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CVPR/2025/A Theory of Learning Unified Model via Knowledge Integration from Label Space Varying Domains/full.md

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+# A Theory of Learning Unified Model via Knowledge Integration from Label Space Varying Domains
+
+Dexuan Zhang<sup>1</sup> Thomas Westfechtel<sup>1</sup> Tatsuya Harada<sup>1,2</sup>  
+<sup>1</sup>The University of Tokyo, 2RIKEN
+
+{dexuan.zhang, thomas, harada}@mi.t.u-tokyo.ac.jp
+
+# Abstract
+
+Existing domain adaptation systems can hardly be applied to real-world problems with new classes presenting at deployment time, especially regarding source-free scenarios where multiple source domains do not share the label space despite being given a few labeled target data. To address this, we consider a challenging problem: multi-source semi-supervised open-set domain adaptation and propose a learning theory via joint error, effectively tackling strong domain shift. To generalize the algorithm into source-free cases, we introduce a computationally efficient and architecture-flexible attention-based feature generation module. Extensive experiments on various data sets demonstrate the significant improvement of our proposed algorithm over baselines.
+
+# 1. Introduction
+
+Generally, a supervised learning algorithm trained on a particular distribution of labeled samples (source domain) often fails to generalize when deployed on a new environment (target domain) in the presence of domain shift. In this regard, Domain adaptation (DA) [1] algorithms address the domain-shift problem by aligning the data distributions of the source and target domains through learning a domain-invariant feature space using statistical or adversarial learning approaches, which have made remarkable success. However, the problem setting still needs to be relaxed for real-world applications when we aim to integrate the knowledge learned from multiple source domains. Current DA methods can hardly cover the case for varying label space across domains, and the corresponding learning theory has yet to be proposed. Moreover, the solution is limited if this heterogeneous domain setting is extended to source-free situations where each source model may be trained on different network architectures and the label space. This work tackles a challenging multi-source semi-supervised open-set domain adaptation paradigm with varying label space, illustrated in Fig. 1.
+
+In general, multi-source DA (MSDA) [45] and semi-supervised DA (SSDA) [41] are regarded as more practical than the single-source DA setup, considering that labeled data may come from various domains. More precisely, in these cases, the labeled samples can be differently distributed among themselves in addition to the usual domain shit between the source and the target domains. One naive approach to MSDA and SSDA is to group all labeled data into a single domain and deploy any unsupervised DA (UDA) method. However, such a trivial solution may lead to sub-optimal classification due to the gaps among labeled data [41].
+
+Most DA techniques assume the same label space in source and target domains, usually called the closed-set setting. The paradigm of closed-set DA has been substantially explored in the literature for UDA [11, 32, 33, 39, 47, 62, 64], SSDA [23, 37, 41, 43, 59, 61], and MSDA [19, 36, 51, 53, 57, 65, 67]. In contrast, the open-set DA (OSDA) [4] setting allows the presence of target-specific classes in addition to the shared classes. Such an open-set arrangement is more challenging due to a huge label shift across domains. The closed-set DA techniques cannot be directly applied in this case since these target-specific open-set samples, in turn, may jeopardize the domain alignment process. This work formalizes a more generalized problem where source domains may not share the label space, and the unlabeled target domain additionally contains novel classes.
+
+Motivated by these, we consider a learning scenario in this work as the multi-source semi-supervised open-set DA (MSODA) where each source domain has a diverse label space, the labeled target domain consists of a few-shot of target data whose label space, i.e., the known class, is a superset of any source label space, and the unlabeled target domain contains data either from the known or a combined unknown class. Under this setup, the task is to classify the unlabeled data either into one of the known categories or a common unknown class. Such a setup invariably holds huge applications in fields relating to real-world visual perception like medical imaging and remote sensing, where acquisition of multi-domain data is feasible, and novel categories may show up abruptly [4]. Nonetheless, this MSODA problem
+
+cannot be effectively solved by directly utilizing the single source open-set paradigm of [2, 10, 20, 27, 34, 40, 54, 63] mainly because of the following factors: i) the varying label space of multiple source domains becomes an obstacle to the traditional OSDA techniques, and ii) the unknown recognition can be non-trivial since the target domain may be related to each source domain in a different degree. Regarding multi-source models, recent works [25, 55] consider various label spaces but require largely shared common classes for all domains to align the features, where we accept source domains with zero overlap.
+
+[36] argued that reducing the domain gap among the source domains leads to a more robust and effective MSDA model. This idea is particularly relevant to our problem setting since aligning the source domains among themselves inherently helps distinguish unknown from known categories in the target domain. Otherwise, the domain shift among the source domains may lead to an unstable alignment of unlabeled target data. Inspired by this idea, we combine the theoretical results from [62, 63] to build a learning theory that can align all source domains with the labeled target domain via joint error, which is crucial to dealing with label shift [66]. Then we introduce PU learning [58] to detect unknowns with an end-to-end algorithm such that the generalization error is guaranteed, unlike those methods applying closed-set DA after unknown separation. Our major contributions can be summarized as:
+
+- We introduce a challenging problem setting of multi-source semi-supervised open-set DA with varying label space and propose a learning theory via joint error.  
+- We design a framework to generate labeled source features via an attention-based mechanism for source-free cases.  
+- We demonstrate the efficacy of our proposal through extensive experiments on two benchmark datasets where we perform thorough robustness analysis.
+
+![](images/9f82b9808ce1520db69fff4acb45a3db7e5889bd39e3c49c8ed40582377becd7.jpg)  
+Figure 1. Knowledge integration from heterogeneous domains can be considered a task for multi-source semi-supervised open-set domain adaptation. Given a few labeled target data as the key, we aim to build a unified target model from multiple source domains with varying label space, which can be applied to query data containing unknown categories.
+
+# 2. Learning Theory of Unified Model
+
+In this section, we present the theory that transfers the knowledge from multiple source domains to the target domain given a few labeled target data under the open-set situation. First, we propose a target error bound via joint error based on the theoretical results from [62, 63]. Then, we derive the generalization error of the proposed learning theory based on the generalized Vapnik-Chervonenkis (VC) complexity [49, 50] of real-valued function space. Finally, we proceed with the empirical objective function as an upper bound of a trivial convex combination with the log-sum-exp trick, which leads to a smoother optimization process.
+
+We consider the unified model (UM) as a solution to multi-source semi-supervised open-set domain adaptation (MSODA) tasks, where the learning algorithm has access to multiple source domains that may have different label spaces. A set of  $n_i$  ( $i = 1,..,N$ ) labeled points  $\{(x_{s_i}^j, y_{s_i}^j) \in (\mathcal{X} \subseteq \mathbb{R}^D \times \mathcal{Y}_i' \subseteq \mathcal{Y}')\}_{j=1}^{n_i}$  sampled i.i.d. from each source domain  $S_i'$ . In addition, a set of  $l$  labeled points (few-shot)  $\{(x_v^j, y_v^j) \in (\mathcal{X} \subseteq \mathbb{R}^D \times \mathcal{Y}' = \{1, ..., K-1\})\}_{j=1}^l$  sampled i.i.d. from the labeled target domain  $V'$  is available during learning. We seek a hypothesis that can classify a set of  $m$  unlabeled points  $\{(x_t^j) \in X \subseteq \mathbb{R}^D\}_{j=1}^m$  sampled i.i.d. from target domain  $T$  where  $\mathcal{Y} = \{1,..,K\}$  containing unknown class  $K$ . Let  $\mathcal{K} = \{k | k \in \mathbb{R}^K : \sum_{y \in \mathcal{Y}} k[y] = 1, k[y] \in [0,1]\}$  denotes output space and  $S_i, V$  indicate the complete domains with label space  $\mathcal{Y}$ .
+
+Theorem 2.1 (Target Error Bound for MSODA via Joint Error<sup>1</sup>). Given source  $(S_i)$ , labeled  $(V)$  and unlabeled target  $(T)$  domains that contain data from the unknown class, let  $f_{S_i}, f_V, f_T: \mathcal{X} \to \mathcal{K}$  be the true labeling functions of  $S_i, V, T$  respectively whose outputs are one-hot vectors denoting the corresponding classes of inputs. Let  $\epsilon: \mathcal{K} \times \mathcal{K} \to \mathbb{R}$  denote a distance metric and  $\epsilon_D(f, f') := \mathbb{E}_{x \sim D} \epsilon(f(x), f'(x))$  measure the expected disagreement between the outputs of  $f, f': \mathcal{X} \to \mathcal{K}$  over a distribution  $D$  on  $\mathcal{X}$ . Regarding the source error of a hypothesis  $h \in \mathcal{H}: \mathcal{X} \to \mathcal{K}$  where  $h(x)[y]$  indicates the probability of  $x \in \mathcal{X}$  labeled as  $y \in \mathcal{Y}$ , we use the shorthand  $\epsilon_{S_i}(h) := \epsilon_{S_i}(h, f_{S_i})$ . Similarly, we use  $\epsilon_V(h), \epsilon_T(h)$  to denote the labeled and unlabeled target error. For  $\forall h, f_{S_i}', f_V', f_T' \in \mathcal{H}: \mathcal{X} \to \mathcal{K}$ , the expected target error is bounded,
+
+$$
+\begin{array}{l} 2 \epsilon_ {T} (h) \leq \epsilon_ {V} (h) + \sum_ {i = 1} ^ {N} \alpha_ {i} U _ {i} (h), \quad s. t. \quad \sum_ {i = 1} ^ {N} \alpha_ {i} = 1 \\ = \epsilon_ {V} (h) + \sum_ {i = 1} ^ {N} \alpha_ {i} \left[ \epsilon_ {S _ {i}} (h) + 2 D _ {S _ {i}, V, T} \left(f _ {S _ {i}} ^ {*}, f _ {V} ^ {*}, f _ {T} ^ {*}, h\right) + 2 \theta_ {i} \right], \tag {1} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} 2 D _ {S _ {i}, V, T} (f _ {S _ {i}} ^ {*}, f _ {V} ^ {*}, f _ {T} ^ {*}, h) \\ = \epsilon_ {T} (f _ {S _ {i}} ^ {*}, f _ {T} ^ {*}) + \epsilon_ {T} (f _ {V} ^ {*}, f _ {T} ^ {*}) + \epsilon_ {T} (h, f _ {S _ {i}} ^ {*}) + \epsilon_ {T} (h, f _ {V} ^ {*}) \\ + \epsilon_ {V} (f _ {S _ {i}} ^ {*}, f _ {V} ^ {*}) + \epsilon_ {S _ {i}} (f _ {V} ^ {*}, f _ {S _ {i}} ^ {*}) - \epsilon_ {V} (h, f _ {S _ {i}} ^ {*}) - \epsilon_ {S _ {i}} (h, f _ {V} ^ {*}) \tag {2} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \theta_ {i} = \underbrace {\epsilon_ {S _ {i}} (f _ {S _ {i}} , f _ {S _ {i}} ^ {*}) / 2 + \epsilon_ {V} (f _ {S _ {i}} , f _ {S _ {i}} ^ {*}) + \epsilon_ {T} (f _ {S _ {i}} , f _ {S _ {i}} ^ {*})} _ {\theta_ {S} ^ {i}} \\ + \underbrace {\epsilon_ {V} (f _ {V} , f _ {V} ^ {*}) / 2 + \epsilon_ {S _ {i}} (f _ {V} , f _ {V} ^ {*}) + \epsilon_ {T} (f _ {V} , f _ {V} ^ {*})} _ {\theta_ {V} ^ {i}} + \underbrace {\epsilon_ {T} (f _ {T} , f _ {T} ^ {*})} _ {\theta_ {T} ^ {i}} \quad (3) \\ \end{array}
+$$
+
+In the following, we discuss the approach to obtain generalization guarantees for multiple source domain adaptation in classification settings by a trivial union-bound argument.
+
+Assumption 2.2 (Substitutes for True Labeling Functions). For finite training data  $\{\hat{S}_i\}_{i = 1}^N,\hat{T},\hat{V}$  , we assume there exist approximated labeling functions  $\{f_{S_i}^*\}_{i = 1}^N,f_T^*,f_V^*$  that can lead the empirical deviation  $\sum_{i}\alpha_{i}\hat{\theta}_{i}$  very close to zero such that it can be ignored during the practical learning process.
+
+Theorem 2.3 (Generalization Error $^1$ ). Let  $\hat{S}_i, \hat{V}, \hat{T}$  denote the empirical distributions generated with  $m$  i.i.d. samples from each domain. Let  $\mathcal{F} = \{f(x) = \epsilon(h(x), h'(x)): \mathcal{X} \to [0, M] | h, h' \in \mathcal{H}\}$  be a function space with complexity measured by uniform covering number  $\mathcal{N}_1(\xi, \mathcal{F}, m)$ . Let  $\alpha_i = \frac{\exp(\nu \hat{U}_i(h))}{\sum_j \exp(\nu \hat{U}_j(h))}$ ,  $\nu > 0$ , given Jensen's & Cauchy's inequality and Assumption 2.2, there exist  $f_{S_i}^* \in \mathcal{H}_{S_i} \subseteq \mathcal{H}$ ,  $f_V^* \in \mathcal{H}_V \subseteq \mathcal{H}$ ,  $f_T^* \in \mathcal{H}_T \subseteq \mathcal{H}$ , such that for  $0 < \delta < 1$ , with probability at least  $1 - \delta$ , for  $\forall h \in \mathcal{H}$ :
+
+$$
+\begin{array}{l} \epsilon_ {T} (h) \leq \frac {1}{2} [ \underbrace {\epsilon_ {\hat {V}} (h)} _ {L _ {c l s} ^ {V} (h)} + \frac {1}{\nu} \log \sum_ {i = 1} ^ {N} \exp (\nu \hat {U} _ {i} (h)) ] \\ + \mathcal {O} \left(\inf  _ {\sqrt {\frac {2}{m}} \leq \gamma \leq M} (\gamma + \int_ {\gamma} ^ {M} \sqrt {\frac {1}{m} \log \frac {2 (1 1 N + 6) \mathcal {N} _ {1} (\frac {\xi}{8} , \mathcal {F} , 2 m)}{\delta}} d \xi)\right) \tag {4} \\ \end{array}
+$$
+
+$$
+\hat {U} _ {i} (h) = \underbrace {\epsilon_ {\hat {S} _ {i}} (h)} _ {L _ {c l s} ^ {S _ {i}} (h)} + 2 \underbrace {D _ {\hat {S} _ {i} , \hat {V} , \hat {T}} \left(f _ {S _ {i}} ^ {*}, f _ {V} ^ {*}, f _ {T} ^ {*}, h\right)} _ {L _ {d i s} ^ {i} \left(f _ {S _ {i}} ^ {*}, f _ {V} ^ {*}, f _ {T} ^ {*}, h\right)} \tag {5}
+$$
+
+The log-sum-exp trick [30] yields an upper bound of the convex combination as Theorem 2.3, where we no longer need to heuristically decide the value of  $\alpha_{i}$  in the unified model. It smooths the objective and provides a principled and adaptive way to combine all the gradients from the  $N$  source domains. This often leads to better generalizations in practice because of the ensemble effect of multiple sources implied by the upper bound [65].
+
+According to [13, 56, 63], for  $\forall f, f': \mathcal{X} \to \mathcal{K}, \epsilon_V(f, f')$  can be approximated by the expectation on  $V'$  based on PU learning. Moreover, source data  $S_i'$  may be unavailable due to privacy concerns (e.g., medical data) during the adaptation phase. To tackle this source-free domain adaptation (SFDA) problem, we propose a source features generation pipeline based on the attention mechanism, which can transfer the knowledge between models with different architectures.
+
+# 3. Methodology
+
+In this section, we first recall several preliminaries crucial to the learning algorithm of open-set domain adaptation. Then, we propose a pipeline to transfer the knowledge between models with different architectures based on the attention mechanism to recover feature space under the source-free setting. Finally, we define constrained hypothesis space to obtain a rigorous objective function.
+
+# 3.1. Discrepancy Measurement
+
+As introduced in [63], we recall the definition of Open-set Margin Discrepancy and Unknown Predictive Discrepancy, which serve as key components to bridging the gap between the theory and algorithm for open-set domain adaptation.
+
+Definition 3.1 (Open-set Margin Discrepancy). Let  $y, y'$  denote outputs of  $f, f': \mathcal{X} \to \mathcal{K}$  where  $y = l(f(x)), l(f'(x)) = y'$  given induced labeling function:
+
+$$
+l \circ f: x \rightarrow \underset {y \in \mathcal {Y}} {\arg \max } f (x) [ y ] \tag {6}
+$$
+
+The Open-set Margin Discrepancy between two functions  $f, f'$  over a distribution  $D$  is given by:
+
+$$
+\begin{array}{l} \epsilon_ {D} (f, f ^ {\prime}) = \mathbb {E} _ {x \sim D} [ \operatorname {o m d} (f (x), f ^ {\prime} (x)) ] (7) \\ \operatorname {o m d} (f (x), f ^ {\prime} (x)) = \max  \left(\left| \log (1 - f (x) [ y ]) - \log (1 - f ^ {\prime} (x) [ y ]) \right|\right), \\ | \log (1 - f (x) [ y ^ {\prime} ]) - \log (1 - f ^ {\prime} (x) [ y ^ {\prime} ]) |) (8) \\ \end{array}
+$$
+
+Definition 3.2 (Unknown Predictive Discrepancy). Let  $v: \mathcal{K} \times \mathcal{K} \to \mathbb{R}$  denote the Unknown Predictive Discrepancy as a distance metric and  $v_{D}(f, f') := \mathbb{E}_{x \sim D} v(f(x), f'(x))$  measure the expected disagreement between the  $K$ -th outputs of  $f, f': \mathcal{X} \to \mathcal{K}$  over a distribution  $D$  on  $\mathcal{X}$ . Let  $e^{K}: \mathcal{X} \to [0, \dots, 1] \in \mathcal{K}$  denote a function that can predict any input as the unknown class. The deviation from  $e^{K}$  for a hypothesis  $h \in \mathcal{H}$  is further referred to as the shorthand  $v_{D}(h) := v_{D}(h, e^{K})$  that measures the probability that samples from  $D$  not classified as unknowns.
+
+$$
+v _ {D} (f, f ^ {\prime}) = \mathbb {E} _ {x \sim D} | \log (1 - f (x) [ K ]) - \log (1 - f ^ {\prime} (x) [ K ]) | \tag {9}
+$$
+
+# 3.2. Inference on Expectation with PU Learning
+
+In this section, we introduce the techniques from PU learning [58] to estimate the expectation over source domain  $S_{i}$  by the incomplete source domain  $S_{i}^{\prime}$  and target domain  $T$  for the open-set scenario. The expectation over  $V$  can be derived analogously.
+
+Assumption 3.3. Let  $S_{i}^{k} = P_{S_{i}}(x|y = k), V^{k} = P_{V}(x|y = k), T^{k} = P_{T}(x|y = k)$  denote class conditional distributions,  $S_{i}^{\backslash K} = P_{S_{i}}(x|y \neq K), V' = P_{V}(x|y \neq K), T' = P_{T}(x|y \neq K)$  indicate incomplete domains that do not contain unknown class  $S_{i}^{K}, V^{K}, T^{K}$ . Given a feature extractor  $g: \mathcal{X} \subseteq \mathbb{R}^{D} \to \mathcal{Z} \subseteq \mathbb{R}^{F}$ , assume that the feature space can be aligned by DA techniques such that  $Z^{K} = P_{S_{i}^{K}}(z) = P_{V^{K}}(z) = P_{T^{K}}(z), Z' = P_{S_{i}^{\backslash K}}(z) = P_{V'}(z) = P_{T'}(z)$ .
+
+Lemma 3.4 (PU Estimation $^1$ ). Let  $g: \mathcal{X} \subseteq \mathbb{R}^D \to \mathcal{Z} \subseteq \mathbb{R}^F$  denote the feature extractor. Let  $h \in \mathcal{H}^F: \mathcal{Z} \to \mathcal{K}$  where  $h \circ g \in \mathcal{H}: \mathcal{X} \to \mathcal{K}$  and  $f_V^* \in \mathcal{H}_V^F, f_T^* \in \mathcal{H}_T^F, f_{S_i}^* \in \mathcal{H}_{S_i}^F: \mathcal{Z} \to \mathcal{K}$  denote the decomposed approximated labeling functions. Let  $\sum_{k=1}^{K} \pi_{S_i}^k = 1, \sum_{k=1}^{K} \pi_V^k = 1, \sum_{k=1}^{K} \pi_T^k = 1$  denote the class priors of each domain. Given Assumption 3.3, the expectation on  $S_i$  can be estimated by expectation on  $S_i^{\backslash K}$  and Unknown Predictive Discrepancy (Definition 3.2) with a mild condition that  $\pi_{S_i}^K = \pi_T^K = 1 - \alpha$ :
+
+$$
+\epsilon_ {S _ {i}} (h \circ g) = \alpha \left[ \epsilon_ {S _ {i} ^ {\backslash K}} (h \circ g) - v _ {S _ {i} ^ {\backslash K}} (h \circ g) \right] + v _ {T} (h \circ g) \tag {10}
+$$
+
+$$
+\begin{array}{l} \epsilon_ {S _ {i}} (f _ {S _ {i}} ^ {*} \circ g, f _ {V} ^ {*} \circ g) = \alpha [ \epsilon_ {S _ {i} \backslash K} (f _ {S _ {i}} ^ {*} \circ g, f _ {V} ^ {*} \circ g) - v _ {S _ {i} \backslash K} (f _ {S _ {i}} ^ {*} \circ g, f _ {V} ^ {*} \circ g) ] \\ + v _ {T} \left(f _ {S _ {i}} ^ {*} \circ g, f _ {V} ^ {*} \circ g\right) (11) \\ \epsilon_ {S _ {i}} (f _ {V} ^ {*} \circ g, h \circ g) = \alpha [ \epsilon_ {S _ {i} ^ {\backslash} K} (f _ {V} ^ {*} \circ g, h \circ g) - v _ {S _ {i} ^ {\backslash} K} (f _ {V} ^ {*} \circ g, h \circ g) ] \\ + v _ {T} \left(f _ {V} ^ {*} \circ g, h \circ g\right) (12) \\ \end{array}
+$$
+
+Assumption 3.5. Given a feature extractor  $g: \mathcal{X} \to \mathcal{Z}$ , assume that the covariate shift between each source and labeled target domain can be addressed for known categories as  $P_{S_i^k}(z) = P_{V^k}(z), k = 1,..K - 1$ .
+
+Corollary 3.6. Let  $\mathcal{Y}_i^{\prime \prime} = \{k|k\notin \mathcal{Y}_i^\prime ,k = 1,..K - 1\}$  denote the label space that is absent from  $S_{i}^{\prime}$ . Given Assumption 3.5, we further decompose the source error as:
+
+$$
+\begin{array}{l} \alpha \epsilon_ {S _ {i} ^ {\backslash K}} (h \circ g) = \sum_ {k \in \mathcal {Y} _ {i} ^ {\prime \prime}} \pi_ {S _ {i}} ^ {k} \epsilon_ {S _ {i} ^ {k}} (h \circ g) + \sum_ {k \in \mathcal {Y} _ {i} ^ {\prime}} \pi_ {S _ {i}} ^ {k} \epsilon_ {S _ {i} ^ {k}} (h \circ g) \\ = \rho_ {i} \sum_ {k \in \mathcal {Y} _ {i} ^ {\prime \prime}} \epsilon_ {V _ {i} ^ {k}} (h \circ g) + (1 - \rho_ {i}) \epsilon_ {S _ {i} ^ {\prime}} (h \circ g), \tag {13} \\ \end{array}
+$$
+
+where  $\rho_{i} = |\mathcal{Y}_{i}^{\prime \prime}| / K$  under a mild condition that  $\pi_{S_i}^k = 1 / K$  for  $k\in \mathcal{Y}_i^{\prime \prime}$
+
+Remark 3.7. According to Definition 3.2, minimizing  $v_{\hat{T}}(h \circ g)$  means mapping target samples to the unknown class. In practice, a multiplier  $\beta < 1$  is applied on  $v_{\hat{T}}(h \circ g)$  to prevent all target samples from being classified as unknown.
+
+# 3.3. Towards Source-Free Knowledge Transfer with Attention-based Feature Generation
+
+Source-free domain adaptation (SFDA) has been considered a means of reducing reliance on source data. As described in [24], the existing SFDA research can generally be categorized into data-centric and model-centric methods. Model-centric methods employ techniques such as self-supervision, while data-centric methods focus on image-based reconstruction. Model-centric methods like [28, 29, 35, 60] require source model fine-tuning, where the generalization to multi-source cases with label shift can be nontrivial since it may fail to fully leverage the few-shot labeled data due to the missing classes in source domains. Meanwhile, for data-centric methods like [6, 26], the pipeline to generate source-like images is generally computationally intensive
+
+![](images/b924781068c2bfcbf6d052fefbd06b27a184ae76704098abb9c471e600243125.jpg)  
+Figure 2. The mechanism of attention-based feature generation for source-free domain adaptation. Given a similarity-based weight estimated by the knowledge preserved in the pre-trained source model consisting of a black-box feature extractor  $g_{i}$  and a visible classifier  $f_{i}$ , labeled features generated with the attention module can be considered a weighted average of unlabeled target features, which serve as the anchor for the target distribution alignment in the adaptation phase.
+
+and time-consuming, which can hardly be applied to highly structured domains. Furthermore, it might violate the intention of SFDA to protect privacy by recovering source-like images. Motivated by this, in this section, we propose a novel attention-based feature generation (AFG) algorithm that can produce labeled anchors for the alignment of unlabeled target data by leveraging the knowledge equipped in source models, which is more computationally efficient and independent from source model fine-tuning.
+
+The SFDA scenario involves two phases: pre-training and adaptation. During pre-training,  $N$  models are trained on labeled data from each source domain  $x_{s_i} \sim S_i'$ ,  $i = 1\dots N$ . Subsequently, the goal of the adaptation stage is to adapt the pre-trained source model to the unlabeled target data  $x_t \sim T$  given few-shot labeled target data  $x_v \sim V'$ . The proposed approach assumes a challenging open-set form, implying that the label spaces among the target and source domains are distinct.
+
+Inspired by [28], which uses a single-layer linear classifier in source models to store the cluster center of source features, we choose the Bayesian linear classifier during pre-training such that source features can be sampled by the re-parameterization trick [21] in the adaptation phase. Let  $g_{i}:\mathcal{X}\rightarrow \mathcal{Z}_{i}\subseteq \mathbb{R}^{F_{i}}$  and  $f_{i}\coloneqq \{\mu_{i},\sigma_{i}\}$  denote each pre-trained source model. As illustrated in Fig. 2, given the source features approximated by the weight samples
+
+of Bayesian linear classifier as  $g_{i}(\hat{S}_{i}^{\prime}) = \left( \begin{array}{c} g_{i}(x_{s_{i}}^{1}) \\ \vdots \\ g_{i}(x_{s_{i}}^{\| \mathcal{Y}_{i}^{\prime}\|}) \end{array} \right) :=$
+
+$$
+\mu_ {i} + \sigma_ {i} \odot \left( \begin{array}{c} \zeta_ {i} ^ {1} \\ \vdots \\ \zeta_ {i} ^ {\| \mathcal {Y} _ {i} ^ {\prime} \|} \end{array} \right),   \zeta_ {i} ^ {j} \sim \mathcal {N} (0, I) \text {w i t h s i z e} \| \mathcal {Y} _ {i} ^ {\prime} \| \times F _ {i}
+$$
+
+(multiple samples can be generated from each class in practice), along with the query and key mapping functions  $w_{q_i}, w_{k_i}: \mathcal{Z}_i \to \mathcal{Z}_i' \subseteq \mathbb{R}^{F_i'}$ , the corresponding labeled anchor defined as  $\{(g(x_{s_i}^j), y_i^j \in \mathcal{Y}_i')\}_{j=1}^{\|\mathcal{Y}_i'\|}, y_i^j \neq y_i^{j'}$  is given
+
+by:
+
+$$
+g (\hat {S} _ {i} ^ {\prime}) = \operatorname {s o f t m a x} \left(\frac {w _ {q _ {i}} \left(g _ {i} \left(\hat {S} _ {i} ^ {\prime}\right)\right) \cdot w _ {k _ {i}} \left(g _ {i} \left(\hat {T} ^ {\prime}\right)\right) ^ {\top}}{\sqrt {F _ {i} ^ {\prime}}}\right) g (\hat {T} ^ {\prime}), \tag {14}
+$$
+
+where  $\hat{T}'$  denotes the estimated known-class data from the target. To produce meaningful features for the distribution alignment in the adaptation phase, we propose two objective functions to learn the query and key mapping  $\{w_{q_i}, w_{k_i}\}_{i=1}^N$  of each source domain. Analogous to [5], we train  $w_{q_i}, w_{k_i}$  by maximizing the similarity between the projections of the same target features extracted by the per-trained source model  $g_i$  while pushing the different target features far apart, which can be achieved with minimizing reconstruction loss  $L_{rec}^i$  such that the output of the attention module can approximate target features  $g(\hat{T}')$  given target data as query and key. To further regularize  $w_{q_i}, w_{k_i}$ , we introduce a cycle-consistency loss  $L_{cyc}^i$  that can bring the features generated by labeled and unlabeled target data  $\hat{V}', \hat{T}'$  close to each other.
+
+$$
+L _ {r e c} ^ {i} = \left| \operatorname {s o f t m a x} \left(\frac {w _ {q _ {i}} \left(g _ {i} \left(\hat {T} ^ {\prime}\right)\right) \cdot w _ {k _ {i}} \left(g _ {i} \left(\hat {T} ^ {\prime}\right)\right) ^ {\top}}{\sqrt {F _ {i} ^ {\prime}}}\right) g \left(\hat {T} ^ {\prime}\right) - g \left(\hat {T} ^ {\prime}\right) \right| \tag {15}
+$$
+
+$$
+L _ {c y c} ^ {i} = | \mathrm {s o f t m a x} (\frac {w _ {q _ {i}} (g _ {i} (\hat {S} _ {i} ^ {\prime})) \cdot w _ {k _ {i}} (g _ {i} (\hat {V} ^ {\prime})) ^ {\top}}{\sqrt {F _ {i} ^ {\prime}}}) g (\hat {V} ^ {\prime}) - g (\hat {S} _ {i} ^ {\prime}) | \quad (1 6)
+$$
+
+Progressive Unknown Rejection (PUR) is additionally proposed to improve the recognition accuracy on unknown class. In the open-set setting, empirical target data  $\hat{T}$  includes the unknown class, while the generated labeled anchors  $g(\hat{S}_i^{\prime})$  should be limited to the known class. According to the generation mechanism defined by Eq. (14), labeled anchors can be considered a similarity-based weighted average of target features, which are not supposed to contain components from irrelevant features of the unknown class. However, it is impractical to learn the ideal results where the weights assigned to those unrelated target features are zero by pure regularization of mapping functions  $w_{q_i}, w_{k_i}$ . To address this problem, we introduce a scheme to gradually reject the target features from the unknown class by removing the target data labeled as unknown given the current hypothesis  $h$  from  $\hat{T}$ . Specifically, at each training iteration during the adaptation stage, for a batch of input target data, we rank the likelihood of the unknown class for each target sample  $p(y = K|x_t) = h(x_t)[K]$  in ascending order. Given a threshold  $0 < \tau < 1$  progressively increasing from zero according to the exponential ramp-up function [22], we select bottom  $1 - \tau$  target samples as  $\hat{T}'$ .
+
+# 3.4. Hypothesis Constraint
+
+Proposition 3.8. If  $\mathcal{H}_{S_i}^F, \mathcal{H}_V^F, \mathcal{H}_T^F$  are sets of functions that can minimize a part of  $\hat{\theta}_S^i, \sum_i \hat{\theta}_V^i, \sum_i \hat{\theta}_T^i$  respectively, then  $f_{S_i}^* \in \mathcal{H}_{S_i}^F, f_V^* \in \mathcal{H}_V^F, f_T^* \in \mathcal{H}_T^F$  must hold such that we can relax  $L_{\text{dis}}$  in Theorem 2.3 by considering maximum w.r.t. functions  $f_{S_i}', f_V', f_T'$  as:
+
+$$
+\begin{array}{l} \log \sum_ {i} \exp \left(\nu \left[ L _ {c l s} ^ {S _ {i}} (h; g) + 2 L _ {d i s} ^ {i} \left(f _ {S _ {i}} ^ {*}, f _ {V} ^ {*}, f _ {T} ^ {*}, h; g\right) \right]\right) \\ \leq \sup _ {\{f _ {S _ {i}} ^ {\prime} \in \mathcal {H} _ {S _ {i}} ^ {F} \} _ {i = 1} ^ {N}, f _ {V} ^ {\prime} \in \mathcal {H} _ {V} ^ {F}, f _ {T} ^ {\prime} \in \mathcal {H} _ {T} ^ {F}} \log \sum_ {i} \exp (\nu [ L _ {c l s} ^ {S _ {i}} (h; g) + 2 L _ {d i s} ^ {i} (f _ {S _ {i}} ^ {\prime}, f _ {V} ^ {\prime}, f _ {T} ^ {\prime}, h; g) ]) \\ \end{array}
+$$
+
+(17)
+
+Definition 3.9 (Approximated Labeling Function Space). Let  $L_{\mathcal{H}_S}^i, L_{\mathcal{H}_V}^i, L_{\mathcal{H}_T}^i$  denote the hypothesis constraints, i.e., a part of the empirical deviation between approximated and true labeling functions  $\hat{\theta}_S^i, \hat{\theta}_T^i, \hat{\theta}_V^i$ . Approximated Labeling Function Space  $\mathcal{H}_{S_i}^F, \mathcal{H}_V^F, \mathcal{H}_T^F$  can be defined as the sets whose members  $f_{S_i}', f_V', f_T' \in \mathcal{H}^F$  can minimize  $L_{\mathcal{H}_S}^i, \sum_i L_{\mathcal{H}_V}^i, \sum_i L_{\mathcal{H}_T}^i$ :
+
+$$
+\begin{array}{l} \left\{\mathcal {H} _ {S _ {i}} ^ {F} = \left\{f _ {S _ {i}} ^ {\prime} | \arg \min  _ {g, f _ {S _ {i}} ^ {\prime} \in \mathcal {H} ^ {F}} \left[ L _ {\mathcal {H} _ {S}} ^ {i} \left(f _ {S _ {i}} ^ {\prime}; g\right) = L _ {c l s} ^ {S _ {i}} \left(f _ {S _ {i}} ^ {\prime}; g\right) / 2 + L _ {c l s} ^ {V} \left(f _ {S _ {i}} ^ {\prime}; g\right) \right] \right\} \right. \\ \left\{\mathcal {H} _ {V} ^ {F} = \left\{f _ {V} ^ {\prime} \mid \arg \min  _ {g, f _ {V} ^ {\prime} \in \mathcal {H} ^ {F}} \sum_ {i} \left[ L _ {\mathcal {H} _ {V}} ^ {i} \left(f _ {V} ^ {\prime}; g\right) = L _ {c l s} ^ {V} \left(f _ {V} ^ {\prime}; g\right) / 2 + L _ {c l s} ^ {S _ {i}} \left(f _ {V} ^ {\prime}; g\right) \right] \right\} \right. \\ \left\{\mathcal {H} _ {T} ^ {F} = \left\{f _ {T} ^ {\prime} \mid \arg \min  _ {g, f _ {T} ^ {\prime} \in \mathcal {H} ^ {F}} \sum_ {i} \left[ L _ {\mathcal {H} _ {T}} ^ {i} \left(f _ {T} ^ {\prime}; g\right) = \left[ L _ {\text {c l s}} ^ {S _ {i}} \left(f _ {T} ^ {\prime}; g\right) + L _ {\text {c l s}} ^ {V} \left(f _ {T} ^ {\prime}; g\right) \right] / 2 + L _ {\text {s s l}} \right] \right\} \right. \\ \end{array}
+$$
+
+(18)
+
+To build a more reliable target function space  $\mathcal{H}_T^F$ , we approximate the target error with the error rate on labeled samples and a semi-supervised regularization term  $L_{ssl}^2$  including entropy minimization [14, 15], pseudo labeling [44, 46] and consistency regularization [22, 42], which has been intensively discussed in [23, 43, 59, 63].
+
+# 3.5. Algorithm
+
+As described in Algorithm 1, we introduce a gradient reversal layer [12] to train the overall objective together. ImageNet [8] pre-trained ResNet-50 [16] is used as feature extractor  $g$  and randomly initialized 2-layer fully-connected networks are used for classifiers  $f_{S_i}^{\prime}, f_V^{\prime}, f_T^{\prime}, h$ . We adopt SGD with a momentum of 0.9 for optimization, where the initial learning rate is empirically set to 0.001. We employ the learning rate annealing strategy proposed in [12]. We use RandomFlip, RandomCrop, and RandAugment [7] as data augmentation with the batch size fixed to 24.
+
+![](images/2368c1f59ad63ea02d5288c77d00848fb6a4405f5d48019ba52a15b5fac7f00b.jpg)  
+Figure 3. Alignment mechanism of UM, where unknown target data  $\hat{T}^K$  (green) is pushed away from labeled data into a separated cluster, while known target data  $\hat{T}'$  is aligned back towards labeled clusters by  $\min_g L_{dis}$ .
+
+# Algorithm 1 UM
+
+Input: source  $\{\hat{S}_i^{\prime}\}_{i = 1}^{N}$ , labeled target  $\hat{V}^{\prime}$ , unlabeled target  $\hat{T}$
+
+Output: updated parameters  $\phi = (\{f_{S_i}^{\prime}\}_{i=1}^{N}, g, h, f_V^{\prime}f_T^{\prime})$ ,  $w = \{w_{q_i}, w_{k_i}\}_{i=1}^{N}$
+
+Parameter: trade-off parameter  $\lambda$ ; learning rate  $\eta$ ; known class ratio estimator  $\alpha$ ; coefficients  $\nu, \beta, \tau$
+
+Notation: gradient reversal operator  $R(\cdot)$
+
+for epoch  $= 1,2,\dots$  do
+
+Estimate known class ratio  $\alpha$  on  $\hat{T}$  with  $g,h$
+
+if source-free then
+
+Estimate  $\hat{T}^{\prime}$  according to PUR and Update  $w$  to optimize AFG:
+
+$$
+w \leftarrow w - \eta \Delta w, \Delta w = \frac {\partial \sum_ {i = 1} ^ {N} \left(L _ {r c c} ^ {i} + L _ {c y c} ^ {i}\right)}{\partial w}
+$$
+
+Generate labeled features  $g(\hat{S}_i^{\prime})$  according to Eq. (14)
+
+# end if
+
+Compute labeled target error  $L_{cls}^{V}(h;g) = L_{V}$ , source error  $L_{cls}^{S_i}(h;g) = L_S^i$ , hypothesis constraints  $L_{\mathcal{H}_S}^i (f_{S_i}';g) + L_{\mathcal{H}_V}^i (f_V';g) + L_{\mathcal{H}_T}^i (f_T';g) = L_H^i$  for  $i = 1,..N$
+
+Compute discrepancy  $L_{dis}^{i}(f_{S_{i}}^{\prime}, f_{V}^{\prime}, f_{T}^{\prime}, R \circ h \circ R; R \circ g) = L_{D}^{i}$  given the gradient reversal layer for  $i = 1,..N$
+
+Update  $\phi$  to minimize the target error bound:
+
+$$
+\phi \leftarrow \phi - \eta \Delta \phi ,
+$$
+
+$$
+\Delta \phi = \frac {\partial (\frac {1}{2} [ L _ {V} + \frac {1}{\nu} \log \sum_ {i = 1} ^ {N} \exp (\nu [ L _ {S} ^ {i} + L _ {H} ^ {i} - \lambda L _ {D} ^ {i} ]) ])}{\partial \phi}
+$$
+
+end for
+
+# 4. Evaluation
+
+We evaluated our proposal using two benchmarks, Office-Home and DomainNet. The trade-off parameter  $\lambda$  is set to 0.01 during the training procedure according to [62, 63]. In addition, we empirically set the PU, scaling, and threshold coefficients  $\beta$  to 0.15,  $\nu$  to 0.1, and  $\tau$  to 0.3 for all experiments. For the semi-supervised setting, we select the same few-shot labeled target data according to [41]. Regarding the open-set setting, we assign a distinct label space for each source domain as a subset of the target label space described below. We quantitatively compare our results against various baselines, including OSBP [40], PGL [34], ANNA [27], PUJE [63], MOSDANET [38], HyMOS [3], and MPU [58].
+
+Evaluation Metrics for the proposed method and baselines are the widely used measures [34, 40], i.e., normalized accuracy for the known class only  $(\mathrm{OS}^{*})$  and harmonic mean  $\mathrm{HOS} = 2(\mathrm{OS}^{*} \times \mathrm{UNK}) / (\mathrm{OS}^{*} + \mathrm{UNK})$  [2, 27, 31, 54, 63].
+
+Office-Home [52] is a widely-used domain adaptation benchmark, which consists of 15,500 images from 65 categories and four domains: Art, Clipart, Product, and RealWorld. We select the first 30 classes alphabetically as the known class and group the rest as the unknown. Each source domain contains 10 classes without overlap, leading to a large label shift scenario.
+
+DomainNet [36] is a more challenging benchmark dataset for large-scale domain adaptation that has 345 classes and
+
+<table><tr><td rowspan="2">METHOD</td><td rowspan="2">TYPE</td><td colspan="2">→Clipart</td><td colspan="2">→Product</td><td colspan="2">→RealWorld</td><td colspan="2">→Art</td><td colspan="2">Avg.</td></tr><tr><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td></tr><tr><td>OSBP</td><td>Source-Combine</td><td>60.4</td><td>62.6</td><td>70.1</td><td>72.3</td><td>69.7</td><td>68.3</td><td>60.7</td><td>64.3</td><td>65.2</td><td>66.9</td></tr><tr><td>PGL</td><td></td><td>59.0</td><td>61.8</td><td>67.7</td><td>69.9</td><td>66.7</td><td>68.9</td><td>61.2</td><td>64.0</td><td>63.7</td><td>66.2</td></tr><tr><td>ANNA</td><td></td><td>65.8</td><td>67.7</td><td>71.0</td><td>73.4</td><td>70.3</td><td>70.3</td><td>61.0</td><td>63.7</td><td>67.0</td><td>68.8</td></tr><tr><td>PUJE</td><td></td><td>65.8</td><td>71.7</td><td>73.3</td><td>74.2</td><td>75.0</td><td>78.1</td><td>65.5</td><td>67.3</td><td>69.9</td><td>72.8</td></tr><tr><td>MOSDANET</td><td>Multi-Source</td><td>61.5</td><td>65.9</td><td>70.0</td><td>73.8</td><td>71.4</td><td>69.6</td><td>61.6</td><td>63.6</td><td>66.1</td><td>68.2</td></tr><tr><td>HyMOS</td><td></td><td>56.6</td><td>64.4</td><td>64.4</td><td>67.3</td><td>66.2</td><td>68.4</td><td>59.0</td><td>62.2</td><td>61.6</td><td>65.6</td></tr><tr><td>UM</td><td></td><td>68.0</td><td>72.1</td><td>79.0</td><td>83.0</td><td>79.4</td><td>80.8</td><td>67.7</td><td>70.3</td><td>73.5</td><td>76.6</td></tr><tr><td>MPU*</td><td>Source-Free</td><td>46.3</td><td>54.4</td><td>59.7</td><td>66.3</td><td>57.8</td><td>60.2</td><td>58.3</td><td>62.5</td><td>55.5</td><td>60.9</td></tr><tr><td>OSBP*</td><td></td><td>44.5</td><td>56.5</td><td>55.6</td><td>65.1</td><td>59.3</td><td>64.3</td><td>55.6</td><td>59.9</td><td>53.8</td><td>61.5</td></tr><tr><td>PUJE*</td><td></td><td>52.2</td><td>58.4</td><td>65.0</td><td>70.3</td><td>66.2</td><td>70.0</td><td>58.7</td><td>62.7</td><td>60.5</td><td>65.4</td></tr><tr><td>UM+AFG</td><td></td><td>61.1</td><td>66.0</td><td>77.0</td><td>80.1</td><td>72.0</td><td>78.8</td><td>60.3</td><td>64.6</td><td>67.6</td><td>72.4</td></tr></table>
+
+Table 1. HOS (%) of ResNet-50 model fine-tuned on Office-Home dataset under 1-shot/3-shot setting  
+
+<table><tr><td rowspan="2">METHOD</td><td rowspan="2">TYPE</td><td colspan="2">→Clipart</td><td colspan="2">→Painting</td><td colspan="2">→Real</td><td colspan="2">→Sketch</td><td colspan="2">Avg.</td></tr><tr><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td></tr><tr><td>OSBP</td><td>Source-Combine</td><td>54.2</td><td>57.4</td><td>49.8</td><td>53.1</td><td>62.6</td><td>64.0</td><td>49.5</td><td>50.1</td><td>54.0</td><td>56.2</td></tr><tr><td>PGL</td><td></td><td>59.8</td><td>62.0</td><td>59.4</td><td>61.4</td><td>67.4</td><td>69.4</td><td>59.7</td><td>61.2</td><td>61.6</td><td>63.5</td></tr><tr><td>ANNA</td><td></td><td>55.6</td><td>61.5</td><td>53.6</td><td>54.3</td><td>67.5</td><td>66.5</td><td>57.9</td><td>58.1</td><td>58.7</td><td>60.1</td></tr><tr><td>PUJE</td><td></td><td>64.4</td><td>66.2</td><td>59.8</td><td>61.7</td><td>67.7</td><td>69.3</td><td>61.2</td><td>64.2</td><td>63.3</td><td>65.4</td></tr><tr><td>MOSDANET</td><td>Multi-Source</td><td>56.4</td><td>55.3</td><td>55.6</td><td>58.2</td><td>68.5</td><td>69.8</td><td>54.1</td><td>54.9</td><td>58.7</td><td>59.6</td></tr><tr><td>HyMOS</td><td></td><td>53.0</td><td>54.4</td><td>54.1</td><td>56.0</td><td>65.1</td><td>67.4</td><td>56.3</td><td>57.1</td><td>57.1</td><td>58.7</td></tr><tr><td>UM</td><td></td><td>70.3</td><td>71.5</td><td>66.0</td><td>68.8</td><td>75.1</td><td>78.5</td><td>66.1</td><td>69.5</td><td>69.4</td><td>72.1</td></tr><tr><td>MPU*</td><td>Source-Free</td><td>54.5</td><td>57.6</td><td>55.0</td><td>60.1</td><td>62.4</td><td>66.4</td><td>48.4</td><td>52.9</td><td>55.1</td><td>59.3</td></tr><tr><td>MOSDANET*</td><td></td><td>58.1</td><td>60.5</td><td>54.3</td><td>59.3</td><td>63.2</td><td>62.5</td><td>49.4</td><td>54.3</td><td>56.3</td><td>59.2</td></tr><tr><td>PUJE*</td><td></td><td>60.5</td><td>62.2</td><td>55.3</td><td>61.4</td><td>64.0</td><td>67.8</td><td>53.1</td><td>56.2</td><td>58.2</td><td>61.9</td></tr><tr><td>UM+AFG</td><td></td><td>64.8</td><td>69.7</td><td>60.0</td><td>64.2</td><td>67.6</td><td>73.4</td><td>60.0</td><td>64.8</td><td>63.1</td><td>68.0</td></tr></table>
+
+Table 2. HOS (%) of ResNet-50 model fine-tuned on DomainNet dataset under 1-shot/3-shot setting
+
+6 domains. Following the protocol established in [41], we pick 4 domains (Real, Clipart, Painting, Sketch) with 126 classes for the experiments. We select the first 60 classes alphabetically as the known class and group the rest as the unknown. Similarly, each source domain contains 20 classes without any overlap.
