Add Batch 32f7544d-2013-414f-8813-a11635fb338a data

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2025/Learning Relative Gene Expression Trends from Pathology Images in Spatial Transcriptomics/36bbfac4-cb79-4f1d-9408-1a75077dbbc8_origin.pdf

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2025/Learning Relative Gene Expression Trends from Pathology Images in Spatial Transcriptomics/full.md

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+# Learning Relative Gene Expression Trends from Pathology Images in Spatial Transcriptomics
+
+Kazuya Nishimura $^{1}$  Haruka Hirose $^{1}$  Ryoma Bise $^{2}$  Kaito Shiku $^{2}$  Yasuhiro Kojima $^{1}$
+
+$^{1}$  Laboratory of Computational Life Science, National Cancer Center Japan
+
+$^{2}$  Department of Advanced Information Technology, Kyushu University, Japan  
+kanishi4@ncc.go.jp
+
+# Abstract
+
+Gene expression estimation from pathology images has the potential to reduce the RNA sequencing cost. Point-wise loss functions have been widely used to minimize the discrepancy between predicted and absolute gene expression values. However, due to the complexity of the sequencing techniques and intrinsic variability across cells, the observed gene expression contains stochastic noise and batch effects, and estimating the absolute expression values accurately remains a significant challenge. To mitigate this, we propose a novel objective of learning relative expression patterns rather than absolute levels. We assume that the relative expression levels of genes exhibit consistent patterns across independent experiments, even when absolute expression values are affected by batch effects and stochastic noise in tissue samples. Based on the assumption, we model the relation and propose a novel loss function called STRank that is robust to noise and batch effects. Experiments using synthetic datasets and real datasets demonstrate the effectiveness of the proposed method. The code is available at https://github.com/naivete5656/STRank.
+
+# 1 Introduction
+
+With the development of spatial transcriptomic techniques (ST), the comprehensive gene expression profile can be captured on a small spot with a spatial location corresponding to the pathology image [16]. Due to the high cost of acquiring spatial transcriptomics (ST) data, there is growing interest in using computer vision techniques to estimate gene expression from pathology images as a more affordable way [19, 30, 4, 7].
+
+One of the main difficulties in estimating gene expression from pathology images is the batch effects and stochastic fluctuations in observed data. As shown in Figure 1 (a), differences in reagent batches, equipment, and other technical factors in the measurement process (i.e., batch effects) can cause variations in data scaling across tissues [11, 18]. Additionally, due to cellular heterogeneity and temporal dynamics, the observed gene expression level stochastically fluctuates even though the appearance of the pathology image is the same as shown in Figure 1 (b).
+
+Although mean squared error (MSE) is commonly used in the previous estimation methods [7, 19, 30, 4], it is hard to capture variations from the data containing batch effects and stochastic noise. MSE loss focuses on predicting the absolute values of gene expression without correcting for batch effects. Consequently, models trained with MSE loss may inadvertently learn patient-specific biases rather than biologically relevant signals. Additionally, since MSE loss does not model stochastic noise explicitly, it can not account for the significance of biological signals from expression count data.
+
+In this paper, we aim to estimate the relative expression relation instead of directly estimating absolute values of gene expression. The key hypothesis of this paper is that relative gene expression trends
+
+(a) Batch effects  
+![](images/9bf4ca9c579e0693ceec4ceacebca8e9eb798b7e4d30488effd21d7d006ce088.jpg)  
+o Temporal dynamics
+
+(b) Stochastic noise  
+![](images/be19dd2db6d83e6240e5c410d18205aa871cbe0eeeeb957885a7e11bd1e80f86.jpg)  
+Heterogeneity of cell
+
+(c) Relative expression trends  
+Figure 1: (a) Illustration of scaling bias due to batch effects, (b) stochastic noise, and (c) our hypothesis: learning relative expression trends. Even in the presence of batch effects and stochastic noise, the relative expression trends between patches are preserved.  
+![](images/f1540584ef24cb0e97e15c998788c3e2a3962f907829747a61ad80f669fc0043.jpg)  
+- Follows count dist.
+
+between image patches are preserved, even when batch effects or stochastic noise are contained in count values. For instance, as illustrated in Figure 1 (c), if we extract cancerous and non-cancerous patches from several tissues, the expression of the cancer cell-specific gene is expected to exhibit higher expression in cancerous patches for all the tissues. Although the absolute expression values and their scales may vary across tissues, we posit that the relative relation of expression between patches remains consistent. In addition, the relative expression relation has been widely used for downstream analysis, such as differential expression analysis, which detects the relative expression difference between clusters. Therefore, capturing relative expression differences between patches within each tissue is more reasonable than directly estimating the absolute value of gene expression.
+
+Learning to rank (i.e., ranking loss) is one of the solutions to learn the relationships between samples [3, 22, 9, 10, 27]. This ranking loss learns which of a given pair of samples has a higher score, and the pairwise learning approach can mitigate batch effects. However, because signal-to-noise ratios of lowly expressed genes tend to be lower than those of highly expressed genes, stochastic noise can alter the relative ranking between samples when gene expression levels are low. Therefore, it is essential to model the relative relationships in a manner that reflects the probabilistic characteristics of gene expression data.
+
+To address the challenge of learning relative relationships in gene expression data affected by stochastic noise, we propose a novel loss function that models gene expression as a discrete probabilistic distribution conditioned on pairwise or listwise input. Specifically, we assume that the expression counts of paired patches are the consequence of the counting process given relative frequencies among the patches; we assume a binomial distribution for pairwise and a multinomial distribution for listwise scenarios. This formulation enables the model to capture relative relationships in a manner that is consistent with the probabilistic nature of observed count data.
+
+To confirm the characteristics of our proposed loss function, we compared a previous loss function with ours using a synthetic dataset. The experiments demonstrated the effectiveness of batch effects and stochastic noise in the low signal situation. Moreover, experiments using real datasets show the generability of our loss function.
+
+Our contributions are highlighted as follows:
+
+- We redefine the task setup of gene expression estimation from pathology images as a rank score estimation setting. The setting is more practical and realistic for the downstream tasks of using gene expression.  
+- We propose a noise-robust loss function designed to handle batch effects and probabilistic noise, which dynamically adjusts its weighting based on the relative magnitude of expression for spatial transcriptomic data (STRank). This allows the model to learn more effectively, even when the expressions are sparse.  
+- We demonstrate the effectiveness of our loss function for batch effect and stochastic noise using synthetic datasets. In addition, we validated the robustness of the proposed method in expression estimation using real-world data, confirming its effectiveness under practical conditions.
+
+# 2 Setup and Notation
+
+Let  $\mathcal{X}^{(n)} = \{x^{n,i}\}_{i=1}^{N^p}$  denote a set of image patches in  $n$ -th tissue ( $n = 1, \ldots, N^{(\mathrm{tissue})}$ ) and  $\mathcal{E}^{(n)} = \{e^{n,i}\}_{i=1}^{N^p}$  be a set of their corresponding gene expression levels at each patch, where  $N^p$  is the number of patches. The  $e^i$  is an  $N^g$ -dimensional vector and indicates gene expression level for each gene, where  $N^g$  is the number of genes. Unlike previous expression estimation setups that directly estimate expression value  $e^{n,i}$  from  $x^{n,i}$  without considering tissue  $n$ , we aim to learn a function  $f: x^{n,i} \to r^{n,i}$  that estimates the rank score [3] for each gene  $g$ , which reflects the relative relation of gene expression between the given patches from the same tissue. The  $r_g^{n,i}$  is the scale-invariant, it reflects relative relation among expression of tissue  $n$ , where if  $e_g^{n,i} > e_g^{n,j}$ , then the rank score should be  $r_g^{n,i} > r_g^{n,j}$ . Since raw gene expression values have some scaling biases introduced by the complexity of the observation technique and experimental conditions, our setup is more intuitive and practical than directly estimating the raw expression value. For readability, we omit the subscript  $n$  in,  $x^{n,i}$  and  $e^{n,i}$ . These values are referred to as  $x^i$  and  $e^i$ , which are samples in tissue  $n$ .
+
+To motivate this approach, we first summarise the two losses most widely adopted in prior work — MSE (pointwise) and Rank (pairwise) — and clarify why they remain vulnerable to either batch effects or stochastic noise.
+
+A mean squared error loss between raw expression value  $e^i$  and the estimated expression has been widely used for gene expression estimation from pathology image [7, 19, 30]. The mean squared error (MSE) loss for a sample  $x^i$  is defined as:
+
+$$
+L _ {\mathrm {M S E}} \left(e ^ {i}, \hat {r} ^ {i}\right) = \frac {1}{N ^ {g}} \left| \left| e ^ {i} - \hat {r} ^ {i} \right| \right| ^ {2}, \tag {1}
+$$
+
+where  $||\cdot ||$  is the L2 norm, and  $\hat{r}^i$  is the output of a function  $f$  that estimates the expression value, such as a neural network,  $\hat{r}^i = f(x^i)$
+
+The loss function is calculated based on one patch (i.e., pointwise input). If there are batch effects between  $\mathcal{E}^{(n)}$  and  $\mathcal{E}^{(m)}$ , the loss function can be influenced by scaling bias due to batch effects because the loss does not consider relations between samples obtained from different tissues. When patient data is imbalanced, the model may disproportionately rely on data from certain individuals. This can cause the model to learn spurious patterns, resulting in experimental bias.
+
+To deal with the batch effects in the target data, the pairwise loss, such as Rank loss [3] or listwise loss, such as PCC loss [25, 2] that learn relative relations from multiple inputs, is one of the countermeasures. Ranking loss is designed to capture the relative ordering between pairwise samples. Given pairwise samples from the same tissue  $n$ ,  $x^i$  and  $x^j$ , such that their gene expression satisfies  $e^i > e^j$ , the loss is computed for each pairwise sample in the batch as follows:
+
+$$
+\mathcal {L} _ {\text {R a n k}} \left(\hat {r} ^ {i}, \hat {r} ^ {j}\right) = \max  \left(0, \hat {r} _ {g} ^ {i} - \hat {r} _ {g} ^ {j} + \varepsilon\right), \tag {2}
+$$
+
+where  $\varepsilon$  is a margin value.
+
+While we could learn pairwise relations in tissue by introducing the Rank loss, since the Rank loss does not consider stochastic fluctuation, it is difficult to capture the signal in low signal conditions. The gene expression profile captured by spatial transcriptomics is very sparse and noisy. Therefore, to capture the signal from such a sparse dataset, we should consider the probabilistic model for the loss function.
+
+# 3 Spatial Transcriptomics Ranking Loss
+
+The motivation of our Spatial Transcriptomics Ranking Loss (STRank) is to learn the relative relationships of gene expression by considering the stochastic noise effect using the distribution of count data by modeling gene expression counts at multiple spots. Similar to the learning to rank setting, we consider two setups: pairwise and listwise loss functions.
+
+# 3.1 Pairwise STRank Loss
+
+Let us consider the pairwise situation similar to learning to rank [3]. Given a pair of patches  $i,j$  obtained from a single tissue, we train the model  $f$  to predict the rank score  $r^i$ , which reflects
+
+the relation of gene expression  $e^{i}, e^{j}$  between the samples from patch images  $x^{i}, x^{j}$ . In contrast to the conventional rank loss function [3] that focuses only on learning ordinal relationships (i.e., which sample is larger), our proposed loss function captures relative differences by incorporating the magnitude of gene expression level.
+
+We assume that the expression count  $e^i$  on the spot  $i$ , given the pairwise patches  $x^i, x^j$  and total expression  $\mathbf{t}^{i,j}$ , follows a Binomial distribution.
+
+$$
+\Pr \left(e ^ {i} \mid x ^ {i}, x ^ {j}, t ^ {i, j}\right) = \prod_ {g = 1} ^ {N ^ {g}} \Pr \left(e _ {g} ^ {i} \mid x ^ {i}, x ^ {j}, t _ {g} ^ {i, j}\right), \Pr \left(e _ {g} ^ {i} \mid x ^ {i}, x ^ {j}, t _ {g} ^ {i, j}\right) = \operatorname {B i n o m i a l} \left(t _ {g} ^ {i, j}, p _ {g} ^ {i}\right), \tag {3}
+$$
+
+where  $N^g$  is the number of genes,  $p_g^i$  is the frequency parameter of Bionomial distribution, which quantify how frequently the gene  $g$  is observed at spot  $i$  given the expression count is derived from either spot  $i$  or  $j$ , and  $t_{g}^{i,j}$  is the total expression level of gene  $g$  across these two spots:  $t_{g}^{i,j} = e_{g}^{i} + e_{g}^{j}$ . This modeling approach accounts for the unique statistical characteristics of count data. It enables adaptive weighting of inter-sample relationships based on the observed count levels, thereby improving the ability to learn from count data distributions which has stochastic fluctuations.
+
+Given pairwise patches,  $x^i$  and  $x^j$ , the model  $f$  output scores,  $\hat{r}^i$  and  $\hat{r}^j$ , where  $\hat{r}^i, \hat{r}^j \in \mathbb{R}^{N^g}$ . A softmax function is then applied between  $\hat{r}^i$  and  $\hat{r}^j$ .
+
+$$
+\hat {p} _ {g} ^ {i} = \frac {\exp (\hat {r} _ {g} ^ {i})}{e x p (\hat {r} _ {g} ^ {i}) + e x p (\hat {r} _ {g} ^ {j})}, \quad \hat {p} _ {g} ^ {j} = \frac {\exp (\hat {r} _ {g} ^ {j})}{e x p (\hat {r} _ {g} ^ {j}) + e x p (\hat {r} _ {g} ^ {i})}, \quad \hat {p} _ {g} ^ {j} = 1 - \hat {p} _ {g} ^ {i}. \qquad (4)
+$$
+
+Our loss function models the predicted probabilities  $p_{g}^{i}$  and  $p_{g}^{j}$  as parameters of a binomial distribution, and the model is trained by minimizing the negative log-likelihood of the binomial distribution, thereby aligning the predicted distributions with the observed count-based outcomes. The negative log-likelihood can be decomposed into the following form:
+
+$$
+\begin{array}{l} - \log \Pr \left(e ^ {i} \mid x ^ {i}, x ^ {j}, t ^ {i, j}\right) = - \log \left(\prod_ {g = 1} ^ {N ^ {g}} \binom {t _ {g} ^ {i, j}} {e _ {g} ^ {i}} p _ {g} ^ {i e _ {g} ^ {i}} p _ {g} ^ {j e _ {g} ^ {j}}\right) (5) \\ = - \sum_ {g = 1} ^ {N ^ {g}} \left(e _ {g} ^ {i} \log p _ {g} ^ {i} + e _ {g} ^ {j} \log p _ {g} ^ {j} + \log \binom {t _ {g} ^ {i, j}} {e _ {g} ^ {i}}\right). (6) \\ \end{array}
+$$
+
+Since  $\log \left( \begin{array}{c} T_{g}^{i,j} \\ e_{g}^{i} \end{array} \right)$  is constant value, our final loss function  $L_{\mathrm{STRank}}^{\mathrm{pair}}$  is as follows:
+
+$$
+L _ {\mathrm {S T R a n k}} ^ {\text {p a i r}} \left(x ^ {i}, x ^ {j}, e ^ {i}, e ^ {j}\right) = - \sum_ {g = 1} ^ {N ^ {g}} \left(e _ {g} ^ {i} \log \hat {p} _ {g} ^ {i} + e _ {g} ^ {j} \log \hat {p} _ {g} ^ {j}\right). \tag {7}
+$$
+
+To construct sample pairs, we randomly select a sample from within the same tissue for each reference sample. Patch pairs are generated per tissue using grouped permutation, and the total loss for a mini-batch  $M$ , randomly sampled reference without considering patients, is defined to integrate relative signals across tissues as follows:
+
+$$
+L _ {\mathrm {S T R a n k}} ^ {\text {p a i r}} (M) = \frac {1}{N ^ {b}} \sum_ {s = 1} ^ {N ^ {b}} L _ {\mathrm {S T R a n k}} ^ {\text {p a i r}} \left(x ^ {i}, x ^ {\pi (i)}, e ^ {i}, e ^ {\pi (i)}\right), \tag {8}
+$$
+
+where  $\pi$  denotes a permutation index obtained by randomly shuffling the sequential sample indices within each tissue  $n$  and  $x^{\pi (i)}$  corresponds to a randomly selected sample from the same tissue as  $x^{i}$ . After training, the  $\hat{r}$  serves as a rank score indicating the relative expression levels across individual spots.
+
+Relation with Ranking Loss Function. Our proposed pairwise loss function can be interpreted as a relaxed variant of a traditional ranking loss, enabling flexible optimization by considering count value while preserving the core objective of learning relative sample orderings. The previous ranking loss
+
+functions focus solely on the relative ordering between pairs of samples, determining whether one is larger than the other. This implicitly assumes that the difference in rank scores is sufficiently large to make the ordering unambiguous. Under this assumption,  $\hat{r}^i$  is sufficiently larger than  $\hat{r}^j$ ,  $\hat{r}^i \gg \hat{r}^j$  and the pairwise probabilities are treated as follows:  $p_g^i = \frac{\exp(\hat{r}_g^i)}{exp(\hat{r}_g^i) + exp(\hat{r}_g^j)} \approx 1$ ,  $p_g^j = \frac{\exp(\hat{r}_g^j)}{exp(\hat{r}_g^i) + exp(\hat{r}_g^j)} \approx \frac{\exp(\hat{r}_g^j)}{exp(\hat{r}_g^i)}$ . Then, our loss function can be transformed as follows:
+
+$$
+L _ {\mathrm {S T R a n k}} ^ {\text {P a i r}} \left(x ^ {i}, x ^ {j}, e ^ {i}, e ^ {j}\right) = - \sum_ {g = 1} ^ {N ^ {g}} \left(e _ {g} ^ {i} \log \frac {\exp \left(\hat {r} _ {g} ^ {j}\right)}{\exp \left(\hat {r} _ {g} ^ {i}\right)}\right) = - \sum_ {g = 1} ^ {N ^ {g}} \left(e _ {g} ^ {i} \left(\hat {r} _ {g} ^ {j} - \hat {r} _ {g} ^ {i}\right)\right) \propto \hat {r} _ {g} ^ {j} - \hat {r} _ {g} ^ {i}. \tag {9}
+$$
+
+Introducing a margin term and a max operation to the difference in rank scores recovers the form of conventional ranking loss functions, such as the hinge-based pairwise ranking loss.
