Title: A Dynamic Learning Strategy for Dempster-Shafer Theory with Applications in Classification and Enhancement
Venue: NeurIPS 2025
Accepted: yes
Keywords: Dempster–Shafer theory

Dynamic learning strategy

Adaptive diffusion probability transformation

Collaborative decision optimization
Abstract: Effective modelling of uncertain information is crucial for quantifying uncertainty. Dempster–Shafer evidence (DSE) theory is a widely recognized approach for handling uncertain information. However, current methods often neglect the inherent a priori information within data during modelling, and imbalanced data lead to insufficient attention to key information in the model. To address these limitations, this paper presents a dynamic learning strategy based on nonuniform splitting mechanism and Hilbert space mapping. First, the framework uses a nonuniform splitting mechanism to dynamically adjust the weights of data subsets and combines the diffusion factor to effectively incorporate the data a priori information, thereby flexibly addressing uncertainty and conflict. Second, the conflict in the information fusion process is reduced by Hilbert space mapping. Experimental results on multiple tasks show that the proposed method significantly outperforms state-of-the-art methods and effectively improves the performance of classification and low-light image enhancement (LLIE) tasks. The code is available at https://anonymous.4open.science/r/Third-ED16.

A Dynamic Learning Strategy for Dempster-Shafer Theory with Applications in Classification and Enhancement

Abstract

Effective modelling of uncertain information is crucial for quantifying uncertainty. Dempster-Shafer evidence (DSE) theory is a widely recognized approach for handling uncertain information. However, current methods often neglect the inherent a priori information within data during modelling, and imbalanced data lead to insufficient attention to key information in the model. To address these limitations, this paper presents a dynamic learning strategy based on nonuniform splitting mechanism and Hilbert space mapping. First, the framework uses a nonuniform splitting mechanism to dynamically adjust the weights of data subsets and combines the diffusion factor to effectively incorporate the data a priori information, thereby flexibly addressing uncertainty and conflict. Second, the conflict in the information fusion process is reduced by Hilbert space mapping. Experimental results on multiple tasks show that the proposed method significantly outperforms state-of-the-art methods and effectively improves the performance of classification and low-light image enhancement (LLIE) tasks. The code is available at https://anonymous.4open.science/r/Third-ED16 .

1 Introduction

Uncertain information is inevitably encountered in the process of modelling data-based complex systems via deep learning. Effective modelling and processing of uncertain information is an important technique that plays a key role in the decision-making process and improves the ultimate decision-making level. Currently, many methods have been proposed to solve this problem, such as evidence-theoretic methods [57] and fuzzy logic methods [10]. and deep learning-based methods [44]. These techniques have been applied in several fields, such as graph clustering [49], classification [56], and target detection [32]. However, how to effectively handle uncertain information from different sources and combine them effectively while avoiding mutual conflicts is still a challenging problem.

In machine learning, classification is the task of making predictions about new sample categories by learning the features of samples from known categories [26]. In the classification task, the reliability of the classifier plays an important role. However, owing to the uncertainty of the data itself and the possible conflict or redundancy of information from different sources, effectively and rationally addressing information from different sources to improve classification accuracy has become an urgent problem. Currently, many classical machine learning methods have been proposed. Denoeux

et al. [15] viewed the neighbors of unclassified samples as evidence for the hypothesis, with support as a function of distance between the vectors, and used Dempster's combination (DC) rule to combine the evidence for the classification task. Freund et al. [19] proposed the classification method of an alternating decision tree, which solved the problem that the original classifier is complicated and difficult to understand. Chang et al. [5] proposed a support vector machine (SVM) and an SVM with radial basis functions to implement classification problems. Xu et al. [60] utilized a normality test and normality transformation to address nonnormal data for classification tasks. Hu et al. [23] investigated Bayesian and mutual information classifiers and applied them in a classification task. Xu et al. [61] extended the classical probabilistic calibration approach to an evidence-theoretic framework when dealing with different sources of information to address the problem that a single probability measure may not adequately express uncertainty when modelling the calibration step. Xiao et al. [58] proposed weighted belief-jensen-shannon divergence for decision-making improvement on the basis of Dempster-Shafer evidence (DSE) theory. Although the above methods can address information from different sources, the importance of the information and the measurement of discrepancies are still not comprehensively considered. These shortcomings are reflected mainly in the following aspects:

In DSE theory, the current method does not consider the a priori information brought by the data itself, nor can it dynamically adjust the tendency of splitting on the basis of the importance between different subsets in the process of splitting, which means that the uncertainty information cannot be well handled.

When fusing multiple basic belief assignments (BBAs), different evidence sources conflict due to uncertainty or data distribution differences. Direct use of DC rules may lead to conflict amplification or even produce unreasonable results. Moreover, traditional methods are usually based on Euclidean space or simple statistical metrics to calculate the differences between evidence, which makes it difficult to capture the nonlinear characteristics of complex data distributions.

In practical tasks, data distributions are often unbalanced, which can lead to insufficient focus on key information in the model. In addition, current methods are unable to customize the degree of attention according to different data, and traditional training methods treat all features equally, which may lead to insufficient learning of key regions.

