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Are Independently Estimated View Uncertainties Comparable? Unified Routing for Trusted Multi-View Classification

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arXiv:2604.09288v1 [cs.LG] 10 Apr 2026

Are Independently Estimated View Uncertainties Comparable? Unified Routing for Trusted Multi-View Classification Yilin Zhang

Cai Xu

Haishun Chen

Xidian University Xi’an, China ylzhang_3@stu.xidian.edu.cn

Xidian University Xi’an, China cxu@xidian.edu.cn

Xidian University Xi’an, China chenhaishun@stu.xidian.edu.cn

Ziyu Guan

Wei Zhao

Xidian University Xi’an, China zyguan@xidian.edu.cn

Xidian University Xi’an, China ywzhao@mail.xidian.edu.cn

Abstract

not only make correct predictions, but also be able to dynamically assess which views are more trusted for the current sample. To this end, recent trusted multi-view learning methods [6, 10, 25, 34] are largely built on evidential learning [3, 27] and subjective logic [16, 28]. A typical paradigm is shown in Fig. 1(a), which let each view-specific branch produce non-negative class evidence, convert it into an uncertainty-aware opinion, and then fuse these opinions according to their branch-wise uncertainty. This line of work is attractive because it is modular, compatible with heterogeneous encoders, and naturally provides an interpretable fusion process through uncertainty-aware weighting. However, it also relies on an assumption that is rarely questioned explicitly: the evidence and self-assessed uncertainty from different views are treated as directly comparable across branches. We find that this assumption is in fact fragile, and it arises from two sources: (1) Data-level heterogeneity. Different views often reside in different feature spaces and exhibit different signal-tonoise ratios, statistical complexity, and levels of semantic abstraction, which naturally induce different evidence distributions. (2) Model-level heterogeneity. Each branch is trained primarily to improve its own classification objective, typically by increasing the evidence assigned to the ground-truth class and suppressing other classes. Such supervision encourages per-branch correctness, but places no explicit constraint on the absolute strength of evidence across views [35]. Moreover, different architectural choices can further amplify this mismatch. Therefore, features of the same quality may still yield evidence with significantly different strengths. A toy experiment, as shown in Fig. 1(b), demonstrates that for the same input features, simply changing the depth of one branch can lead to significant differences in evidence strength and induce different uncertainty values. This issue becomes particularly critical in trusted multi-view fusion, because subjective-logic uncertainty is directly tied to total evidence strength. When the uncertainty used for fusion weighting is dominated by branch-dependent scale rather than true samplewise reliability, it is meaningful only within each branch and loses fairness as a cross-view arbitration signal, as shown in Fig. 1(c). Therefore, the challenge is not merely how to estimate uncertainty within each branch, but how to perform fair cross-view arbitration when evidence scales are not directly comparable. This observation calls for a rethinking of the design principle of trusted multi-view fusion. If raw cross-view evidence cannot

Trusted multi-view classification typically relies on a view-wise evidential fusion process: each view independently produces class evidence and uncertainty, and the final prediction is obtained by aggregating these independent opinions. While this design is modular and uncertainty-aware, it implicitly assumes that evidence from different views is numerically comparable. In practice, however, this assumption is fragile. Different views often differ in feature space, noise level, and semantic granularity, while independently trained branches are optimized only for prediction correctness, without any constraint enforcing cross-view consistency in evidence strength. As a result, the uncertainty used for fusion can be dominated by branch-specific scale bias rather than true sample-level reliability. To address this issue, we propose Trusted Multi-view learning with Unified Routing (TMUR), which decouples view-specific evidence extraction from fusion arbitration. TMUR uses view-private experts and one collaborative expert, and employs a unified router that observes the global multi-view context to generate sample-level expert weights. Soft load-balancing and diversity regularization further encourage balanced expert utilization and more discriminative expert specialization. We also provide theoretical analysis showing why independent evidential supervision does not identify a common cross-view evidence scale, and why unified global routing is preferable to branch-local arbitration when reliability is sample-dependent. Extensive experiments on 14 datasets and comparisons with 15 recent baselines demonstrate that TMUR consistently improves both classification performance and reliability. The code will be released upon publication.

Keywords Trusted Multi-view Classification, Uncertainty-aware Deep Learning, Evidential Deep Learning.

Introduction

Multi-view classification [17, 22, 36] aims to improve prediction by integrating complementary information from multiple views. In many real-world scenarios, different views may exhibit distinct noise levels [32], semantic granularity, or even conflicting clues [11] for the same sample, causing their quality to vary dynamically across samples. As a result, a reliable multi-view model should 1

Are Independently Estimated View Uncertainties Comparable?

(a) Conventional assumption 𝑢1



𝒆1 𝑢2

Text Classifier

Fusion

Audio Classifier

k =1

(e1k + 1)

𝒆

Branch 1 (deeper)



𝑢3 Class-wise Evidence 𝒆3

Weighted by Self-assessed Uncertainty

𝑢1

𝒆2

Reliability Diagram on Caltech K K

K k =1

(e + 1) 2 k

= 𝑢2

Accuracy

Image Classifier

(b) Why the assumption fails

Same uncertainty, Different accuracy

𝒆2

Branch 2 (shallow)

Uncertainy

Same input features, different evidence scale induce different uncertainty values

Real multi-view data show inconsistent reliabllity across views

Figure 1: (a) Conventional trusted multi-view fusion uses branch-local self-assessed uncertainty for weighting. (b) However, heterogeneous branches can produce different evidence scales, so even the same input may yield different uncertainty values. (c) On Caltech dataset, different views show inconsistent reliability under the same uncertainty level. This suggests that branch-local uncertainty is not directly comparable across views and may be unreliable for cross-view arbitration.

2 Related Work 2.1 Multi-View Learning

be compared reliably in magnitude, then fusion weights should no longer be determined solely by each branch’s own uncertainty. Instead, they should be assigned by an independent arbitration module that has access to the global multi-view context. Motivated by this, we propose Trusted Multi-view learning with Unified Routing (TMUR), a framework that explicitly decouples viewspecific evidence extraction from fusion arbitration. Specifically, TMUR equips each view, together with one global view, with its own expert network, and employs a unified router that observes the global multi-view context to generate sample-wise weights over these experts. In this way, fusion authority is no longer inferred only from branch-local uncertainty, but is instead determined through global, sample-aware routing. This design preserves the trusted nature of evidential prediction while mitigating the fusion bias induced by raw cross-view evidence-scale mismatch. We also provide a theoretical analysis explaining why independent evidence supervision cannot determine a universal evidence scale across views, and when view reliability depends on the sample, global dynamic routing is better grounded than branch-local self-weighting. In summary, this work makes the following contributions:

Multi-view learning aims to exploit complementary yet heterogeneous views for stronger prediction. Recent work improves multiview reasoning through stronger cross-view alignment [21], trustaware pairwise modeling [26], and uncertainty-refined representation learning [12]. In the classification setting, trusted multi-view methods such as TMC [9] and its dynamic evidential extension [10] formulate the problem through view-wise evidence extraction and decision-level fusion. Subsequent studies further enhance this line by modeling inter-view conflict [34], leveraging trust-aware learning under conflicting views [25], introducing fuzzy evidence for reliable classification [6], addressing ambiguity and debiasing in open-set settings [7], and incorporating expert knowledge constraints [20]. These efforts clearly demonstrate the importance of reliability-aware multi-view learning. Our work follows this trusted multi-view learning line, but focuses on a different bottleneck: whether the evidence produced by different branches is numerically comparable enough to support direct fusion.

