GAUGE: Granularity-Adaptive Counterfactual Gating of Evidence for Incomplete Multimodal Classification

# Granularity-Adaptive Counterfactual Gating of Evidence for Incomplete Multimodal Classification
Source: [https://arxiv.org/html/2608.05608](https://arxiv.org/html/2608.05608)
###### 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 proposeGAUGE, 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\(Lianget al\.[2024](https://arxiv.org/html/2608.05608#bib.bib7); Xuet al\.[2023](https://arxiv.org/html/2608.05608#bib.bib44); Zhanget al\.[2025a](https://arxiv.org/html/2608.05608#bib.bib1)\)\. However, they are typically developed under the idealized assumption that all required modalities are available for each input\(Maet al\.[2021](https://arxiv.org/html/2608.05608#bib.bib9); Duet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib17); Yanget al\.[2025](https://arxiv.org/html/2608.05608#bib.bib2)\)\. In practice, modalities may be missing or inaccessible because of sensor failures, heterogeneous acquisition protocols, privacy constraints, or transmission errors\(Sunet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib35); Yaoet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib16)\)\. Reliable prediction under such incomplete inputs is therefore critical in high\-stakes applications such as medical diagnosis\(Yaoet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib16)\)and autonomous driving\(Parket al\.[2025](https://arxiv.org/html/2608.05608#bib.bib43)\)\.

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\(Maet al\.[2021](https://arxiv.org/html/2608.05608#bib.bib9); Wuet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib12); Zhanget al\.[2025b](https://arxiv.org/html/2608.05608#bib.bib37)\), avoiding imputation\-induced errors but discarding potentially recoverable cross\-modal cues\. In contrast,*recovery\-based*approaches explicitly impute missing modalities\(Zhanget al\.[2022](https://arxiv.org/html/2608.05608#bib.bib14); Wanget al\.[2023](https://arxiv.org/html/2608.05608#bib.bib15)\)to restore complementary cues, but the reconstructed content may contain low\-fidelity, noisy, or semantically inconsistent evidence\. Recent quality\-aware dynamic fusion\(Zhanget al\.[2023](https://arxiv.org/html/2608.05608#bib.bib22); Caoet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib23)\)and modality\-robustness methods\(Wanget al\.[2024](https://arxiv.org/html/2608.05608#bib.bib21)\)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\(Zhuet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib46)\)\. 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\(Denget al\.[2021](https://arxiv.org/html/2608.05608#bib.bib26); Fanet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib47)\)\. 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—NNadditional passes forNNcomponents\(Fong and Vedaldi[2017](https://arxiv.org/html/2608.05608#bib.bib45)\)\. Scalable control therefore requires estimating all component\-wise effects using a fixed number of forward–backward scoring passes independent ofNN\.

To address these questions, we proposeGAUGE\(Granularity\-Adaptive coUnterfactualGating ofEvidence\), 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 units*111*Evidence 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 proposeGAUGE, 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\(Neverovaet al\.[2016](https://arxiv.org/html/2608.05608#bib.bib8); Maet al\.[2021](https://arxiv.org/html/2608.05608#bib.bib9); Leeet al\.[2023](https://arxiv.org/html/2608.05608#bib.bib11); Wuet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib12); Sunet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib35); Wanget al\.[2018](https://arxiv.org/html/2608.05608#bib.bib4)\), 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\(Wanget al\.[2023](https://arxiv.org/html/2608.05608#bib.bib15); Daiet al\.[2025](https://arxiv.org/html/2608.05608#bib.bib48); Yuet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib5); Liuet al\.[2025](https://arxiv.org/html/2608.05608#bib.bib39); Zhaoet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib41); Huanget al\.[2026](https://arxiv.org/html/2608.05608#bib.bib42)\), alongside recent advances in retrieval\-augmented prompts\(Langet al\.[2025](https://arxiv.org/html/2608.05608#bib.bib38)\), feature disentangling\(Yaoet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib16)\), pre\-training\(Duet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib17)\), and inference\-time selection\(Duet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib18)\)\. 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\(Xue and Marculescu[2023](https://arxiv.org/html/2608.05608#bib.bib19)\), quality\-aware weighting\(Zhanget al\.[2023](https://arxiv.org/html/2608.05608#bib.bib22)\), predicted modality contributions\(Caoet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib23)\), gradient\-based decoupling\(Wanget al\.[2024](https://arxiv.org/html/2608.05608#bib.bib21)\), and cross\-modal enhancement\(Chenet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib40)\)\. 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\(Sundararajanet al\.[2017](https://arxiv.org/html/2608.05608#bib.bib24); Shrikumaret al\.[2017](https://arxiv.org/html/2608.05608#bib.bib20); Anconaet al\.[2018](https://arxiv.org/html/2608.05608#bib.bib36); Liet al\.[2025](https://arxiv.org/html/2608.05608#bib.bib3)\)\. 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\(Denget al\.[2021](https://arxiv.org/html/2608.05608#bib.bib26)\)\. 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\.

![Refer to caption](https://arxiv.org/html/2608.05608v1/x1.png)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\.Let𝒟=\{\(Xi,yi\)\}i=1\|𝒟\|\\mathcal\{D\}=\\\{\(X\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{\|\\mathcal\{D\}\|\}be a dataset for incomplete multimodal classification, whereyi∈\{1,…,K\}y\_\{i\}\\in\\\{1,\\ldots,K\\\}is the class label amongKKtarget classes\. For each sampleiiunder the incomplete\-modality setting, only a subset of modalities𝒪i⊆\{1,…,M\}\\mathcal\{O\}\_\{i\}\\subseteq\\\{1,\\ldots,M\\\}is observed, whereMMdenotes the total number of modalities\. The incomplete input is thus represented asXi=\{xi\(m\)\}m∈𝒪iX\_\{i\}=\\\{x\_\{i\}^\{\(m\)\}\\\}\_\{m\\in\\mathcal\{O\}\_\{i\}\}, wherexi\(m\)x\_\{i\}^\{\(m\)\}is the observation from modalitymm, while the missing modalities are indexed by the set𝒰i=\{1,…,M\}∖𝒪i\\mathcal\{U\}\_\{i\}=\\\{1,\\ldots,M\\\}\\setminus\\mathcal\{O\}\_\{i\}\. A pre\-trained multimodal imputerΓ\\Gammais employed to reconstruct the missing inputs\. Specifically, each missing modalityu∈𝒰iu\\in\\mathcal\{U\}\_\{i\}is recovered asx~i\(u\)=Γu​\(Xi\)\\tilde\{x\}\_\{i\}^\{\(u\)\}=\\Gamma\_\{u\}\(X\_\{i\}\)\. The formulation also extends to feature\-level missingness within a partially observed modality, as detailed in Appendix[4](https://arxiv.org/html/2608.05608#A2.T4)\. Merging the observed and reconstructed inputs yields the completed inputX¯i=\{x¯i\(m\)\}m=1M\\bar\{X\}\_\{i\}=\\\{\\bar\{x\}\_\{i\}^\{\(m\)\}\\\}\_\{m=1\}^\{M\}, wherex¯i\(m\)=xi\(m\)\\bar\{x\}\_\{i\}^\{\(m\)\}=x\_\{i\}^\{\(m\)\}ifm∈𝒪im\\in\\mathcal\{O\}\_\{i\}, andx¯i\(m\)=x~i\(m\)\\bar\{x\}\_\{i\}^\{\(m\)\}=\\tilde\{x\}\_\{i\}^\{\(m\)\}otherwise\. The final goal is to predictyiy\_\{i\}given the completed inputX¯i\\bar\{X\}\_\{i\}\.

### Overview of GAUGE

To achieve this goal, we proposeGAUGE, a principled and scalable framework for granularity\-adaptive evidence control\. As illustrated in Fig\.[1](https://arxiv.org/html/2608.05608#Sx2.F1)and summarized in Algorithm[1](https://arxiv.org/html/2608.05608#alg1), GAUGE consists of three phases:

Phase1– Complete & Encode\.A frozen imputerΓ\\Gammareconstructs missing modalities to produce the completed multimodal inputX¯i\\bar\{X\}\_\{i\}\. Modality\-specific encodersh\(m\)h^\{\(m\)\}then map each completed modalityx¯i\(m\)\\bar\{x\}\_\{i\}^\{\(m\)\}to one or more evidence units, which are concatenated with a learnable\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]token to form the multimodal sequenceT~i\\tilde\{T\}\_\{i\}\.

