
+ "type": "text",
+ "text": "where $\\mathcal{X}^{\\mathrm{T}}(:, :, i_3)$ is the $i_3$ -th frontal slice of $\\mathcal{X}^{\\mathrm{T}}$ .",
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+ },
+ {
+ "type": "text",
+ "text": "Definition 3 (Orthogonal tensor) [15] A tensor $\\mathcal{X}$ is called orthogonal when it satisfies",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {X} * \\mathcal {X} ^ {\\mathrm {T}} = \\mathcal {X} ^ {\\mathrm {T}} * \\mathcal {X} = \\mathcal {I}, \\tag {4}\n$$\n",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "where $\\mathcal{I}$ is the identity tensor.",
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+ "type": "text",
+ "text": "Definition 4 (f-diagonal tensor) [16] A tensor $\\mathcal{X} \\in \\mathbb{R}^{I_1 \\times I_2 \\times I_3}$ is called f-diagonal when each frontal slice $\\mathcal{X}^{(i_3)}, i_3 = 1, \\dots, I_3$ is a diagonal matrix.",
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+ "type": "text",
+ "text": "Definition 5 (tensor Singular Value Decomposition(tSVD) [2, 15, 25]) For a tensor $\\mathcal{X}\\in \\mathbb{R}^{I_1\\times I_2\\times I_3}$ , its t-SVD is represented as:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathcal {X} = \\mathcal {U} * \\mathcal {S} * \\mathcal {V} ^ {\\mathrm {T}}, \\tag {5}\n$$\n",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "where $\\mathcal{U}\\in \\mathbb{R}^{I_1\\times I_1\\times I_3}$ and $\\mathcal{V}\\in \\mathbb{R}^{I_2\\times I_2\\times I_3}$ are orthogonal tensors, and $\\mathcal{S}\\in \\mathbb{R}^{I_1\\times I_2\\times I_3}$ is an $f$ -diagonal tensor.",
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+ "page_idx": 1
+ },
+ {
+ "type": "page_number",
+ "text": "26214",
+ "bbox": [
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+ "page_idx": 1
+ },
+ {
+ "type": "image",
+ "img_path": "images/db9f7d055ed2ad31b7190f79bae4ca3ad7576642014311e81574dda271791946.jpg",
+ "image_caption": [
+ "(a)"
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "image",
+ "img_path": "images/4c9d93ca5b6f778271fbb65facc3957bcfbe10daffd6ee54c2d2a65164a873a8.jpg",
+ "image_caption": [
+ "(b)",
+ "Figure 2. The framework of $\\mathbf{S}^2\\mathrm{MVTC}$ . (a) The process involves mapping each pre-given anchor graph $\\phi (\\mathbf{X}_v)$ using $\\mathbf{U}_v$ to obtain the corresponding embedding feature $\\mathbf{B}_v$ . These embedding features, $\\mathbf{B}_v$ , are then formed into a tensor $\\mathcal{B}$ . This tensor, along with the newly defined $\\mathcal{B}_{\\mathrm{TLFA}}$ and $\\sum_{v = 1}^{V}\\| \\mathbf{B}_v - \\tilde{\\mathbf{B}}\\|_{\\mathrm{F}}$ , are utilized to explore intra-view correlations and semantic consistency, respectively, where $\\tilde{\\mathbf{B}} = \\sum_{v = 1}^{V}\\frac{1}{V}\\mathbf{B}_v$ . (b) The tensor low-frequency approximation $\\mathcal{B}_{\\mathrm{TLFA}}$ ."
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "code",
+ "sub_type": "algorithm",
+ "code_caption": [
+ "Algorithm 1 t-SVD"
+ ],
+ "code_body": "Input: $\\mathcal{X}\\in \\mathbb{R}^{I_1\\times I_2\\times I_3}$ \nOutput: $\\mathcal{U}\\in \\mathbb{R}^{I_1\\times I_1\\times I_3},\\mathcal{S}\\in \\mathbb{R}^{I_1\\times I_2\\times I_3},\\mathcal{V}\\in$ $\\mathbb{R}^{I_2\\times I_2\\times I_3}$ . \n $\\hat{\\mathcal{X}}\\gets \\mathrm{fft}(\\mathcal{X},[],3)$ $[\\mathbf{U},\\mathbf{S},\\mathbf{V}] = \\mathbf{svd}(\\widehat{\\mathcal{X}}(:, :, 1))$ \nfor $i_{3} = 1$ to $I_{3}$ do \n $[\\mathbf{U},\\mathbf{S},\\mathbf{V}] = \\mathbf{svd}(\\widehat{\\mathcal{X}}(:, :, i_3));$ $\\widehat{\\mathcal{U}}(:, :, i_3) = \\mathbf{U};\\widehat{\\mathcal{S}}(:, :, i_3) = \\mathbf{S};\\widehat{\\mathcal{V}}(:, :, i_3) = \\mathbf{V}.$ \nend for \n $\\mathcal{U} = \\mathrm{ifft}(\\widehat{\\mathcal{U}},[],3);\\mathcal{S} = \\mathrm{ifft}(\\widehat{\\mathcal{S}},[],3);\\mathcal{V} = \\mathrm{ifft}(\\widehat{\\mathcal{V}},[],3).$",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "According to the T-product operation in Definition 1, this decomposition can be obtained by Algorithm 1, where fft and ifft are the fast Fourier transform (FFT) operator and the inverse FFT, respectively.",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "2.3. Related works",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "2.3.1 Anchor learning for large-scale MVC",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "The general framework of an anchor-learning-based large-scale MVC is constructed as follows:",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\left\\{\\mathbf {Z} _ {v} \\right\\} _ {v = 1} ^ {V}} \\sum_ {v = 1} ^ {V} \\left\\| \\mathbf {X} _ {v} - \\mathbf {A} _ {v} \\mathbf {Z} _ {v} \\right\\| _ {\\mathrm {F}} ^ {2} + \\lambda \\Psi (\\mathbf {Z} _ {v})\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathrm {s . t .} \\mathbf {Z} _ {v} ^ {\\mathrm {T}} \\mathbf {1} = 1, \\mathbf {Z} _ {v} \\geq 0, v = 1, \\dots , V, \\tag {6}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "where $\\mathbf{A}_v\\in \\mathbb{R}^{D_v\\times M}$ is the learned anchor matrix in the $v$ -th view and $M$ is the number of anchors. $\\mathbf{Z}_v\\in \\mathbb{R}^{M\\times N}$ is the anchor graph, which is used to describe the relationship among samples. In addition, $\\Psi (\\mathbf{Z}_v)$ represents the regularization terms. Kang et al. [14] consider Frobenius norm",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "on $\\mathbf{Z}_v$ to find a stable solution for large-scale MVC. However, the anchor graph in each view is learned separately, failing to explore the complementary information. To solve this, many fusion strategies are considered to find the consensus representation among views. For example, Liu et al. [39] considered the respective projection matrix to find the consistent graph. Li et al. [21] considered the joint graph across multiple views via a self-supervised weighting manner. Chen et al. [5] introduced the Fast Self-guided Multi-view Subspace Clustering algorithm, which effectively combines view-shared anchor learning and global-guided-local self-guidance learning into a comprehensive model. Huang et al. [9] considered three levels of diversity: features, anchors, and neighbors. By leveraging these levels of diversity, view-sharing bipartite graphs are constructed, enhancing the effectiveness of the clustering process. In addition, some methods based on tensor anchor graphs have been proposed [12, 24, 42]. These methods stack each anchor graph into a tensor and use low-rank tensor approximation to integrate the consistency information among anchor graphs across views, achieving good performance on large-scale data.",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "2.3.2 Embedding feature learning for large-scale MVC",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Unlike large-scale MVC methods based on anchor learning, another approach focuses on learning embedding features through the pre-defined anchor graphs in the pre-processing step. Typically, these anchor graphs are constructed using kernel mapping, for instance, through nonlinear Radial Basis Function (RBF) mapping, as follows:",
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+ "page_idx": 2
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\phi \\left(\\mathbf {x} _ {v} ^ {n}\\right) = \\left[ \\exp \\left(- \\frac {\\left\\| \\mathbf {x} _ {v} ^ {n} - \\mathbf {z} _ {v} ^ {1} \\right\\| ^ {2}}{\\sigma}\\right), \\dots , \\exp \\left(- \\frac {\\left\\| \\mathbf {x} _ {v} ^ {n} - \\mathbf {z} _ {v} ^ {M} \\right\\| ^ {2}}{\\sigma}\\right) \\right] ^ {\\mathrm {T}}, \\tag {7}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "page_number",
+ "text": "26215",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "where $\\sigma$ is the kernel width, $\\mathbf{z}_v^m\\in \\mathbb{R}^{D_v}$ $m = 1,\\dots ,M$ are anchors, which randomly chooses from the column of $v$ -th view $\\mathbf{X}_v\\in \\mathbb{R}^{D_v\\times N}$ $\\phi (\\mathbf{x}_v^n)\\in \\mathbb{R}^M$ represents the nonlinear relationship between $M$ anchors and the $n$ -th samples. In this way, Zhang et al. [45] projected non-linear anchor graphs from different views to a common Hamming feature space. This shared feature space is then fed into binary clustering structure learning to derive the final clustering results. However, this method solely explores pairwise correlations between views. To uncover global correlations among views, Zhang et al. [44] considered consistent projections, obtained by low-rank tensor approximation, to find a better binary feature representation for large-scale MVC tasks. Additionally, Wang et al. [35] considered autoencoder learning techniques to find consistent projections for generating binary features used in clustering.",
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "3. Methods",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "As mentioned above, anchor-based methods mostly focus on learning high-order information between anchor graphs or projection matrices. However, their ultimate goal is to obtain shared embedding features across views.",
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+ "page_idx": 3
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+ {
+ "type": "text",
+ "text": "3.1. Model development",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "In this section, we will focus on how to quickly learn the consistency among embedding features well, which involves the exploration of intra-view graph similarity (IGS) and ensuring inter-view semantic consistency (ISC). With anchor graphs $\\phi (\\mathbf{X}_v)\\in \\mathbb{R}^{M\\times N},v = 1,\\dots ,V$ obtained in advance, we can first learn $V$ embedding features by projection matrices $\\mathbf{U}_v,v = 1,\\dots ,V$ , as follows:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\min _ {\\left\\{\\mathbf {U} _ {v}, \\mathbf {B} _ {v} \\right\\} _ {v = 1} ^ {V}} \\sum_ {v = 1} ^ {V} \\frac {\\lambda}{2} \\left| \\left| \\mathbf {U} _ {v} \\right| \\right| _ {\\mathrm {F}} ^ {2} + \\frac {1}{2} \\left| \\left| \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (\\mathbf {X} _ {v}) \\right| \\right| _ {\\mathrm {F}} ^ {2} \\tag {8} \\\\ \\begin{array}{l} \\text {s . t .} \\mathbf {B} _ {v} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N, v = 1, \\dots , V, \\end{array} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "where $\\lambda$ serves as a weight aiming to attain a stable solution for the mapping matrix $\\mathbf{U}_v$ . $\\mathbf{B}_v(:,n)\\in \\mathbb{S}$ represents the $z$ -score normalization constraint, ensuring that each feature from different views has a mean of 0 and a standard deviation of 1. Here $\\mathbf{b}_v^n = \\mathbf{B}_v(:,n)$ , and $\\mathbb{S} = \\{\\mathbf{b}_v^n:\\sum_{k = 1}^K\\mathbf{b}_v^n (k) = 0,\\frac{\\sum_{k = 1}^K(\\mathbf{b}_v^n(k))^2}{K - 1} = 1,n = 1,\\dots ,N\\}$ .",
+ "bbox": [
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "To efficiently explore the intra-view information of embedding features, we initially stack the embedding features from each view into a tensor. Subsequently, we introduce a newly tensor low-frequency approximation $\\| \\mathcal{B}\\|_{\\mathrm{TLFA}}$ , applied to its rotated form $\\mathcal{B} \\in \\mathbb{R}^{K\\times V\\times N}$ , aiming to capture the IGS.",
