Structured Latent Modeling for Supervised Multimodal Information Decomposition

1 INTRODUCTION

• The model addresses the lack of explicit latent disentanglement in prior supervised approaches and the limits of assuming all task-relevant information is shared.A structured source–target model must accommodate nonlinear features while distinguishing source–target relationships from other cross-source associations.Source: §1 · pp. 1–2
• S2MLVM separates shared predictive, modality-specific predictive, and shared nuisance factors in a supervised latent variable model.Marginalizing the factors yields a low-rank-plus-diagonal covariance that accommodates shared source variation not directly predictive of the target.Source: §1 · p. 2
• S2MLVM-MASK and S2MLVM-CONTRAST pair neural encoders and dimension-preserving flows with task, density, and auxiliary representation objectives.The variants use within-source masked reconstruction or contrastive learning, respectively, without early cross-modal fusion.Source: §1 · p. 2
• The framework connects its latent structure to Partial Information Decomposition and analyzes mechanisms that generate synergistic information.The analysis includes cooperative nuisance suppression and V-structure exploitation.Source: §1 · p. 2
• Across diverse real-world datasets, both S2MLVM variants mitigate unimodal bias and outperform strong representation-learning baselines in two- and three-modality settings.In controlled settings, the framework also accurately recovers task-relevant subspaces.Source: §1 · p. 2

2 METHOD

• Each modality follows a source-specific encoder and invertible flow, and the resulting features are jointly modeled with the target using S2MLVM.The model separates shared predictive, modality-specific predictive, and target-irrelevant variation; the formulation extends to arbitrary numbers of modalities.Source: §2 · p. 2

2.1 STRUCTURED LATENT VARIABLE MODELING

• S2MLVM models the two source representations and target with shared predictive, private predictive, and shared nuisance latent factors.The target does not depend on the shared nuisance factor, which absorbs task-irrelevant cross-modal correlations; without a target, shared predictive and nuisance variation cannot be distinguished.Source: §2.1 · p. 3

2.2 FLOW-BASED DISTRIBUTION MODELING

• Source-wise flows transform nonlinear encoder features into a space modeled jointly with the target under the structured Gaussian distribution.The joint negative log-likelihood combines the Gaussian structured-dependence term with flow Jacobian terms accounting for volume changes.Source: §2.2 · pp. 3–4

2.3 AUXILIARY REPRESENTATION LEARNING

• An auxiliary representation objective prevents encoder degeneracy after dimension reduction by retaining information in the encoder representations.Each variant uses either masked reconstruction or contrastive learning, not both.Source: §2.3 · p. 4

2.4 TOTAL REPRESENTATION LEARNING OBJECTIVE

• Posterior means of the predictive factors are concatenated and passed to an MLP trained with the task loss.The total objective weights this discriminative loss with the Flow–S2MLVM maximum-likelihood loss and auxiliary representation loss, and parameters are jointly optimized end to end.Source: §2.4 · p. 4

3 CONNECTIONS TO OTHER MULTIMODAL LEARNING FRAMEWORKS

• The optimal PID adversary correlates private noise across modalities to align with the combined task signal and reduce joint predictive information.For the Gaussian model, the adversary manipulates the cross-source covariance while preserving source–target marginals.Source: §3 · p. 5
• A conditional-independence assumption provides a tractable analytic proxy for the intractable PID optimization.The proxy is defined by maximizing conditional entropy.Source: §3 · p. 5
• The conditional-entropy adversary cancels three synergy-related mechanisms: nuisance suppression, noise averaging of shared signal, and a private-signal V-structure.The proxy yields analytic bounds, underestimating synergy and redundancy while overestimating unique information.Source: §3 · p. 5
• S2MLVM separates private predictive factors from shared redundancy, covering highly redundant and highly unique settings without discarding synergistic interactions.Its latent factors do not map one-to-one onto PID information atoms.Source: §3 · p. 5

Unsupervised multimodal representation learning

• Unsupervised methods align paired views or separate shared and private content, while some explicitly encourage complementary interactions through complex augmentations or masking.The paper frames its contribution as extending likelihood-based separation to supervised source–target dependencies.Source: §4 · p. 6

Balancing supervised multimodal optimization

• Supervised balancing methods adjust gradients, reconcile competing objectives, or use game-theoretic and mutual-information formulations.S2MLVM instead seeks balance through its generative objective, without explicit gradient modulation.Source: §4 · p. 6

