Navigating Conflicting Views: Harnessing Trust for Learning

## Abstract

Abstract Resolving conflicts is essential to make the decisions of multi-view classification more reliable.
Much research has been conducted on learning consistent informative representations among different views, assuming that all views are identically important and strictly aligned.
However, real-world multi-view data may not always conform to these assumptions, as some views may express distinct information.
To address this issue,
we develop a computational trust-based discounting method to enhance the existing trustworthy framework in scenarios where conflicts between different views may arise.
Its belief fusion process considers
the trustworthiness of predictions made by individual views
via an instance-wise
probability-sensitive trust discounting mechanism.
We evaluate our method on six real-world datasets,
using Top-1 Accuracy,
AUC-ROC for Uncertainty-Aware Prediction,
Fleiss’ Kappa,
and a new metric called Multi-View Agreement with Ground Truth that takes into consideration the ground truth labels.
The experimental results show that computational trust can effectively resolve conflicts, paving the way for more reliable multi-view classification models in real-world applications. 1 1 1 We will release our codes once our paper is accepted.

## 1 Introduction

Multi-View Classification (MVC) plays a critical role in deep learning by greatly enhancing the ability to make accurate decisions through integrating the multi-source information.
Its effectiveness has been verified with the successful application in many domains such as autonomous driving [] and AI-assisted medical diagnostic systems [].
Most of the existing studies on MVC rely on the assumption that data from different views are consistently of good quality and informative with regard to the ground truth [, , ].
Nevertheless, this assumption may not always be valid in real-world scenarios.
Substantial variations in the quality of data from different views can produce contradictory results, thereby undermining the reliability of the model’s predictions.

A possible solution for resolving conflicts is to project data from different views into a latent common space [, , , ],
and then draw a combined representation from the latent space for the classification.
This is achieved by integrating essential features via weighting schemes, such as attention mechanisms [] and weighted fusion [, ].
These methods typically assign higher weights to more informative views or features, thus reducing the impact of potential conflicting or noisy information.
Although these methods have achieved promising results in MVC,
their focus on the combined joint representation can be a limitation. Solely relying on the joint representation
hinders the capacity to thoroughly grasp information provided by different views.
In contexts such as medical diagnosis, characterized by heterogeneous data sources from various views of medical imaging (e.g., lateral, frontal and scaphoid views in X-rays (Figure 1) ),
it’s crucial to thoroughly analyze and comprehend each view before establishing a comprehensive diagnosis, as each view often offers unique information [].

Figure: Figure 1: Conflict in Medical Imaging. Fracture (red box) is evident in one view, but not evident in another two. Adapted from a case study by [].
Refer to caption: extracted/5638815/imgs/fracture-conflict-case.png

Existing approaches to resolve conflicts
build neural networks to generate view-specific predictions
and then combine view-specific predictions together.
As a prime example,
the evidence-based framework is emerging as a promising approach,
offering a reliable means for the final fusion stage.
Within this framework, evidence acts as a metric of endorsement for the associated predicted label,
and the evidence is collected through view-specific neural networks.
Subsequently, evidence from diverse viewpoints is fused, considering their respective epistemic uncertainties.
However, there may exist cases where the view-specific information is not well aligned with the ground truth,
resulting in misleading predictions with high confidence (low uncertainty).
For example, as shown in Figure  1,
Fracture is evident in the scaphoid view but not evident in others.
A pioneering work [] based on the evidence framework tried to resolve conflicts by minimizing the degree of conflict loss.
However, this approach may inadvertently converge towards incorrect predictions, potentially leading to unstable performance.

In this work, we take a significant step further:
leveraging the evidence-based framework,
we propose a new computational trust based opinion fusion method to resolve potential conflicts in MVC.
Specifically, the computational trust is modelled through an evidence network that operates on a view-specific and sample-wise basis.
Drawing upon the principle of trust discounting in subjective logic, it evaluates the trustworthiness of view-specific predictions generated by existing evidence-based frameworks,
such as Evidential Deep Learning (EDL) [].
Within the proposed method,
each view-specific evidence is transformed into a degree of trust using the Binomial opinion theory [].
These degrees of trust are then utilized to establish uncertainty and a trust-aware opinion, ultimately facilitating the generation of reliable predictions.
In summary, the contributions of this paper include:

- 1.
We present a novel learnable trust-discounting mechanism to extend the widely-used evidence-based trustworthy MVC framework, enhancing its conflict resolution capabilities.
Drawing from the Binomial opinion theory within subjective logic, it operates on a view-specific and instance-wise basis, adeptly resolving conflicts among views through a probability-sensitive trust discounting rule;
- 2.
We develop a stage-wise training strategy to optimize the parameters of the proposed mechanism, which works robustly on different datasets;
- 3.
We conduct extensive experiments on six real-world datasets, showing that our method outperforms the existing trusted MVC methods based on evidential deep learning,
particularly on the datasets exhibiting large discrepancy among view-specific predictions.
In addition, our method can also enhance the consistency among opinions derived from different views.

## 2 Trust-discounted Fusion for Conflict Resolving

Given training data $\mathcal{D}={\{\{\bm{x}^{v}_{i}\}_{v=1}^{V},y_{i}\}}_{i=1}^{N}$ where $N$ is the number of training data, each instance $\bm{x}_{i}$ has $V$ views, ground truth label $y_{i}$ and an one-hot encoded label $\mathbf{y}_{i}$ (i.e.,
for a $K$-class classification problem, $\mathbf{y}_{i,k}$ is 1 if $k$ is the index of ground truth label for $i$-th instance, otherwise it is 0).
The task of MVC is to learn a function $f$ that maps $\{\bm{x}_{i}^{v}\}_{v=1}^{V}$ to $\mathbf{y}_{i}$.

### 2.1 Preliminaries on Evidence Theory for MVC

The evidence-based framework applies Subjective Logic (SL) to the $K$-class classification problem
by assigning belief masses to individual class labels and computing epistemic uncertainty for the generated belief masses.
The formulation links the evidence collected from instance view-specific observation to the concentration parameter of the Dirichlet Distribution.
Let $f_{\theta}^{v}(\cdot)$ denote the view-specific neural network for evidence generation,
where the view-specific evidence for an instance is $\bm{e}^{v}=f_{\theta}^{v}(\bm{x}^{v})$,
the association between the evidence and the Dirichlet parameters is simply $\alpha_{k}=e_{k}+1$ [, ] .
The belief mass on class label $k$, denoted as $b_{k}$, and uncertainty $u$ are subject to the additive requirement, i.e., $u+\sum_{k=1}^{K}b_{k}=1$.
With respect to MVC, the view-specific belief mass $b_{k}^{v}$ and uncertainty $u^{v}$ can then be computed as

$$ $S^{v}=\sum_{k=1}^{K}\alpha^{v}_{k},\quad b_{k}^{v}=\frac{e_{k}^{v}}{S^{v}}= \frac{\alpha_{k}^{v}-1}{S^{v}},\quad u^{v}=1-\sum_{k=1}^{K}b_{k}^{v}=\frac{K}{ S^{v}}$ (1) $$

To generate the final prediction,
SL models the view-specific predictions as multinomial opinions,
denoted as $\omega^{v}=[\bm{b}^{v},u^{v},\bm{a}^{v}]$,
with $\bm{a}^{v}$ being the base rate (i.e., a prior probability distribution over classes, generally a discrete uniform distribution),
and then combine them together with an appropriate belief fusion rules based on the context [].
The Belief Constraint Fusion (BCF) [],
an extension of Dempher-Shafer combination rule [],
was first adopted by [] in trusted MVC.
Other fusion rules, such as Aleatory Cumulative Belief Fusion (A-CBF) [] and Averaging Belief Fusion (ABF) []
have also been explored.
We stay with BCF in our experiments based on the empirical results.
For fusing different opinions, i.e., $\omega=\omega^{1}\oplus\omega^{2}\oplus\cdots\omega^{V}$ from $V$ views,
the fusion rule of BCF
among two views can be formulated as follows:

$$ $b_{k}=\frac{1}{1-C}(b^{1}_{k}b^{2}_{k}+b^{1}_{k}u^{2}+b^{2}_{k}u^{1}),\quad u= \frac{1}{1-C}u^{1}u^{2}$ (2) $$

where $C=\sum_{i\neq j}b^{1}_{i}b^{2}_{j}$ is the normalization factor.
Applying Eq. (2) iteratively will derive the final fused opinion from the $V$ views,
and the order does not affect the combined result [].
For the fused opinion $\omega$, we can derive the parameters of the Dirichlet $\alpha_{k}$ by reversing the computation of Eq. (1).

An alternative representation for BCF is based on combining the evidence, from which the opinions $[\bm{b},u,\bm{a}]$ can be derived:

$$ $e_{k}=e^{1}_{k}+e^{2}_{k}+\frac{e^{1}_{k}e^{2}_{k}}{K}$ (3) $$

Note that A-CBF corresponds to the case where the third term is removed, and thus for the Dirichlet likelihood can be interpreted as Bayesian updating assuming independent evidence.