+ "bbox": [
+ 75,
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "Definition 6 (Tensor low-frequency approximation (TLFA)) According to the $t$ -SVD computation outlined in Algorithm 1, the TLFA of a tensor $\\mathcal{B} \\in \\mathbb{R}^{K \\times V \\times N}$ is",
+ "bbox": [
+ 75,
+ 854,
+ 468,
+ 900
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "defined as the frontal slices in the low-frequency domain. Mathematically, it is expressed as:",
+ "bbox": [
+ 498,
+ 90,
+ 888,
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\mathcal {Y}} \\frac {1}{2} \\| \\mathcal {B} - \\mathcal {Y} \\| _ {\\mathrm {F}} ^ {2}, s. t. \\mathcal {Y} = \\operatorname {i f t} (\\widehat {\\mathcal {Y}}, [, 3),\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 558,
+ 131,
+ 828,
+ 160
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "where $\\widehat{\\mathcal{Y}} = \\widehat{\\mathcal{B}}(:, :, 1) + \\sum_{n=2}^{L} (\\widehat{\\mathcal{B}}(:, :, n) + \\widehat{\\mathcal{B}}(:, :, N+2-n))$ . $\\widehat{\\mathcal{B}} = \\mathrm{fft}(\\mathcal{B}, [], 3)$ , means applying the fast Fourier transform along the sample (3-rd) dimension for the embedding features from different views. The obtained low-frequency components incorporate graph similarity into embedding feature learning, resulting in a smooth representation.",
+ "bbox": [
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "Besides, the term $\\sum_{v=1}^{V} \\|\\tilde{\\mathbf{B}} - \\mathbf{B}_v\\|_F^2$ is included to ensure ISC [40], where $\\tilde{\\mathbf{B}}$ is the average embedding feature.",
+ "bbox": [
+ 498,
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "Overall, the framework of $\\mathbf{S}^2\\mathrm{MVTC}$ is formulated as:",
+ "bbox": [
+ 517,
+ 297,
+ 874,
+ 311
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\min _ {\\left\\{\\mathbf {B} _ {v}, \\mathbf {U} _ {v} \\right\\} _ {v = 1} ^ {V}, \\tilde {\\mathbf {B}}} \\sum_ {v = 1} ^ {v} \\left(\\frac {\\lambda}{2} | | \\mathbf {U} _ {v} | | _ {\\mathrm {F}} ^ {2} + \\frac {1}{2} | | \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (\\mathbf {X} _ {v}) | | _ {\\mathrm {F}} ^ {2} \\right. \\\\ + \\frac {\\beta}{2} \\left\\| \\tilde {\\mathbf {B}} - \\mathbf {B} _ {v} \\right\\| _ {\\mathrm {F}} ^ {2}) + \\tau \\| \\mathcal {B} \\| _ {\\mathrm {T L F A}} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 511,
+ 321,
+ 867,
+ 393
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\text {s . t .} \\tilde {\\mathbf {B}} (:, n), \\mathbf {B} _ {v} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N, v = 1, \\dots , V. \\tag {9}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 508,
+ 395,
+ 890,
+ 426
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "Here $\\mathcal{B} = \\Omega (\\mathbf{B}_1,\\dots ,\\mathbf{B}_V)$ , where $\\Omega$ represents the operation of stacking each feature and rotating and the inverse operator $\\mathbf{B}_v = \\Omega_v^{-1}(\\mathcal{B})$ .",
+ "bbox": [
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "To make the above optimization problem separable, we introduce an auxiliary variable $\\mathcal{V}$ , leading to the reformulation as follows:",
+ "bbox": [
+ 498,
+ 472,
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+ 516
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\min _ {\\left\\{\\mathbf {B} _ {v}, \\mathbf {U} _ {v} \\right\\} _ {v = 1} ^ {V}, \\tilde {\\mathbf {B}}} \\sum_ {v = 1} ^ {v} \\left(\\frac {\\lambda}{2} | | \\mathbf {U} _ {v} | | _ {\\mathrm {F}} ^ {2} + \\frac {1}{2} | | \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (\\mathbf {X} _ {v}) | | _ {\\mathrm {F}} ^ {2} \\right. \\\\ + \\frac {\\beta}{2} | | \\tilde {\\mathbf {B}} - \\mathbf {B} _ {v} | | _ {\\mathrm {F}} ^ {2} + \\frac {1}{2} | | \\mathcal {B} - \\mathcal {Y} | | _ {F} ^ {2} + \\tau | | \\mathcal {Y} | | _ {\\mathrm {T L F A}} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 511,
+ 525,
+ 867,
+ 595
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\text {s . t .} \\tilde {\\mathbf {B}} (:, n), \\mathbf {B} _ {v} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N, v = 1, \\dots , V. \\tag {10}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "3.2. Solutions",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "This optimal problem in Eq. (10) can be solved through an alternating optimization approach, where each parameter is updated individually while keeping the others fixed.",
+ "bbox": [
+ 498,
+ 662,
+ 890,
+ 705
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "Update $\\mathbf{U}_v$ . Fixing other variables, the subproblem of $\\mathbf{U}_v$ can be rewritten as:",
+ "bbox": [
+ 498,
+ 707,
+ 890,
+ 736
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\mathbf {U} _ {v}} \\frac {1}{2} \\| \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (\\mathbf {X} _ {v}) \\| _ {\\mathrm {F}} ^ {2} + \\frac {\\lambda}{2} \\| \\mathbf {U} _ {v} \\| _ {\\mathrm {F}} ^ {2}. \\tag {11}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 562,
+ 744,
+ 890,
+ 775
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "By taking the derivative of the equation and setting it equal to zero, we can obtain:",
+ "bbox": [
+ 498,
+ 786,
+ 890,
+ 815
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbf {U} _ {v} = \\left(\\mathbf {B} _ {v} \\phi^ {\\mathrm {T}} \\left(\\mathbf {X} _ {v}\\right)\\right) \\left(\\phi \\left(\\mathbf {X} _ {v}\\right) \\phi^ {\\mathrm {T}} \\left(\\mathbf {X} _ {v}\\right) + \\lambda \\mathbf {I}\\right) ^ {\\dagger}, \\tag {12}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 532,
+ 825,
+ 890,
+ 844
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "where $\\phi (\\mathbf{X}_v)\\phi^{\\mathrm{T}}(\\mathbf{X}_v)$ can be pre-calculated outside the main loop to decrease the computational cost. Therefore, the storage and computational complexity are $O(MN)$ and",
+ "bbox": [
+ 498,
+ 854,
+ 890,
+ 901
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "page_number",
+ "text": "26216",
+ "bbox": [
+ 478,
+ 944,
+ 519,
+ 955
+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "$O(\\max (KMN,M^3))$ , respectively.",
+ "bbox": [
+ 76,
+ 90,
+ 318,
+ 106
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Update $\\mathbf{B}_v$ . The subproblem of $\\mathbf{B}_v$ can be rephrased as:",
+ "bbox": [
+ 76,
+ 106,
+ 450,
+ 122
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\mathbf {B} _ {v}} \\frac {1}{2} \\| \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (x _ {v}) \\| _ {F} ^ {2} + \\frac {\\beta}{2} \\| \\tilde {\\mathbf {B}} - \\mathbf {B} _ {v} \\| _ {F} ^ {2} + \\frac {1}{2} \\| \\mathcal {B} - \\mathcal {Y} \\| _ {F} ^ {2},\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 78,
+ 128,
+ 475,
+ 161
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\text {s . t .} \\mathbf {B} _ {v} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N. \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 76,
+ 164,
+ 294,
+ 179
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "(13)",
+ "bbox": [
+ 437,
+ 179,
+ 467,
+ 193
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Similarly, by taking the derivative of the equation and setting it equal to zero, we can obtain",
+ "bbox": [
+ 76,
+ 194,
+ 467,
+ 224
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbf {B} _ {v} = \\text {n o r m a l i z e} \\left(\\beta \\tilde {\\mathbf {B}} + \\Omega_ {v} ^ {- 1} (\\mathcal {Y}) + \\mathbf {U} _ {v} \\phi \\left(\\mathbf {x} _ {v}\\right)\\right) / (\\beta + 2), \\tag {14}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 88,
+ 233,
+ 467,
+ 263
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "where $\\Omega_v^{-1}(\\mathcal{Y}) = \\mathbf{Y}_v \\in \\mathbb{R}^{K \\times N}$ . normalize is the $z$ -score normalization function, which can be achieved by the Matlab command \"normalize\". In this case, both the storage and computational complexity of updating $\\mathbf{B}_v$ are $O(KN)$ . Update $\\mathcal{Y}$ . The subproblem of $\\mathcal{Y}$ is rewritten as:",
+ "bbox": [
+ 76,
+ 263,
+ 468,
+ 340
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\mathcal {Y}} \\tau \\| \\mathcal {Y} \\| _ {\\mathrm {T L F A}} + \\frac {1}{2} \\| \\mathcal {B} - \\mathcal {Y} \\| _ {\\mathrm {F}} ^ {2}. \\tag {15}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 168,
+ 349,
+ 468,
+ 378
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "According to the definition of $\\| \\mathcal{V}\\|_{\\mathrm{TLFA}}$ , a closed-form solution for $\\mathcal{V}$ can be obtained by choosing the low-frequency component in the Fourier domain. The details are outlined in Algorithm 2. The computational and storage complexity are $O(KVN\\log (N))$ and $O(KNV)$ , respectively.",
+ "bbox": [
+ 76,
+ 388,
+ 468,
+ 464
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Update $\\bar{\\mathbf{B}}$ . The subproblem of updating $\\bar{\\mathbf{B}}$ is:",
+ "bbox": [
+ 76,
+ 464,
+ 380,
+ 479
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\tilde {\\mathbf {B}}} \\sum_ {v = 1} ^ {V} \\frac {\\beta}{2} \\| \\tilde {\\mathbf {B}} - \\mathbf {B} _ {v} \\| _ {\\mathrm {F}} ^ {2} \\text {s . t .} \\tilde {\\mathbf {B}} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N. \\tag {16}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 91,
+ 488,
+ 468,
+ 541
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "The solution of $\\tilde{\\mathbf{B}}$ is:",
+ "bbox": [
+ 76,
+ 542,
+ 218,
+ 556
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\tilde {\\mathbf {B}} = \\text {n o r m a l i z e} \\left(\\frac {1}{V} \\sum_ {v = 1} ^ {V} \\mathbf {B} _ {v}\\right), \\tag {17}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 174,
+ 564,
+ 468,
+ 604
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "which needs computational and storage complexity of $O(KN)$ and $O(KNV)$ , respectively.",
+ "bbox": [
+ 76,
+ 614,
+ 468,
+ 645
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "After obtaining the consistent fused embedding feature $\\tilde{\\mathbf{B}}$ . The final clustering results can be obtained by the following clustering structure learning framework:",
+ "bbox": [
+ 76,
+ 645,
+ 468,
+ 690
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\mathbf {D}, \\mathbf {G}} \\| \\tilde {\\mathbf {B}} - \\mathbf {D} \\mathbf {G} \\| _ {\\mathrm {F}} ^ {2} \\text {s . t .} \\sum_ {c = 1} ^ {C} \\mathbf {G} (c,:) = \\mathbf {1}, \\mathbf {G} \\in \\{0, 1 \\}, \\tag {18}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 84,
+ 699,
+ 468,
+ 741
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "where $\\mathbf{D}$ is the clustering center and $\\mathbf{G}$ is the clustering indicator matrix.",
+ "bbox": [
+ 76,
+ 750,
+ 468,
+ 777
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Overall, the method $\\mathbf{S}^2\\mathrm{MVTC}$ is summarized in Algorithm 3. According to the parameters analysis in the algorithm, the main storage and computational complexities of $\\mathbf{S}^2\\mathrm{MVTC}$ are $O(\\max(KVN, MN))$ and $O(\\max(KMNT, M^3T))$ , respectively, where $T$ represents the total number of iterations in the $\\mathbf{S}^2\\mathrm{MVTC}$ .",
+ "bbox": [
+ 76,
+ 779,
+ 468,
+ 869