Phase2– Score & Gate\.GivenT~i\\tilde\{T\}\_\{i\}, GAUGE performs an ungated forward pass to obtain the logitsziz\_\{i\}and the ungated predicted classy^i\\hat\{y\}\_\{i\}\. Withy^i\\hat\{y\}\_\{i\}fixed, a single backward pass on the saliency objectiveℒsal​\(T~i\)=−zi,y^i\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)=\-z\_\{i,\\hat\{y\}\_\{i\}\}yields a magnitude\-based*Taylor evidence score*TEi,j\\mathrm\{TE\}\_\{i,j\}for each evidence unit along the replacement directiondi,j=ei,j′−ei,jd\_\{i,j\}=e^\{\\prime\}\_\{i,j\}\-e\_\{i,j\}\. GAUGE then applies anNN\-adaptive score normalization to obtainsi,js\_\{i,j\}and maps it to a continuous gategi,jg\_\{i,j\}using two shared learnable scalars\(τ,ρ\)\(\\tau,\\rho\)\.

Phase3– Predict & Update\.The resulting gatesgi,jg\_\{i,j\}are converted into an additive attention\-logit bias matrix𝐁i\\mathbf\{B\}\_\{i\}, applied at every Transformer block without altering the underlying backbone architecture\. A gated forward pass on the same sequenceT~i\\tilde\{T\}\_\{i\}then produces the final logitszigatedz\_\{i\}^\{\\mathrm\{gated\}\}and the final predicted classy^ifinal\\hat\{y\}\_\{i\}^\{\\mathrm\{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 backboneFθF\_\{\\theta\}, the classifierπϕ\\pi\_\{\\phi\}, and the gate parameters\(τ,ρ\)\(\\tau,\\rho\); the imputerΓ\\Gammaremains 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 modalitymm, the encoderh\(m\)h^\{\(m\)\}maps the completed modalityx¯i\(m\)\\bar\{x\}\_\{i\}^\{\(m\)\}toLmL\_\{m\}evidence units,Ei\(m\)=h\(m\)​\(x¯i\(m\)\)=\[ei,1\(m\),…,ei,Lm\(m\)\]∈ℝLm×CE\_\{i\}^\{\(m\)\}=h^\{\(m\)\}\(\\bar\{x\}\_\{i\}^\{\(m\)\}\)=\[\\,e\_\{i,1\}^\{\(m\)\},\\ldots,e\_\{i,L\_\{m\}\}^\{\(m\)\}\\,\]\\in\\mathbb\{R\}^\{L\_\{m\}\\times C\}, whereCCdenotes the shared hidden dimension\. WhenLm\>1L\_\{m\}\>1, the units may correspond to image patches, text tokens, or tabular feature embeddings; whenLm=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 asTi=\[Ei\(1\);…;Ei\(M\)\]=\[ei,1,…,ei,N\]∈ℝN×CT\_\{i\}=\[\\,E\_\{i\}^\{\(1\)\};\\ldots;E\_\{i\}^\{\(M\)\}\\,\]=\[\\,e\_\{i,1\},\\ldots,e\_\{i,N\}\\,\]\\in\\mathbb\{R\}^\{N\\times C\}, whereei,je\_\{i,j\}denotes thejj\-th unit in the flattened multimodal sequence andN=∑m=1MLmN=\\sum\_\{m=1\}^\{M\}L\_\{m\}\. Because eachLmL\_\{m\}is fixed by the encoder configuration,NNis constant within each dataset\. We prepend a learnable\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]token to formT~i=\[e\[𝙲𝙻𝚂\];Ti\]∈ℝ\(N\+1\)×C\\tilde\{T\}\_\{i\}=\[\\,e\_\{\[\\mathtt\{CLS\}\]\};T\_\{i\}\\,\]\\in\\mathbb\{R\}^\{\(N\+1\)\\times 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 wheneverLm=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 onT~i\\tilde\{T\}\_\{i\}and uses the resulting predicted class as the fixed target for evidence scoring\. The Transformer backboneFθF\_\{\\theta\}processes the input sequenceT~i\\tilde\{T\}\_\{i\}, and the classifier headπϕ\\pi\_\{\\phi\}maps the resulting\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]representation to class logits, with the corresponding class probabilities defined as:

zi=πϕ​\(Fθ​\(T~i\)\[𝙲𝙻𝚂\]\),pi=softmax​\(zi\),z\_\{i\}=\\pi\_\{\\phi\}\\\!\\left\(F\_\{\\theta\}\(\\tilde\{T\}\_\{i\}\)\_\{\[\\mathtt\{CLS\}\]\}\\right\),\\qquad p\_\{i\}=\\mathrm\{softmax\}\(z\_\{i\}\),\(1\)wherezi∈ℝKz\_\{i\}\\in\\mathbb\{R\}^\{K\}denotes the logit vector over theKKclasses andzi,kz\_\{i,k\}denotes itskk\-th entry\. The ungated predicted class isy^i=arg⁡maxk⁡zi,k\\hat\{y\}\_\{i\}=\\arg\\max\_\{k\}z\_\{i,k\}\. We define the prediction\-aware saliency objective asℒsal​\(T~i\)=−zi,y^i\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)=\-z\_\{i,\\hat\{y\}\_\{i\}\}, wherey^i\\hat\{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](https://arxiv.org/html/2608.05608#A1.SS6)\.

Algorithm 1GAUGE: Minibatch Training0:Minibatch

\{\(Xi,yi\)\}i∈ℬ\\\{\(X\_\{i\},y\_\{i\}\)\\\}\_\{i\\in\\mathcal\{B\}\}; frozen imputer

Γ\\Gamma; trainable parameters

Θ=\{\{h\(m\)\}m=1M,Fθ,πϕ,τ,ρ\}\\Theta=\\bigl\\\{\\\{h^\{\(m\)\}\\\}\_\{m=1\}^\{M\},F\_\{\\theta\},\\pi\_\{\\phi\},\\tau,\\rho\\bigr\\\}\.

0:Updated parameters

Θ\\Theta\.

1:for

i∈ℬi\\in\\mathcal\{B\}do

2:Form the completed input

X¯i\\bar\{X\}\_\{i\}from

XiX\_\{i\}using frozen

Γ\\Gamma\.

3:Compute

Ei\(m\)←h\(m\)​\(x¯i\(m\)\)E\_\{i\}^\{\(m\)\}\\leftarrow h^\{\(m\)\}\(\\bar\{x\}\_\{i\}^\{\(m\)\}\)for

m=1,…,Mm=1,\\ldots,M\.

4:

T~i←\[e\[𝙲𝙻𝚂\];Ei\(1\);…;Ei\(M\)\]\\tilde\{T\}\_\{i\}\\leftarrow\[\\,e\_\{\[\\mathtt\{CLS\}\]\};E\_\{i\}^\{\(1\)\};\\ldots;E\_\{i\}^\{\(M\)\}\\,\]in fixed modality order\.

5:Compute

ziz\_\{i\}\(Eq\.[1](https://arxiv.org/html/2608.05608#Sx3.E1)\) and set

y^i←arg⁡maxk⁡zi,k\\hat\{y\}\_\{i\}\\leftarrow\\arg\\max\_\{k\}z\_\{i,k\}\. \{

y^i\\hat\{y\}\_\{i\}fixed\}

6:endfor

7:

ℒsalℬ←−∑i∈ℬzi,y^i\\mathcal\{L\}\_\{\\mathrm\{sal\}\}^\{\\mathcal\{B\}\}\\leftarrow\-\\sum\_\{i\\in\\mathcal\{B\}\}z\_\{i,\\hat\{y\}\_\{i\}\}\.

8:Differentiate

ℒsalℬ\\mathcal\{L\}\_\{\\mathrm\{sal\}\}^\{\\mathcal\{B\}\}once to obtain all

∇ei,jℒsal​\(T~i\)\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)for

i∈ℬi\\in\\mathcal\{B\}and

j=1,…,Nj=1,\\ldots,N; do not accumulate gradients for

Θ\\Theta\.