**Table 1: Illustration of fusing conflicting opinions with BCF, A-CBF and ABF**
|  | Source Opinions | Fused Opinions | Trust Fusion |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | $\acute{\omega}^{1}$ | $\acute{\omega}^{2}$ | BCF | A-CBF | ABF | $\ddot{\omega}^{1}$ | $\ddot{\omega}^{2}$ | TD+BCF |
| $b_{1}$ | 0.67 | 0.1 | 0.387 | 0.391 | 0.385 | 0.9 | 0.0 | 0.56 |
| $b_{2}$ | 0.2 | 0.2 | 0.222 | 0.203 | 0.200 | 0.0 | 0.9 | 0.19 |
| $b_{3}$ | 0.1 | 0.67 | 0.387 | 0.391 | 0.385 | NA | NA | 0.16 |
| u | 0.03 | 0.03 | 0.004 | 0.015 | 0.030 | 0.1 | 0.1 | 0.09 |

### 2.2 Conflict Resolving by Trust Discounting

We realize conflicts can happen when view-specific opinions express conflicting preferences with similar level of confidence, leading to inaccuracies in the combined opinion.
Based upon this, we define the conflict problem as follows:

###### Definition 1 (Conflicts within Multi-view Learning) .

For a $K$-class multi-class classification problem,
the view-specific models give confident yet opposite opinions for a given data instance.

Table 1 presents a scenario with a two-view instance for 3-class classification, illustrating the concept of conflict views. Conflict arises for such a data instance $\{\bm{x}^{v}\}_{v=1}^{2}$ with the ground label $y=1$, when the opinions are $\bm{b}^{1}=[0.67,0.2,0.1]$ and $u^{1}=0.03$, alongside $\bm{b}^{2}=[0.1,0.2,0.67]$ and $u^{2}=0.03$. Existing fusion approaches, as depicted in Table 1, struggle to provide a clear and accurate final prediction, as they all generate the same belief mass for labels 1 and 3 due to the confidence of both opinions with low uncertainty.

Instead, we utilize the principle of Trust Fusion (TF) by Trust Discounting (TD) [] to resolve conflicts.
The basic idea of TD is to discount information provided by an individual view as a function of trust in that view.
It can be used to weigh the current view-specific opinion according to the degree of trust,
thus guiding the fusion process to generate more reliable prediction.
Here we present a Probability-sensitive Trust Discounting rule, as show in Eq. (4),
and use it in an instance-wise manner in our experiments,
This rule computes the trust degree by the
expect value of projected Beta distribution $\beta(\alpha_{1},\alpha_{2})$ (i.e., $p_{t}=\frac{\alpha_{2}}{\alpha_{1}+\alpha_{2}}$)
for each individual instance on each view,
following Binomial opinion theory.
It is noteworthy that other choices to compute the trust degree are also possible, such as
Uncertainty Favouring Discounting and Opposite Belief Favouring Discounting [].
They can be easily integrated into our framework.

To distinguish different types of opinions,
we modify notations used in the previous section.
Concretely, the opinions before applying trust fusion,
is often referred to as Functional (Trust) Opinion, denoted as $\acute{\omega}$,
in the form of multinomial opinion discussed in Section 2.1.
Meanwhile, the opinion used for discounting is known as the Referral (Trust) Opinion, denoted as a Binomial opinion $\ddot{\omega}$, and
$\breve{\omega}$ is used
to indicate
the trust-discounted opinion.
From [], the discounting rule is defined as:

###### Definition 2 (Instance-wise Probability-Sensitive Trust Discounting) .

For each view of each individual instance, the trust-discounted opinion is defined as

$$ $\breve{\omega}=\ddot{\omega}\otimes\acute{\omega}=\left\{\begin{array}[]{lr} \breve{\bm{b}}=\ddot{p}_{t}*\acute{\bm{b}},\\ \breve{u}=1-\ddot{p}_{t}*\left(\sum_{k=1}^{K}\acute{b}_{k}\right),\\ \breve{\bm{a}}=\acute{\bm{a}}.\end{array}\right.$ (4) $$

where $\otimes$ indicates the trust discounting operator,
and the scalar probability $\ddot{p}_{t}$ denotes the degree of trust,
representing how much we are confident with the information given by the view-specific model.
In contrast, $\ddot{p}_{d}=\frac{\ddot{\alpha}_{1}}{\ddot{\alpha}_{1}+\ddot{\alpha}_{2}}$ denotes the degree of distrust, and is subject to the constraint $\ddot{p}_{t}+\ddot{p}_{d}=1$.

Note this corresponds to updating the Dirichlet evidence by $\breve{e}_{k}=\frac{\ddot{p}_{t}\acute{u}}{1-\ddot{p}_{t}+\ddot{p}_{t}\acute{u
}}\,\acute{e}_{k}$.
Given Eq. (4), we fuse the trust-discounted opinions from $V$
views of $i$-th instance with BCF by:

$$ $\bar{\omega}_{i}=\breve{\omega}^{1}_{i}\oplus\breve{\omega}^{2}_{i}\oplus \cdots\oplus\breve{\omega}^{V}_{i}=\left(\ddot{\omega}^{1}_{i}\otimes\acute{ \omega}^{1}_{i}\right)\oplus\left(\ddot{\omega}^{2}_{i}\otimes\acute{\omega}^{ 2}_{i}\right)\oplus\cdots\oplus\left(\ddot{\omega}^{V}_{i}\otimes\acute{\omega }^{V}_{i}\right)$ (5) $$

In the presence of conflicting opinions, the estimated uncertainty is expected to increase, even if the uncertainty associated with the prediction of each individual view is small.
However,
the uncertainty of fused opinion given by the fusion rules mentioned above
is always under-estimated, thus inconsistent with the generated beliefs, as shown by the example in Table 1.
With the trust-discounting rule, conflicts can be effectively solved within the evidential learning framework,
ensuring theoretically-guaranteed better fusion,
as demonstrated by Proposition 1 and Proposition 2 together with
the example in Table 1.
Specifically,
Proposition 1 illustrates that our TD rule consistently enhance classification accuracy,
while Proposition 2 establishes that TD corrects and adjusts the under-estimated uncertainty.
The following propositions provide theoretical analysis of the proposed TD rule,
and their detailed proof can be found in Appendix A.4.

###### Proposition 1 .

Probability Sensitive Trust-Discounting maximizes the belief mass of the Ground truth label after BCF, under the assumption that at least one view’s prediction is correct.

###### Proposition 2 .

The combined opinion generated by BCF with trust discounting for conflicting views, will exhibit greater uncertainty than obtained through fusion with non-discounted functional opinions.

###### Definition 1 (Conflicts within Multi-view Learning) .

###### Definition 2 (Instance-wise Probability-Sensitive Trust Discounting) .

###### Proposition 1 .

###### Proposition 2 .

### 2.3 Learning to Form Opinions

Figure: Figure 2: The Trust Discounting Mechanism based Multi-view Evidential Framework
Refer to caption: extracted/5638815/imgs/tf-module.png

Similar to the generation process of a functional opinion,
we construct another view-specific evidence network parameterized by $\ddot{\theta}$,
for outputs referral evidence $\ddot{\bm{e}}$,
i.e., $\ddot{\bm{e}}^{v}_{i}=f_{\ddot{\theta}}^{v}([\bm{x}^{v}_{i},\acute{\bm{b}}^{v}
_{i}])$(^2^22We used Bi-Linear layer instead of Dense/Linear Layer in our experiments.).
Figure 2 depicts our TF rule along with entire evidence-based
learning framework.

For optimizing network parameters,
we propose a stage-wise training algorithm (Algo. 1),
drawing inspiration from [],
to avoid any potential numerical instability problems during training.
In the proposed training algorithm,
we follow [, , , ],
and use the following loss term $L(\bm{\alpha})$ (^3^33The closed form of $ace$ loss and KL-divergence loss will be shown in Appendix A.5.)
to optimize the parameters of the Dirichlet distribution for where it applies,

$$ $L(\bm{\alpha})=L_{ace}(\bm{\alpha})+\lambda_{o}\text{D}_{KL}[\text{Dir}( \mathbf{p}|\bm{\tilde{\alpha}})||\text{Dir}(\mathbf{p}|\bm{1})]\,$ (6) $$

where $\tilde{\bm{\alpha}}=\mathbf{y}+(1-\mathbf{y})\odot\bm{\alpha}$ is the Dirichlet parameters after removing non-misleading evidence from predicted parameters $\bm{\alpha}$ , and $\mathbf{p}$ is the projected probability, i.e., $\mathbf{p}=\frac{\bm{\alpha}}{S}$.
$\odot$ is the element-wise product and $\lambda_{o}$ is the annealing factor.
Specifically, in the proposed algorithm, we apply Eq. (6) to the Dirichlet parameters with regard to all functional opinions (i.e., $\{\acute{\bm{\alpha}}^{v}_{i}\}_{v=1}^{V}$) at stage-2a.
At stages 2b and 3, we apply this loss term only to the parameters of the Dirichlet distribution concerning the BCF combined opinion (i.e., $\bar{\bm{\alpha}}_{i}$).
Please note that for the BCF combined opinion, both the functional opinion and the referral opinion are required. Therefore, when applying back-propagation (BP) with regard to the combined opinion, there will be gradients for both the referral network and the functional network.
We update different types of networks at different stages. For example, we only update the functional network at stage-2b and only update the referral network at stage-3.