+
+# 3.2 Listwise STRank Loss
+
+Similar to the pairwise approach, listwise estimation over multiple  $N^k$  samples can be formulated by modeling the expression level associated with each sample. This allows the model to handle group-wise comparisons within a unified probabilistic framework.
+
+We assume that the relationship between the list of patch images,  $\mathbf{X}^{(n)} = [x^1,\dots,x^{N^k}]$ , extracted from the same tissue and their associated gene expression values,  $\mathbf{E}^{(n)} = [e^1,\dots,e^{N^k}]$ , follows a multinomial distribution. This probabilistic formulation enables modeling the joint contribution of individual patches to the overall expression profile in a listwise manner.
+
+$$
+\Pr \left(\mathbf {E} ^ {(n)} \mid \mathbf {X} ^ {(n)}, \mathbf {T} ^ {(n)}\right) = \prod_ {g = 1} ^ {N ^ {g}} \Pr \left(\mathbf {E} _ {g} ^ {(n)} \mid \mathbf {X} ^ {(n)}, T _ {g} ^ {(n)}\right), \Pr \left(\mathbf {E} _ {g} ^ {(n)} \mid \mathbf {X} ^ {(n)}, T _ {g} ^ {(n)}\right) = \text {M u l t i n o m i a l} \left(T _ {g} ^ {(n)}, p _ {g} ^ {i}\right), \tag {10}
+$$
+
+where  $T_{g}^{(n)} = \sum_{i=1}^{N^{k}} e_{g}^{i}$  is the total number of gene expression count, and  $\mathbf{E}_{g}^{(n)} = [e_{g}^{1}, \dots, e_{g}^{N^{k}}]$ . The probabilities for the multinomial distribution  $p_{g}^{i}$  are obtained using the softmax operation, similar to the pairwise case:  $p_{g}^{i} = \frac{\exp(\hat{r}_{g}^{i})}{\sum_{j=1}^{N^{k}} \exp(\hat{r}_{g}^{j})}$ .
+
+The negative log likelihood of the multinomial distribution is transformed
+
+$$
+\begin{array}{l} - \log \Pr \left(\mathbf {E} ^ {(n)} \mid \mathbf {X} ^ {(n)}, T\right) = - \log \left(\prod_ {g = 1} ^ {N ^ {g}} \Pr \left(\mathbf {E} _ {g} ^ {(n)} \mid \mathbf {X} ^ {(n)}, T _ {g}\right)\right) (11) \\ = - \sum_ {g} ^ {N ^ {g}} \sum_ {i} ^ {N ^ {k}} \left(e _ {g} ^ {i} \log p _ {g} ^ {i} + \log \frac {T _ {g} !}{e _ {g} ^ {1} ! e _ {g} ^ {2} ! \cdots e _ {g} ^ {N ^ {b}} !}\right). (12) \\ \end{array}
+$$
+
+Since the second term is independent of the model parameters, it can be omitted during optimization. The resulting listwise rank loss for spatial transcriptomics, referred to as ListWiseSTRank, is defined as follows:
+
+$$
+L _ {\mathrm {S T R a n k}} ^ {\text {L i s t}} \left(\mathbf {X} ^ {(n)}, \mathbf {E} ^ {(n)}\right) = - \sum_ {g} ^ {N ^ {g}} \sum_ {i} ^ {N ^ {k}} e _ {g} ^ {i} \log p _ {g} ^ {i}. \tag {13}
+$$
+
+Similarly to the pairwise loss, we define the total loss for a mini-batch  $M$ , which is randomly sampled without considering patients, as follows:
+
+$$
+L _ {\mathrm {S T R a n k}} ^ {\text {L i s t}} (M) = \sum_ {n = 1} ^ {N ^ {(\text {t i s s u s e})}} L _ {\mathrm {S T R a n k}} ^ {\text {L i s t}} \left(\mathbf {X} ^ {(n)} (M), \mathbf {E} ^ {(n)} (M)\right) \tag {14}
+$$
+
+where  $X^{(n)}(M)$  and  $E^{(n)}(M)$  are the lists of  $x$  and  $e$  derived from tissue  $n$  in mini-batch  $M$ .
+
+Correction Using Expression for Each Spot. Gene expression levels can vary in detectability across spatial spots, so the total count per spot often normalizes expression data. However, such normalization converts the inherently discrete count data into continuous values, which may compromise loss
+
+functions that rely on count-based statistical properties. To mitigate this, we introduce a correction based on the total expression level  $l^i$  at each spot, enabling the model to account for inter-spot variability while preserving the count data structure.  $p_g^i = \frac{\exp(\hat{r}_g^i)l^i}{\sum_{j=1}^{N^k}\exp(\hat{r}_g^j)l^j}$ , where  $l^i = \sum_{g} e_g^i$ .
+
+# 4 Experiments
+
+We evaluated our methods using two types of datasets: synthetic datasets to confirm the hypothesis and characteristics of the proposed loss function, and real datasets to confirm practicality.
+
+Comparisons. We compared our loss function with five loss functions: 1) Mean Squared Error loss (MSE) measures the squared difference between the predicted value and the ground truth on a per-sample basis (widely used on gene expression estimation), 2) Poisson loss (Poisson) models the output as a Poisson-distributed count and minimizes the corresponding negative log-likelihood for each sample, 3) Negative Binomial loss (NB) extends Poisson loss by incorporating a dispersion parameter to handle overdispersed count data at the individual sample level, 4) Rank loss (Rank) operates on pairs of samples and penalizes incorrect relative ordering, encouraging proper ranking, 5) Pearson Correlation Coefficient loss (PCC) is a listwise loss that maximizes the linear correlation between predicted and true values across the full batch. To examine the effectiveness of our proposed learning relative expression trends strategies, we compare the loss functions of PairSTRank (Section 3.1) and ListSTRank (Section 3.2), which are based on pairwise and listwise learning, respectively.
+
+The Spearman Correlation Coefficient (SCC) was used as a metric to evaluate the performance.
+
+# 4.1 Hypothesis Analysis on Synthetic Dataset.
+
+We simulated 1D synthetic data to evaluate the effects of batch effects and stochastic noise. The reason for using synthetic data is that it is difficult to obtain ground truth from raw gene expression datasets since the observed data already contains bias and noise.
+
+In these experiments, each input variable  $x^{i}$  was defined as a one-dimensional scalar constrained to the interval [0, 1]. The corresponding gene expression level  $e^{i}$  was modeled using a negative binomial distribution, consistent with prior work in transcriptomic data analysis [11]. Specifically,  $e^{i}$  is also 1D data and was sampled from the distribution NB  $(\alpha \mu (x^{i}) + \beta ,r)$ , where  $\mu (x^i)$  is the mean response function of the input,  $r$  is the dispersion parameter,  $\alpha$  is scaling parameter,  $\beta$  is bias parameter.
+
+The objective of the experiments is to accurately estimate the mean function  $\mu(x^i)$  from a given dataset  $D = \{\mathcal{X}^{(n)}, \mathcal{E}^{(n)}\}$ ,  $\mathcal{X}^{(n)} = \{x^i\}_{i=1}^{N^n}$ ,  $\mathcal{E}^{(n)} = \{e^i\}_{i=1}^{N^n}$ . It corresponds to finding the meaningful signal from observed data. The  $\mu(x^i)$  is a nonlinear function:  $\mu(x^i) = a \sin(c x^i) + b \sin(dx^i) + a + b$  which is the same with [21]. We prepared four types of functions with different parameters (cf. Supplementary material A). The mean SCC for the four types of functions is calculated.
+
+We assumed a training dataset comprising two patients  $(n = 2)$ , and each patient's gene expression is affected by experimental batch effects, which stem from sources such as differences in imaging protocols or acquisition equipment. We assumed that gene expression measurements for each patient are affected by distinct batch parameters  $\alpha$  (scaling) and  $\beta$  (offset). For simplicity, we set  $\alpha = 1$  and  $\beta = 0$  for the first patient,  $\alpha = 10$  and  $\beta = 10$ , for the second patient (Other conditions are shown in supplementary material A). For training, 50,000 samples were independently sampled from each patient. The validation and test sets, each consisting of 10,000 samples, are sampled from a uniform distribution over the interval [0,1] in both setups. We compared a uniform situation, where input samples are sampled from a uniform distribution (Figure 2 (a)), and an imbalanced situation, where the sample of tissue 2 is sampled in the specific section (Figure 2 (b)). Figure 2 shows an example of synthetic data. The color indicates tissue, the dotted line indicates the mean function, and each plot represents an observed value  $e^i$ , which is sampled with the negative binomial distribution.
+
+A simple MLP (multi-layer perceptron) with 3 linear layers  $[1 \times 128], [128 \times 128], [128 \times 1]$  with ReLU was used for the model. The epoch was 2000 using AdamW [13] with a learning rate  $1e - 3$  with mini batch size  $= 256$ . For the scheduler, we used CosineAnnealing [12]. Details regarding computational resources and related settings are provided in the supplementary material B.
+
+Table 1 shows the comparative performance of various loss functions used for gene expression estimation. Traditional pointwise losses, including widely used Mean Squared Error (MSE), optimize
+
+![](images/b04e1d4d94b396a94dfe77d463d69b457101e70a8ac96de380c8a991fcba7d11.jpg)  
+(a) Uniform
+
+![](images/432c6ce32ac9f588b83f312edc3dbc39ca5e0201fbd869578fa1cf43354278ee.jpg)  
+(b) Imbalanced
+
+![](images/7db037cc9c7ea20f2591fc69f191aece42766af5763ac14b1b50f94ec2404d26.jpg)  
+Figure 2: Example of synthetic data for validating the batch effect. Colors indicate patients; the dashed line represents the mean function to be learned; and the dots show observations. (a) Uniform setting: Each patient's data is drawn from a uniform distribution. (b) Imbalanced setting: Observed data is skewed.
+
+Table 1: Performance comparison across multiple patient conditions. Results are based on synthetic data. Bold faces indicate the best performance in each setting, while underlined values denote the second-best.  
+
+<table><tr><td></td><td></td><td>Uniform</td><td>Imbalanced</td></tr><tr><td rowspan="4">Point</td><td>MSE</td><td>0.748</td><td>0.583</td></tr><tr><td>Po</td><td>0.777</td><td>0.603</td></tr><tr><td>NB</td><td>0.788</td><td>0.601</td></tr><tr><td>Rank</td><td>0.835</td><td>0.738</td></tr><tr><td rowspan="2">Pair</td><td>PairSTRank</td><td>0.907</td><td>0.818</td></tr><tr><td>PCC</td><td>0.858</td><td>0.560</td></tr><tr><td>List</td><td>ListSTRank</td><td>0.945</td><td>0.828</td></tr></table>
+
+prediction accuracy by minimizing the absolute difference in expression levels across individual samples. In contrast, loss functions based on pairwise and listwise learning paradigms, which capture relative expression relationships between genes or samples, demonstrate superior performance overall. These results demonstrate that learning relative expression is effective in situations where batch effects are present. Notably, the proposed methods—PairSTRank and ListSTRank—consistently outperform both conventional pairwise and listwise approaches, indicating their enhanced capacity to model the structured dependencies inherent in gene expression data in both uniform and imbalanced situations. Empirical results show that ListSTRank outperforms its pairwise counterpart. We attribute this improvement to ListSTRank's ability to capture global expression patterns across entire batches, as opposed to the localized comparisons used in pairwise learning. This suggests that leveraging broader relational context is advantageous under batch-affected conditions.
+
+# 4.2 Evaluation on Real Datasets
+
+Dataset. To evaluate the effectiveness of our proposed method, we performed experiments using seven datasets from the benchmark of the HEST-1k dataset [8]: IDC, PRAD, PAAD, COAD, READ, ccRCC, and IDC-LymphNode. SKCM and LUAD datasets were excluded from the analysis because they contain only two patients and do not align with the assumptions of our study. Each dataset contains samples from three individual patients. The IDC, PAAD, and COAD datasets were acquired using the Xenium platform, whereas other datasets were obtained via the Visium platform. To avoid train/test patient-level data leakage, we used patient-stratified splits and one patient for validation and testing data, respectively, and the other patients were used for training data. The motivation for the experiment is not to compare models, but to compare loss functions. Therefore, we do not use PCC or regularization, and simply train a regression model with each loss function.
+
+We used 50 genes with highly variable genes. To assess the influence of the loss function, we kept the feature extractor fixed and trained only a single fully connected (fc) layer, as shown in Figure 3. The feature extractor was CONCH [14], which is a vision and language foundation model for pathology. Model optimization employed the AdamW optimizer with a learning rate of  $5e - 5$  and a batch size of 256. We trained the model for up to 1000 epochs, with early stopping implemented using a patience threshold of 30 epochs.
+
+Table 2 summarizes the performance comparison across all datasets. Overall, the proposed method outperforms conventional loss functions on average. Although STRank demonstrated superior performance on synthetic data, STRank did not consistently outperform alternatives across all conditions in real datasets. Because real data evaluations are based on observations that inherently include stochastic and measurement noise, it remains essential to assess whether the evaluations reliably reflect true model performance. Even under such conditions, STRank remained relatively stable and was able to demonstrate superior performance on average.
+
+Table 2: Real dataset which is obtained from HEST-1k [8]. Bold faces indicate the best performance in each setting, while underlined values denote the second-best. Ave. is average performance.  
+
+<table><tr><td></td><td>Loss</td><td>IDC</td><td>PRAD</td><td>PAAD</td><td>COAD</td><td>READ</td><td>ccRCC</td><td>IDC-L</td><td>Ave.</td></tr><tr><td rowspan="3">Point</td><td>MSE</td><td>0.393</td><td>0.484</td><td>0.307</td><td>0.556</td><td>0.140</td><td>0.093</td><td>0.168</td><td>0.306</td></tr><tr><td>Po</td><td>0.314</td><td>0.485</td><td>0.336</td><td>0.524</td><td>0.172</td><td>0.091</td><td>0.134</td><td>0.293</td></tr><tr><td>NB</td><td>0.199</td><td>0.491</td><td>0.119</td><td>0.538</td><td>0.160</td><td>0.075</td><td>0.126</td><td>0.244</td></tr><tr><td rowspan="2">Pair</td><td>Rank</td><td>0.317</td><td>0.317</td><td>0.181</td><td>0.566</td><td>0.047</td><td>0.059</td><td>0.110</td><td>0.228</td></tr><tr><td>PairSTRank</td><td>0.494</td><td>0.458</td><td>0.346</td><td>0.613</td><td>0.136</td><td>0.127</td><td>0.228</td><td>0.343</td></tr><tr><td rowspan="2">List</td><td>PCC</td><td>0.472</td><td>0.459</td><td>0.307</td><td>0.640</td><td>0.105</td><td>0.102</td><td>0.198</td><td>0.326</td></tr><tr><td>ListSTRank</td><td>0.510</td><td>0.459</td><td>0.343</td><td>0.597</td><td>0.140</td><td>0.125</td><td>0.238</td><td>0.345</td></tr></table>
+
+![](images/def2f96e487a774a45e11d58473e345023299f8096650d07daac9b2a5c7deb56.jpg)  
+Figure 3: Illustration of our framework for the real dataset. To assess the loss function, we only update the classifier head for this evaluation.
+
+![](images/fb044ebf097c5990dfa25abcb4f35c4153e0561db510314aff53b37bb1277ff6.jpg)  
+Figure 4: Effectiveness for sparsity. The x-axis is on a log scale.
+
+# 4.3 Effectiveness for Sparcity
+
+To further assess the robustness of STRank for the sparsity, we conducted performance variability for varying sparsities in a modified real dataset. One way to assess robustness for the sparsity is by assessing the performance on low-expressed genes. However, as discussed in [17], sparsity evaluation becomes challenging in low-expression data due to the lack of known ground-truth signals. To address this, we simulated the sparsity-enhanced expression data for genes with the top 50 highest mean expression levels. For a count of each gene in each cell, we conducted binomial sampling using a specified down-sampling rate to acquire down-sampled count data. We varied levels of expression by downsampling each gene expression count using probabilities  $p = 0.01, 0.05, 0.1, 0.2, 0.5, 0.8, 1$ .
+
+Figure 4 shows the performance of the pairwise and listwise loss functions on each downsampling rate. Our Pair and List STRank loss outperforms Rank and PCC on each down-sampling rate. The difference becomes significant when the rate is 0.01, effectiveness on highly sparse and weak signals. Since gene expression is inherently sparse, these results suggest that the STRank is well-suited for capturing gene expression signals.
+
+# 4.4 Effect for Parameter  $N^k$  of Our Loss Function
+
+We varied  $N^k$  and assessed the effect of  $N^k$ , which is the number of samples to calculate our ListSTRank. A larger  $N^k$  is expected to be generally preferable, as it facilitates the capture of global trends. However, in the presence of noise or distortion in the global structure, pairwise learning may offer improved performance.
+
+Table 3 shows the performance on each  $N^k$ . The results indicate that increasing  $k$  beyond 4 leads to improved performance. However, in practice, increasing  $N^k$  is not always beneficial due to numerical instability and computational overhead in both situations. For  $k \geq 4$ , the model demonstrates robustness, with stable performance observed at  $k = 8$  and  $k = 16$ .
+
+Table 3: Performance of STRank for  $N^k$  on three conditions.  