To address the above problems, we propose a dynamic learning strategy based on nonuniform splitting mechanism and Hilbert space mapping. First, the framework dynamically adjusts the weights of different subsets through the nonuniform splitting mechanism and uses the a priori information of the data in combination with the diffusion factor to flexibly address uncertainties and conflicts. Second, the data are mapped into the Hilbert space for computation to alleviate the information conflict problem that may occur during the information fusion process. Third, a targeted training strategy is proposed to enhance the model's ability to learn important features and regions, which achieves results in both classification and enhancement tasks. In summary, the main contributions of our work are as follows:

Wepropose a nonuniform splitting mechanism. This mechanism can be dynamically adjusted according to the importance between different subsets, giving more weight to some subsets and less weight to others. The a priori information of the data can be utilized by introducing a diffusion factor, and this splitting mechanism, which is based on a priori information, can be more flexible in addressing uncertainty and conflicting information.

We propose a scheme for fusing different BBAs. This scheme maps the data into Hilbert space for computation, which is more responsive to the differences in the true distributions of complex data. A specific way to compute the differences before fusing different BBAs is used to reduce the conflicts between different information.

We propose an effective targeted training strategy (TTS). This strategy enhances the model's ability to learn specific information and regions. Higher weights are assigned to important features to increase attention, thus improving the overall performance of the task. Accuracy is improved in classification tasks, and data imbalance is alleviated in low-light image enhancement (LLIE) tasks.

The rest of the paper is organized as follows. In section 2, related work is briefly introduced. section 3 describes the proposed method. section 4 describes the experiments and analysis of the results. Finally, section 5 provides a discussion and conclusion.

2 Related Work

2.1 Modelling of uncertain information

In classification tasks, information uncertainty often leads the classifier to make incorrect decisions. To address this problem effectively, many methods based on statistical and distance metrics have been proposed with the aim of improving classification performance by means of different mathematical models. Cover and Hart [12] proposed a method based on a distance metric function, which is inferred by selecting nearest neighbor samples. Cortes and Vapnik [11] effectively differentiate between different data distributions by constructing a separating hyperplane that can correctly divide the training dataset and has a maximum geometric interval. Xanthopoulos et al. [55] proposed discriminant analysis on the basis of a statistical approach that uses the grouping information of known samples and their corresponding multivariate variable characteristics to infer the group to which new samples belong. Quinlan et al. [41] designed a model that is based on a tree structure, where each internal node represents a judgment on a feature and each branch represents a possible output of the judgment. However, the above methods may have difficulty making accurate inferences because of their own limitations when dealing with data with conflicting or redundant information, thus affecting inference efficiency.

2.2 Evidence theory-based modelling of uncertain information

Uncertain information can be classified into either the empty set ∅ or the whole set Ω on the basis of the modelling of the uncertain information, thus addressing the information uncertainty. Zhao et al. [71] obtained the final classification results by evaluating the reliability of single and multiple sources through independent and combined reliability assessments, respectively. Liu et al. [38] combined the inferred results from multiple models and used the average as the final output. Jousselme et al. [30] introduced the distance calculation method of the similarity measure to generate more reliable inference results. Zhang et al. [69] proposed a multisource information fusion algorithm based on belief χ 2 scatter for inference tasks in complex data scenarios. However, owing to the uncertainty of the data itself and the possible conflicts or redundant information from different sources, effectively combining information from different sources by taking their importance into account remains a challenging research problem.

2.3 Deep learning-based modelling of uncertain information

Deep learning models often face the problem of decreased prediction reliability due to uncertain information. Chen et al. [8] proposed a radial basis function network learning algorithm based on orthogonal least squares to solve the underperformance problem caused by randomly selecting the centroid method to improve the performance of the task network. Castro et al. [4] investigated the problem of biased results due to data imbalance based on a multilayer perceptron (MLP) neural network and statistically improved the classification performance of the MLP. Sensoy et al. [45] used the prediction output from the network as subjective information for modelling, which in turn served as data support for the deterministic neural network to accomplish the subsequent classification task. Zaidi et al. [67] proposed two methods for automatically building a collection of different network architectures that can weigh the advantages of different structures well and use architectural variations as a source of diversity. However, these methods rely mostly on specific assumptions, and when there are multiple sources of conflicting contradictions in the input data, these models lack an effective dynamic processing framework, which may lead to bias in the final results.

3 Method

3.1 Motivation and overview

In DSE theory, when dealing with highly conflicting information, existing methods cannot adequately consider the lack of precision due to ambiguity or uncertainty in BBAs. First, the existing allocation methods based on the splitting idea assume that the allocation is based on the premise of uniform splitting, which assumes that all subsets are equally proportioned to distribute the quality and cannot be dynamically adjusted according to the quality among different subsets. Thus, the differences between different subsets cannot be adequately considered. Therefore, a nonuniform splitting mechanism is introduced to give more weight to some subsets and less weight to others. This splitting mechanism, which is based on a priori information, namely the inherent characteristics of the data's own structure and the initial evidence distribution, can handle uncertainties and conflicting information more flesxibly. The mechanism is adjustable to assign different split weights to different subsets through the diffusion coefficient. Second, conflicts arise from different evidence sources due to uncertainty or data distribution differences. To avoid the occurrence of the conflict phenomenon in the process of fusing different BBAs, high-order dynamic maximum mean difference (HODMMD) is proposed, which maps the data into the Hilbert space for computation and is more responsive to the differences in the real distributions of complex data. A specific way to calculate the difference before fusing different BBAs is used to reduce the conflict between different information. Third, in practical applications, the imbalance problem inherent in the data leads to insufficient attention of the model to key information. Therefore, an effective TTS is needed to assign higher weights to important features and increase attention, thus improving the overall performance of the task.