Since the parameters of the referral network are randomly initialised at the beginning,
we adopt the ace loss from above with a new smoothed label to warm-up those parameters.
For each $i$-th instance on $v$-th view, the hard label $z^{v}_{i}$ is generated by

$$ $z^{v}_{i}=\begin{cases}1&\text{if }\hat{y}_{i}^{v}={y}_{i}\\ 0&\text{otherwise}\end{cases}$ (7) $$

Following [], we apply label smoothing with smoothing factor $\eta=0.9$ to the hard label.
The association between one-hot encoded hard label $\mathbf{z}^{v}_{i}$ of target $z^{v}_{i}$ and smooth label is $\mathring{\mathbf{z}}^{v}_{i}=\mathbf{z}^{v}_{i}\odot\eta+\eta/2$.
since the smoothed label could provide training signals for neurons of both target and non-target labels, we simply adopt $\sum_{v}L_{ace}(\ddot{\bm{\alpha}}_{i}^{v})$ with smoothed labels $\{\mathring{\mathbf{z}}_{i}^{v}\}_{v=1}^{V}$ in the first warm-up stage, and

$$ $L_{ace}(\ddot{\bm{\alpha}}_{i}^{v})=\sum_{j=1}^{2}\mathring{\mathbf{z}}_{ij}^{ v}(\psi(\ddot{\bm{\alpha}}_{i1}^{v}+\ddot{\bm{\alpha}}_{i2}^{v})-\psi(\ddot{ \bm{\alpha}}_{ij}^{v}))$ (8) $$

where $\ddot{\bm{\alpha}}_{i}^{v}$ is the parameters of Beta distribution of referral opinion and $\psi$ is the digamma function.

## 3 Related Work

Figure: Algorithm 1 Algorithm for Training and Testing

Multi-View Classification leverages multiple data sources, offering varied perspectives on the same object,
to enhance the classification performance.
Recent advancements in MVC have focused on generating noise-robust representations through cluster-based [, , ], self-representation-based [], and partially view-aligned [, ] methods,
harnessing the expressive power of deep neural networks.
However, noise-robust representations may not fully resolve conflicts in opinions for a given data instance, as they lack a mechanism to quantify the trustworthiness of each view.
Our method addresses this limitation by introducing an evidence network that evaluates the reliability of view-specific predictions and adjusts the final predictions according to the degree of trust.

Trusted Multi-View Classification has emerged as a crucial area and a pivotal domain within Multi-View Learning.
This research area aims to enhance the accuracy and dependability of classification models by integrating data from multiple views, guided by their prediction confidence and epistemic uncertainty.
The seminal work, Trusted Multi-View Classification (TMC) [], introduced the fusion of different views from an opinion perspective using the Dempher-Shafer Combination rule.
Building upon TMC, [] extended the approach by incorporating the pseudo-view, a concatenation of all other views, resulting in improved performance.
Subsequent studies by [] and [] explored alternative opinion fusion methods.
Concurrent research efforts, such as those by [] and [],
focus on multiview uncertainty estimation, enhancing the model’s reliability.

Conflictive Multi-View Classification argues that existing work primarily focusing on either learning joint aligned representations or better quantifying uncertainty overlook the problem of potential contradictory in the prediction space.
Recognizing this gap, the pioneer work by [] highlighted this issue and introduced the Degree of Conflict loss.
This loss quantifies the disparity between different predictions in the prediction space while accounting for uncertainty, aiming to mitigate conflict-related challenges.
However, this approach may inadvertently lead correct predictions to converge towards incorrect ones, potentially jeopardizing model stability.
In contrast, our method can generate more accurate predictions with properly estimated uncertainty.

## 4 Experiments

### 4.1 Experimental Setup

Datasets. Following [, , , ],
we conducted experiments on six benchmark datasets:
Handwritten,(^4^44https://archive.ics.uci.edu/ml/datasets/Multiple+Features)
Caltech101 [],
PIE,(^5^55http://www.cs.cmu.edu/afs/cs/project/PIE/MultiPie/Multi-Pie/Home.html)
Scene15 [],
HMDB [] and CUB [] with train-test split of 80% vs. 20%.
As we utilized the exact same datasets, we direct readers to [] for further details regarding these datasets.
A summary of these datasets is provided in the Appendix A.7.

Baseline Methods.
We aim to resolve conflicts among predictions of different views,
so we consider the methods that generate view-specific predictions which could have potential conflicts,
and thus included following baselines:
Fusion by Majority Voting (F-Mode)
and Fusion by Probability Averaging (F-Avg),
which are two commonly used fusion methods in most MVC methods.
We also consider existing trust-worthy baselines,
MGP [] TMC [], and the conflict resolution pioneering work ECML [].
Our method can be extended to leverage the pseudo view, as demonstrated by its application to ETMC [], an extended version of TMC that incorporates pseudo views.
All methods were run on a single 24GB RTX3090 card for fair comparison.

### 4.2 Experiment Results and Analysis

We compared our method with baselines with different metrics.
For each individual metric, mean and standard deviation from ten runs with different random seeds are reported.
In all tables, the best-performing method is highlighted in bold, and the second-best method is underlined.

**Table 2: Top-1 accuracy on test split.**
| Dataset | Handwritten | Caltech101 | PIE | Scene15 | HMDB | CUB | AVG |
| --- | --- | --- | --- | --- | --- | --- | --- |
| F-Mode | 99.10±0.20 | 94.13±0.08 | 79.41±0.00 | 62.45±0.11 | 51.70±0.41 | 70.25±0.38 | 76.13 |
| F-Avg | 99.25±0.00 | 95.59±0.06 | 90.59±0.29 | 76.21±0.09 | 71.49±0.35 | 92.75±0.53 | 87.65 |
| MGP | 99.60±0.10 | 94.42±0.20 | 90.13±0.87 | 74.30±0.41 | 73.97±0.15 | 90.79±1.03 | 87.03 |
| ECML | 99.57±0.11 | 94.25±0.08 | 91.40±0.47 | 64.34±0.11 | 72.90±0.11 | 92.58±0.25 | 85.84 |
| TMC | 99.63±0.13 | 94.30±0.13 | 87.43±0.90 | 73.99±0.19 | 73.30±0.18 | 92.50±0.37 | 86.60 |
| TF(ours) | 99.68±0.11 | 95.26±0.10 | 93.31±0.40 | 77.83±0.32 | 74.35±0.09 | 93.33±0.75 | 88.96 |
| ETMC | 99.75±0.00 | 94.41±0.11 | 91.69±0.47 | 78.41±0.20 | 74.01±0.19 | 93.67±0.41 | 88.74 |
| ETF(ours) | 99.98±0.07 | 95.07±0.08 | 94.63±0.34 | 82.01±0.17 | 75.55±0.15 | 94.08±0.38 | 90.22 |

Prediction Accuracy.
Similar to [, , , ],
we first evaluated the model performance on the test split by Top-1 Classification Accuracy,
as shown in Table 2.
Building on the strengths of pseudo view, our method (ETF) consistently outperforms all trustworthy-based methods,
and gains the best average performance over six datasets compared with all baselines.
For example, on the PIE and Scene15 datasets, the use of trust boosts the accuracy of ETMC by 2.94% and 3.60%, respectively.
Moreover, ETF surpasses the pioneering conflict resolving method ECML by a substantial margin of 3.23% on PIE, 9.66% on Scene15 and 2.65% on HMDB, highlighting better power of conflicts handling of our method.
It is worth noting that Caltech101 inherently has lower level of conflicts, as corroborated by high accuracy and Fleiss’ Kappa scores (Table 3) of all baselines.
Nevertheless, ETF maintains the compatible performance with the best one, F-Avg (a minor decrease of 0.52%), and still outperforms other trustworthy methods, e.g., improve the accuracy of ETMC by 0.66%.

When compared to well-established trustworthy methods like TMC, MGP, and ECML without pseudo views,
our method TF consistently demonstrates superior performance across all datasets.
For example, our proposed trust discounting method enhance TMC’s performance by 3.84% on Scene15 and 5.88% on PIE,
while also achieving the highest Top-1 accuracy on other datasets.
Notably, our method TF, even without incorporating pseudo views,
exhibits comparable performance to ETMC with pseduo views.
For instance, TF outperforms ETMC on three datasets (Caltech101, PIE, and HMDB) out of a total of six.