• We identify cross-view evidence-scale incomparability as an important yet underexplored bottleneck in trusted multi-view classification, and trace it to both data-level and model-level heterogeneity. • We propose TMUR, a unified-routing framework that decouples evidential prediction from fusion arbitration and reduces scale-induced bias in cross-view fusion. • We provide theoretical analysis clarifying why independent evidential supervision cannot guarantee cross-view evidence comparability, and why dynamic global routing is preferable when view reliability is sample-dependent. • We validate TMUR on extensive multi-view benchmarks, reducing average ECE by 8.27 points and achieving consistently strong classification performance.

2.2

Evidential Fusion and Subjective Logic

Evidential Deep Learning maps network outputs to Dirichlet parameters and provides a natural interface for predictive uncertainty [27], with later extensions modeling richer forms of evidential uncertainty [18]. In multi-view settings, these outputs are often combined through subjective logic or Dempster–Shafer style fusion [16, 28]. A related line estimates branch quality from uncertainty, total evidence, calibration, or belief discrepancy, and then reweights the resulting opinions accordingly [1, 7, 8, 10, 30]. However, these signals are still generated independently by each branch, so their magnitudes can remain entangled with branch-specific data and model biases rather than true sample-level reliability. In other words, reweighting branch-local opinions by branch-local 2

Are Independently Estimated View Uncertainties Comparable?

Í Let Dirichlet strength 𝑆 = 𝐾𝑗=1 𝛼 𝑗 . The corresponding predictive probability and subjective-logic uncertainty are 𝐾 𝛼𝑘 𝑢= . (3) 𝑝𝑘 = , 𝑆 𝑆 Equivalently, the associated opinion can be written as O = (𝒃, 𝑢) with belief masses 𝑒𝑘 𝑏𝑘 = , 𝑘 = 1, . . . , 𝐾 . (4) 𝑆 This formulation makes the uncertainty directly depend on the total evidence magnitude. As a result, if different views produce evidence on different numerical scales, then their induced uncertainties need not be directly comparable across views, even when their semantic support patterns are similar.

quality estimates does not by itself make cross-view evidence numerically comparable. Classical evidence theory has long warned that fusion behavior becomes unreliable when the compatibility assumptions behind the combination rule are not justified [37]. Our method is motivated by this gap: instead of continuing to infer fusion authority directly from branch-local evidence quality, we delegate arbitration to a separate global router.

2.3

Dynamic Routing and Expert-Based Arbitration

Sample-dependent fusion has a long history in early expert-routing models [15] and modern routing architectures [29]. In multimodal learning, dynamic fusion has been used to adapt fusion weights to changing modality quality or sample context [38], and predictive fusion methods also exploit sample-wise uncertainty signals to guide fusion decisions [1]. At the same time, recent theory shows that dynamic fusion is not automatically robust: poorly designed weighting rules can even exacerbate modality greediness [4]. This makes the arbitration mechanism itself a first-class design issue. Recent routing-based expert models further show that expert specialization is useful across diverse multi-view or multimodal applications, including multimodal interaction modeling [33], molecular property prediction [31], time-series forecasting [14], and multi-view clustering [39]. Our use of routing is different. We introduce a unified router not as a generic capacity-expansion module, but as a dedicated arbitration mechanism that reduces the dependence of fusion authority on direct comparison of raw view-wise evidence.

3.2

Unified Routing and Evidential Fusion

To address this issue, TMUR explicitly separates two roles that are often entangled in existing trusted multi-view fusion methods: view-wise evidence extraction and cross-view fusion arbitration. The first role is to let each view-specific branch produce trusted evidence while preserving its own semantics. The second role is to decide, under the joint multi-view context of the current sample, how much each expert should contribute to the final decision. Based on this separation, TMUR maintains an expert pool composed of 𝑉 private experts and one collaborative expert, and introduces a unified router that assigns sample-wise fusion weights from the joint multi-view context. The overall architecture is shown in Fig. 2. 3.2.1 Router and Experts Settings. Specifically, We first map all views into a dimension-aligned space:

Method

𝒉 (𝑣) = LN(𝑃 𝑣 (𝒙 (𝑣) )),

We propose a unified-routing framework for trusted multi-view classification. The core principle is to decouple view-wise evidence extraction from multi-view fusion process. Instead of letting each view-specific branch explicitly determine its own fusion authority through its local evidence magnitude or uncertainty, we assign this role to a specialized router that observes the joint multi-view context. In this way, the model retains the advantages of trusted multi-view learning, dynamic fusion and uncertainty awareness, while avoiding direct reliance on the comparability of raw crossview evidence strengths.

𝑯 = [𝒉 (1) ; 𝒉 (2) ; · · · ; 𝒉 (𝑉 ) ].

Problem Definition

(1)

where 𝒙𝑛(𝑣) ∈ R𝑑 𝑣 is the feature of the 𝑣-th view of the 𝑛-th sample, and 𝑦𝑛 ∈ {1, . . . , 𝐾 } is its class label. For simplicity, we omit the sample index 𝑛 below when no confusion arises. In trusted multi-view learning [5, 23], each branch is expected to produce both class evidence and an uncertainty estimate. Following evidential learning, a predictor outputs a non-negative evidence vector 𝒆 ∈ R𝐾≥0 , from which the Dirichlet parameters are defined as: 𝜶 = 𝒆 + 1.

(6)

𝑣 = 1, . . . , 𝑉 ,

(7)

while the collaborative expert receives the concatenated multi-view representation, 𝒛 (𝑉 +1) = 𝐸𝑉 +1 (𝑯 ). (8) This design allows the private experts to express view-specific evidence, while the collaborative expert may captures complementary cues that emerge only after combining multiple views. To provide the router with richer cross-view context, we further construct interaction-enhanced features through cross-view attention: 𝒉˜ (𝑣) = Attn 𝒉 (𝑣) , {𝒉 ( 𝑗 ) }𝑉 , {𝒉 ( 𝑗 ) }𝑉 , (9)

We consider a 𝑉 -view classification problem with 𝐾 classes. The training set is denoted by: 𝑁 D = {(𝒙𝑛(1) , 𝒙𝑛(2) , . . . , 𝒙𝑛(𝑉 ) , 𝑦𝑛 )}𝑛=1 ,

(5)

The expert pool contains 𝑉 private experts and one collaborative expert. Each expert is implemented as a MLP. The 𝑣-th private expert receives only the aligned feature of the 𝑣-th view, 𝒛 (𝑣) = 𝐸 𝑣 (𝒉 (𝑣) ),

3.1

𝑣 = 1, . . . , 𝑉 ,

where 𝑃 𝑣 (·) is a view-specific projection layer and LN(·) denotes LayerNorm. This step improves cross-view feature compatibility while retaining view-specific information. We then concatenate the aligned view features as:

𝑗=1

where the query is the aligned feature of the current view and the keys and values are the aligned features from all views. We then concatenate the enhanced features as: 𝒈 = [𝒉˜ (1) ; 𝒉˜ (2) ; · · · ; 𝒉˜ (𝑉 ) ]. (10)

(2) 3

Are Independently Estimated View Uncertainties Comparable?