+
+<table><tr><td>Nk</td><td>Uniform</td><td>Imbalanced</td><td>B Xenium</td></tr><tr><td>2</td><td>0.907</td><td>0.818</td><td>0.447</td></tr><tr><td>4</td><td>0.938</td><td>0.837</td><td>0.462</td></tr><tr><td>8</td><td>0.958</td><td>0.839</td><td>0.455</td></tr><tr><td>16</td><td>0.943</td><td>0.833</td><td>0.458</td></tr><tr><td>32</td><td>0.938</td><td>0.818</td><td>0.463</td></tr><tr><td>64</td><td>0.926</td><td>0.827</td><td>0.462</td></tr><tr><td>128</td><td>0.941</td><td>0.845</td><td>0.457</td></tr><tr><td>256</td><td>0.945</td><td>0.828</td><td>0.459</td></tr></table>
+
+# 5 Related work
+
+Gene expression estimation from pathological image. Estimating gene expression from pathology images has the potential to reduce sequencing costs and help understand diseases. Deep learning has been introduced in this field, and the deep learning models are trained with patch and gene expression pairs captured by spatial transcriptomics. ST-Net [7] has introduced a transfer learning approach and estimates gene expression by a convolutional neural network pre-trained on ImageNet [5]. To utilize global information of patches, graph convolution neural networks and transformers have been introduced in Hist2gene [19] and Hist2st [30]. To effectively combine local and global information, M2OST [24] and TRIPLEX [4] effectively combined multiple features that are extracted from multiple resolutions. By focusing on the difficulty of directly estimating multiple-dimensional gene expression, exemplar-guided estimation [26, 28, 29], which utilizes retrieved gene expression, has been proposed. BLEEP [26] has trained a model with image and gene expression in a contrastive learning manner and retrieves gene expression based on the image. EGN [29] has refined the retrieved gene expression with a transformer block.
+
+MSE loss has been mainly used as the loss function for these methods. In contrast, we focus on the relative relation among tissues and propose a novel noise-robust loss function with pairwise learning.
+
+Learning to rank. Learning to rank is the field that learns the ranking function from ground-truth rankings [3]. Sculley has adapted the Stochastic Gradient Descent method for learning to rank [22], allowing models to be trained on large datasets. The ranking loss has been integrated with neural networks, which have been widely utilized in applications including Image Quality Assessment [9] and crowd counting [27, 10].
+
+In contrast to these works, which only consider ranking, our loss function considers the size of the count value and adaptively weights depending on the relation of input samples. Since the gene expression is sparse and has a low signal, taking into account the count value helps to learn the relation in the low signal situation.
+
+# 6 Conclusion
+
+In this paper, we tackled gene expression estimation from pathology images by reconsidering the objective. In contrast to the previous method, which estimates the absolute expression value, we aim to learn the relative gene expression relation. In addition, we propose a novel loss function (STRank) designed to capture the relative gene expression across spatial patches by modeling the relative expression relation. Through comprehensive experiments on both synthetic and real-world datasets, we demonstrated that our method achieves more stable and reliable performance compared to traditional point-wise approaches. These results suggest that exploiting relative gene expression patterns is a promising strategy for enhancing robustness in gene expression prediction.
+
+# 7 Limitations
+
+We hypothesized that STRank would perform well under sparse conditions, and we evaluated its performance by varying the number of target genes from 50 to all possible genes. As shown in Supplementary D, we could not confirm the effectiveness of STRank in a sparse situation of a real dataset. Furthermore, as shown in Table 2, rigorous evaluation on real data remains an open challenge,
+
+and the fundamental reasons why STRank fails to outperform conventional loss functions in some datasets have yet to be elucidated.
+
+Although STRank shows effectiveness across multiple patients, it converges more slowly than PCC when evaluated on data from a single patient. This behavior is likely due to STRank's conservative weight updates under low-sample conditions. As a result, PCC may offer better convergence performance on simpler datasets with limited observations.
+
+Another challenge is performance in multi-cohort settings, where the sample is obtained from different experimental conditions (e.g., different hospitals and procedures). As detailed in Supplementary D, our proposed loss functions exhibit limitations when applied to such multi-cohort datasets with a large number of genes. In such cases, batch effects in gene expression are amplified, and pathology images are likewise influenced by experimental batch effects due to variations in imaging conditions. While addressing this issue is beyond the scope of our study, it represents an open question for this field. Future research should aim to develop more robust and generalizable approaches to mitigate these effects.
+
+# Acknowledgments and Disclosure of Funding
+
+Funding in direct support of this work: JSPS KAKEN JP24KJ2205, JPMJBS2406, JP23K18509, and JP25K22846, Development (AMED) grant 24ama221609h0001(P-PROMOTE) (to YK), National Cancer Center Research and Development Fund 2024-A-6 (to YK). We used ABCI 3.0 provided by AIST and AIST Solutions.
+
+# References
+
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+[2] Chaoyu Chen, Xin Yang, Ruobing Huang, Xindi Hu, Yankai Huang, Xiduo Lu, Xinrui Zhou, Mingyuan Luo, Yinyu Ye, Xue Shuang, et al. Fine-grained correlation loss for regression. In MICCAI, pages 663-672. Springer, 2022.  
+[3] Wei Chen, Tie-Yan Liu, Yanyan Lan, Zhi-Ming Ma, and Hang Li. Ranking measures and loss functions in learning to rank. Neurips, 22, 2009.  
+[4] Youngmin Chung, Ji Hun Ha, Kyeong Chan Im, and Joo Sang Lee. Accurate spatial gene expression prediction by integrating multi-resolution features. In CVPR, pages 11591-11600, 2024.  
+[5] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In CVPR, pages 248–255. IEEE, 2009.  
+[6] William Falcon and The PyTorch Lightning team. PyTorch Lightning, March 2019.  
+[7] Bryan He, Ludvig Bergenstråhle, Linnea Stenbeck, Abubakar Abid, Alma Andersson, Åke Borg, Jonas Maaskola, Joakim Lundeberg, and James Zou. Integrating spatial gene expression and breast tumour morphology via deep learning. Nature biomedical engineering, 4(8):827-834, 2020.  
+[8] Guillaume Jaume, Paul Doucet, Andrew Song, Ming Yang Lu, Cristina Almagro Pérez, Sophia Wagner, Anurag Vaidya, Richard Chen, Drew Williamson, Ahrong Kim, et al. Hest-1k: A dataset for spatial transcriptomics and histology image analysis. Neurips, 37:53798–53833, 2024.  
+[9] Xialei Liu, Joost Van De Weijer, and Andrew D Bagdanov. Rankija: Learning from rankings for no-reference image quality assessment. In ICCV, pages 1040-1049, 2017.  
+[10] Xialei Liu, Joost Van De Weijer, and Andrew D Bagdanov. Leveraging unlabeled data for crowd counting by learning to rank. In CVPR, pages 7661-7669, 2018.
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+[11] Romain Lopez, Jeffrey Regier, Michael B Cole, Michael I Jordan, and Nir Yosef. Deep generative modeling for single-cell transcriptomics. Nature methods, 15(12):1053-1058, 2018.  
+[12] Ilya Loshchilov and Frank Hutter. SGDR: Stochastic gradient descent with warm restarts. In ICLR, 2017.  
+[13] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In  $ICLR$ , 2019.  
+[14] Ming Y Lu, Bowen Chen, Drew FK Williamson, Richard J Chen, Ivy Liang, Tong Ding, Guillaume Jaume, Igor Odintsov, Long Phi Le, Georg Gerber, et al. A visual-language foundation model for computational pathology. Nature Medicine, 30(3):863-874, 2024.  
+[15] Ming Y Lu, Drew FK Williamson, Tiffany Y Chen, Richard J Chen, Matteo Barbieri, and Faisal Mahmood. Data-efficient and weakly supervised computational pathology on whole-slide images. Nature Biomedical Engineering, 5(6):555-570, 2021.  
+[16] Vivien Marx. Method of the year: spatially resolved transcriptomics. Nature methods, 18(1):9-14, 2021.  
+[17] Gabriel Mejia, Daniela Ruiz, Paula Cardenas, Leonardo Manrique, Daniela Vega, and Pablo Arbeláez. Enhancing gene expression prediction from histology images with spatial transcriptomics completion. In MICCAI, pages 91-101. Springer, 2024.  
+[18] Kazuya Nishimura, Ryoma Bise, and Yasuhiro Kojima. Towards spatial transcriptomics-guided pathological image recognition with batch-agnostic encoder. In ISBI, pages 1–5. IEEE, 2025.  
+[19] Minxing Pang, Kenong Su, and Mingyao Li. Leveraging information in spatial transcriptomics to predict super-resolution gene expression from histology images in tumors. *BioRxiv*, pages 2021–11, 2021.  
+[20] A Paszke. Pytorch: An imperative style, high-performance deep learning library. arXiv preprint arXiv:1912.01703, 2019.  
+[21] Silvia L Pintea, Yancong Lin, Jouke Dijkstra, and Jan C van Gemert. A step towards understanding why classification helps regression. In ICCV, pages 19972-19981, 2023.  
+[22] D Sculley. Large scale learning to rank. In NIPS Workshop on Advances in Ranking, volume 3, 2009.  
+[23] Ryousei Takano, Shinichiro Takizawa, Yusuke Tanimura, Hidemoto Nakada, and Hirotaka Ogawa. Abci 3.0: Evolution of the leading ai infrastructure in japan, 2024.  
+[24] Hongyi Wang, Xiuju Du, Jing Liu, Shuyi Ouyang, Yen-Wei Chen, and Lanfen Lin. M2ost: Many-to-one regression for predicting spatial transcriptomics from digital pathology images. arXiv preprint arXiv:2409.15092, 2024.  
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+[26] Ronald Xie, Kuan Pang, Sai Chung, Catia Perciani, Sonya MacParland, Bo Wang, and Gary Bader. Spatially resolved gene expression prediction from histology images via bi-modal contrastive learning. Neurips, 36:70626-70637, 2023.  
+[27] Zheng Xiong, Liangyu Chai, Wenxi Liu, Yongtuo Liu, Sucheng Ren, and Shengfeng He. Glance to count: Learning to rank with anchors for weakly-supervised crowd counting. In WACV, pages 343-352, 2024.  
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+
+[30] Yuansong Zeng, Zhuoyi Wei, Weijiang Yu, Rui Yin, Bingling Li, Zhonghui Tang, Yutong Lu, and Yuedong Yang. Spatial transcriptomics prediction from histology jointly through transformer and graph neural networks. bioRxiv, 2022.
+
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+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: All methods employed in this study are appropriately cited. Additional details, including licensing information, are provided in the supplementary material.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
+
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: We will release the source code upon publication.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: Our paper does not involve crowdsourcing.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: Our experiments used only publicly available datasets.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
+
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: We used LLMs for only writing, editing, and coding.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Details of Experiments on Synthetic Dataset
+
+![](images/b29efba5d3a334f3ee5cc5aad4d73a9f876f1942c2a0588980517214ffc647d3.jpg)
+
+![](images/64f4e69c8b55b093d1ef3dcecb02d78e63a8937259aa949d2531484c6b730231.jpg)  
+Figure 5: Visualization of  $\mu (x)$
+
+![](images/3f508e8e4a24953a96011028d53789330d013df8b0c1009a089de837079171d2.jpg)
+
+![](images/089886fa4970ffe12348692f511f26fb0a5a5e3a9b6ee95856f08a6cf3b0837b.jpg)
+
+Figure 5 shows four types of mean functions for our synthetic data. A nonlinear function was chosen to generate waveforms characterized by varying frequencies and slope gradients. This property allows the function to model complex, non-uniform signal behavior, which is relevant in representing heterogeneous patterns observed in the gene expression data.
+
+Figure 6 shows the performance of each loss function under various parameters in the synthetic dataset. We changed scale  $\alpha$ , bias  $\beta$ , dispersion parameter  $r$ , scale for tissue  $2\alpha$ , bias for tissue  $2\beta$ . Overall, our loss function outperforms all comparisons on each condition. The proposed loss function demonstrates robustness under low-scale conditions. Furthermore, its effectiveness improves as the variability in intensity scales across patients increases.
+
+![](images/9f4d98f364b1dc306cf7e27f9edd75da7ad40a5395e95124cc5cafefa926eaea.jpg)
+
+![](images/80e0f9a8ad657237b2b5983394f4cfac07a31a0f189ce15ac9efe42615f1f60d.jpg)
+
+![](images/ecf55ce74674da9032aa69f1a6f5a557bde89fe284943616f0ae8a20e4f4b09f.jpg)
+
+![](images/bbb755571796e6f1edb4da623e5d58a95ba190eafda9093f0dabdf67b715e0a0.jpg)  
+Figure 6: Visualization of  $\mu (x)$
+
+![](images/da3244200cc2ca439501b304f21bb267340e2a98a7b191e2f30644650fb84bfe.jpg)
+
+![](images/87fb14e1281e475fece096e750b4de1c68c5c85cf0323e3dc9537d5f7562a6af.jpg)
+
+# B Computer Resources
+
+We used the Cloud Environment [23] for the experiment on synthetic data, and an internal desktop computer for Experiment 2.
+
+Experiment 1 (Cloud environment)
+
+- CPU: 16 assigned physical CPU cores
+
+Table 4: Performance on HER2ST dataset with 50-9385 gene sets.  
+
+<table><tr><td>Method</td><td>50</td><td>250</td><td>1000</td><td>5000</td><td>9385</td></tr><tr><td>MSE</td><td>0.193</td><td>0.181</td><td>0.172</td><td>0.162</td><td>0.132</td></tr><tr><td>NB</td><td>0.020</td><td>0.154</td><td>0.098</td><td>0.083</td><td>0.074</td></tr><tr><td>Po</td><td>0.009</td><td>0.150</td><td>0.095</td><td>0.076</td><td>0.069</td></tr><tr><td>Rank</td><td>0.095</td><td>0.052</td><td>0.041</td><td>0.042</td><td>0.018</td></tr><tr><td>PairSTRank</td><td>0.244</td><td>0.194</td><td>0.176</td><td>0.177</td><td>0.173</td></tr><tr><td>PCC</td><td>0.189</td><td>0.173</td><td>0.165</td><td>0.171</td><td>0.152</td></tr><tr><td>ListSTRank</td><td>0.260</td><td>0.175</td><td>0.110</td><td>0.085</td><td>0.087</td></tr></table>
+
+Table 5: Performance on COAD Visium dataset with 50-2000 gene sets.  
+
+<table><tr><td>Method</td><td>50</td><td>250</td><td>1000</td><td>2000</td></tr><tr><td>MSE</td><td>0.2712</td><td>0.2090</td><td>0.2125</td><td>0.2084</td></tr><tr><td>NB</td><td>0.3043</td><td>0.1682</td><td>0.0899</td><td>0.0868</td></tr><tr><td>Po</td><td>0.2572</td><td>0.1440</td><td>0.0858</td><td>0.0829</td></tr><tr><td>Rank</td><td>0.1118</td><td>0.0714</td><td>0.0260</td><td>0.0599</td></tr><tr><td>PairSTRank</td><td>0.3456</td><td>0.1957</td><td>0.1383</td><td>0.1319</td></tr><tr><td>PCC</td><td>0.2615</td><td>0.2000</td><td>0.2035</td><td>0.1951</td></tr><tr><td>ListSTRank</td><td>0.3398</td><td>0.1973</td><td>0.1404</td><td>0.1375</td></tr></table>
+
+- GPU: None  
+Memory: 320 GB
+
+Experiment 2 (Internal desktop environment)
+
+- CPU: 12th Gen Intel(R) Core(TM) i9-12900KS, Physical Cores: 16  
+GPU: NVIDIA RTX A6000  
+Memory: 128 GB
+
+# C Licenses for Existing Assets
+
+We implemented our method with Pytorch [20] with modified BSD LICENSE, PytorchLightning [6] with Apache-2.0 LICENSE. For the feature extraction from whole slide images, we modified the CLAM implementation [15]. We used Hest 1k [8] with CC BY-NC-SA 4.0 for the real datasets.
+
+# D Experiments on More Realistic Scenarios
+
+We conducted experiments using the HER2ST datasets [1] and the COAD Visium (multi-cohort) with large gene sets to evaluate the robustness of our loss function on more realistic gene expression estimation scenarios. For HER2ST, we used gene sets of size 50, 250, 1000, 5000, and the full set of 9,385 genes. For COAD, we evaluated the loss functions on gene sets of size 50, 250, 1000, and 2000.
+
+The results are shown in Tables 4 and 5. Our loss function demonstrated robustness on the HER2ST dataset, which is a highly sparse spatial transcriptomics dataset, outperforming other loss functions. This suggests that the proposed loss function is effective for sparse real-world data and for mitigating batch effects within the same cohort. However, for the COAD dataset, our loss function underperformed MSE and PCC when using multi-cohort data with large gene sets (250-2000 genes).
+
+In multi-cohort settings (e.g., COAD setting in Table 5) with many target genes, we assume that the distribution of stochastic noise varies substantially across cohorts; therefore, for low-signal genes where the noise component is dominant, further extensions that account for cohort-specific noise characteristics are needed. To effectively handle multi-cohort data, additional factors beyond batch
+
+Table 6: Results of different methods on Hest2gene  
+
+<table><tr><td>Method</td><td>Hest2gene</td></tr><tr><td>MSE</td><td>0.039</td></tr><tr><td>NB</td><td>0.005</td></tr><tr><td>Po</td><td>-0.014</td></tr><tr><td>Rank</td><td>0.042</td></tr><tr><td>PairSTrank</td><td>0.052</td></tr><tr><td>PCC</td><td>0.039</td></tr><tr><td>ListSTrank</td><td>0.046</td></tr></table>
+
+effects should be considered, including potential differences in biological signals and domain shifts in image features. This is one of the open problems and is essential for the practical application of gene expression prediction.
+
+# E Integration with Previous Method
+
+To evaluate the generalizability of the proposed loss function, we examined its performance when integrated into existing gene expression estimation methods. We evaluated the Spearman correlation coefficient (SCC) for predicting 250 genes using the HER2ST dataset [1]. In contrast to the original study, we adopted a patient-level data split to evaluate generalizability. We then assessed the performance of HisToGene [19] trained with various loss functions. As summarized in the Table 6, the proposed PairSTrank loss achieved the highest performance. In conclusion, our loss functions demonstrated effectiveness on this setup.
+
+# F Exploring the Impact of Mini-Batch Sampling Strategies
+
+To investigate the effect of a mini-batch sampling strategy, we compared three settings: the original implementation (Default), a setting where the mini-batch contains samples from only one tissue (Intra-tissue), and a setting where samples are evenly sampled from all tissues (Inter-tissue).