HODMMD

…

……

……

M1(A)

M2(A)

M3(A)

Probabilistic reliability assessment

Collaborative decision optimization

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M

M1(A)

Fusion

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……

M2(A)

M3(A)

M4(A)

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classification

Image

classification

Low- light image

enhancement

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classification

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enhancement

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M1(A)

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Build BBA

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dataset

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dataset

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dataset

Feature generation

Adaptive transformation

Adaptive diffusion probability transformation

……

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M1(A)

M2(A)

M3(A)

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Mn-1(A)

Mn(A)

Build BBA

Pattern

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Image

classification

dataset

Low-light

dataset

Feature generation

Adaptive transformation

M1(A)

M1(A)

Mn-1(A)

Mn(A)

田

田

田

O

O

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田

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田

田

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O

田

Given a set of data I , after feature extraction, different BBAs ̂ m 1 , ̂ m 2 , · · · , ̂ m n are obtained via adaptive diffusion probability transformation (ADPT) ( Tadpt ):

After that, the reliability of the different BBAs is calculated in the probabilistic reliability assessment (PRA) ( Tpra ) stage:

These reliabilities are utilized as discount factors to perform collaborative decision optimization (CDO) ( Tcdo ) operations on the BBAs, which are fused to obtain the final decision result:

where ˙ ϕ 1 , ˙ ϕ 2 , · · · , ˙ ϕ i , · · · ˙ ϕ n are discount factors. The final decision result is obtained after fusion.

3.2 Adaptive diffusion probability transformation

On the basis of the definition of DSE theory, the proposed ADPT model is as follows:

where A i , A j ∈ 2 Ω \{∅} , τ is the number of iterative splits. ˙ m τ g is τ order of ˙ m g , and ˙ m 0 g is the initial basic belief assignment (BBA). D ( A i , A j ) = ( | A i | | A j | ) 1 τ is a diffusion function used to control the quality of the distribution from subset A i to A j , which regulates the process in which the mass function is transferred from a fuzzy hypothesis A j to a more specific hypothesis A i , effectively quantifying and integrating prior information. When τ tends to ∞ , the result of D ( A i , A j ) tends to 1, and Equation (4) degenerates into a uniform distribution [25], which is no longer affected by the importance of different regions. | · | is the number of elements contained, Ω = { θ 1 , θ 2 , θ 3 , . . . , θ N } is the frame of discernment, and 2 Ω = {∅ , { θ 1 } , { θ 2 } , . . . , { θ N } , { θ 1 ∪ θ 2 } , . . . , Ω } is the power set. After normalization processing,

3.3 Probabilistic reliability assessment and its properties

When different quality functions are obtained, if they are simply combined according to the DC rule [48], the integration between different BBAs may be hindered because of conflicting information. For this reason, the idea of discounting techniques [46] is introduced. Defined HODMMD:

where H is the regenerated kernel Hilbert space and the kernel function is computed via a Gaussian kernel function. The properties of the HODMMD are as follows.

Property 1. When τ →∞ , the HODMMD is equivalent to measuring the difference between the pignistic probability transformations Bet P with ̂ m 1 and ̂ m 2 of the maximum mean difference.

Property 2. When ̂ m 1 and ̂ m 2 degenerate into probability distributions, that is, U = ( u 1 , u 2 , . . . , u N ) and V = ( v 1 , v 2 , . . . , v N ) , the proposed HODMMD degenerates into a maximum mean difference.

Property 3. HODMMD τ ( ̂ m 1 , ̂ m 2 ) and HODMMD τ ( ̂ m 2 , ̂ m 1 ) are equivalent.

Property 4. When ̂ m 1 = ̂ m 2 , the value of HODMMD is always equal to 0.

Proofs of these properties can be found in the technical appendix.

3.4 Collaborative decision optimization

When ADPT is used to obtain multiple BBAs, the reliability of different BBAs needs to be calculated to measure the impact of different sources of evidence. Let ϕ = [ ϕ 1 , ϕ 2 , ..., ϕ K ] be the reliability of

Input:

Training data X training and testing set X testing .

Output:

Category probabilistic decision results

1

for

i

= 1

to

K

do

2

Generate values on the basis of the data attribute characteristics of

X

training

3

end

4

Reliability of different classifiers obtained via decision optimization scheme Eq. (11)

5

Calculate the discount factor via Eq. (12) and normalize it

6

for

i

= 1

to

K

do

7

Obtain the results of

K

classifiers

8

Discounting the different classification results via Eq. (13)

9 Fuse different BBAs via Eq. (14)

10

Test the

k

th basic classifiers

11

end

12 Use the decision results for subsequent tasks

different query patterns or BBAs that satisfy ϕ k ∈ [0 , 1] and ∑ K 1 ϕ k = 1 . To measure the reliability of the different BBAs, the HODMMD is used for the calculation:

where l is the index of the different query patterns. G l = [ G l (1) , G l (2) , ..., G l ( K )] is the ground truth. m l k is the possibility of the query pattern belonging to the class θ i . ̂ m l k = [ ̂ m l k (1) , ̂ m l k (2) , . . . , ̂ m l k (Ω)] , which also satisfies m l k ∈ [0 , 1] and ∑ 2 N -1 k =1 m l k = 1 . The reliability vector that minimizes the BBA error is calculated via sequential least squares programming (SLSQP). Therefore, the discount factor for the k th BBA is defined as:

According to the idea of discounting techniques [46], the discounted BBA is as follows:

Fusing of different BBAs according to the DC rule:

The probabilities of different patterns obtained after fusion are prepared for subsequent classification and LLIE tasks. In LLIE tasks, the image is divided into different image blocks and then input into the proposed decision framework T rans to obtain the degradation degree representation of different features for each image block. The lower the quality is, the greater the weight assigned to the region. Given low-light image I low and normal light image I normal ,

where I pro represents the representation of the degree of degradation in different regions of the image. Next, the LLIE task can be formalized as:

where Loss is the loss function of the original network and where T net ξ is the network with parameter ξ , · represents the dot product. During this process, the network focuses more intensely on regions of the image that have undergone greater degradation, effectively integrating an understanding of the image content into the learning process. The above process of our method leads to Algorithm 1, which contains three modules: the ADPT, PRA, and CDO of classifiers. These three modules are co-optimized to work together. In this process, initial values are first generated on the basis of the features of the training data. Next, adaptive targeted iterations are performed via ADPT to achieve the generation of BBAs under different classifiers. After that, the reliability of each classifier is calculated as a discount factor using HODMMD. Finally, fusion is performed with the help of a discount factor to obtain the final decision.

