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GAUGE: Granularity-Adaptive Counterfactual Gating of Evidence for Incomplete Multimodal Classification
Yunping Shi, En Yu, Kairui Guo, Jie Lu
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
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Summary
The paper introduces GAUGE, a lightweight counterfactual gating framework designed for incomplete multimodal classification. It addresses the limitation of existing methods that operate at a coarse modality level by proposing fine-grained evidence control. GAUGE imputes missing modalities using a frozen imputer, encodes inputs into fine-grained evidence units, and computes prediction-aware Taylor evidence scores via a single forward-backward pass. These scores are mapped to continuous gates that act as additive attention-logit biases, modulating the influence of each evidence unit without altering the backbone architecture. Theoretical analysis provides an upper bound on the error of the first-order Taylor approximation relative to the exact counterfactual effect.
Entities (6)
Relation Signals (6)
GAUGE → solves → incomplete multimodal classification
confidence 95% · GAUGE is proposed as a framework for incomplete multimodal classification.
GAUGE → uses → Taylor evidence scores
confidence 92% · GAUGE scores the counterfactual effect... through prediction-aware Taylor evidence scores
Taylor evidence scores → aremappedto → continuous gates
confidence 90% · These scores are mapped to continuous gates
GAUGE → modulates → Evidence units
confidence 90% · GAUGE... encodes observed and recovered inputs uniformly as fine-grained evidence units... unit-wise evidence modulation
continuous gates → areconvertedto → Attention-logit biases
confidence 88% · continuous gates, which are converted into additive attention-logit biases
GAUGE → outperforms → strong baselines
confidence 85% · Experiments across six benchmarks demonstrate that GAUGE outperforms strong baselines
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Abstract
Abstract:Multimodal classification typically assumes all modalities are available, yet real-world inputs are often incomplete. Imputation and dynamic fusion can mitigate such incompleteness, but existing methods operate at a coarse modality level and thus cannot retain reliable components while suppressing misleading ones within the same recovered modality, compromising prediction reliability. To address this issue, we propose GAUGE, a lightweight counterfactual gating framework for incomplete multimodal classification. GAUGE first imputes missing modalities with a frozen imputer and encodes observed and recovered inputs uniformly as fine-grained evidence units. Rather than intervening on each unit explicitly, GAUGE scores the counterfactual effect of replacing every unit with a reference representation through prediction-aware Taylor evidence scores, all obtained in a single forward-backward pass. These scores are mapped to continuous gates, which are converted into additive attention-logit biases for unit-wise evidence modulation without altering the backbone architecture. Experiments across six benchmarks demonstrate that GAUGE outperforms strong baselines across diverse incomplete-input settings. Furthermore, a Taylor remainder theoretical analysis characterizes the error of the first-order approximation relative to the exact counterfactual effect, establishing GAUGE as a principled and scalable framework for fine-grained evidence control under modality incompleteness.
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- Source: https://arxiv.org/abs/2608.05608v1
- Canonical: https://arxiv.org/abs/2608.05608v1
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GAUGE: Granularity-Adaptive Counterfactual Gating of Evidence for Incomplete Multimodal Classification Yunping Shi, En Yu, Kairui Guo, Jie Lu Abstract Multimodal classification typically assumes all modalities are available, yet real-world inputs are often incomplete. Imputation and dynamic fusion can mitigate such incompleteness, but existing methods operate at a coarse modality level and thus cannot retain reliable components while suppressing misleading ones within the same recovered modality, compromising prediction reliability. To address this issue, we propose GAUGE, a lightweight counterfactual gating framework for incomplete multimodal classification. GAUGE first imputes missing modalities with a frozen imputer and encodes observed and recovered inputs uniformly as fine-grained evidence units. Rather than intervening on each unit explicitly, GAUGE scores the counterfactual effect of replacing every unit with a reference representation through prediction-aware Taylor evidence scores, all obtained in a single forward–backward pass. These scores are mapped to continuous gates, which are converted into additive attention-logit biases for unit-wise evidence modulation without altering the backbone architecture. Experiments across six benchmarks demonstrate that GAUGE outperforms strong baselines across diverse incomplete-input settings. Furthermore, a Taylor remainder theoretical analysis characterizes the error of the first-order approximation relative to the exact counterfactual effect, establishing GAUGE as a principled and scalable framework for fine-grained evidence control under modality incompleteness. Introduction Multimodal models have achieved remarkable success in complex perception and reasoning tasks by leveraging complementary information from diverse sources, such as images, text, and sensor data (Liang et al. 2024; Xu et al. 2023; Zhang et al. 2025a). However, they are typically developed under the idealized assumption that all required modalities are available for each input (Ma et al. 2021; Du et al. 2024; Yang et al. 2025). In practice, modalities may be missing or inaccessible because of sensor failures, heterogeneous acquisition protocols, privacy constraints, or transmission errors (Sun et al. 2024; Yao et al. 2024). Reliable prediction under such incomplete inputs is therefore critical in high-stakes applications such as medical diagnosis (Yao et al. 2024) and autonomous driving (Park et al. 2025). Existing methods for learning with incomplete modalities can be broadly categorized into two paradigms: recovery-free and recovery-based. Recovery-free methods learn directly from the available modalities (Ma et al. 2021; Wu et al. 2024; Zhang et al. 2025b), avoiding imputation-induced errors but discarding potentially recoverable cross-modal cues. In contrast, recovery-based approaches explicitly impute missing modalities (Zhang et al. 2022; Wang et al. 2023) to restore complementary cues, but the reconstructed content may contain low-fidelity, noisy, or semantically inconsistent evidence. Recent quality-aware dynamic fusion (Zhang et al. 2023; Cao et al. 2024) and modality-robustness methods (Wang et al. 2024) mitigate this unreliability through weighting, gradient-guided decoupling, or availability-conditioned parameter switching. However, these mechanisms operate only at the modality level, applying a shared control decision to the entire encoder output. This coarse-grained approach overlooks within-modality heterogeneity in the reliability of individual components. The limitation is especially pronounced for recovered modalities, where uneven reconstruction errors can cause informative and misleading signals to coexist. Therefore, effective post-recovery control should operate at a finer granularity within each modality, independently modulating its representations according to the contributions to the current prediction. However, realizing such fine-grained post-recovery control raises three interrelated questions: 1) At what granularity should control operate? Modern multimodal architectures employ heterogeneous encoders that produce representation sets with different structures across modalities. Existing mechanisms disregard this encoder-defined structure by assigning a shared control value to the entire representation of each modality (Zhu et al. 2026). Effective control should therefore operate at the native granularity of each encoder. 