9:for

i∈ℬi\\in\\mathcal\{B\}do

10:for

j=1,…,Nj=1,\\ldots,Ndo

11:

ei,j′←𝟎e^\{\\prime\}\_\{i,j\}\\leftarrow\\mathbf\{0\};

di,j←ei,j′−ei,jd\_\{i,j\}\\leftarrow e^\{\\prime\}\_\{i,j\}\-e\_\{i,j\}\.

12:

TEi,j←sg⁡\(‖∇ei,jℒsal​\(T~i\)⊙di,j‖1\)\\mathrm\{TE\}\_\{i,j\}\\leftarrow\\operatorname\{sg\}\\\!\\left\(\\left\\\|\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\\odot d\_\{i,j\}\\right\\\|\_\{1\}\\right\)\.

13:endfor

14:Compute

si,js\_\{i,j\}and

gi,jg\_\{i,j\}for

j=1,…,Nj=1,\\ldots,Nby Eqs\.[6](https://arxiv.org/html/2608.05608#Sx3.E6)and[7](https://arxiv.org/html/2608.05608#Sx3.E7)\.

15:Set

gi,0←1g\_\{i,0\}\\leftarrow 1and construct

𝐁i\\mathbf\{B\}\_\{i\}\(Eq\.[9](https://arxiv.org/html/2608.05608#Sx3.E9)\)\.

16:

zigated←πϕ​\(Fθ​\(T~i;𝐁i\)\[𝙲𝙻𝚂\]\)z\_\{i\}^\{\\mathrm\{gated\}\}\\leftarrow\\pi\_\{\\phi\}\\\!\\left\(F\_\{\\theta\}\(\\tilde\{T\}\_\{i\};\\mathbf\{B\}\_\{i\}\)\_\{\[\\mathtt\{CLS\}\]\}\\right\)\.

17:

pigated←softmax⁡\(zigated\)p\_\{i\}^\{\\mathrm\{gated\}\}\\leftarrow\\operatorname\{softmax\}\(z\_\{i\}^\{\\mathrm\{gated\}\}\)\.

18:endfor

19:Compute

ℒtaskℬ\\mathcal\{L\}\_\{\\mathrm\{task\}\}^\{\\mathcal\{B\}\}\(Eq\.[10](https://arxiv.org/html/2608.05608#Sx3.E10)\), backpropagate, and update

Θ\\Theta\.

#### 2\) Representation\-level counterfactual replacement\.

For thejj\-th evidence unitei,je\_\{i,j\}in the flattened multimodal sequenceTiT\_\{i\}, wherej∈\{1,…,N\}j\\in\\\{1,\\ldots,N\\\}, we construct a representation\-level counterfactual by replacingei,je\_\{i,j\}with a reference representationei,j′e^\{\\prime\}\_\{i,j\}\. Specifically, we setei,j′=𝟎∈ℝCe^\{\\prime\}\_\{i,j\}=\\mathbf\{0\}\\in\\mathbb\{R\}^\{C\}\. The resulting evidence sequence is denoted byTi\(j←ei,j′\)=\[ei,1,…,ei,j′,…,ei,N\]T\_\{i\}^\{\(j\\leftarrow e^\{\\prime\}\_\{i,j\}\)\}=\[\\,e\_\{i,1\},\\ldots,e^\{\\prime\}\_\{i,j\},\\ldots,e\_\{i,N\}\\,\], and its\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]\-prepended counterpart byT~i\(j←ei,j′\)=\[e\[𝙲𝙻𝚂\];Ti\(j←ei,j′\)\]\\tilde\{T\}\_\{i\}^\{\(j\\leftarrow e^\{\\prime\}\_\{i,j\}\)\}=\[\\,e\_\{\[\\mathtt\{CLS\}\]\};T\_\{i\}^\{\(j\\leftarrow e^\{\\prime\}\_\{i,j\}\)\}\\,\]\. The\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]token is never replaced\. The exact counterfactual effect of unitjjis then defined as

CEffi,jexact=ℒsal​\(T~i\(j←ei,j′\)\)−ℒsal​\(T~i\)\.\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}=\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\\\!\\left\(\\tilde\{T\}\_\{i\}^\{\(j\\leftarrow e^\{\\prime\}\_\{i,j\}\)\}\\right\)\-\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\.\(2\)However, evaluating Eq\. \([2](https://arxiv.org/html/2608.05608#Sx3.E2)\) for allNNevidence units requiresNNadditional 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 unitei,je\_\{i,j\}, letdi,j=ei,j′−ei,jd\_\{i,j\}=e^\{\\prime\}\_\{i,j\}\-e\_\{i,j\}denote the replacement direction fromei,je\_\{i,j\}to its reference representationei,j′e^\{\\prime\}\_\{i,j\}\. A first\-order Taylor expansion ofℒsal\\mathcal\{L\}\_\{\\mathrm\{sal\}\}aroundei,je\_\{i,j\}alongdi,jd\_\{i,j\}yields

CEffi,jexact=⟨∇ei,jℒsal​\(T~i\),di,j⟩\+Ri,j,\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}=\\left\\langle\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\),d\_\{i,j\}\\right\\rangle\+R\_\{i,j\},\(3\)whereRi,jR\_\{i,j\}denotes the Taylor remainder\. We denote the signed first\-order term byai,j=⟨∇ei,jℒsal​\(T~i\),di,j⟩a\_\{i,j\}=\\langle\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\),d\_\{i,j\}\\rangle\. Thus, Eq\. \([3](https://arxiv.org/html/2608.05608#Sx3.E3)\) can be written asCEffi,jexact=ai,j\+Ri,j\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}=a\_\{i,j\}\+R\_\{i,j\}\. Assuming thatℒsal\\mathcal\{L\}\_\{\\mathrm\{sal\}\}is twice continuously differentiable along the replacement segment and that the spectral norm of its Hessian with respect toei,je\_\{i,j\}is bounded byHi,jH\_\{i,j\}throughout this segment, the Taylor remainder satisfies\|Ri,j\|≤Hi,j2​‖di,j‖22\|R\_\{i,j\}\|\\leq\\tfrac\{H\_\{i,j\}\}\{2\}\\\|d\_\{i,j\}\\\|\_\{2\}^\{2\}\. The full derivation is provided in Appendix[A](https://arxiv.org/html/2608.05608#A1)\.

#### 2\) Taylor evidence scoring\.

The signed first\-order termai,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 theℓ1\\ell\_\{1\}norm, inspired by gradient\-based attribution methods\(Sundararajanet al\.[2017](https://arxiv.org/html/2608.05608#bib.bib24); Shrikumaret al\.[2017](https://arxiv.org/html/2608.05608#bib.bib20); Denget al\.[2021](https://arxiv.org/html/2608.05608#bib.bib26)\):

TEi,j\\displaystyle\\mathrm\{TE\}\_\{i,j\}=sg⁡\(∑c=1C\|∂ℒsal​\(T~i\)∂ei,j,c​di,j,c\|\)\\displaystyle=\\operatorname\{sg\}\\\!\\left\(\\sum\_\{c=1\}^\{C\}\\left\|\\frac\{\\partial\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\}\{\\partial e\_\{i,j,c\}\}\\,d\_\{i,j,c\}\\right\|\\right\)\(4\)=sg⁡\(‖∇ei,jℒsal​\(T~i\)⊙di,j‖1\)\.\\displaystyle=\\operatorname\{sg\}\\\!\\left\(\\left\\\|\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\\odot d\_\{i,j\}\\right\\\|\_\{1\}\\right\)\.
Here,sg⁡\(⋅\)\\operatorname\{sg\}\(\\cdot\)denotes the stop\-gradient operator: it acts as the identity in the forward pass and has zero derivative during backpropagation\. It therefore leaves bothTEi,j\\mathrm\{TE\}\_\{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\}\|\\leq\\mathrm\{TE\}\_\{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\.\|\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}\|\\leq\\mathrm\{TE\}\_\{i,j\}\+\\frac\{H\_\{i,j\}\}\{2\}\\\|d\_\{i,j\}\\\|\_\{2\}^\{2\}\.\(5\)This bound is established in Corollary[1](https://arxiv.org/html/2608.05608#Thmcorollary1)of Appendix[A\.5](https://arxiv.org/html/2608.05608#A1.SS5)\. Accordingly,TEi,j\\mathrm\{TE\}\_\{i,j\}quantifies the aggregate magnitude of the channel\-wise first\-order contributions induced by replacing unitjjalongdi,jd\_\{i,j\}\. AllNNgradients\{∇ei,jℒsal​\(T~i\)\}j=1N\\\{\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\\\}\_\{j=1\}^\{N\}are obtained simultaneously from a single ungated backward pass, enabling GAUGE to compute all evidence\-unit scores without theNNadditional intervention forward passes required for exact counterfactual evaluation\.