**Table 3: Fleiss’ Kappa on test splits.**
| Dataset | Handwritten | Caltech101 | PIE | Scene15 | HMDB | CUB | AVG |
| --- | --- | --- | --- | --- | --- | --- | --- |
| F-Mode | 0.63±0.04 | 0.97±0.00 | 0.38±0.00 | 0.42±0.00 | 0.56±0.00 | 0.71±0.01 | 0.61 |
| F-Avg | 0.54±0.03 | 0.97±0.00 | 0.37±0.01 | 0.42±0.00 | 0.55±0.01 | 0.58±0.06 | 0.57 |
| MGP | 0.59±0.05 | 0.94±0.00 | 0.21±0.01 | 0.33±0.00 | 0.51±0.00 | 0.43±0.07 | 0.50 |
| ECML | 0.42±0.05 | 0.95±0.00 | 0.40±0.01 | 0.26±0.00 | 0.53±0.01 | 0.44±0.07 | 0.50 |
| TMC | 0.54±0.07 | 0.94±0.01 | 0.23±0.02 | 0.30±0.01 | 0.52±0.01 | 0.37±0.19 | 0.48 |
| TF(ours) | 0.65±0.02 | 0.95±0.00 | 0.36±0.01 | 0.39±0.00 | 0.54±0.00 | 0.51±0.10 | 0.57 |
| ETMC | 0.66±0.01 | 0.84±0.00 | 0.28±0.04 | 0.37±0.00 | -0.15±0.04 | 0.45±0.10 | 0.41 |
| ETF(ours) | 0.76±0.02 | 0.95±0.00 | 0.48±0.01 | 0.48±0.01 | 0.65±0.00 | 0.64±0.03 | 0.66 |

Reliable Prediction. To further validate the effectiveness of our proposed method, we evaluate it with two additional metrics, Fleiss’ Kappa [] and our proposed metric, Multi-View Agreement with Ground Truth (MVAGT).
Fleiss’ Kappa is a statistical measure for assessing the reliability of agreement between different raters,
with scores closer to 1 indicating higher agreement among the different predictions.
As depicted in Table 3,
our method (ETF) achieves the highest Fleiss’ Kappa score on four datasets (Handwritten, PIE, Scene15, HMDB and CUB).
Even through ETF does not rank first on the remaining two datasets (the third on Caltech101 and the second on CUB),
it remains the most generalizable model with the highest average Fleiss’ Kappa (0.66).
It’s worth noting that while our method assumes the existence of conflicts, Caltech101 is a dataset with fewer conflicts, which explains the performance discrepancy in Table 2.
Nevertheless, ETF still outperforms other trustworthiness-based methods and enhances the robustness of ETMC with an improvement of approximately 13% on Caltech101.
Moreover, it’s essential to highlight that ETMC exhibits extremely poor agreement on HMDB with a negative value of -0.15.
However, by applying our method, ETF significantly improves performance by an absolute value of 0.8. This underscores the relative robustness of our method across different datasets.

**Table 4: MVAGT on test split.**
| Dataset | Handwritten | Caltech101 | PIE | Scene15 | HMDB | CUB |
| --- | --- | --- | --- | --- | --- | --- |
| F-Mode | 88.87±1.73 | 94.13±0.08 | 79.41±0.00 | 62.54±0.11 | 51.70±0.41 | 70.25±0.38 |
| F-Avg | 18.78±5.89 | 93.89±0.24 | 17.06±1.22 | 27.70±0.36 | 51.18±0.51 | 59.50±5.25 |
| MGP | 81.37±5.73 | 91.55±0.29 | 63.20±2.31 | 52.10±0.41 | 50.43±0.42 | 42.50±9.26 |
| ECML | 74.08±0.61 | 91.05±0.27 | 78.46±1.19 | 41.91±0.31 | 50.95±0.48 | 48.58±5.36 |
| TMC | 81.58±6.57 | 90.27±0.38 | 51.54±3.00 | 51.42±0.46 | 50.37±0.45 | 43.25±14.8 |
| TF(ours) | 88.97±0.61 | 92.01±0.22 | 80.59±0.75 | 60.41±0.52 | 52.47±0.35 | 54.33±7.54 |
| ETMC | 98.10±0.17 | 92.41±0.32 | 75.15±4.13 | 73.75±0.45 | 8.45±1.09 | 91.08±1.06 |
| ETF(ours) | 98.53±0.08 | 94.47±0.12 | 90.37±0.40 | 79.18±0.38 | 71.43±0.32 | 91.17±0.67 |

While a multi-view classification (MVC) classifier with both high accuracy and high Fleiss’ Kappa score generally suggests good reliability,
high Fleiss’ Kappa scores without reference to the ground truth label might be misleading, particularly in cases where the majority of views agree on the same incorrect class.
Therefore, we propose a new evaluation metric (MVAGT),
specifically tailored for conflict MVC scenarios.
MVAGT assesses correctness on the test split by verifying that more than half of the views make correct decisions.
Since majority agreement is often more reasonable for final decisions in real-world scenarios, evaluating methods using MVAGT ensures the reasonableness of the fused decision.
Further details about MVAGT are provided in Appendix A.6.
As depicted in Table. 4,
ETF demonstrates superior performance compared to other methods.
Moreover, ETF exhibits good generalizability across different datasets,
where ETMC experiences significant decreases (e.g., HMDB) or other methods alternately occupy the second-best position.

AUC-ROC for Uncertainty-Aware Prediction.
Following prior work [], we assess uncertainty by approaching it as the task of determining whether to trust the model’s prediction, evaluating the model’s ability to identify incorrect predictions based on uncertainty scores.
Specifically, we employed AUC-ROC to measure the model’s discriminate power in distinguishing incorrect predictions using uncertainty scores.
As shown in Table 5, TF and ETF consistently demonstrate the best performance in uncertainty-based classification across five out of the six datasets, showcasing their robust generalizability.
Despite a slight performance decrease on the CUB dataset, our method (ETF) still maintains the second-best result, outperforming other approaches, whether incorporating pseudo views or not.
One possible reason for the decrease is that our method may increase the belief mass of ground truth label for some incorrectly predicted instances, thus correcting the incorrect predictions.
However, the decrease in uncertainty is relatively small (e.g., 0.5 for incorrect changes to 0.4 for correct). An example is shown in the Appendix B.2.

**Table 5: AUC-ROC scores for identifying incorrect predictions using uncertainty scores.**
| Dataset | Handwritten | Caltech101 | PIE | Scene15 | HMDB | CUB |
| --- | --- | --- | --- | --- | --- | --- |
| MGP | 99.29±0.30 | 87.62±0.90 | 88.43±0.67 | 63.92±1.96 | 82.87±0.60 | 58.20±11.4 |
| ECML | 79.05±5.62 | 86.31±0.50 | 87.51±0.49 | 60.50±0.25 | 81.63±0.15 | 57.30±8.50 |
| TMC | 99.23±0.22 | 87.33±0.47 | 90.16±0.99 | 62.60±0.54 | 82.63±0.48 | 64.80±10.5 |
| TF(ours) | 99.32±0.35 | 88.99±0.54 | 95.90±0.08 | 64.56±2.02 | 83.59±0.23 | 53.52±14.3 |
| ETMC | 99.30±0.19 | 88.35±0.63 | 93.02±1.40 | 66.49±0.44 | 85.42±0.34 | 72.56±8.11 |
| ETF(ours) | 99.90±0.30 | 88.70±0.54 | 92.47±1.19 | 70.44±1.10 | 86.23±0.49 | 64.41±3.54 |

Figure: Figure 3: Performance comparison of simulated conflict predictions.
Refer to caption: extracted/5638815/imgs/conflict-hw.png

Simulating Conflicting Predictions with Noisy Instances.
Following a similar approach as [, , ],
we introduce varying levels of Gaussian noise (e.g., with standard deviations of 10 and 100) to randomly selected half of the views for each instance in the dataset, creating noisy views.
The original views are then replaced by these noisy views, and the model is trained using the updated dataset.
Consequently, the model’s predictions based on the noisy views may conflict with the predictions from the views without explicitly introducing noise.
As illustrated in Figure 3 and Appendix B.3, our methods (ETF and TF) consistently outperform others with or without pseudo views, demonstrating the effectiveness of TD and our training algorithm’s ability to handle conflicting views.

## 5 Conclusion

In this paper, we introduced a theoretically-funded approach for resolving conflicts in Multi-View Classification.
This approach is built on top of the principle of trust discounting in Subjective Logic, where the computational trust, aka referral trust, is represented as a Binomial opinion with a Beta probability density function.The functional trust is
then discounted by the amount computed as a function of the degree of trust.
We demonstrated through extensive experiments that the proposed trust discounting method not only benefits classification accuracy but also increases consistency among different views,
providing a new reliable approach to handling conflicts in MVC.
A possible limitation of our work is that the warm-up loss is not optimal solution. We leave it for future work.

## Appendix A Proofs and Derivations

### A.1 Calculation of Predictive Probability

According to Subjective Logic (SL) [],
the predictive probability $p_{k}$ for class $k$, can be calculated by

$$ $p_{k}=b_{k}+a_{k}*u$ (9) $$

where $b_{k}$ is the belief mass for $k$-th label, $u$ is the predictive uncertainty or epistemic uncertainty [].
We usually assume the prior $a_{k}$ conforms to a uniform discrete distribution, i.e., $a_{k}=1/K$.

Above equation is identical to

$$ $p_{k}=\frac{\alpha_{k}}{S}$ (10) $$

where $\alpha_{k}$ is the Dirichlet concentration parameter for $k$-th label, and $S$ is the Dirichlet strength, i.e., $S=\sum_{k}\alpha_{k}$.

###### Proof.

$$ $\displaystyle p_{k}$ $\displaystyle=$ $\displaystyle b_{k}+a_{k}*u$ $\displaystyle=$ $\displaystyle b_{k}+\frac{1}{K}*\frac{K}{S}$ $\displaystyle=$ $\displaystyle\frac{e_{k}}{S}+\frac{1}{S}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{k}}{S}$ $$

∎

Since Beta Distribution is 2-dimensional Dirichlet Distribution,
above equations for calculating probabilities of multinomial opinions
could also be applied to binomial opinions.