2) How should control strength be quantified? The contribution of an individual representation component depends on the current multimodal context and predicted class (Deng et al. 2021; Fan et al. 2026). Its control strength should be assessed in a prediction-aware manner. A direct criterion is counterfactual replacement, which replaces a component with a reference representation while keeping the remaining representation fixed and measures the magnitude of the resulting change in the predicted-class response. 3) How can control strength be estimated efficiently? Although counterfactual replacement yields a decision-aligned score for each component, exact evaluation requires one intervention forward pass per component—N additional passes for N components (Fong and Vedaldi 2017). Scalable control therefore requires estimating all component-wise effects using a fixed number of forward–backward scoring passes independent of N. To address these questions, we propose GAUGE (Granularity-Adaptive coUnterfactual Gating of Evidence), a lightweight counterfactual-attribution gating framework for incomplete multimodal classification. Given an incomplete input, GAUGE recovers the missing modalities with a frozen imputer and encodes each observed or recovered modality into fine-grained evidence units111Evidence unit denotes the finest representation emitted by an encoder. It covers fine-grained representations, e.g., image patches, text tokens, and tabular feature embeddings, as well as modality-level embeddings when an encoder emits a single representation.. This abstraction supports unit-level modulation for multi-unit modalities and reduces to modality-level gating for single-representation modalities. Furthermore, GAUGE defines a representation-level counterfactual effect for each unit by measuring how the predicted-class response changes when that unit is replaced by a reference representation. To avoid exact per-unit interventions, GAUGE derives a signed first-order Taylor approximation and aggregates its channel-wise terms into a non-canceling magnitude score. All scores are obtained with a single ungated forward–backward pass and mapped by a two-scalar gate into additive attention-logit biases, reducing attention to low-scoring units while leaving the Transformer backbone architecture and training objective unchanged. Our contributions are summarized as follows: • We formulate a granularity-adaptive evidence control paradigm for incomplete multimodal classification, representing both observed and reconstructed modalities as encoder-emitted evidence units. This abstraction enables fine-grained evidence modulation at the finest available representation level while naturally retaining compatibility with modality-level settings. • We propose GAUGE, a lightweight counterfactual-attribution gating framework that modulates evidence according to prediction-aware evidence scores. GAUGE formalizes an interpretable unit-wise counterfactual replacement objective and derives a first-order Taylor scoring formulation that obtains all unit-level evidence scores in a single forward–backward pass, bypassing the prohibitive overhead of explicit per-unit interventions. • Extensive experiments across diverse incomplete-modality settings demonstrate the effectiveness of GAUGE. We further provide a Taylor remainder analysis for the signed first-order approximation and derive an upper bound relating the Taylor evidence score to the magnitude of the exact counterfactual effect. Related Work Incomplete Multimodal Learning. Existing methods for incomplete multimodal learning broadly follow two paradigms: recovery-free and recovery-based. Recovery-free approaches predict directly from available modalities via modality dropout, missing-aware training, prompt tuning, or contrastive learning (Neverova et al. 2016; Ma et al. 2021; Lee et al. 2023; Wu et al. 2024; Sun et al. 2024; Wang et al. 2018), avoiding fabricated content but discarding potentially recoverable cues. In contrast, recovery-based approaches explicitly reconstruct missing modalities through cross-modal generation, diffusion models, disentanglement, or context-guided completion (Wang et al. 2023; Dai et al. 2025; Yu et al. 2026; Liu et al. 2025; Zhao et al. 2026; Huang et al. 2026), alongside recent advances in retrieval-augmented prompts (Lang et al. 2025), feature disentangling (Yao et al. 2024), pre-training (Du et al. 2024), and inference-time selection (Du et al. 2026). However, these methods predominantly treat each observed or imputed modality as a monolithic decision unit, lacking prediction-aware control at the encoder-emitted feature level. GAUGE departs from this paradigm by dynamically re-evaluating both observed and recovered inputs as fine-grained evidence units, rather than treating a reconstructed modality as a static substitute. Dynamic Multimodal Fusion. Dynamic multimodal fusion studies sample-specific modality utility at inference time, such as input-conditioned routing (Xue and Marculescu 2023), quality-aware weighting (Zhang et al. 2023), predicted modality contributions (Cao et al. 2024), gradient-based decoupling (Wang et al. 2024), and cross-modal enhancement (Chen et al. 2026). However, these methods operate exclusively at the coarse modality level, applying a uniform scalar weight, routing score, or gating decision to each modality. This formulation conflicts with Transformer encoders, whose fine-grained internal units exhibit heterogeneous prediction relevance that becomes especially restrictive when handling recovered modalities. Our research advances this by shifting dynamic fusion from modality-level weighting to fine-grained unit-level modulation. Feature Attribution and Counterfactual Reasoning. Gradient-based attribution methods characterize reference-based or local changes in model outputs with respect to input features (Sundararajan et al. 2017; Shrikumar et al. 2017; Ancona et al. 2018; Li et al. 2025). In particular, a first-order Taylor expansion around the current representation provides a local approximation of the prediction change induced by replacing a feature with a reference (Deng et al. 2021). Unlike the post-hoc explainers, GAUGE repurposes such attribution signals as an in-the-loop mechanism, computing unit-level Taylor evidence scores in a single backward pass against a prediction-aware objective to directly drive the attention bias of the underlying model. Figure 1: Overview of the GAUGE framework: (1) complete & encode; (2) score & gate, where Taylor evidence scores are computed for all evidence units in a single backward pass; and (3) predict & update via additive attention-bias injection. Methodology Task Description. Let =(Xi,yi)i=1||D=\(X_i,y_i)\_i=1^|D| be a dataset for incomplete multimodal classification, where yi∈1,…,Ky_i∈\1,…,K\ is the class label among K target classes. For each sample i under the incomplete-modality setting, only a subset of modalities i⊆1,…,MO_i \1,…,M\ is observed, where M denotes the total number of modalities. The incomplete input is thus represented as Xi=xi(m)m∈iX_i=\x_i^(m)\_m _i, where xi(m)x_i^(m) is the observation from modality m, while the missing modalities are indexed by the set i=1,…,M∖iU_i=\1,…,M\ _i. A pre-trained multimodal imputer Γ is employed to reconstruct the missing inputs. Specifically, each missing modality u∈iu _i is recovered as x~i(u)=Γu(Xi) x_i^(u)= _u(X_i). The formulation also extends to feature-level missingness within a partially observed modality, as detailed in Appendix 4. Merging the observed