Non-target Divergence Hypothesis: Toward Understanding Domain Gaps in Cross-Modal Knowledge Distillation

KD is a general technique for transferring knowledge from a teacher network to a student network, widely applied in various vision tasks [1, 2, 3, 4]. Although progress has been made in improving distillation techniques and exploring new application areas, research on the mechanisms underlying KD remains limited [17, 18, 19, 20]. For example, Cho et al. [18] and Mirzadeh et al. [21] investigate KD in the context of model compression, focusing on the performance of student and teacher networks when their model sizes differ. They point out that a mismatch in capacity between the student and teacher networks can lead to KD failure. Ren et al. [20] analyze KD in vision transformers and find that the inductive bias of the teacher model is more important than its accuracy in improving the performance of the student model. However, the aforementioned methods mainly focus on the factors affecting unimodal KD, while research on cross-modal KD remains limited.

With the widespread use of the internet and the increasing application of multimodal sensors, there is a surge of interest in multimodal learning [22]. In line with this trend, KD extends to cross-modal KD, which commonly applies in three scenarios. First, Knowledge Expansion: Cross-modal KD compensates for insufficient data in a single modality. For instance, Ahmad et al. [23] use multimodal MRI data to overcome clinical data scarcity, while Li et al. [24] transfer RGB-trained models to infrared data, addressing its data limitations. Similar approaches include using Galvanic Skin Response signals to offset Electroencephalogram data collection challenges [25], supplementing Synthetic Aperture Radar images with RGB data [26], and transferring English-trained models to other languages [27]. Second, Multimodal Knowledge Fusion: Cross-modal KD fuses complementary information from different modalities into a unimodal network, requiring multimodal data only during training and simplifying deployment [28], such as using Lidar for RGB images [29], RGB for point clouds [30], facial images for voice data [31], and RGB to enhance thermal imaging [32]. Large language models also aid in cross-modal retrieval [33]. Third, Additional Constraints: In some cases, significant differences in network structures for different modalities make feature layer fusion highly challenging. The loss function of cross-modal knowledge distillation then acts as a regularization term, enforcing consistency constraints on outputs to enhance the performance of individual modalities. For instance, Sarkar et al. [34] use KL divergences to align audio and video outputs, while another method, MCKD [35], applies multimodal contrastive KD for video-text retrieval. Although these methods show potential in cross-modal KD, most widely used approaches still rely on single-modal techniques, raising questions about their effectiveness and limitations. This paper, therefore, analyzes key factors influencing cross-modal KD to enhance its application.

In cross-modal KD, domain gaps critically impact performance. To reduce these gaps, modality alignment methods are often employed to unify different modalities from various domains into a common [36, 37, 31, 34, 38, 16] or intermediate modality [39]. However, this approach may result in the loss of modality-specific characteristics. To address this, some methods incorporate decoupling strategies to preserve these characteristics, such as using independent detection heads [40] for KD or employing feature partitioning to effectively transfer knowledge from the teacher modality while preserving unique features of the student modality [41]. Additionally, other approaches focus on minimizing domain gaps by filtering out features with significant discrepancies. For example, Zhuang et al. [42] and Wang et al. [43] focus on evaluating and filtering significant domain discrepancies, while Huo et al. [44] selectively filter out misaligned samples to avoid modality imbalance. Although these methods have made some progress, they primarily address multimodal feature fusion and are not directly applicable to dual-branch networks in cross-modal KD. To address this issue, this paper proposes a mask-based approach to mitigate the impact of domain shifts on cross-modal KD.

We provide a comprehensive overview of the fundamentals of KD and introduce the symbols used in this study. Although our focus is primarily on C-class classification problems, the concepts discussed are also applicable to regression tasks. To maintain the generality of the framework, we consider a two-modal setting, where the data is represented as ${{x}^{a}}$ and ${{x}^{b}}$, corresponding to the data from modalities ‘A’ and ‘B’, respectively, as shown in Eq. (1).

where $(x_{i}^{a},{{y}_{i}})$ and $(x_{i}^{b},{{y}_{i}})$ respectively represent the feature-label pair of modality ‘A’ and modality ‘B’, and ${{\Delta}^{c}}$ represents the set of c-dimensional probability vectors.