4 Experimental Results

4.1 Numerical experiment

To better understand the working mechanism of the ADPT, in this section, a discussion of the proposed ADPT is presented. by several concrete examples.

Split 0

Split 1

Split 3

Split 5

10%

20%

30%

40%

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60%

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The value of

iteration of

m

1({

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})

m

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1({

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m

1({

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2})

Example 5.1 Let a frame of discernment be Ω = { θ 1 , θ 2 } . The BBA is as follows:

When split once, when τ = 1 is substituted into Equation (4), three scenarios exist: m 1 g ( { a 1 } ) = 0 . 3 , m 1 g ( { a 2 } ) = 0 . 5 , and m 1 g ( { a 1 , a 2 } ) = 0 . 2 . When the second, τ = 2 , is substituted into Equation (4), there can be three scenarios: m 2 g ( { a 1 } ) = 0 . 3741 , m 2 g ( { a 2 } ) = 0 . 5839 , and m 2 g ( { a 1 , a 2 } ) = 0 . 0420 . The process was plotted as a radar chart and a line graph, and the results are shown in Figure 2. The results show that as τ increases, the values of the mass function belonging to { θ 1 , θ 2 } gradually shift to the mass functions of { θ 1 } and { θ 2 } . The mass function with a larger set of initial values still maintains a larger weight.

Example 5.2 Let a frame of discernment be Ω = { θ 1 , θ 2 } . The BBAs are as follows:

The variation of the proposed HODMMD is further explored by changing the values of x and y and plotting Figure 3. As τ increases, the difference between ̂ m 1 and ̂ m 2 gradually decreases, verifying that the uncertainty information of the BBAs gradually decreases with increasing ADPT.

x

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4.2 Pattern classification

For each dataset, the same testing method as in [58] was adopted to perform fivefold cross-validation according to a training set-to-test machine ratio of 4:1. The classification accuracy was evaluated, and then all the results were averaged to arrive at the final result, which ensures a fair comparison between the different methods. The experimental results are shown in Table 1. The proposed method achieves the optimal performance except for the Parkinsons dataset because the features of the different samples in this dataset are too close to each other, resulting in poor differentiation of importance when performing ADPT. On the basis of the evaluation metric of classification accuracy, improvements of 2.67%, 11.9%, 1.48%, 1.66% and 0.22% are obtained over those of the advanced DMA method on

the Iris, Heart, Hepatitis, Australian and Segment datasets, respectively. This is because our method better handles the contribution of different features to the classification and better solves the conflict problem when fusing different BBAs.

Dataset

Iris[18]

Heart[27]

Hepatitis[1]

Parkinsons[36]

Australian[42]

Segment[2]

CBench [16]

NaB[23]

94.67%

82.59%

76.76%

68.75%

79.56%

80.22%

67.68%

95.33%

57.78%

65.71%

83.02%

67.40%

96.93%

99.02%

REPTree[19]

92.00%

70.74%

71.64%

80.94%

80.59%

95.11%

65.45%

SVM[5]

94.67%

83.70%

79.96%

70.13%

80.29%

64.50%

88.48%

SVM-RBF[5]

94.67%

82.96%

76.76%

81.03%

79.86%

81.73%

68.18%

MLP[4]

93.33%

75.19%

74.93%

74.39%

82.32%

95.93%

80.40%

RBFN[8]

92.67%

81.85%

81.32%

82.05%

82.61%

87.58%

83.64%

95.33%

76.30%

80.57%

78.01%

78.41%

93.37%

94.42%

NDC[60]

94.00%

82.59%

79.40%

70.26%

80.01%

79.61%

43.13%

EvC[61]

94.67%

83.70%

79.88%

81.64%

80.60%

95.90%

76.33%

DMA[58]

96.00%

84.07%

83.04%

75.03%

84.14%

99.74%

100.00%

ours

98.67%

95.97%

84.52%

82.50%

85.80%

99.96%

100.00%

4.3 Image classification

For the image classification task, we examine the performance on the CIFAR-10 and CIFAR-100 datasets, which contain 10 and 100 categories, respectively. To verify the effectiveness of the proposed fusion framework, which is based on a nonuniform splitting mechanism and Hilbert space mapping more comprehensively, we perform feature extraction via ResNet-18 and the CNN framework proposed in [48] before adopting the proposed method for decision making. In this task, prior information is the initial probability distribution obtained by performing feature extraction on the input samples for the CNN and ResNet-18, with the last linear layer removed. The comparison results of the different methods are shown in Table 2. Our method achieves better results, which once again validates the effectiveness of the proposed method in addressing uncertain information.