and reconstructed inputs yields the completed input X¯i=x¯i(m)m=1M X_i=\ x_i^(m)\_m=1^M, where x¯i(m)=xi(m) x_i^(m)=x_i^(m) if m∈im _i, and x¯i(m)=x~i(m) x_i^(m)= x_i^(m) otherwise. The final goal is to predict yiy_i given the completed input X¯i X_i. Overview of GAUGE To achieve this goal, we propose GAUGE, a principled and scalable framework for granularity-adaptive evidence control. As illustrated in Fig. 1 and summarized in Algorithm 1, GAUGE consists of three phases: Phase 1 – Complete & Encode. A frozen imputer Γ reconstructs missing modalities to produce the completed multimodal input X¯i X_i. Modality-specific encoders h(m)h^(m) then map each completed modality x¯i(m) x_i^(m) to one or more evidence units, which are concatenated with a learnable [][ CLS] token to form the multimodal sequence T~i T_i. Phase 2 – Score & Gate. Given T~i T_i, GAUGE performs an ungated forward pass to obtain the logits ziz_i and the ungated predicted class y^i y_i. With y^i y_i fixed, a single backward pass on the saliency objective ℒsal(T~i)=−zi,y^iL_sal( T_i)=-z_i, y_i yields a magnitude-based Taylor evidence score TEi,jTE_i,j for each evidence unit along the replacement direction di,j=ei,j′−ei,jd_i,j=e _i,j-e_i,j. GAUGE then applies an N-adaptive score normalization to obtain si,js_i,j and maps it to a continuous gate gi,jg_i,j using two shared learnable scalars (τ,ρ)(τ,ρ). Phase 3 – Predict & Update. The resulting gates gi,jg_i,j are converted into an additive attention-logit bias matrix iB_i, applied at every Transformer block without altering the underlying backbone architecture. A gated forward pass on the same sequence T~i T_i then produces the final logits zigatedz_i^gated and the final predicted class y^ifinal y_i^final. During training, we minimize the cross-entropy loss on the gated prediction, which updates the encoders h(m)m=1M\h^(m)\_m=1^M, the backbone FθF_θ, the classifier πϕ _φ, and the gate parameters (τ,ρ)(τ,ρ); the imputer Γ remains frozen. Granularity-Adaptive Evidence Units To instantiate the fine-grained evidence control, GAUGE first represents each observed or recovered modality at the finest granularity exposed by its encoder. For modality m, the encoder h(m)h^(m) maps the completed modality x¯i(m) x_i^(m) to LmL_m evidence units, Ei(m)=h(m)(x¯i(m))=[ei,1(m),…,ei,Lm(m)]∈ℝLm×CE_i^(m)=h^(m)( x_i^(m))=[\,e_i,1^(m),…,e_i,L_m^(m)\,] ^L_m× C, where C denotes the shared hidden dimension. When Lm>1L_m>1, the units may correspond to image patches, text tokens, or tabular feature embeddings; when Lm=1L_m=1, the modality is represented by a single global embedding. The evidence units from all modalities are concatenated in a fixed modality order as Ti=[Ei(1);…;Ei(M)]=[ei,1,…,ei,N]∈ℝN×CT_i=[\,E_i^(1);…;E_i^(M)\,]=[\,e_i,1,…,e_i,N\,] ^N× C, where ei,je_i,j denotes the j-th unit in the flattened multimodal sequence and N=∑m=1MLmN= _m=1^ML_m. Because each LmL_m is fixed by the encoder configuration, N is constant within each dataset. We prepend a learnable [][ CLS] token to form T~i=[e[];Ti]∈ℝ(N+1)×C T_i=[\,e_[ CLS];T_i\,] ^(N+1)× C; modality and positional embeddings are incorporated but omitted from the notation for brevity. Each encoder-emitted representation thus forms a separate evidence unit, allowing units from the same modality to receive different Taylor evidence scores and gates, while the formulation reduces to modality-level gating whenever Lm=1L_m=1. Prediction-Aware Counterfactual Formulation 1) Ungated prediction and saliency objective. To determine each unit’s evidence strength in a prediction-aware manner, GAUGE performs an ungated forward pass on T~i T_i and uses the resulting predicted class as the fixed target for evidence scoring. The Transformer backbone FθF_θ processes the input sequence T~i T_i, and the classifier head πϕ _φ maps the resulting [][ CLS] representation to class logits, with the corresponding class probabilities defined as: zi=πϕ(Fθ(T~i)[]),pi=softmax(zi),z_i= _φ\! (F_θ( T_i)_[ CLS] ), p_i=softmax(z_i), (1) where zi∈ℝKz_i ^K denotes the logit vector over the K classes and zi,kz_i,k denotes its k-th entry. The ungated predicted class is y^i=argmaxkzi,k y_i= _kz_i,k. We define the prediction-aware saliency objective as ℒsal(T~i)=−zi,y^iL_sal( T_i)=-z_i, y_i, where y^i y_i is fixed during evidence-unit replacement and differentiation. We adopt this logit-based objective rather than cross-entropy because the latter may yield weak attribution gradients for confident predictions; a detailed comparison is provided in Appendix A.6. Algorithm 1 GAUGE: Minibatch Training 0: Minibatch (Xi,yi)i∈ℬ\(X_i,y_i)\_i ; frozen imputer Γ ; trainable parameters Θ=h(m)m=1M,Fθ,πϕ,τ,ρ = \\h^(m)\_m=1^M,F_θ, _φ,τ,ρ \. 0: Updated parameters Θ . 1: for i∈ℬi do 2: Form the completed input X¯i X_i from XiX_i using frozen Γ . 3: Compute Ei(m)←h(m)(x¯i(m))E_i^(m)← h^(m)( x_i^(m)) for m=1,…,Mm=1,…,M. 4: T~i←[e[];Ei(1);…;Ei(M)] T_i←[\,e_[ CLS];E_i^(1);…;E_i^(M)\,] in fixed modality order. 5: Compute ziz_i (Eq. 1) and set y^i←argmaxkzi,k y_i← _kz_i,k. y^i y_i fixed 6: end for 7: ℒsalℬ←−∑i∈ℬzi,y^iL_sal^B←- _i z_i, y_i. 8: Differentiate ℒsalℬL_sal^B once to obtain all ∇ei,jℒsal(T~i) _e_i,jL_sal( T_i) for i∈ℬi and j=1,…,Nj=1,…,N; do not accumulate gradients for Θ . 9: for i∈ℬi do 10: for j=1,…,Nj=1,…,N do 11: ei,j′←e _i,j 0; di,j←ei,j′−ei,jd_i,j← e _i,j-e_i,j. 12: TEi,j←sg(‖∇ei,jℒsal(T~i)⊙di,j‖1)TE_i,j \! ( \| _e_i,jL_sal( T_i) d_i,j \|_1 ). 13: end for 14: Compute si,js_i,j and gi,jg_i,j for j=1,…,Nj=1,…,N by Eqs. 6 and 7. 15: Set gi,0←1g_i,0← 1 and construct iB_i (Eq. 9). 16: zigated←πϕ(Fθ(T~i;i)[])z_i^gated← _φ\! (F_θ( T_i;B_i)_[ CLS] ). 17: pigated←softmax(zigated)p_i^gated (z_i^gated). 18: end for 19: Compute ℒtaskℬL_task^B (Eq. 10), backpropagate, and update Θ . 2) Representation-level counterfactual replacement. For the j-th evidence unit ei,je_i,j in the flattened multimodal sequence TiT_i, where j∈1,…,Nj∈\1,…,N\, we construct a representation-level counterfactual by replacing ei,je_i,j with a reference representation ei,j′e _i,j. Specifically, we set ei,j′=∈ℝCe _i,j=0 ^C. The resulting evidence sequence is denoted by Ti(j←ei,j′)=[ei,1,…,ei,j′,…,ei,N]T_i^(j← e _i,j)=[\,e_i,1,…,e _i,j,…,e_i,N\,], and its [][ CLS]-prepended counterpart by T~i(j←ei,j′)=[e[];Ti(j←ei,j′)] T_i^(j← e _i,j)=[\,e_[ CLS];T_i^(j← e _i,j)\,]. The [][ CLS] token is never replaced. The exact counterfactual effect of unit j is then defined as CEffi,jexact=ℒsal(T~i(j←ei,j′))−ℒsal(T~i).CEff_i,j^exact=L_sal\! ( T_i^(j← e _i,j) )-L_sal( T_i). (2) However, evaluating Eq. (2) for all N evidence units requires N additional forward passes per sample, motivating the pass-efficient Taylor scoring formulation introduced next. Efficient Taylor Gating via Attention Bias 1) First-order counterfactual approximation. To overcome this bottleneck, GAUGE avoids per-unit intervention forward passes by approximating each exact counterfactual effect with a signed first-order Taylor term. For evidence unit ei,je_i,j, let di,j=ei,j′−ei,jd_i,j=e _i,j-e_i,j denote the replacement direction from ei,je_i,j to its reference representation ei,j′e _i,j. A first-order Taylor expansion of ℒsalL_sal around ei,je_i,j along di,jd_i,j yields CEffi,jexact=⟨∇ei,jℒsal(T~i),di,j⟩+Ri,j,CEff_i,j^exact= _e_i,jL_sal( T_i),d_i,j +R_i,j, (3) where Ri,jR_i,j denotes the Taylor remainder. We denote the signed first-order term by ai,j=⟨∇ei,jℒsal(T~i),di,j⟩a_i,j= _e_i,jL_sal( T_i),d_i,j . Thus, Eq. (3) can be written as CEffi,jexact=ai,j+Ri,jCEff_i,j^exact=a_i,j+R_i,j. Assuming that ℒsalL_sal is twice continuously differentiable along the replacement segment and that the spectral norm of its Hessian with respect to ei,je_i,j is bounded by Hi,jH_i,j throughout this segment, the Taylor remainder satisfies |Ri,j|≤Hi,j2‖di,j‖22|R_i,j|≤ H_i,j2\|d_i,j\|_2^2. The full derivation is provided in Appendix A. 2) Taylor evidence scoring. The signed first-order term ai,ja_i,j is the sum of the channel-wise gradient–direction products, so contributions with opposite signs can cancel. To obtain a non-canceling magnitude score, we instead aggregate the absolute channel-wise products using the ℓ1 _1 norm, inspired by gradient-based attribution methods (Sundararajan et al. 2017; Shrikumar et al. 2017; Deng et al. 2021): TEi,j _i,j =sg(∑c=1C|∂ℒsal(T~i)∂ei,j,cdi,j,c|) =sg\! ( _c=1^C | _sal( T_i)∂ e_i,j,c\,d_i,j,c | ) (4) =sg(‖∇ei,jℒsal(T~i)⊙di,j‖1). =sg\! ( \| _e_i,jL_sal( T_i) d_i,j \|_1 ). Here, sg(⋅)sg(·) denotes the stop-gradient operator: it acts as the identity in the forward pass and has zero derivative during backpropagation. It therefore leaves both TEi,jTE_i,j and the bound below numerically unchanged, while preventing task-loss gradients from propagating through the gradient computation used to form the score during training. Applying the triangle inequality to their sum yields |ai,j|≤TEi,j|a_i,j| _i,j. Combined with the second-order remainder bound, this leads to the following upper bound on the magnitude of the exact counterfactual effect: |CEffi,jexact|≤TEi,j+Hi,j2‖di,j‖22.