Private Expert 1

𝒉(2)

Private Expert 2

𝒉(1)

⋮

𝒉(V)

Private Expert V

𝒉(2)

𝑯

Collabrative Expert

𝓞 𝑓𝑢𝑠𝑒𝑑

𝒆𝑓𝑢𝑠𝑒𝑑 = ෍ 𝜋𝑖 𝒆(𝑖)

𝑏3

View-Specific Projection

(2)

𝒉

𝑊𝑄

𝑸

𝑊𝐾 𝑲

⋮ 𝒉(V)

Data flow Training loss

𝑊𝑉

𝑏2

Fused Opinion

𝑽

෩ (1) 𝒉

Add

෩ (2) 𝒉 ෩ (V) 𝒉

𝓛𝑏𝑎𝑙

Router

⋮

𝒉(1)

𝑏1

⋮

𝒉(V)

Cross-View Attention

View V

𝑢 Expert-specific Opinions

Unified Awareness Routing

Aligned View Features

Projector V

𝑗=1

⋮

𝑖

⋮

𝐾

𝑘=1

𝜶(𝒊) = 𝒆(𝒊) + 1

⋮

LayerNorm

Projector 2

𝐾

𝓛𝑐𝑙𝑠 = ෍ 𝑦𝑗 𝜓(෍ 𝛼𝑗(𝑖) ) − 𝜓(𝛼𝑘(𝑖) )

⋮

𝓛𝑑𝑖𝑣 Evidential Head

𝒉(1) Projector 1

Unified Evidential Fusion

Evidence Expert Prediction Pool

View Alignment

⋮ Expert weights 𝝅

Prevent Weight Collapse

Enhanced Features

Figure 2: Overview of TMUR. Aligned per-view features are sent to view-private experts for view-specific evidence extraction, while their concatenation is processed by a collaborative expert for joint multi-view evidence. In parallel, a unified router performs router-side cross-view interaction and predicts sample-wise expert weights from the global multi-view context. The final prediction is obtained by weighted evidence aggregation, so fusion authority is assigned by the router rather than by directly comparing branch-local evidence or uncertainty. These enhanced features are used to form the routing context, whereas the private experts still operate on their own aligned view features. In this way, cross-view interaction affects how experts are weighted, rather than replacing the view-specific evidence produced by the private experts themselves. Each expert outputs a non-negative evidence vector through an evidential classification head:

The final fused evidence is obtained by weighted aggregation: 𝒆 fused =

(14)

The corresponding fused opinion is denoted by O fused = (𝒃 fused, 𝑢 fused ),

𝑖 ∈ {1, . . . , 𝑉 + 1},

(11) 𝑏𝑘fused =

where 𝜙 (·) is a non-negative activation function. We use a softplusbased implementation in practice. The corresponding Dirichlet paÍ𝐾 rameters are 𝜶 (𝑖 ) = 𝒆 (𝑖 ) + 1. Let 𝑆 (𝑖 ) = 𝑘=1 𝛼𝑘(𝑖 ) . Then the opinion associated with expert 𝑖 is denoted by O

𝜋𝑖 𝒆 (𝑖 ) .

𝑖=1

(15)

where

𝒆 (𝑖 ) = 𝜙 (𝒛 (𝑖 ) ),

(𝑖 )

𝑉 +1 ∑︁

= (𝒃

(𝑖 )

,𝑢

(𝑖 )

𝑅(𝒈) ∈ R𝑉 +1, 𝜏

, 𝑆 fused

𝑢 fused =

𝐾 .

(16)

𝑆 fused

The predictive class probabilities can still be obtained from the fused Dirichlet mean: 𝑝𝑘fused =

𝛼𝑘fused

. (17) 𝑆 fused This fusion mechanism preserves the evidential output form while reassigning the arbitration role. Existing trusted multi-view fusion methods often let each branch both produce evidence and implicitly influence its own fusion weight through branch-local uncertainty. In contrast, TMUR keeps evidence generation inside the experts and lets a unified router assign fusion authority from the joint multi-view context. Consequently, the final weights are guided primarily by sample-level cross-view context, which reduces their dependence on branch-specific evidential scale.

(12)

3.2.2 Router-Driven Evidential Fusion. If raw evidential magnitude is not a reliable cross-view arbitration signal, then fusion weights should not be inferred directly from branch-local evidence or uncertainty alone. Instead, they should be assigned by a dedicated routing mechanism that can observe all views jointly. Based on the enhanced multi-view context 𝒈, we compute the sample-wise expert weights as: 𝝅 = softmax

𝑒𝑘fused

3.3

(13)

Training Objective

Our training objective is composed of several complementary terms that respectively supervise the fused prediction, maintain expert discriminability, and regularize the routing behavior.

where 𝑅(·) denotes the routing network and 𝜏 is the routing temperature. 4

Are Independently Estimated View Uncertainties Comparable?

Fused evidential classification loss. We take the fused output as the primary supervision target and optimize it with the evidential digamma loss [2]: # " 𝐾 ! 𝐾 ∑︁ ∑︁ fused fused Lfused = 𝑦𝑘 𝜓 𝛼𝑗 − 𝜓 𝛼𝑘 , (18) 𝑗=1

𝑘=1

where 𝜓 (·) is the digamma function and 𝒚 is the one-hot label vector.

4.1

Auxiliary expert supervision. Although the final prediction is produced after fusion, each expert should still learn to generate meaningful class evidence. We therefore apply the same evidential supervision to every expert: " 𝐾 ! # 𝑉 +1 𝐾 ∑︁ 1 ∑︁ ∑︁ Lview = 𝑦𝑘 𝜓 𝛼 𝑗(𝑖 ) − 𝜓 𝛼𝑘(𝑖 ) . (19) 𝑉 + 1 𝑖=1 𝑗=1

𝜶 = 𝒆 + 1,

𝑢=

𝛼𝑘 ,

𝒆(𝑡) = 𝑡 𝒓,

𝐾 . 𝑆

(25)

𝑡 > 0.

(26)

Then 𝜶 (𝑡) = 1 + 𝑡 𝒓, where 𝑅 =

𝐵

𝑢 (𝑡) =

𝐾 , 𝐾 + 𝑡𝑅

(27)

Í𝐾

𝑘=1 𝑟 𝑘 .

(20) Theorem 4.1 (Scale changes uncertainty without changing support direction). For the above family,

𝑏=1

be the average routing weight of expert 𝑖 over a mini-batch of size 𝐵, and define the routing concentration as ∑︁ ¯ = Conc( 𝝅) 𝜋¯𝑖2 . (21)

𝑑 𝑢 (𝑡) 𝐾𝑅 =− < 0. 𝑑𝑡 (𝐾 + 𝑡𝑅) 2

(28)

Moreover, if 𝑟 𝑦 > 𝑅/𝐾 for the ground-truth class 𝑦, then the predictive probability of the true class increases with 𝑡 as well. Therefore, along the same class-support pattern, increasing the evidence scale decreases the induced uncertainty even though the underlying support direction is unchanged.

𝑖

The load-balancing loss is 𝜌 , 0 , 𝑉 +1

𝐾 ∑︁

Thus, uncertainty is inversely proportional to the total evidence scale. Consider a fixed non-negative support pattern 𝒓 ∈ R𝐾≥0 and define a one-parameter evidence family

Load-balancing regularization. Since multi-view data can contain persistently strong and weak views, we do not force the router to use all experts equally. Instead, we only penalize excessive concentration of expert usage within a mini-batch, so that routing remains adaptive while avoiding collapse. Let

¯ − Lbal = max Conc( 𝝅)

𝑆=

𝑘=1

This auxiliary supervision stabilizes expert learning and prevents the router from compensating for poorly trained experts.

1 ∑︁ 𝜋𝑏,𝑖 𝐵

Why branch-local uncertainty is scale-sensitive

For an expert outputting non-negative evidence 𝒆 ∈ R𝐾≥0 , define

𝑘=1

𝜋¯𝑖 =

Theoretical Analysis

This section formalizes two points that directly support our design. First, branch-local uncertainty is scale-sensitive: changing evidence magnitude changes subjective-logic uncertainty even when the underlying class-support pattern is unchanged. Second, when optimal fusion authority depends on joint cross-view context, branch-local self-weighting is fundamentally limited, whereas a unified router can exploit the required joint information.