+
+As reported in Tabs. 1 and 2, under the same setting given 1-shot/3-shot labeled target data (1/3 samples per class), we observe that our method UM consistently outperforms the state-of-the-art results, improving HOS by  $3.6\% / 3.8\%$  and  $6.1\% / 6.7\%$  on the benchmark datasets of Office-Home and DomainNet respectively, when source data is available. Furthermore,  $\mathrm{UM + AFG}$  enhances HOS by  $7.1\% / 7.0\%$  in Office-Home and  $4.9\% / 6.1\%$  in DomainNet under the challenging source-free setting. Note that our proposed approach provides significant performance gains for the more complex datasets like DomainNet, which requires knowledge transfer across different modalities, regardless of covariate or label shift. We group all source domains with labeled target data as a single domain for the baselines that require the source-combine strategy. For the source-free cases, we introduce a few confident target data labeled by pre-trained models as pseudo-source data to enable several algorithms denoted by  $*$  under this problem setting since none of the existing methods can directly address the open-set task under the source-free condition with a huge label shift across source domains.
+
+# 4.1. Feature Space Visualization
+
+To intuitively visualize the effectiveness of different approaches, we extracted features from the baseline models and our proposed model on the  $\rightarrow$  Art task (Office-Home) and  $\rightarrow$  Real task (DomainNet) with the ResNet-50 backbone
+
+<table><tr><td rowspan="2">METHOD</td><td rowspan="2">TYPE</td><td colspan="3">Office-Home →RealWorld</td><td colspan="3">DomainNet →Clipart</td></tr><tr><td>UNK</td><td>OS*</td><td>HOS</td><td>UNK</td><td>OS*</td><td>HOS</td></tr><tr><td>DEFAULT</td><td>Source-Free</td><td>73.7</td><td>70.4</td><td>72.0</td><td>72.6</td><td>58.6</td><td>64.8</td></tr><tr><td>w/o Lsim</td><td></td><td>70.9</td><td>70.6</td><td>70.7</td><td>69.3</td><td>59.1</td><td>63.8</td></tr><tr><td>w/o Lcyc</td><td></td><td>74.0</td><td>68.5</td><td>71.1</td><td>72.8</td><td>56.9</td><td>63.9</td></tr><tr><td>w/o PUR</td><td></td><td>39.6</td><td>87.0</td><td>54.4</td><td>47.3</td><td>72.9</td><td>57.4</td></tr></table>
+
+[16]. The feature distributions were processed with t-SNE [48] afterward. As shown in Fig. 4, compared with baselines, our method UM achieves a better alignment between source and target distributions, especially when the domain shift is large. Benefiting from our joint error-based adversarial alignment mechanism, the extracted feature space, including the cluster of unknown target data (green), has a more discriminative class-wise decision boundary.
+
+# 4.2. Ablation Study
+
+Self-supervised learning methods have shown that, by relying only on unlabeled data, it is still possible to obtain classification performance close to those of the supervised approaches [5, 17, 18]. In the source-free setting, we adopt the typical SimCLR [5] to help group the feature of unknown target data into a single cluster. As expected in Tab. 3,  $L_{sim}^2$  can slightly improve the accuracy of the unknown class for a higher HOS. Furthermore, Progressive Unknown Rejection (PUR), a denoising of generated labeled features, is crucial to detecting unknowns in source-free cases. As also illustrated in Fig. 7d, generally, a larger threshold  $\tau$  will lead to a higher UNK at the cost of low OS*, characterized as the trade-off between recognizing known and unknown data for open-set tasks. In addition, we verify the effectiveness of cycle-consistency regularization  $L_{cyc}$  and find it helps maintain the normalized accuracy of the known class.
+
+# 4.3. Robustness against Varying Openness
+
+To verify the robustness of the proposed method, we conducted experiments on the  $\rightarrow$  Painting task (DomainNet) with the openness varying in  $\{0.25, 0.5, 0.75\}$ . Here, openness is defined by the ratio of unknown samples in the entire target data. PGL approach heuristically sets the hyperparameter according to the true unknown ratio to control the openness, while PUJE and UM automatically estimate the weight  $\alpha$  during the training procedure. From Fig. 5a, we observe that our proposal consistently outperforms baselines by a large margin, which confirms its robustness to the change in openness.
+
+# 4.4. Stabel Coverage
+
+In Fig. 5b, we illustrate the recognition performance of UM over training steps on the  $\rightarrow$ Art task of the Office-Home
+
+Table 3. Ablation study verified with ResNet-50 model on OfficeHome & DomainNet dataset  
+
+<table><tr><td rowspan="3">METHOD</td><td rowspan="3">TYPE</td><td rowspan="3">BACKBONE</td><td colspan="6">Office-Home</td><td colspan="6">DomainNet</td><td rowspan="2">Avg. 
+O*</td><td rowspan="2">HOS</td><td></td></tr><tr><td colspan="3">→Art</td><td colspan="3">→Product</td><td colspan="3">→Painting</td><td colspan="3">→Real</td><td></td></tr><tr><td>UNK</td><td>OS*</td><td>HOS</td><td>UNK</td><td>OS*</td><td>HOS</td><td>UNK</td><td>OS*</td><td>HOS</td><td>UNK</td><td>OS*</td><td>HOS</td><td>UNK</td><td>OS*</td><td>HOS</td></tr><tr><td>UM</td><td>Multi-Source</td><td>ResNet-50</td><td>72.8</td><td>63.3</td><td>67.7</td><td>78.7</td><td>79.3</td><td>79.0</td><td>77.9</td><td>57.3</td><td>66.0</td><td>74.9</td><td>75.2</td><td>75.1</td><td>76.1</td><td>68.8</td><td>72.0</td></tr><tr><td>UM+AFG</td><td>Source-Free</td><td>ResNet-50</td><td>66.1</td><td>55.5</td><td>60.3</td><td>83.3</td><td>71.6</td><td>77.0</td><td>63.9</td><td>56.5</td><td>60.0</td><td>76.6</td><td>60.6</td><td>67.6</td><td>72.5</td><td>61.1</td><td>66.2</td></tr><tr><td>UM+AFG</td><td></td><td>ViT-16</td><td>77.7</td><td>59.8</td><td>67.5</td><td>87.6</td><td>80.9</td><td>84.1</td><td>68.5</td><td>57.4</td><td>62.5</td><td>82.8</td><td>73.2</td><td>77.7</td><td>79.2</td><td>67.8</td><td>73.0</td></tr></table>
+
+Table 4. Accuracy of ViT-B/16 model fine-tuned on Office-Home & DomainNet dataset under 1-shot setting
+
+dataset. OS* experiences a downward while the UNK keeps improving, which characterizes a trade-off between the accuracy of knowns and the accuracy of unknowns. We further observe that some previous works [27, 34] do not converge at the optimum. In contrast, our method always reaches a reliable convergence without suffering from a severe performance drop in recognizing known classes.
+
+# 4.5. Flexibility in Backbone Architecture
+
+As presented in Sec. 3.3, AFG allows the target model to use a different backbone architecture from the pre-trained source models. Therefore, unlike those model-centric methods whose performance is deeply limited by source model architecture, our method can be effectively applied to real-world problems where each source model is trained using various networks by leveraging the power of advanced backbones like ViT [9] for the target model. Tab. 4 reveals a clear advantage of AFG when changing the target backbone to ViT-B/16 as the HOS scores under the source-free condition approach and even outperform the source data results. The same ResNet-50 backbone is used for pre-trained source models across different experiments.
+
+# 4.6. Advantage in Increasing Labeled Target Data
+
+Sec. 4.6 shows the behavior of different methods when the number of labeled examples in the target domain increases from 1 to 10 per class on DomainNet using ResNet50 backbone. Cluster-based methods like OSBP, MOSDANET, and HyMOS will finally be caught up by a simple multi-class PU learning (MPU) when the sample size increases. On the contrary, our method consistently outperforms the most competitive baseline PUJE for various sizes of labeled target data. Furthermore, along with the growth in the size of  $\hat{V}$ , the HOS score achieved by UM+AFG in the source-free setting gradually approaches, even surpasses those methods using source data.
+
+# 4.7. Sensitivity to PU, Scaling, and Threshold Coefficients
+
+We show the sensitivity of our approach to varying PU coefficient  $\beta$ , scaling factor  $\nu$ , and threshold  $\tau$  in Sec. 4.7. We can draw two observations from this: the OS* score is relatively stable, and the unknown recognition achieves a more reliable performance for a larger coefficient  $\beta$ ; generally, a larger  $\nu$  means focusing on the source domain that contributes more error and ignoring others, while a smaller  $\nu$  will equalize
+
+![](images/baead833bffaee584e19426d1998eeead018567ac396ee5e16b320c5b2abac07.jpg)  
+(a) OSBP
+
+![](images/d2c42b2a171394521b9bf8602debd0a11083dadfd7c65d32b226e4d545974e23.jpg)  
+(b) HyMOS
+
+![](images/f15e018d85423071d5e641ae072ae0e60c6ba0d2528ecb2c32ee97ab6e125b84.jpg)  
+(c) PUJE
+
+![](images/6ec085147406cebbe3d592a8c7a65096c4e7d31086a9c5de0aa4cc22b480166b.jpg)
+
+![](images/7223fd9ce990de23916bc3816c1301a5e021cae6e1c49bb981d2921d50924a21.jpg)  
+(e) MOSDANET
+
+![](images/3f9362d1ac223f824da343dcedf9a097dfb6bc69ee51c5583e18eef0313724f7.jpg)  
+(f) PGL
+
+![](images/dd384efbc1b66ee6ffd33a478ba5bc330de776d7c1b837d36e0facd77a7d23eb.jpg)  
+(g) ANNA
+
+![](images/229d3e1d4f2e004257d80251515d4af69f89ddd3649ecec6018adea93c384ae3.jpg)  
+(d) UM  
+(h) UM
+
+![](images/9f9bbd10253eda6ef4040bce1e232b300a2c7f34eed873b250f6a8e023b35dbf.jpg)  
+Figure 4. T-SNE visualization of feature distributions in (a)-(d)  $\rightarrow$  Art task (Office-Home dataset); (e)-(h)  $\rightarrow$  Real task (DomainNet dataset).  
+(a) robust against openness  
+Figure 5. (a) Performance comparisons w.r.t. varying openness of the  $\rightarrow$  Painting task from DomainNet dataset; (b) Convergence analysis of the  $\rightarrow$  Art task from Office-Home dataset compared to other baselines with confidence intervals
+
+![](images/8788f58a6b02ee9cf8f99ba6e949cce3625a03ac7f02f767a2ac7316ee280f69.jpg)  
+(b) stable convergence
+
+![](images/910dc367fec1548161cea36b899c04a7a077d125ae14d606c26f5d6fe125312a.jpg)  
+(a)  $\rightarrow$  Clipart task  
+Figure 6. Accuracy vs the number of labeled target samples on DomainNet using ResNet50 backbone. Our method maintains a high level of performance for different sample sizes of the labeled target data.
+
+![](images/4674bd3fa5c9b459b0e1b137015581139b0181d2b1023e114cefbc8320254cbc.jpg)  
+(b)  $\rightarrow$  Sketch task
+
+the importance of each domain, which can harm the performance when a remarkable label shift exists among source domains implied by Fig. 7c (the imbalance setting indicates a case where one source contains 20 classes while the other two sources take 5 classes respectively).
+
+![](images/797e41538c541eec8477180d0c59dd7dda8051c6221e3cd0a25f29e3f626c2e0.jpg)
+
+![](images/7874ae66c62a8388641219cdc0de361c3cbc0776ee47813622ef4eccd1aab535.jpg)
+
+![](images/212f295cdfc1c696e682eaa82a99ff0decf163f161fdf18f0d73bfabf7838eee.jpg)  
+(a) sensitivity to  $\beta$  
+(c) sensitivity to  $\nu$  
+Figure 7. (a)-(d) Sensitivity to varying loss coefficient  $\beta, \nu, \tau$  verified in Office-Home dataset.
+
+![](images/74305a12e89c6566e6ac7d20ad9e0e21e0845f102794138569e075a441842f7b.jpg)  
+(b) sensitivity to  $\beta$  
+(d) sensitivity to  $\tau$
+
+# 5. Conclusion
+
+In this work, we addressed the semi-supervised open-set domain shift problem in multi-source cases with inconsistent label space by introducing a novel learning theory based on joint error and multi-class PU learning that can reduce the open-set risk, where the generalization error is bounded by the extension of VC learning theory based on uniform covering number. We generalize our method into source-free scenarios by attention-based feature generation, which is computationally efficient with reliable performance. We conduct extensive experiments on multiple domain adaptation benchmarks. Our model achieves the best performance regardless of source data, compared with recent baseline methods, proving our proposed approach's efficacy.
+
+# Acknowledgements
+
+# References
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CVPR/2025/A Unified Approach to Interpreting Self-supervised Pre-training Methods for 3D Point Clouds via Interactions/full.md

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+# A Unified Approach to Interpreting Self-supervised Pre-training Methods for 3D Point Clouds via Interactions
+
+Qiang Li, Jian Ruan, Fanghao Wu, Yuchi Chen, Zhihua Wei*, and Wen Shen* Tongji University, Shanghai, China
+
+{qli, jianruan, 2432055, 1953721, zhihua_wei, wenshen}@tongji.edu.cn
+
+# Abstract
+
+Recently, many self-supervised pre-training methods have been proposed to improve the performance of deep neural networks (DNNs) for 3D point clouds processing. However, the common mechanism underlying the effectiveness of different pre-training methods remains unclear. In this paper, we use game-theoretic interactions as a unified approach to explore the common mechanism of pre-training methods. Specifically, we decompose the output score of a DNN into the sum of numerous effects of interactions, with each interaction representing a distinct 3D substructure of the input point cloud. Based on the decomposed interactions, we draw the following conclusions. (1) The common mechanism across different pre-training methods is that they enhance the strength of high-order interactions encoded by DNNs, which represent complex and global 3D structures, while reducing the strength of low-order interactions, which represent simple and local 3D structures. (2) Sufficient pre-training and adequate fine-tuning data for downstream tasks further reinforce the mechanism described above. (3) Pre-training methods carry a potential risk of reducing the transferability of features encoded by DNNs. Inspired by the observed common mechanism, we propose a new method to directly enhance the strength of high-order interactions and reduce the strength of low-order interactions encoded by DNNs, improving performance without the need for pre-training on large-scale datasets. Experiments show that our method achieves performance comparable to traditional pre-training methods.
+
+# 1. Introduction
+
+Self-supervised pre-training methods for 3D point clouds have developed rapidly in recent years [1, 9, 13, 16, 21, 23, 30, 32, 38, 43]. Pre-training methods first train DNNs on large-scale unlabeled datasets, then fine-tune the DNNs on downstream tasks, generally enhancing their performance.
+
+However, the common mechanism underlying different pretraining methods remains unclear, posing challenges for gaining insights into effective model training strategies.
+
+In this paper, we aim to explore the common mechanism behind the performance improvements of different pre-training methods, thereby providing insights into pretraining, and offering better guidance for the training process. Recent studies have employed interactions to explain the reasoning processes of DNNs [20, 25, 41]. Inspired by these studies, we use interactions to provide a unified interpretation of different pre-training methods.
+
+Specifically, given a point cloud  $x$  with  $n$  regions indexed by  $N = \{1,2,\dots,n\}$ , an interaction represents the collaborations among regions within a specific 3D structure  $S \subseteq N$ , where each interaction has a numerical effect  $I(S)$  on the network output. For example, as shown in Fig. 1, the interaction between the regions in  $S_{1} = \{\text{wingtip}, \text{wing root}\}$  form a concept of "wing", contributing  $I(S_{1})$  to push the classification result toward the class "airplane". It has been proven by [6, 44] that the output score of a DNN consistently equals the sum of the effects of all activated interactions, regardless of how the input regions are masked. In this way, interactions can be seen as the detailed inference patterns encoded by the DNN.
+
+Based on interactions, we conduct comparative experiments to explore the common reasons behind the performance improvements of different pre-training methods. Specifically, we explore the impact of pre-training methods on the complexity of interactions encoded by DNNs. Here, the complexity refers to the number of regions contained in an interaction, i.e., the order of the interaction. A high-order interaction, e.g.,  $S_{3}$  in Fig. 1, captures collaborations among massive point regions, representing complex and global 3D structures. In contrast, a low-order interaction, e.g.,  $S_{1}$ , measures collaborations between a few regions, representing simple and local 3D structures. From the experiments, we draw the following key conclusions.
+
+- The common mechanism across different pre-training
+
+![](images/df7338f800ae385b4dddd39cd83328592a92a96ae7466c162041d4f73216074f.jpg)
+
+![](images/571d909e20d4adaf0184bed55e90c7839287c07b105246f692cd9db6857ef0af.jpg)  
+Figure 1. (a) Illustration of how interactions can be used to explain a DNN. Given an input point cloud with  $n$  regions, the output score of the DNN can be decomposed into the sum of the numerical effects of  $2^{n}$  interactions, where each interaction  $S$  encodes the collaborations among the point cloud regions in the set  $S$ . (b) Comparing the strength of interactions across different orders encoded by the DGCNN trained from scratch (scr) and the DGCNN using a pre-training method (pt). Results show that the pre-trained DGCNN encodes stronger high-order interactions and weaker low-order interactions than the DGCNN trained from scratch.
+
+methods is that they enhance the strength of high-order interactions encoded by DNNs while reducing the strength of low-order interactions. This common mechanism indicates that pre-training methods enhance the DNNs' ability to capture global 3D structures, while reducing their reliance on local 3D structures.
+
+- Sufficient pre-training and adequate fine-tuning data for downstream tasks further reinforce the mechanism described above. We observe that the strength of high-order interactions increases with the number of pretraining epochs while the strength of low-order interactions decreases. Additionally, increasing the amount of data for downstream tasks also amplifies this effect.  
+- Pre-training methods carry a potential risk of reducing the transferability of features encoded by DNNs. We observe that the performance of the pre-trained DNNs may decrease on unseen test datasets, possibly due to pretraining methods causing the DNNs to encode high-order interactions with excessively high strength.
+
+Building on the common mechanism we identified, we propose a new method to directly enhance the strength of high-order interactions encoded by DNNs while reducing the strength of low-order interactions. Experimental results on classification and semantic segmentation benchmarks show that our method achieves performance comparable to pre-training methods, without the need for pre-training on large-scale unlabeled datasets.
+
+# 2. Related work
+
+Self-supervised learning (SSL) of 3D point clouds. Recently, 3D point cloud processing has developed rapidly [5, 11, 14, 15, 24, 27, 35, 37, 40], with many self-supervised methods proposed to learn representations from individual 3D objects [1, 9, 13, 16, 21, 23, 30, 32, 38, 43]. The goal of SSL is to design pretext tasks to help the model learn the
+
+data distribution and features in advance, preparing it for downstream tasks. In this paper, we explore the common mechanism behind the performance improvement of the following five widely used open-source pre-training methods.
+
+- Occlusion Completion (OcCo) [32]. OcCo masks occluded points from a camera view and trains an encoder-decoder model to reconstruct these missing points.  
+- Jigsaw [21]. Jigsaw trains a model to reconstruct point clouds with parts rearranged in random order.  
+- Implicit Auto-encoder (IAE) [38]. IAE trains the model as an encoder to map the point clouds to a high-dimensional space and uses a decoder to reconstruct the encoder's outputs back into 3D geometry.  
+- Spatio-Temporal Representation Learning (STRL) [9]. STRL captures spatio-temporal information from 3D sequences by using two temporally correlated frames to learn invariant representations.  
+- CrossPoint [1]. CrossPoint learns transferable representations by maximizing the agreement between 3D point clouds and corresponding 2D images.
+
+Using game-theoretical interactions to explain DNNs. Game-theoretical interactions provide a solid theoretical foundation for explaining DNNs. Ren et al. [17] proposed a mathematical formulation for the concepts encoded by a DNN, while Ren et al. [18] further leveraged these concepts to define optimal baseline values for Shapley values. Li et al. [10] provided a theoretical guarantee that interactions accurately capture the true concepts encoded by a DNN. At the application level, interactions have been widely used to explain the representation capacity of DNNs from various perspectives, including adversarial robustness [17, 34], adversarial transferability [33], and generalization power [41, 45]. In this paper, we use interactions to investigate the common mechanism underlying different pre-training methods for 3D point clouds.
+
+![](images/3289a2b5798488eb7ebe8ac75c333eef9e141c31637aeebe7028b38bc5fad491.jpg)  
+Figure 2. Process of dividing an input point cloud into  $n$  regions.
+
+# 3. Interactions in 3D point cloud processing
+
+Preliminaries: interactions. As a new explanatory metric, interaction has been used to clarify the inference logic [7], generalization power [34], and robustness of a DNN [45]. It can be viewed as a universal measure due to its close theoretical connections with other metrics. As proven by [19], the Harsanyi interaction serves as the basis for existing game-theoretic attributions and interactions, including the Shapley value [22], the Shapley interaction index [8], and the Shapley Taylor interaction index [28]. Please see the supplementary material for additional details.
+
+Quantifying interactions for 3D point cloud processing. We extend interactions to 3D point clouds. Considering an point cloud  $x \in \mathbb{R}^{P \times 3}$ , we divide it into  $n$  regions, as shown in Fig. 2. First, we apply the farthest point sampling (FPS) algorithm to select  $n$  points from the point cloud as the centers of each region. Then, we use the  $k$ -dimensional tree (KDTree) algorithm to assign the remaining points to their nearest region. By doing so, we divide the input point cloud  $x$  into  $n$  regions, indexed by  $N = \{1, 2, \dots, n\}$ .
+
+Given a trained DNN  $v: \mathbb{R}^{P \times 3} \to \mathbb{R}$ , we follow [10, 20, 25] to define the DNN's output score as  $v(x) = \log \frac{p}{1 - p}$  to represent the classification confidence, where  $p$  is the output probability of the ground truth class. Then, the output score can be rewritten as the sum of the numerical effects of all  $2^n$  interactions between the point regions, as follows.
+
+$$
+v (x) = \sum_ {S \subseteq N} I (S). \tag {1}
+$$
+
+Here,  $I(S)$  represents the numerical effect of the interaction among the point regions in  $S \subseteq N$ , defined as follows.
+
+$$
+I (S) \triangleq \sum_ {T \subseteq S} (- 1) ^ {| S | - | T |} \cdot v \left(x _ {T}\right), \tag {2}
+$$
+
+where  $x_{T}$  represents the input point cloud with the regions in  $T \subseteq N$  unchanged, while the regions in  $N \backslash T$  are masked by replacing them with the centroid of the point cloud.
+
+Understanding interactions in 3D point cloud processing. The interaction extracted from the input point cloud  $x$  encodes an AND relationship among the point regions in  $S$ , with the numerical effect  $I(S)$  representing the combined contribution of these regions to the output score  $v(x)$ . As shown in Fig. 1, when the point regions in the set  $S_{1} = \{wingtip, wing root\}$  are unmasked, they form a "wing" pattern and contribute a numerical effect  $I(S_{1})$  that pushes the output score  $v(x)$  towards the "airplane" category. Masking any region in  $S_{1}$  will deactivate this AND
+
+interaction and remove  $I(S_{1})$  from  $v(x)$ . In fact, Tang et al. [29] has proven that interaction satisfies the universal matching property, which states that the DNN's inference score  $v(x_{T})$  can always be faithfully explained as the sum of the numerical effects of all activated interactions, regardless of how the point cloud regions are masked.
+
+Theorem 1 (Universal matching property, proven by [29]). Given an input sample  $x$  with  $n$  variables indexed by  $N = \{1,2,\dots,n\}$ , we can generate  $2^n$  masked samples  $x_{T}$  where  $T \subseteq N$ . Let us construct the following surrogate logical model  $\phi(\cdot)$  to use interactions for inference, which are extracted from the DNN  $v(\cdot)$  on the sample  $x$ . Then, the output of the surrogate logical model  $\phi(\cdot)$  can always match the output of the DNN  $v(\cdot)$ , regardless of how the input sample is masked.
+
+$$
+\begin{array}{l} \forall T \subseteq N, \phi (x _ {T}) = v (x _ {T}), \\ \phi \left(x _ {T}\right) = v \left(x _ {\emptyset}\right) + \sum_ {S \subseteq N} I (S) \cdot \mathbb {1} \binom {x _ {T} \text {t r i g g e r s}} {\text {A N D r e l a t i o n} S} \tag {3} \\ = v (x _ {\emptyset}) + \sum_ {\emptyset \neq S \subseteq T} I (S). \\ \end{array}
+$$
+
+Defining and quantifying the representation complexity of DNNs. The order of an interaction is defined as  $m = |S|$ , which reflects the representation complexity of DNNs. High-order interactions measure the effects of collaborations among massive point cloud regions, representing global and complex 3D structures, while low-order interactions measure the effects of collaborations between a few point regions, representing simple and local 3D structures. We introduce a new metric for measuring the representation complexity of DNNs, as follows.
+
+$$
+\kappa^ {(m)} \triangleq \frac {\mathbb {E} _ {x} \mathbb {E} _ {S \subseteq N , | S | = m} [ | I (S) | ]}{Z}, \tag {4}
+$$
+
+where  $\mathbb{E}$  denotes the mathematical expectation, and  $Z = \mathbb{E}_x\mathbb{E}_{S\subseteq N}[|I(S)|]$  is a normalization term to ensure fair comparisons across different DNNs. Here,  $\kappa^{(m)}$  measures the normalized average strength of the  $m$ -th order interactions. If the value of  $\kappa^{(m)}$  of a high-order is larger than that of a low-order, the DNN's representation complexity is enough to capture global and complex 3D structures. Otherwise, the DNN's representation complexity is limited to encoding only local and simple 3D structures.
+
+We further propose the following metrics to measure the strength of high-order interactions and low-order interactions encoded by the DNN.
+
+$$
+\begin{array}{l} \kappa^ {\text {h i g h}} = \sum_ {m \in \Omega^ {\text {h i g h}}} \kappa^ {(m)}, s. t. \Omega^ {\text {h i g h}} \stackrel {{\text {d e f}}} {{=}} \left\{m \mid \lceil \frac {2}{3} n \rceil <   m \leq n \right\}, \\ \kappa^ {\text {l o w}} = \sum_ {m \in \Omega^ {\text {l o w}}} \kappa^ {(m)}, s. t. \Omega^ {\text {l o w}} \stackrel {{\text {d e f}}} {{=}} \left\{m \mid 1 \leq m \leq \lceil \frac {1}{3} n \rceil \right\}, \end{array} \tag {5}
+$$
+
+where  $\Omega^{\mathrm{high}}$  and  $\Omega^{\mathrm{low}}$  denote the ranges of high-order and low-order interactions, respectively.
+
+![](images/67782b0baf6715b310e74bd4a3d170c23d7160ddc09fccb8198fea6e40ff40e5.jpg)
+
+![](images/7bd74d7397421b56e077a1cf24ea8db8769d47f46f8c4c5ef7afe2e6847ef828.jpg)
+
+![](images/2d4ad121797fb8d91330c85b0aa60c8b6db70c00e96fc0b651c0a4a601792d2e.jpg)
+
+![](images/766c06325e4154bea1e38a0f39b207067288d0d6955c7cf7c8cdf979b57fde53.jpg)
+
+![](images/5cffe96ae4f6bd91c1e2184252f0c73e72a1f8e7e956f170434a17bcd890a18a.jpg)
+
+![](images/2224f7471994640d18c96b588f8ca20940dc565deddc8597e39b0b3ce677752c.jpg)
+
+![](images/06db17dd0b0b4c5e9c0c318dddeb42bcd5217b19031bc05f0ec0541021418599.jpg)  
+Figure 3. [Conclusion 1] (a) Comparing the normalized average strength of interactions encoded by different DNNs, including DNNs trained from scratch and DNNs trained with different pre-training methods. Results show that the DNNs using pre-training methods consistently encode stronger high-order interactions and weaker low-order interactions than the DNNs trained from scratch. (b) The relationship between the strength of high-order interactions encoded by different DNNs and their corresponding classification accuracy. Results show that DNNs encoding stronger high-order interactions tend to exhibit higher accuracy.
+
+![](images/254d67983fdf6831316bb09ceed5141568c4ed51c6ec7dbf0ed8cb2eab5a12cc.jpg)
+
+![](images/de395489ef92dbdb43b95b11a7d31e7ef78124b86d6c6d378c96946f3f6b100f.jpg)
+
+![](images/076650b651e6aa4430796a6f9f858dfef69848080a46eb6b562fc16c0fb90619.jpg)
+
+![](images/e2be93cf1a71bff0a1240735116ef1c0ea2645678d7d602e2b6cd91f0e07c5a4.jpg)
+
+![](images/2b0a90577a7f2b4cd573d46ab5d832b246c463ed621f16630a3d65b902a1901b.jpg)
+
+# 4. Interpreting different pre-training methods using interactions
+
+# 4.1. Comparative study setup
+
+For a given network architecture, we compare the interactions encoded by the model trained from scratch with those encoded by models trained using various pre-training methods. This comparison aims to explore whether these pretraining methods share a common underlying reason for performance improvement, which we define as the common mechanism across these methods. To provide a unified explanation for most pre-training methods, we conduct experiments on five widely used open-source pre-training methods, including IAE [38], STRL [9], CrossPoint [1], OcCo [32] and JigSaw [21], as detailed in Sec. 2.
+
+Networks and datasets. We conduct experiments on three network architectures: DGCNN [35], PointNet [14], and PCN [40]. For DGCNN, we utilize all five pre-training methods, while for PointNet and PCN, we focus on OcCo [32] and Jigsaw [21], depending on the accessibility of open-source implementations for each pre-training method.
+
+To compare the interactions encoded by different DNNs, we use three benchmark datasets for 3D classification task: ModelNet40 [36], ShapeNet² [3], and ScanObjectNN [31]. Tab. 1 shows the statistics of these datasets. We randomly select 10 samples per class from each dataset and use the
+
+<table><tr><td>Name</td><td>Type</td><td># Class</td><td># Training / Testing</td></tr><tr><td>ModelNet</td><td>synthesized</td><td>40</td><td>9,843 / 2,468</td></tr><tr><td>ShapeNet</td><td>synthesized</td><td>16</td><td>12,137 / 4,744</td></tr><tr><td>ScanObjectNN</td><td>real world</td><td>15</td><td>2,304 / 576</td></tr></table>
+
+Table 1. Statistics of datasets for classification.
+
+method in Sec. 3 to divide each point cloud sample into  $n$  regions for quantifying the interactions encoded by DNNs.
+
+# 4.2. Exploring the common mechanism of different pre-training methods
+
+Conclusion 1. The common mechanism across different pre-training methods is that they enhance the strength of high-order interactions encoded by DNNs, while reducing the strength of low-order interactions.
+
+Fig. 3 (a) shows the normalized average strength of the interactions encoded by different DNNs, including DNNs trained from scratch and DNNs using different pre-training methods. Results show that the strength of high-order interactions encoded by the DNNs using pre-training methods is consistently greater than that of the DNNs trained from scratch, across all datasets and network architectures. Conversely, the DNNs using pre-training methods typically encode weaker low-order interactions than the DNNs trained from scratch. Fig. 3 (b) further illustrates the relationship between the strength of high-order interactions and the clas
+
+![](images/3dda22ba09d99ba14a40a722fa9072e88e2e08d51ddd3ca13441f0b237d9a780.jpg)  
+Figure 4. Visualization of interactions encoded by the DGCNN trained from scratch (scr) and the DGCNN pre-trained (pt) with IAE. The pre-trained DGCNN typically encodes stronger high-order interactions and weaker low-order interactions compared to the DGCNN trained from scratch.
+
+![](images/20f4fd5661286d9a212e95d4a2b4c514ec0522bee551dcc0bb8bb8d1903d9b35.jpg)
+
+sification accuracy across different DNNs. We observe that DNNs encoding stronger high-order interactions tend to exhibit higher accuracy. Thus, we regard this shared phenomenon as the common mechanism behind the performance improvement of different pre-training methods, i.e., different pre-training methods generally enhance the strength of high-order interactions encoded by DNNs, while reducing the strength of low-order interactions, as summarized in Conclusion 1.
+
+Conclusion 1 reveals that pre-training methods enhance the ability of DNNs to encode complex and global 3D structures, while reducing their reliance on simple and local 3D structures. As simple and local 3D structures (e.g., a curve, a corner) can appear across different categories, they often lack sufficient classification information, so an over-reliance on them may lead to incorrect classifications. For example, as shown in Fig. 4, the DNN trained from scratch incorrectly classifies a "plant" sample as a "stool". This misclassification may occur because the local structures the DNN learns for the plant, such as the "stem" and the "leaf", are similar to some local structures of a stool, such as the "legs". However, the DNN still encodes a high strength for these local structures (i.e., low-order interactions), which results in an incorrect classification. In contrast, pre-training methods improve the modeling of complex and global 3D structures, allowing DNNs to get a more comprehensive understanding of the input, which in turn enhances their performance. Thus, beyond traditional accuracy metrics, interactions can help identify the potential reasons for classification errors by revealing which 3D structures modeled by the DNN have inappropriate weights, offering a new perspective for debugging.
+
+Comparison with transformer-based pre-training methods. We also measure interactions encoded by transformer-
+
+![](images/2b18729678ddf6c40e4c4532bb609d7399d56e1ef4fd93b55a6b755cc2577e1f.jpg)  
+Figure 5. Comparing the normalized average strength of interactions encoded by (1) transformer-based models, (2) traditional DNNs (e.g., DGCNN and PointNet) trained from scratch, and (3) traditional DNNs using pre-training methods (e.g., DGCNN with IAE, and PointNet with OcCo). Results show that transformer-based models also encode stronger high-order interactions and weaker low-order interactions, exhibiting a similar pattern to traditional DNNs using pre-training methods.
+
+based models, including PointBERT [39], PointMAE [12], PointM2AE [42], and PointGPT [4]. These models integrate pre-training methods into the model architecture, making them incompatible with traditional DNNs (e.g., DGCNN). Therefore, we directly compare the interactions encoded by transformer-based models with the interactions encoded by traditional DNNs, including the DNNs trained from scratch and the DNNs trained with pre-training methods. As shown in Fig. 5, transformer-based models also encode stronger high-order interactions and weaker low-order interactions than traditional DNNs trained from scratch, which exhibit a similar pattern to the interactions encoded by traditional DNNs using pre-training methods. This further supports Conclusion 1.
+
+# 4.3. Exploring the impact of different factors on the common mechanism
+
+We further explore two factors that impact the common mechanism: (a) the extent of pre-training, and (b) the amount of fine-tuning data used for downstream tasks.
+
+Conclusion 2(a). The pre-training process progressively enhances the strength of high-order interactions encoded while weakening the strength of low-order interactions as the extent of pre-training increases.
+
+In this subsection, we first investigate the relationship between the extent of pre-training and the strength of interactions encoded by DNNs. Here, the extent of pre-training refers to the number of pre-training epochs, i.e., the range of epochs from the start of pre-training to the epoch at which pre-training converges. To this end, we conduct experiments on DGCNN with two pre-training methods, including IAE and CrossPoint. For each pre-training method, let  $T_{\mathrm{max}}$  denote the total number of epochs at which the pre-training process of the DNN converges. We select the DNNs at training epochs  $0, 0.2T_{\mathrm{max}}, 0.4T_{\mathrm{max}}, \ldots, T_{\mathrm{max}}$ , covering six different stages of the pre-training process. Then, for all
+
+![](images/a87161cf4e1679b3d5dad2935f636cc2f3b4ffb672ab7dd1d6f0a10ba275ac30.jpg)  
+Figure 6. [Conclusion 2(a)] Comparing the normalized average strength of interactions encoded by DGCNNs pre-trained for different extents, ranging from initial pre-training (0%) to full convergence (100%). As the extent of pre-training increases, the strength of high-order interactions encoded by the DNNs typically rises, while the strength of low-order interactions generally decreases.
+
+![](images/8aa174ef63623ca8ad69b2f8f26b8407fd901be9a1d01012708f253c838129c1.jpg)  
+Figure 7. [Conclusion 2(b)] Comparing the normalized average strength of interactions encoded by DNNs fine-tuned with varying amounts of data. Results show that as the amount of fine-tuning data increases from  $1\%$  to  $100\%$ , the strength of high-order interactions encoded by the DNNs generally increases, while the strength of low-order interactions generally decreases.
+
+DNNs at different pre-training extents, we fine-tune them on the same downstream task and quantify the interactions encoded by these fine-tuned DNNs.
+
+Fig. 6 presents the experimental results. We observe that as the extent of pre-training increases, the strength of high-order interactions encoded by the DNNs generally increases, while the strength of low-order interactions typically decreases. We summarize this relationship between the extent of pre-training and the interactions encoded by DNNs in Conclusion 2(a). This conclusion suggests that sufficient pre-training enhances the model's ability to capture complex and global 3D contexts, further validating the common mechanism outlined in Conclusion 1.
+
+Conclusion 2(b). Increasing the amount of fine-tuning data further enhances the strength of high-order interactions encoded by DNNs, while weakening the strength of low-order interactions.
+
+To investigate the relationship between the amount of fine-tuning data for downstream tasks and the interactions encoded by DNNs, we construct seven training sets of varying sizes from the ModelNet40 dataset, containing  $1\%$ ,  $10\%$ ,  $20\%$ ,  $30\%$ ,  $50\%$ ,  $70\%$ , and  $100\%$  of the original ModelNet40 training data, respectively. Note that we ensure
+
+![](images/037f4d02f5874a1f526c7bb0b41b8a9fd67377bf8b99ad52f1b4877cba002600.jpg)  
+Figure 8. Comparing the classification accuracy and the strength of high-order interactions encoded by different DNNs fine-tuned with varying amounts of data. As the amount of data increases, the accuracy gap between the DNN trained from scratch and the DNN pre-trained with IAE narrows, while the gap in the strength of high-order interactions encoded by these DNNs widens.
+
+at least one sample from each class is included, allowing the model to learn from all categories. We then use the different-sized training sets to fine-tune DGCNNs, including those pre-trained using the IAE method and the Cross-Point method. As shown in Fig. 7, as the amount of fine-tuning data increases, the strength of high-order interactions encoded by DNNs gradually increases, while the strength of low-order interactions decreases. We summarize this relationship between the amount of fine-tuning data and the interactions encoded by DNNs in Conclusion 2(b).
+
+# 4.4. Exploring the potential risk of pre-training methods in reducing DNN's transferability
+
+Conclusion 3. Pre-training methods carry a potential risk of reducing the transferability of features encoded by DNNs.
+
+When exploring the relationship between the amount of fine-tuning data and the interactions encoded by DNNs, we observe the following anomalous phenomenon. As shown in Fig. 8, the gap in classification accuracy between the pretrained DNN and the DNN trained from scratch becomes marginal as the fine-tuning data increases. For example, when the fine-tuning data reaches  $100\%$ , the accuracy gap is only  $0.2\%$ . However, the gap in the strength of high-order interactions between the two DNNs gradually increases, indicating that high-order interactions with excessively high strength are not necessary for performance improvement.
+
+Since high-order interactions generally carry a greater risk of overfitting [41], we investigate the potential risk of pre-training methods in reducing the transferability of features encoded by DNNs. Here, the transferability of features refers to the generalization ability of the features. For example, if the features learned from one dataset (e.g., the features of the airplane class in ModelNet) can be applied to another unseen dataset (e.g., identifying the airplane class in ShapeNet), we consider these features to have high transferability. To this end, we use ShapeNet as the unseen
+
+![](images/524e5286dfb6e543842f6239f5a62352866fbed12fbf6488c0615028f8db39c5.jpg)  
+Figure 9. [Conclusion 3] Comparing the zero-shot classification accuracy of DNNs with and without pre-training, followed by fine-tuning with varying amounts of data. Results show that the zero-shot accuracy of the pre-trained DNN initially exceeds that of the DNN trained from scratch when the fine-tuning data is limited  $(e.g., 1\%)$ , but falls below that of the DNN trained from scratch as the fine-tuning data becomes sufficient  $(e.g., 100\%)$ .
+
+dataset and compare the classification accuracy of different DNNs, including DGCNNs trained with varying amounts of data from ModelNet, as well as DGCNNs pre-trained and then fine-tuned with varying amounts of data. Since the category labels in the two datasets do not completely align, we identify eight common categories. Please see the supplementary material for more implementation details.
+
+Fig. 9 shows the results. We find that when the amount of fine-tuning data is limited (e.g.,  $1\%$ ), pre-trained DNNs, such as the DNN pre-trained with CrossPoint, achieve higher zero-shot accuracy  $(+8.9\%)$  compared to the DNN trained from scratch. In contrast, when the fine-tuning data is sufficient (e.g.,  $100\%$ ), the accuracy of the DNN pretrained with CrossPoint significantly lags behind that of the DNN trained from scratch  $(-14.7\%)$ . We attribute this to pre-training methods causing DNNs to encode high-order interactions with excessively high strength, which in turn reduces the transferability of the features encoded by the DNNs. Note that we are not criticizing the use of pretraining methods to enhance the strength of high-order interactions encoded by DNNs as inherently negative. Rather, we are proposing this potential risk and offering new insights for the design of pre-training methods.
+
+# 5. Guiding the training process using the common mechanism
+
+Traditional pre-training methods, while improving performance, inevitably require extensive pre-training on large-scale unlabeled datasets, which demands considerable time and computational resources. As discussed above, we find that the common mechanism underlying different pre-training methods is that they universally enhance the strength of high-order interactions encoded by DNNs while reducing the strength of low-order interactions. Building on this insight, we propose a new method that directly enhances the strength of high-order interactions encoded by DNNs while reducing the strength of low-order interactions. In this way, our method achieves performance
+
+![](images/b54690b9075dd82112437b300e73e7480e077e9978c3ea391be11a238f70d379.jpg)  
+Figure 10. (a) Curves showing the values of the proposed loss term  $\mathcal{L}_{\text{interaction}}$  for different values of  $\alpha$  throughout the training process. (b) Comparison of the normalized average strength of interactions encoded by DNNs for various  $\alpha$  values in the loss term.
+
+comparable to traditional pre-training methods while avoiding the need for pre-training on large-scale unlabeled datasets. Specifically, we introduce a new heuristic loss term defined as follows.