|CEff_i,j^exact| _i,j+ H_i,j2\|d_i,j\|_2^2. (5) This bound is established in Corollary 1 of Appendix A.5. Accordingly, TEi,jTE_i,j quantifies the aggregate magnitude of the channel-wise first-order contributions induced by replacing unit j along di,jd_i,j. All N gradients ∇ei,jℒsal(T~i)j=1N\ _e_i,jL_sal( T_i)\_j=1^N are obtained simultaneously from a single ungated backward pass, enabling GAUGE to compute all evidence-unit scores without the N additional intervention forward passes required for exact counterfactual evaluation. 3) N-adaptive score normalization. Raw Taylor evidence scores can differ substantially in scale across samples because their magnitudes depend jointly on the prediction gradient and the replacement direction, both of which vary across samples. Since the gate uses a threshold and temperature shared across units and samples, this variation can hinder consistent gate calibration across samples. GAUGE therefore normalizes the N Taylor evidence scores of each sample using the following N-adaptive rule before gate construction: si,j=TEi,j,N≤3,TEi,j−μimax(σi,ϵ),N>3,s_i,j= casesTE_i,j,&N≤ 3,\\[3.0pt] TE_i,j- _i ( _i,ε),&N>3, cases (6) where for N>3N>3, μi=1N∑ℓ=1NTEi,ℓ _i= 1N _ =1^NTE_i, , σi2=1N−1∑ℓ=1N(TEi,ℓ−μi)2, _i^2= 1N-1 _ =1^N(TE_i, - _i)^2, and ϵ>0ε>0 is a small constant for numerical stability. For N>3N>3, this per-sample standardization removes the sample-specific offset and scale, making si,js_i,j a relative score within the sample. For N≤3N≤ 3, statistics based on at most three scores can yield highly constrained standardized values and remove informative absolute magnitude, so GAUGE retains the raw scores. Because N is fixed by the encoder layout, each model uses the same normalization branch throughout training and inference. Appendix C.2 evaluates this normalization rule. 4) Continuous gate construction. Given the evidence score si,js_i,j, GAUGE maps it to a continuous gate gi,j∈(0,1)g_i,j∈(0,1): gi,j=sigmoid(si,j−τexp(ρ)).g_i,j=sigmoid\! ( s_i,j-τ (ρ) ). (7) Here, τ∈ℝτ is a learnable threshold, with gi,j=0.5g_i,j=0.5 at si,j=τs_i,j=τ, and ρ∈ℝρ parameterizes the positive temperature exp(ρ) (ρ), which controls the transition sharpness. This monotonic mapping assigns larger gates to higher-scoring units, enabling graded modulation rather than hard selection. The scalars (τ,ρ)(τ,ρ) are shared across all samples and evidence units, adding only two parameters regardless of the modality or unit count. The [][ CLS] token, indexed by j=0j=0, serves as the aggregation anchor rather than an evidence unit and is therefore kept ungated (gi,0=1g_i,0=1). 5) Key-wise attention-bias injection. GAUGE converts each gate into an additive attention-logit bias at the key position in each Transformer block, augmenting standard scaled dot-product attention (Vaswani et al. 2017): AttnGAUGE(,,)=softmax(⊤dh+i),Attn_GAUGE(Q,K,V)=softmax\! ( QK d_h+B_i )V, (8) where dhd_h denotes the key dimension of each attention head, and i∈ℝ(N+1)×(N+1)B_i ^(N+1)×(N+1) is defined as (i)q,r=logmaxgi,r,ϵg,q,r∈0,…,N,(B_i)_q,r= \g_i,r, _g\, q,r∈\0,…,N\, (9) where q and r index the query and key positions, respectively, and ϵg∈(0,1) _g∈(0,1) is a small constant that prevents log0 0 and keeps the bias finite. Index 0 corresponds to the [][ CLS] token, whose gate is fixed to gi,0=1g_i,0=1, yielding zero attention bias. Because (i)q,r(B_i)_q,r depends only on the key index r, the same unit-level bias is applied at every query position and shared across attention heads. Since 0<maxgi,r,ϵg≤10< \g_i,r, _g\≤ 1, all entries of iB_i are non-positive. Adding logmaxgi,r,ϵg \g_i,r, _g\ multiplies the unnormalized attention weight of key r by maxgi,r,ϵg \g_i,r, _g\. Smaller gates therefore more strongly down-weight the corresponding keys, while preserving the sequence length and backbone architecture. Training and Inference GAUGE follows the same imputation-and-gating pipeline during training and inference, with the imputer Γ kept frozen. During training, given a minibatch ℬB, we minimize the cross-entropy loss on the gated prediction pigated=softmax(zigated)p_i^gated=softmax(z_i^gated): ℒtaskℬ=−1|ℬ|∑i∈ℬlogpigated(yi).L_task^B=- 1|B| _i p_i^gated(y_i). (10) This objective updates the modality encoders h(m)m=1M\h^(m)\_m=1^M, Transformer backbone FθF_θ, classifier πϕ _φ, and gate parameters (τ,ρ)(τ,ρ). The Taylor evidence scores are detached before normalization and gate construction, so task optimization does not backpropagate through the saliency-gradient computation or require second-order differentiation. Each training step comprises one ungated forward–backward scoring pass and one gated forward–backward optimization pass. At inference, the same ungated forward–backward scoring pass is followed by one gated forward pass to produce the prediction. Model PolyMNIST Acc (%) ↑ MST Acc (%) ↑ CelebA Acc (%) ↑ DVM Acc (%) ↑ CAD AUC (%) ↑ Infarction AUC (%) ↑ Missing Rate η Missing Modalities Missing Tabular Rate γ 0 0.6 0.8 ∅ S,T M,T ∅ I T 0 0.9 1 0 0.9 1 0 0.9 1 (a) Recovery-based Methods for Missing Modality MultiAE 99.77 95.36 84.39 99.96 97.00 81.60 89.98 15.51 89.71 - - - - - - - - - MultiAE† 99.94 97.50 89.86 99.87 98.33 83.44 88.66 72.04 87.44 - - - - - - - - - MoPoE 99.79 93.94 79.84 99.62 90.86 79.01 38.97 13.91 34.84 - - - - - - - - - MoPoE† 99.63 96.81 87.06 99.39 96.50 82.54 68.22 56.90 65.75 - - - - - - - - - M3Care 99.93 56.66 40.53 99.99 16.03 9.34 92.33 99.92 51.75 98.44 - 11.92 85.62 - 64.99 70.61 - 70.53 M3Care† 99.99 97.27 87.92 99.98 98.27 85.16 98.73 97.14 91.32 98.94 - 93.43 72.48 - 72.48 83.27 - 68.44 OnlineMAE 100.00 98.29 90.09 99.90 98.14 84.14 86.67 86.67 86.67 90.92 - 89.90 85.22 - 70.96 84.05 - 61.39 (b) Recovery-free Methods for Missing Modality ModDrop 99.97 97.66 88.44 100.00 98.21 82.47 99.93 99.93 87.32 99.02 89.80 87.97 85.10 70.77 69.18 84.76 72.06 72.16 MTL 99.97 98.43 91.14 99.96 98.60 84.37 99.69 99.26 89.38 99.44 93.38 92.32 84.87 73.08 70.23 83.59 69.82 69.90 MAP 99.86 43.00 23.19 100.00 9.83 10.13 99.98 99.93 85.33 98.86 74.88 63.15 84.39 71.39 70.11 84.62 68.47 69.17 MAP† 99.99 96.74 76.20 99.99 97.84 11.36 100.00 99.99 86.06 99.37 92.43 91.15 85.26 70.36 68.76 85.49 71.38 70.81 MUSE 99.93 94.73 77.56 99.86 97.14 35.96 99.93 99.86 88.35 96.86 - 1.64 83.47 - 53.23 84.40 - 66.78 (c) Dynamic Recovery Method for Missing Modality DyMoc 100.00 99.79 96.74 99.99 97.61 84.11 100.00 100.00 87.26 99.34 93.48 92.87 84.12 71.29 68.31 86.96 73.70 71.36 !15 GAUGE 100.00 99.67 97.61 99.99 97.86 85.26 100.00 100.00 93.16 99.30 94.79 94.42 84.68 74.20 70.41 87.12 74.37 72.86 Table 1: Comparison of incomplete-modality methods across six benchmarks. Bold indicates the best result, and underline the