Methods

Architecture

Methods

Architecture

DIR-Net [40] IJCV'2023

ResNet-18

92.80%

Dspike[34] NeurIPS'2021

ResNet-19

73.12%

MST[51] ICCV'2023

ResNet-18

93.20 %

GLIF[66] NeurIPS' 2022

ResNet-19

77.05%

ReSTE[54] ICCV'2023

ResNet-18

92.63 %

Diet-SNN[43] TNNLS'2021

VGG-16

69.67%

ADMM[9] TIP'2023

ResNet-18

95.40 %

ResNet-14

72.63%

SML[14] ICML'2023

ResNet-19

95.12 %

MPBN [22] ICCV'2023

ResNet-19

74.40%

UDSP[20] CVPR'2024

ResNet-56

93.78%

MS-ResNet[24] ICCV'2023

MS-ResNet18

75.39%

ResNet-20

93.75%

BiPer [50] CVPR'2024

ResNet-18

93.75%

BKDSNN[63] ECCV'2024

ResNet-19

74.95%

VGG-small

92.11%

TAB[28] ICIL'2024

VGG-9

93.41%

TAB[28] ICIL'2024

VGG-11

76.31%

APL[70] TPAMI'2023

ResNet-18

96.00%

APL [70] TPAMI'2024

ResNet-18

78.90%

ESNN[47] EAAI'2025

VGG-16

93.55%

ESNN[47] EAAI'2025

VGG-16

76.55%

CNN

95.67%

CNN

79.71%

CIFAR-10

ours

ResNet-18

95.61%

CIFAR-100

ours

ResNet-18

79.78%

4.4 Low-light image enhancement

In LLIE, dark areas may contain critical information that is often difficult for models to adequately learn and focus on [35]. Traditional training strategies, which apply uniform processing across the entire image, are inherently limited in their ability to specifically enhance model learning for these critical regions. This often results in suboptimal detail recovery in the target areas during enhancement.

Therefore, a TTS is proposed in conjunction with the proposed method, which explicitly guides the model to increase the level of attention to low-quality regions. The method is a plug-and-play module. In this task, prior information refers to the degree of degradation of different regions in the image. We used the LLIE network of the last few years as a baseline network, and the results are shown in Table 3. A consistent improvement in performance can be seen after using the TTS. To obtain a more intuitive sense of the enhancement, we present the results of the baseline method and our method, and the results are shown in Figure 4. Our method effectively improves the quality of images. For the LOL-v1 dataset, our method enhances the texture details on the glass; for the LOL-v2-real dataset, our method recovers the text on the wall more clearly; for the LOL-v2-real dataset, our method recovers the color of the flower petals closer to the ground truth. This finding verifies the validity of the proposed method as well as the idea and further shows that the proposed ADPT can characterize the data well. As seen through the above experiments, applying different learning strategies to different regions can effectively enhance the model's focus on the target region. By adaptively adjusting the learning weights of different regions, the model's ability to focus on low-quality regions is enhanced, the data distribution imbalance problem is alleviated, and the overall performance of the model is improved. To the best of our knowledge, this is the first time that DSE theory has been introduced into LLIE.

More experiments can be found in the technical appendices.

5 Conclusion

This paper proposes a dynamic learning strategy based on nonuniform splitting mechanism and Hilbert space mapping, which is based on DSE theory, as an efficient method for processing uncertain information. The current method cannot dynamically adjust the tendency of splitting on the basis of the importance between different subsets in the process of splitting, and directly using the DC rule in fusion will produce unreasonable results owing to the conflict problem. First, the nonuniform splitting mechanism proposed in this paper takes the data's inherent a priori information into account and thus can handle uncertainty and conflict information more flexibly while improving the accuracy of the task. Second, mapping the data into the Hilbert space for computation is more responsive to the differences in the true distributions of complex data, thus providing an effective strategy for the subsequent fusion of different BBAs. We conducted experiments on multiple tasks as well as multiple publicly available datasets. The experimental results show that our method significantly outperforms existing machine learning methods and deep learning methods. To the best of our knowledge, we are the first to introduce DSE theory to LLIE and provide effective performance enhancement for LLIE tasks.

(a) GT/Low

(b) SNR

(c) LLFlow-L

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A Appendix/supplemental material

A.1 More Details of Preparatory Knowledge

A.1.1 Dempster-Shafer evidence theory

Dempster-shafer evidence theory is a well-established general framework for uncertainty reasoning that was first proposed by Arthur P. Dempster [13] on the basis of statistical inference and later formalized and significantly extended by Glenn Shafer into a framework for simulating epistemic uncertainty[46]. Owing to its effectiveness in dealing with uncertain information, DSE theory is widely used in various fields.

Definition 1. Frame of discernment

Define a set of classes called discriminant frames:

where θ i ( i = 1 , 2 , ..., N ) are mutually exclusive. The power set is defined as:

where Ω is the whole set, ∅ is the empty set and the one containing ∪ is a multielement set [13, 46].

Definition 2. Basic belief assignment

The basic belief assignment (BBA), also known as the mass function, denoted m ( · ) is defined as a mapping of 2 Ω to the interval [0 , 1] [13, 46].

and is satisfied:

where A ∈ 2 Ω \{∅} , ∅ is the empty set.

Definition 3. Focal set

If m ( A ) > 0 , A is called a focal set, and the value of m ( · ) indicates the level of support of the model[13, 46].

Definition 4. Dempster's combination rule

Different fusion algorithms can be used to fuse information from different sources, one of which is Dempster's combination rule. As it can process the fusion of different sources of evidence represented by a BBA, it is assumed that m 1 and m 2 are two mutually independent BBAs defined on the same recognition frame 2 Ω . Dempster's combination rule aims to derive a combined BBA, usually denoted as m = m 1 ⊕ m 2 . The combination frame is defined as follows[13]:

where A i , A j , A k ∈ Ω and R are the conflict coefficients of m 1 and m 2 .