Suppose that our goal is to train a student network that takes ${{x}^{b}}$ as input. In the case of cross-modal KD, the teacher network takes the training data ${{x}^{a}}$ as input and minimizes the training objective as follows:

where ${{\mathcal{F}}_{t}}$ is a class of functions from ${{\mathbb{R}}^{{{d}_{a}}}}$ to ${{\mathbb{R}}^{c}}$ , the function $\sigma$ : ${{\mathbb{R}}^{c}}\to{{\Delta}^{c}}$ is the softmax operation

For all $1\leq k\leq c$ , the function $\mathcal{L}:{{\Delta}^{c}}\times{{\Delta}^{c}}\to\mathbb{R}$ is Kullback-Leibler(KL) Divergence [47], and $\Omega:\mathbb{R}\to\mathbb{R}$ is an increasing function which serves as a regularizer.

After training the teacher model ${{f}_{t}}$ using the data $x_{i}^{a}$ from modality ‘A’, our goal is to transfer the knowledge acquired by the teacher network to the student network operating in modality ‘B’. Therefore, in addition to minimizing the KL loss between the student output and one-hot label, it is also required to minimize the KL loss between the teacher and student outputs. The objective of optimizing the student network is as follows:

where ${{s}_{i}}=\sigma({{f}_{t}}(x_{i}^{a})/T)\in{{\Delta}^{c}}$ represents the soft predictions obtained from ${{f}_{t}}$ about the training on modality ‘A’. The temperature parameter $T$ ($T>0$) controls the level of softening or smoothing of the class-probability predictions from ${{f}_{t}}$ and the imitation parameter $\lambda\in[0,1]$ determines the balance between imitating the soft predictions ${{s}_{i}}$ and predicting the true hard labels ${{y}_{i}}$.

Given that the primary focus of this paper is the impact of domain gaps in cross-modal KD, we have listed the following assumptions conditions to control variables and exclude the interference of model capacity and modality strength. These assumptions form the foundational conditions for the discussion in this paper and the basis for the theoretical reasoning in Sec. III-C.

• •

  Assumption 1: ${\mathcal{F}_{s}}$ and ${\mathcal{F}_{t}}$ have the same model capacity, meaning they have the same ability to fit or learn complexity or accommodate information.

• •

  Assumption 2: $x_{i}^{a}$ and $x_{i}^{b}$ have the same modality strength, meaning that when the same model is trained using $x_{i}^{a}$ and $x_{i}^{b}$ as data separately, the difference in model prediction accuracy is not significant.

Under the assumption in Sec. III-A and based on VC theory [46], the upper and lower bounds of the cross-modal KD approximation error have been derived (see Sec. III-C for the omitted proof). According to this conclusion, it can be inferred that the divergence in non-target class prediction distributions between the teacher and student networks is a key factor affecting the effectiveness of cross-modal KD. This divergence stems from inherent domain discrepancies between modalities, which hinder effective guidance from the teacher to the student. This leads to the non-target divergence hypothesis.

Non-target Divergence Hypothesis (NTDH): For cross-modal KD, the performance of KD is determined by the distribution divergence of non-target classes. The smaller this divergence, the better the student network performs.

This hypothesis posits that cross-modal KD benefits from the consistency in the distribution of non-target classes. Furthermore, it explains the observation that, in certain circumstances, the performance of the teacher network is not directly correlated with the performance of the student network.

To intuitively understand our hypothesis, we use the Scikit-learn toolkit to simulate six sets of multimodal data, where $\gamma\in\left[0,1\right]$ represents the degree of domain discrepancy between modalities, with higher values indicating greater domain differences, as shown in Fig. 3. To satisfy Assumptions 1 and 2, both the teacher and student networks use the same architecture and maintain consistency in data intensity (see Sec. III-C for details). We conduct experiments on these six datasets and record the Jensen-Shannon Divergence between the teacher and student networks in both target and non-target distributions during stable training, denoted as TCJSD and NCJSD, respectively:

where ${{m}_{i}}=\frac{1}{2}\cdot\left(\hat{p}_{i}^{ta}+\hat{p}_{i}^{sb}\right)$, ${{m}_{k}}=\frac{1}{2}\cdot\left(p_{k}^{ta}+p_{k}^{sb}\right)$,

The experimental results indicate that as the domain discrepancy between modalities increases (i.e., as $\gamma$ increases from 0 to 1), the accuracy of the teacher network remains stable, while the performance of the student network declines significantly. Concurrently, there is a significant increase in the distance of the non-target distribution, whereas the distance of the target distribution remains almost unchanged, as illustrated in Fig. 3. These findings suggest that in cross-modal KD, the greater the distribution divergence of non-target classes, the lower the effectiveness of KD. In contrast, a smaller discrepancy leads to better distillation outcomes. This is a qualitative observation; further validation of these conclusions will be provided through theoretical derivations in Sec. III-C.