#### 3\)NN\-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 theNNTaylor evidence scores of each sample using the followingNN\-adaptive rule before gate construction:

si,j=\{TEi,j,N≤3,TEi,j−μimax⁡\(σi,ϵ\),N\>3,s\_\{i,j\}=\\begin\{cases\}\\mathrm\{TE\}\_\{i,j\},&N\\leq 3,\\\\\[3\.0pt\] \\dfrac\{\\mathrm\{TE\}\_\{i,j\}\-\\mu\_\{i\}\}\{\\max\(\\sigma\_\{i\},\\epsilon\)\},&N\>3,\\end\{cases\}\(6\)where forN\>3N\>3,μi=1N​∑ℓ=1NTEi,ℓ\\mu\_\{i\}=\\frac\{1\}\{N\}\\sum\_\{\\ell=1\}^\{N\}\\mathrm\{TE\}\_\{i,\\ell\},σi2=1N−1​∑ℓ=1N\(TEi,ℓ−μi\)2,\\sigma\_\{i\}^\{2\}=\\frac\{1\}\{N\-1\}\\sum\_\{\\ell=1\}^\{N\}\(\\mathrm\{TE\}\_\{i,\\ell\}\-\\mu\_\{i\}\)^\{2\},andϵ\>0\\epsilon\>0is a small constant for numerical stability\. ForN\>3N\>3, this per\-sample standardization removes the sample\-specific offset and scale, makingsi,js\_\{i,j\}a relative score within the sample\. ForN≤3N\\leq 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\. BecauseNNis fixed by the encoder layout, each model uses the same normalization branch throughout training and inference\. Appendix[C\.2](https://arxiv.org/html/2608.05608#A3.SS2)evaluates this normalization rule\.

#### 4\) Continuous gate construction\.

Given the evidence scoresi,js\_\{i,j\}, GAUGE maps it to a continuous gategi,j∈\(0,1\)g\_\{i,j\}\\in\(0,1\):

gi,j=sigmoid⁡\(si,j−τexp⁡\(ρ\)\)\.g\_\{i,j\}=\\operatorname\{sigmoid\}\\\!\\left\(\\frac\{s\_\{i,j\}\-\\tau\}\{\\exp\(\\rho\)\}\\right\)\.\(7\)Here,τ∈ℝ\\tau\\in\\mathbb\{R\}is a learnable threshold, withgi,j=0\.5g\_\{i,j\}=0\.5atsi,j=τs\_\{i,j\}=\\tau, andρ∈ℝ\\rho\\in\\mathbb\{R\}parameterizes the positive temperatureexp⁡\(ρ\)\\exp\(\\rho\), which controls the transition sharpness\. This monotonic mapping assigns larger gates to higher\-scoring units, enabling graded modulation rather than hard selection\. The scalars\(τ,ρ\)\(\\tau,\\rho\)are shared across all samples and evidence units, adding only two parameters regardless of the modality or unit count\. The\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]token, indexed byj=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\(Vaswaniet al\.[2017](https://arxiv.org/html/2608.05608#bib.bib25)\):

AttnGAUGE​\(𝐐,𝐊,𝐕\)=softmax​\(𝐐𝐊⊤dh\+𝐁i\)​𝐕,\\mathrm\{Attn\}\_\{\\mathrm\{GAUGE\}\}\(\\mathbf\{Q\},\\mathbf\{K\},\\mathbf\{V\}\)=\\mathrm\{softmax\}\\\!\\left\(\\frac\{\\mathbf\{Q\}\\mathbf\{K\}^\{\\top\}\}\{\\sqrt\{d\_\{h\}\}\}\+\\mathbf\{B\}\_\{i\}\\right\)\\mathbf\{V\},\(8\)wheredhd\_\{h\}denotes the key dimension of each attention head, and𝐁i∈ℝ\(N\+1\)×\(N\+1\)\\mathbf\{B\}\_\{i\}\\in\\mathbb\{R\}^\{\(N\+1\)\\times\(N\+1\)\}is defined as

\(𝐁i\)q,r=log⁡max⁡\{gi,r,ϵg\},q,r∈\{0,…,N\},\(\\mathbf\{B\}\_\{i\}\)\_\{q,r\}=\\log\\max\\\{g\_\{i,r\},\\epsilon\_\{g\}\\\},\\qquad q,r\\in\\\{0,\\ldots,N\\\},\(9\)whereqqandrrindex the query and key positions, respectively, andϵg∈\(0,1\)\\epsilon\_\{g\}\\in\(0,1\)is a small constant that preventslog⁡0\\log 0and keeps the bias finite\. Index0corresponds to the\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]token, whose gate is fixed togi,0=1g\_\{i,0\}=1, yielding zero attention bias\. Because\(𝐁i\)q,r\(\\mathbf\{B\}\_\{i\}\)\_\{q,r\}depends only on the key indexrr, the same unit\-level bias is applied at every query position and shared across attention heads\. Since0<max⁡\{gi,r,ϵg\}≤10<\\max\\\{g\_\{i,r\},\\epsilon\_\{g\}\\\}\\leq 1, all entries of𝐁i\\mathbf\{B\}\_\{i\}are non\-positive\. Addinglog⁡max⁡\{gi,r,ϵg\}\\log\\max\\\{g\_\{i,r\},\\epsilon\_\{g\}\\\}multiplies the unnormalized attention weight of keyrrbymax⁡\{gi,r,ϵg\}\\max\\\{g\_\{i,r\},\\epsilon\_\{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 imputerΓ\\Gammakept frozen\. During training, given a minibatchℬ\\mathcal\{B\}, we minimize the cross\-entropy loss on the gated predictionpigated=softmax⁡\(zigated\)p\_\{i\}^\{\\mathrm\{gated\}\}=\\operatorname\{softmax\}\(z\_\{i\}^\{\\mathrm\{gated\}\}\):

ℒtaskℬ=−1\|ℬ\|​∑i∈ℬlog⁡pigated​\(yi\)\.\\mathcal\{L\}\_\{\\mathrm\{task\}\}^\{\\mathcal\{B\}\}=\-\\frac\{1\}\{\|\\mathcal\{B\}\|\}\\sum\_\{i\\in\\mathcal\{B\}\}\\log p\_\{i\}^\{\\mathrm\{gated\}\}\(y\_\{i\}\)\.\(10\)This objective updates the modality encoders\{h\(m\)\}m=1M\\\{h^\{\(m\)\}\\\}\_\{m=1\}^\{M\}, Transformer backboneFθF\_\{\\theta\}, classifierπϕ\\pi\_\{\\phi\}, and gate parameters\(τ,ρ\)\(\\tau,\\rho\)\. 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\.

Table 1:Comparison of incomplete\-modality methods across six benchmarks\.Boldindicates the best result, andunderlinethe second best; ties are both marked\. Except for the reproduced DyMoc, baseline results are fromDuet al\.\([2026](https://arxiv.org/html/2608.05608#bib.bib18)\)under the same missingness protocol;†\\daggerdenotes their incomplete\-simulation variants, and “\-” marks unsupported settings \(Appendix[B\.2](https://arxiv.org/html/2608.05608#A2.SS2)\)\.