+
+Table 7 shows the results of these experiments on the 250 and 1000 gene sets. Even when we use different mini-batch strategies, the performance of ListSTRank is not improved. PairSTRank loss also does not consider the intra-tissue and inter-tissue, but it still outperforms ListSTRank loss. This suggests that the performance degradation of ListSTRank is not due to the mini-batch setting. Based on these results, we suspect that the performance degradation of ListSTRank may be due to numerical effects.
+
+Table 7: Effect of mini-batch sampling on 250 and 1000. We compare three settings: Default, Intra-tissue (mini-batch contains samples from only one tissue), and Inter-tissue (samples are evenly sampled from all tissues).  
+
+<table><tr><td>Method</td><td>250</td><td>1000</td></tr><tr><td>Default</td><td>0.175</td><td>0.110</td></tr><tr><td>Intra-tissue</td><td>0.177</td><td>0.105</td></tr><tr><td>Inter-tissue</td><td>0.173</td><td>0.105</td></tr></table>

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+# Learning Simple Interpolants for Linear Integer Arithmetic
+
+Minchao Wu
+
+The University of Tokyo
+
+minchaow@g.ecc.u-tokyo.ac.jp
+
+Naoki Kobayashi
+
+The University of Tokyo
+
+koba@is.s.u-tokyo.ac.jp
+
+# Abstract
+
+Craig interpolation plays a central role in formal verification tasks such as model checking, invariant generation, and abstraction refinement. In the domain of linear integer arithmetic (LIA), interpolants are crucial for deriving inductive invariants that characterize unreachable or safe program states, enabling scalable and precise reasoning about software and hardware correctness. Despite progress in interpolation algorithms, generating concise and interpretable interpolants remains a key challenge. We propose a lightweight learning-based approach to generating simple interpolants for LIA. Our model learns to lazily sample input problems directly and is complementary to existing logical methods. We show that when Z3 is guided by our learned model, the complexity of the interpolants it produces can be reduced by up to  $47.3\%$ . For older solvers, the reduction rate can reach up to  $69.1\%$ .
+
+# 1 Introduction
+
+Craig interpolation is a foundational technique in formal verification, instrumental in applications such as abstraction refinement, invariant generation, and compositional reasoning [20, 14, 21, 22, 16, 15, 23, 33, 1]. In the realm of linear integer arithmetic (LIA), interpolants serve as crucial intermediaries, connecting preconditions and postconditions within program verification frameworks. However, generating interpolants in LIA presents significant challenges due to the discrete nature of integers and the complexity of associated arithmetic operations. Traditional methods often produce interpolants with large coefficients and intricate structures, which can hinder the efficiency and scalability of verification tools.
+
+Recent advancements in machine learning have shown promise in complementing formal methods by providing heuristics that guide complex decision-making processes. Despite this progress, the integration of learning-based techniques into the domain of LIA interpolation remains relatively unexplored. Existing interpolation procedures primarily rely on symbolic reasoning and decision procedures, which, while sound and complete, may not always yield the most concise interpolants [1].
+
+In this work, we introduce a novel learning-based framework aimed at simplifying interpolants in LIA. Our approach leverages a lightweight neural architecture that combines a message-passing graph neural network (GNN) [40] with a self-attention-based [34] selector mechanism. The GNN encodes the structural and semantic information of disjunction-free LIA formulas, while the selector network tries to identify minimal subsets of the input formulas that are sufficient for generating valid interpolants. This design enables a single invocation of the satisfiability modulo theories (SMT) solver during the computation, enhancing efficiency and scalability.
+
+To facilitate this learning process, we require a dataset of challenging interpolation problems for both training and evaluation. As no such dataset exists in the literature to our knowledge, we construct one comprising approximately 66,000 interpolation problems in LIA. The dataset is generated through a
+
+systematic process that ensures realistic and practical coverage of problem instances. We hope it will serve as a valuable resource for advancing learning-based interpolation methods.
+
+Empirical evaluations demonstrate the effectiveness of our approach. When integrated with state-of-the-art interpolation solvers such as Z3 [9], our model achieves a reduction in interpolant complexity of up to  $47.3\%$ . When applied to older solvers, the reduction rate increases to  $69.1\%$ , demonstrating the potential of our method to significantly enhance existing tools.
+
+Our contributions can be summarized as follows:
+
+- We propose a novel learning-based approach that guides the generation of simpler interpolants in linear integer arithmetic, complementing traditional logic-based methods. Motivated by the demands of downstream verification workloads, our model architecture emphasizes both fast evaluation and high-quality interpolant generation.  
+- We introduce a dataset containing approximately 66,000 challenging interpolation problems in linear integer arithmetic. We detail the generation process to ensure reproducibility and coverage.  
+- We demonstrate that our approach effectively reduces interpolant complexity, achieving significant improvements when integrated with both modern and legacy interpolation solvers.
+
+# 2 Preliminaries
+
+Craig interpolation is a foundational concept in logic and program verification, widely used in constructing inductive invariants and enabling efficient model checking. Given two logical formulas  $A$  and  $B$  such that their conjunction  $A \land B$  is unsatisfiable, a Craig interpolant is a formula  $I$  satisfying the following properties:
+
+1.  $A\Rightarrow I$  
+2.  $I\Rightarrow \neg B$  
+3.  $\operatorname{Var}(I) \subseteq \operatorname{Var}(A) \cap \operatorname{Var}(B)$ , where  $\operatorname{Var}(\cdot)$  denotes the set of variables in a formula.
+
+The existence of an interpolant is guaranteed by Craig's interpolation theorem for any two unsatisfiable (or disjoint) formulas  $A$  and  $B$  in LIA. In what follows, we refer to the complexity of a formula as the number of logical connectives it contains. Since our domain is integer arithmetic, negation  $\neg$  is unnecessary and we count only  $\land$  and  $\lor$ .
+
+As an example, consider the following formulas  $A$  and  $B$ :
+
+$$
+A = (x = 0 \wedge y = 0) \vee (x = 1 \wedge y = 2) \vee (x = 2 \wedge y = 4)
+$$
+
+$$
+B = \neg (x = 1 0 0 \Rightarrow y = 2 0 0)
+$$
+
+One can verify that all of
+
+$$
+I _ {1} = (x = 0 \wedge y = 0) \vee (x = 1 \wedge y = 2) \vee (x = 2 \wedge y = 4)
+$$
+
+$$
+I _ {2} = x = 1 0 0 \Rightarrow y = 2 0 0
+$$
+
+$$
+I = y = 2 x
+$$
+
+are interpolants between  $A$  and  $B$ .
+
+# 2.1 Applications to Program Verification and Why Simple Interpolants Matter
+
+In program verification, Craig interpolants are often used to find an invariant, which certifies that error states are unreachable. Let  $A$  be a formula describing an approximation of states reachable from initial states, and  $B$  be a formula describing an approximation of states reachable to error states. Then, an interpolant  $I$  between  $A$  and  $B$  serves as a candidate invariant.
+
+The example interpolation problem above may be extracted from the following program.
+
+```latex
+$\begin{array}{rl}{\mathrm{x}}&{=0;\quad\mathrm{y}=0;}\\{\mathrm{while}(*)\quad\{\mathrm{x}++;\quad\mathrm{y}+=2\}}\\{\mathrm{assert}(\mathrm{x}=100\quad=>\quad\mathrm{y}=200)}\end{array}$
+```
+
+The formula  $A \equiv (x = 0 \wedge y = 0) \vee (x = 1 \wedge y = 2) \vee (x = 2 \wedge y = 4)$  describes the states reachable within two iterations of the loop, and the formula  $B \equiv \neg (x = 100 \Rightarrow y = 200)$  describes an error state. To prove that an error state is unreachable, we need to find an (inductive) invariant  $I$  such that  $A \Rightarrow I$  and  $I \Rightarrow \neg B$ , i.e.,  $I$  is an interpolant between  $A$  and  $B$  (with an additional condition that  $I$  is closed under the transition relation).
+
+In practice, the complexity of the interpolant  $I$  plays a crucial role as simplicity often leads to generality. A formula  $I$  being an interpolant between reachable states and error states is merely a necessary condition for  $I$  to be an inductive invariant —  $I$  must additionally be closed under the transition relation — and that multiple interpolants can exist between the same pair of formulas. A simpler interpolant is more likely to constitute a true invariant, as it is less prone to overfitting to specific approximations  $A$  and  $B$  of reachable and error states. In the example above, both  $I_{1}$  and  $I_{2}$  are valid interpolants, but it is not difficult to see that only  $I$ , the simplest of the three, is an inductive invariant of the program.
+
+This is analogous to machine learning, where a simple model that captures the core pattern will generalize better than an overly complex one that overfits to training data. In the verification context, a complex interpolant can "overfit" particular approximations of reachable and error states, whereas a concise interpolant focuses on the essential relationship and is less prone to depend on such approximations.
+
+The importance of simple interpolants in program verification has been widely recognized. Albarghouthi and McMillan [1] emphasized this point and empirically demonstrated that simpler interpolants can indeed improve the efficiency of verification.
+
+# 2.2 Lazy Sampling Method
+
+The lazy sampling method described in Albarghouthi and McMillan's compositional approach [1] aims to generate simpler interpolants through compositional reasoning. It proceeds as follows:
+
+1. Given an interpolation problem  $(A, B)$ , the goal is to construct an interpolant  $I$  satisfying the above conditions. Two initially empty sets, known as sample sets, are maintained for feasible disjuncts of  $A$  and  $B$ .  
+2. The algorithm lazily samples disjuncts from  $A$  and  $B$  using an SMT solver, and iteratively attempts to find a simple interpolant  $I$  separating these samples.  
+3. If the candidate interpolant fails, additional disjuncts contradicting this interpolant are sampled and incorporated. The method then restarts with these newly updated sample sets, iteratively refining the interpolant.
+
+As an illustrative example, consider the following interpolation problem from [1]:
+
+$$
+A = (x \leq 1 \wedge y \leq 3) \vee (1 \leq x \leq 2 \wedge y \leq 2) \vee (2 \leq x \leq 3 \wedge y \leq 1)
+$$
+
+$$
+B = (x \geq 2 \wedge y \geq 3) \vee (x \geq 3 \wedge 2 \leq y \leq 3)
+$$
+
+These formulas represent distinct convex regions in two-dimensional space, and their conjunction  $A \wedge B$  is unsatisfiable, as these regions do not intersect.
+
+Applying the lazy sampling method, one might initially sample the disjoint  $(1\leq x\leq 2\land y\leq 2)$  from  $A$  and  $(x\geq 2\land y\geq 3)$  from  $B$ . A candidate interpolant between the sets
+
+$$
+S _ {A} = \{1 \leq x \leq 2 \wedge y \leq 2 \} \quad S _ {B} = \{x \geq 2 \wedge y \geq 3 \}
+$$
+
+might be  $y \leq 2$ . However, since  $(x \leq 1 \land y \leq 3)$  from  $A$  is a counterexample<sup>1</sup> (because this interpolant fails the first condition  $A \Rightarrow I$ ), this disjoint is subsequently sampled as well. Now, considering the updated
+
+$$
+S _ {A} = \{1 \leq x \leq 2 \wedge y \leq 2, x \leq 1 \wedge y \leq 3 \} \quad S _ {B} = \{x \geq 2 \wedge y \geq 3 \}
+$$
+
+a new valid interpolant  $x + y \leq 4$  between them may be found. In this case, the interpolant also successfully separates the original formulas  $A$  and  $B$ .
+
+Note that the counterexample-guided selection of new disjuncts does not guarantee simplicity. It is possible to end up with unnecessary disjuncts in the final sets  $S_A$  and  $S_B$ , especially when multiple disjuncts can serve as counterexamples simultaneously. In such cases, additional heuristics may be required to rank and prioritize the formulas.
+
+# 3 Models
+
+We now describe our approach to generating simple interpolants, following the lines described in Section 2.2. Since every LIA formula can be converted to its disjunctive normal form, below we always treat an input formula of an interpolation problem as a set (or sequence) of conjunctive formulas. Each formula in this set is referred to as a disjunct.
+
+# 3.1 Motivation
+
+A naive way to improve upon the logical methods in Section 2.2 is to replace the counterexample-guided selection of formulas with learned heuristics. There are however two major drawbacks with this approach. The first is that it would require multiple invocations of both the model and the SMT solver throughout the computation. This is because we are essentially constructing the subsets of the two input formulas from the bottom up. Each time the SMT solver provides feedback on the validity of a candidate interpolant, the model is called to make a prediction. A new candidate is then composed, and the process is repeated. This becomes increasingly expensive as the input formulas grow larger. The second drawback is the lack of a clear definition of ground truth in this setting. Given any timestep  $t$  and the selected subsets  $(S_A)_t$  and  $(S_B)_t$ , it is possible that no disjunct from the two input formulas, if added to  $(S_A)_t$  or  $(S_B)_t$ , will immediately yield a valid interpolant between  $A$  and  $B$ . In such cases, no definitive signal can be provided to properly train the model.
+
+A better way to achieve a similar improvement is to lazily sample the disjuncts in a top-down manner. That is, we want the model to directly construct subsets  $S_A \subset A$  and  $S_B \subset B$  such that the interpolant between them is also a valid interpolant between  $A$  and  $B$ . Once these subsets are identified, we only need to run the interpolation solver once on them.
+
+This is a challenging task. Mathematically, any interpolant between  $A$  and  $B$  is also a valid interpolant between any  $S_A \subset A$  and  $S_B \subset B$ . However, it is not generally true that there exist  $S_A \subset A$  and  $S_B \subset B$  such that any interpolant between them is a valid interpolant between  $A$  and  $B$ .
+
+In practice, lazily sampling a small number of disjuncts from  $A$  and  $B$  often suffices to find an interpolant for  $A$  and  $B$  [1]. However, this heavily depends on the underlying mechanics of the interpolation solver. Given the same  $S_A \subset A$  and  $S_B \subset B$ , different solvers may produce different interpolants, and not all of them are necessarily valid interpolants for  $A$  and  $B$ .
+
+This implies that an ideal model will inevitably need to internalize certain aspects of the target solver's behavior. Modern SMT solvers such as Z3 have complex codebases and decades of carefully tuned optimizations. It would be fascinating if even a portion of such a solver could be approximated by a relatively small model.
+
+We emphasize the importance of a lightweight approach and intentionally avoid relying on large language models (LLMs). This choice is motivated by the fact that both the speed and accuracy of interpolation solving are critical for downstream verification tasks.
+
+To achieve this, we develop a model with two components: an encoder that learns representations of disjunction-free LIA formulas using a message-passing graph neural network, followed by an attention-based selector network that selects appropriate disjuncts from the original input formulas. The high-level computation flow is illustrated in Figure 1.
+
+# 3.2 Encoding LIA Formulas
+
+Given a LIA formula, we first obtain its graph representation by converting it into the corresponding abstract syntax tree (AST). Each node in the formula graph is represented by a feature vector  $\mathbf{x}_i\in \mathbb{R}^d$  where  $d = 1 + d_{t} + 1 + 1$ , comprising:
+
+- A discrete node type indicator  $t_i \in \{0,1,2\}$  representing operators, variables, and constants, respectively
+
+![](images/b93fb199973e4a11a526abe7994cefceff29402bfada6ece722ca05a1545a078.jpg)  
+Figure 1: Computation flow. The upper part illustrates the embedding process for a single LIA formula, as described in Sections 3.2 to 3.5. The lower part shows the disjunct selection mechanism, described in Section 3.6.
+
+- A learnable token embedding vector  $\mathbf{e}_i\in \mathbb{R}^{d_t}$  capturing semantic information  
+- A depth value  $h_i \in \mathbb{N}$  indicating the node's position in the formula tree  
+- A raw value  $v_{i} \in \mathbb{Z}$  for constant nodes (zero-padded for non-constants)
+
+A LIA formula is thus represented by a set of node features along with the corresponding edge information, which are then processed by a message-passing graph neural network to compute feature embeddings.
+
+We use a message-passing neural network architecture specifically designed to learn representations of integer arithmetic formulas. The model employs a hierarchical feature embedding scheme followed by multiple rounds of message passing to capture both local arithmetic constraints and the global structure of the formula.
+
+# 3.3 Feature Embedding
+
+The raw node features are transformed into a latent representation through a multi-component embedding network:
+
+$$
+\mathbf {z} _ {i} ^ {t y p e} = \operatorname {E m b} _ {t y p e} (t _ {i}) \in \mathbb {R} ^ {d _ {h} / 4}
+$$
+
+$$
+\mathbf {z} _ {i} ^ {\text {t o k e n}} = \operatorname {M L P} _ {\text {t o k e n}} \left(\mathbf {e} _ {i}\right) \in \mathbb {R} ^ {d _ {h} / 2}
+$$
+
+$$
+\mathbf {z} _ {i} ^ {\text {d e p t h}} = \mathrm {M L P} _ {\text {d e p t h}} \left(h _ {i}\right) \in \mathbb {R} ^ {d _ {h} / 8}
+$$
+
+$$
+\mathbf {z} _ {i} ^ {\text {v a l u e}} = \mathrm {M L P} _ {\text {v a l u e}} (v _ {i}) \in \mathbb {R} ^ {d _ {h} / 8}
+$$
+
+where  $d_h$  is the hidden dimension. These embeddings are concatenated and passed through a feature combination network:
+
+$$
+\mathbf {h} _ {i} ^ {(0)} = \operatorname {L a y e r N o r m} \left(\operatorname {M L P} _ {\text {c o m b}} \left(\left[ \mathbf {z} _ {i} ^ {\text {t y p e}}; \mathbf {z} _ {i} ^ {\text {t o k e n}}; \mathbf {z} _ {i} ^ {\text {d e p t h}}; \mathbf {z} _ {i} ^ {\text {v a l u e}} \right]\right)\right) \tag {2}
+$$
+
+The resulting node embeddings are then refined through multiple rounds of message passing: in each round, a message is computed for every edge based on the concatenation of the source and target node embeddings, passed through a multi-layer perceptron (MLP), and then aggregated additively at the target node.