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "In addition, the theoretical convergence can be well guaranteed. Our problem is bounded due to the summation of",
+ "bbox": [
+ 76,
+ 869,
+ 468,
+ 900
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Algorithm 2 Updating Y",
+ "bbox": [
+ 501,
+ 90,
+ 669,
+ 106
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Input: B, low frequency parameter $L$ .",
+ "bbox": [
+ 516,
+ 109,
+ 772,
+ 125
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Output: $y$",
+ "bbox": [
+ 517,
+ 126,
+ 599,
+ 138
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\widehat {\\mathcal {B}} \\leftarrow \\operatorname {f f t} (\\mathcal {B}, [, 3);\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 517,
+ 140,
+ 632,
+ 155
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\hat {\\mathcal {Y}} (:,: 1)) = \\hat {\\mathcal {B}} (:,: 1));\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 517,
+ 156,
+ 669,
+ 172
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "for $n = 2$ to $L$ do",
+ "bbox": [
+ 517,
+ 172,
+ 638,
+ 186
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\widehat {\\mathcal {Y}} (:,: n) = \\widehat {\\mathcal {B}} (:,: n);\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 534,
+ 186,
+ 678,
+ 202
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\widehat {\\mathcal {Y}} (:,: N + 2 - n) = \\operatorname {c o n j} (\\widehat {\\mathcal {Y}} (:,: n)).\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 534,
+ 202,
+ 784,
+ 219
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "end for",
+ "bbox": [
+ 517,
+ 220,
+ 571,
+ 232
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathcal {Y} = \\operatorname {i f t} (\\widehat {\\mathcal {Y}}, [, 3).\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 517,
+ 232,
+ 635,
+ 250
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Algorithm 3 S²MVTC",
+ "bbox": [
+ 501,
+ 271,
+ 656,
+ 286
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Input: Anchor graphs $\\phi (\\mathbf{X}_v), v = 1,\\dots ,V$ , low frequency parameter $L$ , regularization parameters $\\beta, \\lambda$ .",
+ "bbox": [
+ 516,
+ 291,
+ 890,
+ 321
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Initialize: $\\tau = 1$ , Maximum iterations $T = 7$",
+ "bbox": [
+ 517,
+ 323,
+ 808,
+ 335
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "for $t = 1$ to $T$ do",
+ "bbox": [
+ 517,
+ 338,
+ 635,
+ 349
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "for $v = 1$ to $V$ do",
+ "bbox": [
+ 534,
+ 353,
+ 655,
+ 366
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Update $\\mathbf{U}_v$ via Eq. (12);",
+ "bbox": [
+ 550,
+ 367,
+ 712,
+ 382
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Update $\\mathbf{B}_v$ via Eq. (14);",
+ "bbox": [
+ 550,
+ 383,
+ 710,
+ 398
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "end for",
+ "bbox": [
+ 534,
+ 398,
+ 586,
+ 411
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Update $\\mathcal{V}$ via Eq. (15);",
+ "bbox": [
+ 534,
+ 412,
+ 686,
+ 428
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Update $\\tilde{\\mathbf{B}}$ via Eq. (17);",
+ "bbox": [
+ 534,
+ 428,
+ 687,
+ 443
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\tau = \\tau * 1. 1.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 534,
+ 444,
+ 616,
+ 455
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "end for",
+ "bbox": [
+ 517,
+ 459,
+ 571,
+ 470
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Apply Eq. (18) to find the clustering indicator matrix $\\mathbf{G}$ .",
+ "bbox": [
+ 517,
+ 473,
+ 890,
+ 502
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Output: Clustering result.",
+ "bbox": [
+ 517,
+ 503,
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "norms with positive penalty parameters. Furthermore, the exact minimum points of each subproblem can be achieved, implying that each subproblem exhibits a monotone decrease. The proposed algorithm converges according to the convergence theorem in [28] (theorem 7.29).",
+ "bbox": [
+ 496,
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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. Experimental settings",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Multi-view Datasets: Six well-known large-scale multi-view datasets, including CCV [13], NUS-WIDE-OBJ [7], Caltech102 [19], AwA [18], Cifar-10 [17], Youtube-Face_sel [41] evaluate the effectiveness of $S^2$ MVTC, where the statistical information is presented in Tab. 2. Detailed descriptions can be found in ??",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Compared Clustering Algorithms. To compare clustering performance, we select $k$ -means and seven state-of-the-art methods: binary multi-view clustering (BMVC) [2018, TP-MAI] [45], fast multi-View anchor-correspondence clustering (FMVACC) [2022, NeurIPS] [38], one-pass multi-view clustering (OPMC) [2021, ICCV] [23], fast multi-view clustering via ensembles (FastMICE) [2023, TKDE] [10], scalable and parameter-free multiview graph clustering",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "page_number",
+ "text": "26217",
+ "bbox": [
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+ },
+ {
+ "type": "table",
+ "img_path": "images/557a97c8ef65be10a3990bebc2f909924bc034bc17607413301a7a9cfb2c0001.jpg",
+ "table_caption": [],
+ "table_footnote": [],
+ "table_body": "| Symbol | Definition |
| x, x, X, and X | A scalar, a vector, a matrix, and a tensor |
| v = 1,···, V | Indices range from 1 to their capital version |
| R, C | Fields of real numbers and complex numbers |
| conj(X) | the complex conjugate of X |
| [x] | the one greater than or equal to x |
| X ∈ R1×I2×I3 | A 3-rd order tensor |
| X(:, :, i3) | the i3-th frontal slice of X |
| N, V, C, M | Number of samples, views, clusters, anchors |
| φ(Xv) ∈ RM×N | Anchor graph in the v-th view |
| Uv ∈ RM×K | Projection matrix in the v-th view |
| Bv ∈ RK×N | embedding feature in the v-th view |
| B ∈ RK×V×N | The rotated embedding feature tensor |
"
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "2.2. Preliminaries on t-SVD"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.406,
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+ ],
+ "angle": 0,
+ "content": "Definition 1 (T-product [27]) The T-product of two third-order tensors \\(\\mathcal{X} \\in \\mathbb{R}^{I_1 \\times I_2 \\times I_3}\\) and \\(\\mathcal{Y} \\in \\mathbb{R}^{I_2 \\times J \\times I_3}\\) is defined as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ 0.466
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {Z} = \\mathcal {X} * \\mathcal {Y} \\in \\mathbb {R} ^ {I _ {1} \\times J \\times I _ {3}}, \\tag {1}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.47,
+ 0.892,
+ 0.501
+ ],
+ "angle": 0,
+ "content": "which can be computed by multiple matrix multiplication operations in the Fourier transform domain."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.51,
+ 0.892,
+ 0.556
+ ],
+ "angle": 0,
+ "content": "Definition 2 (Tensor transpose [15]) Under the Fourier transform, the conjugate transpose of a tensor \\(\\mathcal{X} \\in \\mathbb{C}^{I_1 \\times I_2 \\times I_3}\\) is denoted as \\(\\mathcal{X}^{\\mathrm{T}} \\in \\mathbb{C}^{I_2 \\times I_1 \\times I_3}\\). It satisfies"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.604,
+ 0.56,
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+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {X} ^ {\\mathrm {T}} (:,: 1) = \\left(\\mathcal {X} (:,: 1)\\right) ^ {\\mathrm {T}}, \\tag {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.586,
+ 0.533,
+ 0.599
+ ],
+ "angle": 0,
+ "content": "and"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.51,
+ 0.606,
+ 0.892,
+ 0.626
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {X} ^ {\\mathrm {T}} (:,: i _ {3}) = \\left(\\mathcal {X} (:,: I _ {3} + 2 - i _ {3})\\right) ^ {\\top}, i _ {3} = 2, \\dots , I _ {3}, \\tag {3}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "where \\(\\mathcal{X}^{\\mathrm{T}}(:, :, i_3)\\) is the \\(i_3\\)-th frontal slice of \\(\\mathcal{X}^{\\mathrm{T}}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.655,
+ 0.892,
+ 0.686
+ ],
+ "angle": 0,
+ "content": "Definition 3 (Orthogonal tensor) [15] A tensor \\(\\mathcal{X}\\) is called orthogonal when it satisfies"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.611,
+ 0.692,
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+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {X} * \\mathcal {X} ^ {\\mathrm {T}} = \\mathcal {X} ^ {\\mathrm {T}} * \\mathcal {X} = \\mathcal {I}, \\tag {4}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.5,
+ 0.717,
+ 0.697,
+ 0.732
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathcal{I}\\) is the identity tensor."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.74,
+ 0.892,
+ 0.786
+ ],
+ "angle": 0,
+ "content": "Definition 4 (f-diagonal tensor) [16] A tensor \\(\\mathcal{X} \\in \\mathbb{R}^{I_1 \\times I_2 \\times I_3}\\) is called f-diagonal when each frontal slice \\(\\mathcal{X}^{(i_3)}, i_3 = 1, \\dots, I_3\\) is a diagonal matrix."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.794,
+ 0.892,
+ 0.839
+ ],
+ "angle": 0,
+ "content": "Definition 5 (tensor Singular Value Decomposition(tSVD) [2, 15, 25]) For a tensor \\(\\mathcal{X}\\in \\mathbb{R}^{I_1\\times I_2\\times I_3}\\), its t-SVD is represented as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ 0.845,
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+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {X} = \\mathcal {U} * \\mathcal {S} * \\mathcal {V} ^ {\\mathrm {T}}, \\tag {5}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.869,
+ 0.892,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathcal{U}\\in \\mathbb{R}^{I_1\\times I_1\\times I_3}\\) and \\(\\mathcal{V}\\in \\mathbb{R}^{I_2\\times I_2\\times I_3}\\) are orthogonal tensors, and \\(\\mathcal{S}\\in \\mathbb{R}^{I_1\\times I_2\\times I_3}\\) is an \\(f\\) -diagonal tensor."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "26214"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
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+ "angle": 0,
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+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "(a)"
+ },
+ {
+ "type": "image",
+ "bbox": [
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+ "angle": 0,
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+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "(b)"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 2. The framework of \\(\\mathbf{S}^2\\mathrm{MVTC}\\). (a) The process involves mapping each pre-given anchor graph \\(\\phi (\\mathbf{X}_v)\\) using \\(\\mathbf{U}_v\\) to obtain the corresponding embedding feature \\(\\mathbf{B}_v\\). These embedding features, \\(\\mathbf{B}_v\\), are then formed into a tensor \\(\\mathcal{B}\\). This tensor, along with the newly defined \\(\\mathcal{B}_{\\mathrm{TLFA}}\\) and \\(\\sum_{v = 1}^{V}\\| \\mathbf{B}_v - \\tilde{\\mathbf{B}}\\|_{\\mathrm{F}}\\), are utilized to explore intra-view correlations and semantic consistency, respectively, where \\(\\tilde{\\mathbf{B}} = \\sum_{v = 1}^{V}\\frac{1}{V}\\mathbf{B}_v\\). (b) The tensor low-frequency approximation \\(\\mathcal{B}_{\\mathrm{TLFA}}\\)."