+
+$$
+\mathcal {L} _ {\text {i n t e r a c t i o n}} = \mathbb {E} _ {x} \left[ \mathbb {E} _ {| S | \in \Omega^ {\mathrm {l o w}}} [ | I (S) | ] - \mathbb {E} _ {| S | \in \Omega^ {\mathrm {h i g h}}} [ | I (S) | ] \right], \tag {6}
+$$
+
+where  $\Omega^{\mathrm{high}}$  and  $\Omega^{\mathrm{low}}$  define the ranges of high-order and low-order interactions, as detailed in Sec. 3. Minimizing the loss term  $\mathcal{L}_{\mathrm{interaction}}$  forces the DNN to weaken the strength of low-order interactions, i.e., decreasing  $\mathbb{E}_x\mathbb{E}_{|S|\in \Omega^{\mathrm{low}}}[|I(S)|]$ , while enhancing the strength of high-order interactions, i.e., increasing  $\mathbb{E}_x\mathbb{E}_{|S|\in \Omega^{\mathrm{high}}}[|I(S)|]$ .
+
+However, computing Eq. (6) is NP-hard. To overcome this challenge, we approximate  $\mathcal{L}_{\mathrm{interaction}}$  using a sampling-based approach. Specifically, given a point cloud  $x$  with  $n$  regions indexed by  $N = \{1,2,\dots,n\}$ , we sample three disjoint subsets  $S_{1}, S_{2}, S_{3} \subseteq N$  where the orders of the subsets  $|S_1|, |S_2|, |S_3| \in \Omega^{\mathrm{low}}$ , with each subset representing a low-order interaction encoded by the DNN. We consider the union  $S_{\mathrm{union}} = S_{1} \cup S_{2} \cup S_{3}$  as a relatively high-order interaction. Then, we can approximate the interaction loss  $\mathcal{L}_{\mathrm{interaction}}$  as follows.
+
+$$
+\mathcal {L} _ {\text {i n t e r a c t i o n}} ^ {\prime} = \mathbb {E} _ {S _ {1}, S _ {2}, S _ {3} \subseteq N} \left[ \mathbb {E} _ {i \in \{1, 2, 3 \}} [ | I (S _ {i}) | ] - | I (S _ {\text {u n i o n}}) | \right]. \tag {7}
+$$
+
+Given a traditional DNN, we incorporate the interaction loss into the training process using the following loss function for the classification task, without the need for additional pre-training on large-scale unlabeled datasets.
+
+$$
+\mathcal {L} = \mathcal {L} _ {\text {c l a s s i f i c a t i o n}} + \alpha \mathcal {L} _ {\text {i n t e r a c t i o n}}, \tag {8}
+$$
+
+where  $\mathcal{L}_{\mathrm{classification}}$  denotes the standard classification loss function (e.g., cross-entropy loss), and  $\alpha > 0$  is the hyperparameter controlling the strength of the interaction loss. Please see Tab. 4 for the effects of varying  $\alpha$ . As shown in Fig. 10 (b), the strength of high-order interactions encoded by the DNN with  $\alpha > 0$  is generally higher than the result when  $\alpha = 0$ , but it does not increase indefinitely as  $\alpha$  grows. This shows the effectiveness of our interaction loss.
+
+Experiments and results analysis. To evaluate the effectiveness of the proposed loss term, we conduct experi-
+
+<table><tr><td>Method</td><td>ModelNet40</td><td>ScanObjectNN</td><td>No Pre-train</td></tr><tr><td>PointNet</td><td>89.2</td><td>68.0</td><td>✓</td></tr><tr><td>PointNet + JigSaw</td><td>89.6</td><td>-</td><td>✗</td></tr><tr><td>PointNet + OcCo</td><td>90.1</td><td>-</td><td>✗</td></tr><tr><td>PointNet + Linteraction (Ours)</td><td>90.1</td><td>69.0</td><td>✓</td></tr><tr><td>DGCNN</td><td>92.5</td><td>78.1</td><td>✓</td></tr><tr><td>DGCNN + JigSaw</td><td>92.6</td><td>83.5</td><td>✗</td></tr><tr><td>DGCNN + OcCo</td><td>93.0</td><td>84.3</td><td>✗</td></tr><tr><td>DGCNN + STRL</td><td>93.1</td><td>-</td><td>✗</td></tr><tr><td>DGCNN + CrossPoint</td><td>92.8</td><td>-</td><td>✗</td></tr><tr><td>DGCNN + IAE</td><td>94.2</td><td>85.6</td><td>✗</td></tr><tr><td>DGCNN + Linteraction (Ours)</td><td>93.3</td><td>79.4</td><td>✓</td></tr><tr><td>CurveNet</td><td>92.8</td><td>79.2</td><td>✓</td></tr><tr><td>CurveNet + Linteraction (Ours)</td><td>93.1</td><td>82.0</td><td>✓</td></tr><tr><td>GDANet</td><td>92.3</td><td>78.7</td><td>✓</td></tr><tr><td>GDANet + Linteraction (Ours)</td><td>92.8</td><td>80.0</td><td>✓</td></tr></table>
+
+Table 2. Classification accuracy (\%) on ModelNet40 and ScanObjectNN datasets. The best results are shown in bold and the second-best results are underlined. Our method achieves results comparable to pre-training methods, while not requiring pretraining on large-scale datasets.
+
+ments on 3D point cloud classification and semantic segmentation tasks. For the classification task, we use the ModelNet40 and ScanObjectNN datasets, as described in Sec. 4.1. Specifically, for the ScanObjectNN dataset, we conduct experiments using the PB_T50_RS variant, which is the most challenging variant. We train PointNet and DGCNN using the proposed loss term and set  $\alpha$  to 0.0005. As shown in Tab. 2, our proposed loss term consistently improves the performance of PointNet, DGCNN, CurveNet [11], and GDANet [37] on both the ModelNet40 and the ScanObjectNN testing splits, compared to their original versions. Moreover, our method demonstrates performance comparable to pre-training methods, without the need for pre-training on large-scale datasets.
+
+For the semantic segmentation task, we conduct experiments on the Stanford Large-Scale 3D Indoor Spaces (S3DIS) dataset [2]. The S3DIS consists of 3D point clouds collected from six distinct large-scale indoor environments, with each point cloud annotated with per-point categorical labels. We randomly subsample 4,096 points from the original point cloud and apply 6-fold cross-validation during fine-tuning. Since the proposed interaction loss is specifically designed for 3D classification, it cannot be directly applied to segmentation tasks. Instead, we adopt a two-stage training approach: first, we train a DNN on the classification task with our interaction loss, and then fine-tune the model on the semantic segmentation task. As shown in Tab. 3, the DGCNN using the proposed loss term achieves  $86.8\%$  overall accuracy and  $59.0\%$  mIoU, outperforming the majority of pre-training methods. Additionally, our loss term also improves the performance of the PointNet.
+
+Effects of the hyper-parameter  $\alpha$ . We train the
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">S3DIS 6-Fold</td></tr><tr><td>OA</td><td>mIoU</td></tr><tr><td>PointNet</td><td>78.5</td><td>47.6</td></tr><tr><td>PointNet + Linteraction (Ours)</td><td>82.1</td><td>50.8</td></tr><tr><td>DGCNN</td><td>84.1</td><td>56.1</td></tr><tr><td>DGCNN + JigSaw</td><td>84.4</td><td>56.6</td></tr><tr><td>DGCNN + OcCo</td><td>85.1</td><td>58.5</td></tr><tr><td>DGCNN + STRL</td><td>84.2</td><td>57.1</td></tr><tr><td>DGCNN + IAE</td><td>85.9</td><td>60.7</td></tr><tr><td>DGCNN + Linteraction (Ours)</td><td>86.8</td><td>59.0</td></tr></table>
+
+Table 3. Semantic segmentation on S3DIS. We report Overall Accuracy (OA) and mean Intersection over Union (mIoU) across six folds. Our method surpasses most pre-training methods.
+
+<table><tr><td>α</td><td>ModelNet40</td><td>ScanObjectNN</td></tr><tr><td>0.0</td><td>92.5</td><td>78.1</td></tr><tr><td>0.0001</td><td>93.0</td><td>79.0</td></tr><tr><td>0.0005</td><td>93.3</td><td>79.4</td></tr><tr><td>0.001</td><td>91.3</td><td>78.1</td></tr></table>
+
+Table 4. Classification accuracy (\%) for DGCNNs trained with varying hyper-parameters  $\alpha$  for the interaction loss.
+
+DGCNN with various interaction loss weights  $\alpha$  and evaluate the testing accuracy, as shown in Tab. 4. The accuracy initially increases and then decreases as  $\alpha$  rises. We attribute this to the loss term enhancing the strength of high-order interactions encoded by the DNN. At lower  $\alpha$  values, the interaction loss improves the DNN's modeling of global 3D structures. However, excessively high values of  $\alpha$  lead to excessively high strength of high-order interactions, increasing the risk of overfitting, as discussed in Conclusion 3. With an appropriately chosen  $\alpha$ , the interaction loss effectively enhances the training process, further supporting the common mechanism outlined in Conclusion 1.
+
+# 6. Conclusion
+
+In this paper, we use interactions to investigate the common mechanism underlying the effectiveness of different pretraining methods for 3D point clouds. Specifically, these methods generally enhance the strength of high-order interactions encoded by DNNs, while reducing the strength of low-order interactions. We then explore the impact of various factors on the mechanism and find that sufficient pretraining and adequate fine-tuning data further reinforce this mechanism. Additionally, we identify a potential risk that pre-training may reduce the transferability of DNNs. Based on the common mechanism, we propose a new method that directly enhances the strength of high-order interactions encoded by DNNs while weakening the strength of low-order interactions. Experiments show that our method achieves performance comparable to pre-training methods, without the need for pre-training on large-scale datasets.
+
+# Acknowledgments
+
+This work is partially supported by the National Nature Science Foundation of China (No. 62206170, 62376199).
+
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+# A Unified Framework for Heterogeneous Semi-supervised Learning
+
+Marzi Heidari*, Abdullah Alchihabi*, Hao Yan*, Yuhong Guo*†  
+*School of Computer Science, Carleton University, Ottawa, Canada†Canada CIFAR AI Chair, Amii, Canada
+
+{marziheidari@cmail., abdullahalchihibi@cmail., haoyan6@cmail., yuhong.guo@}carleton.ca
+
+# Abstract
+
+In this work, we introduce a novel problem setup termed as Heterogeneous Semi-Supervised Learning (HSSL), which presents unique challenges by bridging the semi-supervised learning (SSL) task and the unsupervised domain adaptation (UDA) task, and expanding standard semi-supervised learning to cope with heterogeneous training data. At its core, HSSL aims to learn a prediction model using a combination of labeled and unlabeled training data drawn separately from heterogeneous domains that share a common set of semantic categories. This model is intended to differentiate the semantic categories of test instances sampled from both the labeled and unlabeled domains. In particular, the labeled and unlabeled domains have dissimilar label distributions and class feature distributions. This heterogeneity, coupled with the assorted sources of the test data, introduces significant challenges to standard SSL and UDA methods. Therefore, we propose a novel method, Unified Framework for Heterogeneous Semi-supervised Learning (Uni-HSSL), to address HSSL by directly learning a fine-grained classifier from the heterogeneous data, which adaptively handles the inter-domain heterogeneity while leveraging both the unlabeled data and the inter-domain semantic class relationships for cross-domain knowledge transfer and adaptation. We conduct comprehensive experiments and the experimental results validate the efficacy and superior performance of the proposed Uni-HSSL over state-of-the-art semi-supervised learning and unsupervised domain adaptation methods.
+
+# 1. Introduction
+
+Deep learning models, owing to their hierarchical learned representations and intricate architectures, have monumentally advanced the state-of-the-art across a myriad of tasks [18]. Nonetheless, the success of deep learning has been often contingent on the availability of copious amounts of labeled data. Data annotation, especially in specialized domains, is not only resource-intensive but can also entail exorbitant costs [32]. Consequently, semi-supervised learn
+
+ing (SSL) has been popularly studied, aiming to successfully utilize the free available unlabeled data to help train deep models in an annotation efficient manner [35].
+
+However, current SSL methods assume that the unlabeled and labeled data are sampled from similar (homogeneous) distributions [25]. Such an assumption presents substantial practical limitations to applying traditional SSL methods to a wide range of application domains, where labeled and unlabeled data can have different distributions. For example, in the field of medical imaging, it is common for labeled MRI scans to be sourced from state-of-the-art research hospitals, while an influx of unlabeled scans could emanate from a myriad of rural clinics, each with its distinct scanning equipment and calibration idiosyncrasies. Similar heterogeneity patterns manifest in domains like aerial imagery, wildlife monitoring, and retail product classification. In such settings, the challenge lies in leveraging the unlabeled data given its dissimilarity with its labeled counterpart.
+
+Therefore, to address the current limitations of the traditional SSL, we propose a novel heterogeneous semi-supervised learning (HSSL) task, where the training data consist of labeled and unlabeled data sampled from different distribution domains. The two domains contain a common set of semantic classes, but have different label and class feature distributions. The goal of HSSL is to train a model using the heterogeneous training data so that it can perform well on a held-out test set sampled from both the labeled and unlabeled domains. Without posing distribution similarity assumptions between the labeled and unlabeled data, HSSL is expected to be applicable to a broader range of real-world scenarios compared to standard SSL. This novel heterogeneous semi-supervised learning task however is much more challenging due to the following characteristics: (1) The domain gap, expressed as divergence between class feature distributions across the labeled and unlabeled domains, presents a significant impediment to model generalization and learning. (2) The absence of annotated samples from the unlabeled domain during training further compounds the complexity of the task. (3) Considering that the test set comprises samples from both domains, the devised so
+
+lution methods need to accurately model the distributions inherent to each domain. It is imperative for the models to discern not only the domain from which a sample originates but also the specific semantic class it belongs to. This requires either an explicit or implicit methodology to categorize samples accurately with respect to both domain origin and semantic class categories, distinguishing the task from both conventional SSL and unsupervised domain adaptation (UDA)—traditional SSL overlooks the domain heterogeneity within both the training and testing data, whereas UDA exclusively concentrates on the unlabeled domain as the target domain [11, 21]. Therefore, traditional SSL and UDA methods are not readily applicable or effective in addressing the proposed HSSL task. A recent work [14] has made an effort to expand the traditional SSL task beyond its homogeneous assumptions. However, the proposed solution method learns separately in different domains using distinct components where an off-the-shelf UDA technique is employed to generate pseudo-labels for the unlabeled samples, bypassing the opportunity to train a unified cohesive model that could harness insights from both domains. Furthermore, their test set is confined to a labeled domain, while HSSL aims to train a model that generalizes across labeled and unlabeled domains. HSSL presents a more complex challenge, requiring the model to adapt and perform accurately across heterogeneous test data.
+
+In this work, we propose a novel method, named as Unified framework for Heterogeneous Semi-Supervised Learning (Uni-HSSL), to address the HSSL problem. The proposed method learns a fine-grained classification model cohesively under a unified framework by amalgamating the labeled and unlabeled class categories within an extended and precisely doubled label space. The framework consists of three technical components designed to tackle the HSSL challenges: a weighted moving average pseudo-labeling component, a cross-domain prototype alignment component, and a progressive inter-domain mixup component. The pseudo-labeling component leverages a weighted moving average strategy to assign and update pseudo-labels for the unlabeled data. In this manner, it generates smooth and adaptive assignment of pseudo-labels, reducing the potential pitfalls of oscillating updates or noisy label assignments, which is crucial given the significant domain gap between labeled data and unlabeled data. The cross-domain prototype alignment ensures that the inherent semantic structures of similar classes across the labeled and unlabeled domains are aligned. This alignment of class-centric prototypes between domains leverages inter-domain semantic class relationships, enabling knowledge transfer from the labeled domain to the unlabeled domain. The progressive inter-domain mixup component generates new synthetic instances by interpolating between labeled and unlabeled samples and bridges the gap between the two domains. By adopting a progressive
+
+augmentation schedule, it gradually adapts the model to the distribution of the unlabeled domain, facilitating a steady and reliable knowledge transfer. Comprehensive experiments are conducted on several benchmark datasets. The empirical results demonstrate the efficacy and superior performance of our proposed unified framework compared to multiple state-of-the-art SSL and unsupervised domain adaptation baselines for HSSL.
+
+# 2. Related Works
+
+# 2.1. Semi-Supervised Learning
+
+Conventional Semi-Supervised Learning (SSL) In conventional SSL, the labeled and unlabeled segments of the dataset encompass identical classes, sharing consistent class and feature distributions. SSL methods are primarily classified into three categories: regularization-based techniques, teacher-student models, and pseudo-labeling strategies. Regularization-based techniques like II-model [16] modify the loss function with additional terms for model refinement. Teacher-student models like MT [33] and ICT [37] involve training a student network to mimic a teacher model using unlabeled data. Pseudo-labeling strategies like Pseudo-Label [19], FixMatch [31], FlexMatch [39], and SimMatch [41] expand labeled datasets using unlabeled data with pseudo-labels in various ways.
+
+Open-Set Semi-Supervised Learning (OS-SSL) OS-SSL deals with unknown or additional classes present in the unlabeled data but absent in the labeled set. OS-SSL assumes the same feature distribution over labeled and unlabeled sets. This is different from HSSL, which operates under the assumption that labeled and unlabeled data come from separate domains with different feature distributions. The concept of OS-SSL, introduced in [25], focuses on class distribution mismatches in open-set scenarios. Methods for OS-SSL like UASD [6] use self-distillation to exclude outliers from unlabeled data. DS3L [12] and MTCF [38] employ diverse weighting strategies for subset mismatches, minimizing the impact of private data in unlabeled sets. OpenMatch [3] utilizes one-vs-all classifiers for outlier detection but faces difficulties with unseen categories. While OS-SSL has advanced SSL towards practical use, it lacks capacity to handle feature distribution mismatches between labeled and unlabeled data.
+
+Universal Semi-Supervised Learning (USSL) Universal SSL [13] involves both shared and unique classes across the labeled and unlabeled sets, with the test set matching the labeled set's class distribution. HSSL, however, assumes shared classes across the labeled and unlabeled domains and tests on samples from both domains without their domain identities, adding complexity.
+
+Similar to our work, bidirectional Adaptation [14] addresses the disparity between limited labeled and abundant unlabeled data, but it tests only within the labeled domain's feature distribution. It uses UDA techniques for pseudolabeling, avoiding the complexities and benefits of cross-domain modeling. In contrast, HSSL aims for effective generalization across both domains, posing a more intricate challenge in model adaptation and generalization.
+
+# 2.2. Unsupervised Domain Adaptation
+
+Unsupervised domain adaptation aims at learning a target model given labeled data from a source domain and unlabeled data from a target domain. Typical deep UDA approaches can be categorized into three types: alignment-based, regularization-based, and self-training-based methods. Alignment-based methods aim to reduce the cross-domain feature discrepancy with adversarial alignment [11, 22] and distance-based methods [4, 21, 27, 29]. Regularization-based methods utilize regularization terms to leverage knowledge from the unlabeled target data. Typical regularization terms include entropy minimization [30], virtual adversarial training [30], batch spectral penalization [5], batch nuclear-norm maximization [9], and mutual information maximization [17]. Self-training-based methods explore effective pseudo-labeling for unlabeled target data fitting, including confidence threshold [2, 42] and cycle self-training [20].
+
+# 3. Method
+
+# 3.1. Problem Setup
+
+We consider the following Heterogeneous Semi-Supervised Learning (HSSL) setup. The training data consist of a set of labeled instances  $\mathcal{D}_L = \{(\mathbf{x}_i^l,\mathbf{y}_i^l)\}_{i = 1}^{N_l}$ , where each instance  $\mathbf{x}_i^l$  is annotated with a one-hot label indicator vector  $\mathbf{y}_i^l$  with length  $C$ , and a set of unlabeled instances  $\mathcal{D}_U = \{\mathbf{x}_i^u\}_{i = 1}^{N_u}$ . The labeled data and unlabeled data are from two different domains that have dissimilar label distributions such that  $p_L(\mathbf{y})\neq p_U(\mathbf{y})$  and heterogeneous feature distributions such that  $p_L(\mathbf{x}|\mathbf{y})\neq p_U(\mathbf{x}|\mathbf{y})$ , but share the same set of  $C$  semantic classes. The goal is to train a prediction model using both the labeled set  $\mathcal{D}_L$  and unlabeled set  $\mathcal{D}_U$  so that the trained model would generalize well on a held-out test set that is indistinguishably sampled from both the labeled and unlabeled domains.
+
+# 3.2. Proposed Method
+
+In this section, we present the proposed Uni-HSSL method, which tackles the  $C$ -class HSSL problem by combining the labeled and unlabeled class categories to a doubled label space and learning a fine-grained  $2C$ -class classification model under a unified framework, aiming to adaptively handle the heterogeneous distributions across domains and gain better generalization over test instances randomly sampled
+
+from both the labeled and unlabeled domains. The core idea centers on simultaneously facilitating effective knowledge transfer from the labeled domain to the unlabeled domain while harnessing the information within the unlabeled data.
+
+We start by first pre-training a feature encoder and a  $C$ -class semantic classifier on the labeled dataset, which can be used to produce the initial pseudo-labels of the unlabeled training data and provide partial initialization for our Uni-HSSL model. Then the  $2C$ -class Uni-HSSL model, which consists of a feature encoder  $f$  and a  $2C$ -class classifier  $h$ , will be learned within the proposed unified semi-supervised framework shown in Figure 1. The framework introduces three technical components to facilitate heterogeneous SSL. The weighted-moving-average (WMA) based pseudo-labeling component is deployed to support the effective exploitation of the unlabeled data, while the cross-domain prototype alignment component and progressive inter-domain mixup component are designed to promote information sharing and efficient and steady knowledge transfer from the labeled domain to the unlabeled domain. Further elaboration will be provided in the following sections.
+
+# 3.2.1. Supervised Pre-training
+
+The initial challenge in training a  $2C$ -class classification model with the given heterogeneous data is the absence of labeled instances entirely in the unlabeled domain. To tackle this problem, we exploit the assumption that the labeled and unlabeled domains share the same set of  $C$  semantic class categories, and pre-train a  $C$ -class classification model in the labeled domain to provide initial pseudo-labels for the training instances in the unlabeled domain.
+
+Specifically, we pre-train a  $C$ -class model, which consists of a feature encoder  $f$  and a  $C$ -class probabilistic classifier  $g$ , on the labeled data  $\mathcal{D}_L$  by minimizing the following supervised cross-entropy loss:
+
+$$
+\mathcal {L} _ {c e} ^ {L} = \mathbb {E} _ {\left(\mathbf {x} _ {i} ^ {l}, \mathbf {y} _ {i} ^ {l}\right) \in \mathcal {D} _ {L}} \left[ \ell_ {c e} \left(\mathbf {y} _ {i} ^ {l}, g (f (\mathbf {x} _ {i} ^ {l}))\right) \right] \tag {1}
+$$
+
+where  $\ell_{ce}$  denotes the cross-entropy function. Then we deploy the pre-trained classification model to make predictions on the unlabeled training instances in  $\mathcal{D}_U$  to generate their initial pseudo-labels:
+
+$$
+\bar {\mathbf {y}} _ {i} ^ {0} = g \left(f \left(\mathbf {x} _ {i} ^ {u}\right)\right), \quad \forall \mathbf {x} _ {i} ^ {u} \in \mathcal {D} _ {U} \tag {2}
+$$
+
+where  $\bar{\mathbf{y}}_i^0$  denotes the predicted class probability vector with length  $C$  for the unlabeled instance  $\mathbf{x}_i^u$ . To provide initial labels on the unlabeled data for training the  $2C$ -class model, we further expand each  $\bar{\mathbf{y}}_i^0$  by concatenating it with a zero vector with length  $C$ ,  $\mathbf{0}_C$ :
+
+$$
+\hat {\mathbf {y}} _ {i} ^ {0} = \operatorname {c o n c a t} \left(\mathbf {0} _ {C}, \bar {\mathbf {y}} _ {i} ^ {0}\right) \tag {3}
+$$
+
+This results in the first set of  $C$  classes out of the  $2C$  classes corresponding to the classes in the labeled domain, with
+
+![](images/b2dc7b8e254ecc5e7a924b0093c898cef1b0802eae32003cf24e947489c5e0c3.jpg)  
+Figure 1. An overview of the proposed Uni-HSSL training framework. The classification model consists of a feature encoder  $f$  and a  $2C$ -class classifier  $h$ . After initialization with pre-training, the model is trained by jointly minimizing the combination of a supervised loss  $\mathcal{L}_{cl}^{L}$  on the labeled data, a WMA pseudo-labeling loss  $\mathcal{L}_{pl}^{U}$  on the unlabeled data, a cross-domain prototype alignment loss  $\mathcal{L}_{pa}$ , and a prediction loss  $\mathcal{L}_{\mathrm{Mixup}}$  on the augmentation data produced via progressive inter-domain mixup.
+
+the remaining set of  $C$  classes corresponding to the classes in the unlabeled domain. Moreover, the parameters of the pre-trained  $C$ -class model  $(g \circ f)$  can also be utilized to initialize the feature encoder  $f$  and part of the classifier  $h$  corresponding to the first  $C$  classes in the  $2C$ -class model, while the other part of  $h$  will be randomly initialized.
+
+# 3.2.2. Semi-Supervised Training with Adaptive Pseudo-Labeling
+
+After initialization, the proposed  $2C$ -class classification model (feature encoder  $f$  and probabilistic classifier  $h$ ) will be trained by leveraging both the labeled set  $\mathcal{D}_L$  and the unlabeled set  $\mathcal{D}_U$  within a pseudo-labeling based SSL framework. On the labeled set  $\mathcal{D}_L$ , the following standard supervised cross-entropy loss will be used as the training objective:
+
+$$
+\mathcal {L} _ {c l} ^ {L} = \mathbb {E} _ {\left(\mathbf {x} _ {i} ^ {l}, \mathbf {y} _ {i} ^ {l}\right) \in \mathcal {D} _ {L}} \left[ \ell_ {c e} \left(h \left(f \left(\mathbf {x} _ {i} ^ {l}\right)\right), \operatorname {c o n c a t} \left(\mathbf {y} _ {i} ^ {l}, \mathbf {0} _ {C}\right)\right) \right] \tag {4}
+$$
+
+where the concatenated label vector,  $\operatorname{concat}(\mathbf{y}_i^l, \mathbf{0}_C)$ , expands the ground-truth label vector  $\mathbf{y}_i^l$  into the  $2C$ -class label space by appending a zero vector with length  $C$  to it.
+
+Although we have obtained initial pseudo-labels for the unlabeled set  $\mathcal{D}_U$  by utilizing the pre-trained  $C$ -class classifier, those initial labels are unavoidably noisy due to the existence of domain gap between the labeled and unlabeled domains. In order to effectively leverage the unlabeled data, we update the pseudo-label for each unlabeled instance  $\mathbf{x}_i^u$  during each training iteration in a weighted moving average (WMA) fashion as follows:
+
+$$
+\hat {\mathbf {y}} _ {i} ^ {t} = \beta \hat {\mathbf {y}} _ {i} ^ {t - 1} + (1 - \beta) h \left(f \left(\mathbf {x} _ {i} ^ {u}\right)\right) \tag {5}
+$$
+
+where  $\beta \in (0,1)$  is a hyper-parameter that controls the rate of update, and  $\hat{\mathbf{y}}_i^t$  is the updated pseudo-label for  $\mathbf{x}_i^u$  at the  $t$ -th training iteration. This weighted moving average update strategy can yield a smooth and adaptive assignment of pseudo-labels by promptly incorporating the progress in the classification model and mitigating the risk of oscillatory updates. Moreover, to further mitigate the adverse impact of noisy pseudo-labels, we deploy the following cross-entropy loss on the unlabeled set during training, selectively utilizing only instances with more reliable pseudo-labels:
+
+$$
+\mathcal {L} _ {p l} ^ {U} = \mathbb {E} _ {\mathbf {x} _ {i} ^ {u} \in \mathcal {D} _ {U}} [ \mathbb {1} (\max  (\hat {\mathbf {y}} _ {i} ^ {t}) > \epsilon) \ell_ {c e} (h (f (\mathbf {x} _ {i} ^ {u})), \hat {\mathbf {y}} _ {i} ^ {t}) ] \tag {6}
+$$
+
+where  $\mathbb{1}(\cdot)$  denotes an indicator function;  $\epsilon \in (0,1)$  is a predefined confidence threshold to ensure that only unlabeled instances with the maximum prediction probabilities larger than  $\epsilon$  are used for the current training iteration.
+
+By treating semantic classes in distinct domains as separate categories, the  $2C$ -class classification model serves as a strategic choice to differentiate samples across domains. This approach avoids the additional complexity associated with a dedicated domain classifier and naturally handles the divergence in class-feature distributions across domains. It also simplifies the process and has the potential to enhance domain generalization through a shared feature encoder.
+
+# 3.2.3. Cross-Domain Semantic Prototype Alignment
+
+Given that the labeled domain and unlabeled domain are comprised of the same set of  $C$  semantic classes, there is a one-to-one correspondence relationship between each cross-domain class pair for the same semantic concept. In order to facilitate knowledge sharing and transfer across domains,
+
+we propose to align each semantic class from the labeled domain with its corresponding semantic class in the unlabeled domain within the learned feature embedding space. To this end, we represent each class using a class-prototype vector and design a cross-domain semantic class-prototype alignment component to enforce the corresponding semantic class pairs across the domains are more similar in the feature embedding space than non-corresponding class pairs.
+
+Specifically, we compute the prototype vector for the  $k$ -th class in the labeled set as the average feature embedding of the labeled instances belonging to the class:
+
+$$
+\mathbf {p} _ {k} = \mathbb {E} _ {\left(\mathbf {x} _ {i} ^ {l}, \mathbf {y} _ {i} ^ {l}\right) \in \mathcal {D} _ {L}} \left[ \mathbb {1} \left(\arg \max  _ {j} \mathbf {y} _ {i j} ^ {l} = k\right) f \left(\mathbf {x} _ {i} ^ {l}\right) \right] \tag {7}
+$$
+
+where  $\mathbf{y}_{ij}^{l}$  denotes the  $j$ -th entry of the label vector  $\mathbf{y}_i^l$ . The corresponding  $k$ -th semantic class in the unlabeled set is the  $(C + k)$ -th class in the  $2C$ -class label space. We compute the class prototype vectors in the unlabeled set based on the instances with reliable pseudo-labels, such that:
+
+$$
+\mathbf {p} _ {C + k} = \mathbb {E} _ {\mathbf {x} _ {i} ^ {u} \in \mathcal {D} _ {U}} \left[ \mathbb {1} \left(\max  (\hat {\mathbf {y}} _ {i} ^ {t}) > \epsilon \wedge \right. \left. \arg \max  _ {j} \hat {\mathbf {y}} _ {i j} ^ {t} = C + k\right) f \left(\mathbf {x} _ {i} ^ {u}\right) \right] \tag {8}
+$$
+
+Then for each semantic class  $k \in \{1, \dots, C\}$ , we align the prototypes of the corresponding class pairs from the labeled and unlabeled domains,  $(\mathbf{p}_k, \mathbf{p}_{C + k})$ , by employing a cross-domain contrastive prototype alignment loss as follows:
+
+$$
+\begin{array}{l} \mathcal {L} _ {p a} = - \sum_ {k = 1} ^ {C} \left[ \log \frac {\exp \left(\cos \left(\mathbf {p} _ {k} , \mathbf {p} _ {C + k}\right) / \tau\right)}{\sum_ {k ^ {\prime} = 1} ^ {C} \mathbb {1} \left(k ^ {\prime} \neq k\right) \exp \left(\cos \left(\mathbf {p} _ {k} , \mathbf {p} _ {C + k ^ {\prime}}\right) / \tau\right)} \right. \\ \left. + \log \frac {\exp \left(\cos \left(\mathbf {p} _ {k} , \mathbf {p} _ {C + k}\right) / \tau\right)}{\sum_ {k ^ {\prime} = 1} ^ {C} \mathbb {1} \left(k ^ {\prime} \neq k\right) \exp \left(\cos \left(\mathbf {p} _ {k ^ {\prime}} , \mathbf {p} _ {C + k}\right) / \tau\right)} \right] \tag {9} \\ \end{array}
+$$
+
+where  $\tau$  is a temperature hyper-parameter, and  $\cos (\cdot ,\cdot)$  denotes the cosine similarity function. This contrastive loss promotes the sharing of predictive information between the labeled and unlabeled domains by encouraging the corresponding class prototype pairs to be closer to each other while simultaneously pushing the non-corresponding cross-domain class prototype pairs farther apart.
+
+# 3.2.4. Progressive Inter-Domain Mixup
+
+In order to bridge the gap between the labeled domain and the unlabeled domain, we propose a progressive inter-domain mixup mechanism to augment the training set by dynamically generating synthetic instances between the labeled set and unlabeled set, with the objective of facilitating steady and efficient knowledge transfer from the labeled domain to the unlabeled domain.
+
+Specifically, we generate an inter-domain synthetic instance  $(\mathbf{x}^m,\mathbf{y}^m)$  by mixing a labeled instance  $(\mathbf{x}^l,\mathbf{y}^l)$  from the labeled set  $\mathcal{D}_L$  with a pseudo-labeled instance  $(\mathbf{x}^u,\hat{\mathbf{y}}^t)$
+
+from the unlabeled set  $\mathcal{D}_U$  through linear interpolation:
+
+$$
+\mathbf {x} ^ {m} = \lambda \mathbf {x} ^ {u} + (1 - \lambda) \mathbf {x} ^ {l}, \tag {10}
+$$
+
+$$
+\mathbf {y} ^ {m} = \lambda \hat {\mathbf {y}} ^ {t} + (1 - \lambda) \operatorname {c o n c a t} (\mathbf {y} ^ {t}, \mathbf {0} _ {C}),
+$$
+
+where  $\lambda \in [0,1]$  is the mixing coefficient. To fully utilize the available data in both domains, we can generate  $N^{m} = \max (N^{l},N^{u})$  synthetic instances to form a synthetic set  $\mathcal{D}_{\mathrm{Mixup}}$  by mixing each instance in the larger domain with a randomly selected instance in the other domain.
+
+In the standard mixup [40], the mixing coefficient  $\lambda$  is sampled from a fixed  $\mathrm{Beta}(\alpha, \alpha)$  distribution with hyperparameter  $\alpha$ . To facilitate a steady and smooth adaptation from the labeled domain to the unlabeled domain for HSSL, we propose to dynamically generate the mixup data in each training iteration  $t$  by deploying a progressive mixing up strategy that samples  $\lambda$  from a shifted  $\mathrm{Beta}(\alpha, \alpha)$  distribution based on a schedule function  $\psi(t)$ , such that:
+
+$$
+\lambda \sim \psi (t) \times \operatorname {B e t a} (\alpha , \alpha), \quad \psi (t) = 0. 5 + \frac {t}{2 T} \tag {11}
+$$
+
+where  $T$  denotes the total number of training iterations. Following this schedule, at the beginning of the training process, we have  $\psi(0) \approx 0.5$  and  $\lambda$  is sampled from the approximate interval [0, 0.5) as the model prioritizes the labeled domain, guarding against noisy pseudo-label predictions from unlabeled data. As the training progresses, the model gradually increases its reliance on the unlabeled data, and the interval [0,  $\psi(t)$ ] from which  $\lambda$  is sampled is expanded gradually towards [0, 1] (with  $\psi(T) = 1$ ), allowing it to adapt seamlessly between domains.
+
+Following previous works on using mixup data [1], we employ the mixup set  $\mathcal{D}_{\mathrm{Mixup}}$  for model training by minimizing the following mean squared error:
+
+$$
+\mathcal {L} _ {\text {M i x u p}} = \mathbb {E} _ {\left(\mathbf {x} _ {i} ^ {m}, \mathbf {y} _ {i} ^ {m}\right) \in \mathcal {D} _ {\text {M i x u p}}} \left[ \left\| h \left(f \left(\mathbf {x} _ {i} ^ {m}\right)\right) - \mathbf {y} _ {i} ^ {m}\right) \right\| ^ {2} \tag {12}
+$$
+
+# 3.2.5. Training Objective
+
+By integrating the classification loss terms on the labeled set, the unlabeled set, and the mixup set, with the class prototype alignment loss, we obtain the following joint training objective for the Uni-HSSL model:
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {c l} ^ {L} + \lambda_ {p l} \mathcal {L} _ {p l} ^ {U} + \lambda_ {p a} \mathcal {L} _ {p a} + \lambda_ {\text {M i x u p}} \mathcal {L} _ {\text {M i x u p}} \tag {13}
+$$
+
+where  $\lambda_{pl},\lambda_{pa}$  and  $\lambda_{\mathrm{Mixup}}$  are trade-off hyper-parameters.
+
+# 4. Experiments
+
+# 4.1. Experimental Setup
+
+Datasets We conducted comprehensive experiments to evaluate the performance of our proposed framework on four image classification benchmark datasets: Office-31,
+
+<table><tr><td></td><td>Supervised</td><td>FlexMatch</td><td>FixMatch</td><td>SimMatch</td><td>CDAN+Sup</td><td>MCC+Sup</td><td>BiAdapt</td><td>Uni-HSSL</td></tr><tr><td>A/C</td><td>53.1(0.7)</td><td>51.1(1.2)</td><td>51.9(1.5)</td><td>57.8(1.6)</td><td>47.0(0.5)</td><td>54.9(1.2)</td><td>55.1(1.8)</td><td>60.1(0.9)</td></tr><tr><td>C/A</td><td>66.0(1.2)</td><td>68.1(1.3)</td><td>63.8(0.7)</td><td>69.7(0.9)</td><td>63.9(0.7)</td><td>70.5(0.3)</td><td>65.1(1.2)</td><td>72.0(0.7)</td></tr><tr><td>C/R</td><td>77.5(0.9)</td><td>72.1(0.9)</td><td>79.5(0.5)</td><td>78.5(0.5)</td><td>67.1(0.8)</td><td>75.4(0.5)</td><td>75.2(1.2)</td><td>80.5(0.4)</td></tr><tr><td>R/C</td><td>63.9(1.2)</td><td>67.8(1.6)</td><td>66.2(0.7)</td><td>64.3(0.8)</td><td>67.0(1.2)</td><td>69.3(0.5)</td><td>61.2(1.8)</td><td>72.8(0.6)</td></tr><tr><td>R/A</td><td>72.6(0.9)</td><td>59.0(1.2)</td><td>74.1(0.5)</td><td>70.5(0.5)</td><td>74.6(0.9)</td><td>75.1(0.8)</td><td>69.1(0.9)</td><td>75.8(0.6)</td></tr><tr><td>A/R</td><td>75.1(0.7)</td><td>73.5(0.9)</td><td>70.4(0.6)</td><td>75.8(0.5)</td><td>66.5(0.8)</td><td>77.3(0.8)</td><td>72.1(1.4)</td><td>78.3(0.5)</td></tr><tr><td>A/P</td><td>67.4(1.5)</td><td>64.0(0.9)</td><td>62.7(0.6)</td><td>68.9(0.6)</td><td>56.5(0.5)</td><td>71.8(0.4)</td><td>64.9(1.3)</td><td>70.9(0.8)</td></tr><tr><td>P/A</td><td>69.1(1.0)</td><td>64.1(1.2)</td><td>62.8(0.8)</td><td>69.7(0.9)</td><td>74.9(1.2)</td><td>76.1(0.2)</td><td>64.1(0.8)</td><td>78.7(0.4)</td></tr><tr><td>C/P</td><td>69.1(0.9)</td><td>65.6(1.0)</td><td>65.1(1.1)</td><td>70.0(0.4)</td><td>65.5(1.2)</td><td>71.2(0.5)</td><td>69.1(1.4)</td><td>72.8(0.7)</td></tr><tr><td>P/C</td><td>64.6(0.9)</td><td>64.3(1.1)</td><td>65.2(1.5)</td><td>68.5(0.8)</td><td>66.8(0.6)</td><td>68.0(0.5)</td><td>67.7(0.9)</td><td>69.9(0.9)</td></tr><tr><td>P/R</td><td>80.0(0.5)</td><td>73.3(0.7)</td><td>78.1(0.4)</td><td>78.1(0.2)</td><td>89.5(0.4)</td><td>82.1(0.6)</td><td>76.2(1.2)</td><td>82.9(0.4)</td></tr><tr><td>R/P</td><td>77.9(0.1)</td><td>68.1(1.2)</td><td>74.7(0.3)</td><td>74.0(0.7)</td><td>78.2(1.3)</td><td>77.0(1.2)</td><td>74.1(1.4)</td><td>82.1(0.5)</td></tr><tr><td>Avg.</td><td>69.7</td><td>65.9</td><td>67.9</td><td>70.5</td><td>67.4</td><td>72.3</td><td>67.8</td><td>74.7</td></tr></table>
+
+Table 1. Mean classification accuracy (standard deviation is within parentheses) on the Office-Home dataset using the ResNet-50 backbone. The first domain in each row indicates the labeled domain while the second domain indicates the unlabeled domain.
+
+Office-Home, VisDA, and ISIC-2019. In all four datasets, we split the samples of each domain into 90/10 train/test data. Office-31 [28] is comprised of a collection of 4,652 images spanning 31 different categories. The images are sourced from 3 distinct domains: Amazon (A), DSLR (D), and Webcam (W) with different image resolutions, quality, and lighting conditions. Office-Home [36] is a large collection of over 15,500 images spanning 65 categories. The images are sourced from 4 diverse domains: Artistic images (A), Clip Art (C), Product images (P), and Real-World images (R). VisDA-2017 [26] is a large-scale dataset tailored specifically for the visual domain adaptation task. This dataset includes images of 12 distinct categories from two domains, Synthetic (S) and Real (R). With the significant domain shift between the synthetic and real images, VisDA highlights the difficulties associated with bridging significant domain gaps. ISIC-2019 is a comprehensive repository of skin cancer research images sourced from 4 different sources: BCN-20000 (BCN) [8], Skin Cancer MNIST (HAM) [34], MSK4 [7], and an undefined source. We only utilize BCN and HAM sources as they include samples from all eight distinct classes.
+
+Implementation Details For all baselines we compared our Uni-HSSL against, we strictly followed the implementation details and hyper-parameters specified in the corresponding original papers. In order to ensure consistent comparisons with a multitude of earlier studies across various benchmark datasets, we employed two common backbone networks: ResNet-50 and ResNet-101 which are pre-trained on the ImageNet [10] dataset. We utilized ResNet-101 for VisDA dataset experiments and ResNet-50 for all the other benchmark datasets. The supervised pre-training stage is made up of 10 epochs while the semi-supervised training stage is made up of 100 epochs. In both stages, we em
+
+ployed an SGD optimizer with a learning rate of  $5e^{-4}$  and Nesterov momentum [24] of 0.9. In the semi-supervised training stage, the learning rate is adjusted using a cosine annealing strategy [23, 37]. We set the L2 regularization coefficient to  $1e^{-3}$  and the batch size to 32 for all datasets. The trade-off hyper-parameters  $\lambda_{pl},\lambda_{pa},\lambda_{\mathrm{Mixup}}$  take the values 1,  $1e^{-2}$  and 1 respectively, while  $\tau$  and  $\epsilon$  take the value 0.5 and  $\beta$  is set to 0.8. Furthermore, similar to [1], we apply random translations and horizontal flips to the input images prior to applying the Progressive Inter-Domain Mixup. We report the mean classification accuracy and the corresponding standard deviation over 3 runs in each experiment.