(22)

Proof. The expression for 𝑢 (𝑡) follows directly from the Dirichlet definition, and differentiating with respect to 𝑡 gives the stated derivative. If 𝑟 𝑦 > 𝑅/𝐾, the predictive probability of the true class is also increasing in 𝑡. □ Theorem 4.1 explains why branch-local uncertainty is vulnerable to scale bias. Two branches may encode essentially the same relative class preference, yet the one with a larger evidence scale will appear less uncertain and hence more trusted.

where 𝜌 > 1 controls the tolerated concentration. Expert diversity regularization. Because the private experts are optimized jointly, they may still drift toward redundant hidden representations. To encourage specialization, we regularize their hidden features to be decorrelated. Let 𝒛ˆ (𝑣) be the ℓ2 -normalized hidden feature of private expert 𝑣. We define 2 ∑︁ 2 Ldiv = 𝒛ˆ (𝑖 )⊤ 𝒛ˆ ( 𝑗 ) . (23) 𝑉 (𝑉 − 1) 1≤𝑖< 𝑗 ≤𝑉

4.2

Why unified routing is preferable to branch-local self-weighting

The squared cosine form penalizes both positive and negative alignment, thereby encouraging different private experts to capture different view-conditioned directions. The overall training objective is

Let L𝑥 (𝒘) denote the sample-wise fusion loss as a function of the expert-weight vector

L = Lfused + 𝜆Lview + 𝛽Lbal + 𝛾 Ldiv,

where Δ𝑉 is the probability simplex over the 𝑉 + 1 experts. Define

𝒘 ∈ Δ𝑉 ,

(24)

(29)

the oracle routing rule where 𝜆, 𝛽, and 𝛾 control the strengths of auxiliary expert supervision, load balancing, and diversity regularization, respectively.

𝒘 ★ (𝑥) = arg min L𝑥 (𝒘). 𝒘 ∈Δ𝑉

(30)

Are Independently Estimated View Uncertainties Comparable?

Table 1: Five-seed benchmark accuracy on the first seven reported datasets. Each entry reports mean ± standard deviation over five seeds. Method names include citation tags, while the ‘References’ column lists venue and year.

Method

Ref.

HandWritten

Scene

LandUse

NUS

Caltech-6V

PIE

WebKB

TMC [9] TMDLO [24] ETMC [10] RCML [34] ETF [25] FUML [6] TMCEK [20] TUNED [13] TEF [19] SAEML [35] RTMC [40] RCMCL [11]

ICLR’21 AAAI’22 PAMI’23 AAAI’24 ICML’25 ICML’25 ICML’25 AAAI’25 ICLR’25 MM’25 WWW’25 PAMI’26

98.00±0.88 97.20±1.03 98.30±0.48 98.00±0.79 96.65±1.12 98.90±0.54 98.30±0.84 98.90±0.34 98.80±0.51 98.80±0.53 84.55±1.61 97.15±1.06

75.76±0.95 66.87±2.97 78.10±1.14 76.03±1.35 68.21±0.79 80.65±1.03 77.37±1.20 78.15±0.69 78.00±0.48 79.67±1.42 68.92±1.31 70.81±1.37

60.95±2.25 52.67±3.36 65.48±1.27 65.86±1.72 19.52±8.86 76.62±1.24 70.10±1.18 72.95±1.80 80.48±1.41 74.05±1.23 54.38±0.36 45.24±1.15

44.70±0.36 43.09±0.42 48.53±0.34 44.43±0.28 44.12±0.69 48.07±0.34 43.20±0.32 43.35±0.31 47.14±0.52 42.94±0.25 35.12±0.69 37.75±0.28

93.72±0.30 86.15±2.42 94.06±0.47 94.60±0.38 93.97±0.48 95.61±0.37 95.40±0.51 95.15±0.75 94.85±0.28 93.81±0.51 94.60±0.33 94.14±0.70

91.32±2.43 73.09±1.89 94.85±1.14 95.29±1.10 95.59±1.40 96.91±1.35 97.79±0.93 89.56±2.56 97.65±1.27 95.88±1.78 92.79±1.99 96.18±1.18

82.44±2.84 83.41±2.84 82.93±2.67 82.93±2.67 80.49±3.09 72.20±18.09 85.37±2.67 80.00±2.39 84.88±2.39 84.88±2.39 78.54±2.39 81.95±2.49

NLC [36] MAMC [21] BCM [17]

AAAI’25 ICLR’25 MM’25

98.30±0.73 98.75±0.47 99.00±0.42

80.38±1.56 81.50±0.70 80.71±0.60

70.29±1.15 79.33±0.59 78.81±1.86

47.85±0.49 48.14±0.34 47.25±0.46

92.34±0.81 96.03±0.30 94.35±0.84

89.71±2.42 97.06±1.04 94.56±1.95

71.71±9.58 84.39±1.19 85.37±3.09

TMUR

Ours

99.10±0.46

82.88±1.36

80.48±0.78

50.07±0.26

96.69±0.36

97.94±0.86

88.29±0.98

5 Experiments 5.1 Experimental Setup

Suppose branch-local self-weighting rules can only access local statistics 𝒔 (𝑥), such as branch-wise evidence or uncertainty, and therefore take the form 𝒘local (𝑥) = 𝜓 (𝒔 (𝑥)).

Dataset. We evaluate TMUR on fourteen public multi-view classification datasets1 : HandWritten, Scene, LandUse, NUS, Caltech6V, PIE, WebKB, Caltech, UCI, CUB, Animal, MSRCV1, BBC, and Leaves. Detailed dataset statistics, view descriptions are deferred to the appendix.

(31)

By contrast, a unified router can access a joint representation 𝒈(𝑥) and output 𝒘global (𝑥) = 𝜙 (𝒈(𝑥)).

Compared methods. We compare TMUR with two groups of baselines. The trusted baselines include TMC [9], TMDLO [24], and ETMC [10], which formulate multi-view predictions as evidence or opinions and perform uncertainty-aware fusion. They also include more recent trusted methods for conflict handling and reliability modeling, including RCML [34], ETF [25], FUML [6], and RCMCL [11], which focus on conflict-aware aggregation, trust discounting, fuzzy uncertainty modeling, or robust collaborative learning under conflictive views. In addition, we compare with recent refinements of trusted fusion, including TMCEK [20], TUNED [13], TEF [19], SAEML [35], and RTMC [40], which introduce expert knowledge constraints, feature-neighborhood dynamics, evolutionary fusion architecture search, strength-adaptive evidence weighting, and a refined treatment of confusion and ignorance, respectively. The non-trusted baselines include NLC [36], which addresses noisy-label multi-view classification via label calibration, MAMC [21], which combines multi-scale alignment with expanded decision boundaries, and BCM [17], which performs multi-view classification in Hamming space with a bit-level calibration mechanism.

(32)

Theorem 4.2 (Branch-local self-weighting has an irreducible information gap). Assume that for each sample 𝑥, the fusion loss L𝑥 (𝒘) is 𝜇-strongly convex on Δ𝑉 for some 𝜇 > 0. Let Wlocal := {𝒘 (𝑥) = 𝜓 (𝒔 (𝑥))}

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denote the class of branch-local self-weighting rules. Then 𝜇 E[L𝑥 (𝒘 (𝑥))] −E L𝑥 (𝒘 ★ (𝑥)) ≥ E Var 𝒘 ★ (𝑥) | 𝒔 (𝑥) . 2 (34) In particular, if the optimal authority assignment depends on crossview relations that are not recoverable from branch-local statistics alone, then every branch-local self-weighting rule incurs a strictly positive irreducible excess risk. inf

𝒘 ∈ Wlocal

Proof sketch. Strong convexity lower-bounds excess loss by squared distance to the oracle weight 𝒘 ★ (𝑥). Among all rules measurable with respect to 𝒔 (𝑥), the best predictor of 𝒘 ★ (𝑥) under squared loss is the conditional expectation, and the corresponding minimum residual is the conditional variance. Therefore, if 𝒘 ★ (𝑥) is not measurable from branch-local statistics alone, branch-local self-weighting necessarily leaves a positive irreducible gap. □

Implementation details. All experiments are conducted on NVIDIA RTX 4090 GPUs using PyTorch 2.2.2. Following our benchmark protocol, every method is evaluated with five fixed random 1 https://github.com/JethroJames/Awesome-Multi-View-Learning-Datasets

Are Independently Estimated View Uncertainties Comparable?