second best; ties are both marked. Except for the reproduced DyMoc, baseline results are from Du et al. (2026) under the same missingness protocol; † denotes their incomplete-simulation variants, and “-” marks unsupported settings (Appendix B.2). Experiments Experiment Settings Datasets and Evaluation Metrics. We evaluate GAUGE on six multimodal classification benchmarks, including three simulated datasets (Sutter et al. 2021): PolyMNIST, MST, and bimodal CelebA, and two real-world image–tabular datasets: Deep Visual Marketing (DVM) (Huang et al. 2022) and UK Biobank (UKBB) (Sudlow et al. 2015). For UKBB, we consider two cardiac disease classification benchmarks: coronary artery disease (CAD) and myocardial infarction (Infarction). We report the area under the curve (AUC) on the two UKBB benchmarks and accuracy on the remaining four benchmarks. Dataset details and missingness protocols are provided in Appendix B.1. Baselines. We compare GAUGE with nine baselines: four recovery-based methods (MultiAE (Ngiam et al. 2011), MoPoE (Sutter et al. 2021), M3Care (Zhang et al. 2022), and OnlineMAE (Woo et al. 2023)), four recovery-free methods (ModDrop (Neverova et al. 2016), MTL (Ma et al. 2022), MAP (Lee et al. 2023), and MUSE (Wu et al. 2024)), and the dynamic-recovery baseline DyMoc (Du et al. 2026). Implementation details are provided in Appendix B.3. Figure 2: Unit-level gating and attention redistribution on the Infarction dataset at missing rate γ=1γ=1. Main Results & Analysis Quantitative Analysis. GAUGE achieves the best or tied-best performance in 9 of 12 incomplete-input settings reported in Table 1. Compared with recovery-free and recovery-based baselines, GAUGE’s stronger performance reflects the discarding–imputation dilemma (Du et al. 2026): discarding missing modalities may forfeit recoverable task-relevant cues, whereas indiscriminately incorporating imputed content may introduce task-irrelevant noise from unreliable reconstructions. GAUGE mitigates this trade-off through finer-grained evidence modulation. We further compare GAUGE with DyMoc, which uses the same backbone and frozen imputation pipeline but operates at the coarse modality level. GAUGE outperforms DyMoc in 10 of 12 incomplete-input settings, with larger gains under severe missingness. Specifically, it improves accuracy by 5.905.90 percentage points on CelebA T\T\, with the text modality entirely missing, and yields consistent gains on CAD and Infarction. These gains under severe missingness indicate that, as predictions rely more heavily on reconstructed inputs, unit-wise gating is more effective than modality-level weighting because it down-weights lower-scoring components without uniformly attenuating the recovered modality. Qualitative Analysis. Fig. 2 visualizes the unit-level gates and final-layer [][ CLS]-to-unit attention for an Infarction sample at γ=1γ=1 (1616 observed image units and 7575 imputed tabular units). As shown in Fig. 2(a), observed image units generally receive higher gates, whereas the 7575 imputed tabular units receive lower but nonuniform gates. This within-modality variation demonstrates that GAUGE differentiates encoder-emitted units via their prediction-aware Taylor evidence scores, avoiding the rigidity of modality-level assignment. Fig. 2(b) illustrates the resulting attention redistribution. Because the gates are injected as key-wise log-biases, lower gates introduce more negative biases, down-weighting lower-scoring units relative to higher-scoring ones. Across the test set, the mean gate is 0.690.69 for observed units and 0.430.43 for imputed units (Fig. 2(c)), showing that GAUGE assigns lower gates on average to reconstructed evidence while retaining substantial variation among units within the recovered modality. Crucially, this nonuniform gating highlights a limitation of modality-level control: a shared modality weight cannot down-weight low-scoring recovered components without simultaneously down-weighting higher-scoring ones from the same modality. Method PolyMNIST DVM CAD Infarction η=0.8η=0.8 γ=1γ=1 γ=1γ=1 γ=1γ=1 GAUGEv1 87.21 91.24 68.22 71.34 GAUGEv2 97.35 94.18 69.13 68.32 GAUGEv3 97.23 94.27 68.95 69.52 GAUGE 97.61 94.42 70.41 72.86 Table 2: Ablation of fine-grained evidence control under severe missingness. Ablation Study To evaluate the necessity of fine-grained evidence control, we compare GAUGE with three alternatives shown in Table 2. GAUGEv1 performs inference directly on the completed inputs without fine-tuning or evidence gating. GAUGEv2 uses the same missingness-aware fine-tuning as full GAUGE but fixes all gates to one, yielding zero attention bias; GAUGEv3 shares one gate across all evidence units within each modality; and GAUGE assigns a separate gate to each evidence unit. GAUGE consistently achieves the best performance across all four benchmarks with multi-unit modalities. The advantage over modality-level gating (GAUGEv3) is particularly pronounced on clinical benchmarks with many evidence units in the recovered tabular modality, reaching 3.343.34 AUC points on Infarction. Notably, modality-level gating can underperform GAUGEv1, as a shared modality gate cannot suppress low-scoring units without simultaneously down-weighting high-scoring ones from the same modality; unit-level gating mitigates this limitation by assigning a separate gate to each evidence unit. Further ablations on evidence-unit scoring, score normalization, and evidence-unit granularity are provided in Appendix C. Additional Analysis Efficiency Analysis. We compare GAUGE with exact counterfactual scoring across four configurations with different evidence-unit counts N. Table 3 shows that exact scoring time increases markedly with N, whereas GAUGE remains within 3.113.11–3.603.60 ms. At N=91N=91, GAUGE reduces the per-sample scoring time from 96.4396.43 ms to 3.113.11 ms, yielding a 31.01×31.01× speedup. These results show that GAUGE avoids per-unit intervention overhead and supports scalable fine-grained evidence scoring. Detailed timing protocol is in Appendix D. Dataset N GAUGE (ms) Exact (ms) Speedup MST 3 3.15 4.51 1.43× PolyMNIST 20 3.60 26.30 7.31× DVM 33 3.18 39.06 12.28× CAD 91 3.11 96.43 31.01× Table 3: Per-sample scoring time for GAUGE and exact counterfactual scoring on a Quadro RTX 8000 with batch size 11. Figure 3: Test AUC (%) on CAD and Infarction under varying tabular missing rates γ. Performance under Increasing Missingness. To evaluate robustness across missing rates, we compare GAUGE and DyMoc on CAD and Infarction. As shown in Fig. 3, GAUGE outperforms DyMoc in most scenarios, demonstrating consistent adaptability from mild to complete tabular absence. Crucially, GAUGE’s advantage becomes increasingly pronounced as missingness rises. When missingness is mild (γ≤0.5γ≤ 0.5) and observed features dominate, both methods perform comparably. However, once imputation becomes dominant (γ≥0.7γ≥ 0.7), GAUGE generally exhibits a larger advantage over DyMoc. At extreme missingness (γ=1.0γ=1.0), GAUGE surpasses DyMoc by 2.102.10 and 1.501.50 percentage points in AUC on CAD and Infarction, respectively. This further demonstrates that fine-grained evidence control becomes increasingly important when models rely heavily on reconstructed content. Conclusion and Limitations In this paper, we introduced GAUGE, a principled and scalable framework for granularity-adaptive evidence control in incomplete multimodal classification. By formulating a unit-wise counterfactual gating mechanism driven by pass-efficient Taylor evidence scoring, GAUGE effectively transitions multimodal evidence modulation from coarse modality-level aggregation to the finest granularity exposed by Transformer encoders. Extensive experiments across diverse benchmarks demonstrate that GAUGE consistently outperforms state-of-the-art baselines, exhibiting exceptional robustness under severe modality incompleteness. 