+
+# 4.2. Comparison Results
+
+We evaluate the proposed Uni-HSSL framework on the heterogeneous semi-supervised learning tasks and compare it to four categories of baselines: Supervised Learning baselines, Semi-Supervised Learning (SSL) baselines, Unsupervised Domain Adaptation (UDA) baselines, and Bidirectional Adaptation baselines. The supervised baseline is exclusively trained on the labeled data and does not leverage the unlabeled data during training. We employ a set of representative SSL baselines (FlexMatch [39], FixMatch [31], and SimMatch [41]) and a set of representative UDA baselines (CDAN [22] and MCC [15]). In particular, we also compare our work with the state-of-the-art bidirectional adaptation method (BiAdapt) [14]. As the traditional UDA methods are trained to perform well solely on an unlabeled target domain, to ensure a fair comparison, we equip the UDA methods with a Supervised classifier (Sup) trained on the labeled set and a domain classifier and refer to them as  $\mathrm{MCC} + \mathrm{Sup}$  and  $\mathrm{CDAN} + \mathrm{Sup}$ . At inference time, the domain classifier assigns each test sample to the appropriate classifier in the corresponding domain—either the supervised classifier for samples predicted to originate from the labeled domain or
+
+<table><tr><td></td><td>Supervised</td><td>FlexMatch</td><td>FixMatch</td><td>SimMatch</td><td>CDAN+Sup</td><td>MCC+Sup</td><td>BiAdapt</td><td>Uni-HSSL</td></tr><tr><td>Plane</td><td>93.8(0.2)</td><td>98.3(0.7)</td><td>94.9(0.5)</td><td>93.6(0.8)</td><td>98.4(0.3)</td><td>98.6(0.3)</td><td>90.1(1.4)</td><td>98.2(0.5)</td></tr><tr><td>Bicycle</td><td>74.1(0.5)</td><td>74.8(0.9)</td><td>53.5(0.2)</td><td>81.1(0.8)</td><td>94.4(0.7)</td><td>96.6(0.5)</td><td>79.1(1.2)</td><td>97.5(0.9)</td></tr><tr><td>Bus</td><td>79.4(0.7)</td><td>53.9(1.2)</td><td>79.5(0.8)</td><td>56.9(1.2)</td><td>90.1(0.5)</td><td>88.6(0.7)</td><td>54.7(1.3)</td><td>91.4(0.8)</td></tr><tr><td>Car</td><td>86.2(0.9)</td><td>36.4(2.1)</td><td>88.5(0.3)</td><td>59.6(1.5)</td><td>85.1(0.5)</td><td>84.8(0.9)</td><td>56.1(1.2)</td><td>89.0(0.9)</td></tr><tr><td>Horse</td><td>90.9(0.2)</td><td>97.4(0.5)</td><td>76.0(0.8)</td><td>65.6(1.0)</td><td>96.6(0.1)</td><td>97.6(0.3)</td><td>62.1(1.4)</td><td>98.2(0.3)</td></tr><tr><td>Knife</td><td>87.5(0.7)</td><td>77.2(0.8)</td><td>78.8(0.9)</td><td>71.9(0.5)</td><td>95.0(1.4)</td><td>95.1(0.9)</td><td>68.2(0.1)</td><td>98.9(0.4)</td></tr><tr><td>Motor.</td><td>94.5(0.4)</td><td>66.6(1.2)</td><td>40.8(1.2)</td><td>70.8(0.9)</td><td>96.6(0.5)</td><td>94.2(0.2)</td><td>68.1(1.5)</td><td>97.0(0.6)</td></tr><tr><td>Person</td><td>80.0(0.7)</td><td>80.5(0.8)</td><td>58.9(1.6)</td><td>64.1(0.8)</td><td>94.3(0.6)</td><td>94.6(0.2)</td><td>62.5(1.2)</td><td>95.6(0.7)</td></tr><tr><td>Plant</td><td>91.1(0.7)</td><td>91.8(0.8)</td><td>62.7(0.7)</td><td>65.5(0.9)</td><td>96.5(0.4)</td><td>97.3(0.5)</td><td>63.5(1.9)</td><td>95.7(0.2)</td></tr><tr><td>Skateboard</td><td>81.8(0.9)</td><td>90.0(0.5)</td><td>68.9(1.2)</td><td>57.0(1.7)</td><td>85.5(0.5)</td><td>83.0(0.8)</td><td>59.3(1.7)</td><td>91.5(0.8)</td></tr><tr><td>Train</td><td>96.0(0.3)</td><td>96.8(0.7)</td><td>94.2(0.4)</td><td>74.2(0.9)</td><td>95.7(0.7)</td><td>95.6(0.1)</td><td>71.3(1.5)</td><td>97.0(0.3)</td></tr><tr><td>Truck</td><td>59.8(0.9)</td><td>49.2(1.2)</td><td>49.5(1.2)</td><td>52.1(1.7)</td><td>79.8(0.2)</td><td>80.6(1.0)</td><td>50.1(1.5)</td><td>82.4(0.7)</td></tr><tr><td>Avg.</td><td>84.1</td><td>82.4</td><td>87.3</td><td>80.8</td><td>92.1</td><td>92.0</td><td>79.1</td><td>93.1</td></tr></table>
+
+Table 2. Mean classification accuracy (standard deviation is within parentheses) on the VisDA dataset using the ResNet-101 backbone. The rows correspond to the different classes of the dataset.  
+
+<table><tr><td></td><td>B/H</td><td>H/B</td><td>Avg.</td></tr><tr><td>Supervised</td><td>70.5(0.9)</td><td>65.4(1.2)</td><td>67.9</td></tr><tr><td>FlexMatch</td><td>71.3(1.4)</td><td>68.7(0.8)</td><td>70.0</td></tr><tr><td>FixMatch</td><td>77.5(0.8)</td><td>65.0(0.7)</td><td>71.3</td></tr><tr><td>SimMatch</td><td>75.1(1.5)</td><td>69.2(1.7)</td><td>72.2</td></tr><tr><td>CDAN+Sup</td><td>72.9(1.0)</td><td>65.2(0.4)</td><td>69.1</td></tr><tr><td>MCC+Sup</td><td>60.2(1.8)</td><td>56.7(1.7)</td><td>58.7</td></tr><tr><td>BiAdapt</td><td>74.2(0.7)</td><td>68.3(1.3)</td><td>71.2</td></tr><tr><td>Uni-HSSL</td><td>79.9(0.7)</td><td>71.0(0.9)</td><td>75.4</td></tr></table>
+
+Table 3. Mean classification accuracy (standard deviation is within parentheses) on ISIC-2019 using the ResNet-50 backbone. The first domain in each row indicates the labeled domain the second domain indicates the unlabeled domain.
+
+the UDA classifier for those predicted to come from the unlabeled domain.
+
+The comparison results on the Office-Home, VisDA, ISIC-2019 and Office-31 datasets are reported in Tables 1, 2, 3 and 4, respectively, where the first domain indicates the labeled domain and the second domain indicates the unlabeled domain. In the case of the VisDA dataset, the labeled dataset is sampled from the synthetic domain (S) and the unlabeled dataset is sampled from the real domain (R) and we report the average classification accuracy for each class and the overall average classification accuracy. The tables show that Uni-HSSL consistently outperforms all baselines on all datasets across all setups. The performance gains over the supervised baseline are notable exceeding  $9\%$ ,  $4\%$ ,  $9\%$ , and  $7\%$  on average in the cases of the Office-31, Office-Home, VisDA, and ISIC-2019 datasets, respectively. In the case of the VisDA dataset, the performance improvement over the supervised baseline at the class level is substantial, exceeding  $22\%$  for some classes. Furthermore, Uni-HSSL consistently outperforms all the SSL baselines, achieving
+
+performance gains exceeding  $3\%$ ,  $4\%$ ,  $5\%$ , and  $3\%$  over the most effective SSL baselines on the Office-31, Office-Home, VisDA, and ISIC-2019 datasets, respectively. In some cases, such as A/W on Office-31 and P/A on Office-Home, the performance improvement over SSL baselines is notable, surpassing  $6\%$  and  $8\%$ , respectively, highlighting the limitations of traditional SSL baselines in the proposed HSSL task. In the case of the UDA baselines, Uni-HSSL yields superior performance with all domain setups on all four datasets with performance gains around  $4\%$ ,  $2\%$ ,  $1\%$ , and  $6\%$  on Office-31, Office-Home, VisDA and ISIC-2019 datasets, respectively. Uni-HSSL outperforms the UDA baselines on almost all classes of the VisDA dataset, with the UDA baselines slightly excelling in only two classes. However, Uni-HSSL still maintains superior overall performance compared to the UDA baselines. Furthermore, the MCC+Sup baseline does not perform well on the ISIC-2019 dataset, where it suffers a major drop in performance which can be attributed to the MCC baseline's sensitivity to the class imbalance inherent in this dataset. Moreover, our Uni-HSSL also substantially outperforms BiAdapt, with performance gains surpassing  $5\%$ ,  $6\%$ ,  $14\%$ , and  $4\%$  on the Office-31, Office-Home, VisDA and ISIC-2019 datasets, respectively. These results underscore the robustness of Uni-HSSL and highlight the limitations of BiAdapt in effectively addressing the challenges posed by the proposed HSSL task.
+
+# 4.3. Ablation Study
+
+In order to investigate the contribution of each component of the proposed framework, we conducted an ablation study to compare the proposed Uni-HSSL with its six variants: (1) “-w/o WMA”, which drops the Weighted Moving Average component of the pseudo-label update and simply uses the model predictions to generate pseudo-labels; (2) “-w/o  $\mathcal{L}_{\mathrm{cl}}^{L}$ ”, which drops the cross-entropy classification loss on the la
+
+<table><tr><td></td><td>W/A</td><td>A/W</td><td>A/D</td><td>D/A</td><td>D/W</td><td>W/D</td><td>Avg.</td></tr><tr><td>Supervised</td><td>68.6(1.6)</td><td>82.8(1.2)</td><td>85.1(1.8)</td><td>35.5(0.9)</td><td>96.9(0.4)</td><td>98.2(0.5)</td><td>77.8</td></tr><tr><td>FlexMatch</td><td>68.1(1.8)</td><td>81.3(1.3)</td><td>85.1(1.8)</td><td>63.0(2.1)</td><td>98.5(0.2)</td><td>98.9(0.2)</td><td>82.4</td></tr><tr><td>FixMatch</td><td>69.1(1.3)</td><td>83.4(0.9)</td><td>86.4(0.8)</td><td>53.7(1.3)</td><td>98.1(0.2)</td><td>98.2(0.2)</td><td>81.5</td></tr><tr><td>SimMatch</td><td>71.1(0.9)</td><td>84.1(1.0)</td><td>86.5(0.5)</td><td>68.6(1.1)</td><td>96.8(0.5)</td><td>98.8(0.4)</td><td>84.3</td></tr><tr><td>CDAN+Sup</td><td>61.2(1.2)</td><td>82.5(1.3)</td><td>87.4(2.2)</td><td>58.3(2.6)</td><td>79.2(0.4)</td><td>97.5(0.4)</td><td>77.7</td></tr><tr><td>MCC+Sup</td><td>71.5(2.7)</td><td>88.8(0.7)</td><td>89.1(0.5)</td><td>67.6(1.3)</td><td>81.7(0.7)</td><td>99.5(0.4)</td><td>83.0</td></tr><tr><td>BiAdopt</td><td>70.2(0.9)</td><td>85.0(0.5)</td><td>77.4(0.7)</td><td>67.1(1.0)</td><td>94.2(0.5)</td><td>98.5(0.3)</td><td>82.0</td></tr><tr><td>Uni-HSSL</td><td>73.1(1.0)</td><td>90.2(0.8)</td><td>90.0(0.2)</td><td>72.1(0.7)</td><td>100(0.0)</td><td>100(0.0)</td><td>87.5</td></tr></table>
+
+Table 4. Mean classification accuracy (standard deviation is within parentheses) on the Office-31 dataset using the ResNet-50 backbone. The first domain in each column indicates the labeled domain the second domain indicates the unlabeled domain.  
+
+<table><tr><td></td><td>W/A</td><td>A/W</td><td>A/D</td><td>D/A</td><td>D/W</td><td>W/D</td><td>Avg.</td></tr><tr><td>Uni-HSSL</td><td>73.1(1.0)</td><td>90.2(0.8)</td><td>90.0(0.2)</td><td>72.1(0.7)</td><td>100(0.0)</td><td>100(0.0)</td><td>87.5</td></tr><tr><td>-w/o WMA</td><td>72.8(0.5)</td><td>87.1(0.8)</td><td>88.3(0.9)</td><td>71.0(0.8)</td><td>100(0.0)</td><td>100(0.0)</td><td>86.8</td></tr><tr><td>-w/o \( \mathcal{L}_{cl}^{L} \)</td><td>67.6(1.7)</td><td>85.5(0.8)</td><td>86.1(1.2)</td><td>64.8(2.0)</td><td>93.2(0.5)</td><td>92.9(0.6)</td><td>81.7</td></tr><tr><td>-w/o \( \mathcal{L}_{pl}^{U} \)</td><td>72.5(0.8)</td><td>87.9(0.9)</td><td>88.1(0.7)</td><td>71.0(0.9)</td><td>98.0(0.2)</td><td>98.5(0.2)</td><td>86.1</td></tr><tr><td>-w/o \( \mathcal{L}_{pa} \)</td><td>72.7(0.5)</td><td>88.9(0.7)</td><td>87.2(0.6)</td><td>71.3(0.9)</td><td>99.1(0.0)</td><td>100(0.0)</td><td>86.5</td></tr><tr><td>-w/o \( \mathcal{L}_{Mixup} \)</td><td>71.9(1.2)</td><td>86.7(0.9)</td><td>88.1(0.8)</td><td>71.3(1.1)</td><td>98.0(0.4)</td><td>99.9(0.0)</td><td>86.1</td></tr><tr><td>-w/o Prog. Mixup</td><td>71.3(0.9)</td><td>84.8(0.9)</td><td>88.1(1.0)</td><td>70.0(1.3)</td><td>99.2(0.5)</td><td>99.9(0.0)</td><td>85.6</td></tr></table>
+
+Table 5. Ablation study results in terms of mean classification accuracy (standard deviation is within parentheses) on the Office-31 dataset using the ResNet-50 backbone. The first domain in each column indicates the labeled domain while the second domain indicates the unlabeled domain.
+
+beled set  $\mathcal{D}_L$ ; (3) “ $-\mathrm{w/o}\mathcal{L}_{\mathrm{pl}}^U$ , which drops the cross-entropy pseudo-label classification loss on the unlabeled set  $\mathcal{D}_U$ ; (4) “ $-\mathrm{w/o}\mathcal{L}_{\mathrm{pa}}$ , which drops the Cross-Domain Prototype Alignment component; (5) “ $-\mathrm{w/o}\mathcal{L}_{\mathrm{Mixup}}$ , which drops the Progressive Inter-Domain Mixup component; and (6) “ $-\mathrm{w/o}$  Prog. Mixup”, which drops the progressive component of the Inter-Domain Mixup and uses a simple mixup for inter-domain data augmentation. We compare the proposed UniHSSL with all the six variants on the Office-31 dataset and report the results in Table 5.
+
+From the table, we can see that dropping any component from the proposed unified framework results in performance degradation in all cases. “-w/o  $\mathcal{L}_{\mathrm{cl}}^{L}$ ” variant suffered the largest performance degradation, which highlights the importance of the ground-truth labels of  $\mathcal{D}_L$  in guiding the learning process of the framework. Dropping the WMA from the pseudo-label generation component led to a slight average performance drop to  $86.8\%$ , underscoring its role in obtaining stable and confident pseudo-labels. Similarly, dropping the classification loss on the unlabeled data  $\mathcal{L}_{pl}^{U}$  led to a performance degradation to  $86.1\%$ . Furthermore, the variant “-w/o Prog. Mixup” suffers a larger drop in performance in comparison with the variant “-w/o  $\mathcal{L}_{\mathrm{Mixup}}$ ”, which highlights the importance of progressively generating the augmented samples to ensure the accuracy of their corresponding augmented labels. Generating inter-domain augmented samples without taking into account the domain gap between the labeled domain and unlabeled domain can
+
+lead to a degradation in performance due to the noisy augmented labels of the generated samples. Overall, the consistent performance drops across all the tasks of Office-31 for each variant validate the essential contribution of each corresponding component of the Uni-HSSL framework.
+
+# 5. Conclusion
+
+In this paper, we introduced a challenging heterogeneous semi-supervised learning problem, where the labeled and unlabeled training data come from different domains and possess different label and class feature distributions. To address this demanding setup, we proposed a Unified Framework for Heterogeneous Semi-Supervised Learning (Uni-HSSL), which trains a fine-grained classification model over the concatenated label space by effectively exploiting the labeled and unlabeled data as well as their relationships. Uni-HSSL adopts a WMA pseudo-labeling strategy to obtain stable and confident pseudo-labels for the unlabeled data, while deploying a cross-domain class prototype alignment component to support knowledge transfer and sharing between domains. A novel progressive inter-domain mixup component is further devised to augment the training data and bridge the significant gap between the labeled and unlabeled domains. The experimental results demonstrate the effectiveness and superiority of the proposed Uni-HSSL over state-of-the-art semi-supervised learning methods and unsupervised domain adaptation baselines.
+
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CVPR/2025/A Unified Image-Dense Annotation Generation Model for Underwater Scenes/full.md

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+# A Unified Image-Dense Annotation Generation Model for Underwater Scenes
+
+Hongkai Lin Dingkang Liang Zhenghao Qi Xiang Bai* Huazhong University of Science and Technology {hklin,dkliang,xbai}@hust.edu.cn
+
+![](images/dbc64f5ab09687b03fea39ce38291518f31f2ee783bc816eaaa6bfb11ba196c5.jpg)  
+Figure 1. We present TIDE, a unified underwater image-dense annotation generation model. Its core lies in the shared layout information and the natural complementarity between multimodal features. Our model, derived from the text-to-image model and fine-tuned with underwater data, enables the generation of highly consistent underwater image-dense annotations from solely text conditions.
+
+# Abstract
+
+Underwater dense prediction, especially depth estimation and semantic segmentation, is crucial for gaining a comprehensive understanding of underwater scenes. Nevertheless, high-quality and large-scale underwater datasets with dense annotations remain scarce because of the complex environment and the exorbitant data collection costs. This paper proposes a unified Text-to-Image and DEnse annotation generation method (TIDE) for underwater scenes. It relies solely on text as input to simultaneously generate realistic underwater images and multiple highly consistent dense annotations. Specifically, we unify the generation of
+
+text-to-image and text-to-dense annotations within a single model. The Implicit Layout Sharing mechanism (ILS) and cross-modal interaction method called Time Adaptive Normalization (TAN) are introduced to jointly optimize the consistency between image and dense annotations. We synthesize a large-scale underwater dataset using TIDE to validate the effectiveness of our method in underwater dense prediction tasks. The results demonstrate that our method effectively improves the performance of existing underwater dense prediction models and mitigates the scarcity of underwater data with dense annotations. We hope our method can offer new perspectives on alleviating data scarcity issues in other fields. The code is available at https://github.com/HongkLin/TIDE.
+
+# 1. Introduction
+
+Underwater dense prediction, particularly depth estimation and semantic segmentation, is essential for underwater exploration and environmental monitoring. However, the complex environment and the prohibitive data collection costs result in a scarcity of underwater data with dense annotations. Such conditions severely hinder the advancement of dense prediction technologies in underwater scenes.
+
+Fortunately, the recent success of the image generative technique [14, 28, 42] provides a breakthrough in addressing the scarcity of underwater scene data. In the field of general object understanding, controllable data synthesis [22, 30, 31, 37] demonstrates its effectiveness in few-shot scenarios. A straightforward solution is to apply them to underwater scenes directly. For instance, Atlantis [41], a pioneering controllable generation method for underwater depth data that takes ControlNet as its core, utilizes terrestrial depth maps as conditions. It effectively mitigates the issue of scarce underwater depth data and achieves consistent performance improvements across multiple underwater depth datasets and models.
+
+Despite remarkable progress, there are still challenges in Atlantis, as follows: 1) Atlantis, as shown in Fig. 2(a), generates underwater depth data using terrestrial depth maps as conditions due to the lack of underwater depth maps. It is considered a suboptimal approach since it may not align with natural underwater scenes. Better recreating authentic underwater environments is equally essential. 2) It generates data with only a single type of dense annotations, which is insufficient for understanding complex underwater scenes. Thus, a natural question arises: How can we simultaneously generate highly consistent, one-to-many, and vivid underwater images and dense annotation pairs?
+
+In this paper, we explore the possibility of simultaneously generating highly consistent, realistic underwater scene images and multiple types of dense annotations using only text conditions. Our approach, which we refer to as TIDE, is illustrated in Fig. 2(b), presents a unified Text-to-Image and DEnse annotation generation method. TIDE is an end-to-end training and inference model that integrates denoising models in parallel for both text-to-image generation and text-to-dense annotation generation.
+
+To align the images and multiple type dense annotations generated by parallel denoising models, we propose the Implicit Layout Sharing (ILS) mechanism. Specifically, the cross-attention map as an implicit layout is the key to controlling the image layout in the text-to-image model [6, 28], inspiring us to share the implicit layout for aligning images and dense annotations. ILS effortlessly replaces the cross-attention map in the text-to-dense annotation model with that from the text-to-image model, effectively improving the consistency between the image and dense annotations. Furthermore, considering the intrinsic
+
+![](images/17ad3b11a489e5895d1163ec2e425ab8073bebc39c5a14b448aac4484eadf16e.jpg)  
+Figure 2. The comparison between Atlantis [41] and our method. Unlike Atlantis, which requires text and depth map conditions, our method only needs text as the input condition to generate image-dense annotations (e.g., depth maps and semantic masks).
+
+complementarity between features of different modalities, we introduce a cross-modal interaction method called Time Adaptive Normalization (TAN), a normalization layer that modulates the activations using different modal features. The consistency of the image and dense annotations can further be jointly optimized through cross-modal feature interaction among different dense annotation generation and between image and dense annotation generation.
+
+To verify the effectiveness of our method, we use TIDE to generate a large-scale dataset of underwater images with dense annotations named SynTIDE. Extensive experiments demonstrate the effectiveness of SynTIDE for underwater dense prediction tasks. In the underwater depth estimation task, SynTIDE presents consistent improvements in various fine-tuning models. For example, when adopting representative NewCRFs [40] as the fine-tuning model, our approach achieves significance gains over previous work, particularly in the  $SI_{log}$  and  $\delta_1$  metrics, with improvements of 14.73 and 36% on the D3 and D5 subsets of Sea-thru [3] dataset, respectively. In underwater semantic segmentation, pre-training with SynTIDE yields consistent improvements across different models. For instance, when using ViT-Adapter [7] as the training model, pre-training with the SynTIDE dataset leads to improvements of 2.1% mIoU on the USIS10K [20] dataset.
+
+TIDE demonstrates powerful data generation capabilities for underwater scenes. Using only easily accessible text prompts, TIDE can generate highly consistent and realistic underwater images and multiple types of dense annotations. It holds potential as a mainstream data synthesis method for underwater scenes and offers a promising direction for alleviating data scarcity in other fields. The main contribu
+
+tions of this work are as follows: 1) We propose a novel data synthesis method, TIDE, which uses text as the sole condition to generate images and their corresponding multitype dense annotations simultaneously. To our knowledge, TIDE is the first method capable of simultaneously synthesizing both images and multiple dense annotations from text. 2) To align the images and dense annotations, we introduce the Implicit Layout Sharing mechanism. The text-to-image and text-to-dense annotation models share the same layout information, ensuring proper alignment between the image and dense annotations. Meanwhile, the consistency between image and dense annotations can be further optimized through the cross-modal interaction method called Time Adaptive Normalization.
+
+# 2. Relate Work
+
+# 2.1. Underwater Dense Prediction
+
+Dense prediction tasks in underwater scenes are crucial for comprehensively understanding underwater scenes. The publication of SUIM [16] provides a fundamental dataset and benchmark for the exploration of underwater semantic segmentation. To fill the gap in underwater instance segmentation, WaterMask [19] publishes the UIIS dataset, and a model is designed to cater to the unique characteristics of underwater images, improving the accuracy of underwater instance segmentation. Recently, the rise of general foundational segmentation models [17, 27] drives further development in the field of underwater segmentation [20, 36, 43].
+
+Due to the lack of underwater depth estimation datasets, most underwater depth estimation methods focus on traditional techniques, unsupervised, or self-supervised approaches. Traditional methods [9] mainly rely on statistical priors, such as the dark channel prior [12], to estimate underwater depth. Gupta et al. [10] model the relationship between underwater and above-water hazy appearances to depth estimation. UW-GAN [11] and Atlantis [41] improve the performance of underwater depth estimation by synthesizing training datasets through generative models.
+
+While these methods make notable contributions to underwater dense prediction tasks, the large-scale and high-quality dataset in underwater scenes with only segmentation or depth annotations remains insufficient for achieving comprehensive underwater scene understanding.
+
+# 2.2. Controllable Data Synthesis
+
+Thanks to the success of diffusion models [14] and the availability of large-scale, high-quality text-image training data, text-to-image models [6, 23, 26, 28] and controllable image generation models [21, 35, 42] achieve unprecedented success in image quality, diversity, and consistency.
+
+He et al. [13] are the first to explore and demonstrate the effectiveness of state-of-the-art text-to-image genera
+
+tion models for image recognition. This makes it possible to achieve diverse data collection and accurate annotation at a lower cost. Wu et al. and Nguyen et al. [22, 31] explore the ability of pre-trained diffusion models to enhance real data in few-shot settings for segmentation tasks. Diffumask [32] ingeniously combines text-to-image models with AffinityNet [2], achieving open-vocabulary segmentation data synthesis. Freemask [37] demonstrates that synthetic data can further enhance the performance of semantic segmentation models under fully supervised settings by incorporating freestyle, a controllable image generation method using semantic masks as input conditions. Seggen [39] designs a multi-stage semantic segmentation data synthesis method, text2mask and mask2image, which achieves high semantic consistency semantic segmentation data only using text as the condition. Detdiffusion [30] synthesizes object detection data by incorporating object categories and spatial coordinates into the text.
+
+Unlike the aforementioned single-task data synthesis methods, we propose a novel end-to-end underwater data synthesis approach that simultaneously generates semantic masks and depth maps, relying solely on text conditions.
+
+# 3. Preliminaries
+
+Diffusion Models (DMs) [14] emerge as leading text-to-image (T2I) generation models, recognized for their ability to produce realistic images. DMs can reconstruct data distribution by learning the reverse process of a diffusion process. Denoting  $z_{t}$  as the random variable at  $t$ -th timestep, the diffusion process is modeled as a Markov Chain:
+
+$$
+z _ {t} \sim \mathcal {N} (\sqrt {\alpha_ {t}} z _ {t - 1}, (1 - \alpha_ {t}) I), \tag {1}
+$$
+
+where  $\alpha_{t}$  is the fixed coefficient predefined in the noise schedule, and  $I$  refers to identity matrix. A prominent variant, the Latent Diffusion Model (LDM) [28], innovatively shifts the diffusion process of standard DMs into a latent space. This transition notably decreases computational costs while preserving the generative quality and flexibility of the original model. The resulting efficiency gain primarily arises from the reduced dimensionality of the latent space, which allows for lower training costs without compromising the model's generative capabilities.
+
+Stable Diffusion, an exemplary implementation of LDM, comprises an AutoEncoder [29] and a latent diffusion model. The AutoEncoder  $\varepsilon$  is designed to learn a latent space that is perceptually equivalent to the image space. Meanwhile, the LDM  $\epsilon_{\theta}$  is parameterized as a denoising model with cross-attention and trained on a large-scale dataset of text-image pairs via:
+
+$$
+\mathcal {L} _ {L D M} := \mathbb {E} _ {\varepsilon (x), y, \epsilon \sim N (0, 1), t} [ \| \epsilon - \epsilon_ {\theta} (z _ {t}, t, \tau_ {\theta} (y)) \| _ {2} ^ {2} ], \tag {2}
+$$
+
+where  $\epsilon$  is the target noise.  $\tau_{\theta}$  and  $y$  are the pre-trained
+
+![](images/01bdc75ed160a5f95660aa52c6cb1677e6053d024173d47df59b4017a0df17b8.jpg)  
+(a) Training phase of our TIDE
+
+![](images/f7f5c503325057f5f49b3b485661f5143744fa49061911315aa4ec6d5c742250.jpg)  
+(b) Inference phase  
+Figure 3. Training and Inference. The denoising model of TIDE mainly consists of three transformers, each dedicated to text-to-image, text-to-depth, and text-to-mask. The proposed Implicit Layout Sharing mechanism (ILS) and Time Adaptive Normalization (TAN) are used to align the generated image, depth map, and semantic mask.
+
+text encoder (e.g., CLIP [24], T5 [25]) and text prompts, respectively. This equation represents the mean-squared error (MSE) between the target noise  $\epsilon$  and the noise predicted by the model, encapsulating the core learning mechanism of the latent diffusion model.
+
+# 4. Our Method
+
+An overview of our method, a unified text-to-image and dense annotation generation model (TIDE), is shown in Fig. 3. TIDE is built upon a pre-trained transformer [6] for text-to-image generation, along with two fine-tuned mini-transformers (details provided in Sec. 5.1.1) dedicated to text-to-depth and text-to-mask generation. Simply parallelizing multiple text-to-image processes does not ensure consistency between the images and dense annotations. To enable consistency between them, we propose Implicit Layout Sharing (ILS) and the cross-modal interaction method named Time Adaptive Normalization (TAN). After training, TIDE simultaneously generates images and multiple dense annotations with high consistency using only text as input.
+
+# 4.1. Data Preparation
+
+We aim to generate realistic underwater images, corresponding highly consistent depth maps, and semantic masks. However, existing high-quality, dense annotation data primarily consists of mask annotations. Therefore, we construct training data around these datasets with semantic masks, as shown in Tab. 1. On this basis, we obtain the corresponding depth map and caption for each image using existing foundation models. Specifically, for each underwater image, the corresponding depth map is obtained by pre-trained Depth Anything [38]. Meanwhile, the caption of
+
+each image is obtained from the pre-trained BLIP2 [18]. We construct approximately 14K quadruples {Image, Depth, Mask, Caption} for TIDE training.
+
+Table 1. Segmentation Datasets and Data Splits. \* denotes the training set of TIDE, while the others are used for evaluation.  
+
+<table><tr><td>Datasets</td><td>Seg Task</td><td>Train</td><td>Val</td><td>Test</td></tr><tr><td>SUIM [16]</td><td>Semantic</td><td>1,488*</td><td>110*</td><td>/</td></tr><tr><td>UIIS [19]</td><td>Instance</td><td>3,937*</td><td>691</td><td>/</td></tr><tr><td>USIS10K [20]</td><td>Instance</td><td>7,442*</td><td>1,594</td><td>1,596*</td></tr></table>
+
+# 4.2. Implicit Layout Sharing Mechanism
+
+In advanced text-to-image models [6, 28], the cross-attention map plays a crucial role in controlling the image layout. Existing methods [21, 35] demonstrate that adjusting the cross-attention map during the text-to-image process can effectively control the layout of the generated image. Therefore, the cross-attention map can be considered as the implicit layout information. Intuitively, sharing the implicit layout between text-to-image and text-to-dense annotations may establish a strong correlation between the generated image and dense annotations. To this end, we propose an Implicit Layout Sharing mechanism to align the generated image and dense annotations. Specifically, cross-attention, as a crucial process for generating implicit layouts in text-to-image/mask/depth model, can first be formulated as:
+
+$$
+\operatorname {A t t n} _ {i} \left(\boldsymbol {Q} _ {i}, \boldsymbol {K} _ {i}, \boldsymbol {V} _ {i}\right) = \operatorname {s o f t m a x} \left(\boldsymbol {Q} _ {i} \boldsymbol {K} _ {i} ^ {\top} / \sqrt {c}\right) \boldsymbol {V} _ {i},
+$$
+
+$$
+\operatorname {A t t n} _ {d} \left(\boldsymbol {Q} _ {d}, \boldsymbol {K} _ {d}, \boldsymbol {V} _ {d}\right) = \operatorname {s o f t m a x} \left(\boldsymbol {Q} _ {d} \boldsymbol {K} _ {d} ^ {\top} / \sqrt {c}\right) \boldsymbol {V} _ {d}, \tag {3}
+$$
+
+$$
+\operatorname {A t t n} _ {m} \left(\boldsymbol {Q} _ {m}, \boldsymbol {K} _ {m}, \boldsymbol {V} _ {m}\right) = \operatorname {s o f t m a x} \left(\boldsymbol {Q} _ {m} \boldsymbol {K} _ {m} ^ {\top} / \sqrt {c}\right) \boldsymbol {V} _ {m},
+$$
+
+where  $c$  refers to the feature channel.  $Q_{i} / Q_{d} / Q_{m}$ ,  $K_{i} / K_{d} / K_{m}$ , and  $V_{i} / V_{d} / V_{m}$  represent the query, key, and value within the text-to-image/depth/mask cross-attention module, respectively. Since text-to-image models are pre-trained on high-quality and large-scale image-caption datasets, they exhibit strong controllability and generalization. Therefore, sharing the implicit layouts from the text-to-image model is the optimal choice to ensure the quality of the generated data. As shown in Fig. 3(a), the implicit layouts from the block in the text-to-image model are shared with the cross-attention in the block of text-to-dense annotation models. The implicit layouts refer to:
+
+$$
+\boldsymbol {M} _ {i} = \operatorname {s o f t m a x} \left(\boldsymbol {Q} _ {i} \boldsymbol {K} _ {i} ^ {\top} / \sqrt {c}\right). \tag {4}
+$$
+
+By sharing the implicit layouts from the text-to-image model, the cross-attention of text-to-depth  $(\mathsf{Attn}_d)$  and text-to-mask  $(\mathsf{Attn}_m)$  can be simplified as follows:
+
+$$
+\operatorname {A t t n} _ {d} \left(\boldsymbol {Q} _ {d}, \boldsymbol {K} _ {d}, \boldsymbol {V} _ {d}\right) = \boldsymbol {M} _ {i} \times \boldsymbol {V} _ {d},
+$$
+
+$$
+\operatorname {A t t n} _ {m} \left(\boldsymbol {Q} _ {m}, \boldsymbol {K} _ {m}, \boldsymbol {V} _ {m}\right) = \boldsymbol {M} _ {i} \times \boldsymbol {V} _ {m}, \tag {5}
+$$
+
+where  $\times$  refers to matrix multiplication. Implicit Layout Sharing is an elegant and efficient method that unifies image and dense annotation generation, improving consistency between them. It also reduces the overall generation cost, as there is no need to compute separate cross-attention maps for the text-to-dense annotation models.
+
+# 4.3. Time Adaptive Normalization
+
+Considering the complementary nature of different modality features, we propose a cross-modal feature interaction method called Time Adaptive Normalization (TAN), as shown in Fig. 4.
+
+Specifically, TAN is utilized to adjust the image layout by leveraging the cross-modal features  $\boldsymbol{x}_f$  from different branches. The cross-modal features are mapped to two normalization parameters,  $\gamma$  and  $\beta$ , by MLPs, which are used to control the variation in the image layout. In this context, the features from text-to-depth and text-to-mask serve as cross-modal input features for each other. For instance, in the TAN corresponding to the  $i$ -th text-to-depth block, the outputs from both the  $i$ -th text-to-depth block and the  $i$ -th text-to-mask block serve as the input feature  $\boldsymbol{x}$  and cross-modal input feature  $\boldsymbol{x}_f$ , respectively. A slight difference is that for text-to-image, the features from both text-to-depth and text-to-mask serve as the cross-modal features. In the TAN cross-modal interaction process of text-to-image, two sets of  $\gamma$  and  $\beta$  are obtained, provided by the different modalities features from text-to-depth and text-to-mask. These two sets of parameters are averaged to the  $\bar{\gamma}$  and  $\bar{\beta}$ . Then, time embeddings  $\boldsymbol{x}_t$  is introduced to adaptively control the influence of the cross-modal features. The normalization can be formalized as follows:
+
+$$
+\boldsymbol {x} ^ {\prime} = \alpha \cdot \gamma \boldsymbol {x} + \alpha \cdot \beta , \boldsymbol {x} ^ {*} = \boldsymbol {x} ^ {\prime} + \boldsymbol {x}, \tag {6}
+$$
+
+![](images/3699708a205ef92db41f8131b1aaa0f5d3fe93d139c57d08349f9880048dc5ce.jpg)  
+Sigmoid  $\oplus$  Element-wise add  $\otimes$  Element-wise product  
+Figure 4. In TAN, the cross-modal features are first mapped to the modulation parameters  $\gamma$  and  $\beta$ . Then, a time-adaptive confidence  $\alpha$  is introduced to control the degree of normalization.
+
+where  $\pmb{x}$ ,  $\pmb{x}'$ , and  $\pmb{x}^*$  are the input feature, normalized feature, and output feature, respectively.  $\alpha$  is the time adaptive coefficients obtained from  $\pmb{x}_t$  through linear transformation and sigmoid. The TAN will be applied not only from text-to-dense annotations to text-to-image but also between text-to-depth and text-to-mask to improve the consistency among dense annotations. Implicit Layout Sharing and Time Adaptive Normalization are two complementary methods that construct a joint interaction process, optimizing the consistency between the generated image and dense annotations during training.
+
+# 4.4. Learning Objective
+
+During training, the learnable parameters include only the proposed TAN module and the LoRA [15] used to fine-tune the pre-trained transformer. The overall loss  $\mathcal{L}$  is composed equally of the denoising losses from the three branches: text-to-image, text-to-depth, and text-to-mask:
+
+$$
+\mathcal {L} = \mathcal {L} _ {m s e} ^ {I} + \mathcal {L} _ {m s e} ^ {D} + \mathcal {L} _ {m s e} ^ {M}. \tag {7}
+$$
+
+# 4.5. Data Synthesis
+
+Thanks to the proposed ILS and TAN, TIDE can generate realistic and highly consistent underwater images and dense annotations after training, using only text conditions, as shown in Fig. 3(b).
+
+We filter out redundant parts from the 14K captions obtained in Sec. 4.1, resulting in approximately 5K nonredundant captions as text conditions. For each caption, we generate ten samples to construct a large-scale synthetic dataset named SynTIDE. Some representative examples are shown in Fig. 1. The SynTIDE dataset is utilized to validate the effectiveness of our method in the dense prediction task for underwater scenes.
+
+# 4.6. Analysis
+
+Insights of framework design. In the text-to-image model, the cross-attention map contains the layout information of
+
+Table 2. Quantitative comparisons on real underwater depth estimation datasets.  
+
+<table><tr><td>Method</td><td>Fine-tuning Dataset</td><td>Reference</td><td>\( S I_{log} \)↓</td><td>A.Rel↓</td><td>\( log_{10} \)↓</td><td>RMSE↓</td><td>S.Rel↓</td><td>RMSElog↓</td><td>\( \delta_1 \uparrow \)</td><td>\( \delta_2 \uparrow \)</td><td>\( \delta_3 \uparrow \)</td></tr><tr><td colspan="12">Quantitative comparisons on the D3 and D5 subsets of Sea-thru [3] dataset.</td></tr><tr><td>AdaBins [5]</td><td>Atlantis [41]SynTIDE (Ours)</td><td>CVPR 24-</td><td>38.2426.92(-11.32)</td><td>1.331.31(-0.02)</td><td>0.120.08(-0.04)</td><td>1.411.12(-0.29)</td><td>12.8915.74(+2.85)</td><td>0.390.27(-0.12)</td><td>0.500.71(+0.21)</td><td>0.810.95(+0.14)</td><td>0.920.99(+0.07)</td></tr><tr><td>NewCRFs [40]</td><td>Atlantis [41]SynTIDE (Ours)</td><td>CVPR 24-</td><td>37.1022.37(-14.73)</td><td>1.681.50(-0.18)</td><td>0.120.06(-0.06)</td><td>1.441.24(-0.20)</td><td>14.7622.50(+7.74)</td><td>0.380.23(-0.15)</td><td>0.480.84(+0.36)</td><td>0.840.97(+0.13)</td><td>0.950.99(+0.04)</td></tr><tr><td>PixelFormer [1]</td><td>Atlantis [41]SynTIDE (Ours)</td><td>CVPR 24-</td><td>23.7021.39(-2.31)</td><td>1.341.46(+0.12)</td><td>0.060.05(-0.01)</td><td>1.171.15(-0.02)</td><td>17.2921.79(+4.50)</td><td>0.240.22(-0.02)</td><td>0.810.88(+0.07)</td><td>0.970.98(+0.01)</td><td>0.990.99(+0.00)</td></tr><tr><td>MIM [34]</td><td>Atlantis [41]SynTIDE (Ours)</td><td>CVPR 24-</td><td>37.0122.49(-14.52)</td><td>1.371.27(-0.10)</td><td>0.110.06(-0.05)</td><td>1.511.01(-0.50)</td><td>14.4216.46(+2.04)</td><td>0.380.23(-0.15)</td><td>0.560.85(+0.29)</td><td>0.840.97(+0.13)</td><td>0.940.99(+0.05)</td></tr><tr><td colspan="12">Quantitative comparisons on the SQUID [4] dataset.</td></tr><tr><td>AdaBins [5]</td><td>Atlantis [41]SynTIDE (Ours)</td><td>CVPR 24-</td><td>29.5625.63(-3.93)</td><td>0.280.23(-0.05)</td><td>0.110.09(-0.02)</td><td>2.242.69(+0.45)</td><td>0.690.92(+0.23)</td><td>0.310.27(-0.04)</td><td>0.560.67(+0.11)</td><td>0.860.90(+0.04)</td><td>0.940.97(+0.03)</td></tr><tr><td>NewCRFs [40]</td><td>Atlantis [41]SynTIDE (Ours)</td><td>CVPR 24-</td><td>25.1925.55(+0.36)</td><td>0.230.23(-0.00)</td><td>0.090.09(+0.00)</td><td>2.563.02(+0.46)</td><td>0.831.07(+0.24)</td><td>0.260.27(+0.01)</td><td>0.680.68(+0.00)</td><td>0.900.91(+0.01)</td><td>0.960.97(+0.01)</td></tr><tr><td>PixelFormer [1]</td><td>Atlantis [41]SynTIDE (Ours)</td><td>CVPR 24-</td><td>21.3419.08(-2.26)</td><td>0.180.16(-0.02)</td><td>0.070.07(-0.00)</td><td>1.861.75(-0.11)</td><td>0.430.36(-0.07)</td><td>0.220.19(-0.03)</td><td>0.760.79(+0.03)</td><td>0.940.97(+0.03)</td><td>0.980.99(+0.01)</td></tr><tr><td>MIM [34]</td><td>Atlantis [41]SynTIDE (Ours)</td><td>CVPR 24-</td><td>27.4526.98(-0.47)</td><td>0.260.25(-0.01)</td><td>0.100.09(-0.01)</td><td>2.143.04(+0.90)</td><td>0.681.11(+0.43)</td><td>0.280.28(-0.00)</td><td>0.610.65(+0.04)</td><td>0.880.89(+0.01)</td><td>0.950.96(+0.01)</td></tr></table>
+
+the image. Thus, the cross-attention map can be viewed as an implicit layout. If two text-to-image models share the same implicit layout and undergo proper fine-tuning, the generated images are likely to exhibit strong layout similarity. Therefore, we share the same implicit layout across multiple text-to-image models. Meanwhile, we use LoRA to fine-tune the multiple text-to-image models [6].
+
+Zero-shot generation ability. Thanks to our training strategy, which fine-tunes the pre-trained text-to-image model using only LoRA, the generalization ability of the text-to-image model is retained to some extent. This enables TIDE to generate underwater images during inference that are not seen during training. Furthermore, due to the proposed Implicit Layout Sharing and Time Adaptive Normalization mechanisms, the generated depth maps align well with these images. Therefore, TIDE has the ability to generate zero-shot underwater image-depth map pairs.
+
+# 5. Experiments
+
+# 5.1. Dataset and Evaluation Metrics
+
+Underwater Depth Estimation. We follow the work [41], the D3 and D5 subsets of Sea-thru [3], and the SQUID dataset [4] used to evaluate the depth estimation capability in underwater scenes. These datasets include underwater images with depth maps obtained via the Structure-fromMotion (SfM) algorithm.
+
+The quantitative evaluation metrics include root
+
+mean square error (RMSE) and its logarithmic variant  $(RMSE_{log})$ , absolute error in log-scale  $(\log_{10})$ , absolute relative error (A.Rel), squared relative error (S.Rel), the percentage of inlier pixels  $(\delta_i)$  with thresholds of  $1.25^i$ , and scale-invariant error in log-scale  $(SI_{log})$ :  $100\sqrt{Var(\epsilon_{log})}$ .
+
+Underwater Semantic Segmentation. The UIIS [19] and USIS10K [20] datasets are chosen to validate the effectiveness of our method in underwater semantic segmentation tasks. Instance masks belonging to the same semantic category are merged to construct semantic segmentation annotations for the UIIS and USIS10K datasets.
+
+We calculate the mean Intersection over Union (mIoU) for six categories (i.e., Fish, Reefs, Aquatic Plants, Wrecks, Human Divers, and Robots) to evaluate the accuracy of the segmentation results.
+
+# 5.1.1 Implementation Details
+
+The training process consists of two parts: pre-training the mini-transformer and training TIDE. In the first stage, the mini-transformer is initialized with the first ten layers of the PixArt- $\alpha$  [6] pre-trained transformer. Then, the mini-transformer is trained for 60K iterations on the text-to-image task with all parameters. The training data consists of 14K underwater image-caption pairs from Sec. 4.1. In the second stage, the PixArt- $\alpha$  pre-trained transformer and the mini-transformer are used as initial weights for the text-to-image and text-to-dense annotation models, respectively.
+
+Table 3. Quantitative results of underwater semantic segmentation.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Backbone</td><td colspan="2">Training Data</td><td colspan="2">mIoU</td></tr><tr><td>Real</td><td>SynTIDE</td><td>UIIS</td><td>USIS10K</td></tr><tr><td rowspan="3">Segformer [33] (NeurIPS 21)</td><td rowspan="3">MiT-B4</td><td>✓</td><td></td><td>70.2</td><td>74.6</td></tr><tr><td></td><td>✓</td><td>76.5</td><td>72.8</td></tr><tr><td>✓</td><td>✓</td><td>75.4(+5.2)</td><td>76.1(+1.5)</td></tr><tr><td rowspan="3">Mask2former [8] (CVPR 22)</td><td rowspan="3">Swin-B</td><td>✓</td><td></td><td>72.7</td><td>76.1</td></tr><tr><td></td><td>✓</td><td>74.2</td><td>72.9</td></tr><tr><td>✓</td><td>✓</td><td>74.3(+1.6)</td><td>77.1(+1.0)</td></tr><tr><td rowspan="3">ViT-Adapter [7] (ICLR 23)</td><td rowspan="3">ViT-Adapter-B</td><td>✓</td><td></td><td>73.5</td><td>74.6</td></tr><tr><td></td><td>✓</td><td>75.7</td><td>72.6</td></tr><tr><td>✓</td><td>✓</td><td>75.1(+1.6)</td><td>76.7(+2.1)</td></tr></table>
+
+Meanwhile, they are fine-tuned using LoRA [15] for 200K iterations with a batch size of 4. The LoRA ranks of the text-to-image/depth/mask branches are 32, 64, and 64, respectively. All experiments are conducted on a server with four NVIDIA 4090 24G GPUs.