Table 2: Calibration results on trusted multi-view classification methods (ECE%, lower is better). Method

Ref.

LandUse

PIE

TMC [9] TMDLO [24] ETMC [10] RCML [34] TEF [19] TMCEK [20] TUNED [13] SAEML [35] RTMC [40] RCMCL [11]

ICLR’21 AAAI’22 PAMI’23 AAAI’24 ICLR’25 ICML’25 AAAI’25 MM’25 WWW’25 PAMI’26

13.34±2.13 27.16±4.29 8.69±2.60 64.90±0.81 8.13±5.73 62.25±4.67 48.83±0.67 9.83±1.71 72.39±1.70 51.09±4.38

46.07±1.74 40.29±2.33 23.21±2.39 26.71±2.62 40.03±3.55 41.78±1.53 58.64±1.61 51.92±1.91 10.83±2.06 24.58±2.18 61.33±1.97 58.32±0.99 55.69±1.17 52.62±1.14 12.45±0.99 16.62±1.22 59.47±1.32 47.88±0.36 52.62±1.38 35.62±1.16

TMUR

Ours

5.36±1.66

6.87±1.69 7.35±0.94 52.50±3.93 12.01±4.44

Animal

Caltech-6V

Leaves

74.50±0.53 8.91±2.95 43.31±7.20 58.25±2.28 32.36±11.21 25.60±1.09 84.49±1.53 21.56±6.58 23.93±10.13 87.05±1.80 38.95±4.01 51.11±2.23 70.47±4.63 47.51±8.00 24.91±0.52 92.16±0.73 28.62±5.77 54.89±2.00 81.17±1.55 31.17±2.90 64.18±1.24 60.35±4.20 10.11±3.12 30.68±0.71 90.69±1.98 47.41±3.69 81.50±0.97 86.92±1.12 42.86±3.71 60.70±0.71

6.43±4.11 18.16±2.05 7.76±3.54 47.07±1.47 9.21±2.02 54.25±4.19 52.09±1.42 5.01±1.10 87.16±0.34 42.15±1.99

87.56±2.59 50.23±1.91 87.74±3.86 90.43±1.15 72.84±1.71 93.83±0.67 84.56±1.52 66.96±1.17 84.50±2.54 93.61±0.72

7.55±5.23

44.43±8.60

Density

FUML

RCML

TMC

SAEML

Ours

PIE

R1 R2

92.89 82.35

93.38 89.95

88.73 81.86

93.87 84.80

96.08 91.67

R1 R2

67.41 69.12

61.32 63.43

68.75 71.16

69.75 73.39

71.39 74.99

LandUse Caltech-6V WebKB

R1 R2 R1 R2 R1 R2

45.40 42.14 94.70 93.17 64.23 61.79

57.62 54.68 94.56 94.98 81.30 77.24

52.78 56.90 93.93 93.86 82.11 79.67

49.29 52.78 92.89 88.15 81.30 78.86

0 0.0

0.2

0.4

0.6

0.8

Uncertainty

0 0.0

1.0

0.2

63.97 65.16

95.33 94.84

0.8

1.0

In-Distribution OOD (σ=0.1) OOD (σ=1) OOD (σ=10)

3 2 1

86.18 86.99

0.6

(d) Caltech101-6V

0 0.0

0.4

Uncertainty

In-Distribution OOD (σ=0.1) OOD (σ=1) OOD (σ=10)

0.2

0.4

0.6

Uncertainty

0.8

1.0

0 0.0

0.2

0.4

0.6

Uncertainty

0.8

1.0

Figure 3: Uncertainty distributions of TMUR under increasing Gaussian perturbation on four datasets. As the noise level increases from 𝜎 = 0.1 to 𝜎 = 1 and 𝜎 = 10, the predicted uncertainty shifts upward, demonstrating that TMUR remains uncertainty-aware under degraded inputs.

seeds, and the reported accuracy is the mean ± standard deviation over the five runs. We use Adam with an initial learning rate of 10−3 , cosine learning-rate decay, and a fixed batch size of 128. The auxiliary expert-supervision weight 𝜆 is fixed to 0.3 on all datasets, while the load-balancing and diversity weights 𝛽 and 𝛾 are selected per dataset; their final values are reported in the appendix.

5.2

In-Distribution OOD (σ=0.1) OOD (σ=1) OOD (σ=10)

Density

Scene

(b) WebKB In-Distribution OOD (σ=0.1) OOD (σ=1) OOD (σ=10)

Methods

Ratio

9.79±5.73

(a) Scene

Table 3: View Strength Variation Experiments. The view strength in the multi-view dataset is randomly rescaled.

Dataset

WebKB

Density

Scene

Density

HandWritten

overall pattern: TMUR remains highly competitive and achieves the best or tied-best performance on six datasets, indicating that the proposed routing mechanism improves accuracy without sacrificing generality across diverse multi-view benchmarks. Accuracy alone is insufficient for trustworthy multi-view learning; the fused prediction should also remain well calibrated. Table 2 therefore compares ECE on eight representative datasets against trusted baselines. TMUR achieves the lowest ECE on six of the eight datasets, namely HandWritten, Scene, LandUse, PIE, Animal, and Leaves, and remains competitive on the remaining two datasets. Averaged over all eight datasets, TMUR reduces ECE from 26.50% achieved by the strongest baseline SAEML to 18.23%, yielding an absolute improvement of more than 8 ECE points. These results are important because they show that our gains are not merely due to

Main Results

Table 1 reports the current five-seed accuracy results on seven representative datasets, with each entry shown as mean ± standard deviation. To keep the comparison readable under the enlarged baseline set, the methods are separated into trusted baselines, nontrusted baselines, and our model, with horizontal rules marking the three groups. The remaining are reported in appendix. TMUR achieves the best or tied-best accuracy on all of the seven datasets in Table 1, including LandUse21, NUS, PIE, Caltech1016V, and WebKB. In particular, the gains on NUS and WebKB are clear, suggesting that unified routing is especially helpful when the reliability of different views varies substantially across samples. The appendix results on the remaining seven datasets show the same 7

Are Independently Estimated View Uncertainties Comparable?

$FFXUDF\

Table 4: Ablation study of different components in our method. Variant

HandWritten

Scene

LandUse

PIE

WebKB

Animal

w/o Lbal w/o Ldiv w/o Cross-Attention Full method

98.85±0.46 99.00±0.45 98.95±0.51 99.10±0.46

82.63±1.63 82.23±1.47 82.43±1.70 82.88±1.36

80.05±0.87 79.48±0.73 78.95±0.83 80.48±0.78

97.50±0.88 97.79±0.93 97.65±0.72 97.94±0.86

87.80±1.54 88.29±0.98 87.80±1.54 88.29±0.98

87.62±0.60 96.28±0.48 99.44±0.23 87.55±0.71 96.40±0.45 99.62±0.23 87.58±0.59 96.61±0.24 99.62±0.13 87.83±0.57 96.69±0.36 99.69±0.20

$FFXUDF\

5.4

G/DQG8VH

Ablation study.