###### Proof.

### A.2 Alternative Representation of Belief Constraint Fusion(BCF)

###### Proof.

We the proof for Eq. (3) as follows,

$$ $\displaystyle e_{k}$ $\displaystyle=$ $\displaystyle S*b_{k}$ $\displaystyle=$ $\displaystyle S\,\frac{1}{1-C}(b_{k}^{1}b_{k}^{2}+b_{k}^{1}u^{2}+b_{k}^{2}u^{1})$ $\displaystyle=$ $\displaystyle S\,\frac{1-\sum_{k}b_{k}}{u^{1}u^{2}}(b_{k}^{1}b_{k}^{2}+b_{k}^{ 1}u^{2}+b_{k}^{2}u^{1})$ $\displaystyle=$ $\displaystyle(S-S*\sum_{k}b_{k})\,\frac{1}{u^{1}u^{2}}(b_{k}^{1}b_{k}^{2}+b_{k }^{1}u^{2}+b_{k}^{2}u^{1})$ $\displaystyle=$ $\displaystyle(S-\sum_{k}e_{k})\,\frac{1}{u^{1}u^{2}}(b_{k}^{1}b_{k}^{2}+b_{k}^ {1}u^{2}+b_{k}^{2}u^{1})$ $\displaystyle=$ $\displaystyle K\,\frac{1}{u^{1}u^{2}}(b_{k}^{1}b_{k}^{2}+b_{k}^{1}u^{2}+b_{k}^ {2}u^{1})$ $\displaystyle=$ $\displaystyle K\,\frac{1}{u^{1}u^{2}}(\frac{e_{k}^{1}e_{k}^{2}}{S^{1}S^{2}}+ \frac{e_{k}^{1}u^{2}}{S^{1}}+\frac{e_{k}^{2}u^{1}}{S^{2}})$ $\displaystyle=$ $\displaystyle K(\frac{e_{k}^{1}e_{k}^{2}}{K*K}+\frac{e_{k}^{1}u^{2}}{Ku^{2}}+ \frac{e_{k}^{2}u^{1}}{Ku^{1}})$ $\displaystyle=$ $\displaystyle\frac{e_{k}^{1}e_{k}^{2}}{K}+e_{k}^{1}+e_{k}^{2}$ $$

∎

###### Proof.

### A.3 Dirichlet Evidence Updating by Trust Discounting (TD)

As mentioned earlier, the TD in Definition 2 is corresponds to update Dirichlet evidence using following equation,

$$ $\breve{e}_{k}=\frac{\ddot{p}_{t}\acute{u}}{1-\ddot{p}_{t}+\ddot{p}_{t}\acute{u }}\,\acute{e}_{k}$ (11) $$

where $\ddot{p}_{t}$ is the probability representing trust degree and $\acute{u}$ is the uncertainty for functional opinion. $\acute{e}_{k}$ is Dirichlet evidence of functional opinion, and $\breve{e}_{k}$ is Dirichlet evidence after discounting.

###### Proof.

$$ $\displaystyle\breve{\bm{e}}_{k}$ $\displaystyle=$ $\displaystyle\breve{\bm{b}}_{k}*\breve{S}$ $\displaystyle=$ $\displaystyle\frac{\ddot{p_{t}}\acute{b_{k}}K}{\breve{u}}$ $\displaystyle=$ $\displaystyle\frac{\ddot{p_{t}}\acute{b_{k}}K}{1-\ddot{p_{t}}+\ddot{p_{t}} \acute{u}}$ $\displaystyle=$ $\displaystyle\frac{\ddot{p_{t}}}{1-\ddot{p_{t}}+\ddot{p_{t}}\acute{u}}\frac{ \acute{e_{k}}}{\acute{S}}K$ $\displaystyle=$ $\displaystyle\frac{\ddot{p_{t}}}{1-\ddot{p_{t}}+\ddot{p_{t}}\acute{u}}\frac{K} {\acute{S}}\acute{e_{k}}$ $\displaystyle=$ $\displaystyle\frac{\ddot{p_{t}}\acute{u}}{1-\ddot{p_{t}}+\ddot{p_{t}}\acute{u} }\,\acute{e_{k}}$ $$

∎

###### Proof.

### A.4 Detailed Proof of Propositions

###### Proof.

Proof details of Proposition 1. We use $g$ to denote the index of ground-truth label.
Recall that scalar probability $\ddot{p}_{t}$ represents the degree of trust as mentioned before.
With at least one view’s prediction is correct,
the belief mass of ground truth label of BCF combined opinion,
which integrates through trust-discounted opinions,
is as follows,

$$ $\displaystyle\bar{b}_{g}$ $\displaystyle=$ $\displaystyle\frac{1}{1-\breve{C}}(\breve{b}^{1}_{g}\breve{b}^{2}_{g}+\breve{b }^{1}_{g}\breve{u}^{2}+\breve{b}^{2}_{g}\breve{u}^{1})$ $\displaystyle=$ $\displaystyle\frac{1}{1-\breve{C}}((\acute{b}^{1}_{g}\ddot{p}^{1}_{t})(\acute{ b}^{2}_{g}\ddot{p}^{2}_{t})+\acute{b}^{1}_{g}\ddot{p}^{1}_{t}\breve{u}^{2}+ \acute{b}^{2}_{g}\ddot{p}^{2}_{t}\breve{u}^{1})$ $$

And we have the discounted uncertainty

$$ $\displaystyle\breve{u}$ $\displaystyle=$ $\displaystyle 1-\ddot{p}_{t}(\sum_{k}\acute{b}_{k})$ $\displaystyle=$ $\displaystyle 1-\ddot{p}_{t}(1-\acute{u})$ $\displaystyle=$ $\displaystyle 1-\ddot{p}_{t}+\ddot{p}_{t}*\acute{u}$ $$

In cases where the belief mass of the ground truth label is maximal (i.e., predicted label is the ground truth label), our referral network should generate a high trust value, that is $\ddot{p}_{t}\to 1$ when $\acute{b}_{g}=max(\mathbf{\acute{b}})$, and otherwise $\ddot{p}_{t}\to 0$.
Therefore, $\breve{u}\to\acute{u}$ when $\acute{b}_{g}=max(\mathbf{\acute{b}})$, and $\breve{u}\to 1$ as well. Therefore, the fused term will be dominant by correctly predicted views as $\ddot{p}_{t}\to 1$
as well as $\breve{u}\to 1$ which contributes in cross multiplication.
Subsequently,

$$ $\displaystyle\bar{b}_{g}$ $\displaystyle=$ $\displaystyle\frac{1}{1-\breve{C}}((\acute{b}^{1}_{g}\ddot{p}^{1}_{t})(\acute{ b}^{2}_{g}\ddot{p}^{2}_{t})+\acute{b}^{1}_{g}\ddot{p}^{1}_{t}\breve{u}^{2}+ \acute{b}^{2}_{g}\ddot{p}^{2}_{t}\breve{u}^{1})$ $\displaystyle\geq$ $\displaystyle\frac{1}{1-\breve{C}}((\acute{b}^{1}_{k}\ddot{p}^{1}_{t})(\acute{ b}^{2}_{k}\ddot{p}^{2}_{t})+\acute{b}^{1}_{k}\ddot{p}^{1}_{t}\breve{u}^{2}+ \acute{b}^{2}_{k}\ddot{p}^{2}_{t}\breve{u}^{1}(\text{equality holds iif. }k=z))$ $\displaystyle=$ $\displaystyle\frac{1}{1-\breve{C}}(\breve{b}^{1}_{k}\breve{b}^{2}_{k}+\breve{b }^{1}_{k}\breve{u}^{2}+\breve{b}^{2}_{k}\breve{u}^{1})=\bar{b}_{k}$ $$

∎

###### Proof.

Proof details of Proposition 2. Let $\bar{u}$ and $\bar{u}^{\prime}$ denote the uncertainty of BCF combined opinion with or without Trust Discounting, respectively.

$$ $\displaystyle\bar{u}$ $\displaystyle=$ $\displaystyle\frac{1}{\sum^{K}_{k=1}(\frac{\breve{b}^{1}_{k}\breve{b}^{2}_{k}} {\breve{u}^{1}\breve{u}^{2}}+\frac{\breve{b}^{1}_{k}}{\breve{u}^{1}}+\frac{ \breve{b}^{2}_{k}}{\breve{u}^{2}})+1}$ $\displaystyle=$ $\displaystyle\frac{1}{\sum^{K}_{k=1}(\frac{\acute{b}^{1}_{k}\ddot{p}^{1}_{t} \acute{b}^{2}_{k}\ddot{p}^{2}_{t}}{(\acute{u}^{1}\ddot{p}^{1}_{t}+1-\ddot{p}^{ 1}_{t})(\acute{u}^{2}\ddot{p}^{1}_{t}+1-\ddot{p}^{2}_{t})}+\frac{\acute{b}^{1} _{k}\ddot{p}^{1}_{t}}{\acute{u}^{1}\ddot{p}^{1}_{t}+1-\ddot{p}^{1}_{t}}+\frac{ \acute{b}^{2}_{k}\ddot{p}^{2}_{t}}{\acute{u}^{2}\ddot{p}^{1}_{t}+1-\ddot{p}^{2 }_{t}})+1}$ $\displaystyle=$ $\displaystyle\frac{1}{\sum^{K}_{k=1}(\frac{\acute{b}^{1}_{k}\acute{b}^{2}_{k}} {(\frac{\acute{u}^{1}}{\ddot{p}^{2}_{t}}+\frac{1}{\ddot{p}^{1}_{t}\ddot{p}^{2} _{t}}-\frac{1}{\ddot{p}^{2}_{t}})(\frac{\acute{u}^{2}}{\ddot{p}^{1}_{t}}+\frac {1}{\ddot{p}^{1}_{t}\ddot{p}^{2}_{t}}-\frac{1}{\ddot{p}^{1}_{t}})}+\frac{ \acute{b}^{1}_{k}}{\acute{u}^{1}+\frac{1}{\ddot{p}^{1}_{t}}-1}+\frac{\acute{b} ^{2}_{k}}{\acute{u}^{2}+\frac{1}{\ddot{p}^{2}_{t}}-1})+1}$ $\displaystyle=$ $\displaystyle\frac{1}{\sum^{K}_{k=1}(\frac{\acute{b}^{1}_{k}\acute{b}^{2}_{k}} {(\frac{\acute{u}^{1}}{\ddot{p}^{2}}+\frac{1-\ddot{p}^{1}}{\ddot{p}^{1}\ddot{p }^{2}})(\frac{\acute{u}^{2}}{\ddot{p}^{1}}+\frac{1-\ddot{p}^{2}}{\ddot{p}^{1} \ddot{p}^{2}})}+\frac{\acute{b}^{1}_{k}}{\acute{u}^{1}+\frac{1}{\ddot{p}^{1}}- 1}+\frac{\acute{b}^{2}_{k}}{\acute{u}^{2}+\frac{1}{\ddot{p}^{2}}-1})+1}$ $\displaystyle\geq$ $\displaystyle\frac{1}{\sum^{K}_{k=1}(\frac{\acute{b}^{1}_{k}\acute{b}^{2}_{k}} {\acute{u}^{1}\acute{u}^{2}}+\frac{\acute{b}^{1}_{k}}{\acute{u}^{1}}+\frac{ \acute{b}^{2}_{k}}{\acute{u}^{2}})+1}=\bar{u}^{\prime}$ $$