+
+# 5.2. Main results
+
+# 5.2.1 Underwater Depth Estimation
+
+We train four representative depth estimation models, Adasbin [5], NewCRFs [29], PixelFormer [1], and MIM [34], to present quantitative results, as shown in Tab. 2. Compared to previous underwater data synthesis work Atlantis [41], depth estimation models trained on our SynTIDE dataset show consistent improvements across most quantitative metrics on two evaluated datasets. Especially on MIM [34], a powerful pre-trained model, our method reduces the  $SI_{\text{log}}$  metric from  $37.01 \rightarrow 22.49$  (-14.52) and improves  $\delta_1$  from  $0.56 \rightarrow 0.85$  (+0.29) on the D3 and D5 subsets of the Seathru dataset. Meanwhile, on PixelFormer [1], a depth estimation model with outstanding generalization that also performs best for Atlantis, our method achieves better performance across nearly all quantitative metrics on both evaluated underwater depth estimation datasets.
+
+These results demonstrate that our method achieves highly competitive consistency compared to Atlantis, which uses stronger dense conditions. Furthermore, the data generated by TIDE is closer to natural underwater scenes and shows rich species diversity. Most importantly, TIDE unifies the generation of images and multiple highly consistent dense annotations, capabilities that Atlantis lacks.
+
+# 5.3. Underwater Semantic Segmentation
+
+In the underwater semantic segmentation task, we validate the effectiveness of our method by pre-training with the SynTIDE dataset in three representative semantic segmentation models, Segformer [33], Mask2former [8], and ViT
+
+Table 4. Ablation on the impact of each TIDE component.  
+
+<table><tr><td>ILS</td><td>TAN</td><td>SIlog ↓</td><td>A.Rel ↓</td><td>δ1 ↑</td><td>mIoU</td></tr><tr><td>✓</td><td></td><td>24.46</td><td>1.23</td><td>0.76</td><td>36.8</td></tr><tr><td></td><td>✓</td><td>24.59</td><td>1.40</td><td>0.78</td><td>36.2</td></tr><tr><td>✓</td><td>✓</td><td>23.71</td><td>1.37</td><td>0.79</td><td>42.1</td></tr></table>
+
+Table 5. Ablation on the impact of the component positions. " {Start, End}" indicates the starting and ending positions where the operations are applied, with a step size of 3.  
+
+<table><tr><td>{Start, End}</td><td>SIlog ↓</td><td>A.Rel ↓</td><td>δ1 ↑</td><td>mIoU</td></tr><tr><td>{0,12}</td><td>22.06</td><td>1.46</td><td>0.86</td><td>34.7</td></tr><tr><td>{15,27}</td><td>44.86</td><td>1.09</td><td>0.43</td><td>8.6</td></tr><tr><td>{0,27}</td><td>23.71</td><td>1.37</td><td>0.79</td><td>42.1</td></tr></table>
+
+Table 6. Ablation on the impact of scaling synthetic data for underwater dense prediction tasks.  
+
+<table><tr><td>N Sample</td><td>SIlog ↓</td><td>A.Rel ↓</td><td>δ1↑</td><td>mIoU</td></tr><tr><td>1</td><td>23.49</td><td>1.46</td><td>0.82</td><td>55.3</td></tr><tr><td>3</td><td>22.94</td><td>1.54</td><td>0.85</td><td>60.9</td></tr><tr><td>6</td><td>22.96</td><td>1.56</td><td>0.85</td><td>63.6</td></tr><tr><td>10</td><td>22.37</td><td>1.50</td><td>0.84</td><td>64.2</td></tr></table>
+
+Adapter [7]. Following the work [37], we filter the noise in the generated annotations with 1.5 tolerance.
+
+Pre-training on high-quality synthetic datasets is widely recognized as a way to gain strong prior knowledge. On the UIIS dataset, models trained on the SynTIDE dataset consistently achieve superior results compared to real data. On the other larger USIS10K dataset, by further fine-tuning the model on the UIIS10K train set, we achieve notable improvements. Especially on ViT-Adapter, we enhance the performance of the model from  $74.6\% \rightarrow 76.7\%$ .
+
+These results show that models pre-trained on the SynTIDE dataset exhibit strong prior knowledge in the underwater semantic segmentation task. Additionally, these results demonstrate that the unified image and dense annotation generation model proposed in this paper can generate highly consistent image-dense annotation pairs, making it suitable for various underwater dense prediction tasks.
+
+# 5.4. Ablation Studies
+
+Unless otherwise specified, we conduct ablation studies by training TIDE for 30K iterations. We synthesize three samples for each caption, as described in Sec. 4.5. We conduct ablation studies on the USIS10K dataset with SegFormer-B4 for semantic segmentation and the D3 and D5 subsets of the Sea-thru dataset with NewCRFs for depth estimation.
+
+Ablation on the effectiveness of each component. We first evaluate the contribution of each component within
+
+TIDE, as shown in Tab. 4. When utilizing only the Implicit Layout Sharing (ILS) mechanism or Time Adaptive Normalization (TAN), the former outperforms the latter in depth estimation and semantic segmentation. Combining both methods results in a significant improvement  $(36.8\% \rightarrow 42.1\%)$  in semantic segmentation. These results indicate that ILS and TAN are complementary methods. By combining them for end-to-end training, the consistency between images and dense annotations can be further optimized. Additionally, we further demonstrate the effectiveness of the time-adaptive operation. As shown in the last row of Tab. 4, without time-adaptive parameters, the quality of the generated data will be varying degrees of degradation, especially for the semantic segmentation task.
+
+Ablation on the position of components. We then study the effect of the position of ILS and TAN, as shown in Tab. 5. We find that applying the ILS and TAN mechanisms in the first half of the transformer of text-to-image yields better performance than using them in the second half. This can be attributed to the layout information produced in the first half of the transformer, which is mismatched with the ILS introduced in the latter part. Meanwhile, the results demonstrate that combining both achieves better consistency between the image and dense annotations.
+
+Ablation on data scaling. Finally, we synthesize  $N$  samples for each caption to validate the impact of synthetic data scale on underwater dense prediction tasks, as shown in Tab. 6. It can be observed that as the amount of synthetic data increases, there is no substantial improvement in the underwater depth estimation task. However, for the underwater semantic segmentation task, a significant gain is observed in the early stages as  $N$  increases, but the tendency of improvement begins to flatten after  $N = 6$ .
+
+# 5.5. More Challenging Underwater Data Synthesis
+
+We validate whether TIDE can generate more challenging data by adding extra text prompts about underwater lighting or water quality (e.g., low light, turbidity) to the original underwater scene caption. As shown in Fig. 5, the results demonstrate that TIDE can generate more challenging underwater images. While annotating these underwater images may be extremely difficult for humans, TIDE can effortlessly produce highly consistent and accurate dense annotations, which hold great practical value for real-world underwater applications. In additional, to demonstrate the diversity of generated underwater data, we generate twelve underwater images from the same text prompt, as shown in Fig. 6. It can be observed that, despite sharing the same text prompt, the generated images exhibit rich diversity.
+
+# 5.6. Limitation
+
+Despite the promising results achieved, our method still has some limitations. First, our approach cannot gener-
+
+![](images/f365ef2c20cdc9bd13840b7a564002ebdccc654c8edfb99a69b2dcfceb71171a.jpg)  
+Figure 5. More challenging underwater data generated by TIDE.
+
+![](images/5d98c398f2a8ca0fdebbecb449d4a876f85b15b8d1979a6867739d416b881fa2.jpg)  
+Figure 6. Visualization of generated data diversity.
+
+ate instance-level semantic masks from the generation perspective. Relying on text prompts to guide the generation of instance-level masks with semantic annotations remains challenging. Additionally, although TIDE can leverage the powerful priors of pre-trained T2I models to generate highly challenging underwater images (e.g., low light, turbidity), there is still room for improvement. These will be key directions for future expansion.
+
+# 6. Conclusion
+
+This paper introduces a unified text-to-image and dense annotation generation model for underwater scenes. The model can generate realistic underwater images and multiple highly consistent dense annotations using only text prompts as input. We validate the effectiveness of our method on underwater depth estimation and semantic segmentation tasks by synthesizing a large-scale underwater dataset containing images along with highly consistent depth and semantic segmentation annotations. In the depth estimation task, extensive experimental results show that our method, using only text as input, achieves highly competitive results compared to previous methods that required stronger dense conditions for underwater depth synthesis. Meanwhile, pre-training with data synthesized using our method further improves model performance in the semantic segmentation task. Our study provides a new perspective for alleviating data scarcity in other fields.
+
+Acknowledgement. This work was supported by the NSFC (Grant U234120202 and 62225603).
+
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+# A Unified Latent Schrödinger Bridge Diffusion Model for Unsupervised Anomaly Detection and Localization
+
+Shilhora Akshay $^{1,*}$  Niveditha Lakshmi Narasimhan $^{2}$  Jacob George $^{2}$  Vineeth N Balasubramanian $^{1}$ $^{1}$ Indian Institute of Technology, Hyderabad  $^{2}$ KLA Corporation *shilhora.akshay333@gmail.com
+
+# Abstract
+
+Anomaly detection and localization remain pivotal challenges in computer vision, with applications ranging from industrial inspection to medical diagnostics. While current supervised methods offer high precision, they are often impractical due to the scarcity of annotated data and the infrequent occurrence of anomalies. Recent advancements in unsupervised approaches, particularly reconstruction-based methods, have addressed these issues by training models exclusively on normal data, enabling them to identify anomalies during inference. However, these methods frequently rely on auxiliary networks or specialized adaptations, which can limit their robustness and practicality. This work introduces the Latent Anomaly Schrödinger Bridge (LASB), a unified unsupervised anomaly detection model that operates entirely in the latent space without requiring additional networks or custom modifications. LASB transforms anomaly images into normal images by preserving structural integrity across varying anomaly classes, lighting, and pose conditions, making it highly robust and versatile. Unlike previous methods, LASB does not focus solely on reconstructing anomaly features, but emphasizes anomaly transformation, achieving smooth anomaly-to-normal image conversions. Our method achieves state-of-the-art performance on both the MVTec-AD and VisA datasets, excelling in detection and localization tasks. Our code is available at https://github.com/ShilhoraAkshayPatel/LASB.
+
+# 1. Introduction
+
+Anomaly detection and localization is a critical task in computer vision with several applications in medical [4, 18, 52, 53] and industrial [21, 31, 43, 60], which has attracted significant attention from the research community. The main goal of anomaly detection is to identify and localize abnormal patterns that are unusual from those seen in normal
+
+![](images/348132e01528d7ebd2ebe824dd596fe2dffeeb94379f6c2db810f1dcd844479c.jpg)  
+Figure 1. Illustration of the Gaussian and Bridge diffusion processes, with reverse image trajectory from LASB. LASB learns a direct diffusion bridge between anomaly and normal distributions, enhancing interpretability and anomaly-free transformation.
+
+instances. In the past few years the research community has proposed various supervised anomaly detection methods [15, 24, 30, 44, 50, 59]. The extensive need for annotations is expensive and the infrequent presence of anomalous samples makes these methods unsuitable for practical applications due to their limitations in addressing real-world scenarios.
+
+Recent research has sought to overcome the limitations of supervised anomaly detection methods by relying solely on normal images during training and localizing abnormal patterns at inference time. Among the various categories within the unsupervised paradigm, reconstruction-based methods have garnered significant attention due to their promising results and strong performance in real-world scenarios. The core concept behind these methods is that the model is trained exclusively on normal images and during inference, it reconstructs abnormal samples as normal samples. Thus, doing so benefits in two directions, one can interpret how the model's ability to detect and localize anomalies. Second, it aids the model to perform better while reconstructing the anomalies. Recent advancements have introduced novel approaches lever
+
+aging AutoEncoder-based (AE) [3, 6, 56-58], Generative Adversarial Networks-based (GANs) [1, 26, 45, 62], and diffusion-based approaches [4, 18, 21, 31, 43, 52, 60]. Our research specifically focuses on various diffusion models tailored for anomaly detection tasks.
+
+Existing diffusion-based methods primarily focus on developing novel noise-conditioning techniques, often incorporating an additional discriminative sub-network [60]. Another approach utilizes a score-based diffusion model to identify anomalies by evaluating how effectively samples can return to the normal data distribution after perturbation, though it fails to deliver competitive performance [43]. Additionally, DiAD [21] proposes novel semantic guidance to specifically understand the anomalies and showcase its robustness. However, earlier diffusion-based methods generally depend on auxiliary networks or tailored diffusion processes to extract anomaly features. Moreover, most of these methods prioritize the extraction and reconstruction of anomaly features.
+
+In this work, we introduce an intuitive mechanism that neither requires model-specific adaptations to extract anomaly features nor relies on an additional feature extractor for training or inference, to extract discriminative features. By leveraging the capabilities of Schrödinger Bridge [8, 11, 14, 42, 51], we propose the Latent Anomaly Schrödinger Bridge (LASB), which operates entirely in the latent space, transforming anomalous images into normal ones, regardless of anomaly class, and demonstrating robustness to variations in lighting and pose. In the following, we outline the key contributions of our study.
+
+- We propose LASB, a unsupervised bridge-based anomaly detection model that operates in latent space. LASB transforms anomaly images into normal images, regardless of the anomaly class.  
+- Unified Framework: LASB offers a comprehensive framework for both anomaly detection and localization without the need for auxiliary networks. Unlike conventional methods that rely on Gaussian diffusion, LASB employs a bridge-based diffusion process that inherently preserves structural integrity, marking the first application of such a process in latent diffusion model for anomaly detection.  
+- Efficient and Scalable: By utilizing the Linear Schrödinger Bridge in latent space, LASB significantly reduces training time, memory consumption, and sampling speed, enhancing overall efficiency and scalability.  
+- Our method achieves state-of-the-art performance on the MVtec dataset, with a image-level  $\mathrm{AUROC}_{cls} / \mathrm{AP}_{cls}$  of  $99.2\% / 99.3\%$  and an pixel-level  $\mathrm{AUROC}_{seg} / \mathrm{AP}_{seg}$  of  $98.6\% / 78.2\%$  on the test set, establishing a new benchmark for unsupervised anomaly detection.
+
+# 2. Related Work
+
+Reconstruction-based Methods: Reconstruction-based anomaly detection models operate on the premise that networks trained exclusively on normal images will fail to accurately reconstruct anomalous ones due to their unfamiliarity with the abnormal distribution. Notably, autoencoder [6, 10, 28] and Generative Adversarial Network (GAN) [1, 41] frameworks have been widely utilized for this task, where anomaly scores are computed based on the reconstruction error between the input and its generated counterpart. Nevertheless, such approaches frequently struggle with direct copy issues and elevated false positive rates due to their limited generalization capacity over anomalous regions or due to the overgeneralization capacity of reconstructive methods. To mitigate these generalization challenges, recent innovations have pivoted towards inpainting strategies [35, 57] or integrating supplementary memory modules [17, 47]. Furthermore, the DRAEM [56] model enhances performance by leveraging pseudoanomalies and coupling autoencoders with a segmentation network, though its effectiveness diminishes when faced with substantial deviations between real and pseudo anomalies.
+
+Diffusion-based Methods: Diffusion models, known for their high-fidelity image synthesis, are increasingly applied to anomaly detection in both medical and industrial domains [18, 21, 43, 52, 60]. In the medical field, SANO [18] leverages score-based diffusion models to localize skin conditions such as eczema, while AnoDDPM [52] employs the Denoising Diffusion Probabilistic Model (DDPM) [23] to segment tumors, initiating with simplex noise. These methods primarily utilize standard DDPM [23] or Score-based Generative Models (SGMs) [46], yet they often struggle to capture structural features because they rely on starting from pure Gaussian or simplex noise and frequently require external guidance for complex feature integration. In industrial applications, DiffAD [60] introduces a noise interpolation technique coupled with a discriminative sub-network to enhance detection capabilities. AD-SPR [43] emphasizes the significance of perturbed resilience using SGMs [46] to identify anomalies. Additionally, DiAD [21] presents a novel semantic guidance network that directs a stable diffusion model [37] to effectively detect and localize anomalies in both industrial and selected medical imaging applications. However, these models often require class-specific training or additional networks to guide the diffusion process or to perform further segmentation via a discriminative sub-network.
+
+Unified/Multi-class Methods: Early approaches in the literature developed models on a per-class basis, which limited their generalizability. More recent research has shifted towards unified or multi-class models that are trained on
+
+![](images/4fb7253a32f925bb900a6dde4f0bd8f50e6ac343e6b27b7991abe1f462799f5d.jpg)
+
+![](images/e4ffb779d8ba2249faef708cb0ab8c73765de979b76a5380378ff518a1c2e26e.jpg)  
+Figure 2. The LASB model framework consists of two key stages: training and inference. During training, anomaly augmentations are applied to images, introducing distortions that the LASB model learns to remove, ultimately reconstructing a normal image. This iterative process continues until the model effectively filters out the anomalies. In the inference stage, the model processes real anomalous images, reconstructing normal versions. The anomaly detection is achieved by computing the difference  $(p_B - p_A)$  between the original and reconstructed images, with anomalies visualized via a heatmap.
+
+![](images/2168f79c8e9cbdaec09dbc1aa9abacddcfda286c1ca5b93866ccc3559d979ead.jpg)
+
+entire datasets, demonstrating improved robustness [21, 22, 32, 55]. For instance, UniAD [55] utilizes layer-wise query decoding, neighbor-masked attention, and feature jittering techniques to enhance multi-class anomaly detection. In contrast, HVQ-Trans [32] employs a hierarchical vector quantized prototype-oriented Transformer to improve discriminative capacity across multiple classes. Both methods focus on tackling the "identical shortcut" issue. On the other hand, RLR [22] introduces a novel framework using learnable reference representations with locality constraints to explicitly learn normal patterns and circumvent "learning shortcuts", achieving superior results on standard datasets (MVTec-AD and VisA). Additionally, the DiAD [21] model, designed for multi-class settings and based on diffusion-based reconstruction, outperforms UniAD [55], RLR [22], and HVQ-Trans [32] in terms of localization and detection performance.
+
+# 3. Proposed Methodology
+
+# 3.1. Preliminaries
+
+Before introducing the full methodology, we begin with a preliminary overview of score-based generative models (SGMs). Next, we explain the Schrödinger Bridge concept and discuss its connection to SGMs. Finally, we present the core methodology, detailing how and why the latent Schrödinger Bridge effectively addresses anomaly detection and localization.
+
+Notations. Here we introduce some notations and use them throughout the work. Let  $X_{t}(\in \mathbb{R}^{d})$  denote a stochastic process, where  $t\in [0,1]$  is a continuous time step. The intermediate steps are uniformly distributed in the interval
+
+$(\mathcal{U}(t \sim [0,1]))$ . The initial distribution is the corrupted data distribution is denoted as  $p_A$  and the terminal distribution is clean data distribution and is denoted as  $p_B$ . The Wiener process and its reversed counterpart, adopted from Anderson et al. [2], are represented as  $W_t$  and  $\overline{W}_t$ , respectively, both in  $\mathbb{R}^d$ .  $\mathbb{I} \in \mathbb{R}^{d \times d}$  is the identity matrix.
+
+Score-based Generative Models. Score-based Generative Models (SGMs) perturb the data stochastically across continuous noise scales and then use reverse-time stochastic differential equations (SDEs) to learn the reverse-time diffusion process. This reverse-time SDE relies on a score function enabling the reconstruction of any distribution starting from a Gaussian [46]. Given data  $X_0 \sim p_A$ , forward and backward SDEs are formulated as:
+
+$$
+d X _ {t} = f _ {t} \left(X _ {t}\right) d t + \sqrt {\beta_ {t}} d W _ {t}, \quad X _ {0} \sim p _ {A}, \tag {1}
+$$
+
+$$
+d X _ {t} = \left[ f _ {t} - \beta_ {t} \nabla \log p \left(X _ {t}, t\right) \right] d t + \sqrt {\beta_ {t}} d \bar {W} _ {t}, \tag {2}
+$$
+
+where,  $f(\cdot ,t):\mathbb{R}^n\to \mathbb{R}^n$  and the terminal distributions (i.e., at  $t = 1$ ) approach Gaussian  $(X_{1}\sim \mathcal{N}(0,I))$ . To achieve this, diffusion coefficient  $\beta_{t}\in \mathbb{R}$  is carefully tuned and ensuring that the base drift  $f_{t}$  is linear in  $X_{t}$ . Here,  $p$  is the marginal density of equation (2) at time  $t$ , and  $\nabla \log p$  denotes its score [46].
+
+SGMs as Schrödinger Bridge. The absence of flexibility in SGM's to transport data to desired distribution, demands a versatile strategy. The Schrödinger Bridge (SB) is a strategy often applied in optimal transport to find the optimal path measure between two marginal densities and it is expressed as,
+
+$$
+\min  _ {Q \in P \left(p _ {A}, p _ {B}\right)} \operatorname {K L} (\mathbb {Q} | | \mathbb {P}), \tag {3}
+$$
+
+Here,  $\mathbb{Q}$  represents a path measure within  $\mathbb{P}(p_A,p_B)$ , characterized by having marginal densities  $p_A$  and  $p_B$  at times  $t = 0$  and 1, respectively. We consider  $\mathbb{P}$  as the reference measure, specifically chosen as the path measure in equation (1). We will further elaborate on the conditions that define the optimality of the SB (as shown in equation (3)) with specified boundary conditions. The optimality condition for SB is characterized by solving PDEs [9][8]. Let  $\Psi (t,x)$  and  $\widehat{\Psi} (t,x)$  be the solutions to the following PDEs:
+
+$$
+\frac {\partial \Psi (z , t)}{\partial t} = - \nabla \Psi^ {T} f - \frac {1}{2} \beta \Delta \Psi \tag {4}
+$$
+
+$$
+\frac {\partial \widehat {\Psi} (z , t)}{\partial t} = - \nabla \cdot (\widehat {\Psi} f) + \frac {1}{2} \beta \Delta \widehat {\Psi} \tag {5}
+$$
+
+subject to the conditions,
+
+$$
+\Psi (z, 0) \widehat {\Psi} (z, 0) = p _ {A} (z), \quad \Psi (z, 1) \widehat {\Psi} (z, 1) = p _ {B} (z).
+$$
+
+Then, the solution to optimization (3) can be expressed by the path measure of the following forward equation (6), or equivalently backward equation (7), SDE:
+
+$$
+d X _ {t} = \left[ f _ {t} + \beta_ {t} \nabla \log \Psi (X _ {t}, t) \right] d t + \sqrt {\beta_ {t}} d W _ {t}, \tag {6}
+$$
+
+$$
+d X _ {t} = \left[ f _ {t} - \beta_ {t} \nabla \log \hat {\Psi} \left(X _ {t}, t\right) \right] d t + \sqrt {\beta_ {t}} d \bar {W} _ {t}, \tag {7}
+$$
+
+where  $\nabla \log \Psi$  and  $\nabla \log \widehat{\Psi}$  represent the non-linear optimal forward and backward drifts for the Schrödinger Bridge. This SB has a non-linear behavior and it generalizes the SGMs with variation in prior data. Note that, forward and backward equations (6, 7) of Schrödinger Bridge are same as SGMs forward-backward equations (1, 2) except the forward drift term ( $\nabla \log \Psi$ ). Thus, from the Nelson's Duality [34] we obtain  $\Psi(x,t)\widehat{\Psi}(x,t) = q^{\mathrm{SB}}(x,t)$ . As the backward drift of Schrödinger Bridge does not act as a score function anymore with two indivisible components. Thus, from equation (7) and Nelson's Duality [34] we obtain  $dX_{t} = [f_{t} - \beta_{t}(\nabla \log q^{\mathrm{SB}}(X_{t},t) - \nabla \log \Psi(X_{t},t))]dt + \sqrt{\beta_{t}} d\overline{W}_{t}$ .
+
+Linear-SB. The constraints posed above cause the model to be intractable and to overcome this impediment existing literature suggests various lines of strategies such as Iterative Proportional Fitting [11, 25], likelihood-based training of SB [8] etc. These methods help the SGMs to construct non-linear diffusion bridges. But, we can make them tractable by posing some linear conditions. A more detailed examination of SB theory with SGM framework reveals that the nonlinear drifts specified in equations (6, 7) correspond to the score functions described in equation (2). Assuming that  $\Psi (\cdot ,t)$  and  $\widehat{\Psi} (\cdot ,t)$  function as probability density functions, we reformulate equations (4, 5) to effectively address the solutions of the Fokker-Planck equation [36].
+
+The Schrödinger Bridges in equation (4, 5) are satisfied, the backward and forward drifts,  $\nabla \log \widehat{\Psi}(X_t, t)$  and  $\nabla \log \Psi(X_t, t)$  represent the score functions for the following linear SDEs, respectively [36][27]:
+
+$$
+d X _ {t} = f _ {t} \left(X _ {t}\right) d t + \sqrt {\beta_ {t}} d W _ {t}, \quad X _ {0} \sim \Psi (\cdot , 0), \tag {8}
+$$
+
+$$
+d X _ {t} = f _ {t} \left(X _ {t}\right) d t + \sqrt {\beta_ {t}} d \overline {{W}} _ {t}, \quad X _ {1} \sim \widehat {\Psi} (\cdot , 1). \qquad (9)
+$$
+
+The key characteristic of the linear SDEs presented in equations (8, 9) lies in their differing boundary conditions, resulting in distinct distributions compared to nonlinear SDEs. Sampling is facilitated by parameterizing  $\nabla \log \widehat{\Psi}$  using a score network and applying established SGM methods. However, the computational complexity arises due to the intractability of the boundary conditions  $\Psi(\cdot, 0)$  and  $\widehat{\Psi}(\cdot, 1)$ . To resolve this, we introduce Dirac delta functions as boundary conditions for these previously intractable terms, thus rendering them manageable. Now to address this we impose Dirac delta boundary conditions [36][27]. Assume  $p_A(\cdot) = \delta_\kappa(\cdot)$  is Dirac delta distribution that is centered at  $\kappa \in \mathbb{R}^d$ . Then, the initial distributions of the linear SDEs (equation (8,9)) are:
+
+$$
+p _ {B} = \Psi (\cdot , 1) \widehat {\Psi} (\cdot , 1), \quad \widehat {\Psi} (\cdot , 0) = \delta_ {\kappa} (\cdot), \tag {10}
+$$
+
+The above equation (10) suggests that the optimal backward drift of the process in equation (8) consistently targets the Dirac delta  $\delta_{\kappa}(\cdot)$ , achieving convergence to  $\kappa$  independent of  $p_B$ . This concept is integrated into the loss function, where the score is recalculated as  $\nabla \log p(X_t,t|X_0 = \kappa)$  for each instance  $\kappa$ . This approach enhances computational efficiency and establishes a robust mathematical basis for training  $\nabla \log \widehat{\Psi} (\cdot)$ .
+
+# 3.2. Linear Schrödinger Bridge in the Latent Space
+
+We now describe how the above discussion on linear Schrödinger bridges can be used for anomaly detection and localization.
+
+Overall Pipeline. Figure 2 illustrates the entire pipeline of the proposed method. Initially, anomaly augmentation [56] is applied to a normal image, which is then processed by the Latent Anomaly Schrödinger Bridge (LASB) model. The LASB model learns the anomaly statistics and transforms the augmented anomaly image back into a normal image. Crucially, the aim of this step is not to learn the anomalies themselves, but rather to reconstruct the normal image regardless of the anomalies introduced during training. During training, various masks and noise patterns are applied to augment the images. After training, the LASB model is evaluated on anomaly images from the test set, with its weights frozen. During the inference phase, the model is presented with an anomaly image from the test
+
+Algorithm 1 Training Procedure
+
+1: Input: normal and anomaly latent  $p_A^{(z)}(\cdot), p_B^{(z)}(\cdot | z_0)$  
+2: repeat  
+3:  $z_0\sim p_A^{(z)}(z_0),z_1\sim p_B^{(z)}(z_1|z_0)$  
+4:  $z_{t}\sim q(z_{t}|z_{0},z_{1})$  according to equation (11)  
+5: calculate gradient  $\nabla \epsilon (z_t,t;\theta)$  using equation (12)  
+6: until convergence
+
+set, which it then reconstructs as a normal image. Both the original anomaly image and its reconstructed normal image are subsequently passed through a difference block. In this step, multi-scale features are extracted using a pre-trained ImageNet model, and the discrepancies between these features are computed to pinpoint regions of interest (ROI). These differences are then detected and localized, following a strategy akin to DiAD [21].
+
+Latent Anomaly Schrödinger Bridge. The foundational basis of Latent Anomaly Schrödinger Bridge (LASB) is inspired by linear Schrödinger Bridge's from equation (10) i.e., the optimal backward drift in equation (8) consistently aims for the Dirac delta  $\delta_{\kappa}(\cdot)$ , ensuring convergence to  $\kappa$  regardless of  $p_B$ . That means, the forward process transforms the noisy anomaly image statistics and learns to reconstruct a normal image irrespective of terminal distribution. Now to elucidate the working mechanism of the LASB diffusion model which will consider a Chang et al. [7] design strategy
+
+SGMs and diffusion models operating in pixel space require significant computational resources and memory. To address this challenge, we introduce the LASB model. Before training the Linear-SB, we first train  $E_{VQ}$  and  $D_{VQ}$ , the encoder and decoder of the VQ-VAE model [49], for perceptual compression, following the approach of stable diffusion [37]. During VQ-VAE training, we input either normal or anomaly-augmented images from the training set into  $E_{VQ}$  and aim to reconstruct the input using  $D_{VQ}$ . Once the model converges, its weights are frozen, and we proceed to train the Linear-SB in the latent space. Unlike Gaussian diffusion models, where noise is progressively added at each time step, leading to the deterioration of image structure at later stages, we adopt a semidegradation strategy similar to Saharia et al. [40]. This ensures that, during the forward pass, the LASB model performs a smooth transformation while preserving structural information. Figure 1 illustrates the contrast between the Gaussian diffusion process and the Schrödinger Bridge diffusion process when applied to anomaly-free reconstruction. According to Liu et al. [27] for this smooth structural transformation the posterior for the SB (equation 6, 7)
+
+Algorithm 2 Inference/Sampling Procedure
+
+1: Input:  $z_{N} \sim p_{B}^{(z)}(z_{N})$ , trained  $\epsilon_{\theta}(\cdot, \cdot)$  
+2: for  $n = N$  to 1 do  
+3: Predict  $z_0^\epsilon$  using  $\epsilon_{\theta}(z_n, t_n)$  
+4:  $z_{n - 1}\sim p(z_{n - 1}|z_0^\epsilon ,z_n)$  
+5: end for  
+6: return  $z_0$
+
+given a boundary pair condition  $(X_0, X_1)$  reveals an analytic. Now as we inject the image statistics via perceptual compression using VQ-VAE into latent space, the analytical form can be formulated as,
+
+$$
+q \left(z _ {t} \mid z _ {0}, z _ {1}\right) = \mathcal {N} \left(z _ {t}, \mu_ {t} \left(z _ {0}, z _ {1}\right), \Sigma_ {t}\right) \tag {11}
+$$
+
+where  $\mu_t = \frac{\bar{\sigma}_t^{(z)2}}{\bar{\sigma}_t^{(z)2} + \sigma_t^{(z)2}} z_0 + \frac{\sigma_t^{(z)2}}{\bar{\sigma}_t^{(z)2} + \sigma_t^{(z)2}} z_1$ , and
+
+$$
+\Sigma_ {t} = \frac {\bar {\sigma} _ {t} ^ {(z) 2} \sigma_ {t} ^ {(z) 2}}{\bar {\sigma} _ {t} ^ {(z) 2} + \sigma_ {t} ^ {(z) 2}} \mathbb {I}.
+$$
+
+The accumulated variances are,  $\sigma_t^{(z)2}$ $\begin{array}{r}\int_0^t\beta_\tau d\tau \quad \mathrm{and}\quad \overline{\sigma}_t^2 \coloneqq \int_t^1\beta_\tau d\tau . \end{array}$
+
+We hence train the proposed LASB using our objective function:
+
+$$
+\mathcal {L} _ {L A S B} := \left| \left| \epsilon_ {\theta} \left(z _ {t}, t\right) - \left(\frac {z _ {t} - z _ {0}}{\sigma_ {t} ^ {(z)}}\right) \right| \right| \tag {12}
+$$
+
+where  $\epsilon (\cdot)$  is the network and  $\theta$  are the parameters associated with it. The training and the inference procedures are summarized in Algorithms 1 and 2.
+
+# 4. Experiments and Results
+
+Datasets. To validate the efficacy of the LASB method, we utilize two challenging datasets in industrial anomaly detection: the MVTec-AD [5] dataset and the VisA [63] dataset. The MVTec-AD dataset comprises 15 categories, including 10 object types and 5 texture types, with a total of 5,354 high-resolution images. Of these, 3,629 anomaly-free images are used for training, while the remaining 1,725 test images include both normal and anomalous samples. The VisA dataset consists of 12 distinct objects organized into 12 subsets, categorized into complex Structures, Multiple Instances, and Single Instance types. It includes 10,821 high-resolution images, with 9,621 normal images and 1,200 anomalous images exhibiting 78 distinct anomalies, offering a comprehensive benchmark for anomaly detection and localization methods. Both datasets provide pixel-level ground truth annotations to facilitate the evaluation of anomaly localization performance.
+
+Evaluation Metrics. For a rigorous quantitative evaluation of our experimental results in anomaly detection and localization, we employ AUROC (Area Under the Receiver Operating Characteristic Curve), AP (Average Precision),
+
+![](images/bebe370c000dcd6188296baeead296d86b1951bde9a039801ca88b3425d3a5fb.jpg)  
+Figure 3. Comparison of training time (left), training memory (middle), and sampling time (right) across different image resolutions for SGM, SB, DDPM, and LASB models. The LASB model (green) demonstrates consistently lower training time, memory usage, and sampling time, particularly in contrast to the SB model (red), which exhibits the highest resource consumption.
+
+![](images/e28d21147abc88f15c8b3804ebe7c08beeb9e9b3d821d6fba38cf1c7600684d3.jpg)
+
+![](images/97f2c5035eabf69b86385ba0d9a96a5e665d6104fc16570eb1ad549b60262cdb.jpg)
+
+and F1max (maximum F1-score) as our primary metrics [21]. Here,  $cls$  denotes image-level anomaly detection, while seg pertains to pixel-level anomaly localization. A detailed class-specific comparison and results for all the aforementioned metrics are provided in Section S2 of supplementary material. In Table 1, we present the average performance of each model across the evaluated dataset.
+
+<table><tr><td>Method</td><td>Venue</td><td>Adopted Method</td><td>MvTec</td><td>ViSA</td></tr><tr><td colspan="5">Class-based</td></tr><tr><td>SimpleNet [29]</td><td>CVPR 2023</td><td>Embedding-based</td><td>99.6 / 98.1</td><td>96.2 / 98.5</td></tr><tr><td>PatchCore [38]</td><td>CVPR 2022</td><td>Embedding-based</td><td>99.1 / 98.1</td><td>91.0 / 98.1</td></tr><tr><td>DSR [58]</td><td>ECCV 2022</td><td>-</td><td>98.2 / 70.2</td><td>-</td></tr><tr><td>PaDiM [12]</td><td>ICPR 2021</td><td>Memory Bank</td><td>97.5 / 92.1</td><td>89.1 / 85.9</td></tr><tr><td>CS-Flow [39]</td><td>WACV 2022</td><td>Normalization Flow</td><td>97.2 / 84.5</td><td>-</td></tr><tr><td>CFLOW-AD [19]</td><td>WACV 2022</td><td>Normalization Flow</td><td>96.2 / 97.1</td><td>-</td></tr><tr><td>OCR-GAN [26]</td><td>TIP 2023</td><td>GAN</td><td>98.3 / -</td><td>97.9 / -</td></tr><tr><td>DRAEM [56]</td><td>ICCV 2021</td><td>Encoder</td><td>98.0 / 97.3</td><td>88.7/93.5</td></tr><tr><td>RD4AD [13]</td><td>CVPR 2022</td><td>Embedding-based</td><td>98.5 / 97.8</td><td>96.9 / 98.3</td></tr><tr><td>ADSPR [43]</td><td>ICCV 2023</td><td>Diffusion</td><td>97.67 / 97.36</td><td>-</td></tr><tr><td>DiffAD [60]</td><td>ICCV 2023</td><td>Diffusion</td><td>98.72 / 98.26</td><td>89.79 / 68.28</td></tr><tr><td>D3AD [48]</td><td>CVPR 2024</td><td>Diffusion</td><td>97.15 / 97.44</td><td>95.51 / 94.27</td></tr><tr><td>DDAD [33]</td><td>Arxiv 2023</td><td>Diffusion</td><td>99.84 / 98.05</td><td>98.9 / 97.58</td></tr><tr><td>TransFusion [16]</td><td>ECCV 2024</td><td>Diffusion</td><td>99.24 / 94.33</td><td>98.53 / 86.26</td></tr><tr><td>LASB (Ours)</td><td>-</td><td>Diffusion</td><td>99.66 / 99.15</td><td>98.52 / 99.06</td></tr><tr><td colspan="5">Unified (multi-class)</td></tr><tr><td>DRAEM [56]</td><td>ICCV 2021</td><td>Encoder</td><td>88.1 / 87.2</td><td>79.1 / 91.3</td></tr><tr><td>HVQ-Trans [32]</td><td>NeurIPS 2023</td><td>Non-Diffusion</td><td>98.0 / 97.3</td><td>93.2 / 98.7</td></tr><tr><td>MambaAD [20]</td><td>NeurIPS 2024</td><td>Non-Diffusion</td><td>98.6 / 97.7</td><td>94.3 / 98.5</td></tr><tr><td>OmniAL [61]</td><td>CVPR 2023</td><td>Non-Diffusion</td><td>97.2 / 98.3</td><td>94.2 / 96.0</td></tr><tr><td>GLAD [54]</td><td>ECCV 2024</td><td>Diffusion</td><td>97.5 / 97.4</td><td>91.8 / 97.8</td></tr><tr><td>UniAD [55]</td><td>NeurIPS 2022</td><td>Transformer</td><td>96.52 / 96.8</td><td>85.5 / 95.925</td></tr><tr><td>DIAD [21]</td><td>AAAI 2024</td><td>Diffusion</td><td>97.15 / 96.76</td><td>86.75 / 96.04</td></tr><tr><td>LASB (Ours)</td><td>-</td><td>Diffusion</td><td>99.14 / 98.66</td><td>94.2 / 98.18</td></tr></table>
+
+Table 1. Comparison of state-of-the-art (SOTA) anomaly detection methods categorized into class-based and unified (multi-class) approaches. The table highlights the adopted methodologies and their class-wise average performance in terms of AUROC<sub>cls</sub> / AUROC<sub>seg</sub> metrics on two datasets: MvTec and ViSA respectively. An exhaustive list of metrics along with class specific performance details is comprehensively illustrated in Section S2 of supplementary material.
+
+Implementation Details. For all our experiments, we resized MVTec-AD images to  $256 \times 256$  resolution. To fine-tune the VQ-VAE (Auto-encoder) utilizing the KL-based method, we initialized the network weights from the Stable Diffusion [37] model, where the image is encoded into a latent vector with a size of  $64 \times 64 \times 3$ . We chose this latent vector size due to its preservation of details and low FID generation capabilities. After training the auto-encoder, we froze the weights and trained the latent denoising network.
+
+We used the U-Net model as in [23] as our latent denoising network. During inference, we pass only the anomaly test image as input to the DDPM sampler [23]. Further details on batch size and various other network hyperparameters are in Section S4 of the appendix.
+
+Baselines. To benchmark LASB, we provide a detailed comparison with state-of-the-art (SOTA) reconstruction methods, as summarized in Table 1. These methods are categorized into two distinct groups: class-based and unified (multi-class) approaches. Class-based methods train models specific to each class, with the number of models increasing linearly with the number of classes, making scalability a challenge. In contrast, unified approaches handle multiple classes within a single model, offering a more scalable solution. By separating the class-based and unified approaches, we ensure a fair and comprehensive comparison.
+
+Class-based Methods: Performance Analysis. Among the class-based methods, PaDiM [12], with its memory bank-based architecture, achieves  $97.5\%$  on MVtec-AD and  $89.1\%$  on ViSA. While effective for simpler anomalies, its scalability and ability to model inter-class variations remain limited. In contrast, DiffAD [60] demonstrates strong competitiveness, achieving  $98.72\%$  on MVtec-AD. However, its performance on ViSA drops to  $89.79\%$ , indicating its difficulty in adapting to diverse industrial anomalies. DDAD [33] shows highly competitive results, achieving  $99.84\%$  on MVtec-AD and  $98.9\%$  on ViSA. Its diffusion-based modeling effectively captures structural patterns, making it one of the closest competitors to LASB. However, LASB surpasses DDAD on ViSA by a margin of  $0.38\% \uparrow$ , demonstrating superior adaptability to multi-class industrial scenarios.
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Metrics</td><td colspan="3">Pixel Space</td><td colspan="3">Latent Space</td></tr><tr><td>DDPM [23]</td><td>DDAD [33]</td><td>GLAD [54]</td><td>LDM [37]</td><td>DiAD [21]</td><td>LASB (Ours)</td></tr><tr><td rowspan="3">Detection</td><td>AUROCcls</td><td>71.9</td><td>99.8</td><td>97.5</td><td>76.6</td><td>97.2</td><td>99.2 (-0.6%)</td></tr><tr><td>APcls</td><td>81.6</td><td>99.5</td><td>99.1</td><td>87.6</td><td>99.0</td><td>99.3 (-0.2%)</td></tr><tr><td>F1maxcls</td><td>86.6</td><td>97.9</td><td>96.6</td><td>88.1</td><td>96.5</td><td>98.5 (+2.0%)</td></tr><tr><td rowspan="3">Localization</td><td>AUROCseg</td><td>75.6</td><td>98.0</td><td>97.4</td><td>85.1</td><td>96.8</td><td>98.6 (+0.60%)</td></tr><tr><td>APseg</td><td>13.3</td><td>59.0</td><td>60.8</td><td>27.6</td><td>52.6</td><td>78.2 (+17.4%)</td></tr><tr><td>F1maxseg</td><td>19.5</td><td>59.4</td><td>60.7</td><td>31.0</td><td>55.5</td><td>70.7 (+10.0%)</td></tr></table>
+
+Table 2. Performance comparison of LASB with diffusion-based models on MVTec-AD using AUROC, AP, and F1max. The best results are in bold; the second-best is underline. Improvements are shown as a percentage over the second-best.
+
+![](images/16e30b7fc9635c0159686acf423a06eb1cae9a146bb93bc63681736cf4af9f66.jpg)  
+Figure 4. Visual representation of test samples from the MVtec dataset, depicted through heatmaps for various models for different categories.
+
+Unified Models: Performance Analysis. For unified approaches, DRAEM [56] emerges as one of the least competitive models, with AUROC<sub>cls</sub> scores of  $88.1\%$  on MVTec-AD and  $79.1\%$  on ViSA. Its reliance on autoencoder-based reconstruction, combined with limited augmentation strategies, hampers its ability to generalize to diverse and complex anomaly scenarios, particularly in the multi-class ViSA dataset. Similarly, UniAD [55] performs less competitively, achieving  $96.52\%$  on MVTec-AD and  $85.5\%$  on ViSA. Its transformer-based architecture, while robust for certain tasks, struggles to balance efficiency and accuracy in anomaly detection across diverse classes. Similarly, DIAD [21] achieves  $97.15\%$  on MVTec-AD and  $86.75\%$  on ViSA but falls short in handling the intricate class variations present in the datasets. Among the competitive unified methods, MambaAD [20] achieves  $98.6\%$  on MVTec-AD and  $94.3\%$  on ViSA, showcasing its strength in handling multi-class scenarios using state-space models. However, LASB outperforms MambaAD with a score of  $99.14\%$  on MVTec-AD and  $94.2\%$  on ViSA, highlighting its superior capability to balance computational efficiency with detection accuracy. Furthermore, HVQ-Trans [32], with scores of  $98.0\%$  on MVTec-AD and  $93.2\%$  on ViSA, demonstrates solid performance but remains slightly behind LASB in scalability and robustness. Detailed class-specific results are available in Section S2 of supplementary material, and Figure 4 provides qualitative heat-map visualizations of anomaly regions, further demonstrating LASB's efficacy in both class-based and unified settings.
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Metrics</td><td colspan="2">Non-Diffusion Method</td><td colspan="3">Diffusion-based Method</td></tr><tr><td>DRAEM [23]</td><td>UniAD [55]</td><td>DiffAD [60]</td><td>DiAD [21]</td><td>LASB (Ours)</td></tr><tr><td rowspan="3">Detection</td><td>AUROCcls</td><td>79.1</td><td>85.5</td><td>89.5</td><td>86.8</td><td>94.2</td></tr><tr><td>APcls</td><td>81.9</td><td>85.5</td><td>-</td><td>88.3</td><td>92.2</td></tr><tr><td>F1maxcls</td><td>78.9</td><td>84.4</td><td>-</td><td>85.1</td><td>94.5</td></tr><tr><td rowspan="3">Localization</td><td>AUROCseg</td><td>91.3</td><td>95.9</td><td>71.2</td><td>96.0</td><td>98.2</td></tr><tr><td>APseg</td><td>23.5</td><td>21.0</td><td>-</td><td>26.1</td><td>46.4</td></tr><tr><td>F1maxseg</td><td>29.5</td><td>27.0</td><td>-</td><td>33.0</td><td>52.6</td></tr></table>
+
+Table 3. Results for multi-class anomaly detection and localization on VisA dataset. The best results are indicated in bold, and the second-best results are denoted with an underline.