+ },
+ {
+ "type": "code_caption",
+ "bbox": [
+ 0.079,
+ 0.384,
+ 0.214,
+ 0.399
+ ],
+ "angle": 0,
+ "content": "Algorithm 1 t-SVD"
+ },
+ {
+ "type": "algorithm",
+ "bbox": [
+ 0.092,
+ 0.403,
+ 0.47,
+ 0.566
+ ],
+ "angle": 0,
+ "content": "Input: \\(\\mathcal{X}\\in \\mathbb{R}^{I_1\\times I_2\\times I_3}\\) \nOutput: \\(\\mathcal{U}\\in \\mathbb{R}^{I_1\\times I_1\\times I_3},\\mathcal{S}\\in \\mathbb{R}^{I_1\\times I_2\\times I_3},\\mathcal{V}\\in\\) \\(\\mathbb{R}^{I_2\\times I_2\\times I_3}\\) . \n\\(\\hat{\\mathcal{X}}\\gets \\mathrm{fft}(\\mathcal{X},[],3)\\) \n\\([\\mathbf{U},\\mathbf{S},\\mathbf{V}] = \\mathbf{svd}(\\widehat{\\mathcal{X}}(:, :, 1))\\) \nfor \\(i_{3} = 1\\) to \\(I_{3}\\) do \n\\([\\mathbf{U},\\mathbf{S},\\mathbf{V}] = \\mathbf{svd}(\\widehat{\\mathcal{X}}(:, :, i_3));\\) \n\\(\\widehat{\\mathcal{U}}(:, :, i_3) = \\mathbf{U};\\widehat{\\mathcal{S}}(:, :, i_3) = \\mathbf{S};\\widehat{\\mathcal{V}}(:, :, i_3) = \\mathbf{V}.\\) \nend for \n\\(\\mathcal{U} = \\mathrm{ifft}(\\widehat{\\mathcal{U}},[],3);\\mathcal{S} = \\mathrm{ifft}(\\widehat{\\mathcal{S}},[],3);\\mathcal{V} = \\mathrm{ifft}(\\widehat{\\mathcal{V}},[],3).\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.592,
+ 0.47,
+ 0.655
+ ],
+ "angle": 0,
+ "content": "According to the T-product operation in Definition 1, this decomposition can be obtained by Algorithm 1, where fft and ifft are the fast Fourier transform (FFT) operator and the inverse FFT, respectively."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.663,
+ 0.225,
+ 0.679
+ ],
+ "angle": 0,
+ "content": "2.3. Related works"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.687,
+ 0.391,
+ 0.703
+ ],
+ "angle": 0,
+ "content": "2.3.1 Anchor learning for large-scale MVC"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.712,
+ 0.469,
+ 0.741
+ ],
+ "angle": 0,
+ "content": "The general framework of an anchor-learning-based large-scale MVC is constructed as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.156,
+ 0.751,
+ 0.422,
+ 0.793
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\left\\{\\mathbf {Z} _ {v} \\right\\} _ {v = 1} ^ {V}} \\sum_ {v = 1} ^ {V} \\left\\| \\mathbf {X} _ {v} - \\mathbf {A} _ {v} \\mathbf {Z} _ {v} \\right\\| _ {\\mathrm {F}} ^ {2} + \\lambda \\Psi (\\mathbf {Z} _ {v})\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.156,
+ 0.794,
+ 0.469,
+ 0.814
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathrm {s . t .} \\mathbf {Z} _ {v} ^ {\\mathrm {T}} \\mathbf {1} = 1, \\mathbf {Z} _ {v} \\geq 0, v = 1, \\dots , V, \\tag {6}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.825,
+ 0.47,
+ 0.9
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathbf{A}_v\\in \\mathbb{R}^{D_v\\times M}\\) is the learned anchor matrix in the \\(v\\) -th view and \\(M\\) is the number of anchors. \\(\\mathbf{Z}_v\\in \\mathbb{R}^{M\\times N}\\) is the anchor graph, which is used to describe the relationship among samples. In addition, \\(\\Psi (\\mathbf{Z}_v)\\) represents the regularization terms. Kang et al. [14] consider Frobenius norm"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.385,
+ 0.893,
+ 0.717
+ ],
+ "angle": 0,
+ "content": "on \\(\\mathbf{Z}_v\\) to find a stable solution for large-scale MVC. However, the anchor graph in each view is learned separately, failing to explore the complementary information. To solve this, many fusion strategies are considered to find the consensus representation among views. For example, Liu et al. [39] considered the respective projection matrix to find the consistent graph. Li et al. [21] considered the joint graph across multiple views via a self-supervised weighting manner. Chen et al. [5] introduced the Fast Self-guided Multi-view Subspace Clustering algorithm, which effectively combines view-shared anchor learning and global-guided-local self-guidance learning into a comprehensive model. Huang et al. [9] considered three levels of diversity: features, anchors, and neighbors. By leveraging these levels of diversity, view-sharing bipartite graphs are constructed, enhancing the effectiveness of the clustering process. In addition, some methods based on tensor anchor graphs have been proposed [12, 24, 42]. These methods stack each anchor graph into a tensor and use low-rank tensor approximation to integrate the consistency information among anchor graphs across views, achieving good performance on large-scale data."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.499,
+ 0.735,
+ 0.892,
+ 0.751
+ ],
+ "angle": 0,
+ "content": "2.3.2 Embedding feature learning for large-scale MVC"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.758,
+ 0.892,
+ 0.85
+ ],
+ "angle": 0,
+ "content": "Unlike large-scale MVC methods based on anchor learning, another approach focuses on learning embedding features through the pre-defined anchor graphs in the pre-processing step. Typically, these anchor graphs are constructed using kernel mapping, for instance, through nonlinear Radial Basis Function (RBF) mapping, as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.5,
+ 0.857,
+ 0.903,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "\\[\n\\phi \\left(\\mathbf {x} _ {v} ^ {n}\\right) = \\left[ \\exp \\left(- \\frac {\\left\\| \\mathbf {x} _ {v} ^ {n} - \\mathbf {z} _ {v} ^ {1} \\right\\| ^ {2}}{\\sigma}\\right), \\dots , \\exp \\left(- \\frac {\\left\\| \\mathbf {x} _ {v} ^ {n} - \\mathbf {z} _ {v} ^ {M} \\right\\| ^ {2}}{\\sigma}\\right) \\right] ^ {\\mathrm {T}}, \\tag {7}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.48,
+ 0.99,
+ 0.518,
+ 1.0
+ ],
+ "angle": 0,
+ "content": "26215"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.091,
+ 0.473,
+ 0.334
+ ],
+ "angle": 0,
+ "content": "where \\(\\sigma\\) is the kernel width, \\(\\mathbf{z}_v^m\\in \\mathbb{R}^{D_v}\\) \\(m = 1,\\dots ,M\\) are anchors, which randomly chooses from the column of \\(v\\) -th view \\(\\mathbf{X}_v\\in \\mathbb{R}^{D_v\\times N}\\) \\(\\phi (\\mathbf{x}_v^n)\\in \\mathbb{R}^M\\) represents the nonlinear relationship between \\(M\\) anchors and the \\(n\\) -th samples. In this way, Zhang et al. [45] projected non-linear anchor graphs from different views to a common Hamming feature space. This shared feature space is then fed into binary clustering structure learning to derive the final clustering results. However, this method solely explores pairwise correlations between views. To uncover global correlations among views, Zhang et al. [44] considered consistent projections, obtained by low-rank tensor approximation, to find a better binary feature representation for large-scale MVC tasks. Additionally, Wang et al. [35] considered autoencoder learning techniques to find consistent projections for generating binary features used in clustering."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
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+ ],
+ "angle": 0,
+ "content": "3. Methods"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.374,
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+ 0.435
+ ],
+ "angle": 0,
+ "content": "As mentioned above, anchor-based methods mostly focus on learning high-order information between anchor graphs or projection matrices. However, their ultimate goal is to obtain shared embedding features across views."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
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+ 0.266,
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+ ],
+ "angle": 0,
+ "content": "3.1. Model development"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.468,
+ 0.469,
+ 0.573
+ ],
+ "angle": 0,
+ "content": "In this section, we will focus on how to quickly learn the consistency among embedding features well, which involves the exploration of intra-view graph similarity (IGS) and ensuring inter-view semantic consistency (ISC). With anchor graphs \\(\\phi (\\mathbf{X}_v)\\in \\mathbb{R}^{M\\times N},v = 1,\\dots ,V\\) obtained in advance, we can first learn \\(V\\) embedding features by projection matrices \\(\\mathbf{U}_v,v = 1,\\dots ,V\\), as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.093,
+ 0.586,
+ 0.469,
+ 0.647
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\min _ {\\left\\{\\mathbf {U} _ {v}, \\mathbf {B} _ {v} \\right\\} _ {v = 1} ^ {V}} \\sum_ {v = 1} ^ {V} \\frac {\\lambda}{2} \\left| \\left| \\mathbf {U} _ {v} \\right| \\right| _ {\\mathrm {F}} ^ {2} + \\frac {1}{2} \\left| \\left| \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (\\mathbf {X} _ {v}) \\right| \\right| _ {\\mathrm {F}} ^ {2} \\tag {8} \\\\ \\begin{array}{l} \\text {s . t .} \\mathbf {B} _ {v} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N, v = 1, \\dots , V, \\end{array} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.659,
+ 0.469,
+ 0.754
+ ],
+ "angle": 0,
+ "content": "where \\(\\lambda\\) serves as a weight aiming to attain a stable solution for the mapping matrix \\(\\mathbf{U}_v\\). \\(\\mathbf{B}_v(:,n)\\in \\mathbb{S}\\) represents the \\(z\\)-score normalization constraint, ensuring that each feature from different views has a mean of 0 and a standard deviation of 1. Here \\(\\mathbf{b}_v^n = \\mathbf{B}_v(:,n)\\), and \\(\\mathbb{S} = \\{\\mathbf{b}_v^n:\\sum_{k = 1}^K\\mathbf{b}_v^n (k) = 0,\\frac{\\sum_{k = 1}^K(\\mathbf{b}_v^n(k))^2}{K - 1} = 1,n = 1,\\dots ,N\\}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.755,
+ 0.469,
+ 0.843
+ ],
+ "angle": 0,
+ "content": "To efficiently explore the intra-view information of embedding features, we initially stack the embedding features from each view into a tensor. Subsequently, we introduce a newly tensor low-frequency approximation \\(\\| \\mathcal{B}\\|_{\\mathrm{TLFA}}\\), applied to its rotated form \\(\\mathcal{B} \\in \\mathbb{R}^{K\\times V\\times N}\\), aiming to capture the IGS."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.856,
+ 0.47,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "Definition 6 (Tensor low-frequency approximation (TLFA)) According to the \\(t\\)-SVD computation outlined in Algorithm 1, the TLFA of a tensor \\(\\mathcal{B} \\in \\mathbb{R}^{K \\times V \\times N}\\) is"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.092,
+ 0.89,
+ 0.122
+ ],
+ "angle": 0,
+ "content": "defined as the frontal slices in the low-frequency domain. Mathematically, it is expressed as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.56,
+ 0.132,
+ 0.83,
+ 0.161
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\mathcal {Y}} \\frac {1}{2} \\| \\mathcal {B} - \\mathcal {Y} \\| _ {\\mathrm {F}} ^ {2}, s. t. \\mathcal {Y} = \\operatorname {i f t} (\\widehat {\\mathcal {Y}}, [, 3),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.162,
+ 0.892,
+ 0.258
+ ],
+ "angle": 0,
+ "content": "where \\(\\widehat{\\mathcal{Y}} = \\widehat{\\mathcal{B}}(:, :, 1) + \\sum_{n=2}^{L} (\\widehat{\\mathcal{B}}(:, :, n) + \\widehat{\\mathcal{B}}(:, :, N+2-n))\\). \\(\\widehat{\\mathcal{B}} = \\mathrm{fft}(\\mathcal{B}, [], 3)\\), means applying the fast Fourier transform along the sample (3-rd) dimension for the embedding features from different views. The obtained low-frequency components incorporate graph similarity into embedding feature learning, resulting in a smooth representation."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.265,
+ 0.892,
+ 0.299
+ ],
+ "angle": 0,