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Cited by: Introduction. Y. Zhao, C. Chen, Q. Y. Pang, Y. Fu, Q. Li, C. Tang, B. T. Ang, and Y. Jin (2026) Tackling dual-stage missing modalities in brain tumor segmentation via robust modality reconstruction and prompt-guided modality adaptation. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 40, p. 13314–13322. Cited by: Related Work. A. Zhu, M. Hu, Y. Xing, and Y. Tang (2026) Decoupled sub-feature uncertainty modeling for robust multimodal representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Findings, p. 6921–6931. Cited by: Introduction. Appendix A Theoretical Analysis This appendix analyzes the Taylor evidence score used in GAUGE. It first defines the exact unit-level counterfactual effect and derives a first-order approximation with a remainder bound. It then examines the non-canceling aggregation and explains the choice of a logit-based saliency objective. A.1 Intervention Setup and Notation We first establish the intervention notation used to define the exact counterfactual effect, its signed first-order approximation, and the resulting Taylor evidence score. For sample i, replacing evidence unit ei,je_i,j in T~i=[e[];ei,1,…,ei,N] T_i=[\,e_[ CLS];e_i,1,…,e_i,N\,] with a reference representation ei,j′e _i,j yields T~i(j←ei,j′) T_i^(j← e _i,j), with replacement direction di,j=ei,j′−ei,jd_i,j=e _i,j-e_i,j. Throughout this work, we use the zero reference ei,j′=e _i,j=0, while the [][ CLS] token is never replaced. A.2 Exact Unit-Level Counterfactual Effect The saliency objective is ℒsal(T~i)=−zi,y^iL_sal( T_i)=-z_i, y_i, the negative logit of the ungated predicted class. Here y^i y_i is determined once from the unreplaced sequence T~i T_i and is held constant under both evidence-unit replacement and differentiation: we do not differentiate through the argmax , and if the predicted class changes after replacement, ℒsal(⋅)L_sal(·) still denotes the negative logit of the original ungated predicted class y^i y_i. With y^i y_i fixed, the exact counterfactual effect of unit j is defined as: CEffi,jexact=ℒsal(T~i(j←ei,j′))−ℒsal(T~i),CEff_i,j^exact=L_sal\! ( T_i^(j← e _i,j) )-L_sal( T_i), and its magnitude |CEffi,jexact||CEff_i,j^exact| measures the response change induced by replacing unit j with its reference representation. Computing this for all units requires one counterfactual forward pass per unit, i.e., N additional forward passes after the ungated forward pass. GAUGE instead constructs first-order local surrogates for all N effects using a single ungated forward–backward scoring pass. A.3 First-Order Counterfactual Approximation Because evaluating the exact effect separately for every evidence unit is computationally expensive, we derive a local first-order surrogate whose unit-wise values can be obtained from a single backward pass. Fixing all evidence units except ei,je_i,j, regard the saliency objective as a function of this unit alone: ψi,j(v)=ℒsal(T~i(j←v)). _i,j(v)=L_sal\! ( T_i^(j← v) ). (11) With y^i y_i fixed, ψi,j _i,j is a scalar function of v, with ψi,j(ei,j)=ℒsal(T~i) _i,j(e_i,j)=L_sal( T_i) and ψi,j(ei,j′)=ℒsal(T~i(j←ei,j′)) _i,j(e _i,j)=L_sal( T_i^(j← e _i,j)), so that CEffi,jexact=ψi,j(ei,j′)−ψi,j(ei,j)CEff_i,j^exact= _i,j(e _i,j)- _i,j(e_i,j). A first-order Taylor expansion of ψi,j _i,j at ei,je_i,j gives CEffi,jexact≈ai,j:=⟨∇ei,jℒsal(T~i),di,j⟩,CEff_i,j^exact≈ a_i,j:= _e_i,jL_sal( T_i),\,d_i,j , (12) which is the first-order term of Eq. (3) in the main text; we call ai,ja_i,j the signed first-order term. This formulation makes all unit-wise first-order terms available from a single backward pass. The following analysis first characterizes its approximation error and then addresses the cancellation that can arise when channel-wise contributions are summed. A.4 Taylor Remainder and Approximation Error We now characterize the approximation error of the signed first-order term relative to the exact counterfactual effect under a local smoothness condition. Theorem 1 (First-order Counterfactual Approximation). Let Ωi,j=ei,j+αdi,j:α∈[0,1] _i,j=\\,e_i,j+α\,d_i,j:α∈[0,1]\,\ be the line segment between ei,je_i,j and ei,j′e _i,j. Assume that ψi,j _i,j is twice continuously differentiable on a neighborhood of Ωi,j _i,j and that the unit-wise Hessian is uniformly bounded on this segment: supv∈Ωi,j‖∇2ψi,j(v)‖op≤Hi,j. _v∈ _i,j \|∇^2 _i,j(v) \|_op≤ H_i,j. Then the remainder Ri,j:=CEffi,jexact−ai,jR_i,j:=CEff_i,j^exact-a_i,j satisfies CEffi,jexact=ai,j+Ri,j,|Ri,j|≤Hi,j2‖di,j‖22,CEff_i,j^exact=a_i,j+R_i,j, |R_i,j|≤ H_i,j2\,\|d_i,j\|_2^2, (13) recovering Eq. (3) of the main text with an explicit remainder bound. Consequently, ||CEffi,jexact|−|ai,j||≤Hi,j2‖di,j‖22, |\, |CEff_i,j^exact |-|a_i,j|\, |≤ H_i,j2\,\|d_i,j\|_2^2, (14) i.e., |ai,j||a_i,j| approximates the magnitude of the exact counterfactual effect up to a second-order error. Remark 1 (ReLU and smoothness). Networks with ReLU are not twice differentiable at activation boundaries. The stated remainder bound therefore applies when the replacement segment remains within a single differentiable activation region. If the segment crosses activation boundaries, the C2C^2 assumption of Theorem 1 need not hold. For smooth activations such as GELU or SiLU, a locally bounded Hessian can instead be assumed on the finite replacement segment. The theorem requires only this local bound, not a global Hessian bound. Proof. Let ψi,j _i,j be defined as in Eq. (11). By Taylor’s theorem with Lagrange remainder, there exists a point ξi,j∈Ωi,j _i,j∈ _i,j such that ψi,j(ei,j′) _i,j(e _i,j) =ψi,j(ei,j)+⟨∇ψi,j(ei,j),di,j⟩ = _i,j(e_i,j)+ ∇ _i,j(e_i,j),\,d_i,j (15) +12di,j⊤∇2ψi,j(ξi,j)di,j. + 12\,d_i,j ∇^2 _i,j( _i,j)\,d_i,j. Since ∇ψi,j(ei,j)=∇ei,jℒsal(T~i)∇ _i,j(e_i,j)= _e_i,jL_sal( T_i) and CEffi,jexact=ψi,j(ei,j′)−ψi,j(ei,j)CEff_i,j^exact= _i,j(e _i,j)- _i,j(e_i,j), subtracting ψi,j(ei,j) _i,j(e_i,j) from both sides identifies the remainder as Ri,j=CEffi,jexact−ai,j=12di,j⊤∇2ψi,j(ξi,j)di,j.R_i,j=CEff_i,j^exact-a_i,j= 12\,d_i,j ∇^2 _i,j( _i,j)\,d_i,j. (16) The Hessian bound implies |Ri,j|≤12‖∇2ψi,j(ξi,j)‖op‖di,j‖22≤Hi,j2‖di,j‖22.|R_i,j|≤ 12 \|∇^2 _i,j( _i,j) \|_op\|d_i,j\|_2^2≤ H_i,j2\,\|d_i,j\|_2^2. (17) Finally, since |CEffi,jexact|=|ai,j+Ri,j||CEff_i,j^exact|=|a_i,j+R_i,j|, the reverse triangle inequality ||ai,j+Ri,j|−|ai,j||≤|Ri,j| |\,|a_i,j+R_i,j|-|a_i,j|\, |≤|R_i,j| proves Eq. (14). ∎ The theorem therefore identifies the regime in which the signed first-order term is a faithful local surrogate: its approximation error is controlled by the local curvature and grows quadratically with the replacement distance. A.5 Non-Canceling Taylor Evidence Scoring The magnitude of the signed first-order term can be reduced by cancellation among channel-wise contributions with opposite signs. We therefore motivate GAUGE’s non-canceling aggregation and establish its relation to the exact counterfactual effect. The signed first-order term can be written channel-wise as ai,j=∑c=1C∂ℒsal(T~i)∂ei,j,cdi,j,c.a_i,j= _c=1^C _sal( T_i)∂ e_i,j,c\,d_i,j,c. GAUGE instead sums the absolute channel-wise contributions. This avoids sign cancellation, although the resulting score may exceed the magnitude of the signed first-order term, |ai,j||a_i,j|. Specifically, TEi,j _i,j =sg(∑c=1C|∂ℒsal(T~i)∂ei,j,cdi,j,c|) =sg\! ( _c=1^C | _sal( T_i)∂ e_i,j,c\,d_i,j,c | ) (18) =sg(‖∇ei,jℒsal(T~i)⊙di,j‖1). =sg\! ( \| _e_i,jL_sal( T_i) d_i,j \|_1 ). The stop-gradient operator does not change the numerical value of TEi,jTE_i,j and therefore does not affect the inequalities below; it only prevents the task loss from differentiating through the gradient-based score. Corollary 1. Because sg(⋅)sg(·) does not alter the forward value, the triangle inequality yields |ai,j|≤TEi,j.