A.1.2 Probability transformation methods

Definition 5. Pignistic probability transformation

Let Ω = { θ 1 , θ 2 , θ 3 , . . . , θ N } denote the discriminant framework. Given a corresponding BBA defined on Ω , denoted m ( · ) , the pignistic probabilistic transformation of an element θ i ∈ Ω , denoted P ppt , is defined as follows:

where A belongs to 2 Ω , A = ∅ . | A | is the cardinality of A .

̸

Definition 6. Plausibility transformation method

Let Ω = { θ 1 , θ 2 , θ 3 , . . . , θ N } denote the discriminant framework. Given a corresponding BBA defined on Ω , denoted m ( · ) , the plausibility transformation of an element θ i ∈ Ω , denoted P pt , is defined as follows:

where M ( · ) is the plausibility function:

A.2 More Proof Details for Properties

When different quality functions are obtained, if they are simply combined according to Dempster's rule [48], the integration between different BBAs may be hindered because of conflicting information. For this reason, the idea of discounting techniques [46] is introduced. The high-order dynamic maximum mean difference (HODMMD) is defined as follows:

where H is the regenerated kernel Hilbert space and the kernel function is computed via a Gaussian kernel function, A i ∈ 2 Ω \{∅} , and τ is the number of iterative splits. ̂ m τ g is τ order of ̂ m g . The HODMMD has a variety of properties. The properties of the HODMMD and the corresponding proofs are as follows.

Property 1. When τ →∞ , the HODMMD is equivalent to measuring the difference between the pignistic probability transformations with ̂ m 1 and ̂ m 2 of the maximum mean difference.

Proof. We denote by Bet P t the pignistic probability transformations of ̂ m 1 and ̂ m 2 , t = 1 , 2 . That is,

If A i = θ i , then the cardinality of A i is 1. Thus, when τ →∞ and | A i | = 1 ,

Let

where A i is a subset of A j . When | A i | = 1 , there must be | A j | ≥ 1 ; hence, | A j | -1 ≥ 0 and τ | A j |-1 > 0 . The numerator and denominator are equally divisible by τ | A j |-1

Let ρ = 1 , when τ →∞ , have 1 → 0 , which is ρ → 0 . A Taylor expansion of ( ρ +1) | A j |

where o ( ρ ) is the infinitesimal of ρ , have

Therefore, when τ →∞

Thus,

Property 2. When ̂ m 1 and ̂ m 2 degenerate into probability distributions, that is, U = ( u 1 , u 2 , . . . , u N ) and V = ( v 1 , v 2 , . . . , v N ) , the proposed HODMMD degenerates into a maximum mean difference.

Proof. When ̂ m 1 and ̂ m 2 are probability distribution,

therefore,

Property 3. HODMMD τ ( ̂ m 1 , ̂ m 2 ) and HODMMD τ ( ̂ m 2 , ̂ m 1 ) are equivalent.

Proof.

Thus,

Property 4. When ̂ m 1 = ̂ m 2 , the value of HODMMD is always equal to 0.

Proof. When ̂ m 1 = ̂ m 2

A.3 Broader Impacts

This paper proposes a dynamic learning strategy based on nonuniform splitting mechanism and Hilbert space mapping to promote real-world applications. We apply the proposed method in pattern classification, image classification, and low-light image enhancement encountered in real life, which encourages research on their synergistic combination in real life. In addition, we are the first to introduce DSE theory to low-light image enhancement and achieve effective performance enhancement for low-light image enhancement tasks. This provides a new idea for the low-light image enhancement task. As far as this paper is concerned, we believe that the proposed method does not have any significant negative impact.

A.4 Additional details regarding the experiment

A.4.1 Dataset and experimental settings

To rigorously evaluate the performance of the proposed method, we conducted extensive experiments and applied the proposed algorithm in three experiments. For pattern classification, experiments were conducted on the Iris [18], Heart [27], Hepatitis [1], Parkinsons [36], Australian [42], Segment [2], and Connectionist Bench (CBench)[16] datasets, and the details of these datasets are shown in Table 4. In this evaluation phase, we compare the proposed method with a class of classical classifiers: a Bayes theorem-based classifier (NaB) [23], a k-nearest neighbor classification method (kNN) [12], a decision tree algorithm (REPTree) [19], a support vector machine classifier (SVM) [5], an SVM method with a radial basis function (SVM-RBF) [5], a multilayer perceptron method (MLP) [4], and a radial basis function network (RBFN) [8]. Another class is evidence theory-based classifiers: a DS theory-based kNN method (kNN-DST) [15], a data probability distribution-based method (NDC) [60], an evidence calibration method (EvC) [61], and a generalized divergence-based decision-making method (DMA) [58].

For image classification, experiments were conducted on the publicly available datasets CIFAR10[31] and CIFAR-100[31], and the details of these datasets are shown in Table 5. Furthermore, performance comparisons were made with current state-of-the-art methods, including DIR-Net [40],

In addition, we validated the effectiveness of the proposed method on a low-light image enhancement task and tested it on the publicly available datasets LOL-v1 [53], LOL-v2-real [65] and LOLv2-syn [65]. The allocation of the datasets and related information is shown in Table 6. In this evaluation phase, our method was compared with several state-of-the-art low-light image enhancement methods, including MIRNet[68], FIDE[59], ZeroDCE[21], Sparse[53], DRBN[64], RUAS[37], ZeroDCE++[33], SCI[39] and Restormer[29]. In addition, to realistically demonstrate the superiority of our method, methods from recent years were selected as the baseline network, and only targeted training strategies were added to the original methods, including SNR [62], LLFlow-L [52], LLFlowS [52] and Retinexformer [3]. For fairness, the parameter settings were kept the same as those in the original method. All the experiments were run on NVIDIA RTX 3090 GPUs.