## Experiments

### Experiment Settings

#### Datasets and Evaluation Metrics\.

We evaluate GAUGE on six multimodal classification benchmarks, including three simulated datasets\(Sutteret al\.[2021](https://arxiv.org/html/2608.05608#bib.bib13)\): PolyMNIST, MST, and bimodal CelebA, and two real\-world image–tabular datasets: Deep Visual Marketing \(DVM\)\(Huanget al\.[2022](https://arxiv.org/html/2608.05608#bib.bib30)\)and UK Biobank \(UKBB\)\(Sudlowet al\.[2015](https://arxiv.org/html/2608.05608#bib.bib31)\)\. 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](https://arxiv.org/html/2608.05608#A2.SS1)\.

#### Baselines\.

We compare GAUGE with nine baselines: four recovery\-based methods \(MultiAE\(Ngiamet al\.[2011](https://arxiv.org/html/2608.05608#bib.bib32)\), MoPoE\(Sutteret al\.[2021](https://arxiv.org/html/2608.05608#bib.bib13)\), M3Care\(Zhanget al\.[2022](https://arxiv.org/html/2608.05608#bib.bib14)\), and OnlineMAE\(Wooet al\.[2023](https://arxiv.org/html/2608.05608#bib.bib33)\)\), four recovery\-free methods \(ModDrop\(Neverovaet al\.[2016](https://arxiv.org/html/2608.05608#bib.bib8)\), MTL\(Maet al\.[2022](https://arxiv.org/html/2608.05608#bib.bib10)\), MAP\(Leeet al\.[2023](https://arxiv.org/html/2608.05608#bib.bib11)\), and MUSE\(Wuet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib12)\)\), and the dynamic\-recovery baseline DyMoc\(Duet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib18)\)\. Implementation details are provided in Appendix[B\.3](https://arxiv.org/html/2608.05608#A2.SS3)\.

![Refer to caption](https://arxiv.org/html/2608.05608v1/x2.png)Figure 2:Unit\-level gating and attention redistribution on the Infarction dataset at missing rateγ=1\\gamma=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](https://arxiv.org/html/2608.05608#Sx3.T1)\. Compared with*recovery\-free*and*recovery\-based*baselines, GAUGE’s stronger performance reflects the*discarding–imputation dilemma*\(Duet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib18)\): 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 DyMocin 10 of 12 incomplete\-input settings, with larger gains under severe missingness\. Specifically, it improves accuracy by5\.905\.90percentage points on CelebA\{T\}\\\{\\mathrm\{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](https://arxiv.org/html/2608.05608#Sx4.F2)visualizes the unit\-level gates and final\-layer\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]\-to\-unit attention for an Infarction sample atγ=1\\gamma=1\(1616observed image units and7575imputed tabular units\)\. As shown in Fig\.[2](https://arxiv.org/html/2608.05608#Sx4.F2)\(a\), observed image units generally receive higher gates, whereas the7575imputed 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](https://arxiv.org/html/2608.05608#Sx4.F2)\(b\) 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 is0\.690\.69for observed units and0\.430\.43for imputed units \(Fig\.[2](https://arxiv.org/html/2608.05608#Sx4.F2)\(c\)\), 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\.

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](https://arxiv.org/html/2608.05608#Sx4.T2)\. GAUGEv1performs inference directly on the completed inputs without fine\-tuning or evidence gating\. GAUGEv2uses the same missingness\-aware fine\-tuning as full GAUGE but fixes all gates to one, yielding zero attention bias; GAUGEv3shares 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, reaching3\.343\.34AUC 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](https://arxiv.org/html/2608.05608#A3)\.

### Additional Analysis

#### Efficiency Analysis\.

We compare GAUGE with exact counterfactual scoring across four configurations with different evidence\-unit countsNN\. Table[3](https://arxiv.org/html/2608.05608#Sx4.T3)shows that exact scoring time increases markedly withNN, whereas GAUGE remains within3\.113\.11–3\.603\.60ms\. AtN=91N\{=\}91, GAUGE reduces the per\-sample scoring time from96\.4396\.43ms to3\.113\.11ms, yielding a31\.01×31\.01\\timesspeedup\. These results show that GAUGE avoids per\-unit intervention overhead and supports scalable fine\-grained evidence scoring\. Detailed timing protocol is in Appendix[D](https://arxiv.org/html/2608.05608#A4)\.

Table 3:Per\-sample scoring time for GAUGE and exact counterfactual scoring on a Quadro RTX 8000 with batch size11\.![Refer to caption](https://arxiv.org/html/2608.05608v1/x3.png)Figure 3:Test AUC \(%\) on CAD and Infarction under varying tabular missing ratesγ\\gamma\.
#### Performance under Increasing Missingness\.

To evaluate robustness across missing rates, we compare GAUGE and DyMocon CAD and Infarction\. As shown in Fig\.[3](https://arxiv.org/html/2608.05608#Sx4.F3), GAUGE outperforms DyMocin 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 \(γ≤0\.5\\gamma\\leq 0\.5\) and observed features dominate, both methods perform comparably\. However, once imputation becomes dominant \(γ≥0\.7\\gamma\\geq 0\.7\), GAUGE generally exhibits a larger advantage over DyMoc\. At extreme missingness \(γ=1\.0\\gamma=1\.0\), GAUGE surpasses DyMocby2\.102\.10and1\.501\.50percentage 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\. However, GAUGE currently relies on the pre\-trained backbones and frozen imputers inherited from existing baselines\. Future work will explore extending this pass\-efficient counterfactual gating paradigm to broader multimodal reasoning and generation architectures\.

## Acknowledgments

The work was supported by the Australian Research Council \(ARC\) under Laureate project FL190100149\. This research has been conducted using the UK Biobank Resource under Project Number 105141\.

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## Appendix ATheoretical 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\.1Intervention 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 sampleii, replacing evidence unitei,je\_\{i,j\}inT~i=\[e\[𝙲𝙻𝚂\];ei,1,…,ei,N\]\\tilde\{T\}\_\{i\}=\[\\,e\_\{\[\\mathtt\{CLS\}\]\};e\_\{i,1\},\\ldots,e\_\{i,N\}\\,\]with a reference representationei,j′e^\{\\prime\}\_\{i,j\}yieldsT~i\(j←ei,j′\)\\tilde\{T\}\_\{i\}^\{\(j\\leftarrow e^\{\\prime\}\_\{i,j\}\)\}, with replacement directiondi,j=ei,j′−ei,jd\_\{i,j\}=e^\{\\prime\}\_\{i,j\}\-e\_\{i,j\}\. Throughout this work, we use the zero referenceei,j′=𝟎e^\{\\prime\}\_\{i,j\}=\\mathbf\{0\}, while the\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]token is never replaced\.

### A\.2Exact Unit\-Level Counterfactual Effect

The saliency objective isℒsal​\(T~i\)=−zi,y^i\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)=\-z\_\{i,\\hat\{y\}\_\{i\}\}, the negative logit of the ungated predicted class\. Herey^i\\hat\{y\}\_\{i\}is determined once from the unreplaced sequenceT~i\\tilde\{T\}\_\{i\}and is held constant under both evidence\-unit replacement and differentiation: we do not differentiate through thearg⁡max\\arg\\max, and if the predicted class changes after replacement,ℒsal​\(⋅\)\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\cdot\)still denotes the negative logit of the original ungated predicted classy^i\\hat\{y\}\_\{i\}\. Withy^i\\hat\{y\}\_\{i\}fixed, the exact counterfactual effect of unitjjis defined as:

CEffi,jexact=ℒsal​\(T~i\(j←ei,j′\)\)−ℒsal​\(T~i\),\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}=\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\\\!\\big\(\\tilde\{T\}\_\{i\}^\{\(j\\leftarrow e^\{\\prime\}\_\{i,j\}\)\}\\big\)\-\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\),and its magnitude\|CEffi,jexact\|\|\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}\|measures the response change induced by replacing unitjjwith its reference representation\. Computing this for all units requires one counterfactual forward pass per unit, i\.e\.,NNadditional forward passes after the ungated forward pass\. GAUGE instead constructs first\-order local surrogates for allNNeffects using a single ungated forward–backward scoring pass\.