+
+# 5. Ablation Studies and Analysis
+
+In this section, we first differentiate the proposed LASB method's performance in latent spaces and explain the advantages of utilizing Linear-SB in the latent space. We then assess the LASB model's effectiveness when compared with Stable Diffusion [37] and other models for anomaly localization and detection tasks. Finally, we demonstrate the robustness of our proposed method, demonstrating its ability to deliver precise and consistent outcomes during test-time evaluations.
+
+LASB vs Standard Diffusion Models. In the realm of anomaly detection, standard diffusion models such as DDPM [23] and LDM [37] achieve competitive performance in anomaly detection tasks but exhibit significant limitations in localization, as evidenced in Table 2. To mitigate these limitations, DiAD [21] integrates a semantic guidance network to improve LDM's localization by incorporating additional contextual information. Despite these improvements, DiAD [21] still falls short of achieving state-of-the-art (SOTA) results, indicating persistent challenges in fully leveraging diffusion models for comprehensive anomaly detection tasks. In contrast, our LASB employs a novel approach that leverages Linear-SB within the latent space, avoiding the limitations associated with reconstructing from pure Gaussian noise, a common issue in standard diffusion processes when applied to anomaly-free reconstruction. Moreover, LASB semi-degrades latent space to retain structural integrity, facilitating more effective anomaly detection and localization. This method not only enhances the robustness of the detection process but also significantly improves localization accuracy. Consequently, the LASB model surpasses standard diffusion models across multiple metrics, demonstrating superior performance in multi-class anomaly detection, as evidenced in Table 2.
+
+We compare the performance of our proposed Latent Anomaly Schrödinger Bridge (LASB) method with the DDPM [23], DDAD [33], GLAD [54], LDM [37], and DiAD [21] models to demonstrate the effectiveness and efficiency of our approach. First, we evaluate the models on the MVTec-AD dataset for both anomaly detection and localization tasks. As shown in Table 2, LASB significantly outperforms all the other models in localization tasks for all the metrics. Specifically, LASB achieves a  $0.6\% \uparrow$ ,  $17.4\% \uparrow$ , and  $10.0\% \uparrow$  improvement in AUROC $_{seg}$ , AP $_{seg}$ , F1max $_{seg}$  respectively. Also, it shows  $2.0\% \uparrow$  enhancement in F1max $_{cls}$  for detection, compared to GLAD [54]. These improvements highlight the robustness and effectiveness of the latent space approach specifically for localization. Furthermore, as shown in Figure 3, pixel-space models like DDPM [23], SGM [46], and SB models trained via Iterative Proportional Fitting [11, 25] or likelihood-based methods [8] re
+
+<table><tr><td></td><td>NFE-1</td><td>NFE-2</td><td>NFE-5</td><td>NFE-10</td><td>NFE-100</td><td>NFE-500</td><td>NFE-1000</td></tr><tr><td>MVTec-AD</td><td>97.42 / 96.82</td><td>98.16 / 97.84</td><td>98.94 / 98.16</td><td>99.14 / 98.66</td><td>99.14 / 98.66</td><td>99.14 / 98.66</td><td>99.14 / 98.66</td></tr><tr><td>VisA</td><td>92.74 / 96.71</td><td>93.21 / 97.15</td><td>93.96 / 97.90</td><td>94.2 / 98.18</td><td>94.2 / 98.18</td><td>94.2 / 98.18</td><td>94.2 / 98.18</td></tr><tr><td>Inference Time (secs)</td><td>0.12</td><td>0.25</td><td>0.52</td><td>0.74</td><td>1.5</td><td>7.5</td><td>15</td></tr></table>
+
+Table 4. Performance evaluation across varying numbers of function evaluations (NFEs) on the MVTec-AD and VisA datasets. The tabulated metrics, AUROC<sub>cls</sub> / AUROC<sub>seg</sub>, provide a comprehensive overview of image-level and pixel-level anomaly detection performance at each NFE. Additionally, the inference time, measured in wall-clock seconds on an NVIDIA V100 GPU, underscores the computational trade-offs associated with increasing NFEs, reflecting the balance between model efficiency and performance.  
+
+<table><tr><td>Method</td><td>DR/EM [56]</td><td>PatchCore [38]</td><td>DiffAD [60]</td><td>TransFusion [16]</td><td>DIAD [21]</td><td>GLAD [54]</td><td>ADSPR [43]</td><td>DDAD [33]</td><td>LASB (ours)</td></tr><tr><td>Inference Time (secs)</td><td>0.15</td><td>0.44</td><td>2.6</td><td>1.2</td><td>2.8</td><td>1.8</td><td>1.6</td><td>1.5</td><td>0.74</td></tr></table>
+
+require more training time, memory, and exhibit slower sampling. In contrast, latent-space LASB excels in all areas, achieving up to  $5 \times$  and  $2 \times$  faster training,  $3 \times$  and  $2 \times$  memory reduction, and  $4 \times$  and  $2 \times$  faster sampling compared to SB-based models [8, 11, 25] and DDPM [23], respectively.
+
+Table 5. Inference times for various methods where the inference time, measured in wall-clock seconds on an NVIDIA V100 GPU.  
+
+<table><tr><td>Task</td><td>Metrics</td><td>1×</td><td>2×</td><td>4×</td><td>8×</td></tr><tr><td rowspan="3">Detection</td><td>AUROCcls</td><td>99.05</td><td>99.01</td><td>99.01</td><td>98.98</td></tr><tr><td>APcls</td><td>99.15</td><td>99.14</td><td>99.14</td><td>99.11</td></tr><tr><td>F1maxcls</td><td>98.59</td><td>98.33</td><td>99.83</td><td>98.32</td></tr><tr><td rowspan="3">Localization</td><td>AUROCseg</td><td>98.58</td><td>98.58</td><td>98.59</td><td>98.57</td></tr><tr><td>APseg</td><td>78.17</td><td>78.14</td><td>78.14</td><td>78.12</td></tr><tr><td>F1maxseg</td><td>70.62</td><td>70.59</td><td>70.61</td><td>70.58</td></tr></table>
+
+Table 6. Stability analysis of LASB on the MVTec-AD dataset for detection and localization tasks by performing the sampling multiple times. Results reported are mean values across classes and multiple samplings (see Section S1 of appendix for more detailed results).
+
+Generalization and Sampling Stability. In the field of anomaly detection using diffusion models, achieving consistent outcomes during the sampling or inference stage is notably challenging due to the inherent stochastic nature of generative processes. This inconsistency can be particularly problematic in real-world applications where reliable and stable detection is critical. Therefore, assessing the stability of model outputs across multiple inferences is essential. Our approach involves training the model once and then conducting multiple sampling or inference tests to evaluate if the outcomes remain consistent over time. A critical consideration for real-world applicability is achieving fast inference while minimizing computational overhead. The proposed LASB model strikes a balance between performance and efficiency, ensuring its suitability for deployment in practical scenarios.
+
+Our findings, as detailed in Table 6, reveal that the LASB model exhibits remarkable stability across all evaluation metrics, showing negligible variance across numerous inferences. This consistency is attributed to the model's capability to maintain structural integrity and effectively tran
+
+sition from anomalous to normal latent spaces. The LASB model is designed to reconstruct a normal image regardless of the underlying anomalies, compelling it to disregard anomalous features during reconstruction. This process not only ensures that anomalies of various patterns, sizes, and orientations are effectively handled but also enhances the overall reliability of the model.
+
+Inference Complexity. As shown in Table 4, LASB demonstrates exceptional efficiency with its rapid sampling capabilities, achieving near-optimal performance with as few as 10 NFEs (Number of Function Evaluations) and an inference time of only 0.74 seconds on an NVIDIA V100 GPU. This highlights LASB's practicality for real-world applications, where fast and reliable anomaly detection and localization are crucial. To further illustrate its efficiency, we compare LASB's inference time with existing state-of-the-art (SOTA) models in Table 5. LASB is approximately  $1.6 \times$  faster than TransFusion [16], a leading class-based method, while also outperforming it in both anomaly localization and detection tasks. Compared to DiAD [21], LASB achieves  $2 \times$  faster inference times, delivering significantly stronger performance in both detection and localization metrics. This demonstrates LASB's clear advantage in balancing computational efficiency and robust anomaly detection.
+
+# 6. Conclusions and Future Work
+
+The proposed Latent Anomaly Schrödinger Bridge (LASB) model demonstrates robust performance in anomaly detection and localization tasks. Its unified nature delimits the need for extra guidance or additional network components. LASB also excels in producing stable inferential results, requires less computational memory, and benefits from faster training and sampling rates. Given these advantages, LASB is well-suited for deployment in real-world industrial applications. Looking ahead, future research could focus on enhancing the robust translation mechanisms specific to anomaly detection and localization tasks.
+
+Acknowledgements. We thank the anonymous reviewers for their valuable feedback that improved the presentation of this paper.
+
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+# A Unified Model for Compressed Sensing MRI Across Undersampling Patterns
+
+Armeet Singh Jatyani* Miguel Liu-Schiaffini
+
+Jiayun Wang* Aditi Chandrashekar Zihui Wu Bahareh Tolooshams Anima Anandkumar Ifornia Institute of Technology
+
+{armeet,peterw,ajchandr,zwu2,mliuschi,btoloosh,ana}@caltech.edu
+
+# Abstract
+
+Compressed Sensing MRI reconstructs images of the body's internal anatomy from undersampled measurements, thereby reducing scan time—the time subjects need to remain still. Recently, deep learning has shown great potential for reconstructing high-fidelity images from highly undersampled measurements. However, one needs to train multiple models for different undersampling patterns and desired output image resolutions, since most networks operate on a fixed discretization. Such approaches are highly impractical in clinical settings, where undersampling patterns and image resolutions are frequently changed to accommodate different real-time imaging and diagnostic requirements.
+
+We propose a unified MRI reconstruction model robust to various measurement undersampling patterns and image resolutions. Our approach uses neural operators—a discretization-agnostic architecture applied in both image and measurement spaces—to capture local and global features. Empirically, our model improves SSIM by  $11\%$  and PSNR by 4 dB over a state-of-the-art CNN (End-to-End VarNet), with  $600\times$  faster inference than diffusion methods. The resolution-agnostic design also enables zero-shot super-resolution and extended field-of-view reconstruction, offering a versatile and efficient solution for clinical MR imaging. Our unified model offers a versatile solution for MRI, adapting seamlessly to various measurement undersampling and imaging resolutions, making it highly effective for flexible and reliable clinical imaging. Our code is available at https://armeet.ca/nomri.
+
+# 1. Introduction
+
+Magnetic Resonance Imaging (MRI) is a popular noninvasive imaging technology, used in numerous medical and scientific applications such as neurosurgery [38], clinical oncology [20], diagnostic testing [16], neuroscience [22], and pharmaceutical research [36]. MRI is greatly limited by a
+
+slow data acquisition process, which sometimes requires patients to remain still for an hour [4, 39]. Hence, accelerating MRI scan has garnered tremendous attention [11, 18, 28].
+
+Compressed Sensing (CS) [9] enables MRI at sub-Nyquist rates and reduces acquisition time for greater clinical utility. This is framed as an ill-posed inverse problem [12], where prior knowledge about MR images is crucial for reconstruction. Traditional Compressed Sensing MRI assumes a sparse prior in a transform domain (e.g., wavelets [3]). Recent deep learning methods learn underlying data structures to achieve superior performance [5, 40]. Current state-of-the-art models establish an end-to-end mapping [15, 40] from undersampled measurements to image reconstruction in both image and frequency domains. However, these models often struggle with generalization across varying resolutions, a critical need in clinical practice where flexible resolution adjustments are necessary. A unified model that is agnostic to discretizations would greatly improve efficiency.
+
+Neural Operators (NOs) [21] are a deep learning framework that learns mappings between infinite-dimensional function spaces, making them agnostic to discretizations (resolutions). This property makes them suitable for tasks with data at varying resolutions, such as partial differential equations (PDEs) [21, 25, 34] and PDE-related applications [32, 35]. NOs could also be suitable for compressed sensing MRI due to measurements with multiple undersampling patterns. Various NO architectures [24, 26, 34] have been proposed. Recently, discrete-continuous (DISCO) convolutions [26, 31] have emerged as an efficient neural operator that captures local features and leverages GPU acceleration for standard convolutions. Due to the similarity to standard convolutions, the building blocks of many existing MRI deep learning models [5, 40], DISCO is a good candidate for resolution-agnostic MRI reconstruction.
+
+Our approach: We propose a unified model based on NOs, that is robust to different undersampling patterns and image resolutions in compressed sensing MRI (Fig. 1a). Our model follows an unrolled network design [15, 40] with DISCO [26, 31]. As the image resolution increases, DISCO maintains a resolution-agnostic kernel with a consistent con
+
+![](images/79bd6ab2cc51219f5cbccc1f37da7e35cd83009cc487046729e06a66ec617a2f.jpg)
+
+![](images/fac92d5640d352f6c9f5be132c40c252f7c8960477bf38717cfcc6057fcf75af.jpg)
+
+![](images/4c2e564d9eecb317644849f140a69641654d87616d351ad9b836f7c25fa1567f.jpg)  
+Figure 1. (a) We propose a unified model for MRI reconstruction, called neural operator (NO), which works across various measurement undersampling patterns, overcoming the resolution dependency limit of CNN-based methods like [40] that require a specific model for each pattern. (b) NO achieves consistent performance across undersampling patterns and outperforms CNN architectures such as [40] (for  $2 \times$  acceleration with one unrolled network cascade). (c) NO is resolution-agnostic. As image resolution increases, it maintains a consistent kernel size for alias-free rescaling, unlike CNNs with variable kernel sizes that risk aliasing. (d) NO enhances zero-shot super-resolution MRI reconstruction, outperforming CNNs [40].
+
+![](images/bd4d3c4cd2c6b2889be5b0e7e58493399d38bcf4d297f231bf96b7ad58336866.jpg)
+
+volution patch size, while the regular convolution kernel contracts to a point (Fig. 1c). The DISCO operators learn in both measurement/frequency  $\mathbf{k}$  space  $(\mathrm{NO_k})$  and image space  $(\mathrm{NO_i})$ .  $\mathrm{NO_k}$  makes our framework agnostic to different measurement undersampling patterns, and  $\mathrm{NO_i}$  makes the framework agnostic to different image resolutions. Additionally, the learning in both frequency and image space allows the model to capture both local and global features of images due to the duality of the Fourier transform that connects the frequency and image space. The resolution-agnostic design also enables super-resolution in both frequency and image space, allowing the extended field of view (FOV) and super-resolution of the reconstructed MR images.
+
+We empirically demonstrate that our model is robust to different measurement undersampling rates and patterns (Fig. 1a). Our model performs consistently across these pattern variations, whereas the existing method drops in performance (Fig. 1b). We achieve up to  $4 \times$  lower NMSE and 5 dB PSNR improvement from the baseline when evaluating on different undersampling patterns. The model is efficient and  $600 \times$  faster than the diffusion baseline [5, 17, 43]. We also show that our model outperforms the state-of-the-art in
+
+zero-shot super-resolution inference (Fig. 1d) and extended FOV reconstruction (Fig. 5).
+
+Our work has two main contributions: 1) We propose a unified neural operator model that learns in function space and shows robust performance across different undersampling patterns and image resolutions in compressed sensing MRI. To the best of our knowledge, this is the first resolution-agnostic framework for MRI reconstruction. 2) Our model demonstrates empirical robustness across measurement undersampling rates and patterns, reconstructing MR images with zero-shot higher resolutions and a larger field of view.
+
+# 2. Related Works
+
+Accelerated MRI. One way to accelerate MRI scan speed is parallel imaging, in which multiple receiver coils acquire different views of the object of interest simultaneously, and then combine them into a single image [11, 30, 37]. When MRI reconstruction is paired with compressed sensing, predefined priors or regularization filters can be leveraged to improve reconstruction quality [27, 28]. Recent works have shown that learned deep-learning priors outperform handcrafted priors in reconstruction fidelity. Convolutional neural
+
+networks (CNNs) [8, 15, 18, 40], variational networks (based on variational minimization) [15, 40], and generative adversarial networks (GANs) [7, 18] have all demonstrated superior performance than traditional optimization approach for compressed sensing MRI reconstruction from undersampled measurements. However, unlike conventional compressed sensing which operates in the function space and is agnostic to measurement undersampling patterns, the aforementioned deep learning methods operate on a fixed resolution. As a result, changes in resolution lead to degradation in performance, and multiple models are needed for different settings. We propose a resolution-agnostic unified model.
+
+Discretization-Agnostic Learning and Neural Operators. Empirically, diffusion models have shown relatively consistent performance with different measurement undersampling patterns in accelerated MRI [14]. However, diffusion models usually take more runtime at inference and need extensive hyperparameter tuning for good performance (Section 4.5). Additionally, they are not fundamentally discretization-agnostic by design. Neural operators [1, 21] are deep learning architectures specifically designed to learn mappings between infinite-dimensional function spaces. They are discretization-agnostic, allowing evaluation at any resolution, and converge to a desired operator as the resolution approaches infinity. Neural operators have empirically achieved good performance as surrogate models of numerical solutions to partial differential equations (PDEs) [21, 25, 34] with various applications, such as material science [35], weather forecasting [32], and photoacoustic imaging [13]. The design of neural operators often depends on the application at hand. For example, the Fourier neural operator (FNO) [24], which performs global convolutions, has shown consistent discretization-agnostic performance in various applications [1]. Other designs of neural operators [23, 26] rely on integration with locally-supported kernels to capture local features, which has shown to be useful in applications where local features are important, such as modeling turbulent fluids [23]. Additionally, neural operators with local integrals can be made efficient with parallel computing compared to those requiring global integrals. Our MRI framework, based on neural operators with local integrals, is agnostic to undersampling patterns and output image resolutions.
+
+# 3. Methods
+
+We first discuss the background of compressed sensing MRI and the unrolled network framework we use. We then discuss how we can extend the existing network building block, standard convolution, to resolution-agnostic neural operators. We also introduce DISCO [31], a neural operator design we adopt, and we capture global and local image features with DISCO. We conclude the section with the super-resolution designs. We call the measurement or frequency space  $\mathbf{k}$ -space, and physical or spatial space image space hereafter.
+
+# 3.1. MRI Reconstruction with Unrolled Networks
+
+Background. In MRI, anatomical images  $\mathbf{x}$  of the patient are reconstructed by acquiring frequency-domain measurements  $\mathbf{k}$ , where the relationship is defined as:
+
+$$
+\mathbf {k} := \mathcal {F} (\mathbf {x}) + \epsilon \tag {1}
+$$
+
+where  $\epsilon$  is the measurement noise and  $\mathcal{F}$  is the Fourier transform. In this paper, we consider the parallel imaging setting with multiple receiver coils [19, 44], where each coil captures a different region of the anatomy. The forward process of the  $i^{\mathrm{th}}$  coil measures  $\mathbf{k}_i\coloneqq \mathcal{F}(S_i\mathbf{x}) + \epsilon_i$  where  $S_{i}$  is a position-dependent sensitivity map for the  $i^{\mathrm{th}}$  coil. To speed up the imaging process, measurements are undersampled as  $\tilde{\mathbf{k}} = M\mathbf{k}$  in the compressed sensing MRI setting, where  $M$  is a binary mask that selects a subset of the k-space points. Classical compressed sensing methods reconstruct the image  $\hat{\mathbf{x}}$  by solving an optimization problem
+
+$$
+\hat {\mathbf {x}} = \operatorname {a r g m i n} _ {\mathbf {x}} \frac {1}{2} \sum_ {i} \left\| \mathcal {A} (\mathbf {x}) - \tilde {\mathbf {k}} \right\| _ {2} ^ {2} + \lambda \Psi (\mathbf {x}) \tag {2}
+$$
+
+where  $i$  is the coil index,  $\mathcal{A}(\cdot) \coloneqq MFS_{i}(\cdot)$  is the linear forward operator, and  $\Psi(\mathbf{x})$  is a regularization term. The optimization objective can be considered as a combination of physics constraint and prior. While the above optimization can be solved using classical optimization toolboxes, an increasing line of works uses deep neural networks to learn data priors and show improved reconstruction performance [15, 40]. Among them, unrolled networks [15, 40] have gained popularity as they incorporate the known forward model, resulting in state-of-the-art performance. Unrolling, which started with the nominal work of LISTA [10], proposes to design networks using iterations of an optimization algorithm to solve inverse problems. This approach incorporates domain knowledge (i.e., the forward model) and leverages deep learning to learn implicit priors from data [29, 41]. In the context of MRI and assuming a differential regularization term, the optimization problem is expanded to iterative gradient descent steps with injected CNN-based data priors. Each layer mimics the gradient descent step from  $\mathbf{x}^t$  to  $\mathbf{x}^{t+1}$ :
+
+$$
+\mathbf {x} ^ {t + 1} \leftarrow \mathbf {x} ^ {t} - \eta^ {t} \mathcal {A} ^ {*} (\mathcal {A} (\mathbf {x} ^ {t}) - \tilde {\mathbf {k}}) + \lambda^ {t} \operatorname {C N N} (\mathbf {x} ^ {t}) \tag {3}
+$$
+
+where  $\eta^t$  controls the weight of data consistency term and  $\lambda^t$  controls that of the data-driven prior term. The data consistency term samples the data in the frequency domain, hence it is applicable to any spatial resolution. However, the prior term only operates on a specific resolution with CNNs. This means when changing the undersampling patterns, one needs another CNN trained for that setting, which greatly limits the flexibility of the reconstruction system.
+
+Extending to Neural Operators. We learn the prior in function space via discretization-agnostic neural operators
+
+![](images/1cfb141d7675bddef93de0b6f015a4dc0b22a2793f8b945af4a3bae97a2d8b20.jpg)  
+Figure 2. MRI reconstruction pipeline. NO learns data priors in function space with infinite resolution. Specifically we propose NOs in the k (frequency) space  $\mathrm{NO_k}$  (k space NO) and image space  $\mathrm{NO_i}$  (image space NO), which capture both global and local image features, due to the duality between physical and frequency space.  $\mathcal{F}^{-1}$  refers to the inverse Fourier transform. We provide the framework design details in Section 3.1 and NO design details in Section 3.2.
+
+in  $\mathbf{k}$  space  $(\mathrm{NO_k})$  and image space  $(\mathrm{NO_i})$ . Specifically, we first use a  $\mathbf{k}$  space neural operator  $\mathrm{NO_k}$  to learn  $\mathbf{k}$  space prior and then apply a cascade of unrolled layers, each of which features a data consistency loss and the image space  $\mathrm{NO_i}$  for image prior learning:
+
+$$
+\mathbf {x} ^ {0} \leftarrow \mathcal {F} ^ {- 1} \left(\mathrm {N O} _ {\mathbf {k}} (\tilde {\mathbf {k}})\right) \tag {4}
+$$
+
+$$
+\mathbf {x} ^ {t + 1} \leftarrow \mathbf {x} ^ {t} - \eta^ {t} \mathcal {A} ^ {*} (\mathcal {A} (\mathbf {x} ^ {t}) - \tilde {\mathbf {k}}) + \lambda^ {t} \mathrm {N O} _ {\mathbf {i}} ^ {t} (\mathbf {x} ^ {t}) \tag {5}
+$$
+
+where  $\mathrm{NO}_{\mathbf{i}}^{t}$  refers to the image-space NO at cascade  $t$ . We follow existing works [15, 40] and only have one  $\mathrm{NO}_{\mathbf{k}}$  for the first cascade. Our framework flexibly works for different resolutions with the design details in Section 3.2.
+
+Framework Overview. Fig. 2 depicts the pipeline of our neural operator framework for MRI reconstruction. The undersampled measurement  $\tilde{\mathbf{k}}$  is first fed to a neural operator  $\mathrm{NO_k}$  which operates in measurement  $\mathbf{k}$  space to learn global image features and then inverse Fourier transformed to get an image. Following Eqn. 4 and 5, we iterate a few cascades of unrolled layers, consisting of a neural operator  $\mathrm{NO_i}$  which operates in image  $\mathbf{x}$  space and a data consistency update.
+
+# 3.2. Neural Operator Design
+
+Neural operators, which learn mappings between function spaces, offer a unified approach to discretization-agnostic MRI reconstruction. Given that accurate MRI reconstruction depends on capturing both local and global image features, we propose a neural operator architecture that incorporates both global and local inductive biases. We first discuss how we learn local features with local integration operators.
+
+Local Features via Local Integration Operator. Historically, the most common method of embedding a local inductive bias into deep neural networks has been by using locally-supported convolutional kernels, as in convolutional neural networks (CNNs). However, standard discrete convolutional kernels used in CNNs do not satisfy the resolution-agnostic properties of neural operators. Specifically, Liu et al. [26] show that CNN-style convolutional kernels converge to pointwise linear operators as the resolution is increased, instead of the desired local integration in the limit of infinite
+
+resolution. For a kernel  $\kappa$  and input function  $g$  defined over some compact subset  $D\subset \mathbb{R}^d$ , the local convolution operator in a standard convolution layer, which transforms input  $u$  to output  $v$ , is given by
+
+$$
+(k \star g) (v) = \int_ {D} \kappa (u - v) \cdot g (u) d u. \tag {6}
+$$
+
+Given a particular set of input points  $(u_{j})_{j = 1}^{m}\subset D$  with corresponding quadrature weights  $q_{j}$  and output positions  $v_{i}\in D$ , we adopt the discrete-continuous convolutions (DISCO) framework for operator learning [26, 31] and approximate the continuous convolution (Eqn. 6) as
+
+$$
+(k \star g) (v _ {i}) \approx \sum_ {j = 1} ^ {m} \kappa \left(u _ {j} - v _ {i}\right) \cdot g \left(x _ {j}\right) q _ {j}. \tag {7}
+$$
+
+We follow parameterize  $\kappa$  as a linear combination of predefined basis functions  $\kappa^{\ell}$ :  $\kappa = \sum_{\ell=1}^{L} \theta^{\ell} \cdot \kappa^{\ell}$ , where  $\theta^{\ell}$  are learnable parameters. We choose the linear piecewise basis from [26] as this achieves the greatest empirical results (see Sections B.3 & E of the supplementary). The convolutional kernel is thus parameterized by a finite number of parameters, independently of the grid on which the kernel is evaluated. The kernel is resolution-agnostic because we disentangle the resolution-agnostic basis and discrete learnable parameters. The basis  $\kappa^{\ell}$  is defined in the function space, and will be discretized at the desired resolution; discrete parameters  $\theta^{\ell}$  can be learned with gradient descent. Since we are operating on an equidistant grid on a compact subset of  $\mathbb{R}^2$ , we follow [26] and implement Eqn. 7 using standard convolutional kernels (thus enjoying the benefits of acceleration on GPUs using standard deep learning libraries) with two crucial modifications: 1) the kernel itself is defined as a linear combination of basis functions  $\kappa^{\ell}$ , and 2) the size of the kernel scales with the input resolution so as to remain a fixed size w.r.t. the input domain. We adopt the same basis functions as [26] in our experiments, and we use the local integration operator as the resolution-agnostic building block for the measurement space and image space operators.
+
+DISCO vs Standard 2D Convolution with Varying Resolutions. As the input resolution increases (the discretization
+
+becomes denser), DISCO [31] maintains the kernel size for each convolution and finally converges to a local integral. The standard 2D convolution kernel, however, gets increasingly smaller and finally converges to a point-wise operator (Fig. 1c). Although one could alleviate the issue of standard convolutions by interpolating the convolutional kernel shape to match with corresponding convolution patch sizes for different resolutions, the interpolated kernel will have artifacts that affect performance at new resolutions (Fig. 1d). DISCO, however, is agnostic to resolution changes as the kernel is in the function space.
+
+Global Features. A common neural operator architecture for learning global features is the Fourier neural operator (FNO) [24]. FNO takes the Fourier transform of the input, truncates the result beyond some fixed number of modes, and pointwise multiplies the result with a learned weight tensor, which is equivalent to a global convolution on the input by the convolution theorem. Interestingly, the forward process of MRI is a Fourier transformation, which means that local operations in measurement  $\mathbf{k}$  space are equivalent to global operators in image  $\mathbf{x}$  space and vice versa, due to their duality. Following FNO, we could apply a pointwise multiplication between the measurement  $\mathbf{k}$  and a learned weight tensor to capture global image features. However, FNO truncates high frequencies, which are crucial for MRI reconstruction. To address this, we directly apply the DISCO local integration operator on the measurement space to capture global image features without feature map truncation.
+
+UDNO: the Building Block. Without loss of generality, we make both the image-space  $\mathrm{NO}_{\mathrm{i}}$  and  $\mathbf{k}$  space  $\mathrm{NO}_{\mathbf{k}}$  be local neural operators that capture local features in the corresponding domain. Such a design learns both global and local image features due to domain duality. Motivations for adopting the U-shaped architecture are in Fig. 7 and Section A of the Supplementary. Each operator consists of multiple sub-layers, to which we refer as the U-Shaped DISCO Neural Operator, or UDNO. The motivation is that multi-scale designs have shown great success in capturing features at different scales in images and that U-shaped networks are among the most popular architectures in computer vision, demonstrating strong performance in various applications from medical imaging to diffusion [6, 33, 37]. Further, UDNO makes our framework very similar to an existing state-of-the-art E2E-VN [40], with the difference being standard convolutions replaced by DISCO operators. The UDNO follows the encoder/decoder architecture of the U-Net [37], replacing regular convolutions with DISCO layers.
+
+Loss. The parameters of the proposed neural operator are estimated from the training data by minimizing the structural similarity loss between the reconstruction  $\mathbf{x}$  and the ground truth image  $\mathbf{x}^*$  (the same as the E2E-VN [40]):
+
+$$
+\mathcal {L} (\hat {\mathbf {x}}, \mathbf {x} ^ {*}) = - \operatorname {S S I M} (\hat {\mathbf {x}}, \mathbf {x} ^ {*}), \tag {8}
+$$
+
+where SSIM is the Structural Similarity Index Measure [42].
+
+![](images/16a9f889ff062d7a55042de1d4ad45ad5bd849d690d5af4a4369881e0fcda26e.jpg)  
+Figure 3. Super resolution (denser discretization) in  $\mathbf{k}$  space or image space increases the FOV or resolution of the reconstructed image. With denser discretization, NO maintains a resolution-agnostic kernel while CNN kernels become relatively smaller in size. Empirically our NO outperforms CNNs [40] (Section 4.4).
+
+# 3.3. Super-Resolution
+
+Neural operators enable zero-shot super-resolution. As shown in Fig. 3, increasing resolution corresponds to denser discretization between fixed minimum and maximum values, while the overall domain range remains constant. Due to the dual nature of frequency and image space, enhancing resolution in  $\mathbf{k}$  space extends the field of view (FOV) in the reconstructed image, whereas increasing resolution in image space enhances the image's detail. Our proposed NO framework includes resolution-agnostic neural operators for both  $\mathbf{k}$  space  $(\mathrm{NO_k})$  and image space  $(\mathrm{NO_i})$ , facilitating zero-shot super-resolution in both domains. We present empirical zero-shot super-resolution results in Section 4.4, comparing our NO framework to E2E-VN [40], a CNN-based architecture with a similar design.
+
+# 4. Experiments
+
+We discuss the datasets and experimental setup, followed by comparisons of our and baseline methods with different k undersampling rates and patterns. We conclude the section with zero-shot super-resolution and additional analysis.
+
+# 4.1. Dataset and Setup
+
+Datasets: The fastMRI dataset [44] is a large and open dataset of knee and brain fully-sampled MRIs.
+
+- fastMRI knee: We use the multi-coil knee reconstruction dataset with 34,742 slices for training and 7,135 slices for evaluation. All samples contain data from 15 coils.  
+- fastMRI brain: We use the T2 contrast subset of the multi-coil brain reconstruction dataset with 6,262 training slices and 502 evaluation slices. We filter for samples with data from 16 coils.
+
+Undersampling Patterns and Rates. We use equispaced,
+
+<table><tr><td rowspan="2">Category</td><td rowspan="2">Method</td><td colspan="2">fastMRI knee</td><td colspan="2">fastMRI brain</td></tr><tr><td>PSNR (dB)</td><td>SSIM</td><td>PSNR (dB)</td><td>SSIM</td></tr><tr><td rowspan="2">Learning-free</td><td>Zero-filled</td><td>31.00±3.33</td><td>0.7848±0.0616</td><td>30.86±1.73</td><td>0.8065±0.0376</td></tr><tr><td>ℓ1-Wavelet [27]</td><td>25.67±3.91</td><td>0.5667±0.2001</td><td>28.68±1.31</td><td>0.6783±0.0684</td></tr><tr><td rowspan="3">Diffusion</td><td>CSGM [17]</td><td>26.52±3.21</td><td>0.6789±0.1220</td><td>-</td><td>-</td></tr><tr><td>ScoreMRI [5]</td><td>25.72±1.80</td><td>0.5789±0.0910</td><td>-</td><td>-</td></tr><tr><td>PnP-DM [43]</td><td>26.52±3.14</td><td>0.6383±0.1320</td><td>-</td><td>-</td></tr><tr><td rowspan="3">End-to-end</td><td>U-Net [37]</td><td>37.07±2.47</td><td>0.8803±0.0504</td><td>37.27±1.76</td><td>0.9211±0.0272</td></tr><tr><td>E2E-VN [40]</td><td>38.33±3.06</td><td>0.9048±0.0732</td><td>38.06±2.70</td><td>0.9620±0.0107</td></tr><tr><td>Ours</td><td>39.14±2.93</td><td>0.9219±0.0724</td><td>38.82±2.77</td><td>0.9621±0.0086</td></tr></table>
+
+Table 1. MRI reconstruction performance on  $4 \times$  equispaced undersampling. NO outperforms existing methods (classical, diffusion, and end-to-end). NO also shows consistent performance across k space undersampling patterns (Section 4.3). Zero-filled refers to reconstructing the image from zero-filled k space.
+
+random, magic, Gaussian, radial, and Poisson undersampling patterns [5, 44] and  $2\mathrm{x}$ ,  $4\times$ ,  $6\times$ , and  $8\times$  undersampling rates (visualizations are in Fig. 8 in the Supplementary). Higher rates result in sparser  $\mathbf{k}$  space samples and shorter imaging time at the cost of a more ill-posed/harder inversion process. Section B in the Supplementary provides additional undersampling details along with mask visualizations.
+
+Neural Operator Model. NO follows Fig. 2. The  $\mathrm{NO_k}$  (k space neural operator) and  $\mathrm{NO_i}$  (image space neural operator) are implemented as UDNOs with 2 input and output channels. This is because complex numbers, commonly used in MRI data, are represented using two channels: one for the real part and one for the imaginary part. We provide UDNO details, DISCO kernel basis configurations and training hyper-parameters in Section B of the Supplementary.
+
+Baseline: Compressed Sensing. We compare with a learning-free compressed sensing method with wavelet  $\ell_1$  regularization for a classical comparison [27].
+
+Baselines: Unrolled Networks. We compare with the E2E-VN (End-to-End VarNet) [40], which shares a similar network structure with our approach, but uses the standard CNNs with resolution-dependent convolutions. Since E2E-VN [40] is only trained on specific resolution, we also consider E2E-VN++, where we train [40] with multiple-patterns that match our NO's training data for fair comparisons. Number of cascades  $t$  is set to 12 following [40].
+
+Baselines: Diffusion. Diffusion models have shown strong performance on inverse problems such as MRI reconstruction. We compare our approach to three prominent diffusion-based methods that leverage these capabilities: Score-based diffusion models for accelerated MRI (ScoreMRI) [5], Compressive Sensing using Generative Models (CSGM) [17], and Plug-and-Play Diffusion Models (PnP-DM) [43]. We replicate the experimental settings described in their respective papers. While they report results on MVUE targets, we evaluate metrics on RSS targets at inference for a fair comparison with our methods.
+
+Hardware and Training. While models can be trained on a single RTX 4090 GPU, we accelerate the training of our
+
+model and baselines with a batch size of 16 across 4 A100 (40G) GPUs. We follow baseline settings for comparison.
+
+Evaluation Protocols. We evaluate image reconstruction performance using normalized mean square error (NMSE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM) which are standard for the fastMRI dataset and MRI [44].
+
+# 4.2. Reconstruction with Different k Space Undersampling Patterns
+
+We train our NO model, E2E-VN and E2E-VN++ on  $4 \times$  equispaced samples for 50 epochs. The performance on the single  $4 \times$  equispace undersampling pattern in Table 1. We further fine-tune NO and E2E-VN++ for an additional 20 epochs on a small dataset (3,474 samples) of equispaced, random, magic, Gaussian, radial, and Poisson samples.
+
+fastMRI Knee. We also provide detailed metric results in Table 2, with a line plot in Fig. 6a, where our NO achieves consistent performance across different patterns. Across all patterns, we achieve an average improvement of  $4.17\mathrm{dB}$  PSNR and  $8.4\%$  SSIM over the E2E-VN. On rectilinear patterns (equispaced, magic, random), our performance remains comparable to  $\mathrm{E2E - VN + + }$  (0.3 dB PSNR gain). Across the irregular patterns (radial, Gaussian, Poisson), we achieve a 0.6 dB PSNR improvement over the improved baseline  $(\mathrm{E2E - VN + + })$
+
+fastMRI Brain. On irregular patterns, we achieve an average improvement of 4.7 dB PSNR and  $10\%$  SSIM over the E2E-VN. On rectilinear patterns (equispaced, magic, random), our performance remains comparable to the E2E-VN. Detailed numbers are reported in Table 7 of Supplementary. Visualization. We observe visual improvements in reconstruction integrity (see Fig. 4). Our model is robust to inference across multiple patterns. We highlight important local regions where our NO is better.
+
+The setting here where multiple patterns are trained together is a common clinical setting where the undersampling patterns are known. We also consider the setting where undersampling patterns are unknown. Zero-shot evaluations of the equispaced-trained  $(4\times)$  model across different patterns show that our NO shows 1.8 dB PSNR gain over E2E-VN.
+
+# 4.3. Reconstruction with Different k Space Undersampling Rates
+
+We train our NO model, E2E-VN and E2E-VN++ on  $4 \times$  equispaced samples for 50 epochs. We further fine-tune NO and E2E-VN++ for an additional 20 epochs on a small dataset (3,474 samples) of  $4 \times$ ,  $6 \times$ ,  $8 \times$ , and  $16 \times$  equispaced samples.
+
+For fastMRI Knee, we report the multi-rate performance in Fig. 6b and Table 6 of the Supplementary. For fastMRI Brain, we report the multi-rate performance in Table 8 of the Supplementary. Our neural operator model consistently out
+
+![](images/98e9d07e8faf4877e2f41511c26d15c331bbe65b9534d219ae2fdae8975dafea.jpg)  
+Figure 4. MRI reconstructions with different undersampling patterns of various methods: NO (ours), E2E-VN++, E2E-VN [40], L1-Wavelet (learning-free compressed sensing) [27], and CSGM (diffusion) [17]. NO reconstructs high-fidelity images across various downsampling patterns. Zoom-in view in the lower right of each image. Row 1:  $4 \times$  Equispaced undersampling. Row 2:  $4 \times$  Gaussian 2d undersampling. Row 3:  $4 \times$  Radial 2d undersampling.
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Pattern</td><td colspan="3">PSNR (dB) ↑</td><td colspan="3">SSIM ↑</td><td colspan="3">NMSE ↓</td></tr><tr><td>NO (ours)</td><td>E2E-VN++</td><td>E2E-VN [40]</td><td>NO (ours)</td><td>E2E-VN++</td><td>E2E-VN [40]</td><td>NO (ours)</td><td>E2E-VN++</td><td>E2E-VN [40]</td></tr><tr><td rowspan="3">In-domain</td><td>Equispaced</td><td>37.40 ± 2.61</td><td>37.50 ± 2.79</td><td>38.35 ± 3.05</td><td>0.899 ± 0.072</td><td>0.900 ± 0.072</td><td>0.905 ± 0.073</td><td>0.007 ± 0.006</td><td>0.007 ± 0.006</td><td>0.006 ± 0.006</td></tr><tr><td>Random</td><td>36.66 ± 2.48</td><td>36.79 ± 2.65</td><td>37.34 ± 2.75</td><td>0.891 ± 0.070</td><td>0.892 ± 0.072</td><td>0.897 ± 0.071</td><td>0.008 ± 0.006</td><td>0.008 ± 0.007</td><td>0.007 ± 0.005</td></tr><tr><td>Magic</td><td>38.46 ± 2.99</td><td>38.34 ± 3.06</td><td>38.94 ± 3.55</td><td>0.914 ± 0.070</td><td>0.914 ± 0.069</td><td>0.917 ± 0.071</td><td>0.006 ± 0.006</td><td>0.007 ± 0.006</td><td>0.006 ± 0.006</td></tr><tr><td rowspan="3">OOD</td><td>Radial</td><td>36.23 ± 2.21</td><td>35.50 ± 2.24</td><td>27.02 ± 3.92</td><td>0.900 ± 0.071</td><td>0.892 ± 0.070</td><td>0.764 ± 0.070</td><td>0.009 ± 0.006</td><td>0.011 ± 0.009</td><td>0.069 ± 0.030</td></tr><tr><td>Poisson</td><td>33.42 ± 2.34</td><td>33.01 ± 2.67</td><td>23.61 ± 3.85</td><td>0.878 ± 0.062</td><td>0.873 ± 0.060</td><td>0.687 ± 0.083</td><td>0.016 ± 0.008</td><td>0.017 ± 0.008</td><td>0.152 ± 0.068</td></tr><tr><td>Gaussian</td><td>31.25 ± 2.70</td><td>30.65 ± 2.55</td><td>23.14 ± 3.95</td><td>0.863 ± 0.058</td><td>0.851 ± 0.059</td><td>0.673 ± 0.088</td><td>0.024 ± 0.005</td><td>0.028 ± 0.007</td><td>0.170 ± 0.073</td></tr></table>
+
+Table 2. MRI reconstruction performance across different undersampling patterns. Across multiple patterns, NO maintains reconstruction performance, while baselines do not perform well on out-of-domain (OOD) undersampling patterns (Poisson, radial, Gaussian). Metrics are calculated for the fastMRI knee dataset with a fixed  $4 \times$  acceleration rate. We observe that the E2E-VN overfits to rectilinear patterns, and drops off heavily when evaluated on the irregular patterns (Poisson, radial, Gaussian).
+
+performs the E2E-VN [40], achieving 3.2 dB higher PSNR and  $5.8\%$  higher SSIM on fastMRI knee and 2.0 dB higher PSNR and  $7.5\%$  higher SSIM on fastMRI brain.
+
+# 4.4. Zero-Shot Super-Resolution
+
+We study  $\mathrm{NO_i}$  and  $\mathrm{NO_k}$  zero-shot super-resolution performance and compare them with E2E-VN [40].
+
+Higher MRI Resolution with  $\mathrm{NO_i}$  super-resolution. We train our NO model and the E2E-VN models on  $320\times 320$  knee samples. We then keep the  $\mathrm{NO_k}$  unchanged and use bilinear interpolation to increase the input to  $\mathrm{NO_i}$  to  $640\times$  640. We directly evaluate models without fine-tuning against fully sampled  $640\times 640$  bilinear interpolated ground truth reconstructions. For [40] relying on CNNs, the absolute kernel size stays the same, and the ratio of kernel size over feature map is halved, while the ratio of NO stays the same. Compared to our NO model, the CNN-based E2E-VN [40] produces reconstructions with noticeable artifacts and higher
+
+PSNR and image reconstruction quality (Fig. 1d and Fig. 5b). Larger MRI FOV with  $\mathrm{NO_k}$  super-resolution. k space super-resolution expands the MRI reconstruction field of view (FOV). To validate model performance, we design a proof-of-concept FOV experiment. Our NO model and the E2E-VN [40] train on  $160\times 160$  downsampled k space brain slice samples, where sparse k space sampling results in a reduced FOV in image space. We then perform zero-shot inference on  $320\times 320$  full-FOV k space data. Although neither model encounters data outside the  $160\times 160$  FOV during training, our NO model reconstructs features in this extended region with significantly fewer artifacts compared to E2E-VN [40] (visualizations in Fig. 5a).