+ "content": "Besides, the term \\(\\sum_{v=1}^{V} \\|\\tilde{\\mathbf{B}} - \\mathbf{B}_v\\|_F^2\\) is included to ensure ISC [40], where \\(\\tilde{\\mathbf{B}}\\) is the average embedding feature."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.299,
+ 0.875,
+ 0.313
+ ],
+ "angle": 0,
+ "content": "Overall, the framework of \\(\\mathbf{S}^2\\mathrm{MVTC}\\) is formulated as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.513,
+ 0.323,
+ 0.868,
+ 0.394
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\min _ {\\left\\{\\mathbf {B} _ {v}, \\mathbf {U} _ {v} \\right\\} _ {v = 1} ^ {V}, \\tilde {\\mathbf {B}}} \\sum_ {v = 1} ^ {v} \\left(\\frac {\\lambda}{2} | | \\mathbf {U} _ {v} | | _ {\\mathrm {F}} ^ {2} + \\frac {1}{2} | | \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (\\mathbf {X} _ {v}) | | _ {\\mathrm {F}} ^ {2} \\right. \\\\ + \\frac {\\beta}{2} \\left\\| \\tilde {\\mathbf {B}} - \\mathbf {B} _ {v} \\right\\| _ {\\mathrm {F}} ^ {2}) + \\tau \\| \\mathcal {B} \\| _ {\\mathrm {T L F A}} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.509,
+ 0.396,
+ 0.891,
+ 0.427
+ ],
+ "angle": 0,
+ "content": "\\[\n\\text {s . t .} \\tilde {\\mathbf {B}} (:, n), \\mathbf {B} _ {v} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N, v = 1, \\dots , V. \\tag {9}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.428,
+ 0.891,
+ 0.473
+ ],
+ "angle": 0,
+ "content": "Here \\(\\mathcal{B} = \\Omega (\\mathbf{B}_1,\\dots ,\\mathbf{B}_V)\\), where \\(\\Omega\\) represents the operation of stacking each feature and rotating and the inverse operator \\(\\mathbf{B}_v = \\Omega_v^{-1}(\\mathcal{B})\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.473,
+ 0.892,
+ 0.517
+ ],
+ "angle": 0,
+ "content": "To make the above optimization problem separable, we introduce an auxiliary variable \\(\\mathcal{V}\\), leading to the reformulation as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.512,
+ 0.526,
+ 0.868,
+ 0.597
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\min _ {\\left\\{\\mathbf {B} _ {v}, \\mathbf {U} _ {v} \\right\\} _ {v = 1} ^ {V}, \\tilde {\\mathbf {B}}} \\sum_ {v = 1} ^ {v} \\left(\\frac {\\lambda}{2} | | \\mathbf {U} _ {v} | | _ {\\mathrm {F}} ^ {2} + \\frac {1}{2} | | \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (\\mathbf {X} _ {v}) | | _ {\\mathrm {F}} ^ {2} \\right. \\\\ + \\frac {\\beta}{2} | | \\tilde {\\mathbf {B}} - \\mathbf {B} _ {v} | | _ {\\mathrm {F}} ^ {2} + \\frac {1}{2} | | \\mathcal {B} - \\mathcal {Y} | | _ {F} ^ {2} + \\tau | | \\mathcal {Y} | | _ {\\mathrm {T L F A}} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.51,
+ 0.599,
+ 0.891,
+ 0.63
+ ],
+ "angle": 0,
+ "content": "\\[\n\\text {s . t .} \\tilde {\\mathbf {B}} (:, n), \\mathbf {B} _ {v} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N, v = 1, \\dots , V. \\tag {10}\n\\]"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.64,
+ 0.608,
+ 0.654
+ ],
+ "angle": 0,
+ "content": "3.2. Solutions"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.663,
+ 0.892,
+ 0.707
+ ],
+ "angle": 0,
+ "content": "This optimal problem in Eq. (10) can be solved through an alternating optimization approach, where each parameter is updated individually while keeping the others fixed."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.708,
+ 0.891,
+ 0.737
+ ],
+ "angle": 0,
+ "content": "Update \\(\\mathbf{U}_v\\). Fixing other variables, the subproblem of \\(\\mathbf{U}_v\\) can be rewritten as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.563,
+ 0.746,
+ 0.891,
+ 0.776
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\mathbf {U} _ {v}} \\frac {1}{2} \\| \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (\\mathbf {X} _ {v}) \\| _ {\\mathrm {F}} ^ {2} + \\frac {\\lambda}{2} \\| \\mathbf {U} _ {v} \\| _ {\\mathrm {F}} ^ {2}. \\tag {11}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.787,
+ 0.891,
+ 0.816
+ ],
+ "angle": 0,
+ "content": "By taking the derivative of the equation and setting it equal to zero, we can obtain:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.534,
+ 0.826,
+ 0.891,
+ 0.845
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {U} _ {v} = \\left(\\mathbf {B} _ {v} \\phi^ {\\mathrm {T}} \\left(\\mathbf {X} _ {v}\\right)\\right) \\left(\\phi \\left(\\mathbf {X} _ {v}\\right) \\phi^ {\\mathrm {T}} \\left(\\mathbf {X} _ {v}\\right) + \\lambda \\mathbf {I}\\right) ^ {\\dagger}, \\tag {12}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.855,
+ 0.892,
+ 0.902
+ ],
+ "angle": 0,
+ "content": "where \\(\\phi (\\mathbf{X}_v)\\phi^{\\mathrm{T}}(\\mathbf{X}_v)\\) can be pre-calculated outside the main loop to decrease the computational cost. Therefore, the storage and computational complexity are \\(O(MN)\\) and"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.479,
+ 0.945,
+ 0.52,
+ 0.957
+ ],
+ "angle": 0,
+ "content": "26216"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.078,
+ 0.091,
+ 0.319,
+ 0.107
+ ],
+ "angle": 0,
+ "content": "\\(O(\\max (KMN,M^3))\\), respectively."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.078,
+ 0.107,
+ 0.452,
+ 0.123
+ ],
+ "angle": 0,
+ "content": "Update \\(\\mathbf{B}_v\\). The subproblem of \\(\\mathbf{B}_v\\) can be rephrased as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.08,
+ 0.13,
+ 0.476,
+ 0.162
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\mathbf {B} _ {v}} \\frac {1}{2} \\| \\mathbf {B} _ {v} - \\mathbf {U} _ {v} \\phi (x _ {v}) \\| _ {F} ^ {2} + \\frac {\\beta}{2} \\| \\tilde {\\mathbf {B}} - \\mathbf {B} _ {v} \\| _ {F} ^ {2} + \\frac {1}{2} \\| \\mathcal {B} - \\mathcal {Y} \\| _ {F} ^ {2},\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.078,
+ 0.165,
+ 0.295,
+ 0.18
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\text {s . t .} \\mathbf {B} _ {v} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N. \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.439,
+ 0.18,
+ 0.468,
+ 0.194
+ ],
+ "angle": 0,
+ "content": "(13)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.195,
+ 0.468,
+ 0.225
+ ],
+ "angle": 0,
+ "content": "Similarly, by taking the derivative of the equation and setting it equal to zero, we can obtain"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.089,
+ 0.234,
+ 0.468,
+ 0.265
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {B} _ {v} = \\text {n o r m a l i z e} \\left(\\beta \\tilde {\\mathbf {B}} + \\Omega_ {v} ^ {- 1} (\\mathcal {Y}) + \\mathbf {U} _ {v} \\phi \\left(\\mathbf {x} _ {v}\\right)\\right) / (\\beta + 2), \\tag {14}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.265,
+ 0.469,
+ 0.342
+ ],
+ "angle": 0,
+ "content": "where \\(\\Omega_v^{-1}(\\mathcal{Y}) = \\mathbf{Y}_v \\in \\mathbb{R}^{K \\times N}\\). normalize is the \\(z\\)-score normalization function, which can be achieved by the Matlab command \"normalize\". In this case, both the storage and computational complexity of updating \\(\\mathbf{B}_v\\) are \\(O(KN)\\). Update \\(\\mathcal{Y}\\). The subproblem of \\(\\mathcal{Y}\\) is rewritten as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.169,
+ 0.35,
+ 0.469,
+ 0.38
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\mathcal {Y}} \\tau \\| \\mathcal {Y} \\| _ {\\mathrm {T L F A}} + \\frac {1}{2} \\| \\mathcal {B} - \\mathcal {Y} \\| _ {\\mathrm {F}} ^ {2}. \\tag {15}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.389,
+ 0.469,
+ 0.465
+ ],
+ "angle": 0,
+ "content": "According to the definition of \\(\\| \\mathcal{V}\\|_{\\mathrm{TLFA}}\\), a closed-form solution for \\(\\mathcal{V}\\) can be obtained by choosing the low-frequency component in the Fourier domain. The details are outlined in Algorithm 2. The computational and storage complexity are \\(O(KVN\\log (N))\\) and \\(O(KNV)\\), respectively."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.078,
+ 0.465,
+ 0.382,
+ 0.48
+ ],
+ "angle": 0,
+ "content": "Update \\(\\bar{\\mathbf{B}}\\). The subproblem of updating \\(\\bar{\\mathbf{B}}\\) is:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.092,
+ 0.489,
+ 0.469,
+ 0.542
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\tilde {\\mathbf {B}}} \\sum_ {v = 1} ^ {V} \\frac {\\beta}{2} \\| \\tilde {\\mathbf {B}} - \\mathbf {B} _ {v} \\| _ {\\mathrm {F}} ^ {2} \\text {s . t .} \\tilde {\\mathbf {B}} (:, n) \\in \\mathbb {S}, n = 1, \\dots , N. \\tag {16}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.078,
+ 0.543,
+ 0.219,
+ 0.557
+ ],
+ "angle": 0,
+ "content": "The solution of \\(\\tilde{\\mathbf{B}}\\) is:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.176,
+ 0.565,
+ 0.469,
+ 0.606
+ ],
+ "angle": 0,
+ "content": "\\[\n\\tilde {\\mathbf {B}} = \\text {n o r m a l i z e} \\left(\\frac {1}{V} \\sum_ {v = 1} ^ {V} \\mathbf {B} _ {v}\\right), \\tag {17}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.615,
+ 0.47,
+ 0.646
+ ],
+ "angle": 0,
+ "content": "which needs computational and storage complexity of \\( O(KN) \\) and \\( O(KNV) \\), respectively."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.646,
+ 0.469,
+ 0.691
+ ],
+ "angle": 0,
+ "content": "After obtaining the consistent fused embedding feature \\(\\tilde{\\mathbf{B}}\\). The final clustering results can be obtained by the following clustering structure learning framework:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.085,
+ 0.7,
+ 0.469,
+ 0.742
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\mathbf {D}, \\mathbf {G}} \\| \\tilde {\\mathbf {B}} - \\mathbf {D} \\mathbf {G} \\| _ {\\mathrm {F}} ^ {2} \\text {s . t .} \\sum_ {c = 1} ^ {C} \\mathbf {G} (c,:) = \\mathbf {1}, \\mathbf {G} \\in \\{0, 1 \\}, \\tag {18}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.75,
+ 0.469,
+ 0.779
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathbf{D}\\) is the clustering center and \\(\\mathbf{G}\\) is the clustering indicator matrix."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.78,
+ 0.469,
+ 0.87
+ ],
+ "angle": 0,
+ "content": "Overall, the method \\(\\mathbf{S}^2\\mathrm{MVTC}\\) is summarized in Algorithm 3. According to the parameters analysis in the algorithm, the main storage and computational complexities of \\(\\mathbf{S}^2\\mathrm{MVTC}\\) are \\(O(\\max(KVN, MN))\\) and \\(O(\\max(KMNT, M^3T))\\), respectively, where \\(T\\) represents the total number of iterations in the \\(\\mathbf{S}^2\\mathrm{MVTC}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.871,
+ 0.469,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "In addition, the theoretical convergence can be well guaranteed. Our problem is bounded due to the summation of"
+ },
+ {
+ "type": "code_caption",
+ "bbox": [
+ 0.502,
+ 0.091,
+ 0.671,
+ 0.107
+ ],
+ "angle": 0,
+ "content": "Algorithm 2 Updating Y"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.517,
+ 0.111,
+ 0.773,
+ 0.126
+ ],
+ "angle": 0,
+ "content": "Input: B, low frequency parameter \\( L \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.127,
+ 0.6,
+ 0.14
+ ],
+ "angle": 0,
+ "content": "Output: \\( y \\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.518,
+ 0.141,
+ 0.633,
+ 0.156
+ ],
+ "angle": 0,
+ "content": "\\[\n\\widehat {\\mathcal {B}} \\leftarrow \\operatorname {f f t} (\\mathcal {B}, [, 3);\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.518,
+ 0.157,
+ 0.671,
+ 0.173
+ ],
+ "angle": 0,
+ "content": "\\[\n\\hat {\\mathcal {Y}} (:,: 1)) = \\hat {\\mathcal {B}} (:,: 1));\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.173,
+ 0.64,
+ 0.187
+ ],
+ "angle": 0,