Table 4 shows the full TMUR model achieves the best performance on seven of the eight reported datasets and ties for the best result on WebKB, indicating that the three components are complementary rather than redundant. Removing Lbal leads to visible drops on datasets such as LandUse, PIE, and Leaves, which suggests that the soft load-balancing term helps prevent expert over-concentration while still allowing sample-wise specialization. Removing Ldiv degrades performance on Scene, LandUse, Animal, and Caltech6V, showing that diversity regularization helps the router exploit complementary evidence from different experts. Finally, removing router-side cross-view attention consistently weakens the full model, with the clearest drop appearing on LandUse, which supports our design choice of enhancing the arbitration signal rather than feeding cross-view interaction back into the private experts.

F3,(

$FFXUDF\

E6FHQH

Leaves

Uncertainty response to input corruption. We further inject Gaussian noise into the test features and examine the resulting uncertainty. As shown in Fig. 3, the uncertainty distributions on four datasets shift upward as the noise level increases from 𝜎 = 0.1 to 1 and 10. This shows that TMUR responds to degraded inputs with higher uncertainty rather than overconfident predictions.

D+DQG:ULWWHQ

Caltech-6V

5.5

Hyperparameter sensitivity.

H186

We further study the joint sensitivity of the load-balancing weight 𝛽 and the diversity weight 𝛾 on HandWritten, Scene, PIE, LandUse, NUS, and Animal. As shown in Fig. 4, the accuracy surfaces vary smoothly across a relatively broad hyperparameter region, instead of peaking sharply at a single fragile setting. This indicates that TMUR is reasonably robust to the choices of 𝛽 and 𝛾, and that its gains do not rely on narrow hyperparameter tuning.

I$QLPDO

Figure 4: Hyperparameter sensitivity on six datasets.

higher classification accuracy; instead, the proposed unified routing also produces more reliable confidence estimates under cross-view evidence mismatch. Taken together, Table 1 and Table 2 show that TMUR improves both what the model predicts and how reliably it expresses confidence in that prediction.

5.3

Conclusion

This paper revisits trustworthy multi-view classification from the perspective of evidence comparability. We argue that heterogeneous views and independently optimized branches need not produce evidence on a common numeric scale, making direct cross-view evidence comparison an unreliable basis for fusion arbitration. To address this issue, we propose TMUR, a unified-routing framework in which view-private experts, a collaborative expert, and a global router play explicitly separated roles. The final prediction remains evidential, while fusion authority is reassigned to router-mediated sample-wise arbitration under the joint multi-view context. Extensive experiments on fourteen datasets, together with calibration and robustness analyses, show that TMUR improves both predictive performance and reliability.

Mechanism and Robustness Evaluation

View-strength variation. Following Xu et al. [35], we perturb the feature magnitude of each view at test time by randomly scaling its feature vector. Table 3 reports the results on five datasets. TMUR achieves the best performance in nine out of ten settings, showing stronger robustness when view strength is distorted. This supports our motivation that unified routing provides more stable fusion than branch-local confidence under cross-view scale variation. 8

Are Independently Estimated View Uncertainties Comparable?

References

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Are Independently Estimated View Uncertainties Comparable?

Overview of the Appendix

datasets, such as Caltech101 and CUB, the benchmark release itself uses the commonly adopted subset version rather than the original full dataset, which explains the reduced sample and class counts.

This appendix includes the following supplementary materials: • algorithmic details of TMUR, including the complete training and inference procedure; • dataset descriptions and the dataset-specific hyperparameter settings used in our benchmark; • supplementary experimental results, including additional accuracy tables, calibration analysis, motivation analysis, and hyperparameter sensitivity results; • detailed proofs for the theoretical results in the main paper.

C.2

HandWritten. HandWritten contains 2000 handwritten numerals from “0” to “9”, with 200 samples per class. Following the standard benchmark release, each sample is represented by six views: pixel averages in 2 × 3 windows, Fourier coefficients of the character shape, profile correlations, Zernike moments, Karhunen–Loeve coefficients, and morphological features. Scene. Scene15 contains 4485 indoor and outdoor scene images from 15 categories. Its three views are GIST, PHOG, and LBP descriptors extracted from the same image. LandUse. LandUse21 contains 2100 remote-sensing images from 21 land-use categories. In the benchmark release used in this paper, each aerial image is represented by three visual views with dimensions 20, 59, and 40. NUS. NUS-WIDE-Object contains 30000 images from 31 classes. Each image is described by five visual views, namely color histogram, block-wise color moments, color correlogram, edge direction histogram, and wavelet texture. Caltech-6V. Caltech101-6V is the standard six-view benchmark subset derived from Caltech101. Each image is represented by six complementary visual views with dimensions 48, 40, 254, 1984, 512, and 928. PIE. PIE contains 680 face images from 68 subjects. The three views are intensity features, LBP features, and Gabor features extracted from the same face image. WebKB. WebKB contains 203 webpages from four categories. Each page is described by three text views: the page content, the anchor text of hyperlinks, and the title text. Caltech. The 2-view Caltech101 benchmark selects the top 10 categories from the original 101 classes and uses DECAF and VGG19 to extract two 4096-dimensional image views. UCI. UCI contains handwritten numerals from 10 classes. The three views are the average of pixels in 240 windows, 47 Zernike moments, and 6 morphological features. CUB. CUB is constructed from the first 10 bird categories of the original dataset. Following the standard benchmark setting, GoogleNet features are used as the image view and doc2vec features are used as the text view. Animal. Animal contains 10158 images from 50 animal categories. The two views are deep image features extracted by DECAF and VGG19. MSRCV1. MSRCV1 contains 210 images from 7 classes. The processed version used in this paper provides six visual views for each image, with dimensions 1302, 48, 512, 100, 256, and 210. BBC. BBC contains 685 news documents from 5 categories. Each document is represented by four text views in the standard multiview benchmark release. Leaves. Leaves100 contains 1600 leaf samples from 100 plant species. Its three views correspond to shape descriptors, fine-scale edge features, and texture histograms.

Algorithmic Description of TMUR

Algorithm 1 summarizes the training and inference procedure of TMUR. Algorithm 1 Training and inference procedure of TMUR. /* TRAIN */ Input: Multi-view training set D, epochs 𝑇 , hyperparameters 𝜆, 𝛽, 𝛾. Output: Parameters of view projectors, experts, and router. 𝑉 1: Initialize view projectors {𝑃 𝑣 }𝑉 𝑣=1 , private experts {𝐸 𝑣 } 𝑣=1 , collaborative expert 𝐸𝑐 , and unified router 𝑅. 2: for epoch 𝑡 = 1 : 𝑇 do 3: for mini-batch B ⊂ D do 4: Align each view by 𝒉 (𝑣) = LN(𝑃 𝑣 (𝒙 (𝑣) )) for 𝑣 = 1, . . . , 𝑉 . 5: Form the collaborative input 𝑯 = [𝒉 (1) ; · · · ; 𝒉 (𝑉 ) ]. 6: Build router context 𝒈 by router-side cross-view attention over {𝒉 (𝑣) }𝑉𝑣=1 . 7: Obtain private expert evidence 𝒆 (𝑣) = 𝜙 (𝐸 𝑣 (𝒉 (𝑣) )) for 𝑣 = 1, . . . , 𝑉 . 8: Obtain collaborative evidence 𝒆 (𝑉 +1) = 𝜙 (𝐸𝑐 (𝑯 )). 9: Predict sample-wise gate weights 𝒘 = 𝑅(𝒈). Í 10: Aggregate fused evidence 𝒆 = 𝑉𝑖=1+1 𝑤𝑖 𝒆 (𝑖 ) and Dirichlet parameters 𝜶 = 𝒆 + 1. 11: Compute fused evidential loss Lcls and auxiliary expert loss Laux . Compute load-balancing loss Lbal and expert-diversity 12: loss Ldiv . 13: Update all parameters using L = Lcls + 𝜆Laux + 𝛽Lbal + 𝛾 Ldiv . 14:

Per-dataset View Descriptions

end for

15: end for

/* TEST */ For each test sample, repeat the forward pass, output the fused Dirichlet parameters 𝜶 , the predictive probability 𝒑, and the uncertainty 𝑢.