+
+# 3.4 Message Passing
+
+Our message passing scheme iteratively updates node representations over  $L$  layers. At each layer  $l$ , the message function  $M^{(l)}$  and update function  $U^{(l)}$  are defined as:
+
+$$
+\mathbf {m} _ {i, j} ^ {(l)} = M ^ {(l)} \left(\mathbf {h} _ {i} ^ {(l)}, \mathbf {h} _ {j} ^ {(l)}\right) = \operatorname {M L P} _ {m s g} ^ {(l)} \left(\left[ \mathbf {h} _ {i} ^ {(l)}; \mathbf {h} _ {j} ^ {(l)} \right]\right)
+$$
+
+$$
+\mathbf {a} _ {i} ^ {(l)} = \sum_ {j \in \mathcal {N} (i)} \mathbf {m} _ {i, j} ^ {(l)} \tag {3}
+$$
+
+$$
+\mathbf {h} _ {i} ^ {(l + 1)} = U ^ {(l)} \left(\mathbf {h} _ {i} ^ {(l)}, \mathbf {a} _ {i} ^ {(l)}\right) = \operatorname {L a y e r N o r m} \left(\mathbf {h} _ {i} ^ {(l)} + \operatorname {M L P} _ {u p d} ^ {(l)} \left(\left[ \mathbf {h} _ {i} ^ {(l)}; \mathbf {a} _ {i} ^ {(l)} \right]\right)\right)
+$$
+
+where  $\mathcal{N}(i)$  denotes the neighbors of node  $i$  in the formula graph. Residual connections and layer normalization are applied at each layer to stabilize training.
+
+# 3.5 Formula-Level Representation
+
+After the message-passing phase, node embeddings are aggregated into a single graph-level vector. Specifically, we compute the final latent representation for the formula using the node embeddings in its AST. The final graph representation is obtained by combining global mean and sum pooling operations:
+
+$$
+\mathbf {g} = \frac {1}{2} \left(\operatorname {m e a n p o o l} \left(\left\{\mathbf {h} _ {i} ^ {(L)} \right\} _ {i = 1} ^ {N}\right) + \operatorname {s u m p o o l} \left(\left\{\mathbf {h} _ {i} ^ {(L)} \right\} _ {i = 1} ^ {N}\right)\right) \tag {4}
+$$
+
+followed by a final transformation:
+
+$$
+\mathbf {y} = \operatorname {M L P} _ {\text {o u t}} (\mathbf {g}) \tag {5}
+$$
+
+This architecture explicitly captures both local arithmetic relationships through message passing and global formula structure through hierarchical embedding and pooling mechanisms. The resulting formula-level embedding is then ready for downstream tasks, including—but not limited to—our lazy sampling method for interpolation solving. The process of encoding a (disjunction-free) LIA formula and computing its latent representation is illustrated in the upper part of Figure 1.
+
+# 3.6 Selector Network
+
+By the nature of interpolation solving, the two input formulas must be interdependent when deciding whether a particular disjunct is necessary. Each disjunct in  $A$  should be aware of every disjunct in  $B$ , and vice versa, in order to compute a valid interpolant. Moreover, since we are particularly interested in identifying redundancy within the input formulas, each formula in  $A$  (respectively, in  $B$ ) should also attend to other formulas within  $A$  (or  $B$ ). For this reason, we employ self-attention rather than cross-attention in the selector network.
+
+Given the two sequences of embeddings for formulas in  $A$  and  $B$  computed by the GNN, we first concatenate them to obtain a matrix of formula embeddings  $\mathbf{X} \in \mathbb{R}^{N \times d}$ , where  $N$  is the total number of formulas in  $A \cup B$ , and  $d$  is the embedding dimension.
+
+To preserve set membership information, each embedding in  $\mathbf{X}$  is augmented with a learnable positional encoding. Specifically, a trainable vector  $\mathbf{e}_A \in \mathbb{R}^d$  is added to all embeddings corresponding to formulas in set  $A$ , and a separate vector  $\mathbf{e}_B \in \mathbb{R}^d$  is added to those from set  $B$ . This produces set-aware input embeddings:
+
+$$
+\tilde {\mathbf {X}} = \mathbf {X} + \mathbf {E}, \quad \text {w h e r e} \quad \mathbf {E} _ {i, j,:} = \left\{ \begin{array}{l l} \mathbf {e} _ {A} & \text {i f s e t \_ i d s} _ {i, j} = 0 \\ \mathbf {e} _ {B} & \text {i f s e t \_ i d s} _ {i, j} = 1 \end{array} \right. \text {(i n d i c a t i n g s e t A)}
+$$
+
+This enriched input  $\tilde{\mathbf{X}}$  is passed through a series of self-attention and feedforward blocks, enabling each disjunct to attend to all others in the combined input. In particular, self-attention computes
+
+attention weights  $\alpha_{ij}$  that quantify the influence of formula  $j$  on formula  $i$ , allowing the model to reason about interdependencies and relative importance across  $A \cup B$ . This is especially valuable when certain formulas are redundant or contextually less informative. The combination of set-aware encoding and self-attention allows the selector to make globally informed decisions about which disjuncts are most relevant in context.
+
+The resulting tensor is then passed through a feedforward scoring head, which projects each formula embedding to a scalar score used to rank disjuncts from the input formulas.
+
+# 4 Experiments
+
+# 4.1 Dataset
+
+To demonstrate the effectiveness of our approach, we require a dataset of challenging interpolation problems for both training and evaluation. As no such dataset exists in the literature to our knowledge, we construct one ourselves and describe the generation process in detail.
+
+We begin by creating a bank of disjunction-free LIA formulas that serve as the "atomic" components of our dataset. This is done by running an existing software verification tool on real downstream tasks (e.g., functional program verification), where interpolation solving is used as a subroutine. We then extract the execution traces of the interpolation solver and collect all disjunction-free LIA formulas and subformulas that occur in these traces.
+
+We then scan the entire formula bank and construct a dictionary  $D$ , where each formula serves as a key, and its value is a list of formulas that are all disjoint from the key. Given a sequence of disjunction-free formulas  $\varphi_{1},\ldots ,\varphi_{n}$ , it is not difficult to verify that
+
+$$
+A = \bigvee_ {i = 1} ^ {n} \varphi_ {i} \quad B = \bigvee_ {i = 1} ^ {n} D [ \varphi_ {i} ]
+$$
+
+forms a valid interpolation problem, provided that  $\bigcap_{i=1}^{n} D[\varphi_i]$  is nonempty.
+
+To ensure variety in the dataset, we must randomly access different combinations of keys in  $D$ . However, when  $D$  is large, enumerating all combinations and then sampling from them is computationally infeasible. Instead, we randomly sample an index  $c$  and compute the  $c$ -th combination directly. We omit the implementation details of this procedure here.
+
+The generation of the dataset is controlled by two parameters,  $c_{1}$  and  $c_{2}$ , denoting the maximum length of  $A$  and  $B$ , respectively. Problems where either  $A$  or  $B$  is a singleton are excluded to ensure sufficient complexity. We denote the resulting dataset as  $\mathbf{Interp}(c_{1}, c_{2})$ . In the experiments that follow, we primarily use  $\mathbf{Interp}(4, 4)$ .
+
+Although we use relatively small values of  $c_{1}$  and  $c_{2}$  due to computational constraints, Interp(4, 4) already contains challenging problems for state-of-the-art interpolation solvers. This is partly because we place no restriction on the internal complexity of the disjuncts in the input formulas—only on their count. Table 1 summarizes statistics for Interp(4, 4). The full dataset is split into training and testing sets using an  $80\% - 20\%$  ratio.
+
+Table 1: Dataset Statistics  
+
+<table><tr><td>Statistic</td><td>Value</td><td>Description</td></tr><tr><td>Number of problems</td><td>65,981</td><td>Total number of data points</td></tr><tr><td>Maximum complexity</td><td>31</td><td>Highest observed formula complexity</td></tr><tr><td>Average complexity</td><td>9</td><td>Average over all input formulas</td></tr><tr><td>Constant range</td><td>0–112</td><td>Absolute value range of constants</td></tr><tr><td>Vocabulary size</td><td>355</td><td>Variable names and operators only</td></tr></table>
+
+# 4.2 Ground Truths and Hyperparameters
+
+To generate ground truths, we compute interpolants between all pairs  $(S_A, S_B)$  such that  $S_A \in \mathcal{P}(A)$  and  $S_B \in \mathcal{P}(B)$ , using a target solver (e.g., Z3). A pair  $(S_A, S_B)$  is considered a candidate if the interpolant between  $S_A$  and  $S_B$  is also valid for the original formulas  $A$  and  $B$ . Among all valid candidates, the shortest pair (in terms of set size) is selected as the ground truth.
+
+In practice, we observe that many problems admit multiple valid ground truths. Randomly selecting one can introduce noise and hinder model training. To provide more deterministic supervision, we apply two additional criteria to select among the candidates:
+
+-  $(S_A, S_B)$  has the fewest total tokens among all valid candidates.  
+- The sum of the absolute values of constants occurring in  $(S_A, S_B)$  is minimal.
+
+In the experiments below, we fix the token embedding dimension to 16 and the formula embedding dimension to 32. The hidden feature size of both the GNN encoder and the selector network is set to 128. We use 4 heads in the multi-head attention mechanism. The parameters of the graph encoder, selector network, and token embedding layer are jointly trained with a batch size of 256 using the Adam optimizer [18] for 300 epochs.
+
+The learning rate is initially set to 1e-3 and is reduced by a factor of 0.5 every 50 epochs. Training is performed using PyTorch's Metal backend [27, 2] and takes approximately 1.5 days on a single MacBook Pro with the M1 MAX chip.
+
+# 4.3 Improving Interpolants Found by Z3
+
+We begin by evaluating the improvements our model brings to Z3 in terms of interpolant complexity. Although Z3 is already highly optimized and capable of producing concise interpolants for many problems in our dataset, it does not always yield the simplest solution. To better illustrate the benefit of our approach, we focus on the subset of test problems that are challenging for Z3. We define a problem as challenging if there exists an interpolant whose complexity is at least two units lower than that of the one produced by Z3<sup>3</sup>. This results in a subset of 532 such problems from the test set. Table 2 presents a comparison between the original Z3 and Z3 guided by our model.
+
+Table 2: Performance Gain. Numbers are averaged over valid solutions.  
+
+<table><tr><td>Methods</td><td>Valid</td><td>Complexity</td><td>Length</td><td>Z3 Time (s)</td></tr><tr><td>Original Z3</td><td>100%</td><td>5.38</td><td>186.7</td><td>0.012</td></tr><tr><td>Z3 guided by model</td><td>72.9%</td><td>2.83</td><td>108.6</td><td>0.009</td></tr></table>
+
+As shown in Table 2, the complexity of the interpolants produced by Z3 is reduced by  $47.3\%$  when guided by our model, a result further supported by the corresponding reduction in plain-text length. The execution time of Z3 is also reduced by  $30\%$  under model guidance.
+
+The  $72.9\%$  success rate may appear suboptimal, but this is not a significant limitation in practice—one can simply fall back to the original solver in cases of failure. While this fallback mechanism introduces a small overhead due to model invocation (averaging 20ms in our experiments), the overall impact on performance remains modest.
+
+Importantly, the improvement in interpolant quality can have substantial downstream benefits: a better interpolant can often enable verification tasks to succeed that would otherwise time out completely.
+
+# 4.4 Reviving an Older Solver
+
+As discussed in Section 3.1, the effectiveness of lazy sampling is solver-dependent. Given an interpolation problem  $(A,B)$ , the sampled subsets  $(S_A,S_B)$  that work for Z3 may not generalize to other solvers. In this section, we evaluate how well the trained model transfers to a different solver. Specifically, we experiment with an interpolation solver called CSIsat [4], an older tool whose interpolants on our test problems have an average complexity of 11.5. We assess the performance of
+
+both Z3 and CSIsat when guided by models trained using their respective ground truths, as well as when guided by "exotic" ground truths—i.e., ground truths generated for the other solver.
+
+Table 3: Performance of Solvers Guided by Different Models on All Test Problems  
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">trained with Z3 ground truths</td><td colspan="3">trained with CSIsat ground truths</td></tr><tr><td>Valid</td><td>Complexity</td><td>Solver Time (s)</td><td>Valid</td><td>Complexity</td><td>Solver Time (s)</td></tr><tr><td>Z3</td><td>74.2%</td><td>2.84</td><td>0.009</td><td>52.7%</td><td>2.23</td><td>0.011</td></tr><tr><td>CSIsat</td><td>59.2%</td><td>5.13</td><td>0.007</td><td>73.7%</td><td>3.57</td><td>0.007</td></tr></table>
+
+As shown in Table 3, we observe a roughly symmetric pattern in success rates. Both solvers manage to achieve around  $55\%$  validity when guided by a model trained on exotic ground truths. This suggests that the model captures and transfers some of the underlying interpolation heuristics shared between Z3 and CSIsat.
+
+However, each solver performs significantly better when guided by the model trained specifically for it. This indicates that the dedicated models have acquired solver-specific knowledge during training. Interestingly, when Z3 is guided by the model trained on CSIsat ground truths, the resulting interpolants appear slightly simpler (in terms of complexity) than those generated using the dedicated Z3 model. We do not overinterpret this effect, as it is likely due to the complexity being averaged over a smaller subset of valid solutions.
+
+Lastly, we note that the complexity of the interpolants produced by CSIsat is significantly reduced when guided by either model—by up to  $69.1\%$ . This demonstrates that we can elevate CSIsat's performance to a modern standard without modifying its legacy codebase.
+
+Such a capability is particularly valuable in scenarios where alternative interpolants are sought for difficult problems in downstream verification tasks. In this role, the revived CSIsat can now serve as a practical and efficient tool.
+
+# 5 Related Work
+
+Craig Interpolation in Integer Arithmetic Craig interpolation has been extensively studied in the context of formal verification, particularly for generating intermediate assertions in safety analysis and model checking. Cimatti et al. [6] provides a comprehensive overview of efficient interpolation procedures for satisfiability modulo theories (SMT). Compositional methods [1] have proven especially effective for producing high-quality interpolants. Although learning-based methods for LIA interpolation remains relatively unexplored, traditional statistical machine learning techniques, such as Support Vector Machines (SVMs) [8] have been applied to the problem of finding half-space interpolants [31]. However, such approaches are typically incomplete when the interpolation problem is not linearly separable.
+
+Learning-Based Methods in Theorem Proving The integration of machine learning into automated theorem proving has gained significant traction in recent years. Deep learning has been widely explored in the context of traditional automated theorem proving (ATP), including SAT solving [30, 29, 36, 5, 12]. In contrast to SAT solving, interpolation in LIA involves a richer logical language and requires more sophisticated encoding. While a SAT solution consists of either a satisfying truth assignment or a proof of unsatisfiability, an interpolant is an arbitrary formula within the underlying theory. This additional expressiveness presents unique challenges for applying learning-based techniques.
+
+In the field of interactive theorem proving (ITP) with proof assistants such as Lean [10], Isabelle [26] and Coq [7], large language models (LLMs) have been extensively used for tasks such as automatization [37, 25, 39, 28], proof sketching [17, 11, 32], tactic generation [13, 17, 35] and premise selection [24, 38]. While LLMs are powerful at suggesting formulas, they tend to be slow and imprecise when it comes to symbolic reasoning, making them less suitable for our task, which requires efficient and accurate interpolation.
+
+# 6 Limitations
+
+Characterization of Complexity. In this paper, we primarily characterize the complexity of a linear arithmetic formula by the number of its logical connectives. This metric captures one of the most intuitive and relevant aspects of interpolant simplicity. However, complexity is inherently multi-dimensional. For instance, the magnitude of constants appearing in an interpolant also plays an important role, as large constants can introduce difficulties in downstream verification tasks. Similarly, the number of variables provides another useful perspective: fewer connectives suggest less redundant "regional" information, while fewer variables imply less redundant "dimensional" information. Depending on the downstream application, an interpolant with more connectives but fewer variables may in fact be preferable.
+
+We have incorporated some of this intuition into the design of the deterministic selection criteria described in Section 4.2. Nonetheless, future work could explore dedicated learning-based heuristics to better capture and optimize for these other dimensions of interpolant complexity.
+
+Applications and Separated Learning. Although our model-guided interpolation solvers can produce significantly simpler interpolants, the benefits do not always translate directly to downstream applications. Many verification tools invoke interpolation as a subroutine and may pre-process or decompose complex problems into simpler ones, resulting in trivial interpolation queries. In such cases, our approach offers limited advantage over standard interpolation solvers.
+
+Moreover, interpolants of the same syntactic complexity can yield vastly different outcomes in downstream tasks, depending on the nature of the application and the behavior of the overarching ("master") solver. If the objective is to produce interpolants that are more effective for a specific downstream task, then the training process must incorporate feedback from that task. This would allow the model to internalize the task-specific heuristics and interaction patterns of the broader verification pipeline.
+
+# 7 Conclusion
+
+In this work, we introduced a lightweight learning-based framework for simplifying Craig interpolants in linear integer arithmetic (LIA). Our approach combines a message-passing graph neural network (GNN) with a self-attention-based selector network to directly identify minimal subsets of input formulas sufficient to produce valid interpolants. This design complements existing logical techniques and enables improvements over both modern and legacy interpolation solvers—without modifying their underlying implementations.
+
+Empirical evaluations demonstrate the effectiveness of our approach. When integrated with Z3, a state-of-the-art SMT solver, our model reduces interpolant complexity by up to  $47.3\%$ . When applied to older solvers, the reduction increases to  $69.1\%$ , highlighting the method's potential to substantially enhance existing tools.
+
+Because our method does not rely on linearity, a promising direction for future work is its extension to richer domains such as non-linear integer arithmetic or array theories. Additionally, integrating our framework into downstream tasks—such as invariant synthesis or termination analysis—may further demonstrate its generality and practical value.
+
+# Acknowledgments
+
+This work was supported by JSPS KAKENHI Grant Number JP20H05703. We thank the anonymous reviewers for their insightful comments and suggestions. We thank Ryosuke Sato for adapting MoCHi for this project.