+ "content": "for \\(n = 2\\) to \\(L\\) do"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.535,
+ 0.187,
+ 0.679,
+ 0.203
+ ],
+ "angle": 0,
+ "content": "\\[\n\\widehat {\\mathcal {Y}} (:,: n) = \\widehat {\\mathcal {B}} (:,: n);\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.535,
+ 0.203,
+ 0.785,
+ 0.22
+ ],
+ "angle": 0,
+ "content": "\\[\n\\widehat {\\mathcal {Y}} (:,: N + 2 - n) = \\operatorname {c o n j} (\\widehat {\\mathcal {Y}} (:,: n)).\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.221,
+ 0.572,
+ 0.233
+ ],
+ "angle": 0,
+ "content": "end for"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.518,
+ 0.233,
+ 0.636,
+ 0.25
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {Y} = \\operatorname {i f t} (\\widehat {\\mathcal {Y}}, [, 3).\n\\]"
+ },
+ {
+ "type": "code_caption",
+ "bbox": [
+ 0.502,
+ 0.272,
+ 0.658,
+ 0.287
+ ],
+ "angle": 0,
+ "content": "Algorithm 3 S²MVTC"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.517,
+ 0.292,
+ 0.892,
+ 0.323
+ ],
+ "angle": 0,
+ "content": "Input: Anchor graphs \\(\\phi (\\mathbf{X}_v), v = 1,\\dots ,V\\), low frequency parameter \\(L\\), regularization parameters \\(\\beta, \\lambda\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.324,
+ 0.81,
+ 0.337
+ ],
+ "angle": 0,
+ "content": "Initialize: \\(\\tau = 1\\), Maximum iterations \\(T = 7\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.339,
+ 0.636,
+ 0.351
+ ],
+ "angle": 0,
+ "content": "for \\(t = 1\\) to \\(T\\) do"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.535,
+ 0.354,
+ 0.656,
+ 0.367
+ ],
+ "angle": 0,
+ "content": "for \\(v = 1\\) to \\(V\\) do"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.551,
+ 0.368,
+ 0.713,
+ 0.383
+ ],
+ "angle": 0,
+ "content": "Update \\(\\mathbf{U}_v\\) via Eq. (12);"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.552,
+ 0.384,
+ 0.712,
+ 0.399
+ ],
+ "angle": 0,
+ "content": "Update \\(\\mathbf{B}_v\\) via Eq. (14);"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.535,
+ 0.4,
+ 0.588,
+ 0.412
+ ],
+ "angle": 0,
+ "content": "end for"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.535,
+ 0.414,
+ 0.687,
+ 0.429
+ ],
+ "angle": 0,
+ "content": "Update \\(\\mathcal{V}\\) via Eq. (15);"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.535,
+ 0.429,
+ 0.688,
+ 0.444
+ ],
+ "angle": 0,
+ "content": "Update \\(\\tilde{\\mathbf{B}}\\) via Eq. (17);"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.535,
+ 0.445,
+ 0.617,
+ 0.457
+ ],
+ "angle": 0,
+ "content": "\\[\n\\tau = \\tau * 1. 1.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.46,
+ 0.572,
+ 0.472
+ ],
+ "angle": 0,
+ "content": "end for"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.474,
+ 0.891,
+ 0.503
+ ],
+ "angle": 0,
+ "content": "Apply Eq. (18) to find the clustering indicator matrix \\(\\mathbf{G}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
+ 0.505,
+ 0.694,
+ 0.52
+ ],
+ "angle": 0,
+ "content": "Output: Clustering result."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.551,
+ 0.892,
+ 0.627
+ ],
+ "angle": 0,
+ "content": "norms with positive penalty parameters. Furthermore, the exact minimum points of each subproblem can be achieved, implying that each subproblem exhibits a monotone decrease. The proposed algorithm converges according to the convergence theorem in [28] (theorem 7.29)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.641,
+ 0.633,
+ 0.658
+ ],
+ "angle": 0,
+ "content": "4. Experiments"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.666,
+ 0.704,
+ 0.683
+ ],
+ "angle": 0,
+ "content": "4.1. Experimental settings"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.689,
+ 0.892,
+ 0.78
+ ],
+ "angle": 0,
+ "content": "Multi-view Datasets: Six well-known large-scale multi-view datasets, including CCV [13], NUS-WIDE-OBJ [7], Caltech102 [19], AwA [18], Cifar-10 [17], Youtube-Face_sel [41] evaluate the effectiveness of \\( S^2 \\) MVTC, where the statistical information is presented in Tab. 2. Detailed descriptions can be found in ??"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.781,
+ 0.892,
+ 0.902
+ ],
+ "angle": 0,
+ "content": "Compared Clustering Algorithms. To compare clustering performance, we select \\( k \\)-means and seven state-of-the-art methods: binary multi-view clustering (BMVC) [2018, TP-MAI] [45], fast multi-View anchor-correspondence clustering (FMVACC) [2022, NeurIPS] [38], one-pass multi-view clustering (OPMC) [2021, ICCV] [23], fast multi-view clustering via ensembles (FastMICE) [2023, TKDE] [10], scalable and parameter-free multiview graph clustering"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.479,
+ 0.945,
+ 0.519,
+ 0.957
+ ],
+ "angle": 0,
+ "content": "26217"
+ }
+ ],
+ [
+ {
+ "type": "table",
+ "bbox": [
+ 0.082,
+ 0.089,
+ 0.477,
+ 0.171
+ ],
+ "angle": 0,
+ "content": "| Symbol | Definition |
| x, x, X, and X | A scalar, a vector, a matrix, and a tensor |
| v = 1,···, V | Indices range from 1 to their capital version |
| R, C | Fields of real numbers and complex numbers |
| conj(X) | the complex conjugate of X |
| [x] | the one greater than or equal to x |
| X ∈ R1×I2×I3 | A 3-rd order tensor |
| X(:, :, i3) | the i3-th frontal slice of X |
| N, V, C, M | Number of samples, views, clusters, anchors |
| φ(Xv) ∈ RM×N | Anchor graph in the v-th view |
| Uv ∈ RM×K | Projection matrix in the v-th view |
| Bv ∈ RK×N | embedding feature in the v-th view |
| B ∈ RK×V×N | The rotated embedding feature tensor |
+
+# 2.2. Preliminaries on t-SVD
+
+Definition 1 (T-product [27]) The T-product of two third-order tensors $\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times I_3}$ and $\mathcal{Y} \in \mathbb{R}^{I_2 \times J \times I_3}$ is defined as
+
+$$
+\mathcal {Z} = \mathcal {X} * \mathcal {Y} \in \mathbb {R} ^ {I _ {1} \times J \times I _ {3}}, \tag {1}
+$$
+
+which can be computed by multiple matrix multiplication operations in the Fourier transform domain.
+
+Definition 2 (Tensor transpose [15]) Under the Fourier transform, the conjugate transpose of a tensor $\mathcal{X} \in \mathbb{C}^{I_1 \times I_2 \times I_3}$ is denoted as $\mathcal{X}^{\mathrm{T}} \in \mathbb{C}^{I_2 \times I_1 \times I_3}$ . It satisfies
+
+$$
+\mathcal {X} ^ {\mathrm {T}} (:,: 1) = \left(\mathcal {X} (:,: 1)\right) ^ {\mathrm {T}}, \tag {2}
+$$
+
+and
+
+$$
+\mathcal {X} ^ {\mathrm {T}} (:,: i _ {3}) = \left(\mathcal {X} (:,: I _ {3} + 2 - i _ {3})\right) ^ {\top}, i _ {3} = 2, \dots , I _ {3}, \tag {3}
+$$
+
+where $\mathcal{X}^{\mathrm{T}}(:, :, i_3)$ is the $i_3$ -th frontal slice of $\mathcal{X}^{\mathrm{T}}$ .
+
+Definition 3 (Orthogonal tensor) [15] A tensor $\mathcal{X}$ is called orthogonal when it satisfies
+
+$$
+\mathcal {X} * \mathcal {X} ^ {\mathrm {T}} = \mathcal {X} ^ {\mathrm {T}} * \mathcal {X} = \mathcal {I}, \tag {4}
+$$
+
+where $\mathcal{I}$ is the identity tensor.
+
+Definition 4 (f-diagonal tensor) [16] A tensor $\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times I_3}$ is called f-diagonal when each frontal slice $\mathcal{X}^{(i_3)}, i_3 = 1, \dots, I_3$ is a diagonal matrix.
+
+Definition 5 (tensor Singular Value Decomposition(tSVD) [2, 15, 25]) For a tensor $\mathcal{X}\in \mathbb{R}^{I_1\times I_2\times I_3}$ , its t-SVD is represented as:
+
+$$
+\mathcal {X} = \mathcal {U} * \mathcal {S} * \mathcal {V} ^ {\mathrm {T}}, \tag {5}
+$$
+
+where $\mathcal{U}\in \mathbb{R}^{I_1\times I_1\times I_3}$ and $\mathcal{V}\in \mathbb{R}^{I_2\times I_2\times I_3}$ are orthogonal tensors, and $\mathcal{S}\in \mathbb{R}^{I_1\times I_2\times I_3}$ is an $f$ -diagonal tensor.
+
+
+(a)
+
+
+(b)
+Figure 2. The framework of $\mathbf{S}^2\mathrm{MVTC}$ . (a) The process involves mapping each pre-given anchor graph $\phi (\mathbf{X}_v)$ using $\mathbf{U}_v$ to obtain the corresponding embedding feature $\mathbf{B}_v$ . These embedding features, $\mathbf{B}_v$ , are then formed into a tensor $\mathcal{B}$ . This tensor, along with the newly defined $\mathcal{B}_{\mathrm{TLFA}}$ and $\sum_{v = 1}^{V}\| \mathbf{B}_v - \tilde{\mathbf{B}}\|_{\mathrm{F}}$ , are utilized to explore intra-view correlations and semantic consistency, respectively, where $\tilde{\mathbf{B}} = \sum_{v = 1}^{V}\frac{1}{V}\mathbf{B}_v$ . (b) The tensor low-frequency approximation $\mathcal{B}_{\mathrm{TLFA}}$ .
+
+Algorithm 1 t-SVD
+Input: $\mathcal{X}\in \mathbb{R}^{I_1\times I_2\times I_3}$
+Output: $\mathcal{U}\in \mathbb{R}^{I_1\times I_1\times I_3},\mathcal{S}\in \mathbb{R}^{I_1\times I_2\times I_3},\mathcal{V}\in$ $\mathbb{R}^{I_2\times I_2\times I_3}$ .
+ $\hat{\mathcal{X}}\gets \mathrm{fft}(\mathcal{X},[],3)$ $[\mathbf{U},\mathbf{S},\mathbf{V}] = \mathbf{svd}(\widehat{\mathcal{X}}(:, :, 1))$
+for $i_{3} = 1$ to $I_{3}$ do
+ $[\mathbf{U},\mathbf{S},\mathbf{V}] = \mathbf{svd}(\widehat{\mathcal{X}}(:, :, i_3));$ $\widehat{\mathcal{U}}(:, :, i_3) = \mathbf{U};\widehat{\mathcal{S}}(:, :, i_3) = \mathbf{S};\widehat{\mathcal{V}}(:, :, i_3) = \mathbf{V}.$
+end for
+ $\mathcal{U} = \mathrm{ifft}(\widehat{\mathcal{U}},[],3);\mathcal{S} = \mathrm{ifft}(\widehat{\mathcal{S}},[],3);\mathcal{V} = \mathrm{ifft}(\widehat{\mathcal{V}},[],3).$
+
+According to the T-product operation in Definition 1, this decomposition can be obtained by Algorithm 1, where fft and ifft are the fast Fourier transform (FFT) operator and the inverse FFT, respectively.
+
+# 2.3. Related works
+
+# 2.3.1 Anchor learning for large-scale MVC
+
+The general framework of an anchor-learning-based large-scale MVC is constructed as follows:
+
+$$
+\min _ {\left\{\mathbf {Z} _ {v} \right\} _ {v = 1} ^ {V}} \sum_ {v = 1} ^ {V} \left\| \mathbf {X} _ {v} - \mathbf {A} _ {v} \mathbf {Z} _ {v} \right\| _ {\mathrm {F}} ^ {2} + \lambda \Psi (\mathbf {Z} _ {v})
+$$
+
+$$
+\mathrm {s . t .} \mathbf {Z} _ {v} ^ {\mathrm {T}} \mathbf {1} = 1, \mathbf {Z} _ {v} \geq 0, v = 1, \dots , V, \tag {6}
+$$
+
+where $\mathbf{A}_v\in \mathbb{R}^{D_v\times M}$ is the learned anchor matrix in the $v$ -th view and $M$ is the number of anchors. $\mathbf{Z}_v\in \mathbb{R}^{M\times N}$ is the anchor graph, which is used to describe the relationship among samples. In addition, $\Psi (\mathbf{Z}_v)$ represents the regularization terms. Kang et al. [14] consider Frobenius norm
+
+on $\mathbf{Z}_v$ to find a stable solution for large-scale MVC. However, the anchor graph in each view is learned separately, failing to explore the complementary information. To solve this, many fusion strategies are considered to find the consensus representation among views. For example, Liu et al. [39] considered the respective projection matrix to find the consistent graph. Li et al. [21] considered the joint graph across multiple views via a self-supervised weighting manner. Chen et al. [5] introduced the Fast Self-guided Multi-view Subspace Clustering algorithm, which effectively combines view-shared anchor learning and global-guided-local self-guidance learning into a comprehensive model. Huang et al. [9] considered three levels of diversity: features, anchors, and neighbors. By leveraging these levels of diversity, view-sharing bipartite graphs are constructed, enhancing the effectiveness of the clustering process. In addition, some methods based on tensor anchor graphs have been proposed [12, 24, 42]. These methods stack each anchor graph into a tensor and use low-rank tensor approximation to integrate the consistency information among anchor graphs across views, achieving good performance on large-scale data.