A.4.2 Extended ablation studies

To verify the indispensability of the ADPT and HODMMD modules, we perform ablation studies on the image classification task of the CIFAR-10 dataset with the architecture of ResNet-18, and the results are shown in Table 7.

The above experiments indicate that the model performance decreases to 94.56% when the ADPT module is removed. This shows that ADPT can use the prior information of the data to better address uncertainty information through nonuniform splitting, thereby significantly improving the classification accuracy. The accuracy of the model decreased by 2.06% after the HODMMD module

was removed. The accuracy of the model decreased by 1.3% when the Euclidean distance was used to replace the HODMMD. This finding indicates that our method can more accurately capture the nonlinear difference between complex data distributions, thereby more effectively assessing the conflict between evidence and more reliable fusion decisions.

A.4.3 Hyperparameter sensitivity experiment

To better compare model performance, we have supplemented the experiments with analyses of key hyperparameter sensitivity and model complexity. The numerical experiments are as follows.

By fixing the kernel bandwidth σ = 0 . 5 , we tested the variation in HODMMD in the range of [1,6], and the variation values of the mass function and HODMMD were plotted as a line graph. As τ increases, the value of the mass function of the uncertainty information gradually decrease is assigned to the mass function represents a single category, and the HODMMD value also decreases. In addition, τ = 1 was fixed, and the variation of HODMMD in the range of [0.05, 10] for σ was tested. When σ is between [0.05, 0.1], the HODMMD rapidly decreases, and when σ is between [0.1, 2], it tends to increase and eventually stabilizes.

Second, in the classification and LLIE tasks, τ controls the degree of nonuniform splitting. We tested the effect on task accuracy when τ was 1,2,3,4,5,6. The results show that when τ = 5 , the model performance remains stable and optimal. Similarly, for σ , we took the interval as 0.1 to test the effect on task accuracy between [0.1, 2]. The results show that when σ = 0 . 5 , the model performance remains stable and optimal. In summary, we fix τ = 5 and σ = 0 . 5 .

A.4.4 Analysis of model complexity

We have added parameter (M), FLOPs (G) and FPS comparisons with the baseline method on the low-light image enhancement task. The results are shown in Table 8.

The results indicate that the complexity and time of the model both increase after the introduction of the TTS. However, considering the improvement in model performance, this increase is within the acceptable range, which proves the actual application efficiency of our method.

A.4.5 Large dataset experiment

The experiments on the SID and SMID datasets to verify the effectiveness of the TTS by using the Retinexformer as the baseline networks.The results are shown in Table 9.

A.5 Discussion of Our Method with Machine Learning Methods and Deep Learning Methods

Although different methods are currently available to address decision-making problems, our method has the following advantages over machine learning methods and deep learning methods.

First, our approach has a better ability to handle uncertainty. Machine learning and deep learning usually produce only a single probability estimate and make predictions under the assumption of mapping relationships. This approach does not model uncertainty well in the face of conflicting or uncertain information. However, our method directly models uncertainty through BBA, which allows assigning confidence to composite propositions. Moreover, it can be dynamically adjusted according to the level of importance between different pieces of information, capturing the nonspecificity of the information and quantifying the discrepancies between pieces of evidence in a more flexible way.

Second, our method can fuse information from multiple sources in a more rational way. Machine learning and deep learning approaches tend to use network layers for feature learning and prediction and lack interpretability for information fusion decisions. However, our method can understand the reasons behind the decisions well. For example, the proposed method maps data into Hilbert space for computation, which is more responsive to the differences in the true distributions of complex data, reduces conflicts between different types of information, and is highly interpretable.

Third, our approach remains applicable in the presence of limited data. The training process of deep learning suffers from overfitting on small or unrepresentative data and relies on training complex models. However, our method's splitting mechanism, which is based on a priori information, can be more flexible in dealing with uncertainty and conflicting information and can work effectively with limited data.