+
+# References
+
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+
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+
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+
+# A Variables and Constants in Interpolants
+
+In addition to the number of logical connectives in an interpolant, we also report the number of distinct variables and the largest constant appearing in each interpolant.
+
+Table 4: Variables and constants. Numbers are averaged over all valid solutions across all test problems.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">trained with Z3 GT</td><td colspan="2">trained with CSIsat GT</td><td colspan="2">Original</td></tr><tr><td>Variables</td><td>Constants</td><td>Variables</td><td>Constants</td><td>Variables</td><td>Constants</td></tr><tr><td>Z3</td><td>2.45</td><td>5.97</td><td>2.39</td><td>6.39</td><td>2.45</td><td>4.24</td></tr><tr><td>CSIsat</td><td>2.50</td><td>16386579</td><td>2.52</td><td>460537</td><td>2.60</td><td>7476237725</td></tr></table>
+
+The number of variables in a solution does not change significantly when either solver is guided by the model. However, the largest constants in CSIsat-generated solutions drop by several orders of magnitude when the solver is guided by the model. This may be attributed to CSIsat lacking effective heuristics for optimizing over integer domains.
+
+# B Example Problems from Dataset
+
+Below we include three example problems from Interp(4,4). The included "Z3-solutions" are solutions generated by the original Z3.
+
+The "simp-solutions" are the solutions found by Z3 when guided by the model trained specifically for Z3. These are not part of the original Interp(4, 4) dataset, but we include them here for reference.
+
+```txt
+{ 'input1': [ '1 + v sep_x2 <= v sep_x3', (v sep_x3 <= v sep_x1 and (6 == v sep_x0 and 1 + v sep_x0 <= v sep_x3))', (95 <= v sep_x1 and v sep_x1 <= 100), (1 <= v sep_x0 and 1 + v sep_x0 <= v sep_x1) ], 'input2': [ '(1 == v sep_x1 and (0 <= v sep_x3 and (v sep_x1 <= v sep_x2 and 1 + v sep_x3 <= v sep_x2)))', (0 == v sep_x1 and (v sep_x2 <= v sep_x3 and 1 <= v sep_x3)), (1 + v sep_x3 <= 0 and (1 == v sep_x1 and (v sep_x1 <= v sep_x2 and 1 <= v sep_x2)))', ] 'Z3-solution': ('(not (v sep_x1 <= 1) || not (v sep_x3 + -1* v sep_x2 <= -1)) && (not (v sep_x1 <= 0) || not (v sep_x3 + -1*v sep_x2 <= 0))', 'simp-solution': 'not (v sep_x1 <= 1) || not (v sep_x2 + -1*v sep_x3 >= 0)', 'time': 0.012347221374511719 } { 'input1': [ '1 + v sep_x2 <= v sep_x3', (92 <= v sep_x1 and v sep_x1 <= 97), (2 + v sep_x2 <= v sep_var961 and (3 + v sep_x2 <= v sep_var961 and (2 == v sep_var961 and (v sep_var961 <= v sep_var962 and v sep_var961 <= v sep_var973))))',
+```
+
+```latex
+\(\begin{array}{r l} & {\mathrm{~(~2~ + ~v\_sep\_x2~ <= ~v\_sep\_var702~and~(4~ + ~v\_sep\_x2~ <= ~v\_sep\_var702}}\\ & {\mathrm{~and~(3~ == ~v\_sep\_var702~and~(v\_sep\_var702~<= ~v\_sep\_var703~and~}}\\ & {\mathrm{~v\_sep\_var702~<= ~v\_sep\_var711))})}\end{array}\) 1,   
+'input2': [ \((0 = = v_{\text{sep}}x3\) and \((0 = = 1 + v_{\text{sep}}\text{var} 3004\) and \((1 = = v_{\text{sep}}\text{var} 3005\) and \((2 = = v_{\text{sep}}x1\) and \(2 <   = v_{\text{sep}}x2)))\) , \((1 = = v_{\text{sep}}x1\) and \((0 <   = v_{\text{sep}}x3\) and \((v_{\text{sep}}x1 <   = v_{\text{sep}}x2\) and 1 \(+v_{\text{sep}}x3 <   = v_{\text{sep}}x2)))\) \((1 + v_{\text{sep}}x3 <   = 0\) and \((1 = = v_{\text{sep}}x1\) and \((v_{\text{sep}}x1 = = v_{\text{sep}}x2\) and \(1 <   = v_{\text{sep}}x2)))\) \((0 = = v_{\text{sep}}x1\) and \((v_{\text{sep}}x2 = = v_{\text{sep}}x3\) and \(1 <   = v_{\text{sep}}x3))\) ,   
+],   
+'Z3-solution': \((\mathrm{not}(v_{\text{sep}}x1<   = 1)\quad ||\quad \mathrm{not}(v_{\text{sep}}x3> = 0)\) \(\left|\left|\right|\right|\) not \((v_{\text{sep}}x2 + - 1*v_{\text{sep}}x3 > = 0)\right)\& \& (\mathrm{not}(v_{\text{sep}}x1\) \(\leftarrow 1)\) | | not (v sep x3 <= 1) | | not (v sep x3 <= 0) | | not (v sep x3 >= 2))',   
+'simp-solution': \((\mathrm{not}(v_{\text{sep}}x1<   = 2)\quad ||\quad \mathrm{not}(v_{\text{sep}}x3> = 0)\) | | not (v sep x3 <= 0) | | not (v sep x3 <= 1) | | not (v sep x3 <= -1*v sep x2 <= 0)) ',   
+'time': 0.014998197555541992   
+}   
+{ 'input1': [ \('v_{\text{sep}}x3==1+v_{\text{sep}}x2'\) \(\begin{array}{rl}{\mathrm{(96~<=~v_{sep}x1~and~v_{sep}x1<    = 100)}}&{\mathrm{~(~92~<=~v_{sep}x1~and~v_{sep}var122<    = 100)}}\\ {\mathrm{(1 + v_{sep}x2<    = 0~and~(1 + v_{sep}x3<    = 0~and~v_{sep}x3<    = 1))}}\\ {\mathrm{(0~=~v_{sep}x1~and~(v_{sep}x2~=~v_{sep}x3~and~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~v_{sep}x1~and~(v_{sep}x2~=~v_{sep}x3~and~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~v_{sep}x1~and~(v_{sep}x2~=~v_{sep}x3~and~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep} x3~and~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and ~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and ~1<    = v_{sep}x3))}}\\ {\mathrm{(~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and ~1<    }\right)}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and ~1<    }}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and ~1<    }}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(u_{sep}x2~=~u_{sep}x3~and ~1<    }}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(v_{sep}x2~=~u_{sep}x3~and ~1<    }}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(v_{sep}x2~=~u_{sep}x3~and ~1<    }}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(v_{sep}x2~=~u_{sep}x3~and ~1<    }}}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(v_{sep}x2~=~u_{sep}x3~and ~1<    }}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(v_{sep}x2~=~u_{sep}x3~and ~1<    }}\\ {\mathrm{(~~(~0~=~u_{sep}x1~and~(u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0)"))}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0)"))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0)"))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0)"))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0)"))}}} \\ {\mathrm{(~~(~0~=~u_ {sep} x3<= 0)"))}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathfrak{n}\mathbf{n}\mathbf{n})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{(~~(~0~=~u_\textit{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{(...)}} \\ {\mathrm{~~(~(~0~=~u_\textit{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x}\mathbf{x}\mathbf{\nabla})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x})}}} \\ {\mathrm{~~(~(~0~=~u_\textit{x})}}} \\ {\mathrm{~~(~(~0~=~u_\tag{$i$})}}} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=$ u_\textit{x})}} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~= \( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ {\mathrm{~~(~(~0~=\( u_\textit{x})\) }} \\ \end{array}
+\]
+```
+
+# C Experiments with CVC5
+
+In addition to Z3 and CSIsat, we also report a limited evaluation with CVC5 [3]. Technically, while CVC5 supports interpolation, it struggles with our benchmark suite. In our experiments, CVC5 failed to solve  $85\%$  of the problems within the timeout of 5 seconds. For the remaining  $15\%$ , the problems were so simple that the original interpolants already had complexity 1 and could not be further simplified.
+
+Nonetheless, we observe a significant reduction  $(-28\%$  with a conservative estimation) in solver time which is comparable to the  $30\%$  reduction observed for Z3 in Table 2. Table 5 shows the statistics on guiding CVC5 using the model trained on Z3's ground truths.
+
+Table 5: Solver performance (mean  $\pm  {95}\%$  CI) for solver time.  
+
+<table><tr><td>Method</td><td>Valid</td><td>Complexity</td><td>Solver time (s)</td></tr><tr><td>Original CVC5</td><td>15.3%</td><td>1</td><td>0.301 ± 0.116</td></tr><tr><td>Guided CVC5</td><td>7.3%</td><td>1</td><td>0.107 ± 0.031</td></tr></table>

2025/Learning Simple Interpolants for Linear Integer Arithmetic/images.zip

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2025/Learning Source-Free Domain Adaptation for Visible-Infrared Person Re-Identification/5df06d8f-439d-4c9f-9841-174282b73ab2_origin.pdf

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2025/Learning Source-Free Domain Adaptation for Visible-Infrared Person Re-Identification/full.md

@@ -0,0 +1,563 @@
+# Learning Source-free Domain Adaptation for Visible-Infrared Person Re-Identification
+
+Yongxiang Li $^{1}$ , Yanglin Feng $^{1}$ , Yuan Sun $^{2}$ , Dezhong Peng $^{1,3}$ , Xi Peng $^{1}$ , Peng Hu $^{1*}$
+
+<sup>1</sup>College of Computer Science, Sichuan University, Chengdu, China.
+
+$^{2}$ National Key Laboratory of Fundamental Algorithms and Models for Engineering Numerical Simulation, Sichuan University, Chengdu, China.
+
+$^{3}$ Tianfu Jincheng Laboratory, Chengdu, China.
+
+{rhythmli.scu,penghu.ml}@gmail.com
+
+# Abstract
+
+In this paper, we investigate source-free domain adaptation (SFDA) for visible-infrared person re-identification (VI-ReID), aiming to adapt a pre-trained source model to an unlabeled target domain without access to source data. To address this challenging setting, we propose a novel learning paradigm, termed Source-Free Visible-Infrared Person Re-Identification (SVIP), which fully exploits the prior knowledge embedded in the source model to guide target domain adaptation. The proposed framework comprises three key components specifically designed for the source-free scenario: 1) a Source-Guided Contrastive Learning (SGCL) module, which leverages the discriminative feature space of the frozen source model as a reference to perform contrastive learning on the unlabeled target data, thereby preserving discrimination without requiring source samples; 2) a Residual Transfer Learning (RTL) module, which learns residual mappings to adapt the target model's representations while maintaining the knowledge from the source model; and 3) a Structural Consistency-Guided Cross-modal Alignment (SCCA) module, which enforces reciprocal structural constraints between visible and infrared modalities to identify reliable cross-modal pairs and achieve robust modality alignment without source supervision. Extensive experiments on benchmark datasets demonstrate that SVIP substantially enhances target domain performance and outperforms existing unsupervised VI-ReID methods under source-free settings. Code is available at https://github.com/LYXRhythm/SVIP.
+
+# 1 Introduction
+
+Visible-Infrared Person Re-Identification (VI-ReID) aims to match pedestrian images captured under visible and infrared modalities, achieving consistent cross-modality correspondence across varying lighting conditions, which is particularly valuable in various applications such as nighttime security and smart surveillance [1, 2, 3, 4, 5, 6, 7]. By leveraging complementary visual cues from multiple sensors, VI-ReID systems can maintain consistent performance in both daytime and nighttime environments. However, the majority of existing VI-ReID methods rely heavily on large-scale, well-annotated datasets with aligned visible and infrared images. Constructing such datasets demands extensive human annotation and careful sensor calibration, which is labor-intensive, time-consuming, and increasingly constrained by privacy regulations. These limitations severely hinder the scalability and real-world deployment of VI-ReID systems.
+
+To reduce dependence on labeled data in the target domain, Unsupervised Domain Adaptation (UDA) has emerged as a promising alternative. UDA methods aim to transfer knowledge from a labeled
+
+![](images/ee9d38555e1431eae61b7f7c8058b1f25c4af40d1780faf9f40b992574ea1b79.jpg)  
+Figure 1: Our Observations. Despite recent advances, state-of-the-art unsupervised methods such as SDCL [11] and MMM [12] often suffer from performance limitations due to insufficient supervision in the target domain. Incorporating source-domain knowledge through labeled data (for example, UAN [13]) or pre-trained models (such as our proposed SVIP) can significantly enhance target-domain representation and adaptation. For example, we conduct experiments using SYSU-MM01 [14] as the source domain and LLCM-W under (a)  $Fog$ , (b)  $Frost$ , (c)  $Rain$ , (d)  $Snow$  conditions as target domains (see Section 4.1) to validate this observation.
+
+![](images/fdbadd8c6c6fa7ee949d66532f45dae9b14a0bb23780341c0d1d4ab6856b388d.jpg)
+
+source domain to an unlabeled target domain [8, 9, 10], achieving notable improvements over purely unsupervised methods, as illustrated in Figure 1. Despite their effectiveness, conventional UDA approaches typically assume joint access to source and target data during training. This assumption is often unrealistic in practical VI-ReID applications, where source data may contain sensitive personal information that cannot be shared or retained due to legal and ethical constraints. Moreover, the need to repeatedly load and process large-scale source datasets introduces significant computational overhead, especially in edge computing scenarios or resource-constrained environments. These issues collectively restrict the deployment of conventional UDA pipelines in real-world systems.
+
+To address these practical constraints, Source-Free Domain Adaptation (SFDA) has recently garnered attention [15, 16, 17, 18]. SFDA eliminates the need for source data during adaptation by leveraging a pre-trained source model as the sole knowledge carrier. This setting naturally supports privacy-preserving and lightweight model adaptation, aligning more closely with real-world requirements. However, existing SFDA methods are predominantly designed for unimodal tasks and do not generalize well to VI-ReID, where domain shifts and modality gaps must be addressed simultaneously. In the absence of paired visible-infrared samples or annotated target identities, cross-modal association becomes highly uncertain, often leading to noisy feature learning and suboptimal adaptation. These challenges necessitate a new SFDA framework tailored to the unique characteristics of VI-ReID, capable of learning modality-invariant and identity-discriminative representations without relying on source data or cross-modal supervision.
+
+Motivated by these observations, we propose a novel learning paradigm, termed Source-Free Domain Adaptation for Visible-Infrared Person Re-Identification (SVIP). Specifically, our contributions include: 1) the Source Guided Contrastive Learning (SGCL) mechanism, which improves supervision reliability by integrating dual clustering results from both source and target models on target domain samples. It adaptively fuses supervisory signals weighted by confidence scores to mitigate category inconsistency and label noise across modalities, enhancing the discriminative power of the target model; 2) the Residual Transfer Learning (RTL) mechanism, which introduces a feature residual distillation loss to explicitly align intermediate feature representations of source and target models under identical inputs, preserving the core discriminative structure from the source domain, thus improving structural stability and generalization; 3) the Structural Consistency Guided Cross-modal Alignment (SCCA) mechanism, which exploits the source model's structural semantic knowledge derived from paired visible-infrared images to guide the target model in mining potential cross-modal similar samples within the unpaired target domain, thereby achieving stable and reliable cross-modal feature alignment and further enhancing consistency and robustness of cross-modal representations. Collectively, SVIP effectively transfers discriminative and structural knowledge from the source model to overcome key challenges in the unsupervised target domain, including knowledge transfer difficulty, and insufficient cross-modal alignment, significantly boosting the adaptability and performance of the target model in cross-modal person re-identification.
+
+The main contributions of this work could be summarized as follows:
+
+- We propose a novel SFDA framework, SVIP, for VI-ReID that adapts a pre-trained source model to an unlabeled target domain without requiring access to any source data. To the best of our knowledge, this is among the first works exploring SFDA specifically for VI-ReID.  
+- SVIP incorporates three key mechanisms: Source Guided Contrastive Learning (SGCL) to enhance supervision reliability via confidence-weighted dual clustering; Residual Transfer Learning (RTL) to align intermediate features and preserve source discriminative structures and Structural Consistency Guided Cross-modal Alignment (SCCA) to exploit source structural knowledge for robust cross-modal feature alignment in the unpaired target domain  
+- Extensive experiments under diverse domain adaptation settings demonstrate that SVIP consistently outperforms nine state-of-the-art baselines, validating its effectiveness and superiority.
+
+# 2 Related Works
+
+# 2.1 Source-free Domain Adaptation
+
+Source-free domain adaptation (SFDA) has emerged as a crucial paradigm that addresses the limitations of conventional unsupervised domain adaptation (UDA) by eliminating the requirement to access source domain data during the adaptation process [19, 16, 15, 17, 13]. This characteristic makes SFDA particularly well-suited to practical scenarios that impose strict privacy constraints, involve decentralized data storage, or demand efficient model deployment in distributed computing environments. Existing SFDA methods can generally be divided into two broad categories. 1) Generation-based methods aim to learn domain-invariant representations by synthesizing or transforming target domain samples to approximate the distribution of source domain data. This enables the application of conventional UDA frameworks to reduce domain shifts [20, 18, 21, 17, 22]. 2) Self-training methods adopt an iterative learning strategy in which pseudo-labels generated from model predictions on unlabeled target data serve as supervision signals to progressively refine the model in a fully unsupervised manner [23, 24, 25, 26, 27]. While these approaches have demonstrated effectiveness in unimodal settings, their extension to scenarios involving cross-modal or multi-modal data remains limited. Moreover, they often fall short in fully capturing the rich semantic correlations and intrinsic structure present in heterogeneous target domains, which is a significant challenge in complex tasks such as re-identification.