+
+# 2.3.2 Embedding feature learning for large-scale MVC
+
+Unlike large-scale MVC methods based on anchor learning, another approach focuses on learning embedding features through the pre-defined anchor graphs in the pre-processing step. Typically, these anchor graphs are constructed using kernel mapping, for instance, through nonlinear Radial Basis Function (RBF) mapping, as follows:
+
+$$
+\phi \left(\mathbf {x} _ {v} ^ {n}\right) = \left[ \exp \left(- \frac {\left\| \mathbf {x} _ {v} ^ {n} - \mathbf {z} _ {v} ^ {1} \right\| ^ {2}}{\sigma}\right), \dots , \exp \left(- \frac {\left\| \mathbf {x} _ {v} ^ {n} - \mathbf {z} _ {v} ^ {M} \right\| ^ {2}}{\sigma}\right) \right] ^ {\mathrm {T}}, \tag {7}
+$$
+
+where $\sigma$ is the kernel width, $\mathbf{z}_v^m\in \mathbb{R}^{D_v}$ $m = 1,\dots ,M$ are anchors, which randomly chooses from the column of $v$ -th view $\mathbf{X}_v\in \mathbb{R}^{D_v\times N}$ $\phi (\mathbf{x}_v^n)\in \mathbb{R}^M$ represents the nonlinear relationship between $M$ anchors and the $n$ -th samples. In this way, Zhang et al. [45] projected non-linear anchor graphs from different views to a common Hamming feature space. This shared feature space is then fed into binary clustering structure learning to derive the final clustering results. However, this method solely explores pairwise correlations between views. To uncover global correlations among views, Zhang et al. [44] considered consistent projections, obtained by low-rank tensor approximation, to find a better binary feature representation for large-scale MVC tasks. Additionally, Wang et al. [35] considered autoencoder learning techniques to find consistent projections for generating binary features used in clustering.
+
+# 3. Methods
+
+As mentioned above, anchor-based methods mostly focus on learning high-order information between anchor graphs or projection matrices. However, their ultimate goal is to obtain shared embedding features across views.
+
+# 3.1. Model development
+
+In this section, we will focus on how to quickly learn the consistency among embedding features well, which involves the exploration of intra-view graph similarity (IGS) and ensuring inter-view semantic consistency (ISC). With anchor graphs $\phi (\mathbf{X}_v)\in \mathbb{R}^{M\times N},v = 1,\dots ,V$ obtained in advance, we can first learn $V$ embedding features by projection matrices $\mathbf{U}_v,v = 1,\dots ,V$ , as follows:
+
+$$
+\begin{array}{l} \min _ {\left\{\mathbf {U} _ {v}, \mathbf {B} _ {v} \right\} _ {v = 1} ^ {V}} \sum_ {v = 1} ^ {V} \frac {\lambda}{2} \left| \left| \mathbf {U} _ {v} \right| \right| _ {\mathrm {F}} ^ {2} + \frac {1}{2} \left| \left| \mathbf {B} _ {v} - \mathbf {U} _ {v} \phi (\mathbf {X} _ {v}) \right| \right| _ {\mathrm {F}} ^ {2} \tag {8} \\ \begin{array}{l} \text {s . t .} \mathbf {B} _ {v} (:, n) \in \mathbb {S}, n = 1, \dots , N, v = 1, \dots , V, \end{array} \\ \end{array}
+$$
+
+where $\lambda$ serves as a weight aiming to attain a stable solution for the mapping matrix $\mathbf{U}_v$ . $\mathbf{B}_v(:,n)\in \mathbb{S}$ represents the $z$ -score normalization constraint, ensuring that each feature from different views has a mean of 0 and a standard deviation of 1. Here $\mathbf{b}_v^n = \mathbf{B}_v(:,n)$ , and $\mathbb{S} = \{\mathbf{b}_v^n:\sum_{k = 1}^K\mathbf{b}_v^n (k) = 0,\frac{\sum_{k = 1}^K(\mathbf{b}_v^n(k))^2}{K - 1} = 1,n = 1,\dots ,N\}$ .
+
+To efficiently explore the intra-view information of embedding features, we initially stack the embedding features from each view into a tensor. Subsequently, we introduce a newly tensor low-frequency approximation $\| \mathcal{B}\|_{\mathrm{TLFA}}$ , applied to its rotated form $\mathcal{B} \in \mathbb{R}^{K\times V\times N}$ , aiming to capture the IGS.
+
+Definition 6 (Tensor low-frequency approximation (TLFA)) According to the $t$ -SVD computation outlined in Algorithm 1, the TLFA of a tensor $\mathcal{B} \in \mathbb{R}^{K \times V \times N}$ is
+
+defined as the frontal slices in the low-frequency domain. Mathematically, it is expressed as:
+
+$$
+\min _ {\mathcal {Y}} \frac {1}{2} \| \mathcal {B} - \mathcal {Y} \| _ {\mathrm {F}} ^ {2}, s. t. \mathcal {Y} = \operatorname {i f t} (\widehat {\mathcal {Y}}, [, 3),
+$$
+
+where $\widehat{\mathcal{Y}} = \widehat{\mathcal{B}}(:, :, 1) + \sum_{n=2}^{L} (\widehat{\mathcal{B}}(:, :, n) + \widehat{\mathcal{B}}(:, :, N+2-n))$ . $\widehat{\mathcal{B}} = \mathrm{fft}(\mathcal{B}, [], 3)$ , means applying the fast Fourier transform along the sample (3-rd) dimension for the embedding features from different views. The obtained low-frequency components incorporate graph similarity into embedding feature learning, resulting in a smooth representation.
+
+Besides, the term $\sum_{v=1}^{V} \|\tilde{\mathbf{B}} - \mathbf{B}_v\|_F^2$ is included to ensure ISC [40], where $\tilde{\mathbf{B}}$ is the average embedding feature.
+
+Overall, the framework of $\mathbf{S}^2\mathrm{MVTC}$ is formulated as:
+
+$$
+\begin{array}{l} \min _ {\left\{\mathbf {B} _ {v}, \mathbf {U} _ {v} \right\} _ {v = 1} ^ {V}, \tilde {\mathbf {B}}} \sum_ {v = 1} ^ {v} \left(\frac {\lambda}{2} | | \mathbf {U} _ {v} | | _ {\mathrm {F}} ^ {2} + \frac {1}{2} | | \mathbf {B} _ {v} - \mathbf {U} _ {v} \phi (\mathbf {X} _ {v}) | | _ {\mathrm {F}} ^ {2} \right. \\ + \frac {\beta}{2} \left\| \tilde {\mathbf {B}} - \mathbf {B} _ {v} \right\| _ {\mathrm {F}} ^ {2}) + \tau \| \mathcal {B} \| _ {\mathrm {T L F A}} \\ \end{array}
+$$
+
+$$
+\text {s . t .} \tilde {\mathbf {B}} (:, n), \mathbf {B} _ {v} (:, n) \in \mathbb {S}, n = 1, \dots , N, v = 1, \dots , V. \tag {9}
+$$
+
+Here $\mathcal{B} = \Omega (\mathbf{B}_1,\dots ,\mathbf{B}_V)$ , where $\Omega$ represents the operation of stacking each feature and rotating and the inverse operator $\mathbf{B}_v = \Omega_v^{-1}(\mathcal{B})$ .
+
+To make the above optimization problem separable, we introduce an auxiliary variable $\mathcal{V}$ , leading to the reformulation as follows:
+
+$$
+\begin{array}{l} \min _ {\left\{\mathbf {B} _ {v}, \mathbf {U} _ {v} \right\} _ {v = 1} ^ {V}, \tilde {\mathbf {B}}} \sum_ {v = 1} ^ {v} \left(\frac {\lambda}{2} | | \mathbf {U} _ {v} | | _ {\mathrm {F}} ^ {2} + \frac {1}{2} | | \mathbf {B} _ {v} - \mathbf {U} _ {v} \phi (\mathbf {X} _ {v}) | | _ {\mathrm {F}} ^ {2} \right. \\ + \frac {\beta}{2} | | \tilde {\mathbf {B}} - \mathbf {B} _ {v} | | _ {\mathrm {F}} ^ {2} + \frac {1}{2} | | \mathcal {B} - \mathcal {Y} | | _ {F} ^ {2} + \tau | | \mathcal {Y} | | _ {\mathrm {T L F A}} \\ \end{array}
+$$
+
+$$
+\text {s . t .} \tilde {\mathbf {B}} (:, n), \mathbf {B} _ {v} (:, n) \in \mathbb {S}, n = 1, \dots , N, v = 1, \dots , V. \tag {10}
+$$
+
+# 3.2. Solutions
+
+This optimal problem in Eq. (10) can be solved through an alternating optimization approach, where each parameter is updated individually while keeping the others fixed.
+
+Update $\mathbf{U}_v$ . Fixing other variables, the subproblem of $\mathbf{U}_v$ can be rewritten as:
+
+$$
+\min _ {\mathbf {U} _ {v}} \frac {1}{2} \| \mathbf {B} _ {v} - \mathbf {U} _ {v} \phi (\mathbf {X} _ {v}) \| _ {\mathrm {F}} ^ {2} + \frac {\lambda}{2} \| \mathbf {U} _ {v} \| _ {\mathrm {F}} ^ {2}. \tag {11}
+$$
+
+By taking the derivative of the equation and setting it equal to zero, we can obtain:
+
+$$
+\mathbf {U} _ {v} = \left(\mathbf {B} _ {v} \phi^ {\mathrm {T}} \left(\mathbf {X} _ {v}\right)\right) \left(\phi \left(\mathbf {X} _ {v}\right) \phi^ {\mathrm {T}} \left(\mathbf {X} _ {v}\right) + \lambda \mathbf {I}\right) ^ {\dagger}, \tag {12}
+$$
+
+where $\phi (\mathbf{X}_v)\phi^{\mathrm{T}}(\mathbf{X}_v)$ can be pre-calculated outside the main loop to decrease the computational cost. Therefore, the storage and computational complexity are $O(MN)$ and
+
+$O(\max (KMN,M^3))$ , respectively.
+
+Update $\mathbf{B}_v$ . The subproblem of $\mathbf{B}_v$ can be rephrased as:
+
+$$
+\min _ {\mathbf {B} _ {v}} \frac {1}{2} \| \mathbf {B} _ {v} - \mathbf {U} _ {v} \phi (x _ {v}) \| _ {F} ^ {2} + \frac {\beta}{2} \| \tilde {\mathbf {B}} - \mathbf {B} _ {v} \| _ {F} ^ {2} + \frac {1}{2} \| \mathcal {B} - \mathcal {Y} \| _ {F} ^ {2},
+$$
+
+$$
+\begin{array}{l} \text {s . t .} \mathbf {B} _ {v} (:, n) \in \mathbb {S}, n = 1, \dots , N. \end{array}
+$$
+
+(13)
+
+Similarly, by taking the derivative of the equation and setting it equal to zero, we can obtain
+
+$$
+\mathbf {B} _ {v} = \text {n o r m a l i z e} \left(\beta \tilde {\mathbf {B}} + \Omega_ {v} ^ {- 1} (\mathcal {Y}) + \mathbf {U} _ {v} \phi \left(\mathbf {x} _ {v}\right)\right) / (\beta + 2), \tag {14}
+$$
+
+where $\Omega_v^{-1}(\mathcal{Y}) = \mathbf{Y}_v \in \mathbb{R}^{K \times N}$ . normalize is the $z$ -score normalization function, which can be achieved by the Matlab command "normalize". In this case, both the storage and computational complexity of updating $\mathbf{B}_v$ are $O(KN)$ . Update $\mathcal{Y}$ . The subproblem of $\mathcal{Y}$ is rewritten as:
+
+$$
+\min _ {\mathcal {Y}} \tau \| \mathcal {Y} \| _ {\mathrm {T L F A}} + \frac {1}{2} \| \mathcal {B} - \mathcal {Y} \| _ {\mathrm {F}} ^ {2}. \tag {15}
+$$
+
+According to the definition of $\| \mathcal{V}\|_{\mathrm{TLFA}}$ , a closed-form solution for $\mathcal{V}$ can be obtained by choosing the low-frequency component in the Fourier domain. The details are outlined in Algorithm 2. The computational and storage complexity are $O(KVN\log (N))$ and $O(KNV)$ , respectively.