+
+# 4.5. Additional Analysis
+
+Model Inference and Tuning Time. In Table 3, we compare the model development and inference times of our end-to-end neural operator (NO) with diffusion models. We
+
+![](images/25101edd51d9fb68dfa6f411ff3bd607dcc3f746b05a3e90d3175f2d27b83c82.jpg)  
+Figure 5. Zero-shot super-resolution results in both extended FOV  $(\mathrm{NO_k})$  and high-resolution image space  $(\mathrm{NO_i})$ . (a) Zero-shot extended FOV reconstructions: Our NO model shows fewer artifacts and higher PSNR in the reconstructed brain slices compared to the CNN-based E2E-VN [40] on  $4\times$  Gaussian, despite neither model seeing data outside the initial  $160\times 160$  FOV during training. (b) Zero-shot super-resolution reconstructions in image space on  $2\times$  radial: with input resolution increased to  $640\times 640$  through bilinear interpolation, our NO model preserves reconstruction quality, while E2E-VN [40] produces visible artifacts.
+
+![](images/71aa9ec5616091411c146c9cf0c72104a31cc5d81296e6b8db73d4be9855c4ae.jpg)
+
+![](images/2a7e541d6cb095f704283590550c5580ca00bf0900a101a82cd573f0b0f4a467.jpg)  
+Figure 6. Performance across different undersampling patterns and rates of ours and baseline methods: end-to-end [40], diffusion [17] and learning-free [27]. Our NO remains relatively consistent in performance when evaluated at different undersampling patterns and rates. Note that a high undersampling rate makes the task more difficult and thus a worse score is expected.
+
+![](images/fc1d912141782f9bd93595af6b448723774b8130bc00713c1c085f57cff68ee5.jpg)
+
+<table><tr><td>Category</td><td>Method</td><td>Inference Time (s)</td><td>Tuning* Required</td></tr><tr><td>Learning-free</td><td>\( \ell_1 \)-Wavelet [27]</td><td>5.45</td><td>✓</td></tr><tr><td rowspan="3">Diffusion</td><td>CSGM [17]</td><td>93.84</td><td>✓</td></tr><tr><td>PnP-DM [43]</td><td>84.46</td><td>✓</td></tr><tr><td>ScoreMRI [5]</td><td>96.16</td><td>✓</td></tr><tr><td rowspan="2">Variational</td><td>E2E-VN [40]</td><td>0.104</td><td>✗</td></tr><tr><td>NO (ours)</td><td>0.158</td><td>✗</td></tr></table>
+
+Table 3. Inference and tuning time of methods tested on NVIDIA A100. NO is approximately  $600 \times$  faster than diffusion, and  $35 \times$  faster than the classical baseline based on learning-free compressed sensing methods. *Tuning refers to the  $\mathbf{k}$  undersampling pattern-specific hyperparameter tuning during inference/after model training. Both the  $\ell_{1}$ -Wavelet [27] ( $\sim 0.5$  hrs per pattern) and diffusion methods ( $\sim 6$  hrs per pattern) require pattern-specific tuning, while our NO is trained once for all patterns.
+
+observe that diffusion models require pattern-specific hyperparameter tuning and are over 600 times slower in inference. MRI-diffusion models [5, 17, 43] are unconditionally trained and undersampling patterns are not available during training.
+
+Thus, we empirically tune hyperparameters such as learning rate and guidance scale for each downsampling pattern for approximately 6 hours each time. Traditional learning-free methods like  $\ell_1$ -Wavelet [27] still require hyperparameter tuning for specific  $\mathbf{k}$  undersampling patterns during optimization. Consequently, end-to-end methods, e.g. NO, are significantly more efficient.
+
+Performance Under Same Parameter Size. We show our NO outperforms baseline unrolled network E2E-VN [40] on different patterns and rates with a similar architecture and number of parameters in the Supplementary.
+
+# 5. Conclusion
+
+Our unified model for compressed sensing MRI addresses the need to train multiple models for different measurement undersampling patterns and image resolutions, a common clinical issue. By leveraging discretization-agnostic neural operators, the model captures both local and global features, enabling flexible MRI reconstruction. With extensive experiments on fastMRI knee and brain datasets, our model maintains consistent performance across undersampling patterns and outperforms state-of-the-art methods in accuracy and robustness. It also enhances zero-shot super-resolution and extended FOV (field of view). The work has some limitations: 1) We only explore one neural operator design, DISCO, and future work could explore other operator learning architectures for MRI. 2) We only benchmark the image reconstruction performance without diagnostic accuracy, which is of more clinical relevance.
+
+In short, our approach offers a versatile solution for efficient MRI, with significant utility in clinical settings where flexibility and adaptability to varying undersampling patterns and image resolutions are crucial.
+
+# Acknowledgement
+
+This work is supported in part by ONR (MURI grant N000142312654 and N000142012786). J.W. is supported in part by Schmidt Sciences. A.S.J. and A.C. are supported in part by the Undergraduate Research Fellowships (SURF) at Caltech. Z.W. is supported in part by the Amazon AI4Science Fellowship. B.T. is supported in part by the Swartz Foundation Fellowship. M.L.-S. is supported in part by the Mellon Mays Undergraduate Fellowship. A.A. is supported in part by Bren endowed chair and the AI2050 senior fellow program at Schmidt Sciences.
+
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+# A Unified, Resilient, and Explainable Adversarial Patch Detector
+
+Vishesh Kumar, Akshay Agarwal  
+Trustworthy BiometraVision Lab, IISER Bhopal, India
+
+{vishesh22,akagarwal}@iiserb.ac.in
+
+# Abstract
+
+Deep Neural Networks (DNNs), backbone architecture in 'almost' every computer vision task, are vulnerable to adversarial attacks, particularly physical out-of-distribution (OOD) adversarial patches. Existing defense models often struggle with interpreting these attacks in ways that align with human visual perception. Our proposed AdvPatchXAI approach introduces a generalized, robust, and explainable defense algorithm designed to defend DNNs against physical adversarial threats. AdvPatchXAI employs a novel patch decorrelation loss that reduces feature redundancy and enhances the distinctiveness of patch representations, enabling better generalization across unseen adversarial scenarios. It learns prototypical parts self-supervised, enhancing interpretability and correlation with human vision. The model utilizes a sparse linear layer for classification, making the decision process globally interpretable through a set of learned prototypes and locally explainable by pinpointing relevant prototypes within an image. Our comprehensive evaluation shows that AdvPatchXAI closes the "semantic" gap between latent space and pixel space and effectively handles unseen adversarial patches even perturbed with unseen corruptions, thereby significantly advancing DNN robustness in practical settings<sup>1</sup>.
+
+# 1. Introduction
+
+Besides being dominant in computer vision over the years, deep neural networks (DNNs) have been found vulnerable to adversarial attacks [1, 19]. Early adversarial attacks on DNNs tricked models using slight, barely noticeable noise [18, 52]. While the above attack perturbs an entire image, a few attacks often target the main object in the scene naturally [29, 63]. Another type of adversaries are poisoning attacks, misled models during training by introducing incorrect patterns, such as in poison frogs and backdoor attacks [3, 9, 50]. Surprisingly, many of these minute adversarial attacks are ineffective in the physical world due to
+
+several unconstrained environmental factors, including rotation and translation properties of the objects. This led to the development of adversarial patch attacks, where a patterned sub-image is placed over the input image to deceive the model [8]. Due to expectation over transformation (EoT) constraints while learning the patches, they are highly effective in real-world scenarios. They can fool any possible deep networks, including vision transformers [5, 12, 20, 37, 53, 64].
+
+Therefore, securing DNN-based systems against stealthy and practical attacks is vital, especially in safety-critical domains such as autonomous driving, robotics, smart homes/cities, smart industries, video surveillance, and healthcare. Researchers are constantly developing new defenses or protection strategies [17, 21, 24, 39, 57] to tackle the limitations of DNNs against physical adversarial patches, but understanding their decision-making processes and evaluating the resiliency of the defense algorithms is increasingly important [42, 47]. It is to be noted that physical adversarial patches are visible to the human eye; however, they need extra precautions since the added patch can also be a real-world object. Due to this, automated detection of patches is challenging and leads to several false rejections. It is essential to align human and machine vision to address this issue. One way can be to align the functional properties of human and machine vision [16]. This alignment will help improve the machine's ability to detect and respond to out-of-distribution (OOD) adversarial patches more effectively in several generalized settings, such as unseen patches and unseen perturbation, ensuring better security and performance. "To the best of our knowledge, no existing defense provides both high generalizability and explainability against adversarial patch attacks."
+
+For the first time, we propose a generalized, robust, and explainable adversarial patch detector, namely AdvPatchXAI, by adding the model explainability since the beginning of the development of the network in contrast to the traditional methods, which plug in the explainability module once the model is thoroughly trained. Proposed AdvPatchXAI uses a sparse linear layer that connects learned prototypical parts to classes. This setup allows a user to
+
+interpret the model by inspecting the prototypes and their relation to the classes. The weights of the linear layer are restricted to non-negative, ensuring that the presence of a class-relevant prototype increases the evidence for that class. This layer functions like a scoring sheet: the score for a class is the sum of all present prototypes multiplied by their weights. Using this interpretable and predictive linear layer, AdvPatchXAI ensures a direct relation between the prototypes and the classification. This approach improves the interpretability of the model's decisions and enhances its ability to detect and respond to adversarial patches. The significant strength of the proposed approach is that it can be plugged in with any deep learning architecture, including CNN and transformer-based architecture. In brief, the contributions of this research are:
+
+- We have proposed a generalized, robust, and explainable patch detector (AdvPatchXAI), which effectively detects unseen and out-of-distribution adversarial patches.  
+- For the first time in the literature, we have evaluated the robustness of adversarial patch detectors against several common corruptions, ensuring their practicality in the unconstrained physical world.  
+- Extensive experimental comparison with benchmark and state-of-the-art works demonstrate the proposed defense algorithm's effectiveness, generalizability, and explainability.
+
+# 2. Literature review
+
+We identify three main philosophies for developing robust defenses against adversarial patch attacks. These approaches are not mutually exclusive and can be adopted together.
+
+Adversarial Patched Dataset: The first time Brown et al.[8] introduced the concept of adversarial patches to fool the object detectors. Since then, several advancements have been made to create more effective and stealthy adversarial patches, including LaVAN (focusing on weaknesses) [28], Adversarial QR codes (appearing less suspicious) [10, 11], PS-GAN (improved quality) [34], and DiAP (data-independent) [61]. A survey [51] reveals the vulnerability of various state-of-the-art pre-trained models YOLOv4 [7], ViT-B/16, Unet++ [62], YOLOv3 [45], YOLOv2 [46], YOLOv5 against physical adversarial patch attacks. Our first philosophy lies in the lack of a standardized dataset. While several effective adversarial patch generation algorithms exist [10, 32, 38, 61, 64], a lack of standardized datasets hinders the development of robust defense mechanisms. Recent efforts by Pintor et al. [44] and Ojaswee et al.[43] propose benchmark datasets specifically for adversarial patches but missing the effect of natural noises in real-world scenarios [2, 23]. While Kumar & Agarwal [31], for the first time, explored the combined effect of both adversarial patches and natural noises, they have not
+
+proposed any novel defense algorithm. Our Second philosophy lies with the defense algorithm. It is also to be noted that minimal defense works exist that can effectively detect adversarial patch attacks in several generalized settings such as unseen datasets, unseen adversarial patches, and unseen threat model [25, 27, 32, 43] and also lack explainability. To address these gaps, in our research, we have regenerated a large-scale dataset followed by [31] containing 10 different adversarial patches [44] and three natural noises (gaussian noise, shot noise, and impulse noise); the detailed description of the dataset given in subsection 4.1. Our third philosophy concerns the lack of a generalized and explainable patch defense algorithm. Generalized & Explainable Patch Defense: While several studies have been explored for adversarial example detection in image classification task [6, 15, 59, 60], very few studies talk about adversarial patch detection in image classification. Pintor et al. [44] introduced an ImageNet-Patch dataset to benchmark machine learning models' robustness against adversarial patches. This dataset includes patches optimized to generalize across different models, allowing for a more efficient and transferable robustness evaluation. The dataset's utility has been demonstrated by testing its effectiveness against 127 models, showcasing its potential as a standard benchmark for assessing and improving model robustness against adversarial patches. Further, Ojaswee et al. [43], using subsets of ImageNet [13] and COCO [33] datasets, developed benchmark datasets and generalized the effect of these patches by finetuning several state-of-the-art DNNs in different generalized settings such as (i) seen patch settings (same patches during testing and training), (ii) unseen patch setting (different patches during training and testing), (iii) seen patch + unseen dataset, and (iv) unseen patch + unseen dataset. Again, Kumar & Agarwal [31] extended this work by training traditional machine learning algorithm using the features extracted by state-of-the-art DNNs in several conditions such as (i) seen patch, (ii) seen patch + natural noises, (iii) unseen patch, and (iv) unseen patch + noise. These findings indicate that defending against adversarial patches in unseen settings is challenging, as the effectiveness of defenses is closely tied to the attributes of the patches used during training. This means detectors have a lower detection rate for new, unseen patches. It has been noticed that none of these studies propose novel algorithms for patch detection. Moreover, to our knowledge, existing detectors do not provide sufficient explanations for the alerts they raise, leaving the reasoning behind their decisions unclear. This research primarily focused on unseen settings such as (i) unseen patch setting, (ii) unseen patch + noise, (iii) Unseen patch + unseen dataset, and (iv) unseen patch + unseen dataset + noise. Generalized better than existing defenses and provided detailed explanations through the proposed AdvPatchXAI patch detector.
+
+![](images/9b6d36f85b9743b72767020bbec29aaae3ac69a31a7f32306720f1d190696f12.jpg)  
+Figure 1. Overview of our proposed method. OOD Adversarial patches used in this research show on top of the proposed architecture. Patch Decorrelation: A novel patch decorrelation loss  $L_{PD}$  followed the contrastive learning approach applying to the backbone features along with loss  $L_{A}$  applying on softmax output to assign the same prototype of two representations of patches for an image pair. To avoid trivial solutions and encourage the utilization of all prototypes, a tanh-loss  $L_{T}$  is applied during the self-supervised pretraining phase. Connections between learned part-prototypes and classes are established through a sparse linear layer. The standard loss used is negative log-likelihood, denoted  $L_{C}$ . Model outputs remain unnormalized during testing, allowing them to serve as straightforward scoring metrics.
+
+# 3. Proposed Framework for AdvPatchXAI
+
+This section first briefly describes the binary classification problem of adversarial patch detection, followed by a comprehensive discussion of the proposed explainable adversarial patch detector, i.e., AdvPatchXAI.
+
+# 3.1. Problem Statement & Notations
+
+Given a binary classification problem with training dataset  $D_{tr} = \{(x,y), x \in X_{tr}, y \in Y_{tr}\}$  containing known classes  $C_2 = \{c_1 \text{ (real)}, c_2 \text{ (patched)}\}$  and a testing dataset  $D_{te} = \{(x', y'), x' \in X_{te}, y' \in Y_{te}\}$ , where,  $X_{tr}$  and  $X_{te}$  represent the input images of the training and testing dataset, respectively, while  $Y_{tr}$  and  $Y_{te}$  are the corresponding class labels. We aim to learn interpretable prototypes that can be used as input features for the model explainability. The backbone of our model consists of pre-trained CNNs and ViT, which learn an interpretable, 1-dimensional image encoding  $p$  that indicates the presence or absence of prototypical parts in an image. These prototypical parts (prototypes) are then connected to classes through a sparse linear layer.
+
+Our framework introduces two steps to effectively iden-
+
+tify and classify the given input accurately: (i) data augmentation followed by TrivialAugment [40] to perform self-supervised pretraining of learned prototype patches and (ii) training AdvPatchXAI for effective and explainable adversarial patch detection.
+
+# 3.2. Self-Supervised Pretraining of Prototypes
+
+Following prior self-supervised learning methods [26], we generate a positive pair, denoted as  $x'$  and  $x''$  by using TrivialAugment [40]. This recently introduced augmentation strategy is efficient, requiring no hyperparameter tuning, and applies a single augmentation per image. However, in contrast to the standard approach, we applied TrivialAugment twice, as shown in Figure 1. The first augmentation operation applies location-related transformations or focuses on spatial transformations, including shearing, rotation, and translation. This augmented image is then used as input to another TrivialAugment operation, which involves color alterations, including brightness, sharpness, hue, and contrast. In our approach, we followed grayscale conversion in the second augmentation stage. We also experimented with RGB and YCbCr conversion but found a less
+
+effective strategy than grayscale (see subsection 6.1).
+
+# 3.2.1. Patch Decoration Loss
+
+To reduce redundancy and enhance the distinctiveness of the patch features, we propose a patch decorrelation loss applied to patch features. We adopt a contrastive learning strategy similar to [56] to achieve alignment and uniformity of patch representations. However, unlike their image-level approach, we focus on patch-level alignment. The Patch Decorrelation Module plays a critical role in our method by promoting the similarity between features extracted from different views of the same patch while simultaneously reducing redundancy across different feature dimensions. An input image is first forwarded through the backbone DNN  $f$ . The resulting output  $z = f(x; w_f)$  consists of  $D$  two-dimensional  $(H \times W)$  feature maps, where  $w_f$  denotes the trainable parameters of  $f$ . So, the feature maps for two different views of  $x$ ,  $z' = f(x'; w_f)$  and  $z'' = f(x''; w_f)$ . Since the feature dimensions of each are initially  $(B, D, H, W)$ , where  $B$  denotes the batch size and  $D$  represents the total number of feature maps of dimension  $H \times W$ . We have converted it to  $(B \times H \times W, D)$ , representing each image patch's features within the batch. Compute the cross-correlation matrix using the converted feature maps for both  $x'$  and  $x''$  as  $[D_{cr}] = [d_{ij}]_{(D,D)} = \frac{1}{B \times H \times W} [x']_{(B \times H \times W,D)}^{T} [x'']_{(B \times H \times W,D)}$ , where  $i, j = 1, 2, \dots, D$ . Consider an identity matrix  $I = [I_{ij}]_{(D,D)}$ . Finally, the patch decorrelation loss (i.e.,  $L_{PD}$ ) is defined as:
+
+$$
+L _ {P D} = \frac {\sum_ {i = 1} ^ {D} \left(d _ {i i} - I _ {i i}\right) ^ {2} + \lambda \times \left(\left(d _ {i j} - I _ {i j}\right) _ {i \neq j}\right) ^ {2}}{D} \tag {1}
+$$
+
+Here,  $\lambda = 5 \cdot e^{-3}$  is a trade-off parameter of the loss function. Our objective revolves around diagonalizing this cross-correlation matrix, which mitigates the risk of trivial solutions, such as activating the same prototype for all patches and promoting utilizing the entire prototype space.
+
+# 3.2.2. Alignment Loss on Softmax Outputs
+
+After obtaining raw features aligned using the Patch Decorrelation Module, we further refine the alignment using a softmax-based alignment loss. This alignment loss, denoted as  $L_{A}$  [41], ensures that the softmax-normalized feature vectors of corresponding patches from two views are closely aligned. Apply a softmax over  $D$  such that  $\sum_{i=1}^{D} s_{h,w,i} = 1$ , ensuring that a patch located at  $h, w \in H \times W$  corresponds to prototype  $i$ . Ideally,  $s_{h,w,:}$  is a one-hot encoded vector indicating a perfect assignment to one prototype. To measure the similarity between corresponding patches from two augmented views, we calculate the dot product between their latent representations,  $(s_{h,w,:}^{\prime})$  and  $s_{h,w,:}^{\prime \prime}$ :
+
+$$
+L _ {A} = - \frac {1}{H W} \sum_ {(h, w) \in H \times W} \log \left(s _ {h, w,:} ^ {\prime} \cdot s _ {h, w,:} ^ {\prime \prime}\right) \tag {2}
+$$
+
+Our goal is to identify the presence or absence of prototypical parts within an image, for which a max-pooling operation across each feature map dimension (denoted as  $s::,d$ ) is applied. This results in a presence score tensor,  $p \in [0,1]^D$ , where each element  $p_d$  represents the strength of the  $d$ th prototype's presence in the image. We introduce the tanh-based regularization loss  $L_{T}$  [41] to prevent trivial solutions. The tanh-loss encourages the presence of each prototype at least once in a mini-batch. The tanh loss is defined as:
+
+$$
+L _ {T} (p) = - \frac {1}{D} \sum_ {i = 1} ^ {D} \log (\tanh  (\sum_ {b = 1} ^ {B} p _ {b}) + \epsilon), \tag {3}
+$$
+
+Where tanh and log are element-wise operations,  $B$  is the number of samples in a mini-batch,  $D$  is the number of prototypes, and  $\epsilon$  is a small number for numerical stability. This loss ensures that each prototype is utilized across the mini-batch, preventing any prototype from dominating and promoting a balanced representation across all prototypes.
+
+The combination of patch decorrelation loss, alignment loss, and tanh loss enables our model to achieve robust patch-level alignment and uniformity. The patch decorrelation loss  $(L_{PD})$  enhances the raw feature alignment, the alignment loss  $(L_A)$  ensures the softmax-normalized features are closely matched, and the tanh loss  $(L_T)$  prevents trivial solutions by promoting diversity in prototype assignments. The final objective of our pre-training phase of AdvPatchXAI is:  $\lambda_{PD}L_{PD} + \lambda_{A}L_{A} + \lambda_{T}L_{T}$ .
+
+# 3.3. Training AdvPatchXAI
+
+After the pretraining phase of the prototypes, the patch presence score tensor,  $p$ , is fed into a linear classification layer having non-negative weights,  $w_{c} \in \mathbb{R}^{(D \times C)} \geq 0$ . It looks for only positive or true class evidence in an input image to which the image belongs. These weights connect prototypes to classes  $(C)$  to ensure that the presence of prototypical parts contributes positively to the evidence for their associated class, enhancing interpretability. The bias term adjusts the classifier's decision threshold and is independent of the prototype's contributions. This separation ensures that the model remains interpretable, as each prototype's influence on the decision is straightforward and non-contradictory. The output score for each class is calculated by summing the element-wise product of the presence scores and the corresponding class weights from the linear layer. We incorporate a classification loss term,  $L_{C}$ , to optimize model performance. This loss is calculated as the standard negative log-likelihood between the predicted
+
+class probabilities  $\hat{y}$  and the one-hot encoded ground truth label  $y$ . While  $L_{C}$  primarily influences the weights of the linear layer, it also fine-tunes the prototypes to discriminate features relevant to the classification task better. The overall objective for the second training phase of AdvPatchXAI is:  $\lambda_{PD}L_{PD} + \lambda_{A}L_{A} + \lambda_{T}L_{T} + \lambda_{C}L_{C}$ .
+
+# 4. Experimental Result and Analysis
+
+The dataset used in this research for evaluating the proposed and existing defenses is discussed in subsection 4.1, followed by the implementation details of the proposed algorithm. Finally, the analysis of the proposed defense algorithm is discussed in detail.
+
+# 4.1. Out-of-Distribution Adversarial Patch Dataset
+
+Following the procedure outlined by Kumar & Agarwal [31], this paper introduces two datasets focusing on physical adversarial patches and natural noises using ImageNet and COCO datasets. Images are attacked with 10 different styles of physical adversarial patches and three types of natural noises (Gaussian, Shot, and Impulse noise). For COCO, we randomly selected 2,000 clean images from the validation set. These images served as a clean subset. Another set of 2,000 images from the COCO validation set is selected, and 10 different adversarial patches are applied to each, generating 20,000 adversarial patch images. Clean and patched images are divided into training and testing sets in a 3:2 ratio. For example, 2,000 clean images of COCO and 2,000 images with a single patch are divided into 800 test images and 1200 train images. Also, applying three natural noises resulted in 2,400 noisy test images for both clean and patched sets. Since our primary focus is generalizability, we use ImageNet to test our model when trained on the COCO patched train subset. For ImageNet, we randomly selected 800 clean images from the validation set. Another set of 800 images is selected, and 10 different adversarial patches are applied to each image, generating 8000 adversarial patch images. Again, three different types of natural noise applied to each patch give 24,00 noisy images for clean and patched test sets. In total, the dataset includes 2,800 clean images, 28,000 adversarial patch images (20,000 from COCO and 8,000 from ImageNet), 4,800 noisy test images, and 48,000 images with both adversarial patches and natural noise.
+
+# 4.2. Implementation Details
+
+We utilize three convolutional backbones in AdvPatchXAI: ResNet50 [22] (R), ConvNext-tiny [36] (C), and MobileNetV2 [48] (M). A transformer-based network ViT [14] is also incorporated. The pre-trained models are used, but the strides of the last layers are modified from 2 to 1 to increase the width  $(W)$  and height  $(H)$  of the output feature maps (from  $7 \times 7$  to  $28 \times 28$  for ResNet and MobileNet,
+
+$26 \times 26$  for ConvNext, and reshaped the output feature into  $14 \times 14$  for ViT). This adjustment results in a finer-grained patch grid  $z$ , improving patch similarity optimization. The backbone  $f$  is fine-tuned using Adam with a learning rate 0.0005 and a cosine annealing schedule. The linear layer is trained with a learning rate of 0.05. Loss weights are set to  $\lambda_{C} = \lambda_{T} = 2$  and  $\lambda_{PD} = \lambda_{A} = 5$ . Prototypes are trained for 10 epochs, followed by training AdvPatchXAI for an additional 60 epochs. Images are resized to  $224 \times 224$  and augmented with TrivialAugment [40] in two stages: the first stage is related to location, and the second stage is related to color transformation (grayscale). Experiments are performed with seed value one to ensure reproducibility.
+
+# 4.3. Result Analysis
+
+We evaluated our proposed method, AdvPatchXAI, employed various backbone DNNs using a comprehensive set of experiments. We have trained our network on the COCO training subset for all 10 patches separately and evaluated them under several generalized zero-shot settings. These settings included scenarios with seen datasets with unseen patches and unseen datasets with unseen patches, both with and without natural noises. The effectiveness of AdvPatchXAI can be assessed through two key metrics: the robustness of the chosen backbone DNN and the success of the training patch in identifying unseen patches. The presence of natural noises during the testing ensures the detector's robustness in a black-box setting. To further strengthen the effectiveness of the proposed defense algorithm, we have evaluated its resiliency against adaptive attacks (discussed in supplementary).
+
+# 4.3.1. Zero-Shot OOD Patch Detection in the Absence of Noise
+
+Table 1 and Table 2 present the average classification accuracy (Mean) of our proposed AdvPatchXAI on COCO and ImageNet datasets, respectively, combined with OOD unseen patches in silent (without noise) settings. Based on our findings, it can be seen that AdvPatchXAI-R (i.e., with backbone R) achieved the highest mean accuracy of  $99.33\%$  and  $98.94\%$ , particularly with Patch-9 on COCO and ImageNet datasets, respectively, demonstrating exceptional performance and robustness. AdvPatchXAI-ViT (i.e., with backbone ViT) also performed well, especially for Patch-4 on COCO  $96.42\%$  and Patch-0 on ImageNet  $94.31\%$ , showing reliability with moderate standard deviation (SD) (see supplementary for more detail) but found more vulnerable on Patch-5. However, AdvPatchXAI-M (MobileNet) and AdvPatchXAI-C (ConvNeXt) exhibited higher variability, with significant accuracy drops for specific patches such as Patch-3. Figure 2 gives a clearer picture of the robustness of AdvPatchXAI-R and AdvPatchXAI-ViT on both the COCO and ImageNet datasets. For example, the performance of AdvPatchXAI-R on COCO and ImageNet,  $91.8\%$
+
+Table 1. Adversarial patch detection accuracy of the proposed AdvPatchXAI with different backbone on COCO (seen dataset) subset in several generalized settings such as Silent (unseen patch without any noise), Noisy (unseen patch+noise). The results are reported as mean. Patch-{0-9}\{1} indicates models are trained on Patch-1 and tested on all other patches except Patch-1. R, M, ViT, and C represent ResNet50, MobileNet, Vision Transformer, and ConvNeXt backbones, respectively. The best mean values are highlighted.  
+
+<table><tr><td>Method</td><td>Test</td><td>Patch-{0-9}\{0}</td><td>Patch-{0-9}\{1}</td><td>Patch-{0-9}\{2}</td><td>Patch-{0-9}\{3}</td><td>Patch-{0-9}\{4}</td><td>Patch-{0-9}\{5}</td><td>Patch-{0-9}\{6}</td><td>Patch-{0-9}\{7}</td><td>Patch-{0-9}\{8}</td><td>Patch-{0-9}\{9}</td></tr><tr><td rowspan="2">AdvPatchXAI-R</td><td>Silent</td><td>94.09</td><td>95.57</td><td>84.69</td><td>83.30</td><td>97.99</td><td>79.65</td><td>85.53</td><td>96.91</td><td>94.76</td><td>99.33</td></tr><tr><td>Noisy</td><td>85.81</td><td>88.86</td><td>78.07</td><td>72.01</td><td>96.09</td><td>69.05</td><td>78.95</td><td>94.59</td><td>83.79</td><td>95.92</td></tr><tr><td rowspan="2">AdvPatchXAI-M</td><td>Silent</td><td>94.17</td><td>97.45</td><td>77.33</td><td>64.88</td><td>98.47</td><td>72.28</td><td>93.90</td><td>98.61</td><td>92.90</td><td>97.24</td></tr><tr><td>Noisy</td><td>76.09</td><td>70.82</td><td>56.91</td><td>50.24</td><td>89.63</td><td>51.34</td><td>70.45</td><td>83.65</td><td>71.13</td><td>68.11</td></tr><tr><td rowspan="2">AdvPatchXAI-ViT</td><td>Silent</td><td>95.84</td><td>95.19</td><td>93.20</td><td>81.74</td><td>96.42</td><td>67.27</td><td>89.08</td><td>93.90</td><td>89.00</td><td>94.32</td></tr><tr><td>Noisy</td><td>92.31</td><td>92.61</td><td>88.91</td><td>78.91</td><td>92.95</td><td>64.57</td><td>85.74</td><td>90.52</td><td>80.56</td><td>93.13</td></tr><tr><td rowspan="2">AdvPatchXAI-C</td><td>Silent</td><td>90.08</td><td>85.29</td><td>86.08</td><td>75.94</td><td>96.27</td><td>82.42</td><td>78.63</td><td>87.18</td><td>92.73</td><td>95.28</td></tr><tr><td>Noisy</td><td>79.16</td><td>79.88</td><td>74.62</td><td>64.69</td><td>86.57</td><td>59.72</td><td>72.74</td><td>71.09</td><td>76.41</td><td>84.06</td></tr></table>
+
+and  $90.91\%$ , respectively, is at least ranges in  $[2.2 - 4.81]\%$  better than AdvPatchXAI with other backbones on both datasets. It can be seen that there is only a minor difference  $[0.66 - 1.95]\%$  in performance when both the dataset and patches are unknown to the model.
+
+# 4.3.2. Zero-Shot OOD Patch Detection Under Noise Perturbation
+
+To demonstrate the robustness capabilities of our defense, we evaluated its resiliency when trained on clean images (without exposure to any noise during training) using COCO subsets. This approach is crucial because natural noises are inherent in the environment [4], and training on every possible type of noise is impractical. Therefore, detectors must be resilient enough to handle unseen natural noises.
+
+The resiliency results of AdvPatchXAI in unseen patch detection in noisy (applying all three noise Gaussian, Shot, and Impulse with severity  $= 2$  separately on test subset and taking the average patch wise) settings on COCO and ImageNet datasets are shown in Figure 2. While a drop in detection performance is expected, AdvPatchXAI-ViT exhibits only a marginal reduction. For example, its performance drops from  $89.6\%$  to  $86.02\%$  on COCO and from  $87.65\%$  to  $83.97\%$  on ImageNet. In contrast, other backbones suffer more significant decreases in accuracy. For instance, the performance of AdvPatchXAI-R drops by  $7.49\%$  on COCO and  $7.73\%$  on ImageNet, whereas the accuracy of AdvPatchXAI-M drops by  $19.88\%$  on COCO and  $19.47\%$  on ImageNet. Tables 1 and 2 provide detailed average 10-fold cross-validation performance in the unseen patch noise evaluation setting for COCO and ImageNet datasets, respectively. Notably, in zero-shot evaluations where images (clean and patched) are perturbed, AdvPatchXAI with ViT outperforms other backbones by a significant margin. Patch-4 proves to be more effective in 9 out of 16 evaluations, exhibiting higher mean accuracy. The performance difference between AdvPatch-ViT with the best-performing patch, and Patch-4 is less than  $1.4\%$  on both datasets and in each setting (silent, noisy), yet Patch-4 shows very low SD (see supplementary for more detail), indicating its higher effectiveness. Extensive experimental evaluation reveals that AdvPatchXAI-ViT with Patch-4 generalizes well in unseen patch settings and maintains high resiliency when images
+
+![](images/31943b222b2e80e932f30eb179d708ca89e5be58e983a359f22bb6ea15b3fb6c.jpg)  
+Figure 2. Comparison with SOTA in terms of average adversarial patch detection accuracy for unseen patch and unseen patch + unseen noise detection on both COCO and ImageNet datasets.
+
+are perturbed by noise. Therefore, in real-world applications, we recommend using AdvPatchXAI-ViT with Patch-4 to defend against adversarial patches.
+
+# 5. Explainability and Comparison with SOTA
+
+To demonstrate the effectiveness of the proposed patch detector, we have performed an extensive comparison with state-of-the-art algorithms, which we are going to discuss first. Then, we will discuss the explainability of the proposed approach.
+
+# 5.1. Comparison with SOTA and Baseline
+
+To demonstrate the effectiveness of our proposed model, we compared it with recent state-of-the-art (SOTA) methods: Ojaswee et al. [43] and Kumar & Agarwal [31]. For a fair comparison, we followed the same experimental protocol we used to evaluate the proposed algorithm. As shown in Figure 2, our method outperforms all SOTA methods except in the case of AdvPatchXAI-M on both the COCO and ImageNet datasets when clean and patched images are perturbed with noise. Specifically, our AdvPatchXAI-R exceeds the performance of [43] by  $13.35\%$  and [31] by  $11.9\%$  on the COCO dataset. On the ImageNet dataset, AdvPatchXAI-R outperforms [43] by  $12.03\%$  and [31] by  $14.01\%$  when clean and unseen patched images are used for evaluation. Moreover, when clean and patched images are perturbed with noise, AdvPatchXAI-ViT surpasses [43] by  $13.64\%$  and [31] by  $12.22\%$  on the COCO dataset. On the ImageNet dataset, AdvPatchXAI-ViT outperforms [43] by  $13.33\%$  and [31] by  $12.67\%$ . Our proposed model, based on prototypical parts, is unique in its approach to OOD adver
+
+Table 2. Adversarial patch detection accuracy of the proposed AdvPatchXAI with different backbone on ImageNet (unseen dataset) subset in several generalized settings such as Silent (unseen patch without any noise), Noisy (unseen patch+noise). The results are reported as mean. Patch-{0-9} \{3} indicates models are trained on Patch-3 and tested on all other patches except Patch-3. R, M, ViT, and C represent ResNet50, MobileNet, Vision Transformer, and ConvNeXt backbones, respectively. The best mean values are highlighted.  
+
+<table><tr><td>Method</td><td>Test</td><td>Patch-{0-}\{0}</td><td>Patch-{0-}\{1}</td><td>Patch-{0-}\{2}</td><td>Patch-{0-}\{3}</td><td>Patch-{0-}\{4}</td><td>Patch-{0-}\{5}</td><td>Patch-{0-}\{6}</td><td>Patch-{0-}\{7}</td><td>Patch-{0-}\{8}</td><td>Patch-{0-}\{9}</td></tr><tr><td rowspan="2">AdvPatchXAI-R</td><td>Silent</td><td>92.96</td><td>95.12</td><td>84.49</td><td>82.56</td><td>98.17</td><td>79.40</td><td>84.68</td><td>97.18</td><td>95.60</td><td>98.94</td></tr><tr><td>Noisy</td><td>84.39</td><td>88.51</td><td>77.81</td><td>71.42</td><td>95.67</td><td>68.34</td><td>78.30</td><td>93.72</td><td>78.88</td><td>94.76</td></tr><tr><td rowspan="2">AdvPatchXAI-M</td><td>Silent</td><td>93.68</td><td>95.28</td><td>77.30</td><td>64.13</td><td>95.71</td><td>70.72</td><td>91.93</td><td>96.81</td><td>91.90</td><td>95.92</td></tr><tr><td>Noisy</td><td>75.10</td><td>70.16</td><td>56.88</td><td>50.22</td><td>87.74</td><td>51.15</td><td>69.73</td><td>82.00</td><td>68.55</td><td>67.21</td></tr><tr><td rowspan="2">AdvPatchXAI-ViT</td><td>Silent</td><td>94.31</td><td>92.67</td><td>92.03</td><td>81.37</td><td>92.94</td><td>66.41</td><td>86.71</td><td>91.77</td><td>87.37</td><td>90.94</td></tr><tr><td>Noisy</td><td>90.18</td><td>90.14</td><td>87.98</td><td>78.88</td><td>89.19</td><td>64.22</td><td>83.73</td><td>87.19</td><td>78.16</td><td>90.04</td></tr><tr><td rowspan="2">AdvPatchXAI-C</td><td>Silent</td><td>89.41</td><td>85.26</td><td>85.12</td><td>75.91</td><td>96.23</td><td>81.21</td><td>77.30</td><td>86.67</td><td>91.53</td><td>94.65</td></tr><tr><td>Noisy</td><td>78.14</td><td>79.55</td><td>73.29</td><td>63.66</td><td>85.20</td><td>59.13</td><td>71.54</td><td>70.16</td><td>75.21</td><td>83.41</td></tr></table>
+
+![](images/570ab6025cc4a362d9a1d19834b7d40e6b32f91ac3f74eca5ee847cfa0dc827e.jpg)  
+Figure 3. t-SNE visualization of feature space of proposed AdvPatchXAI with backbone ViT and ConvNeXt trained on the most effective patch, Patch-4, and tested on other patches, Patch-0, Patch-1, and Patch-2. Red and blue represent patched and real class, respectively.
+
+sarial patch detection. To demonstrate that, we compared it to PIP-Net [41], a recent prototypical-parts-based model that is the closest architecture to our proposed method. Since PIP-Net is designed for a convolutional backbone, we integrated ViT into PIP-Net and evaluated results on the COCO subset in an unseen patch (silent) setting. The average patch detection accuracy of PIP-Net with backbone R, M, and ViT are  $87.07\%$ ,  $81.82\%$ , and  $89.32\%$ , respectively. In comparison, our AdvPatchXAI method achieved accuracies of  $91.8\%$ ,  $88.72\%$ , and  $89.6\%$  with the identical respective backbones. In other words, proposed AdvPatchXAI shows  $4.73\%$ ,  $6.9\%$ , and  $0.28\%$  improvement over the baseline PIP-Net with backbone R, M, and ViT, respectively.
+
+Apart from the above datasets, we have benchmarked our proposed model on other standard datasets CUB-200-2011 [55] and Stanford Cars [30] (results discussed in supplementary). These results highlight the robustness and effectiveness of our proposed model in detecting adversarial patches, particularly when utilizing ViT and ResNet backbones.
+
+# 5.2. Explainable AdvPatchXAI
+
+While quantitative analysis is essential for assessing a model's effectiveness and robustness, explainability is
+
+![](images/cf1dc040629defedebb46d9d81b4b42a389e5a3c4e98a56e4df1bacdfd645dab.jpg)  
+Figure 4. Heat map visualization of the proposed AdvPatchXAI trained on Patch-4 and visualized on Patch-0 on the COCO dataset under both silent (without noise) and noisy (gaussian noise with severity=2) settings. The same image has been taken under both silent and noisy settings for a fair comparison.
+
+equally vital in supporting these findings. To address this, we performed several explainability analyses on our proposed AdvPatchXAI, including feature visualization, prototype prediction, prototype visualization, and Grad-CAM [49] visualization. Feature visualization helps in understanding the clusters formed by network features, leading to image classification or misclassification. Figure 3 shows the t-SNE [54] plot of the proposed model using an attention backbone (ViT) and a convolutional backbone (ConvNeXt) when trained on Patch-4 and tested on Patches 0 to 2. The clear separation of feature clusters supports the quantitative effectiveness of our proposed model. We also performed prototype prediction and visualization, along with Grad-CAM visualizations, as shown in Figure 4. The first column shows the unseen patch images and unseen patched+noisy image, and the second column highlights relevant prototypes within the image using yellow boxes, providing a
+
+Table 3. Mean adversarial patch detection accuracy of the proposed AdvPatchXAI across different color channels RGB, YCbCr, and grayscale with backbone ViT and ResNet50 on COCO under silent (unseen patch without any noise) setting.  
+
+<table><tr><td>Model (Channel)</td><td>Patch- \{0-9\} \\{0}</td><td>Patch- \{0-9\} \\{1}</td><td>Patch- \{0-9\} \\{2}</td><td>Patch- \{0-9\} \\{3}</td><td>Patch- \{0-9\} \\{4}</td></tr><tr><td>ViT (RGB)</td><td>64.78</td><td>80.90</td><td>73.13</td><td>62.22</td><td>74.11</td></tr><tr><td>ViT (YCbCr)</td><td>65.57</td><td>90.50</td><td>84.36</td><td>66.72</td><td>78.44</td></tr><tr><td>ViT (Grayscale)</td><td>92.31</td><td>92.61</td><td>88.91</td><td>78.91</td><td>92.95</td></tr><tr><td>R (RGB)</td><td>63.71</td><td>77.49</td><td>61.44</td><td>62.95</td><td>81.26</td></tr><tr><td>R (YCbCr)</td><td>70.56</td><td>87.01</td><td>79.64</td><td>70.33</td><td>78.27</td></tr><tr><td>R (Grayscale)</td><td>85.81</td><td>88.86</td><td>78.07</td><td>72.01</td><td>96.09</td></tr></table>
+
+local explanation. The third column collects these relevant prototypes, further connecting them to the classes with sparse linear layers to predict the correct image class. It can be seen that when noise comes with a patched image, the prototype prediction is compromised, and the extracted prototypes are not as fine-grained and less relevant as clean images. The fourth column presents the Grad-CAM for each prototypical part, illustrating the image regions that the network focuses on while determining its class. This detailed insight into the model's decision-making process supports the robustness and effectiveness of AdvPatchXAI in detecting adversarial patches (further discussed in supplementary).
+
+# 6. Ablation Studies
+
+In this section, we have presented ablation studies highlighting the impact of color channels, real-world robustness, and different terms used in the loss function. To ensure fairness, experiments are conducted using the pre-defined experimental protocol.
+
+# 6.1. Effects of Different Color Augmentations
+
+Table 3 showcases the advantage of converting images into grayscale. Since unseen patches and datasets can have diverse color distributions, having color information makes the system biased toward learning color information rather than focusing on patch information. We assert that suppressing non-useful information can make the system generalized, which can also be visible from the results. For example, when AdvPatchXAI-ViT is trained on Patch 4 using grayscale images, it yields at least  $14.51\%$  higher accuracy than RGB & YCbCr color channel-trained models.
+
+# 6.2. Physical-World Effectiveness and Robustness Against Adversarial Patch Attack
+
+We have further evaluated the robustness of AdvPatchXAI for real-world adaptation by printing and applying the adversarial patches in the real world, as demonstrated by Pinton et al. [44]. Our proposed defense algorithm is robust in handling the attacks in the physical world and yields an average accuracy of  $91.11\%$  and  $88.88\%$  when the proposed ViT defense is trained with Patch 0 and Patch 4, respectively.
+
+Table 4. Ablation study concerning loss terms reflecting the total number of relevant prototypes with at least one non-zero weight present in our proposed AdvPatchXAI algorithm. The mean accuracy demonstrates the advantage of a combined loss function.  
+
+<table><tr><td colspan="2"></td><td colspan="5">Number of Prototypes</td><td></td></tr><tr><td>Method</td><td>Backbone</td><td>Patch-0</td><td>Patch-1</td><td>Patch-2</td><td>Patch-3</td><td>Patch-4</td><td>Mean Acc. ↑</td></tr><tr><td rowspan="2">AdvPatchXAI</td><td>R</td><td>263</td><td>220</td><td>182</td><td>209</td><td>232</td><td>91.13</td></tr><tr><td>ViT</td><td>187</td><td>168</td><td>203</td><td>166</td><td>203</td><td>92.48</td></tr><tr><td>{LPD, LA, LT, LC}</td><td>M</td><td>82</td><td>55</td><td>79</td><td>87</td><td>65</td><td>88.46</td></tr><tr><td rowspan="2">AdvPatchXAI</td><td>R</td><td>370</td><td>284</td><td>251</td><td>252</td><td>347</td><td>86.44</td></tr><tr><td>ViT</td><td>195</td><td>151</td><td>161</td><td>152</td><td>161</td><td>92.66</td></tr><tr><td>{LA, LT, LC}</td><td>M</td><td>67</td><td>61</td><td>100</td><td>91</td><td>101</td><td>77.71</td></tr></table>
+
+We generated perturbation-based PGD patch attacks followed by [35, 58] for a fair comparison to evaluate the adversarial robustness of our proposed defense model. We wanted to highlight that the performance of our proposed AdvPatchXAI-R and AdvPatchXAI-ViT yields  $98.04\%$  and  $95.67\%$  average detection accuracy, respectively, in a silent setting when a patch attack based on PGD perturbation is available for evaluation. Even in noisy settings, our proposed network is better robust against PGD perturbation-based patch attacks with average detection performances of  $95.55\%$  and  $93.05\%$  with the same respective backbone.