### A\.3First\-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 exceptei,je\_\{i,j\}, regard the saliency objective as a function of this unit alone:

ψi,j​\(v\)=ℒsal​\(T~i\(j←v\)\)\.\\psi\_\{i,j\}\(v\)=\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\\\!\\big\(\\tilde\{T\}\_\{i\}^\{\(j\\leftarrow v\)\}\\big\)\.\(11\)Withy^i\\hat\{y\}\_\{i\}fixed,ψi,j\\psi\_\{i,j\}is a scalar function ofvv, withψi,j​\(ei,j\)=ℒsal​\(T~i\)\\psi\_\{i,j\}\(e\_\{i,j\}\)=\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)andψi,j​\(ei,j′\)=ℒsal​\(T~i\(j←ei,j′\)\)\\psi\_\{i,j\}\(e^\{\\prime\}\_\{i,j\}\)=\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}^\{\(j\\leftarrow e^\{\\prime\}\_\{i,j\}\)\}\), so thatCEffi,jexact=ψi,j​\(ei,j′\)−ψi,j​\(ei,j\)\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}=\\psi\_\{i,j\}\(e^\{\\prime\}\_\{i,j\}\)\-\\psi\_\{i,j\}\(e\_\{i,j\}\)\. A first\-order Taylor expansion ofψi,j\\psi\_\{i,j\}atei,je\_\{i,j\}gives

CEffi,jexact≈ai,j:=⟨∇ei,jℒsal​\(T~i\),di,j⟩,\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}\\approx a\_\{i,j\}:=\\big\\langle\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\),\\,d\_\{i,j\}\\big\\rangle,\(12\)which is the first\-order term of Eq\. \([3](https://arxiv.org/html/2608.05608#Sx3.E3)\) in the main text; we callai,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\.4Taylor 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\+α​di,j:α∈\[0,1\]\}\\Omega\_\{i,j\}=\\\{\\,e\_\{i,j\}\+\\alpha\\,d\_\{i,j\}:\\alpha\\in\[0,1\]\\,\\\}be the line segment betweenei,je\_\{i,j\}andei,j′e^\{\\prime\}\_\{i,j\}\. Assume thatψi,j\\psi\_\{i,j\}is twice continuously differentiable on a neighborhood ofΩi,j\\Omega\_\{i,j\}and that the unit\-wise Hessian is uniformly bounded on this segment:

supv∈Ωi,j‖∇2ψi,j​\(v\)‖op≤Hi,j\.\\sup\_\{v\\in\\Omega\_\{i,j\}\}\\big\\\|\\nabla^\{2\}\\psi\_\{i,j\}\(v\)\\big\\\|\_\{\\mathrm\{op\}\}\\leq H\_\{i,j\}\.Then the remainderRi,j:=CEffi,jexact−ai,jR\_\{i,j\}:=\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}\-a\_\{i,j\}satisfies

CEffi,jexact=ai,j\+Ri,j,\|Ri,j\|≤Hi,j2​‖di,j‖22,\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}=a\_\{i,j\}\+R\_\{i,j\},\\qquad\|R\_\{i,j\}\|\\leq\\frac\{H\_\{i,j\}\}\{2\}\\,\\\|d\_\{i,j\}\\\|\_\{2\}^\{2\},\(13\)recovering Eq\. \([3](https://arxiv.org/html/2608.05608#Sx3.E3)\) of the main text with an explicit remainder bound\. Consequently,

\|\|CEffi,jexact\|−\|ai,j\|\|≤Hi,j2​‖di,j‖22,\\Big\|\\,\\big\|\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}\\big\|\-\|a\_\{i,j\}\|\\,\\Big\|\\leq\\frac\{H\_\{i,j\}\}\{2\}\\,\\\|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\.

###### Proof\.

Letψi,j\\psi\_\{i,j\}be defined as in Eq\. \([11](https://arxiv.org/html/2608.05608#A1.E11)\)\. By Taylor’s theorem with Lagrange remainder, there exists a pointξi,j∈Ωi,j\\xi\_\{i,j\}\\in\\Omega\_\{i,j\}such that

ψi,j​\(ei,j′\)\\displaystyle\\psi\_\{i,j\}\(e^\{\\prime\}\_\{i,j\}\)=ψi,j​\(ei,j\)\+⟨∇ψi,j​\(ei,j\),di,j⟩\\displaystyle=\\psi\_\{i,j\}\(e\_\{i,j\}\)\+\\big\\langle\\nabla\\psi\_\{i,j\}\(e\_\{i,j\}\),\\,d\_\{i,j\}\\big\\rangle\(15\)\+12​di,j⊤​∇2ψi,j​\(ξi,j\)​di,j\.\\displaystyle\\quad\+\\frac\{1\}\{2\}\\,d\_\{i,j\}^\{\\top\}\\nabla^\{2\}\\psi\_\{i,j\}\(\\xi\_\{i,j\}\)\\,d\_\{i,j\}\.Since∇ψi,j​\(ei,j\)=∇ei,jℒsal​\(T~i\)\\nabla\\psi\_\{i,j\}\(e\_\{i,j\}\)=\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)andCEffi,jexact=ψi,j​\(ei,j′\)−ψi,j​\(ei,j\)\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}=\\psi\_\{i,j\}\(e^\{\\prime\}\_\{i,j\}\)\-\\psi\_\{i,j\}\(e\_\{i,j\}\), subtractingψi,j​\(ei,j\)\\psi\_\{i,j\}\(e\_\{i,j\}\)from both sides identifies the remainder as

Ri,j=CEffi,jexact−ai,j=12​di,j⊤​∇2ψi,j​\(ξi,j\)​di,j\.R\_\{i,j\}=\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}\-a\_\{i,j\}=\\frac\{1\}\{2\}\\,d\_\{i,j\}^\{\\top\}\\nabla^\{2\}\\psi\_\{i,j\}\(\\xi\_\{i,j\}\)\\,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\}\|\\leq\\frac\{1\}\{2\}\\big\\\|\\nabla^\{2\}\\psi\_\{i,j\}\(\\xi\_\{i,j\}\)\\big\\\|\_\{\\mathrm\{op\}\}\\\|d\_\{i,j\}\\\|\_\{2\}^\{2\}\\leq\\frac\{H\_\{i,j\}\}\{2\}\\,\\\|d\_\{i,j\}\\\|\_\{2\}^\{2\}\.\(17\)Finally, since\|CEffi,jexact\|=\|ai,j\+Ri,j\|\|\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}\|=\|a\_\{i,j\}\+R\_\{i,j\}\|, the reverse triangle inequality\|\|ai,j\+Ri,j\|−\|ai,j\|\|≤\|Ri,j\|\\big\|\\,\|a\_\{i,j\}\+R\_\{i,j\}\|\-\|a\_\{i,j\}\|\\,\\big\|\\leq\|R\_\{i,j\}\|proves Eq\. \([14](https://arxiv.org/html/2608.05608#A1.E14)\)\. ∎

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\.5Non\-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,c​di,j,c\.a\_\{i,j\}=\\sum\_\{c=1\}^\{C\}\\frac\{\\partial\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\}\{\\partial 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\\displaystyle\\mathrm\{TE\}\_\{i,j\}=sg⁡\(∑c=1C\|∂ℒsal​\(T~i\)∂ei,j,c​di,j,c\|\)\\displaystyle=\\operatorname\{sg\}\\\!\\left\(\\sum\_\{c=1\}^\{C\}\\left\|\\frac\{\\partial\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\}\{\\partial e\_\{i,j,c\}\}\\,d\_\{i,j,c\}\\right\|\\right\)\(18\)=sg⁡\(‖∇ei,jℒsal​\(T~i\)⊙di,j‖1\)\.\\displaystyle=\\operatorname\{sg\}\\\!\\left\(\\big\\\|\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\\odot d\_\{i,j\}\\big\\\|\_\{1\}\\right\)\.The stop\-gradient operator does not change the numerical value ofTEi,j\\mathrm\{TE\}\_\{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\.