∎

###### Proof.

###### Proof.

### A.5 Loss Functions and Hyperparameters for Optimization

Recall that the probability density function (pdf) of the Dirichlet distribution, $\text{Dir}(\mathbf{p}\mid\bm{\alpha})$, is given by:

$$ $\text{Dir}(\mathbf{p}\mid\bm{\alpha})=\frac{1}{B(\bm{\alpha})}\prod_{i=1}^{K}p _{i}^{\alpha_{i}-1}$ $$

where:

- •
$\mathbf{p}=(p_{1},p_{2},\ldots,p_{K})$ is a probability vector, such that $\sum_{k=1}^{K}p_{k}=1$ and $p_{k}\geq 0$ for all $k$.
- •
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{K})$ is a vector of concentration parameters, with $\alpha_{k}>0$.
- •
$B(\bm{\alpha})$ is the multivariate Beta function, defined as $B(\bm{\alpha})=\frac{\prod_{k=1}^{K}\Gamma(\alpha_{k})}{\Gamma\left(\sum_{k=1}
^{K}\alpha_{k}\right)}$.
- •
$\Gamma(\cdot)$ is the Gamma function.

Our loss function for Dirichlet Parameters $\bm{\alpha}$ is

$$ $\displaystyle L(\bm{\alpha})=L_{ace}(\bm{\alpha})+\lambda\text{D}_{KL}[\text{ Dir}(\mathbf{p}|\bm{\tilde{\alpha}})||\text{Dir}(\mathbf{p}|\bm{1})]$ (12) $$

Recall that $\tilde{\bm{\alpha}}=\mathbf{y}+(1-\mathbf{y})\odot\bm{\alpha}$ is the Dirichlet parameters after removing non-misleading evidence from predicted parameters $\bm{\alpha}$ and $\mathbf{p}$ is the projected probability, i.e., $\mathbf{p}=\frac{\bm{\alpha}}{S}$.
$\odot$ is the element-wise product and $\lambda_{o}$ is the annealing factor.
We follow [] , and use $\lambda_{o}=min(1.0,o/10)$, where $o$ is the index of the current epoch.

Specifically, the $ace$ loss represents the Bayes risk for Cross-Entropy loss when using a Dirichlet distribution.

$$ $\displaystyle L_{ace}(\bm{\alpha})$ $\displaystyle=$ $\displaystyle\int\left[\sum_{k=1}^{K}-y_{k}log(p_{k})\right]\frac{1}{B(\bm{ \alpha})}\prod_{k=1}^{K}({p_{k}})^{\alpha_{k}-1}d\mathbf{p}$ (13) $\displaystyle=$ $\displaystyle\sum_{k=1}^{K}y_{k}(\psi(S)-\psi(\alpha_{k}))$ $$

Where $\psi$ is the digamma function.

Recall that our referral network will generate the evidence for binomial opinion,
and the evidence will be converted into parameters of Beta Distribution, i.e., $Beta(\alpha_{0},\alpha_{1})$
Subsequently, by replacing the Dirichlet Distribution with Beta Distribution,
and the label $y_{k}$ in above equation with another label, we can have the $ace$ loss for Beta Distribution,
as Eq. (8).

**Table 6: TF and ETF hyper-parameters**
| Hyper-parameter | Handwritten | Caltech101 | PIE | Scene15 | HMDB | CUB |
| --- | --- | --- | --- | --- | --- | --- |
| lr | 3e-3 | 1e-4 | 3e-3 | 1e-2 | 3e-4 | 1e-3 |
| rlr | 3e-4 | 3e-5 | 1e-3 | 3e-3 | 1e-4 | 3e-4 |
| weight-decay | 1e-4 | 1e-4 | 1e-4 | 1e-4 | 1e-4 | 1e-4 |
| warm-up epochs | 1 | 1 | 1 | 1 | 1 | 1 |

The KL divergence loss is

$$ $\displaystyle\text{D}_{KL}\left[\text{Dir}(\mathbf{p}\mid\bm{\alpha})\parallel \text{Dir}(\mathbf{p}\mid\bm{1})\right]$ $\displaystyle=\log\left(\frac{\Gamma\left(\sum_{k=1}^{K}\alpha_{k}\right)}{ \Gamma(K)\prod_{k=1}^{K}\Gamma(\alpha_{k})}\right)+\sum_{k=1}^{K}(\alpha_{k}-1 )\left[\psi(\alpha_{k})-\psi\left(\sum_{j=1}^{K}\alpha_{j}\right)\right]$ (14) $$

The hyper-parameters for training TF and ETF has been shown in in Table 6.
Concretely, "lr" is the learning rate for functional networks, "rlr" indicates the learning rate for referral networks.
For the "lr", we follow ETMC [], and used same strategy to select learning rate for the functional nets.
When tuning the learning rate for referral networks, we follow a basic principle of starting with a value less than or equal to the base learning rate, and then gradually decreasing the learning rate of referral network by a factor of 3.
For fair comparison, we used same learning rate for functional networks for evidence-based methods, except MGP [], where we followed their paper.

We implement our proposed methods (TF and ETF) using Pytorch (version 1.13) [] and used Adam optimizer [] to update network parameters during training on all datasets,
with weight decay 1e-4 following [].

### A.6 Multi-View Agreement with Ground Truth (MVAGT)

The MVAGT (Multi-View Agreement with Ground Truth) is a novel evaluation metric designed specifically for multi-view classification problems with conflicting views. It assesses the model’s performance on the test set by considering the ground truth labels, thus providing a more reliable and realistic measure of the model’s ability to handle view disagreements.
The rationality behind MVAGT lies in its alignment with real-world scenarios, where the majority agreement among multiple views is often considered more reasonable for the final decision. In the presence of view conflicts, a model that can make predictions consistent with the majority of views is deemed more trustworthy and reliable. By evaluating models using MVAGT, we can examine the reasonableness of the fused decision and assess the model’s capability to handle view conflicts effectively.
Mathematically, MVAGT calculates the accuracy of the model on the test set as follows:

$$ $\text{MVAGT}=\frac{1}{M}\sum_{i=1}^{M}\mathbbm{1}\left(\sum_{v=1}^{V}\mathbbm{ 1}((\hat{y}^{v}_{i}=y_{i})>\frac{V}{2}\right)$ (15) $$

where $M$ is the total number of test samples, $V$ is the number of views, $\hat{y}^{v}_{i}$ is the predicted label of the $i$-th sample from the $v$-th view, $y^{i}$ is the ground truth label of the $i$-th sample, and $\mathbbm{1}(\cdot)$ is the indicator function that returns 1 if the condition is satisfied and 0 otherwise.

**Table 7: Summary of Datasets**
| Dataset | Size | K | Dimensions |
| --- | --- | --- | --- |
| HandWritten | 2000 | 10 | 240/76/216/47/64/6 |
| Caltech101 | 8677 | 101 | 4096/4096 |
| PIE | 680 | 68 | 484/256/279 |
| Scene15 | 4485 | 15 | 20/59/40 |
| HMDB | 6718 | 51 | 1000/1000 |
| CUB | 600 | 10 | 1024/300 |

### A.7 Summary of Dataset

We provide the summary of the dataset in Table 7, we direct readers to [] for further details regarding
these datasets.