+
+# 2.2 Domain Adaptation for Person ReID
+
+Person Re-Identification (ReID) faces significant challenges due to large domain shifts caused by variations in environment, camera styles, and modalities [28, 29, 30]. Unsupervised Domain Adaptation (UDA) has attracted much attention for its ability to adapt models to unlabeled target domains without manual annotation [31, 32, 33]. However, UDA methods typically require access to labeled source data during adaptation, which incurs high computational costs and raises privacy concerns, especially in ReID scenarios subject to strict data regulations. These limitations hinder the practical deployment of UDA models. SFDA addresses these issues by adapting pretrained source models to target domains without accessing source data [34, 35, 36]. SFDA aligns well with privacy preservation and deployment constraints. Nonetheless, existing SFDA methods for ReID are primarily designed for unimodal settings and often fail to exploit the intrinsic structure and semantic relationships of the target domain [36, 34], limiting their adaptation effectiveness. In addition, the absence of identity overlap between source and target domains further complicates feature transfer and undermines pseudo-label reliability. These limitations pose significant challenges in mitigating label noise, narrowing modality discrepancies, and achieving effective cross-modal alignment, all of which are crucial for advancing SFDA in ReID.
+
+# 3 Methodology
+
+# 3.1 Problem Statement and Notations
+
+For clarity, we define the key notations and terminology used in this paper. Let  $\Phi_S$  denote the source model trained on a labeled visible-infrared person re-identification (VI-ReID) dataset  $D_S$ . Additionally, let the unlabeled and unpaired VI-ReID dataset in the target domain be denoted as
+
+![](images/30101677d2a7e92a45bf9c65f7e328ef80f10e6dfc01bddb74a7743daeafd35c.jpg)  
+Figure 2: Overview of the proposed SVIP framework. SVIP integrates three core mechanisms: 1) Source Guided Contrastive Learning (SGCL) improves supervision reliability by combining dual clustering from source and target models with confidence-weighted supervision; 2) Residual Transfer Learning (RTL) aligns intermediate features between source and target models via residual distillation to preserve discriminative structures and enhance generalization; 3) Structural Consistency Guided Cross-modal Alignment (SCCA) leverages source domain structural knowledge to guide stable cross-modal feature alignment in the unpaired target domain. Together, these components enable effective source-free domain adaptation for VI-ReID.
+
+$D_{T} = \left\{\{x_{i}^{\mathcal{V}}\}_{i = 1}^{N^{\mathcal{V}}},\{x_{j}^{\mathcal{I}}\}_{j = 1}^{N^{\mathcal{I}}}\right\}$ , where  $N^{\mathcal{V}}$  and  $N^{\mathcal{I}}$  are the numbers of visible images  $x_{i}^{\mathcal{V}}$  and infrared images  $x_{j}^{\mathcal{I}}$ , respectively. For convenience, we denote  $D_{T}$  more generally as  $D_{T} = \left\{\{x_{i}^{\mathcal{P}}\}_{i = 1}^{N^{\mathcal{P}}}\right\}$ , where  $\mathcal{P}\in \{\mathcal{V},\mathcal{I}\}$ . Due to the domain shift between the source and target domains, the performance of  $\Phi_S$  can degrade significantly in practical scenarios, such as those involving environmental changes (e.g., variations in weather conditions). Our objective is to adapt  $\Phi_S$  to the target domain  $D_{T}$  and obtain a target-specific model  $\Phi_T$ , without accessing any source domain samples or ground-truth labels in the target domain. An overview of the SFDA framework for VI-ReID is illustrated in Figure 2. The main motivation of our method lies in the observation that the source model  $\Phi_S$  contains rich discriminative knowledge and cross-modal alignment capabilities. Selectively transferring this knowledge to the target domain can enhance the target model  $\Phi_T$  in terms of both discrimination and cross-modal semantic understanding, thereby overcoming the limitations of existing unsupervised methods.
+
+# 3.2 Source Guided Contrastive Learning
+
+Existing unsupervised methods typically rely on clustering to generate pseudo-labels, which provide pseudo-supervision signals for training samples. However, due to the absence of prior knowledge, the clustering results are often susceptible to modality gaps, leading to a degradation in pseudolabel quality and resulting in label noise issues [7, 37, 38, 39, 40, 41]. To alleviate this problem, we propose a Source Guided Contrastive Learning (SGCL) mechanism that leverages the prior knowledge embedded in the source model  $\Phi_S$  to guide the target model  $\Phi_T$  towards more reliable supervision signals.
+
+Specifically, SGCL adaptively integrates the dual predictions of target domain samples from both the source model and the target model based on their confidence scores. This produces a more trustworthy supervisory signal, enhancing the understanding and adaptability of the target model  $\Phi_T$  to the target domain  $D_T$ . First, we extract features of target domain samples using the source model  $\Phi_S$  and the target model  $\Phi_T$ , and apply DBSCAN clustering to obtain two sets of cluster centers, denoted as  $C_S$  and  $C_T$ :
+
+$$
+C _ {S} = \operatorname {D B S C A N} \left(\Phi_ {S} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right)\right), \quad C _ {T} = \operatorname {D B S C A N} \left(\Phi_ {T} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right)\right). \tag {1}
+$$
+
+To enhance the stability of clustering representations, we introduce two memory banks:  $M_S = \{m_k^S\}_{k=1}^{K_S}$  and  $M_T = \{m_k^T\}_{k=1}^{K_T}$ , which store the cluster center features obtained by the source and target models, respectively, in the current training cycle. Here,  $K_S$  and  $K_T$  denote the number
+
+of clusters for the source and target models, respectively. The memory banks are updated in each training cycle using the exponential moving average (EMA) strategy:
+
+$$
+\boldsymbol {m} _ {k} ^ {S} \leftarrow \eta \boldsymbol {m} _ {k} ^ {S} + (1 - \eta) \cdot \frac {1}{\left| C _ {k} ^ {S} \right|} \sum_ {i \in C _ {k} ^ {S}} \Phi_ {S} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right), \tag {2}
+$$
+
+$$
+\boldsymbol {m} _ {k} ^ {T} \leftarrow \eta \boldsymbol {m} _ {k} ^ {T} + (1 - \eta) \cdot \frac {1}{| C _ {k} ^ {T} |} \sum_ {i \in C _ {k} ^ {T}} \Phi_ {T} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right), \tag {3}
+$$
+
+where  $\eta \in [0,1)$  is the momentum coefficient, and  $C_k^S$  and  $C_k^T$  represent the sample index sets of the  $k$ -th cluster for the source and target models, respectively.
+
+Given a sample  $\boldsymbol{x}_i^P$ , we compute its similarity to all cluster centers from both models and obtain a probability distribution over clusters using the Softmax function:
+
+$$
+p _ {S} (k) = \frac {\exp \left(\left\langle \Phi_ {S} \left(x _ {i} ^ {P}\right) , m _ {k} ^ {S} \right\rangle / \tau\right)}{\sum_ {j = 1} ^ {K _ {S}} \exp \left(\left\langle \Phi_ {S} \left(x _ {i} ^ {P}\right) , m _ {j} ^ {S} \right\rangle / \tau\right)}, \tag {4}
+$$
+
+$$
+p _ {T} (k) = \frac {\exp \left(\left\langle \Phi_ {T} \left(x _ {i} ^ {P}\right) , m _ {k} ^ {T} \right\rangle / \tau\right)}{\sum_ {j = 1} ^ {K _ {T}} \exp \left(\left\langle \Phi_ {T} \left(x _ {i} ^ {P}\right) , m _ {j} ^ {T} \right\rangle / \tau\right)}, \tag {5}
+$$
+
+where  $\tau$  is a temperature parameter that controls the smoothness of the distribution. Then, the prediction confidence is estimated based on the entropy of the distributions, and a fusion weight is computed accordingly:
+
+$$
+\gamma = \frac {\exp (- H (p _ {S}))}{\exp (- H (p _ {S})) + \exp (- H (p _ {T}))}, \tag {6}
+$$
+
+where  $\pmb{p}_S = [p_S(1),\dots ,p_S(k),\dots ,p_S(K_S)]$  and  $\pmb{p}_{T} = [p_{T}(1),\dots ,p_{T}(k),\dots ,p_{T}(K_{T})]$ .  $H(\cdot)$  denotes the Shannon entropy.
+
+Considering the possible inconsistency in the number of clusters between the source and target models (i.e.,  $K_S \neq K_T$ ), directly fusing the full Softmax distributions would lead to dimensional mismatch issues. Therefore, instead of fusing the distributions themselves, we fuse the features based on the respective assigned cluster centers of the sample in each model. This design avoids dimensional inconsistency and enhances fusion accuracy. In other words, for a given sample  $\boldsymbol{x}_i^P$ , the optimization objective is defined as:
+
+$$
+\mathcal {L} _ {\mathrm {d i s}} = \frac {1}{N _ {b}} \sum_ {\mathcal {P} \in \{\mathcal {V}, \mathcal {I} \}} \sum_ {i = 1} ^ {N _ {b}} \gamma \cdot I \left(\Phi_ {S} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right), m _ {+} ^ {S}\right) + (1 - \gamma) \cdot I \left(\Phi_ {T} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right), \boldsymbol {m} _ {+} ^ {T}\right), \tag {7}
+$$
+
+where  $m_{+}^{S}$  and  $m_{+}^{T}$  denote the assigned cluster centers for the sample in the source and target domains, respectively.  $I(\cdot ,\cdot)$  represents the InfoNCE loss [42], and  $N_{b}$  is the batch size.
+
+This fusion strategy fully exploits the multi-view clustering structures of the same sample from both the source and target models. By adaptively balancing their supervision signals based on prediction confidence, it enables the target model to benefit from the prior knowledge embedded in the source model. This, in turn, mitigates the clustering noise caused by modality discrepancies and enhances the generalization and adaptability of the target model in the target domain.
+
+# 3.3 Residual Transfer Learning
+
+Although the source-guided contrastive mechanism introduced in the previous section provides stable supervision for the target model to adapt to the target domain, it remains limited in maintaining the consistency of discriminative behavior. To address this issue, we propose a Residual Transfer Learning (RTL) mechanism to explicitly align source and target domains. Inspired by the principle of preserving core discriminative knowledge in domain adaptation, this mechanism formalizes the discrepancy in discriminative behavior between the source model and the target model under the same input as a form of feature distillation constraint. This constraint encourages the target model to continuously align its intermediate feature representations with those of the source model during the adaptation process, thereby preserving structural knowledge and maintaining consistency in prediction behavior.
+
+Specifically, given a target-domain sample  $\pmb{x}_i^{\mathcal{P}}$ , the residual distillation loss is defined as:
+
+$$
+\mathcal {L} _ {\text {r e s}} = \left\| \Phi_ {S} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right) - \Phi_ {T} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right) \right\| _ {2} ^ {2}. \tag {8}
+$$
+
+$\mathcal{L}_{\mathrm{res}}$  guides the target model to retain the structural knowledge of the source model in the discriminative feature space. Compared to probability distribution alignment approaches, this method does not rely on consistency in the output space of the two models, making it more suitable for the scenarios with label space shifts or mismatched output dimensions. Therefore, it offers greater flexibility and stability. Beyond the SGCL, the RTL mechanism further strengthens the continuous guidance from the source model to the target model, thus achieving a more effective balance between generalization and adaptation capabilities.
+
+# 3.4 Structural Consistency Guided Cross-modal Alignment
+
+In an unlabeled and unpaired target domain, the absence of explicit cross-modal correspondences poses significant challenges to robust alignment. To alleviate this issue, we propose a Structural Consistency Guided Cross-modal Alignment (SCCA) mechanism, which leverages stable and reliable structural semantic knowledge from the source model to assist the target model in discovering potential semantically consistent image pairs, thereby facilitating consistent cross-modal representation learning.
+
+The source model  $\Phi_S$ , trained with supervision on paired visible-infrared images, has acquired strong capabilities in modeling cross-modal structural consistency. To transfer this capability to the unpaired target domain, we jointly exploit the feature modeling of both the source model and the current target model to estimate potential cross-modal similarities. Given the unpaired target-domain samples  $\{\pmb{x}_i^\nu\}_{i=1}^{N^\nu}$  and  $\{\pmb{x}_j^\tau\}_{j=1}^{N^\tau}$ , the structural similarity between a visible image  $\pmb{x}_i^\nu$  and an infrared image  $\pmb{x}_j^\tau$  is defined as:
+
+$$
+S _ {i j} = \frac {1}{2} \left(\left\langle \Phi_ {S} \left(\boldsymbol {x} _ {i} ^ {\mathcal {V}}\right), \Phi_ {S} \left(\boldsymbol {x} _ {j} ^ {\mathcal {I}}\right) \right\rangle + \left\langle \Phi_ {T} \left(\boldsymbol {x} _ {i} ^ {\mathcal {V}}\right), \Phi_ {T} \left(\boldsymbol {x} _ {j} ^ {\mathcal {I}}\right) \right\rangle\right), \tag {9}
+$$
+
+where  $\Phi_S$  provides structural prior guidance, and  $\Phi_T$  reflects the adaptation status of the target model. The similarity score  $S_{ij}$  integrates source-domain knowledge and target-domain modeling capacity, and is used to construct cross-modal similarity vectors  $S_i^{\mathcal{V} \rightarrow \mathcal{I}} = [S_{ij}|S_{i1},\dots,S_{iN^{\mathcal{I}}}]$  and  $S_i^{\mathcal{I} \rightarrow \mathcal{V}} = [S_{ji}|S_{j1},\dots,S_{jN^{\mathcal{V}}}]$ , which are employed to estimate pseudo matching pairs.
+
+To enhance the accuracy and robustness of the pseudo matches, two structural constraints are imposed: 1) Reciprocity Constraint  $(\mathcal{C}_1)$ : Each image pair must be the most similar to each other, i.e., Equation (10). 2) Lower-bound Similarity Constraint  $(\mathcal{C}_2)$ : The structural similarity between the image pair must exceed a predefined threshold  $T$ , i.e., Equation (11).
+
+These constraints are formally defined as:
+
+$$
+\mathcal {C} _ {1} = \left\{ \begin{array}{c c c} \underset {i \in \{1, \dots , N ^ {\mathcal {V}} \}} {\arg \max } S _ {j \in \{1, \dots , N ^ {\mathcal {I}} \}} & S _ {i j i} = i \\ \underset {j \in \{1, \dots , N ^ {\mathcal {I}} \}} {\arg \max } S _ {i \in \{1, \dots , N ^ {\mathcal {V}} \}} & S _ {j i j} = j \end{array} , \right. \tag {10}
+$$
+
+$$
+\mathcal {C} _ {2} = \left(S _ {i j} \geq T\right) \wedge \left(S _ {j i} \geq T\right). \tag {11}
+$$
+
+An image pair  $(\pmb{x}_i^\gamma, \pmb{x}_j^\mathcal{I})$  is regarded as a reliable pseudo match if both  $\mathcal{C}_1$  and  $\mathcal{C}_2$  are satisfied. Such pairs are marked using a binary indicator  $\mathcal{A}_{ij} = \mathbb{1}[(\pmb{x}_i^\gamma, \pmb{x}_j^\mathcal{I})| = \mathcal{C}_1 \wedge \mathcal{C}_2]$ . Under the guidance of the structural priors from the source model, this strategy significantly mitigates the uncertainty caused by the lack of pairing.
+
+On this basis, to further improve the compactness of cross-modal pseudo matches in the feature space, we introduce a Cross-modal Reciprocal Consistency Loss:
+
+$$
+\mathcal {L} _ {c r c} = - \frac {1}{N _ {b}} \sum_ {\mathcal {P} \in \{\mathcal {V}, \mathcal {I} \}} \sum_ {i = 1} ^ {N _ {b}} \mathcal {A} _ {i j} \log \frac {\exp \left(\left\langle \Phi_ {T} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right) , \Phi_ {T} \left(\boldsymbol {x} _ {j} ^ {\mathcal {Q}}\right) \right\rangle / \tau\right)}{\sum_ {t = 1} ^ {N ^ {\mathcal {Q}}} \exp \left(\left\langle \Phi_ {T} \left(\boldsymbol {x} _ {i} ^ {\mathcal {P}}\right) , \Phi_ {T} \left(\boldsymbol {x} _ {j} ^ {\mathcal {Q}}\right) \right\rangle / \tau\right)}, \quad \text {s . t .} \mathcal {Q} \neq \mathcal {P}, \tag {12}
+$$
+
+where  $\tau$  denotes a temperature parameter. This loss employs reliable pseudo matches as supervisory signals and utilizes contrastive learning to enhance semantic consistency between cross-modal images, thereby improving the alignment and discriminative power of cross-modal features.
+
+# 3.5 Overall Training Procedure
+
+Given that source data is unavailable, the proposed method optimizes the target model  $\Phi_T$  in each iteration using only the unlabeled data from the target domain. The entire training process is performed without access to any source domain samples, relying solely on the prior knowledge embedded in the source model  $\Phi_S$  to guide cross-modal semantic transfer. The optimization objective for the target model integrates three key components and is defined as follows:
+
+$$
+\mathcal {L} _ {\text {a l l}} = \mathcal {L} _ {\text {d i s}} + \lambda_ {1} \mathcal {L} _ {\text {r e s}} + \lambda_ {2} \mathcal {L} _ {\text {c r c}}, \tag {13}
+$$
+
+where  $\lambda_{1}$  and  $\lambda_{2}$  are weighting coefficients.
+
+# 4 Experiments
+
+# 4.1 Experiment Settings
+
+Datasets: To systematically investigate domain adaptation in VI-ReID, we conduct experiments on three widely used datasets: SYSU-MM01 [14], RegDB [43], and LLCM [44]. These datasets provide a comprehensive foundation for exploring the challenges and opportunities in cross-modal person re-identification across varying domains.
+
+Domain Adaptation Settings: To comprehensively evaluate the adaptation performance across diverse domains, we define two domain adaptation settings: 1) Basic Setting: This setting is designed to evaluate the model's ability to handle common domain shifts, including variations in illumination, camera viewpoints, and background complexity. Four adaptation scenarios are considered: (i) SYSU-MM01  $\rightarrow$  RegDB, (ii) SYSU-MM01  $\rightarrow$  LLCM, (iii) LLCM  $\rightarrow$  RegDB, and (iv) LLCM  $\rightarrow$  SYSU-MM01. SYSU-MM01 and LLCM are used as source domains, while RegDB, LLCM, and SYSU-MM01 serve as target domains accordingly. Due to its relatively small scale, RegDB is not used as a source domain. 2) Weather Setting: To further evaluate model robustness under realistic environmental variations, we introduce five weather conditions to both RegDB and LLCM, including Sunny, Fog, Frost, Rain, and Snow. Each condition is simulated at three severity levels, resulting in corrupted versions of the datasets, denoted as RegDB-W and LLCM-W. In this setting, the model trained on SYSU-MM01 is adopted as the source model, and domain adaptation is performed on each corrupted target domain. The experiment follows the standard evaluations in VI-ReID [45, 46], including Cumulative Matching Characteristic (CMC), and mean Average Precision (mAP).