+
+Update $\bar{\mathbf{B}}$ . The subproblem of updating $\bar{\mathbf{B}}$ is:
+
+$$
+\min _ {\tilde {\mathbf {B}}} \sum_ {v = 1} ^ {V} \frac {\beta}{2} \| \tilde {\mathbf {B}} - \mathbf {B} _ {v} \| _ {\mathrm {F}} ^ {2} \text {s . t .} \tilde {\mathbf {B}} (:, n) \in \mathbb {S}, n = 1, \dots , N. \tag {16}
+$$
+
+The solution of $\tilde{\mathbf{B}}$ is:
+
+$$
+\tilde {\mathbf {B}} = \text {n o r m a l i z e} \left(\frac {1}{V} \sum_ {v = 1} ^ {V} \mathbf {B} _ {v}\right), \tag {17}
+$$
+
+which needs computational and storage complexity of $O(KN)$ and $O(KNV)$ , respectively.
+
+After obtaining the consistent fused embedding feature $\tilde{\mathbf{B}}$ . The final clustering results can be obtained by the following clustering structure learning framework:
+
+$$
+\min _ {\mathbf {D}, \mathbf {G}} \| \tilde {\mathbf {B}} - \mathbf {D} \mathbf {G} \| _ {\mathrm {F}} ^ {2} \text {s . t .} \sum_ {c = 1} ^ {C} \mathbf {G} (c,:) = \mathbf {1}, \mathbf {G} \in \{0, 1 \}, \tag {18}
+$$
+
+where $\mathbf{D}$ is the clustering center and $\mathbf{G}$ is the clustering indicator matrix.
+
+Overall, the method $\mathbf{S}^2\mathrm{MVTC}$ is summarized in Algorithm 3. According to the parameters analysis in the algorithm, the main storage and computational complexities of $\mathbf{S}^2\mathrm{MVTC}$ are $O(\max(KVN, MN))$ and $O(\max(KMNT, M^3T))$ , respectively, where $T$ represents the total number of iterations in the $\mathbf{S}^2\mathrm{MVTC}$ .
+
+In addition, the theoretical convergence can be well guaranteed. Our problem is bounded due to the summation of
+
+Algorithm 2 Updating Y
+
+Input: B, low frequency parameter $L$ .
+
+Output: $y$
+
+$$
+\widehat {\mathcal {B}} \leftarrow \operatorname {f f t} (\mathcal {B}, [, 3);
+$$
+
+$$
+\hat {\mathcal {Y}} (:,: 1)) = \hat {\mathcal {B}} (:,: 1));
+$$
+
+for $n = 2$ to $L$ do
+
+$$
+\widehat {\mathcal {Y}} (:,: n) = \widehat {\mathcal {B}} (:,: n);
+$$
+
+$$
+\widehat {\mathcal {Y}} (:,: N + 2 - n) = \operatorname {c o n j} (\widehat {\mathcal {Y}} (:,: n)).
+$$
+
+end for
+
+$$
+\mathcal {Y} = \operatorname {i f t} (\widehat {\mathcal {Y}}, [, 3).
+$$
+
+Algorithm 3 S²MVTC
+
+Input: Anchor graphs $\phi (\mathbf{X}_v), v = 1,\dots ,V$ , low frequency parameter $L$ , regularization parameters $\beta, \lambda$ .
+
+Initialize: $\tau = 1$ , Maximum iterations $T = 7$
+
+for $t = 1$ to $T$ do
+
+for $v = 1$ to $V$ do
+
+Update $\mathbf{U}_v$ via Eq. (12);
+
+Update $\mathbf{B}_v$ via Eq. (14);
+
+end for
+
+Update $\mathcal{V}$ via Eq. (15);
+
+Update $\tilde{\mathbf{B}}$ via Eq. (17);
+
+$$
+\tau = \tau * 1. 1.
+$$
+
+end for
+
+Apply Eq. (18) to find the clustering indicator matrix $\mathbf{G}$ .
+
+Output: Clustering result.
+
+norms with positive penalty parameters. Furthermore, the exact minimum points of each subproblem can be achieved, implying that each subproblem exhibits a monotone decrease. The proposed algorithm converges according to the convergence theorem in [28] (theorem 7.29).
+
+# 4. Experiments
+
+# 4.1. Experimental settings
+
+Multi-view Datasets: Six well-known large-scale multi-view datasets, including CCV [13], NUS-WIDE-OBJ [7], Caltech102 [19], AwA [18], Cifar-10 [17], Youtube-Face_sel [41] evaluate the effectiveness of $S^2$ MVTC, where the statistical information is presented in Tab. 2. Detailed descriptions can be found in ??
+
+Compared Clustering Algorithms. To compare clustering performance, we select $k$ -means and seven state-of-the-art methods: binary multi-view clustering (BMVC) [2018, TP-MAI] [45], fast multi-View anchor-correspondence clustering (FMVACC) [2022, NeurIPS] [38], one-pass multi-view clustering (OPMC) [2021, ICCV] [23], fast multi-view clustering via ensembles (FastMICE) [2023, TKDE] [10], scalable and parameter-free multiview graph clustering
+
+| Symbol | Definition |
| x, x, X, and X | A scalar, a vector, a matrix, and a tensor |
| v = 1,···, V | Indices range from 1 to their capital version |
| R, C | Fields of real numbers and complex numbers |
| conj(X) | the complex conjugate of X |
| [x] | the one greater than or equal to x |
| X ∈ R1×I2×I3 | A 3-rd order tensor |
| X(:, :, i3) | the i3-th frontal slice of X |
| N, V, C, M | Number of samples, views, clusters, anchors |
| φ(Xv) ∈ RM×N | Anchor graph in the v-th view |
| Uv ∈ RM×K | Projection matrix in the v-th view |
| Bv ∈ RK×N | embedding feature in the v-th view |
| B ∈ RK×V×N | The rotated embedding feature tensor |
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+ "content": "\\hat{\\mathcal{X}}\\gets \\mathrm{fft}(\\mathcal{X},[],3)"
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+ "content": "I_{3}"
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+ "type": "text",
+ "content": " do \n"
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+ "content": "[\\mathbf{U},\\mathbf{S},\\mathbf{V}] = \\mathbf{svd}(\\widehat{\\mathcal{X}}(:, :, i_3));"
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+ "content": "\\widehat{\\mathcal{U}}(:, :, i_3) = \\mathbf{U};\\widehat{\\mathcal{S}}(:, :, i_3) = \\mathbf{S};\\widehat{\\mathcal{V}}(:, :, i_3) = \\mathbf{V}."
+ },
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+ "type": "text",
+ "content": " \nend for \n"
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+ "type": "inline_equation",
+ "content": "\\mathcal{U} = \\mathrm{ifft}(\\widehat{\\mathcal{U}},[],3);\\mathcal{S} = \\mathrm{ifft}(\\widehat{\\mathcal{S}},[],3);\\mathcal{V} = \\mathrm{ifft}(\\widehat{\\mathcal{V}},[],3)."
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+ "content": "According to the T-product operation in Definition 1, this decomposition can be obtained by Algorithm 1, where fft and ifft are the fast Fourier transform (FFT) operator and the inverse FFT, respectively."
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+ "content": " to find a stable solution for large-scale MVC. However, the anchor graph in each view is learned separately, failing to explore the complementary information. To solve this, many fusion strategies are considered to find the consensus representation among views. For example, Liu et al. [39] considered the respective projection matrix to find the consistent graph. Li et al. [21] considered the joint graph across multiple views via a self-supervised weighting manner. Chen et al. [5] introduced the Fast Self-guided Multi-view Subspace Clustering algorithm, which effectively combines view-shared anchor learning and global-guided-local self-guidance learning into a comprehensive model. Huang et al. [9] considered three levels of diversity: features, anchors, and neighbors. By leveraging these levels of diversity, view-sharing bipartite graphs are constructed, enhancing the effectiveness of the clustering process. In addition, some methods based on tensor anchor graphs have been proposed [12, 24, 42]. These methods stack each anchor graph into a tensor and use low-rank tensor approximation to integrate the consistency information among anchor graphs across views, achieving good performance on large-scale data."
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+ "content": " represents the operation of stacking each feature and rotating and the inverse operator "
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+ "content": ", leading to the reformulation as follows:"
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+ "content": "This optimal problem in Eq. (10) can be solved through an alternating optimization approach, where each parameter is updated individually while keeping the others fixed."
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+ "content": " can be rewritten as:"
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+ "content": "In addition, the theoretical convergence can be well guaranteed. Our problem is bounded due to the summation of"
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+ "content": "L"
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+ "type": "text",
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+ }
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+ }
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+ "bbox": [
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+ "type": "inline_equation",
+ "content": "y"
+ }
+ ]
+ }
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+ "type": "interline_equation",
+ "content": "\\widehat {\\mathcal {B}} \\leftarrow \\operatorname {f f t} (\\mathcal {B}, [, 3);",
+ "image_path": "42353cd941822366c2354027abf2f824b7c7394e9c2f8014afe17f20dd9f62d5.jpg"
+ }
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+ "type": "interline_equation",
+ "content": "\\hat {\\mathcal {Y}} (:,: 1)) = \\hat {\\mathcal {B}} (:,: 1));",
+ "image_path": "d779f825b358ed445967d32ddf05e2b6ad0297abd12f1b8446488d244726dc3e.jpg"
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+ "content": "\\widehat {\\mathcal {Y}} (:,: n) = \\widehat {\\mathcal {B}} (:,: n);",
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+ "content": "\\mathcal {Y} = \\operatorname {i f t} (\\widehat {\\mathcal {Y}}, [, 3).",
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+ "content": "Algorithm 3 S²MVTC"
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+ "content": "\\phi (\\mathbf{X}_v), v = 1,\\dots ,V"
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+ "type": "text",
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+ "type": "text",
+ "content": ", regularization parameters "
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+ "type": "inline_equation",
+ "content": "\\beta, \\lambda"
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+ "type": "text",
+ "content": "."
+ }
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+ }
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+ "content": "Initialize: "
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+ "content": "\\tau = 1"
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+ "type": "text",
+ "content": ", Maximum iterations "
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+ "type": "inline_equation",
+ "content": "T = 7"
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+ "content": "t = 1"
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+ "type": "text",
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+ "type": "inline_equation",
+ "content": "\\mathbf{U}_v"
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+ "type": "text",
+ "content": " via Eq. (12);"
+ }
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+ }
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+ "type": "text",
+ "content": " via Eq. (14);"
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+ }
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+ {
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+ "type": "inline_equation",
+ "content": "\\mathcal{V}"
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+ {
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+ "type": "text",
+ "content": " via Eq. (15);"
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+ "index": 38
+ },
+ {
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+ "content": "Update "
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+ "type": "inline_equation",
+ "content": "\\tilde{\\mathbf{B}}"
+ },
+ {
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+ "type": "text",
+ "content": " via Eq. (17);"
+ }
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+ }
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+ "type": "interline_equation",
+ "content": "\\tau = \\tau * 1. 1.",
+ "image_path": "4de5d10555ef6daab313c1b9e84a351d87f0aba92f92958ce8d2df365538d01f.jpg"
+ }
+ ]
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+ "type": "text",
+ "content": "end for"
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+ "type": "text",
+ "content": "Apply Eq. (18) to find the clustering indicator matrix "
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+ "type": "inline_equation",
+ "content": "\\mathbf{G}"
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+ "type": "text",
+ "content": "."
+ }
+ ]
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+ {
+ "bbox": [
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+ "type": "text",
+ "content": "Output: Clustering result."
+ }
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+ }
+ ],
+ "index": 43
+ },
+ {
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+ "spans": [
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+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "norms with positive penalty parameters. Furthermore, the exact minimum points of each subproblem can be achieved, implying that each subproblem exhibits a monotone decrease. The proposed algorithm converges according to the convergence theorem in [28] (theorem 7.29)."
+ }
+ ]
+ }
+ ],
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+ "type": "text",
+ "content": "4. Experiments"
+ }
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+ }
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+ "type": "text",
+ "content": "4.1. Experimental settings"
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+ "type": "text",
+ "content": "Multi-view Datasets: Six well-known large-scale multi-view datasets, including CCV [13], NUS-WIDE-OBJ [7], Caltech102 [19], AwA [18], Cifar-10 [17], Youtube-Face_sel [41] evaluate the effectiveness of "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "S^2"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " MVTC, where the statistical information is presented in Tab. 2. Detailed descriptions can be found in ??"
+ }
+ ]
+ }
+ ],
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