## Appendix B Supplementary Insights and Additional Analysis

### B.1 Reduce Conflicts by Trust Fusion

Figure: Figure 4: Conflict Ratio on Scene15, Four Methods TMC, TF, ETMC, ETF are compared. GT, Pred, PS are ground-truth view, prediction view, pseudo-view respectively, View 1, 2 and 3 are corresponding to feature view 1, 2 and 3.
Refer to caption: extracted/5638815/imgs/tmc-scene-conflict-hm.png

We calculate the Conflict Ratio (CR) by normalizing the number of times that the $v$-th view prediction is different from $w$-th view, i.e., $\text{CR}(\hat{\bm{y}}^{v},\hat{\bm{y}}^{w})=\frac{1}{M}\sum_{i=1}^{M}\mathbbm
{1}(\hat{y}^{v}_{i}\neq\hat{y}^{w}_{i})$, where $M$ is total number of test instances, $\hat{y}^{w}_{i}$ is the predicted label of $i$-th instance on $w$-th view,
and $\mathbbm{1}$ is the indicator function that returns 1 if the condition is satisfied and 0 otherwise. By applying Trust Discounting, both TMC’s and ETMC’s conflicts between different views are significant reduced.
As an example, the CR on Scene15 is visualized by heatmap, shown in Figure 4. The colors in the heatmap generated by our method are noticeably more blue (or less red) than those of the baselines, indicating that the conflict ratio has been reduced by our method.

### B.2 Explanation for the decrease of AUC-ROC for Uncertainty-Aware Prediction on CUB dataset

Figure: Figure 5: Bar chart for belief mass on each label and predictive uncertainy on one instance of CUB. GT indicates the ground truth label of the selected instance.
Refer to caption: extracted/5638815/imgs/conflict-joint.png

As the error case displayed in Figure 5, ETF corrects the error prediction made by ETMC. However, even though the combined prediction is correct after applying trust discounting, the predictive uncertainty is still relatively high.
If ETF corrects previously incorrect predictions but assigns them relatively high uncertainty scores (e.g., 0.4), it may lead to a decrease in the AUC-ROC for predictive uncertainty.
This is because AUC-ROC evaluates the model’s ability to discriminate between correct and incorrect predictions based on uncertainty scores.
Correcting predictions while maintaining high uncertainty scores can make it more challenging for the model to distinguish between correct and incorrect predictions, resulting in a lower AUC-ROC score, even though the accuracy improves.

Figure: Figure 6: Performance comparison of simulated conflict predictions.
Refer to caption: extracted/5638815/imgs/conflict-hw-2.png

### B.3 Simulating Conflicting Predictions with Noisy Instances

Besides the one we show in Figure 3,
we also plot the model performance for evidence-based methods that do not incorporate pseudo views, as shown in Figure 6.
Our method TF consistently outperforms other methods like TMC and ECML.

Figure: Figure 7: Performance comparison of simulated conflict predictions.
Refer to caption: extracted/5638815/imgs/abl-hw.png

### B.4 Ablation Study of Hyper-parameters

We presented results using a non-over-tuned empirical value of 1 for the hyperparameter ’warm-up epochs’ in all previous experiments. These results indicated our model’s robust performance, as it consistently outperformed baseline models.
Here we demonstrate the effect of this hyperparameter using more finely graded values, starting from 0 (which indicates no warm-up stage) and increasing steadily, for example, to 2, 5, and 10.

From Figure 7, we can find that incorporating warm-up stage (warm-up epochs > 0) can generally results in better model performance.
For some datasets (e.g. HMDB), increasing the number of warm-up epochs further improves performance compared to the results previously reported.
This observation suggests that adjusting this value based on the specific dataset can lead to enhanced performance.

## Appendix C Technical Requirements and Execution

**Table 8: Execution Time for ETF and TF on 24GB RTX3090 (Single run)**
| Method | Handwritten | Caltech101 | PIE | Scene15 | HMDB | CUB |
| --- | --- | --- | --- | --- | --- | --- |
| TF | 4min30s | 8min40sec | 55s | 4min50sec | 5min30s | 40sec |
| ETF | 5min | 11min | 1min20sec | 7min30sec | 8min | 50sec |

### C.1 Execution Time

We report the execution time of single run of the training process, which is shown in
Table 8.
For example, 1min20sec in the table for ETF on PIE means training ETC on PIE dataset using one 24GB RTX3090 GPU card will cost about 1 minute and 20 seconds. In our experiment, we will repeat this step 10 time for calculating the mean and standard error.

## Peer Reviews

confidence: 2
contribution: 3
initial_rating: 6
presentation: 3
questions: It's unclear how this work scales in dynamic situations, such as the autonomous driving example cited early on, where there may be a variety of instances that differ significantly. The appendix has characteristics of said datasets, but the paper does not discuss how similar or varied the instances are, apart from a brief reference to Caltech101 having lower conflict. Could such data be provided?

The information that Table 1 provides is not clear. Is there real data being used to generate the table?  If not, could you clarify what information the table is meant to convey?

Could you clarify the impact of the warm-up epochs on the overall performance? Based on Figure 3, there appears to be limited improvement at best. If only 1 dataset shows improvement, the inclusion of the ablation study seems suspect. Are there simpler procedures that might be used if warm-up epochs do not significantly enhance results?

Could you provide additional context on how MVAGT complements standard MVC metrics? It's explained in the appendix but I would appreciate further detail.
soundness: 3
strengths: The paper is well-written, clearly setting up the problem and explaining the original proposed method, with mathematical justifications provided. The approach introduces a dynamic weighting mechanism that adjusts the influence of each view based on the instance, a relevant adaptation in MVC contexts where view reliability can vary. This instance-wise adaptability demonstrates originality, improving upon  to static weighting systems commonly used in MVC. The method performs consistently well against several baselines, underscoring its robustness and flexibility across diverse scenarios. By prioritizing trustworthy views, this approach has significant potential to improve decision-making reliability in settings where the importance of each view shifts with changing conditions.  The paper mentions domains such as autonomous driving as examples where this technique could improve accuracy.
summary: Not all views can be trusted to the same degree in the field of Multi-View Classification. Trust Discounting-Based Trust Fusion (TDTF) is proposed to model computational trust, emphasizing trustable views in each instance. This trust discounting mechanism aims to prioritize views that are more likely to align with the ground truth while discounting views with greater uncertainty or variable accuracy.

The method is tested across a variety of datasets and compare it to several baseline models. In addition to traditional MVC methods, TDTF is compared with baselines that incorporate static and weighted fusion techniques to asses its relative performance.

The results show TDTF based methods outperforming baselines in most scenarios. An ablation study also examines the effects of warm-up epochs on stabilizing trust scores, and the paper emphasizes the importance and necessity of these warm-up epochs.
weaknesses: The first weakness that stands out is the placement of the algorithm in the appendix. Understanding the method, as well as details about the training process are deferred to the end of the paper, making comprehension somewhat difficult. The process is key to understanding the paper and its results, and placing it in the appendix is a significant flaw, in my view.

The placement of the related work interrupts the flow of the paper - in my view, the context for MVCs and where this paper's contributions fit should come earlier.

The ablation study, while welcome, feels out of place without the inclusion of the algorithm within the body of the paper. The process loses significant clarity in doing so, and much of the results become harder to parse without it.

The original methods add significant time - with the exception of MGP, the author's methods take significantly longer to train.

The diversity of the instances of the datasets is not mentioned, making it difficult to know how well the trust mechanism works in diverse contexts.

There are minor spelling/grammatical errors, but none that affect understanding.

---

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: The paper does not provide information on the number of views in each of the datasets used. How does the trust discounting mechanism perform as the volume of data and the number of views increase?

Furthermore, the Caltech101 dataset in the appendix, which has the highest number of categories among the tested datasets, also happens to be the one where your model underperforms. This raises a concern about whether the model may struggle with datasets that have a large number of categories.

The others can see the weakness part
soundness: 2
strengths: Originality

The paper introduces a trust-based discounting mechanism that innovatively integrates subjective logic with instance-wise probability-sensitive trust discounting to enhance the reliability of multi-view predictions. This approach offers a perspective in the field of multi-view learning, especially in dealing with conflicting information from various views.

Quality

The paper conducts robust experiments on six real-world datasets and uses a comprehensive set of evaluation metrics, including a new metric for multi-view agreement with ground truth, to validate the method's effectiveness and generalizability.

Clarity

The paper is organized with explanations of technical concepts that are generally clear. The use of figures and tables to support the key points is adequate. 

Significance

The paper addresses conflicting information in multi-view classification and improve the reliability and accuracy of predictions, with broad applicability in the field of multi-view learning.
summary: The paper presents a trust-based discounting method to improve multi-view classification (MVC) by addressing conflicts between different views. It extends the Evidential Multi-view framework using subjective logic, allowing for more reliable predictions. The approach involves an instance-wise, probability-sensitive trust discounting mechanism to evaluate and integrate view-specific predictions. The method is validated on six real-world datasets, showing improved performance in scenarios with significant discrepancies among views. Key contributions include a trust-discounting mechanism, a stage-wise training strategy, and comprehensive experimental validation demonstrating enhanced conflict resolution and prediction reliability.
weaknesses: 1.	The paper presents an incremental innovation in the field of multi-view learning by introducing a trust-based discounting mechanism that leverages subjective logic for uncertainty estimation. While the approach offers a novel perspective, the methodological contribution appears to be primarily theoretical, with the actual implementation relying on existing techniques such as the loss functions from the Trusted Multi-View Classification. This suggests that the paper may not offer substantial improvements over current methods in terms of loss function optimization.

2.	In terms of experimental validation, while the paper does provide a comprehensive set of evaluation metrics and tests the method on six real-world datasets, there is room for expansion. The datasets used may not fully capture the challenges associated with uncertainty in multi-view learning. For example, Caltech101 dataset, which has the highest number of categories among the tested datasets, also happens to be the one where your model underperforms. This raises a concern about whether the model may struggle with datasets that have a large number of categories. It would be beneficial to include larger and more diverse datasets to better reflect the complexities and uncertainties inherent in multi-view classification tasks. This would provide a more robust assessment of the method's effectiveness and its ability to handle increased data volume and the number of views.