|a_i,j| _i,j. Thus, TEi,jTE_i,j upper-bounds the magnitude of the signed first-order term and serves as a non-canceling channel-wise magnitude surrogate. Combining this with Theorem 1 gives |CEffi,jexact|≤TEi,j+Hi,j2‖di,j‖22, |CEff_i,j^exact | _i,j+ H_i,j2\,\|d_i,j\|_2^2, (19) which is Eq. (5) of the main text. The Hessian term appears only in the theoretical error bound; GAUGE neither estimates Hi,jH_i,j nor uses it during training. A.6 Choice of Saliency Objective The quality of the resulting evidence score also depends on the saliency objective. We therefore compare the predicted-class logit with negative log-likelihood and show why the former avoids softmax-induced gradient saturation. One alternative is to use predicted-class cross-entropy, equivalently the negative log-likelihood (NLL), as the saliency target: ℒNLL(T~i)=−logpi(y^i).L_NLL( T_i)=- p_i( y_i). (20) Its gradient with respect to the logits is ∇ziℒNLL=pi−y^i. _z_iL_NLL=p_i-1_ y_i. (21) Here, y^i1_ y_i is the one-hot vector of the ungated predicted class. By the chain rule, the evidence-unit gradient is ∇ei,jℒNLL=(∂zi∂ei,j)⊤(pi−y^i). _e_i,jL_NLL= ( ∂ z_i∂ e_i,j )^\! (p_i-1_ y_i ). (22) For a confident model, pi≈y^ip_i 1_ y_i. When the logits–evidence-unit Jacobian is bounded, the factor pi−y^ip_i-1_ y_i can make the evidence-unit gradient very small. NLL-based saliency may therefore underestimate the contribution of informative evidence units due to softmax saturation. In contrast, the gradient of ℒsal=−zi,y^iL_sal=-z_i, y_i with respect to the logits is the constant −y^i-1_ y_i, bypassing the softmax and avoiding saturation. This is consistent with attribution methods that operate at the pre-softmax level (Shrikumar et al. 2017; Sundararajan et al. 2017). Accordingly, GAUGE combines a signed Taylor term for pass-efficient local approximation, ℓ1 _1 aggregation to prevent channel-wise cancellation, and the predicted-class logit to avoid softmax saturation. Appendix B Experimental Configuration B.1 Datasets and Input Configurations Datasets. Table 4 summarizes the six multimodal classification benchmarks derived from five datasets. PolyMNIST, MST (MNIST–SVHN–Text), and bimodal CelebA (Sutter et al. 2021) are three simulated multimodal benchmarks. PolyMNIST contains five image views of the same digit. MST consists of three heterogeneous modalities: MNIST (M), SVHN (S), and synthetic text (T). CelebA pairs face images (I) with attribute-based text descriptions (T). We further evaluate GAUGE on DVM and two cardiac classification benchmarks derived from UK Biobank. DVM (Huang et al. 2022) combines vehicle images with 1717 tabular features, including 44 categorical and 1313 continuous features. The coronary artery disease (CAD) and myocardial infarction (Infarction) benchmarks (Sudlow et al. 2015) each combine cardiac MR images with 7575 disease-related tabular features, including 2626 categorical and 4949 continuous features. Because both diseases have low prevalence, class-balanced subsets are used for training. The dataset splits follow DyMo (Du et al. 2026). Dataset/Task Modalities Train Validation Test Classes PolyMNIST Five image views 60,000 3,000 7,000 10 MST MNIST, SVHN, text 1,121,360 60,000 140,000 10 CelebA Image, text 162,770 19,962 19,867 2 DVM Image, tabular 70,565 17,642 88,207 283 CAD MR image, tabular 3,482 6,510 3,617 2 Infarction MR image, tabular 1,552 6,510 3,617 2 Table 4: Dataset statistics for the six benchmarks, using the splits of Du et al. (2026). CAD and Infarction are constructed from UK Biobank. Missingness Protocols. Following DyMo (Du et al. 2026), we use the same mask-generation procedure and benchmark-specific missingness protocols. For PolyMNIST, each sample randomly misses η×100%η× 100\% of its five image modalities, and Table 1 reports η∈0,0.6,0.8η∈\0,0.6,0.8\. For MST, we report the complete-input setting (∅ ) and the missing-modality subsets S,T\S,T\ and M,T\M,T\. For CelebA, we report ∅ , I\I\, and T\T\. For both benchmarks, each set denotes the modalities removed from the input. For DVM, CAD, and Infarction, the imaging modality remains observed, while each sample randomly misses γ×100%γ× 100\% of its tabular features. Table 1 reports γ∈0,0.9,1γ∈\0,0.9,1\ for all three benchmarks. For CAD and Infarction, Fig. 3 additionally includes γ∈0.1,0.3,0.5,0.7γ∈\0.1,0.3,0.5,0.7\. Here, η=0η=0 or γ=0γ=0 corresponds to complete input, and γ=1γ=1 indicates that the entire tabular modality is missing. To connect the feature-level protocol above with the formulation in the main text, let t denote the tabular modality. For 0<γ<10<γ<1, t remains partially observed and therefore belongs to iO_i. Its missing entries are reconstructed by the frozen TIP imputer used in the DyMo pipeline: x~i(t)=Γt(xi(m)m∈i∖t,xi,obs(t)), x_i^(t)= _t\! (\x_i^(m)\_m _i \t\,\,x_i,obs^(t) ), where xi,obs(t)x_i,obs^(t) denotes the observed tabular entries. The completed modality x¯i(t) x_i^(t) retains the observed entries and fills the missing positions with the corresponding imputed values. When γ=1γ=1, the tabular modality belongs to iU_i, reducing to whole-modality imputation. In all cases, GAUGE operates only on the completed input X¯i X_i produced by the frozen DyMo/TIP pipeline. B.2 Baselines We compare GAUGE with nine baselines and summarize their sources and configurations. The recovery-based baselines are MultiAE (Ngiam et al. 2011), MoPoE (Sutter et al. 2021), M3Care (Zhang et al. 2022), and OnlineMAE (Woo et al. 2023); the recovery-free baselines are ModDrop (Neverova et al. 2016), MTL (Ma et al. 2022), MAP (Lee et al. 2023), and MUSE (Wu et al. 2024). We also include the dynamic-recovery baseline DyMo (Du et al. 2026) with the cosine-distance reward, denoted as DyMoc. GAUGE and our reproduced DyMoc share the same backbone and frozen imputation pipeline, making DyMoc the closest architecture-controlled baseline. Except for our reproduced DyMoc, the results in Table 1 are taken from Tables 1, S3, and S4 of Du et al. (2026) under the same missingness protocols; † marks their incomplete-simulation variants (MultiAE, MoPoE, M3Care, and MAP). Methods are reported only for the settings they support; unsupported entries are marked “-”. Per-baseline configurations (e.g., the drop probability p=0.5p=0.5 of ModDrop and MUSE, MAP prompt lengths, and the encoder initialization of OnlineMAE) are likewise unchanged from that work. Table 1 focuses on methods explicitly designed or adapted for incomplete multimodal learning. Dynamic fusion methods such as QMF, DynMM, and PDF, which were evaluated by Du et al. (2026) on observed-plus-recovered inputs, are not repeated in this table. B.3 Implementation Details Model Initialization. We adopt the dataset-specific backbones and frozen imputers from DyMo (Du et al. 2026). A MoPoE multimodal VAE is used to reconstruct missing modalities on PolyMNIST, MST, and CelebA, while the TIP imputer (Du et al. 2024) reconstructs missing tabular features on DVM, CAD, and Infarction. The modality-specific encoders, multimodal Transformer, and classifier are initialized from our reproduced DyMoc checkpoints using the architectures and hyperparameters reported in Table S2 of Du et al. (2026). During fine-tuning, the encoders, Transformer, classifier, and gate parameters (τ,ρ)(τ,ρ) are updated, whereas all imputer parameters remain frozen. We use the same dataset preprocessing as DyMo. Evidence-Unit Layouts. The evidence-unit layout is fixed by the inherited dataset-specific encoders. Each PolyMNIST view emits 44 units, giving N=20N=20 across its five views. Each modality in MST and CelebA emits one global embedding, giving N=3N=3 and N=2N=2, respectively. DVM emits 1616 image units and 1717 tabular units (N=33N=33), while CAD and Infarction each emit 1616 image units and 7575 tabular units (N=91N=91). These layouts determine the branch used by the N-adaptive normalization rule (Appendix C.2). Because each modality in MST and CelebA