Recall our three actors: the student function ${{f}_{s}}\in{{\mathcal{F}}_{s}}$ (trained on $x_{i}^{b}$ ), the teacher function ${{f}_{t}}\in{{\mathcal{F}}_{t}}$ (trained on $x_{i}^{a}$ or $x_{i}^{b}$), and the real target function of interest to both the student and the teacher, $f\in\mathcal{F}$ . For simplicity, consider pure distillation, where the imitation parameter is set to $\lambda=1$.

According to VC theory [46], the classification error of the classifier, $f_{s}^{b}$ can be expressed as:

where the $O(\cdot)$ and ${{\varepsilon}_{sb}}$ terms are the estimation and approximation error, respectively. The former refers to the performance gap between a model on training data and its theoretical best performance. The latter refers to the difference between a model output and the true target function. It reflects whether the model representational capacity is sufficiently powerful to accurately approximate the true target function. If the hypothesis space of a model cannot capture the complexity of the target function, the approximation error will be large. Here, $R$ represents the error, ${{\left|\cdot\right|}_{C}}$ denotes a measure of the capacity of some function class, and $n$ represents the number of data points.

Let $f_{t}^{a}\in\mathcal{F}_{t}^{a}$ and $f_{t}^{b}\in\mathcal{F}_{t}^{b}$ be the teacher function trained on $x_{i}^{a}$ and $x_{i}^{a}$ , then:

Then, we can transfer the knowledge of the teacher separately from data ‘A’ or ‘B’ to the student. Let $f_{t}^{a}$ serve as the teacher function in cross-modal KD, and $f_{t}^{b}$ in Unimodal KD, then:

Cross modal KD

where ${{\varepsilon}_{m}}$ is the approximation error of the teacher function class $\mathcal{F}_{s}^{b}$ with respect to $f_{t}^{a}\in\mathcal{F}_{t}^{a}$ , and $\frac{1}{2}\leq\alpha\leq 1$.

Unimodal KD

where ${{\varepsilon}_{l}}$ is the approximation error of the teacher function class $\mathcal{F}_{s}^{b}$ with respect to $f_{t}^{b}\in\mathcal{F}_{t}^{b}$.

Then, if we employ cross-modal KD, combining Eqs. (8) and (10), we can obtain an alternative expression for the student learning the real function $f$, as follows:

Alternative Expression Through Cross-Modal KD

Similarly, by employing unimodal KD and combining Eqs. (9) and (11), we can obtain an alternative expression for the student’s learning of the real function $f$, as follows:

Alternative Expression Through Unimodal KD

Combining Eqs. (12) and (13), it is necessary to satisfy Eq. (14); otherwise, cross-modal KD would not outperform Unimodal KD.

Observing Eq. (14), we can see that it consists of two components: estimation and approximation error. Next, we will analyze these two parts separately.

Regarding estimation error, it is primarily determined by model capacity and data pattern strength. According to the assumptions in Sec. III-A, when the model capacity and data strength are the same, $O(\frac{{{\left|F_{t}^{a}\right|}_{C}}}{n})$ and $O(\frac{{{\left|F_{t}^{b}\right|}_{C}}}{n})$ are equivalent. Therefore, we can eliminate $O(\frac{{{\left|F_{t}^{a}\right|}_{C}}}{n})$ and $O(\frac{{{\left|F_{t}^{b}\right|}_{C}}}{n})$ from both sides of Eq. (14) without altering its outcome.

Regarding the approximation error term, it reflects the difference between the output of the neural network or the KD model and the target. In a neural network model, the model has only one objective: to minimize the difference between the network’s predictions and the one-hot encoded target. Here, ${{\varepsilon}_{ta}}$ and ${{\varepsilon}_{tb}}$ represent the approximation errors when training the neural network model under the modality $a$ and the modality $b$, respectively. Under the conditions of Assumption 2, when the intensities of the data modalities are the same, the error between the prediction results trained under modality a and modality b and the ground truth are the same. Therefore, we can further eliminate ${{\varepsilon}_{ta}}$ and ${{\varepsilon}_{tb}}$ on both sides of Eq. (14). In the KD model, the model has two objectives: first, to minimize the discrepancy between the student output and the one-hot target; second, to reduce the difference between teacher and student outputs. ${{\varepsilon}_{l}}$ represents the unimodal KD model, while ${{\varepsilon}_{m}}$ represents the multimodal KD model. For the error generated by the first objective, regardless of whether the prediction results come from the unimodal or multimodal model, they are compared with the ground truth. When both Assumption 1 and Assumption 2 are satisfied, the approximation errors of the models is the same and can therefore be offset against each other. For the error generated by the second objective, due to the different targets of the unimodal and multimodal models (the unimodal model aims to approximate the prediction results under modality $a$, while the multimodal model aims to approximate the prediction results under modality $b$), they cannot be offset against each other. For the unimodal model, if the training time is sufficient and Assumption 1 is met, the error will approach zero because the input modalities of the teacher and student networks are the same. In contrast, due to the modality differences in the multimodal model, the error is larger and cannot be ignored.