Becausesg⁡\(⋅\)\\operatorname\{sg\}\(\\cdot\)does not alter the forward value, the triangle inequality yields

\|ai,j\|≤TEi,j\.\|a\_\{i,j\}\|\\leq\\mathrm\{TE\}\_\{i,j\}\.Thus,TEi,j\\mathrm\{TE\}\_\{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](https://arxiv.org/html/2608.05608#Thmtheorem1)gives

\|CEffi,jexact\|≤TEi,j\+Hi,j2​‖di,j‖22,\\big\|\\mathrm\{CEff\}\_\{i,j\}^\{\\mathrm\{exact\}\}\\big\|\\leq\\mathrm\{TE\}\_\{i,j\}\+\\frac\{H\_\{i,j\}\}\{2\}\\,\\\|d\_\{i,j\}\\\|\_\{2\}^\{2\},\(19\)which is Eq\. \([5](https://arxiv.org/html/2608.05608#Sx3.E5)\) of the main text\.

The Hessian term appears only in the theoretical error bound; GAUGE neither estimatesHi,jH\_\{i,j\}nor uses it during training\.

### A\.6Choice 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​\(T~i\)=−log⁡pi​\(y^i\)\.\\mathcal\{L\}\_\{\\mathrm\{NLL\}\}\(\\tilde\{T\}\_\{i\}\)=\-\\log p\_\{i\}\(\\hat\{y\}\_\{i\}\)\.\(20\)Its gradient with respect to the logits is

∇ziℒNLL=pi−𝟏y^i\.\\nabla\_\{z\_\{i\}\}\\mathcal\{L\}\_\{\\mathrm\{NLL\}\}=p\_\{i\}\-\\mathbf\{1\}\_\{\\hat\{y\}\_\{i\}\}\.\(21\)Here,𝟏y^i\\mathbf\{1\}\_\{\\hat\{y\}\_\{i\}\}is the one\-hot vector of the ungated predicted class\. By the chain rule, the evidence\-unit gradient is

∇ei,jℒNLL=\(∂zi∂ei,j\)⊤​\(pi−𝟏y^i\)\.\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{NLL\}\}=\\left\(\\frac\{\\partial z\_\{i\}\}\{\\partial e\_\{i,j\}\}\\right\)^\{\\\!\\top\}\\left\(p\_\{i\}\-\\mathbf\{1\}\_\{\\hat\{y\}\_\{i\}\}\\right\)\.\(22\)For a confident model,pi≈𝟏y^ip\_\{i\}\\approx\\mathbf\{1\}\_\{\\hat\{y\}\_\{i\}\}\. When the logits–evidence\-unit Jacobian is bounded, the factorpi−𝟏y^ip\_\{i\}\-\\mathbf\{1\}\_\{\\hat\{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 ofℒsal=−zi,y^i\\mathcal\{L\}\_\{\\mathrm\{sal\}\}=\-z\_\{i,\\hat\{y\}\_\{i\}\}with respect to the logits is the constant−𝟏y^i\-\\mathbf\{1\}\_\{\\hat\{y\}\_\{i\}\}, bypassing the softmax and avoiding saturation\. This is consistent with attribution methods that operate at the pre\-softmax level\(Shrikumaret al\.[2017](https://arxiv.org/html/2608.05608#bib.bib20); Sundararajanet al\.[2017](https://arxiv.org/html/2608.05608#bib.bib24)\)\. Accordingly, GAUGE combines a signed Taylor term for pass\-efficient local approximation,ℓ1\\ell\_\{1\}aggregation to prevent channel\-wise cancellation, and the predicted\-class logit to avoid softmax saturation\.

## Appendix BExperimental Configuration

### B\.1Datasets and Input Configurations

Datasets\.Table[4](https://arxiv.org/html/2608.05608#A2.T4)summarizes the six multimodal classification benchmarks derived from five datasets\. PolyMNIST, MST \(MNIST–SVHN–Text\), and bimodal CelebA\(Sutteret al\.[2021](https://arxiv.org/html/2608.05608#bib.bib13)\)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\(Huanget al\.[2022](https://arxiv.org/html/2608.05608#bib.bib30)\)combines vehicle images with1717tabular features, including44categorical and1313continuous features\. The coronary artery disease \(CAD\) and myocardial infarction \(Infarction\) benchmarks\(Sudlowet al\.[2015](https://arxiv.org/html/2608.05608#bib.bib31)\)each combine cardiac MR images with7575disease\-related tabular features, including2626categorical and4949continuous features\. Because both diseases have low prevalence, class\-balanced subsets are used for training\. The dataset splits follow DyMo\(Duet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib18)\)\.

Table 4:Dataset statistics for the six benchmarks, using the splits ofDuet al\.\([2026](https://arxiv.org/html/2608.05608#bib.bib18)\)\. CAD and Infarction are constructed from UK Biobank\.Missingness Protocols\.Following DyMo\(Duet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib18)\), we use the same mask\-generation procedure and benchmark\-specific missingness protocols\. For PolyMNIST, each sample randomly missesη×100%\\eta\\times 100\\%of its five image modalities, and Table[1](https://arxiv.org/html/2608.05608#Sx3.T1)reportsη∈\{0,0\.6,0\.8\}\\eta\\in\\\{0,0\.6,0\.8\\\}\. For MST, we report the complete\-input setting \(∅\\emptyset\) and the missing\-modality subsets\{S,T\}\\\{\\mathrm\{S\},\\mathrm\{T\}\\\}and\{M,T\}\\\{\\mathrm\{M\},\\mathrm\{T\}\\\}\. For CelebA, we report∅\\emptyset,\{I\}\\\{\\mathrm\{I\}\\\}, and\{T\}\\\{\\mathrm\{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 missesγ×100%\\gamma\\times 100\\%of its tabular features\. Table[1](https://arxiv.org/html/2608.05608#Sx3.T1)reportsγ∈\{0,0\.9,1\}\\gamma\\in\\\{0,0\.9,1\\\}for all three benchmarks\. For CAD and Infarction, Fig\.[3](https://arxiv.org/html/2608.05608#Sx4.F3)additionally includesγ∈\{0\.1,0\.3,0\.5,0\.7\}\\gamma\\in\\\{0\.1,0\.3,0\.5,0\.7\\\}\. Here,η=0\\eta=0orγ=0\\gamma=0corresponds to complete input, andγ=1\\gamma=1indicates that the entire tabular modality is missing\. To connect the feature\-level protocol above with the formulation in the main text, letttdenote the tabular modality\. For0<γ<10<\\gamma<1,ttremains partially observed and therefore belongs to𝒪i\\mathcal\{O\}\_\{i\}\. Its missing entries are reconstructed by the frozen TIP imputer used in the DyMo pipeline:

x~i\(t\)=Γt​\(\{xi\(m\)\}m∈𝒪i∖\{t\},xi,obs\(t\)\),\\tilde\{x\}\_\{i\}^\{\(t\)\}=\\Gamma\_\{t\}\\\!\\big\(\\\{x\_\{i\}^\{\(m\)\}\\\}\_\{m\\in\\mathcal\{O\}\_\{i\}\\setminus\\\{t\\\}\},\\,x\_\{i,\\mathrm\{obs\}\}^\{\(t\)\}\\big\),wherexi,obs\(t\)x\_\{i,\\mathrm\{obs\}\}^\{\(t\)\}denotes the observed tabular entries\. The completed modalityx¯i\(t\)\\bar\{x\}\_\{i\}^\{\(t\)\}retains the observed entries and fills the missing positions with the corresponding imputed values\. Whenγ=1\\gamma=1, the tabular modality belongs to𝒰i\\mathcal\{U\}\_\{i\}, reducing to whole\-modality imputation\. In all cases, GAUGE operates only on the completed inputX¯i\\bar\{X\}\_\{i\}produced by the frozen DyMo/TIP pipeline\.