C Dataset Details C.1 Benchmark Statistics Table 5 reports the detailed statistics of all fourteen datasets used in the main paper. We follow the standard processed multi-view releases adopted by prior trusted multi-view studies. For some 10

Are Independently Estimated View Uncertainties Comparable?

Table 5: Detailed statistics of the fourteen datasets used in the main paper.

Dataset

#Samples

#Views

#Classes

2000 4485 2100 30000 2386 680 203 2908 2000 600 10158 210 685 1600

6 3 3 5 6 3 3 2 3 2 2 6 4 3

10 15 21 31 20 68 4 10 10 10 50 7 5 100

HandWritten Scene LandUse NUS Caltech-6V PIE Webkb Caltech UCI CUB Animal MSRCV1 BBC Leaves

Per-view Dimensions {240, 76, 216, 47, 64, 6} {20, 59, 40} {20, 59, 40} {65, 226, 145, 74, 129} {48, 40, 254, 1984, 512, 928} {484, 256, 279} {1703, 230, 230} {4096, 4096} {6, 240, 47} {1024, 300} {4096, 4096} {1302, 48, 512, 100, 256, 210} {4659, 4633, 4665, 4684} {64, 64, 64}

Table 6: Dataset-specific TMUR hyperparameters used in the benchmark experiments.

Dataset

𝜷

𝜸

Dataset

𝜷

𝜸

HandWritten LandUse Caltech-6V Webkb UCI Animal BBC

0.1 0.05 0.05 0.05 0.05 0.01 0.0

0.0 0.0001 0.05 0.01 0.1 0.02 0.05

Scene NUS PIE Caltech CUB MSRCV1 Leaves

0.0 0.05 0.02 0.05 0.05 0.05 0.01

0.2 0.05 0.02 0.1 0.05 0.01 0.002

D Additional Experimental Details D.1 Training Protocol

E Supplementary Results E.1 Accuracy on the Remaining Seven Datasets

Unless otherwise stated, TMUR is trained for 300 epochs with Adam, an initial learning rate of 10−3 , cosine learning-rate decay, and a batch size of 128. All reported results are averaged over the same five random seeds {3407, 7, 601, 101, 503}. The train/test partitions follow the stratified benchmark split implemented in our released codebase, with an 80/20 split ratio. For calibration reporting, the final ECE values discussed in both the main paper and this appendix refer to uncertainty ECE computed from predictive uncertainty 𝑢 (U-ECE), unless otherwise stated.

To keep the main paper readable, Table 7 reports the remaining seven accuracy columns that are not shown in Table 1 of the main paper. The same five-seed evaluation protocol and baseline grouping are used here. TMUR remains competitive on these seven supplementary datasets. In particular, it achieves the best mean accuracy on Caltech, CUB, and MSRCV1, and matches the best result on Leaves. The gains are especially clear on CUB, where TMUR improves over the strongest baseline by about one point, suggesting that unified routing is particularly helpful when the two views are semantically heterogeneous. On UCI, Animal, and BBC, TMUR is not the top-performing method, but it stays in the leading group without the instability observed in several competing trusted fusion baselines.

D.2

Dataset-specific Hyperparameter Settings

The auxiliary expert loss weight is fixed to 𝜆 = 0.3 on all datasets, while the load-balancing weight 𝛽 and the diversity weight 𝛾 are selected per dataset. Table 6 lists the final settings used by TMUR in the current benchmark package.

E.2

Detailed Calibration Summary for TMUR

The main paper compares ECE across methods on eight representative datasets. Table 8 complements that comparison by reporting 11

Are Independently Estimated View Uncertainties Comparable?

Table 7: Supplementary five-seed accuracy results on the remaining seven datasets that are omitted from the main-paper benchmark table for space. Each entry reports mean ± standard deviation over five seeds.

Method

Ref.

Caltech

UCI

CUB

Animal

MSRCV1

BBC

Leaves

TMC [9] TMDLO [24] ETMC [10] RCML [34] ETF [25] FUML [6] TMCEK [20] TUNED [13] TEF [19] SAEML [35] RTMC [40] RCMCL [11]

ICLR’21 AAAI’22 PAMI’23 AAAI’24 ICML’25 ICML’25 ICML’25 AAAI’25 ICLR’25 MM’25 WWW’25 PAMI’26

99.52±0.07 83.37±8.99 99.59±0.14 96.12±0.84 99.11±0.48 99.76±0.08 74.19±16.97 99.59±0.18 99.76±0.08 87.18±5.11 99.79±0.13 91.92±0.52

97.90±1.04 91.60±6.41 98.10±0.75 98.35±0.64 96.90±1.11 98.80±0.90 98.75±0.65 98.90±0.78 99.00±0.57 98.90±0.51 96.15±1.32 97.40±1.10

92.83±1.87 85.17±7.18 93.67±1.72 93.00±1.72 92.33±2.07 93.33±1.90 94.00±1.93 94.67±1.25 94.83±1.33 93.83±1.55 94.00±0.82 94.17±1.83

87.48±0.28 75.37±2.01 88.30±0.43 87.57±0.53 88.40±0.29 87.21±1.06 88.19±0.53 82.41±1.33 87.78±0.53 87.42±0.51 85.80±0.94 87.97±0.38

95.24±3.69 95.24±3.37 98.10±1.78 90.95±1.78 93.81±1.17 98.10±2.33 91.43±2.43 99.05±1.17 99.05±1.17 99.52±0.95 80.95±6.02 97.14±2.33

94.45±1.19 94.16±1.85 94.31±1.49 93.43±1.22 92.12±1.26 79.71±13.28 96.35±0.00 96.93±1.17 96.79±0.99 97.23±0.55 93.14±1.76 94.16±1.31

96.13±0.81 70.25±2.18 98.88±0.25 95.63±0.97 42.94±7.41 99.88±0.15 97.81±0.66 94.25±1.00 89.69±1.15 97.75±0.85 85.94±2.55 97.12±0.61

NLC [36] MAMC [21] BCM [17]

AAAI’25 ICLR’25 MM’25

98.49±1.28 95.22±0.61 99.52±0.25

98.50±0.74 98.80±0.58 99.10±0.46

87.83±2.92 94.50±0.67 93.83±0.85

85.92±0.56 86.23±0.73 87.20±0.64

70.48±6.14 93.81±5.13 98.10±2.33

78.39±5.47 95.47±1.49 91.82±0.72

99.44±0.41 99.56±0.15 99.69±0.34

TMUR

Ours

99.79±0.17

98.90±0.46

95.83±1.75

87.53±0.62

99.52±0.95

95.04±1.26

99.69±0.40

Table 8: Detailed TMUR calibration summary on the eight datasets used in the main-paper calibration benchmark. “ProbECE” measures calibration of predictive probabilities, “U-ECE” measures calibration of uncertainty scores, and 𝑢 is the mean predictive uncertainty.

Dataset

𝜷

𝜸

Acc.