Based on the above analysis, we simplify Eq. (14) by canceling $O(\frac{{{\left|F_{t}^{a}\right|}_{C}}}{n})$ with $O(\frac{{{\left|F_{t}^{b}\right|}_{C}}}{n})$ and ${{\varepsilon}_{ta}}$ with ${{\varepsilon}_{tb}}$ on both sides, ignoring ${{\varepsilon}_{l}}$, and retaining ${{\varepsilon}_{m}}$, ultimately reducing Eq. (14) to:

Since ${{\varepsilon}_{m}}$ uses the KL divergence for optimization in cross-modal KD, and according to Gibbs’ inequality, the KL divergence is nonnegative, ${{\varepsilon}_{m}}$ must also satisfy the following:

By comparing Eq. (15) with Eq. (16), we arrive at a contradictory conclusion, indicating that the condition for cross-modal KD to outperform unimodal KD cannot hold. In other words, when both Assumption 1 and Assumption 2 are satisfied, the performance of cross-modal KD cannot exceed that of unimodal KD. Thus, we obtain the lower bound for the approximation error in cross-modal KD, which corresponds to unimodal KD. In this case, unimodal KD can be regarded as a special case of cross-modal KD, where there are no modality differences.

Next, we will further derive the upper bound of the approximation error ${{\varepsilon}_{m}}$ in cross-modal KD. According to [14], KL divergence can be decomposed into a part related to the target class distribution and a part related to the non-target class distribution, denoted by $KL({{p}^{ta}}||{{p}^{sb}})$ and $KL({{\hat{p}}^{ta}}||{{\hat{p}}^{sb}})$ respectively. The decomposed result is represented as:

where

Based on Eq. (17), the upper bound of the approximation error in cross-modal KD can be obtained as follows:

According to Eq. (18), the upper bound of the approximation error in cross-modal KD is composed of two parts: one part is determined by the distribution error of the target classes, referred to as TCKL; the other part is determined by the distribution error of the non-target classes, referred to as NCKL. Compared to TCKL, NCKL involves the probability distributions of more classes, thereby increasing its complexity and uncertainty. It can be proven that when the number of non-target classes exceeds two, NCKL becomes significantly larger than TCKL. To demonstrate this, consider the expressions for TCKL and NCKL:

The ratio of NCKL to TCKL can be derived from Eqs. (19) and (20), as shown in

To estimate $\mathrm{NCKL/TCKL}$, we perform a Taylor series expansion and approximate by neglecting higher-order infinitesimal terms. Let $\hat{p}_{i}^{sb}=\hat{p}_{i}^{ta}+{\varepsilon}_{i}$ and $p_{k}^{sb}=p_{k}^{ta}+{\varepsilon}_{t}$, where ${\varepsilon}_{i}$ and ${\varepsilon}_{t}$ are small quantities. After the Taylor series expansion, the result of $\log\left(\frac{\hat{p}{i}^{ta}}{\hat{p}_{i}^{sb}}\right)$ is as follows:

Therefore, NCKL and TCKL can be approximated by:

From Eqs. (23) and (24), $\mathrm{NCKL/TCKL}$ can be approximated by:

Based on Assumptions 1 and 2, the prediction errors of the teacher and student networks on the target class are comparable, so ${\varepsilon}_{t}$ is much smaller than ${\varepsilon}_{i}$. When $C-1$ is sufficiently large, $\mathrm{NCKL/TCKL}$ will be much greater than 1. In summary, we have proven that in cross-modal KD, when the number of classes $C$ is greater than 2, NCKL is significantly larger than TCKL. Therefore, the distribution differences among the non-target classes are the key factors that influence the effectiveness of cross-modal KD. When the distribution differences among non-target classes decrease, the upper bound of the approximation error will correspondingly reduce, thereby enhancing the effectiveness of cross-modal KD.