### B\.2Baselines

We compare GAUGE with nine baselines and summarize their sources and configurations\. The recovery\-based baselines are MultiAE\(Ngiamet al\.[2011](https://arxiv.org/html/2608.05608#bib.bib32)\), MoPoE\(Sutteret al\.[2021](https://arxiv.org/html/2608.05608#bib.bib13)\), M3Care\(Zhanget al\.[2022](https://arxiv.org/html/2608.05608#bib.bib14)\), and OnlineMAE\(Wooet al\.[2023](https://arxiv.org/html/2608.05608#bib.bib33)\); the recovery\-free baselines are ModDrop\(Neverovaet al\.[2016](https://arxiv.org/html/2608.05608#bib.bib8)\), MTL\(Maet al\.[2022](https://arxiv.org/html/2608.05608#bib.bib10)\), MAP\(Leeet al\.[2023](https://arxiv.org/html/2608.05608#bib.bib11)\), and MUSE\(Wuet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib12)\)\. We also include the dynamic\-recovery baseline DyMo\(Duet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib18)\)with the cosine\-distance reward, denoted as DyMoc\. GAUGE and our reproduced DyMocshare the same backbone and frozen imputation pipeline, making DyMocthe closest architecture\-controlled baseline\. Except for our reproduced DyMoc, the results in Table[1](https://arxiv.org/html/2608.05608#Sx3.T1)are taken from Tables 1, S3, and S4 ofDuet al\.\([2026](https://arxiv.org/html/2608.05608#bib.bib18)\)under the same missingness protocols;†\\daggermarks 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 \(e\.g\., the drop probabilityp=0\.5p\{=\}0\.5of ModDrop and MUSE, MAP prompt lengths, and the encoder initialization of OnlineMAE\) are likewise unchanged from that work\. Table[1](https://arxiv.org/html/2608.05608#Sx3.T1)focuses on methods explicitly designed or adapted for incomplete multimodal learning\. Dynamic fusion methods such as QMF, DynMM, and PDF, which were evaluated byDuet al\.\([2026](https://arxiv.org/html/2608.05608#bib.bib18)\)on observed\-plus\-recovered inputs, are not repeated in this table\.

### B\.3Implementation Details

Model Initialization\.We adopt the dataset\-specific backbones and frozen imputers from DyMo\(Duet al\.[2026](https://arxiv.org/html/2608.05608#bib.bib18)\)\. A MoPoE multimodal VAE is used to reconstruct missing modalities on PolyMNIST, MST, and CelebA, while the TIP imputer\(Duet al\.[2024](https://arxiv.org/html/2608.05608#bib.bib17)\)reconstructs missing tabular features on DVM, CAD, and Infarction\. The modality\-specific encoders, multimodal Transformer, and classifier are initialized from our reproduced DyMoccheckpoints using the architectures and hyperparameters reported in Table S2 ofDuet al\.\([2026](https://arxiv.org/html/2608.05608#bib.bib18)\)\. During fine\-tuning, the encoders, Transformer, classifier, and gate parameters\(τ,ρ\)\(\\tau,\\rho\)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 emits44units, givingN=20N\{=\}20across its five views\. Each modality in MST and CelebA emits one global embedding, givingN=3N\{=\}3andN=2N\{=\}2, respectively\. DVM emits1616image units and1717tabular units \(N=33N\{=\}33\), while CAD and Infarction each emit1616image units and7575tabular units \(N=91N\{=\}91\)\. These layouts determine the branch used by theNN\-adaptive normalization rule \(Appendix[C\.2](https://arxiv.org/html/2608.05608#A3.SS2)\)\. Because each modality in MST and CelebA emits a single embedding, GAUGE reduces to modality\-level gating on these two benchmarks\.

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](https://arxiv.org/html/2608.05608#A2.T5)\. A new missingness mask is sampled for each minibatch following the protocol in Appendix[B\.1](https://arxiv.org/html/2608.05608#A2.SS1)\. We use the zero referenceei,j′=𝟎e^\{\\prime\}\_\{i,j\}=\\mathbf\{0\}\. The gate thresholdτ\\tauis initialized to0for PolyMNIST, DVM, CAD, and Infarction, which usezz\-score normalization, and to0\.50\.5for MST and CelebA, which use raw scores\. For all benchmarks,ρ\\rhois initialized to0, such thatexp⁡\(ρ\)=1\\exp\(\\rho\)=1, and we setϵ=10−6\\epsilon=10^\{\-6\}andϵg=10−9\\epsilon\_\{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 CSupplementary Experiments

### C\.1Ablation on Evidence\-Unit Scoring

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\.Boldmarks 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 sameNN\-adaptive normalization, continuous gate, attention\-bias injection, initialization, and fine\-tuning settings, differing only in the raw evidence\-unit score\. For evidence unitei,je\_\{i,j\},*random score*samples an independent value from𝒩​\(0,1\)\\mathcal\{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\\\|\\nabla\_\{e\_\{i,j\}\}\\mathcal\{L\}\_\{\\mathrm\{sal\}\}\(\\tilde\{T\}\_\{i\}\)\\\|\_\{1\}\.*Attention score*uses the final\-layer\[𝙲𝙻𝚂\]\[\\mathtt\{CLS\}\]\-to\-unit attention from the ungated forward pass, averaged across attention heads\. The Taylor evidence score is the default score defined in Eq\. \([4](https://arxiv.org/html/2608.05608#Sx3.E4)\), using the zero referenceei,j′=𝟎e^\{\\prime\}\_\{i,j\}=\\mathbf\{0\}\. As shown in Table[6](https://arxiv.org/html/2608.05608#A3.T6), the Taylor evidence score consistently outperforms all four alternatives across all six benchmarks, achieving the highest average performance of85\.6285\.62, which exceeds the strongest competing scorer, gradient norm, by0\.900\.90points\. 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\.2Ablation on Score Normalization

GAUGE adopts anNN\-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, whenN=2N\{=\}2, any two distinct scores are mapped to−1/2\-1/\\sqrt\{2\}and1/21/\\sqrt\{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 defaultNN\-adaptive rule in Eq\. \([6](https://arxiv.org/html/2608.05608#Sx3.E6)\) with two fixed strategies:*Alwayszz\-score*and*Always raw*\. The comparison includes two small\-NNbenchmarks, MST \(N=3N\{=\}3\) and CelebA \(N=2N\{=\}2\), and two large\-NNbenchmarks, CAD and Infarction \(N=91N\{=\}91\)\. All other model components and training settings are kept unchanged\. As shown in Table[7](https://arxiv.org/html/2608.05608#A3.T7), applyingzz\-score normalization uniformly degrades performance in the small\-NNregime, reducing accuracy by0\.230\.23points on MST and0\.580\.58points 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 by1\.941\.94points on CAD and1\.671\.67points on Infarction, showing that normalization is important for calibrating the scores of large evidence\-unit sets\. These results support theNN\-adaptive rule: raw scores are preferable for smallNN, whereas per\-sample standardization is beneficial for largeNN\.

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\.Boldmarks the best result\.
### C\.3Ablation 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 atγ=1\\gamma=1\. Input resolutions of96×9696\\times 96,128×128128\\times 128,160×160160\\times 160,192×192192\\times 192, and256×256256\\times 256pixels yield3×33\\times 3,4×44\\times 4,5×55\\times 5,6×66\\times 6, and8×88\\times 8evidence\-unit grids, respectively\. The native128×128128\\times 128input corresponds to a4×44\\times 4grid\. 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](https://arxiv.org/html/2608.05608#A3.F4)shows that finer grids generally improve test AUC on both benchmarks\. CAD AUC increases from69\.1069\.10at3×33\\times 3to70\.8770\.87at8×88\\times 8, while Infarction reaches its highest AUC of73\.9073\.90at6×66\\times 6, with no further gain at8×88\\times 8\. Because bilinear upsampling adds no new source\-pixel information, the gains beyond the native4×44\\times 4grid suggest that finer evidence\-unit granularity can improve localized scoring and gating\. On Infarction, performance plateaus beyond6×66\\times 6, indicating limited benefit from further grid refinement\.

![Refer to caption](https://arxiv.org/html/2608.05608v1/x4.png)Figure 4:Effect of image resolution and the resulting evidence\-unit granularity on CAD and Infarction atγ=1\\gamma=1\.

## Appendix DScoring Cost Comparison

Table[3](https://arxiv.org/html/2608.05608#Sx4.T3)reports the runtime of the scoring stage, measured from the encoded sequenceT~i\\tilde\{T\}\_\{i\}to theNNevidence\-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 byNNsequential intervention forward passes, with batch size11\. In contrast, GAUGE computes allNNTaylor evidence scores through a single ungated forward–backward pass\. For each configuration, runtime is averaged over500500test samples after2020warm\-up iterations\. We calltorch\.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\.