emits a single embedding, GAUGE reduces to modality-level gating on these two benchmarks. Dataset LR Batch Epochs Metric PolyMNIST 10−310^-3 256 100 Acc. MST 10−410^-4 256 20 Acc. CelebA 10−310^-3 256 20 Acc. DVM 10−410^-4 256 300 Acc. CAD 10−310^-3 128 300 AUC Infarction 3×10−43×10^-4 128 300 AUC Table 5: Dataset-specific fine-tuning settings for GAUGE. Optimization and Gate Settings. We fine-tune GAUGE using Adam with zero weight decay and the dataset-specific learning rates, batch sizes, and maximum numbers of epochs listed in Table 5. A new missingness mask is sampled for each minibatch following the protocol in Appendix B.1. We use the zero reference ei,j′=e _i,j=0. The gate threshold τ is initialized to 0 for PolyMNIST, DVM, CAD, and Infarction, which use z-score normalization, and to 0.50.5 for MST and CelebA, which use raw scores. For all benchmarks, ρ is initialized to 0, such that exp(ρ)=1 (ρ)=1, and we set ϵ=10−6ε=10^-6 and ϵg=10−9 _g=10^-9. We select the checkpoint with the best validation accuracy on PolyMNIST, MST, CelebA, and DVM or the best validation AUC on CAD and Infarction. Appendix C Supplementary Experiments C.1 Ablation on Evidence-Unit Scoring Score Function PolyMNIST MST CelebA DVM CAD Infarction Avg. η=0.8η=0.8 M,T\M,T\ T\T\ γ=1γ=1 γ=1γ=1 γ=1γ=1 Random score 97.14 85.01 92.56 94.22 66.74 70.37 84.34 Feature norm 97.16 84.89 92.71 94.18 68.81 69.26 84.50 Gradient norm 96.93 84.97 92.84 94.14 68.64 70.77 84.72 Attention score 96.87 85.03 92.76 94.29 67.92 69.33 84.37 Taylor evidence score 97.61 85.26 93.16 94.42 70.41 72.86 85.62 Table 6: Ablation of evidence-unit scoring under the most severe missingness setting for each benchmark. Accuracy (%) is reported for PolyMNIST, MST, CelebA, and DVM, while AUC (%) is reported for CAD and Infarction. All variants use the same score-to-gate pipeline and differ only in the raw scoring function. Bold marks the best result. To examine whether GAUGE benefits specifically from its reference-based Taylor evidence score, we compare it with four alternative scoring functions: random score, feature norm, gradient norm, and attention score. For a controlled comparison, all variants use the same N-adaptive normalization, continuous gate, attention-bias injection, initialization, and fine-tuning settings, differing only in the raw evidence-unit score. For evidence unit ei,je_i,j, random score samples an independent value from (0,1)N(0,1) at each scoring pass; feature norm uses ‖ei,j‖2\|e_i,j\|_2; and gradient norm uses ‖∇ei,jℒsal(T~i)‖1\| _e_i,jL_sal( T_i)\|_1. Attention score uses the final-layer [][ CLS]-to-unit attention from the ungated forward pass, averaged across attention heads. The Taylor evidence score is the default score defined in Eq. (4), using the zero reference ei,j′=e _i,j=0. As shown in Table 6, the Taylor evidence score consistently outperforms all four alternatives across all six benchmarks, achieving the highest average performance of 85.6285.62, which exceeds the strongest competing scorer, gradient norm, by 0.900.90 points. These results support the effectiveness of GAUGE’s reference-based Taylor scoring mechanism, showing that combining the prediction-aware gradient with the unit-to-reference displacement yields a stronger gating signal than the four alternative scores. C.2 Ablation on Score Normalization GAUGE adopts an N-adaptive normalization rule because per-sample standardization behaves differently across encoder layouts. When only a few evidence units are available, the sample mean and variance are determined by too few scores, causing the standardized values to become highly constrained and discarding their absolute magnitudes. For example, when N=2N=2, any two distinct scores are mapped to −1/2-1/ 2 and 1/21/ 2, regardless of their original scale. In contrast, when many units are present, raw Taylor evidence scores can exhibit substantial sample-dependent offsets and scales, making the shared gate threshold and temperature difficult to calibrate consistently. To verify whether these two regimes require different treatments, we fix the raw score to the Taylor evidence score and compare the default N-adaptive rule in Eq. (6) with two fixed strategies: Always z-score and Always raw. The comparison includes two small-N benchmarks, MST (N=3N=3) and CelebA (N=2N=2), and two large-N benchmarks, CAD and Infarction (N=91N=91). All other model components and training settings are kept unchanged. As shown in Table 7, applying z-score normalization uniformly degrades performance in the small-N regime, reducing accuracy by 0.230.23 points on MST and 0.580.58 points on CelebA relative to raw scores. This confirms that standardization based on only two or three units can over-constrain their relative values and remove useful magnitude information. Conversely, using raw scores uniformly reduces AUC by 1.941.94 points on CAD and 1.671.67 points on Infarction, showing that normalization is important for calibrating the scores of large evidence-unit sets. These results support the N-adaptive rule: raw scores are preferable for small N, whereas per-sample standardization is beneficial for large N. Normalization MST CelebA CAD Infarction N=3N=3 N=2N=2 N=91N=91 N=91N=91 Always z-score 85.03 92.58 70.41 72.86 Always raw 85.26 93.16 68.47 71.19 N-adaptive (default) 85.26 93.16 70.41 72.86 Table 7: Ablation of score normalization under each benchmark’s most severe missingness setting. Accuracy (%) is reported for MST and CelebA, while AUC (%) is reported for CAD and Infarction. Bold marks the best result. C.3 Ablation on Evidence-Unit Granularity We examine whether finer-grained image evidence units improve GAUGE’s unit-level control. A finer spatial grid exposes more localized regions as separate evidence units, each with its own Taylor evidence score and gate. We test this by varying the input resolution on CAD and Infarction at γ=1γ=1. Input resolutions of 96×9696× 96, 128×128128× 128, 160×160160× 160, 192×192192× 192, and 256×256256× 256 pixels yield 3×33× 3, 4×44× 4, 5×55× 5, 6×66× 6, and 8×88× 8 evidence-unit grids, respectively. The native 128×128128× 128 input corresponds to a 4×44× 4 grid. We keep the model architecture, optimization setup, and gating configuration fixed across all resolution settings. Only the image positional embeddings are reinitialized, while all other pre-trained weights are retained and fine-tuned. Fig. 4 shows that finer grids generally improve test AUC on both benchmarks. CAD AUC increases from 69.1069.10 at 3×33× 3 to 70.8770.87 at 8×88× 8, while Infarction reaches its highest AUC of 73.9073.90 at 6×66× 6, with no further gain at 8×88× 8. Because bilinear upsampling adds no new source-pixel information, the gains beyond the native 4×44× 4 grid suggest that finer evidence-unit granularity can improve localized scoring and gating. On Infarction, performance plateaus beyond 6×66× 6, indicating limited benefit from further grid refinement. Figure 4: Effect of image resolution and the resulting evidence-unit granularity on CAD and Infarction at γ=1γ=1. Appendix D Scoring Cost Comparison Table 3 reports the runtime of the scoring stage, measured from the encoded sequence T~i T_i to the N evidence-unit scores. We exclude imputation, modality encoding, and the final gated forward pass, as these operations are identical for both methods. Exact counterfactual scoring performs one ungated forward pass followed by N sequential intervention forward passes, with batch size 11. In contrast, GAUGE computes all N Taylor evidence scores through a single ungated forward–backward pass. For each configuration, runtime is averaged over 500500 test samples after 2020 warm-up iterations. We call torch.cuda.synchronize() before and after each measurement to account for asynchronous GPU execution. All timing measurements are conducted on a single NVIDIA Quadro RTX 8000.