3.	The motivation behind the paper is somewhat broad, and it does not clearly differentiate itself from existing methods that also address view conflicts, such as ECML in comparative methods. The paper could benefit from a more detailed discussion on how its approach differs from and potentially outperforms these methods, especially in scenarios with a large number of categories or high-volume data.

4.	The paper lacks a detailed description of the view-specific neural network's architecture and implementation, which is crucial for understanding the model's performance and reliability. Without this information, the reproducibility of the method is compromised, and it is difficult to assess the specific contributions of the proposed method to the overall performance.

5.	The ablation study is limited to hyperparameter tuning and does not provide a thorough analysis of the key components of the proposed method. A more comprehensive ablation study that systematically removes or modifies key components would provide clearer insights into their individual contributions to the model's performance. This could include varying the trust discounting factor or the influence of the evidence network on the final predictions.

---

confidence: 5
contribution: 2
initial_rating: 5
presentation: 3
questions: See in the weakness.
soundness: 2
strengths: 1. Addressing conflicts among different views is crucial for enhancing model reliability. 
2. Empirical and theoretical analyses provide an evaluation of the proposed method’s effectiveness.
summary: This paper aims to address conflicting predictions among different views through a method based on computational trust discounting. Specifically, computational trust is modeled through an evidence network that operates on a view-specific and instance-wise basis. Extensive experiments on six real-world datasets show that the proposed method achieves better performance.
weaknesses: **Major Issues**

### Method
1. Proposition 2 introduces an important point: the prediction uncertainty for conflicting views is higher than that obtained through traditional evidence-based fusion, which can help flag incorrect predictions in multi-view fusion with conflicting views. However, there are some issues. In equation (7), the authors use KL divergence as a regularization term for incorrect evidence. During optimization, particularly on the feature-based datasets used in this study, this term may overlook the influence of correct predictions. This often drives evidence for all classes towards zero, resulting in a high uncertainty score even for trusted predictions. The slight increase in uncertainty (as shown in Table 2) does little to improve the identification of incorrect predictions.

2. If I understand correctly, this work aims to assess the trustworthiness of each view’s prediction using referral networks. When distrust is indicated, it implies a conflict with the ground truth, rather than with other views. How can the method theoretically guarantee an improvement in classification performance when dealing with conflicting views, especially if the referral networks themselves have poor performance?

3. Based on the points above, I would like to discuss with the authors how these two propositions theoretically guarantee the effectiveness of the proposed method.

### Experiments
The experimental analysis needs to be more comprehensive. Specifically:

1. In Table 2, the results show that the proposed approach enhances the reliability of evidence-based fusion methods (i.e., BCF, A-CBF, and ABF). However, the experiments lack a direct comparison of these methods with and without the proposed approach in identical settings.

2. The selection of comparison methods is limited. It is recommended to include additional evidence-based multi-view methods, such as TMDOA [1], CCML [2], and TMNR [3].

3. All six datasets used are feature-based. To introduce more challenging tasks, consider adding multi-view datasets with text or image data. Additionally, incorporating real-world cases could visually demonstrate conflicts among different views and showcase the effectiveness of this method.

### Clarifications
1. Line 322 states: “However, this approach may inadvertently lead correct predictions to converge towards incorrect ones, potentially jeopardizing model stability.” Please provide support for this point.

**Minor Issues**

1. In Table 1, the trust score for Dolphin (referral) is 0.90, but Line 179 states, “we know the trust score for Dolphin’s opinion is 0.95.” Please clarify this discrepancy.

[1] Liu W, Yue X, Chen Y, et al. Trusted multi-view deep learning with opinion aggregation[C]//Proceedings of the AAAI Conference on Artificial Intelligence. 2022, 36(7): 7585-7593.

[2] Liu Y, Liu L, Xu C, et al. Dynamic Evidence Decoupling for Trusted Multi-view Learning[J]. arXiv preprint arXiv:2410.03796, 2024.

[3] Xu C, Zhang Y, Guan Z, et al. Trusted Multi-view Learning with Label Noise[J]. arXiv preprint arXiv:2404.11944, 2024.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: 1. lines 178-179: the last row in Table 1 is Polar Bear rather then Dolphin; also, please explain where is the trust score of 0.95 coming from?

2. line 408: please explain the statement "Caltech101 is a dataset with fewer conflicts ...." How did you quantify it? Please be specific about the performance gap that you are are talking about.  Also, please quantify the conflicts in each of the six datasets, then analyze the results (eg, how does this "measure of conflicts" explain/correlate-with the results on other datasets?)

3. line 423: In Table 4, the performance gap between ETF and F-Mode is a lot larger on CUB compared to Caltech101. What is the explication of this gap? Also add an intuitive explanation of why does ETF massively outperforms F-Mode on CUB in Table 3, but situation is flipped in Table 4.

4. Overall, you should choose the most important results in Section 4 and discuss them in depth (for example, AUROC & ABLATION could easily be moved to the APPENDICES)

5. It would be interesting to discuss (intuitively & in-depth) why, for Handwritten, all accuracies are above 99%, while Fleiss' Kappa and MVAGT are spread over a much wider range. Please provide/discuss/analyze specific hypotheses for explaining this phenomenon; if necessary, run additional experiments to confirm/infirm them. Feel free to use "the 5-WHYs rule" that requires you to repeatedly ask WHY? until you identify the true root-cause(s).
soundness: 3
strengths: The paper introduces a novel approach to improving learning in the multi-view scenario, which has many real-world applications, from self-driving cars to medical diagnosis. The work appears to be original, and its empirical evaluation shows that it may have a significant practical impact. The paper's quality and clarity are the focus of the next section.
summary: The paper introduces a novel approach to resolving conflicts between the various views in a multi-view learning scenario. The authors 
propose a computational, trust-based discounting method that enhances the existing Evidential Multi-view framework when the various views conflict with each other. The paper also introduces a new metric, Multi-View Agreement with Ground Truth, for measuring the reliability of the predictions.
weaknesses: The main weakness of the paper the overall quality and clarity of the presentation: in trying to turn a 22-page paper into a 12-page one without removing/compressing any content, the authors made some choices/trade-offs that negatively impact the paper's readability. In its current form, after a reasonable Abstract and Introduction, the paper goes straight into 3.5-page theory section, followed by related work (which should precede it) and then the experimental section. The main contributions of the paper (Algorithms 1 & 2, and even the definition of the novel metric, MVAGT) are relegated to the 10-page APPENDICES, which also include many other details.

In this reviewer's opinion, the paper's quality & readability would be greatly improved by moving most of Section 2 to the APPENDICES. The new structure would be: Introduction, Related Work, Intuitive end-2-end running example (which will re-user some parts of the current Section 2), Algorithms & Metrics, and Experiments. In this form, the authors will have all the space required to INTUITIVELY explain the key concepts and to compare & contrast with existing approaches. Even though this requires adding a new section, it is worth doing, as it will greatly impact the paper's impact & penetration.

For the Related Work section:
- 1st pgf: please explain intuitively (i) why don't noise-robust representation "fully resolve conflicts", and (ii) the main ideas behind cluster-based, self-representation-based, and partially view-alligned approaches
- 2nd pgf: please compare-and-contrast TMC vs your approach
- 3rd pgf: please explain intuitively why (i) CMVC "may inadvertently lead incorrect predictions converge to incorrect ones", and (ii) why this is NOT the case for your approach 

For the new "Intuitive end-2-end running example" section:
- please modify the title/term of TDTF, as it is confusing because it uses the word TRUST twice in a 5-gram; may change to "Trust Fusion based on Trust Discounting" or to something else that is easy to parse & understand.
- the new section could be based on the existing Titanic domain, but it should have at least three examples/instances: (i) one in which there is no conflict, (ii) one in which your approach solves the conflict, and (iii) one in which your approach fails to solve the conflict. If (ii) and (iii) have more then one major use-case, you should add one example for each of them
- Table 1 is a good start for a single example, but it should be expanded to all the examples mentioned in the bullet above. It should also explain (rather than rely on the reader to deduce) how the top part of the table is created from Figure 1 (one sentence would suffice), and how the bottom part is computed.
- please provide an intuitive explanation of BCF and how it is applied to the expanded version of Table 1 (which will include 3+ examples)
- please do NOT intermix the BCF story with A-CBF and ABF. If it is important to compare-and-contrast, do it at the end of this new section or add it to the related work section (in which case it should follow rather than precede this new section). Otherwise, relegate it to the Appendix.  
- please explain how the values in the to-be-expanded Table 2 (to 3+ examples) are computed
- you should apply algorithms 1 & 2 to the examples in tables 1 & 2, and discus the results

Meta-review: The authors study learning from multiple views. The technical contribution uses an existing fusion method [1] to deal with conflicting views which has also been explored in [2] but it does not really become obvious what the advantage/difference over [2] is. Also the empirical evaluation offers room for improvement. The data is rather standard and may not be well-suited for the studied problem of conflicting views (e.g., in contrast to the validation in [2]). In sum, the paper addresses an interesting and important topic but is not yet matured enough for publication at ICLR.

Decision: Reject