Prob-ECE

U-ECE

𝑢

HandWritten Scene LandUse Caltech-6V PIE Webkb Animal Leaves

0.1 0.0 0.05 0.05 0.02 0.05 0.01 0.01

0.0 0.2 0.0001 0.05 0.02 0.01 0.02 0.002

99.05 82.36 79.14 96.65 97.65 88.29 87.81 99.69

6.19 5.67 14.32 10.47 58.81 16.74 14.34 52.15

5.36 6.87 7.35 7.55 52.50 14.01 9.79 44.43

0.059 0.131 0.254 0.104 0.547 0.220 0.192 0.447

E.3

the detailed TMUR calibration summary on the same eight datasets, including both probability ECE and uncertainty ECE. Consistent with the main paper, the final ECE values we report are U-ECE, i.e., the calibration error computed from predictive uncertainty 𝑢. These numbers are taken from the dedicated calibration runs used to build the main-paper calibration benchmark. Overall, TMUR shows relatively well-behaved uncertainty calibration on HandWritten, Scene, LandUse, Caltech-6V, and Animal, where U-ECE stays below 10. In addition, U-ECE is lower than probability ECE on most datasets in the table, indicating that the evidential uncertainty score is generally more stable than raw predictive confidence under our setting, although challenging datasets such as PIE and Leaves still leave substantial room for improvement.

Additional Hyperparameter Sensitivity on the Remaining Eight Datasets

Figure 5 reports the joint sensitivity of 𝛽 and 𝛾 on the remaining eight datasets beyond the six shown in the main paper. Consistent with the main-paper observation, the accuracy surfaces remain relatively smooth across a broad hyperparameter region, indicating that TMUR does not rely on fragile tuning of the load-balancing or diversity weights. For MSRCV1, the surface appears nearly flat because the accuracy stays close to 1.0 throughout this hyperparameter range.

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E.4

Motivation Analysis via Cross-view Calibration Mismatch

F.1

Proof of Theorem 1

Recall that for an evidential output 𝒆(𝑡) = 𝑡 𝒓 ∈ R𝐾≥0 with 𝜶 (𝑡) = 𝒆(𝑡) + 1, the total evidence is

Figure 6 provides additional evidence for the motivation of TMUR using four representative datasets: HandWritten, UCI, NUS, and Caltech-6V. These plots are produced from the saved calibration outputs of TMC and RCML. In each dataset block, the left column shows the uncertainty density of different views, the middle column shows the reliability curves obtained by binning the maximum predicted probability, and the right column shows the reliability curves obtained by binning predictive uncertainty. The first row corresponds to TMC and the second row corresponds to RCML. Two observations are consistent across these datasets. First, the uncertainty distributions differ substantially from one view to another, even within the same sample population. Second, for both confidence-based and uncertainty-based binning, the same confidence or uncertainty range can lead to clearly different empirical accuracies across views. Therefore, branch-local confidence or uncertainty is not directly comparable across heterogeneous views, and using it as the fusion authority can introduce systematic mismatch. This observation supports the central design choice of TMUR: instead of letting each branch self-assign fusion importance according to its own evidence scale, we use a unified router to infer cross-view fusion weights from the joint multi-view context.

𝑆 (𝑡) =

𝐾 ∑︁

𝛼𝑘 (𝑡) = 𝐾 + 𝑡𝑅,

𝑅=

𝑘=1

𝐾 ∑︁

(35)

Hence the subjective-logic uncertainty is 𝑢 (𝑡) =

𝐾 𝐾 = . 𝑆 (𝑡) 𝐾 + 𝑡𝑅

(36)

Differentiating with respect to 𝑡 gives 𝑑 𝑢 (𝑡) 𝐾𝑅 =− < 0, 𝑑𝑡 (𝐾 + 𝑡𝑅) 2

(37)

which shows that increasing the evidence scale strictly decreases uncertainty. For the ground-truth class 𝑦, the predictive probability is 𝑝 𝑦 (𝑡) =

1 + 𝑡𝑟 𝑦 . 𝐾 + 𝑡𝑅

(38)

Its derivative is 𝑑 𝑝 𝑦 (𝑡) 𝐾𝑟 𝑦 − 𝑅 = . 𝑑𝑡 (𝐾 + 𝑡𝑅) 2 𝑑 𝑝 (𝑡 )

𝑟𝑘 .

𝑘=1

(39)

𝑦 Therefore, whenever 𝑟 𝑦 > 𝑅/𝐾, we have 𝑑𝑡 > 0. Thus, along the same class-support direction, enlarging the evidence scale simultaneously decreases uncertainty and increases the true-class predictive probability, even though the relative support pattern is unchanged. This proves that subjective-logic uncertainty is directly affected by evidence scale.

Proofs for Theoretical Analysis

This section provides the detailed proofs omitted from the main paper. 13

Are Independently Estimated View Uncertainties Comparable?

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Figure 6: Motivation analysis using saved calibration results of TMC and RCML on four representative datasets. Within each dataset block, the left column shows per-view uncertainty density distributions, the middle column shows reliability curves based on maximum predicted probability, and the right column shows reliability curves based on predictive uncertainty. The first row is TMC and the second row is RCML. Across datasets, different views exhibit noticeably different calibration behavior, indicating that the same confidence or uncertainty value does not imply the same reliability across views.

F.2

Proof of Theorem 2

L𝑥 (𝒘) is 𝜇-strongly convex, for any 𝒘 ∈ Wlocal we have 𝜇 L𝑥 (𝒘 (𝑥)) − L𝑥 (𝒘 ★ (𝑥)) ≥ ∥𝒘 (𝑥) − 𝒘 ★ (𝑥) ∥ 22 . 2 Taking expectation and infimum over Wlocal yields inf E[L𝑥 (𝒘 (𝑥))] − E L𝑥 (𝒘 ★ (𝑥))

We first use the following standard lemma. Lemma F.1. For any square-integrable random vector 𝒚 and statistic 𝒔, the best 𝐿2 predictor among all measurable functions of 𝒔 is the conditional expectation E[𝒚 | 𝒔], and the minimum squared error is inf E ∥𝒚 − 𝑓 (𝒔)∥ 22 = E ∥𝒚 − E[𝒚 | 𝒔] ∥ 22 . (40)

𝒘 ∈ Wlocal

≥

𝑓

Proof. For any measurable 𝑓 (𝒔), 𝒚 − 𝑓 (𝒔) = 𝒚 − E[𝒚 | 𝒔] + E[𝒚 | 𝒔] − 𝑓 (𝒔) .

𝜇 inf E ∥𝒘 (𝑥) − 𝒘 ★ (𝑥)∥ 22 . 2 𝒘 ∈ Wlocal

(45)

Because every 𝒘 ∈ Wlocal is measurable with respect to 𝒔 (𝑥), the lemma implies h i 2 inf E ∥𝒘 (𝑥) − 𝒘 ★ (𝑥)∥ 22 = E 𝒘 ★ (𝑥) − E[𝒘 ★ (𝑥) | 𝒔 (𝑥)] 2 .

(41)

𝒘 ∈ Wlocal

Expanding the squared norm and taking expectation, the cross term vanishes because E[𝒚 − E[𝒚 | 𝒔] | 𝒔] = 0.

(44)

(46) Recognizing the right-hand side as the conditional variance term, we obtain 𝜇 inf E[L𝑥 (𝒘 (𝑥))] −E L𝑥 (𝒘 ★ (𝑥)) ≥ E Var 𝒘 ★ (𝑥) | 𝒔 (𝑥) . 2 𝒘 ∈ Wlocal (47) Therefore, if the oracle routing decision is not measurable with respect to the branch-local statistic 𝒔 (𝑥), every branch-local selfweighting rule incurs a strictly positive irreducible excess risk.

(42)

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Are Independently Estimated View Uncertainties Comparable? Unified Routing for Trusted Multi-View Classification

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