The five multimodal datasets encompass three types: a simulated dataset (Scikit-learn), a synthetic dataset (MNIST/MNIST-M), and three real-world datasets (RAVDESS, SemanticKITTI, and NYU Depth V2). To ensure that the experimental datasets adhere to the assumptions outlined in Section III-A and to mitigate the effects of data modality strength and model capacity on the results, we perform preprocessing. The adjustments to model capacity and modality strength are summarized in Table I.

We fine-tune the existing network models to serve as the teacher and student models for our experiments. Specifically, for the Scikit-learn, both the teacher and student networks use a 2-layer fully connected network; for the MNIST/MNIST-M, both employ a 3-layer fully connected network; for the RAVDESS, due to the differing shapes of audio and image data, both the teacher and student networks utilize a 2D convolutional network with the same number of layers, and the kernel sizes are adjusted to accommodate the different data modalities. In the SemanticKITTI, the overall network architecture proposed by [42] is adopted, with the intermediate fusion layer removed; similarly, in the NYU Depth V2, the network architecture proposed by [55] is used, and the fusion layer is also removed.

To verify the validity of our hypothesis, we design a detailed experimental plan. We employ a divide-and-conquer approach by splitting the hypothesis into two parts and validating each separately. For the first part, we use a weight adjustment method to verify that the distribution differences among non-target classes are the primary factors affecting the effectiveness of KD. This is done by adjusting the weight coefficients in the loss function that account for the distribution differences between target and non-target classes. For the second part, we apply a masking method to remove features or samples that have a significant impact on the distribution differences among non-target classes, observing whether this can improve the performance of cross-modal KD. The experimental plan is illustrated in Fig. 5, with the detailed steps as follows:

Beyond traditional KD [49], various improved algorithms have been proposed. To evaluate the generalization ability of NTDH, we apply the feature masking method to several of these improved KD algorithms, including FitNet [9], Contrastive Representation Distillation (CRD) [10], Relational Knowledge Distillation (RKD) [11], Probabilistic Knowledge Transfer for deep representation learning (PKT) [12] , Similarity-Preserving KD (SP) [13] and Decoupled Knowledge Distillation (DKD) [14].

Table II presents the experimental results on the MNIST/MNIST-M dataset. The results show that none of the methods improve student performance, indicating that KD methods designed for unimodal scenarios are not suitable for cross-modal applications. However, when we use Algorithm 1 to remove portions with significant non-target class distribution divergence, especially at smaller removal proportions (e.g., 25% and 50%), all algorithms show significant performance improvements. In contrast, random and false masking lead to substantial performance declines, highlighting the critical role of non-target class differences in cross-modal KD.

Table III presents the experimental results on the RAVDESS dataset. The results show that cross-modal KD fails to improve student performance, highlighting the limitations of these methods in cross-modal applications. However, using Algorithm 1 to remove portions with significant modal differences, particularly at lower removal proportions (e.g., 25% and 50%), leads to performance improvements in most algorithms. In contrast, control experiments with Random and False masking result in significant performance declines, underscoring the decisive role of non-target class distribution divergence in cross-modal KD.

This section further explores the practical application of NTDH. According to this hypothesis, in cross-modal KD, if the non-target class distribution difference between Teacher (a) and the student is smaller than that between Teacher (b) and the student, then we expect the student guided by Teacher (a) to outperform the student guided by Teacher (b). To reduce the non-target class distribution difference between the teacher and student networks, the method in Algorithm 2 can be applied. To validate the effectiveness of this method, we apply it to existing cross-modal KD algorithms, such as 2DPASS [57] and PMF [42], and conduct tests on the SemanticKITTI dataset.

Table IV shows the experimental results, indicating that appropriately masking samples with significant non-target distribution divergence (e.g., 25% and 50%) can improve performance. For 2DPASS, the maximum improvement reached 2.0%. By comparing the change curves of NCJSD and TCJSD distances before and after masking, as shown in Fig. 13(a), it is evident that the non-target distribution divergence significantly decreases, while the target distribution divergence changes only slightly. This confirms that the performance improvement is due to the reduction of non-target distribution differences. In contrast, the improvement in PMF is smaller because the Perception-Aware Loss in the original paper has already removed samples with large modality differences, leading to insignificant changes in NCJSD and TCJSD distances before and after masking, as shown in Fig. 13(b).