+
+Implementation Details: During training, pedestrian images are resized to  $288 \times 144$  pixels. The AGW [47] network is employed as the feature extractor. The parameters of the source model remain fixed throughout training. The target model is optimized using the Adam optimizer with a weight decay of  $8 \times 10^{-4}$ . The initial learning rate is set to  $5 \times 10^{-4}$  and is reduced by a factor of 10 every 20 epochs. The batch size  $N_{b}$  is 128, and training proceeds for 50 epochs. The momentum parameter  $\eta$  is fixed at 0.1, while the temperature hyperparameter  $\tau$  is set to 0.05. The clustering algorithm DBSCAN is applied with an epsilon value of 0.6 and a minimum sample size of 4. The threshold  $T$  is maintained at 0.5. The trade-off parameters  $\lambda_{1}$  and  $\lambda_{2}$  are further analyzed in the Supplementary Material. All experiments are performed on an Ubuntu 20.04 system equipped with four NVIDIA RTX 3090 GPUs.
+
+Training of the Source Model: For the source model, our method could use the model trained by any VI-ReID method. Without loss of generality, we follow [23] and present a simple yet representative approach with the triplet loss function [45, 48] to train a source model for our experiments:
+
+$$
+\mathcal {L} _ {t r i} = \sum_ {\mathcal {P} \in \{\mathcal {V}, \mathcal {I} \}} \sum_ {i = 1} ^ {N _ {b}} \max  \left(\| \Phi_ {S} (\boldsymbol {x} _ {i} ^ {a}) - \Phi_ {S} (\boldsymbol {x} _ {i} ^ {p}) \| _ {2} ^ {2} - \| \Phi_ {S} (\boldsymbol {x} _ {i} ^ {a}) - \Phi_ {S} (\boldsymbol {x} _ {i} ^ {n}) \| _ {2} ^ {2} + \tau_ {t r i}, 0\right), \tag {14}
+$$
+
+where  $\pmb{x}_i^a \in D_S$  represents anchor in source training set.  $\pmb{x}_i^p \in D_S$  and  $\pmb{x}_i^n \in D_S$  are positive (same identity) and negative (different identity) samples of  $\pmb{x}_i^a$ .  $\tau_{tri}$  is the margin, which is set to 0.3.
+
+Table 1: Comparisons with the state-of-the-art methods under Basic Setting. R-1 (\%), mAP (\%) and mINP (\%) are reported. The best results are marked in bold, and the second-best results are underlined.  
+
+<table><tr><td rowspan="3"></td><td rowspan="3">Methods</td><td colspan="3">SYSU-MM01 → LLCM</td><td colspan="3">SYSU-MM01 → RegDB</td><td colspan="3">LLCM → SYSU-MM01</td><td colspan="3">LLCM → RegDB</td></tr><tr><td colspan="2">V2I</td><td>I2V</td><td colspan="2">V2T</td><td>T2V</td><td colspan="2">All Search</td><td colspan="2">Indoor Search</td><td colspan="2">T2V</td></tr><tr><td>R-1</td><td>mAP</td><td>R-1</td><td>mAP</td><td>R-1</td><td>mAP</td><td>R-1</td><td>mAP</td><td>R-1</td><td>mAP</td><td>R-1</td><td>mAP</td></tr><tr><td></td><td>Source Only</td><td>12.47</td><td>12.91</td><td>12.21</td><td>13.13</td><td>31.26</td><td>25.71</td><td>28.95</td><td>25.92</td><td>17.89</td><td>20.35</td><td>20.91</td><td>20.56</td></tr><tr><td rowspan="6">UVI-ReID</td><td>OTAL [49]</td><td>17.88</td><td>20.46</td><td>14.97</td><td>18.66</td><td>32.90</td><td>29.70</td><td>32.10</td><td>28.60</td><td>29.90</td><td>27.10</td><td>29.80</td><td>38.80</td></tr><tr><td>CCLNet [50]</td><td>30.33</td><td>27.12</td><td>30.11</td><td>26.67</td><td>69.94</td><td>65.53</td><td>70.17</td><td>66.66</td><td>54.03</td><td>50.19</td><td>56.68</td><td>65.12</td></tr><tr><td>PGM [51]</td><td>30.95</td><td>33.68</td><td>37.82</td><td>43.52</td><td>69.48</td><td>65.41</td><td>69.85</td><td>65.17</td><td>57.27</td><td>51.78</td><td>56.23</td><td>62.74</td></tr><tr><td>GUR [52]</td><td>31.47</td><td>34.77</td><td>29.68</td><td>33.38</td><td>73.91</td><td>70.23</td><td>75.00</td><td>69.94</td><td>60.95</td><td>56.99</td><td>64.22</td><td>69.49</td></tr><tr><td>SDCL [11]</td><td>38.86</td><td>45.19</td><td>46.34</td><td>43.35</td><td>86.91</td><td>78.92</td><td>85.76</td><td>77.25</td><td>64.49</td><td>63.24</td><td>71.37</td><td>76.90</td></tr><tr><td>MMM [12]</td><td>35.39</td><td>42.57</td><td>45.05</td><td>46.26</td><td>89.70</td><td>80.50</td><td>85.80</td><td>77.00</td><td>65.90</td><td>61.80</td><td>70.30</td><td>74.90</td></tr><tr><td rowspan="4">SFDA</td><td>CMT [53]</td><td>31.22</td><td>33.53</td><td>37.37</td><td>40.77</td><td>72.06</td><td>65.01</td><td>72.12</td><td>62.98</td><td>42.15</td><td>38.20</td><td>45.33</td><td>41.87</td></tr><tr><td>LEAD [15]</td><td>29.37</td><td>32.73</td><td>36.16</td><td>38.02</td><td>75.99</td><td>66.38</td><td>70.91</td><td>62.90</td><td>40.88</td><td>36.54</td><td>43.17</td><td>40.12</td></tr><tr><td>DRU [54]</td><td>28.48</td><td>33.80</td><td>36.55</td><td>38.85</td><td>76.26</td><td>68.54</td><td>71.68</td><td>64.43</td><td>41.72</td><td>37.89</td><td>44.65</td><td>41.03</td></tr><tr><td>SVIP (Ours)</td><td>40.50</td><td>46.20</td><td>47.85</td><td>47.30</td><td>90.10</td><td>81.20</td><td>86.50</td><td>77.80</td><td>66.80</td><td>63.50</td><td>71.60</td><td>77.40</td></tr></table>
+
+Table 2: Comparisons with state-of-the-art methods under the Weather Setting (Severity Level 3), reporting R-1 (\%) accuracy. The best results are marked in bold, and the second-best results are underlined.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Methods</td><td colspan="6">LLCM-W (Visible to Infrared)</td><td colspan="6">RegDB-W (Visible to Thermal)</td></tr><tr><td>Sunny</td><td>Fog</td><td>Frost</td><td>Rain</td><td>Snow</td><td>Avg.</td><td>Sunny</td><td>Fog</td><td>Frost</td><td>Rain</td><td>Snow</td><td>Avg.</td></tr><tr><td></td><td>Source Only</td><td>8.07</td><td>8.20</td><td>9.82</td><td>8.19</td><td>8.70</td><td>8.60</td><td>16.06</td><td>20.86</td><td>16.33</td><td>20.18</td><td>11.55</td><td>16.06</td></tr><tr><td rowspan="6">UVI-ReID</td><td>OTAL [49]</td><td>12.17</td><td>13.56</td><td>10.01</td><td>9.72</td><td>12.90</td><td>11.67</td><td>28.14</td><td>30.96</td><td>24.44</td><td>30.84</td><td>17.98</td><td>26.47</td></tr><tr><td>CCLNet [50]</td><td>23.05</td><td>25.86</td><td>21.88</td><td>27.04</td><td>19.82</td><td>23.53</td><td>42.27</td><td>46.84</td><td>36.95</td><td>45.15</td><td>27.26</td><td>39.69</td></tr><tr><td>PGM [51]</td><td>25.16</td><td>26.29</td><td>25.30</td><td>22.52</td><td>18.00</td><td>23.45</td><td>49.77</td><td>54.63</td><td>43.39</td><td>53.55</td><td>32.09</td><td>46.69</td></tr><tr><td>GUR [52]</td><td>26.39</td><td>29.61</td><td>22.89</td><td>21.76</td><td>17.68</td><td>23.67</td><td>42.12</td><td>57.87</td><td>42.41</td><td>54.09</td><td>33.13</td><td>45.93</td></tr><tr><td>SDCL [11]</td><td>26.93</td><td>31.17</td><td>27.11</td><td>27.58</td><td>23.07</td><td>27.17</td><td>50.87</td><td>61.08</td><td>49.21</td><td>64.26</td><td>39.01</td><td>52.89</td></tr><tr><td>MMM [12]</td><td>25.63</td><td>32.32</td><td>24.87</td><td>25.88</td><td>22.76</td><td>26.29</td><td>49.17</td><td>64.28</td><td>47.57</td><td>64.04</td><td>39.68</td><td>52.95</td></tr><tr><td rowspan="4">SFDA</td><td>CMT [53]</td><td>23.24</td><td>29.74</td><td>20.49</td><td>21.16</td><td>15.97</td><td>22.12</td><td>48.03</td><td>60.36</td><td>43.63</td><td>62.01</td><td>32.78</td><td>49.36</td></tr><tr><td>LEAD [15]</td><td>22.34</td><td>30.16</td><td>19.79</td><td>19.75</td><td>17.51</td><td>21.91</td><td>45.89</td><td>63.23</td><td>42.06</td><td>60.81</td><td>33.73</td><td>49.15</td></tr><tr><td>DRU [54]</td><td>20.57</td><td>31.48</td><td>23.33</td><td>22.80</td><td>18.16</td><td>23.27</td><td>46.46</td><td>64.74</td><td>43.30</td><td>61.34</td><td>39.82</td><td>51.13</td></tr><tr><td>SVIP (Ours)</td><td>28.02</td><td>36.13</td><td>32.10</td><td>32.19</td><td>27.48</td><td>30.87</td><td>54.32</td><td>65.92</td><td>49.54</td><td>67.46</td><td>46.05</td><td>57.81</td></tr></table>
+
+# 4.2 Comparison with the State-of-the-Arts
+
+We compare our SVIP method with nine state-of-the-art approaches, categorized into two groups. The first group comprises six Unsupervised Visible-Infrared Person Re-Identification (UVI-ReID) methods: OTAL [49], CCLNet [50], PGM [51], GUR [52], SDCL [11], and MMM [12], which are trained exclusively on the target domain. The second group consists of three Source-Free Domain Adaptation (SFDA) methods: CMT [53], LEAD [15], and DRU [54], which involve training a source model and performing domain adaptation following the authors' recommended procedures. Based on the results shown in Tables 1 and 2, several important observations can be made:
+
+- SVIP consistently achieves superior performance across all benchmarks under both the Basic Setting and Weather Setting, highlighting its strong generalization capability. In some scenarios, the R-1 accuracy exceeds that of the second-best method by over  $7\%$ , reflecting its robustness in handling challenging conditions.  
+- Effective exploitation of source domain knowledge plays a crucial role in enhancing the understanding of the target domain, which largely contributes to SVIP's superiority over existing UVI-ReID methods that rely solely on target domain data.  
+- SFDA baseline methods are generally constrained by the assumption of shared classes between the source and target domains, which hinders their ability to address the challenge of non-overlapping classes (identities) in VI-ReID. This limitation often results in suboptimal performance, and in certain cases, even inferior outcomes compared to UVI-ReID.
+
+# 4.3 Ablation Study
+
+We conduct comprehensive ablation studies to assess the individual contributions of the key components in SVIP, namely SGCL, RTL, and SCCA, to the overall adaptation performance. Specifically, we progressively remove each component and evaluate the resulting performance degradation on
+
+the target domain. As reported in Table 3, the exclusion of any single component leads to a notable drop in re-identification accuracy, which clearly demonstrates the importance of each mechanism in enhancing domain adaptation.
+
+Table 3: Ablation studies on Basic Setting, where RegDB dataset as target domain.  
+
+<table><tr><td rowspan="3">SGCL</td><td rowspan="3">RTL</td><td rowspan="3">SCCA</td><td colspan="3">SYSU-MM01 → RegDB</td><td colspan="4">LLCM → RegDB</td></tr><tr><td colspan="2">V2T</td><td colspan="2">T2V</td><td colspan="2">V2T</td><td>T2V</td></tr><tr><td>R-1</td><td>mAP</td><td>R-1</td><td>mAP</td><td>R-1</td><td>mAP</td><td>R-1</td></tr><tr><td>✓</td><td></td><td></td><td>85.26</td><td>75.64</td><td>82.13</td><td>72.45</td><td>84.32</td><td>76.68</td><td>81.05</td></tr><tr><td></td><td>✓</td><td></td><td>83.71</td><td>73.82</td><td>80.59</td><td>70.28</td><td>82.47</td><td>74.13</td><td>79.24</td></tr><tr><td></td><td></td><td>✓</td><td>81.95</td><td>71.36</td><td>78.43</td><td>68.17</td><td>80.63</td><td>71.89</td><td>77.52</td></tr><tr><td>✓</td><td>✓</td><td></td><td>87.39</td><td>77.25</td><td>84.16</td><td>74.83</td><td>86.21</td><td>78.04</td><td>83.17</td></tr><tr><td></td><td>✓</td><td>✓</td><td>88.17</td><td>78.06</td><td>85.04</td><td>75.91</td><td>87.35</td><td>78.97</td><td>84.26</td></tr><tr><td>✓</td><td></td><td>✓</td><td>86.82</td><td>76.43</td><td>83.27</td><td>73.85</td><td>85.74</td><td>77.32</td><td>82.39</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>90.13</td><td>81.26</td><td>86.92</td><td>77.89</td><td>90.20</td><td>81.64</td><td>86.54</td></tr></table>
+
+# 4.4 Visualization on the Domain Adaptation
+
+To systematically evaluate our method's capability in mitigating domain shift, we conduct t-SNE visualization under the Basic Setting with RegDB as target domain. As shown in Figure 3, we analyze feature distributions across four configurations using five identities (20 images each) from source and target domains. Based on the experimental results, we can make the following observations: 1) Source Only exhibits severe domain gap with separated source (red) and target (blue) clusters; 2) SGCL partially reduces domain discrepancy through confidence-weighted dual clustering but retains scattered distributions due to insufficient cross-modal alignment; 3) SGCL+RTL enhances structural stability by aligning feature residuals, though noise persists from imperfect sample filtering; 4) The full framework (SGCL+RTL+SCCA) achieves tight cross-domain clustering through SCCA-guided structural consistency constraints and confidence-aware filtering. This is clearly shown in the t-SNE visualization, where corresponding blue circles and triangles are closely clustered, reflecting accurate alignment between matched samples. The progressive improvements across different stages further validate the effectiveness of our method in establishing stable cross-modal correspondences and effectively mitigating domain shift.
+
+![](images/8c283b372386a981bd80ad2cf6468138cb90b3118585374ddd7da2c1a3f97157.jpg)  
+(a) Source Only.
+
+![](images/afd4a8d03c3da0a50c5034150634c11c66554218ce6c28f544e1c61ee264979f.jpg)  
+(b) SGCL.
+
+![](images/5a921eb6a5c8a82727e153ef4b5d2f6d79f6f27e7230ff1ce063853f10e93eaf.jpg)  
+Figure 3: The Red and Blue denote source and target domains, respectively, while  $\bullet$  and  $\triangle$  denote visible and infrared images (best viewed in color).
+
+![](images/2999e58a51e954c21728dc4ba6767967cb2df7a2811b447a534e54a3d6899f0d.jpg)  
+(c) SGCL+RTL.  
+(d) SGCL+RTL+SCCA.
+
+# 5 Conclusion
+
+This paper introduces a novel framework for source-free domain adaptation in visible-infrared person re-identification (VI-ReID), termed SVIP. Without access to source data, SVIP effectively leverages the pretrained source model through three tailored components: 1) a Source Guided Contrastive Learning (SGCL) mechanism that improves supervision quality via dual-model clustering and confidence-weighted supervision; 2) a Residual Transfer Learning (RTL) mechanism that distills transferable knowledge through intermediate feature alignment; and 3) a Structural Consistency Guided Cross-modal Alignment (SCCA) mechanism that exploits semantic structure from the source model to guide cross-modal association in the unpaired target domain. These designs collectively enhance target model adaptation under the SFDA setting. Future extensions include generalizing SVIP to other multi-modal retrieval tasks (e.g., video-text ReID), and exploring continual adaptation in streaming settings with evolving target domains.
+
+# Acknowledgments
+
+This work was supported in part by the National Natural Science Foundation of China under Grants 62472295, 62176171, U24B20174, and 62372315; in part by the Fundamental Research Funds for the Central Universities under Grants CJ202303 and CJ202403; in part by the Sichuan Science and Technology Planning Project under Grants 24NSFTD0130, 2024ZDZX0004, and 2024NSFTD0049; in part by the Chengdu Science and Technology Project under Grant 2023-XT00-00004-GX; and in part by the TCL Science and Technology Innovation Fund.
+
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+
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+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
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+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
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+# Answer: [Yes]
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+Justification: The document will release.
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+# Guidelines:
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+# 14. Crowdsourcing and research with human subjects
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+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
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+# Answer: [NA]
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+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
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+# Answer: [NA]
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+Justification: This work does not involve crowdsourcing nor research with human subjects.
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+# Guidelines:
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+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
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+# Answer: [Yes]
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+Justification: We have fully disclosed the details of the use of the adopted LLMs in our supplementary material.
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