We propose a dynamic learning strategy based on nonuniform splitting mechanism and Hilbert space mapping enhances the interpretability of the decision. This will promote the application of deep learning technology in a wider range of fields. In addition to classification and low-light image enhancement, our ideas can be applied in all uncertainty tasks, for example, image segmentation (uncertainty of the object boundary) and automatic driving (uncertainty in the fusion of multisensor information). Owing to time constraints, this work requires a large amount of computing resources and time. We will continue to explore the performance of this method in other fields.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper aims to address the limitations in model performance that arise from insufficient prior information, imbalanced data distribution, and conflicting evidence fusion in Dempster-Shafer Evidence Theory (DSE). The main contribution of this work is a novel learning strategy that combines a non-uniform splitting mechanism with Hilbert space mapping. This approach enables prior knowledge fusion through dynamic basic belief assignments（BBAs） guided by data-driven diffusion factors, resolves nonlinear conflicts using Hilbert space feature mapping, and incorporates targeted training strategies to enhance the learning of critical features. Ultimately, it facilitates effective modeling of uncertain information, thereby improving performance in both classification tasks and image enhancement applications. Experimental results show that the proposed method outperforms existing approaches in these domains and provides new insights into the modeling of uncertain information.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: 1. This article proposes a non-uniform splitting mechanism that dynamically assigns weights based on the prior information of different subset importance to handle uncertainty and conflict information, achieving more efficient and flexible uncertainty modeling. 2. The high-order dynamic maximum mean difference (HODMMD) is used to map the data to Hilbert space for calculation, to reducing the conflict \nduring the fusion process of different BBAs. 3. The method can be used as a plug-and-play module for focusing on important areas and has been validated in the field of low light image enhancement. \nWeaknesses: 1. The paper provides a solid theoretical foundation but lacks sufficient experimental validation. 2. The organization and presentation of content could be improved to enhance readability and logical flow. 3. Several parts of the paper suffer from unclear expressions, especially in the descriptions of the methodology and experimental procedures. These sections need clarification and refinement for better understanding.\n\nQUESTIONS:\n1. There is some confusion in the content presentation. The proposed method is referred to as a \"dynamic learning strategy\" in the section title, a \"fusion framework\" in the abstract, and a \"learning guidance strategy\" in the main text. Additionally, the plug-and-play descriptions of LGS and ADPT are somewhat ambiguous and may cause confusion. Consistent terminology should be used for clarity.\n2. key implementation details—such as the Gaussian kernel bandwidth in HODMMD and iteration settings in ADPT—are insufficiently described; clarification on whether they are empirically or systematically determined is needed.\n3. The third contribution mentioned in the paper is the design of a targeted training strategy. It is, however, unclear whether this refers to the collaborative decision optimization described in Section 3.4 or the loss function introduced in Section 4.4. If the loss in Section 4.4 is indeed part of the proposed method, it should be moved to Section 3 for better organizational clarity.\n4. Compared to other methods such as DMA, this work takes into account the importance and differences among information sources in its relevance measurement. However, the experimental validation lacks a comparison with existing methods in terms of uncertainty modeling. Additionally, there is insufficient comparative analysis — such as against the Dempster's rule or Euclidean distance — for BBAs fusion. Further experiments are needed to substantiate the effectiveness of HODMMD.\n5. It would be helpful to evaluate the method on larger datasets (e.g., SID, SMID) and compare it with recent models like Retinexformer, especially for important region focusing. In addition, its performance in other domains is worth exploring.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a sophisticated information fusion framework based on evidence theory, designed to address the challenges of processing uncertain and conflicting information. The method presents theoretical novelty by incorporating a non-uniform partitioning mechanism (ADPT) for dynamic weight allocation and a Hilbert space metric (HODMMD) to effectively resolve evidence conflicts. The authors successfully apply this framework to tasks such as pattern classification, image classification, and low-light image enhancement, with the latter application demonstrating the potential of using evidence theory as a “learning guidance strategy,” offering a novel perspective on solving practical problems.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. The framework ingeniously connects abstract evidence theory with concrete application needs. A key example is its use in low-light image enhancement, where the complex theory is transformed into a practical, plug-and-play “learning guidance strategy” (LGS). This cross-domain application highlights the flexibility and practical value of the proposed ideas.\n2. A core innovation lies in the dynamism of the method. Through the adaptive diffusion probability transformation (ADPT) mechanism, the framework moves beyond the static or uniform assumptions of traditional methods regarding information importance. This allows the model to more intelligently handle uncertainty by leveraging the data’s own prior information and the importance of different subsets.\n3. By introducing the kernel-based High-Order Dynamic Maximum Mean Discrepancy (HODMMD), the method surpasses traditional Euclidean-based metrics for handling evidence conflicts. It is better equipped to capture non-linear dissimilarities in complex data distributions, thereby enabling a more accurate assessment and mitigation of conflicts before fusion.\n\nWeakness\n1. The paper frequently mentions leveraging the “prior information” of the data. However, it remains unclear what this “prior information” specifically refers to in the context of different tasks (e.g., image classification vs. LLIE). The authors are requested to provide a clearer, more concrete explanation of how this information is defined, quantified, and integrated into the model for each application.\n2. The proposed framework is composed of several novel components (ADPT, HODMMD, etc.). A significant weakness is the lack of ablation studies. Without them, it is impossible to discern the individual contribution of each component to the overall performance improvement. The authors should conduct thorough ablation experiments to demonstrate the effectiveness of each proposed module.\n3. There is insufficient discussion on the model’s complexity and its sensitivity to key hyperparameters. For instance, the selection criteria and impact of the diffusion coefficient τ and the kernel bandwidth σ (for HODMMD) are not detailed. Additionally, a discussion on computational complexity, including a comparison of training times against baseline methods, should be provided to assess the practical efficiency of the framework.\n4. The results in Table 3 show that applying the LGS module sometimes leads to a decrease in the SSIM metric (e.g., for LLFlow-L and LLFlow-S on the LOL-v1 dataset). These negative results are not addressed in the discussion. The authors should provide an analysis or explanation for these occurrences to offer a more balanced and complete evaluation of their method’s impact.\n\nQUESTIONS:\nsee weakness\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

comments: SUMMARY:
This paper is proposed to quantify the uncertainty based on Dampster-Shafer theory. It presents a fusion framework based on a nonuniform splitting mechanism and Hilbert space mapping to incorporate the prior information.

STRENGTHS AND WEAKNESSES:
First, this paper is composed with a wrong template. It does not have any line numbers.

Overall, this paper is very hard to follow. The method section includes too many concepts and equations without further discussion (even without meaningful references to the background). The experiments section, especially 4.2 and 4.3, only shows the improvements of performance. It lacks the discussion on why the proposed method is better.

QUESTIONS:
See Strengths And Weaknesses

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 1
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-02