+
+# 6.3. Effect of Proposed Loss Function
+
+We further examine the impact of the proposed algorithm's different linear combinations of loss functions. The corresponding outcomes presented in Table 4 reveal that the linear combination of our proposed patch decorrelation loss  $L_{PD}$  with other loss functions improves interpretability. In other words, it reduces the number of relevant prototypes for a class and enhances the detection accuracy.
+
+# 7. Conclusion
+
+In this research, we present AdvPatchXAI to significantly advance the development of a generalized, robust, and explainable adversarial patch detector. By incorporating prototypical parts and a novel patch decorrelation module, our model achieves unprecedented accuracy in physical adversarial patch detection, particularly in zero-shot settings with and without unseen noise perturbations. Furthermore, the explainable nature of AdvPatchXAI provides deep insights into the decision-making process, enabling better understanding and trust in AI systems. Future work will focus on enhancing the scalability of this approach and exploring its applicability to other forms of adversarial attacks, aiming to fortify AI defenses in increasingly complex applications.
+
+# Acknowledgement
+
+V. Kumar is partially supported through the Visvesvaraya PhD Fellowship. A. Agarwal is partially funded through the ANRF PMECRG grant of Govt. of India.
+
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+# A Universal Scale-Adaptive Deformable Transformer for Image Restoration across Diverse Artifacts
+
+Xuyi He
+
+Yuhui Quan
+
+$\mathbf{R}$  uotao  $\mathrm{Xu^{2,3}*}$
+
+Hui Ji
+
+$^{1}$ School of Computer Science and Engineering, South China University of Technology  $^{2}$ Institute for Super Robotics, South China University of Technology  $^{3}$ Key Laboratory of Large-Model Embodied-Intelligent Humanoid Robot  $^{4}$ Department of Mathematics, National University of Singapore
+
+csxuyihe@mail.scut.edu.cn, csyhquan@scut.edu.cn, rtxu@superobots.com, matjh@nus.edu.sg
+
+# Abstract
+
+Structured artifacts are semi-regular, repetitive patterns that closely intertwine with genuine image content, making their removal highly challenging. In this paper, we introduce the Scale-Adaptive Deformable Transformer, an network architecture specifically designed to eliminate such artifacts from images. The proposed network features two key components: a scale-enhanced deformable convolution module for modeling scale-varying patterns with abundant orientations and potential distortions, and a scale-adaptive deformable attention mechanism for capturing long-range relationships among repetitive patterns with different sizes and non-uniform spatial distributions. Extensive experiments show that our network consistently outperforms state-of-the-art methods in diverse artifact removal tasks, including image deraining, image demoiring, and image debanding.
+
+# 1. Introduction
+
+Structured artifacts, in contrast to random noise, are repetitive patterns with similar appearance. Examples include Moiré patterns caused by overlapping pattern interference, rain streaks formed during image acquisition in rainy conditions, and banding effects resulting from color quantization. See Figure 1 for an illustration. These artifacts often display similar appearance that repeat in a quasi-periodic way over large image regions. However, they can differ in size, intensity, orientation, and may exhibit certain shape distortions. Additionally, their characteristics can vary across different images due to changes in image content and capturing.
+
+Removing structured artifacts has many applications. Eliminating moiré patterns improves digital photography
+
+![](images/18490bcb9b39b556463d9de3aff215467f7618319c3a6d439533beca73b749a7.jpg)
+
+![](images/27b3199a8bdeb0a4f74b5c24a3e6ece83d9938a1ad95db60b6d44d2f2727a682.jpg)
+
+![](images/f3f4ae252b636f6bb503193f28a39639a40ecec227b60b97af9aff387b63015b.jpg)
+
+![](images/13e0c606cbef0d8ed6d80aa3e24beb844257e273d8c4ad2523b9759e31aa7fab.jpg)  
+Figure 1. Structured artifacts with varying orientations, scales, and may involve warping effect. From left to right: moiré, rain, band.
+
+![](images/d7781c63dc311025e1dd22964f9b94bee05f07bb835318904eb3cf5687c57596.jpg)
+
+![](images/cbf684e07eb7050c2158b2e3123ed491437caf1152495299b503b95199bce24b.jpg)
+
+and screen captures. Rain streak removal improves the reliability of outdoor vision systems in rainy conditions, i.e., those used in autonomous driving and video surveillance. Addressing banding artifacts is necessary in professional imaging and printing to achieve smooth gradients and high dynamic range. Additionally, structured artifact removal finds applications in medical imaging, industrial quality inspection, and consumer electronics, enhancing image quality and reliability. All these highlight a broad impact of structured artifacts removal across many real-world applications.
+
+Structured artifact removal presents significant challenges that differ from random noise removal. The primary difficulties arise due to the quasi-periodic nature of these artifacts and their resemblance to authentic repeating image patterns (e.g., textures). These artifacts often intertwine with natural image features, creating complex overlapping structures that are difficult to disentangle from genuine image content. For instance, moiré patterns can closely mimic textile textures, and banding effects may align with natural linear features. Additionally, these artifacts often display noticeable variations in pattern size and geometric shape, further complicating their identification and removal.
+
+# 1.1. Challenges and Existing Works
+
+Traditional approaches, relying on handcrafted priors to distinguish structured artifacts from genuine image features, often struggle to handle variations in these artifacts, resulting in limited effectiveness for artifacts with noticeable variations. While deep learning methods have shown promising performance in removing these artifacts, they also face notable challenges. Early approaches are based on convolutional neural networks (CNNs) [1, 12, 23, 28, 29, 33, 48], which have limited capacity on capturing long-range relationships due to their local receptive fields. However, the quasi-periodicity of local patterns with certain similarities is a crucial cue for identifying structured artifacts. Thus, the design of the NN for removing structured artifacts removal must be capable of effectively exploiting long-range relationships among local structures to accurately identify these artifacts.
+
+Recent works addressed this limitation by leveraging transformers with self-attention mechanisms [5, 6, 51]. However, challenges remain. The significant variations in the geometric shapes and pattern sizes of structured artifacts make it challenging to synthesize sufficient training data to have a comprehensive coverage of all possible variations. In addition, training NNs on very large datasets can be also computationally expensive. Consequently, relying solely on extensive datasets may not lead to robust generalization. To conclude, an effective transformer-based NN for structured artifacts removal should be designed with deformable capacity and scale adaptability, enabling them to generalize effectively to unseen data even with limited training samples.
+
+# 1.2. Main Idea
+
+An effective NN for structured artifact removal requires specialized mechanisms that can efficiently adapt to the varying scales and distortions of these artifacts, as well as exploit long-range dependencies among local structures. In this paper, we propose a transformer-based NN designed with specific optimization for handling scale variations and distortions of structured artifacts. Our proposed architecture, the Scale-Adaptive Deformable Transformer (SADT), leverages the transformer's capacity to capture long-range dependencies while incorporating new modules for scale adaptation and deformable transformations, enabling effective removal of structured artifacts with noticeable variations.
+
+Scale-enhanced deformable convolution (SEDC): Structured artifacts are semi-regular, exhibiting repetitive patterns with similar appearances but noticeable variations in size, orientation, and shape distortion. Classic convolution with fixed geometric structures struggle to capture such variable patterns effectively. Existing deformable convolutions offer some flexibility through spatial adaptations but remain ineffective in handling large-scale and orientation variations. To address these issues, we introduce the SEDC module, a convolution module designed to better manage these variations.
+
+The cascaded deformable convolutions are used to generate intermediate features from different perspectives, scales, and orientations, enabling flexible sampling and effective modeling of warped patterns. Furthermore, the subsequent Spatial-Channel Mixing Convolution (SCMC) with large-scale convolution kernel is introduced, which helps capture broader contextual information while preserving positional cues. Finally, the parallel features are adaptively aggregated for modeling scale-varying and deformable patterns.
+
+Scale-adaptive deformable attention (SADA): Structured artifacts are repetitive patterns spanning over large image regions. These patterns are quasi-periodic with non-uniform spatial distribution and size variation. While transformer is effective in capturing such repetitive patterns through attention mechanisms, the significant variations in size and non-uniform spatial distribution of these patterns demand better scale adaptability and more effective sampling within the attention mechanisms.
+
+This motivates us to propose Scale-Adaptive Deformable Attention (SADA), which assigns multi-scale multi-head mechanism for capturing long-range dependencies within a semantic layer. Moreover, SADA introduces deformable-sampling offset to effectively handle the non-uniform spatial distribution of repetitive patterns. These two techniques enable local self-attention to better capture long-range quasi-periodic patterns through selective information aggregation.
+
+# 1.3. Our Contributions
+
+In summary, our contributions are listed as follows:
+
+- We propose the SEDC module to model scale-varying artifacts with abundant orientations and potential distortions.  
+- We introduce the SADA module, enhancing local selfattention by incorporating a multi-scale multi-head mechanism and deformable technique, to capture long-range contextual information of quasi-periodic patterns.  
+- Leveraging on SEDC and SADA, we construct the transformer-based SADT for structured artifact removal.
+
+Extensive experiments on diverse artifact removal tasks, including image demoiring, deraining, and debanding, showed that SADT achieves the state-of-the-art performance.
+
+# 2. Related Work
+
+Non-learning structured artifact removal method: Traditional methods rely on pre-defined priors on structured artifacts to distinguish them from genuine image content. For moiré pattern removal from texture images, Liu et al. [22] proposed using a low rank prior for textures and a sparse prior of Moiré patterns in the discrete cosine transform to separate two. For removing rain streak, Luo et al. [27] separates the rain streak and background layers via discriminative sparse coding. Li et al. [19] introduces a patch-based approach leveraging Gaussian mixture model priors to separate
+
+rain streak and background layers. These handcrafted priors are often overly simplistic and tend to fail in removing artifacts with large variations in scale, orientation, and shape.
+
+Multi-scale deep learning methods for structured artifact removal: There is extensive literature on deep learning for structured artifact removal. Here, we focus exclusively on the most relevant works employing multi-scale strategies to address artifact patterns of varying scales. To remove Moiré patterns from images, Sun et al. [33] proposed a multiresolution encoder-decoder architecture designed to model Moiré patterns at varying scales. Yu et al. [48] introduced a semantic-aligned scale-aware module that combines multiscale features through an attention mechanism to reduce scale discrepancies in Moiré patterns. Nguyen et al. [28] developed a multiscale guided restoration block targeting both low- and high-frequency noise. For deraining, Wang et al. [38] proposed a wavelet-inspired multi-level module for rain removal. For debanding, Liu et al. [23] introduced a dual-branch depthwise group fusion module to capture both inter-scale and intra-scale correlations of banding patterns. Quan et al. [30] introduced a cross-Scale invertible NN with deformable convolutions to handle scale variations of banding artifacts. For shadow removal, built upon the Retinex decomposition model, Huang et al. [15] proposed a neural network with a multi-scale structure. These methods have shown promising results in structured artifact removal tasks. However, these methods neither fully exploit the repetitive nature of artifacts for accurate separation from genuine image structures, nor do they effectively address the significant orientation or shape variations of artifacts in their designs.
+
+Transformer for removing structured artifacts: Transformer is an effective architecture for modeling global and local relationship of local structures. In image processing, an image is partitioned into small patches to fit the transformer architecture [3]. Many transformer methods have been proposed for different structured artifact removal tasks. For deraining, Xiao et al. [44] introduced a combination of window-based and global self-attention mechanisms. Chen et al. [5] developed sparse channel self-attention to selectively retain key values. For image debanding, Wen et al. [42] utilized local self-attention with varying window sizes across different heads to capture features at multiple scales.
+
+However, these transformer methods have not fully leveraged self-attention's ability to capture long-range dependencies of repetitive patterns. Such weakness arises from the high computational costs of standard transformers due to quadratic scaling with patch numbers. To reduce costs, these transformers rely on techniques such as window-based self-attention [21, 41], channel-only self-attention in Restormer [51], and local self-attention (LSA) [18, 31]. While computationally efficient, these methods' limited spatial perceptive fields weaken their ability to capture long-range information. To address these limitations, the proposed SADT extends the
+
+LSA with scale-adaptive deformable sampling to effectively capture long-range dependencies of repetitive artifacts with varying scales, orientations, and distortions. This design enables SADT to model structured artifacts more effectively.
+
+Deformable convolution and attention: Deformable convolutions [10, 40, 45, 56] introduce learnable offsets to convolution kernels, enabling adaptive feature sampling to handle geometric variations and spatial transformations. Deformable attention [2, 43, 57] extends this adaptability to the self-attention mechanism, allowing transformers to focus on relevant spatial locations and better capture complex patterns. Zhu et al. [57] proposed a multi-scale deformable attention module for generating a feature pyramid and allocating several keys per query to capture multi-scale features. Xia et al. [43] introduced the deformable attention transformer, which learns shared deformed points for efficient computations. Cao et al. [2] developed a reference-based deformable attention module to enhance low-resolution feature representations using multiple relevant features.
+
+The design of deformable self-attention in our work differs significantly from that of [43]. Xia et al. [43] combines self-attention with deformable convolution, computing the relevance between input and reference images for offset prediction. In contrast, our method integrates deformable sampling within self-attention by first predicting offsets and then calculating the similarity between predicted keys and the query. It is more effective in modeling long-range dependence of patterns with varying appearances. Furthermore, we enhance the scale adaptability of deformable convolution and attention for handle scale variations among artifacts.
+
+# 3. Methodology
+
+In this section, we introduce the two key components of the proposed transformer. SEDC, an advanced deformable convolution module designed to capture local patterns with large-scale and orientation variations of local patterns. SADA, a deformable attention module that exploits long-range relationships among repetitive patterns with varying sizes and non-uniform spatial distributions. Subsequently, detail present the proposed transformer, SADT, which integrates SEDC and SADA within a multi-in multi-out framework, together with the training loss.
+
+# 3.1. Scale-Enhanced Deformable Convolution
+
+SEDC is for effectively capturing scale-varying and deformable patterns. The module consists of two key components: the cascaded DCNs [56] for modeling complex patterns with distortions, and the parallel Spatial-Channel Mixing Convolution (SCMC) layers for capturing patterns with large scale variations. The pipeline of SEDC is illustrated in Figure 2.
+
+To capture patterns of large size, we introduce the SCMC layer for the expansion of receptive field. For each  $i$ -th
+
+![](images/e12412d60de57c41d2a4242b11273d89a189d572ddabc87593d66f7b1eed0f6b.jpg)  
+Figure 2. SEDC configuration.
+
+channel of an input feature  $\mathbf{X} \in \mathbb{R}^{H \times W \times C}$ , SCMC first partitions each feature map into several non-overlapping patches of size  $K \times K$ , and subsequently reshape it into a matrix  $\mathbf{X}_i \in \mathbb{R}^{K^2 \times \frac{HW}{K^2}}$ . Denote such an operation by
+
+$$
+\Psi : \boldsymbol {X} \in \mathbb {R} ^ {H \times W \times C} \to \left\{\boldsymbol {X} _ {i} \in \mathbb {R} ^ {K ^ {2} \times \frac {H W}{K ^ {2}}}, \forall i \right\},
+$$
+
+Then, a learnable linear transform  $\mathbf{W} \in \mathbb{R}^{K^2 \times K^2}$  is applied to each  $\mathbf{X}_i$ : for each  $\mathbf{X}_i$ ,
+
+$$
+\Phi_ {\boldsymbol {W}}: \boldsymbol {X} _ {i} \in \mathbb {R} ^ {K ^ {2} \times \frac {H W}{K ^ {2}}} \to W \boldsymbol {X} _ {i} \in \mathbb {R} ^ {K ^ {2} \times \frac {H W}{K ^ {2}}},
+$$
+
+where  $W$  is shared across different  $X_{i}$  s. Afterward, the output of  $\Phi_W$  is reshaped and concatenated to the original dimension  $(H,W,C)$ , using the inversion of  $\Psi$ . A gating mechanism is then applied, performing an element-wise product between the original input and processed feature:
+
+$$
+\operatorname {S C M C} (\boldsymbol {X}) = \boldsymbol {X} \odot \Psi^ {- 1} \left(\Phi_ {\boldsymbol {W}} \left(\Psi (\boldsymbol {X})\right)\right). \tag {1}
+$$
+
+SCMC can effectively capture large pattern with minimal computation overhead. However, similar to other MLP-based architectures (e.g., MLP Mixer [35]), its non-overlapping partitions may struggle with patterns that span across regions, particularly when intersected by partition boundaries. This issue will be mitigated by the cascaded DCNs, which helps the aggregation of relevant non-local information into local windows, and provides features with different scales into parallel SCMCs. The efficient DCN utilizes a PConv operation [4] followed by a pointwise convolution to generate deformable offsets and modulation scalars.
+
+The complete pipeline of SEDC is shown in Fig. 2. In the main branch, input feature channels are first reduced by a factor of 4. The cascaded DCN processes corresponding features, while parallel SCMCs handle the original input features. Finally, features with different receptive field shapes are merged via pointwise convolution. Overall, this design effectively captures local patterns with varying sizes, orientations, geometric distortions, and warping effects.
+
+# 3.2. Scale-Adaptive Deformable Attention
+
+SADA integrates the multi-scale multi-head mechanism with deformable-sampled offsets to effectively model long-range dependencies in repetitive artifact patterns. Given the input feature map  $\mathbf{X} \in \mathbb{R}^{H \times W \times C}$ , we first generate query  $Q$ , key  $K$  and value  $V$  projections, enriched with local context
+
+by applying pointwise convolutions to aggregate pixel-wise cross-channel context, followed by  $3 \times 3$  depth-wise convolutions to encode channel-wise spatial context. A multi-head mechanism is then used, where  $Q, K, V$  are split into  $S$  parts along channels for subsequent process.
+
+Revisiting standard LSA: Omitting head index, let  $V_{u}$ ,  $Q_{u}$ ,  $K_{u}$  denote the  $u$ -th feature vector in  $V$ ,  $Q$ ,  $K$ , and  $u$  denote the related spatial location. In the standard LSA [31], for query  $Q_{u}$ , the output  $Y_{u}$  is defined as
+
+$$
+\boldsymbol {Y} _ {u} = \sum_ {v \in \mathcal {N} (u)} \omega_ {u \rightarrow v} \boldsymbol {V} _ {u}, \omega_ {u \rightarrow v} = \frac {\exp \left(\boldsymbol {Q} _ {u}\right) ^ {\top} \boldsymbol {K} _ {v}}{\sum_ {w \in \mathcal {N} (u)} \exp \left(\boldsymbol {Q} _ {u}\right) ^ {\top} \boldsymbol {K} _ {w}}, \tag {2}
+$$
+
+where  $\mathcal{N}(u) = u + \{-1,0,1\}^2$  denotes the index set of neighboring points of the position  $u$ , the size of  $\mathcal{N}(u)$  is rather small and results in poor capability to capture long-range dependencies with varying locations.
+
+Multi-scale multi-head mechanism for LSA: To leverage the long-range dependencies of patterns, we introduce multi-dilatation initial grids, denoted as  $\Delta^{(s)} = \{-M_s,0, + M_s\} ^2$  , to construct a multi-scale neighborhood. Specifically, for the  $s$  -th head, the neighborhood is denoted as  $\mathcal{N}^s (u) = u + \Delta^s$  . By assigning different sampling scales to each attention head, a broad and sparse receptive field, as shown in Figure 3(c), is formed through the aggregation of information across heads, enabling the layer to effectively capture long-range contextual information more effectively.
+
+Deformable-sampling offset for LSA: To handle non-uniform distribution of structured artifacts, we integrate deformable sampling into LSA by predicting offsets  $\delta_u^s$  for different position  $u$  and different scale  $s$ . The neighboring set is given by
+
+$$
+\bar {\mathcal {N}} ^ {s} (u) = u + \Delta^ {(s)} + \delta_ {u} ^ {s},
+$$
+
+Let  $D_u^s$  denote  $\delta_u^s$  in vector form, and  $D^s$  combines all  $D_u^s$ . Then, the offsets  $D^s$  are extracted from  $Q$  as
+
+$$
+\boldsymbol {D} ^ {s} = \phi^ {s} (\boldsymbol {Q}) \in \mathbb {R} ^ {H \times W \times 2 | \Delta^ {(s)} |}, \tag {3}
+$$
+
+Here,  $\phi^s$  is implemented using PConv [4] followed by a pointwise convolution. Instead of a single head  $Q_s$ , the full  $Q$  is fed to  $\phi^s$  for a holistic query view, enabling consistent head-specific adaptations. Learnable deformable offsets  $D^s$  adjust sampling locations within a head, enhancing the relevance of sampled key and value to the central query and improving the layer's handling of complex patterns.
+
+Finally, the  $s$ -th head output of SADA is calculated by
+
+$$
+\boldsymbol {Y} _ {u} ^ {s} = \sum_ {v \in \tilde {\mathcal {N}} ^ {s} (u)} \omega_ {u \rightarrow v} \boldsymbol {V} _ {u} ^ {s}. \tag {4}
+$$
+
+To mitigate the negative effects caused by statistical-based offset generator  $\phi^s$ , the low-relevance points, referred to
+
+![](images/26969461c2da252e50c7dc2d873906a68b609d3aad27c7861c0c1a3cf449eef7.jpg)  
+(a) Overall SADA.
+
+![](images/366547a4407cb33e037e3dbe47b51e0544fe3af5a66f089ad586396273162cb8.jpg)  
+(b) Deformable sampling process of the  $s$ -th head.
+
+![](images/780f209f31a2ec7d82446a21bb5d6000393155851241e63cb4cd702a586b842f.jpg)
+
+![](images/7c0e7d3ca0817fa01603efc28366f9e06617052167e5bd6ba41df3116162ee16.jpg)  
+(c) Initial sampling grids  $\{\Delta^{(s)}\}$
+
+![](images/0b220c97b6564a4020272459b3f394be20e6f26d481b3ac2dad432dd4f0a1683.jpg)  
+(d) The process instance for one head.  
+Figure 3. Overview of SADA: (a) SADA employs multi-scale multi-head mechanism to aggregate (c) a broad, sparse receptive field for capturing long-range context, while utilizing (b) deformable sampling offsets to model complex patterns. (d) A process instance of SADA within a head.
+
+invalid points, are filtered out in the weighted sum with a low  $\omega_{u\rightarrow v}$  in Eq. 4. The remaining points are considered valid. The output  $Y^{s}$  s are combined into  $\pmb{Y}$  along channels and fed into a feed-forward network (FFN) for subsequent processing. The FFN is implemented by FRFN [55]. See Fig. 3 for an illustration.
+
+# 3.3. SADT for Structured Artifact Removal
+
+Building on SEDC and SADA, we propose a transformer, called SADT, for structured artifact removal. SADT employs a multi-input multi-output (MIMO) strategy, as shown in Fig. 4. Given a degraded image  $\mathbf{Y} \in \mathbb{R}^{W \times H \times 3}$ , SADT initially extracts shallow features of size  $H \times W \times C$  via a convolution layer. These features then pass through 4 encoder blocks, capturing fine to coarse scales, with each block comprising  $N_{i}$  cascaded SADA layers to flexibly model long-range dependencies of repetitive artifacts and a pair of SEDC layers in the end for capturing scale-varying features.
+
+Within the encoder, channels are expanded and spatial resolution is reduced. Following [8, 9], features from downsampled degraded images are merged into the main path through a fusion module and subsequently processed by 3 decoders to progressively reconstruct the image from coarse to fine scales. Features in the decoder are concatenated with those from the encoder via skip connections, followed by a convolutional layer for channel dimension adjustment. The convolutional layer followed the decoder generates a residual image, which is added to the corresponding downsampled input to restore the image. The image at the finest scale
+
+serves as the final output, while other scales also contribute to the loss function during training.
+
+# 3.4. Loss Function for NN Training
+
+A multi-scale loss is used for training. Let  $O_{t}$  be the output from the  $t$ -th scale decoder block, while  $X^{gt}$  be the ground truth. Besides standard  $L_{1}$  fitting loss, we also include an auxiliary loss function that minimizes the distance between input and output in feature space, tailed for each task.
+
+Image demoiréing: The perceptual loss [16]  $\mathcal{L}_p$  is used as an auxiliary loss, and the overall loss is
+
+$$
+\mathcal {L} = \sum_ {t = 1} ^ {3} \mathcal {L} _ {1} \left(\boldsymbol {O} _ {t}, \boldsymbol {X} _ {\downarrow 2 ^ {t - 1}} ^ {g t}\right) + \lambda_ {p} \cdot \mathcal {L} _ {p} \left(\boldsymbol {O} _ {t}, \boldsymbol {X} _ {\downarrow 2 ^ {t - 1}} ^ {g t}\right), \tag {5}
+$$
+
+where  $\lambda_p > 0$  is a hyper-parameter, set to 1 in experiments.
+
+Image debanding/deraining: Unlike moiré patterns, band effects and rain streaks exhibit more pronounced stripe-like appearances, yielding strong responses in the frequency domain. Thus, the frequency loss  $\mathcal{L}_f$  is introduced as an additional loss term, and the overall loss is
+
+$$
+\mathcal {L} = \sum_ {t = 1} ^ {3} \mathcal {L} _ {1} \left(\boldsymbol {O} _ {t}, \boldsymbol {X} _ {\downarrow 2 ^ {t - 1}} ^ {g t}\right) + \lambda_ {f} \cdot \mathcal {L} _ {f} \left(\boldsymbol {O} _ {t}, \boldsymbol {X} _ {\downarrow 2 ^ {t - 1}} ^ {g t}\right), \tag {6}
+$$
+
+where  $\lambda_{f} > 0$  is a hyper-parameter, set to 0.1 in experiments.
+
+![](images/3a9cacf7fc8ba0171f708675a83945402a6028153ea077f50e4f02ae7094cc26.jpg)  
+Figure 4. The proposed network employs a multiscale hierarchical encoder-decoder architecture. Each block comprises  $N$  cascaded SADA layers while utilizing a pair of SEDC layers at the ends.
+
+# 4. Experiments
+
+In this section, we first discuss the experimental setting, followed by the evaluation of the effectiveness of our SADT for several structured artifact removal tasks, including image demoiring, debanding and deraining. Ablation studies are then conducted to assess each component of SADT. More results can be found in supplementary material.
+
+# 4.1. Experimental Settings
+
+In our SADT, The number of channels is set to  $C = 32$  while  $\{N_0, N_1, N_2, N_3\}$  are set to  $\{0, 4, 4, 4\}$ . The partition size in SCMC is  $K = 8$ . 4 heads are used in SADA, and  $M_s$  is set as  $\{1, 3, 5, 7\}$ . Model training employs the Adam optimizer [17] with  $\beta_1 = 0.9$  and  $\beta_2 = 0.999$ . Code will be released upon acceptance. Experimental datasets and implementation details are listed below:
+
+Image demoiréing: Three datasets are used for benchmarking: TIP2018 [33], FHDMi [13], and LCDMoiré [49]. The training employs cyclic cosine annealing [26], with an initial learning rate of  $2 \times 10^{-4}$  decaying to  $10^{-6}$  over each cycle. For two high-definition datasets, FHDMi and LCDMoiré, we randomly crop  $512 \times 512$  patches from the images, and train the model for 150 epochs with the annealing cycle 50 and batch size of 2. For TIP2018 dataset, the model is trained for 70 epochs with annealing cycle 10. Consistent with [48], no data augmentation is utilized in training.
+
+Image debanding: DID dataset [54] is used for benchmarking, which contains 51,490 pairs of banded and latent image patches with size  $256\times 256$  (30,829 training pairs, 10,237 validation pairs and 10,354 testing pairs) The initial learning rate is 0.0002, and decreased to  $10^{-6}$  for 300K iter
+
+ations with the cosine annealing scheme [26]. Batch size is 4. Data augmentations include horizontal/vertical flipping and rotations of  $0^{\circ}$ ,  $90^{\circ}$ ,  $180^{\circ}$  and  $270^{\circ}$ .
+
+Image deraining: The large-scale real-world dataset, SPAD [39], is used for benchmarking the task of rain streak removal. It contains 638,492 image pairs for training and 1,000 for testing. Batch size is 32 and th total iteration number is  $300\mathrm{K}$ . The initial learning rate is fixed as  $3\times 10^{-4}$  for the first 92K iterations, and then decreased to  $1\times 10^{-6}$  for 208K iterations with the cosine annealing scheme [26]. Random vertical/horizontal flips are used in data augmentation.
+
+Two common quantitative evaluation metrics are used for all tasks: PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity Index). Consistent with existing deraining methods [5, 44], PSNR/SSIM scores are computed using the Y channel in YCbCr color space. For image demoiring and debanding where periodic patterns significantly impact perception, the LPIPS metric [52] is also used.
+
+# 4.2. Quantitative and Qualitative Evaluation
+
+Image demoiréing: See Tab. 1 for the comparison of different methods. SADT outperforms all compared methods on the PSNR metric and achieves the best SSIM scores on both TIP2018 and FHDMi datasets. While many methods, including SADT, achieve comparable SSIM scores on the LCDmoiré dataset, SADT maintains its superiority with the highest PSNR performance. The substantial performance gains on these datasets demonstrate SADT's effectiveness in handling moiré artifacts. Qualitative analysis also reveals that SADT exhibits remarkable improvement in moiré pattern removal, particularly in challenging scenarios such as patterns intertwined with hair details and those distributed
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">TIP2018 [33]</td><td colspan="3">FHDMi [13]</td><td colspan="3">LCDMoiré [49]</td></tr><tr><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td></tr><tr><td>MopNet [12]</td><td>-</td><td>27.75</td><td>0.895</td><td>0.1794</td><td>22.76</td><td>0.7958</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FHDe2Net [13]</td><td>-</td><td>27.78</td><td>0.896</td><td>0.1688</td><td>22.93</td><td>0.7885</td><td>-</td><td>41.40</td><td>-</td></tr><tr><td>WDNet [24]</td><td>-</td><td>28.08</td><td>0.904</td><td>-</td><td>-</td><td>-</td><td>-</td><td>29.66</td><td>0.9670</td></tr><tr><td>MBCNN [53]</td><td>-</td><td>30.03</td><td>0.893</td><td>0.1980</td><td>22.31</td><td>0.8095</td><td>-</td><td>44.04</td><td>0.9948</td></tr><tr><td>ESDNet [48]</td><td>0.0816</td><td>29.81</td><td>0.916</td><td>0.1354</td><td>24.50</td><td>0.8351</td><td>0.0097</td><td>44.83</td><td>0.9963</td></tr><tr><td>CDDF [37]</td><td>-</td><td>28.87</td><td>0.894</td><td>0.1610</td><td>23.63</td><td>0.8040</td><td>-</td><td>44.10</td><td>-</td></tr><tr><td>RVDNet [7]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>24.29</td><td>0.8352</td><td>-</td><td>44.54</td><td>0.9932</td></tr><tr><td>RRID [46]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>24.39</td><td>0.8300</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Ours</td><td>0.0608</td><td>30.77</td><td>0.926</td><td>0.1238</td><td>24.96</td><td>0.8463</td><td>0.0065</td><td>46.43</td><td>0.9923</td></tr></table>
+
+<table><tr><td>Method</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>FCDR [14]</td><td>25.73</td><td>0.7170</td><td>0.3766</td></tr><tr><td>FFmpeg [11]</td><td>35.33</td><td>0.9352</td><td>0.0622</td></tr><tr><td>AdaDeband [36]</td><td>35.35</td><td>0.9392</td><td>0.0639</td></tr><tr><td>BitNet [1]</td><td>38.24</td><td>0.9633</td><td>0.0505</td></tr><tr><td>ADNet [34]</td><td>38.29</td><td>0.9612</td><td>0.0499</td></tr><tr><td>MWCNN [25]</td><td>39.24</td><td>0.9688</td><td>0.4854</td></tr><tr><td>MPRNet [50]</td><td>39.42</td><td>0.9697</td><td>0.0461</td></tr><tr><td>Restormer [51]</td><td>39.50</td><td>0.9709</td><td>0.0478</td></tr><tr><td>Ours</td><td>39.78</td><td>0.9729</td><td>0.0453</td></tr></table>
+
+![](images/09e4b09302f65d399bb59c1de75c81247fdd7ac51fd43a7afd0f0df09fcca9fc.jpg)  
+Figure 5. Visual inspection of the results from different image demoiring methods on sample images; see zoom-in box for details inspection.
+
+across flat clothing regions, as shown in Fig. 5.
+
+Image debanding: Given the limited availability of open-source deep learning methods specifically for debanding, we extended our evaluation to include methods adapted from related tasks. As shown in Tab.2, SADT demonstrates superior performance across all evaluation metrics. See Fig. 6 for visual comparison, where SADT achieves optimal results with minimal banding artifacts and superior color fidelity.
+
+Image deraining: The SPAD dataset [39] is used for benchmarking. As shown in Tab. 3, SADT outperformed all other methods, in PSNR and SSIM. Notably, SADT achieved a  $0.23\mathrm{dB}$  improvement over the second best performer, NeRD-Rain-S [6], with  $42\%$  fewer parameter count. (See Tab. 4). As shown in Fig. 7, the visual comparison further validate our method's effectiveness. While exhibits residual streaks and NeRD-Rain-S produces discontinuous blocks in their outputs, SADT effectively removes rain streaks while maintaining high visual quality.
+
+Complexity comparison: See Tab. 4 for the comparison of different methods in terms of FLOPs and parameter numbers. Our model, SADT, maintains relatively low FLOPs and a small number of parameters, while achieving SOTA performance as evidenced in Tab.  $1\sim 3$ . This shows that the effectiveness of our model is from its design, not model size.
+
+# 4.3. Ablation Study
+
+This study evaluates the contribution of key designs, SADA, SEDC, and MIMO architecture, toward performance gain
+
+Table 1. Quantitative results for image demoiring. The best and second-best results are Table 2. Quantitative results of image deboldfaced and underlined, respectively. banding on the DID dataset [54].  
+
+<table><tr><td>Method</td><td>Source</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>PReLU [32]</td><td>CVPR&#x27;19</td><td>40.16</td><td>0.9816</td></tr><tr><td>RCDNet [38]</td><td>CVPR&#x27;20</td><td>43.36</td><td>0.9831</td></tr><tr><td>SPDNet [47]</td><td>ICCV&#x27;21</td><td>43.55</td><td>0.9875</td></tr><tr><td>MPRNet [50]</td><td>CVPR&#x27;21</td><td>45.00</td><td>0.9897</td></tr><tr><td>ECNet [20]</td><td>WACV&#x27;22</td><td>44.32</td><td>0.9913</td></tr><tr><td>IDT [44]</td><td>TPAMI&#x27;22</td><td>47.34</td><td>0.9929</td></tr><tr><td>Uformer-B [41]</td><td>CVPR&#x27;22</td><td>47.84</td><td>0.9925</td></tr><tr><td>Restormer [51]</td><td>CVPR&#x27;22</td><td>47.98</td><td>0.9921</td></tr><tr><td>DRSformer [5]</td><td>CVPR&#x27;23</td><td>48.53</td><td>0.9924</td></tr><tr><td>NeRD-Rain-S [6]</td><td>CVPR&#x27;24</td><td>48.90</td><td>0.9936</td></tr><tr><td>Ours</td><td>-</td><td>49.13</td><td>0.9939</td></tr></table>
+
+Table 3. Image deraining results on the SPAD dataset [39]. The best and second-best results are boldfaced and underlined, respectively.  
+
+<table><tr><td>Task</td><td>Method</td><td>#Params(M)</td><td>#FLOPs(G)</td></tr><tr><td rowspan="2">Demoiréing</td><td>ESDNet [48]</td><td>5.93</td><td>17.6</td></tr><tr><td>MBCNN [53]</td><td>13.9</td><td>-</td></tr><tr><td rowspan="2">Debanding</td><td>Restormer [51]</td><td>26.13</td><td>150.0</td></tr><tr><td>MPRNet [50]</td><td>20.13</td><td>777.0</td></tr><tr><td rowspan="2">Deraining</td><td>NeRD-Rain-S [6]</td><td>10.53</td><td>79.2</td></tr><tr><td>DRSformer [5]</td><td>33.65</td><td>242.9</td></tr><tr><td></td><td>Ours</td><td>6.13</td><td>39.7</td></tr></table>
+
+Table 4. Complexity comparison with SOTA methods in different tasks. The test image is of size  $256 \times 256$ .
+
+of our model. For SADA, we removed the multi-scale head mechanism and deformable sampling offset, reducing it to multi-head LSA. For SEDC, we substituted it with a single
+
+![](images/d67476ce008b6e90254367f5b920789fb20385f4661dc52e9b60162aefaf1bc0.jpg)  
+Degraded
+
+![](images/d691ae42742b29842e54b1ccbbb3c974c519c71e6701947492c1e7047b82148f.jpg)  
+Reference
+
+![](images/c833da0ab580a2d86d53810a244d54d169308991c112581edf1020cb5e8288e9.jpg)  
+FFmpeg
+
+![](images/831fc324be9d9600d55e81fae8bc6e030c271a484ab588f89aaad949967ee31a.jpg)  
+BitNet
+
+![](images/476347858e28260e4c3cb099275709896347ac097ea0a1281ba1cfd7ce90e429.jpg)  
+ADNet
+
+![](images/308722b10de25de6940078184cbd85d0fdd10b2f66790cda603f15ffc611d5d2.jpg)  
+MWCNN
+
+![](images/9fc412018d2df07dc4983bc954290d12761dac8eee708cefa2aef5bbe09b7368.jpg)  
+MPRNet
+
+![](images/ee0ccb03a30074b1912753b00796a13c6b83dbe013effe654294fb47b065aad3.jpg)  
+Restormer
+
+![](images/51caff34d730129e89291a9db1e8e6b8930a7ad08b9100a4ab4fbe28705f1665.jpg)  
+Ours
+
+![](images/ecfe37fee3dcaf0a42e963ab705ad66f1c3f2d3fac6f0279c8bdc5d8c1302290.jpg)  
+Figure 6. Visual inspection of the results from different image debanding methods on sample images; see zoom-in box for details inspection  
+Degraded  
+Figure 7. Visual inspection of the results from different image deraining methods on sampled images.
+
+![](images/000aa278105d67f7a69eb4a7bc1f826e098f25661e4e789987ff51bc309fe272.jpg)  
+Reference
+
+![](images/e15a3c2430e39fd7123886b5a1710ae9ba1da945d5fb288896c4355e5607ee8f.jpg)  
+SPDNet
+
+![](images/a049252f83c7c347e0e67fa0e7a302e4aec2ee6636b18d23b4a0409918faf8b3.jpg)  
+Restormer
+
+![](images/62cfc5aadba427b59bf2a9f566536de2746a6d7030ddf4a0a4850b7c7b62825f.jpg)  
+DRSformer
+
+![](images/5d9797410ae695ae22fb6444ec2d21f72bc0478f1a0746ada3611a1140555cde.jpg)  
+NeRD-Rain-S
+
+![](images/992b7a7a872e1f181b9a0d38b0728510fdee449bdacd466939f2ff0b3110199b.jpg)  
+Ours
+
+SCMC. To ensure a fair comparison, all models were adjusted for comparable sizes by modifying channel numbers, and are trained on the DID [54] dataset for 300K iterations. As shown in Tab. 5, each component makes significant contribution to performance improvement. Notably, SADA contributes a PSNR gain of  $0.78\mathrm{dB}$  and an SSIM improvement of 0.003, while the inclusion of SEDC yields an additional PSNR increase of  $0.17\mathrm{dB}$ .
+
+<table><tr><td>SADA</td><td>SEDC</td><td>MIMO</td><td>PSNR(dB)↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>-</td><td>-</td><td>-</td><td>38.62</td><td>0.9663</td><td>0.0587</td></tr><tr><td>-</td><td>-</td><td>✓</td><td>38.82</td><td>0.9678</td><td>0.0533</td></tr><tr><td>-</td><td>✓</td><td>✓</td><td>39.05</td><td>0.9692</td><td>0.0521</td></tr><tr><td>✓</td><td>-</td><td>✓</td><td>39.61</td><td>0.9712</td><td>0.0480</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>39.78</td><td>0.9729</td><td>0.0453</td></tr></table>
+
+Impact of the grid dilation: In this study, we investigate the performance benefits of the multi-head mechanism in SADA by comparing different settings of  $M_{s}$  on the DID dataset. Tab. 6 shows that the multi-dilation offset grid improves performance with similar model size. Compared to assigning each head same small dilation (1,1,1,1), our setting (1,3,5,7) better exploits long-range dependencies, resulting in a PSNR gain of 0.2dB and a LPIPS reduction of 0.0039.
+
+Visualization of deformable offsets: Refer to Fig. 8 for a comparison of the sampling points between the standard LSA and our SADA. The purple star indicates the central point, while the yellow points and red circles represent the initial and final sampling positions, respectively. This shows SADA can better capture long-range and multi-scale context, which proves advantageous for the ensuing weighted aggregation.
+
+Table 5. Results of ablation studies. Boldfaced: best values.  
+
+<table><tr><td>Ms</td><td>PSNR(dB)↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>{1,1,1,1}</td><td>39.58</td><td>0.9708</td><td>0.0492</td></tr><tr><td>{1,2,3,4}</td><td>39.70</td><td>0.9723</td><td>0.0463</td></tr><tr><td>{1,3,5,7}</td><td>39.78</td><td>0.9729</td><td>0.0453</td></tr></table>
+
+![](images/bfd953a627a2abc4954138fa3debdc2328a09d9327ee929ea53cae079735db6f.jpg)  
+Table 6. Results using different setting of dilation  $M_{s}$ .
+
+![](images/6d76e442b18716801e65ee1cd73101a25e62b04883e1cf54f954d2eb3ce53086.jpg)  
+0-th head  
+Figure 8. Visualization of sampling points for LSA and SADA.
+
+![](images/73d98c9d86f7edf79ab57523a9982fd5945b3bbc2a7af6ed44c4a69f34719144.jpg)  
+1-st head  
+(c) Sampling points for each SADA head.
+
+![](images/dd646ab5fd9eefd96601511c31171cc032884f794f17d8c47f51ad61d9b7c460.jpg)
+
+(b) Sampling points for each LSA head.
+
+![](images/2a0b1a3470f6a21ffa7906f139fa5646057af9a95b030c7e5c441f09400aeeeb.jpg)  
+2-nd head
+
+![](images/59a0015c909e693453bcbbf9c95f4fea10a30d1297d9446ee47f6ea50a835e12.jpg)  
+(a) Corresponding down-sampled image. (b) Sampling points for each LSA head.  
+3-rd head
+
+# Conclusion
+
+In this paper, we presented SADT, a universal transformer-based architecture for image restoration across diverse artifacts. Our approach integrates the SEDC module to capture scale-varying patterns with abundant orientations and potential distortions, and the SADA module to model long-range relationships among repetitive patterns with diverse sizes and non-uniform distributions. Extensive experiments showed that SADT consistently outperforms SOTA methods in the tasks including image demoiréing, debanding, and deraining.
+
+# Acknowledgements.
+
+This work was supported by National Key R&D Pogram of China (2023YFA1011601), the Basic and Applied Basic Research Foundation of Guangdong Province (2024A1515012287), Science and Technology Key Program of Guangzhou, China (2023B03J1388), National Natural Science Foundation of China (62372186), Natural Science Foundation of Guangdong Province (2023A1515012841), Fundamental Research Funds for the Central Universities (x2jsD2230220), National Natural Science Foundation of China (62106077), Natural Science Foundation of Guangdong Province (2022A1515011087) and Singapore MOE AcRF Tier 1 (Grant No. A-8000981-00-00).
+
+# References
+
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+[2] Jiezhang Cao, Jingyun Liang, Kai Zhang, Yawei Li, Yulun Zhang, Wenguan Wang, and Luc Van Gool. Reference-based image super-resolution with deformable attention transformer. In European conference on computer vision, pages 325-342. Springer, 2022. 3  
+[3] Hanting Chen, Yunhe Wang, Tianyu Guo, Chang Xu, Yiping Deng, Zhenhua Liu, Siwei Ma, Chunjing Xu, Chao Xu, and Wen Gao. Pre-trained image processing transformer. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 12299-12310, 2021. 3  
+[4] Jierun Chen, Shiu-hong Kao, Hao He, Weipeng Zhuo, Song Wen, Chul-Ho Lee, and S-H Gary Chan. Run, don't walk: chasing higher flops for faster neural networks. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 12021-12031, 2023. 4  
+[5] Xiang Chen, Hao Li, Mingqiang Li, and Jinshan Pan. Learning a sparse transformer network for effective image deraining. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5896-5905, 2023. 2, 3, 6, 7  
+[6] Xiang Chen, Jinshan Pan, and Jiangxin Dong. Bidirectional multi-scale implicit neural representations for image deraining. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 25627-25636, 2024. 2, 7  
+[7] Yijia Cheng, Xin Liu, and Jingyu Yang. Recaptured raw screen image and video demoiring via channel and spatial modulations. Advances in Neural Information Processing Systems, 36, 2024. 7  
+[8] Sung-Jin Cho, Seo-Won Ji, Jun-Pyo Hong, Seung-Won Jung, and Sung-Jea Ko. Rethinking coarse-to-fine approach in single image deblurring. In Proceedings of the IEEE/CVF international conference on computer vision, pages 4641-4650, 2021. 5  
+[9] Yuning Cui, Wenqi Ren, Sining Yang, Xiaochun Cao, and Alois Knoll. Irnext: Rethinking convolutional network design
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