Target-relative information decomposition

• PID characterizes unique, redundant, and synergistic information, but estimating it on continuous, high-dimensional neural representations presents computational challenges.Existing continuous estimators use variational, neural, convex Gaussian, or latent-Gaussian approaches.Source: §4 · p. 6
• The paper builds on latent-Gaussian analysis rather than proposing a new PID definition.Its shared and modality-specific predictive factors provide covariance structures for analyzing synergistic information.Source: §4 · p. 6

5.1 SUBSPACE RECOVERY ON SYNTHETIC DATA

• S2MLVM consistently recovers the task-relevant subspace across tested dimensions and noise levels, with canonical correlation remaining high but degrading as estimation becomes more challenging.The synthetic experiments vary ambient dimension, observation noise, and latent-block dimension.Source: §5.1 · p. 7, Table 1 · p. 7

5.2 EXPERIMENTS ON TRIFEATURE

• On TriFeatures, supervised methods MCR and S2MLVM outperform unsupervised methods; S2MLVM-CONTRAST's corresponding factors retain most accuracy for R, U1, and U2, but no individual factor suffices for synergy.TriFeatures tests shared, source-specific, and cross-source predictive information.Source: §5.2 · pp. 7–8, Table 2 · p. 8

5.3 REAL-WORLD MULTIMODAL BENCHMARKS

• S2MLVM-MASK leads on V&T EE, MIMIC, MOSI, and MUSTARD, while S2MLVM-CONTRAST leads on UR-FUNNY, CREMA-D, and UCF101; S2MLVM outperforms prior methods across two-modality tasks.The reported best auxiliary objective varies by dataset.Source: Table 3 · p. 8, §5.3 · p. 9
• Removing the invertible flows or separate nuisance component significantly degrades performance, and loss ablations support both the auxiliary representation objective and likelihood.Hyperparameters are selected on validation data; final metrics are evaluated on held-out test data.Source: §5 · p. 6, §5.3 · p. 9, Table 5 · p. 9
• Across UR-FUNNY, V&T contact prediction, and MOSEI, S2MLVM outperforms baselines on three-modality benchmarks.The tasks use visual, textual, and acoustic features, or image, force, and proprioceptive observations.Source: §5.3 · p. 9, Table 6 · p. 9

6 CONCLUSION

• S2MLVM separates shared predictive, modality-specific predictive, and task-irrelevant nuisance factors to address modality competition in multimodal fusion.The conclusion describes its structured likelihood model as integrated with representation learning through dimension-preserving flows.Source: §6 · p. 9
• Complete separation of all PID components remains open, and handling partially missing modalities is identified as future work.The conclusion also reports that the framework scales to more than two modalities and outperforms strong baselines across diverse real-world benchmarks.Source: §6 · pp. 9–10

AI USE STATEMENT

• AI assisted with drafting; the authors state that all AI-assisted content was subsequently reviewed, verified, and edited.Source: AI USE STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

• The main paper and appendix provide model, learning-objective, protocol, preprocessing, architecture, hyperparameter, and experimental-setting details.Source: REPRODUCIBILITY STATEMENT · p. 10

• MOSI evaluates binary sentiment from paired visual, acoustic, and textual observations, while MOSEI includes two- and three-modality evaluation settings.MOSI contains 2,199 opinion segments; MOSEI has approximately 23,000 monologue clips.Source: §A.1 · p. 14
• UR-FUNNY supports humor recognition in two- and three-modality configurations, while MUSTARD evaluates sarcasm recognition on 690 balanced dialogue utterances.UR-FUNNY contains 16,514 examples from 1,866 TED-talk videos.Source: §A.1 · p. 14
• Vision&Touch includes end-effector regression from two modalities and contact classification from image, force, and proprioceptive observations.Its 150 robotic manipulation trajectories contain 1,000 time steps each; results use MSE×10−4 for regression and accuracy for classification.Source: §A.1 · p. 14
• CREMA-D tests six-way emotion recognition, UCF101 tests audio–visual action recognition, and MIMIC tests binary diagnostic-group prediction.CREMA-D uses face video and speech; MIMIC combines 24-hour clinical measurements with static patient descriptors.Source: §A.1 · p. 14

A.2 ARCHITECTURE

• Each source pathway encodes its input independently, sends clean CLS representations through its own flow, and uses S2MLVM predictive posterior means for downstream task prediction.The task MLP has two linear layers with an intervening ReLU, and the source encoders use no cross-modal attention or target inputs.Source: §A.2 · p. 14, Figure 1 · p. 15
• Dataset-specific pathways include temporal Transformers for several language datasets, MLP and GRU encoders for MIMIC, and separate image and force pathways for Vision&Touch.CREMA-D and UCF101 use independent visual and acoustic encoders with the ResNet-18 backbone configuration of MCR.Source: §A.2 · pp. 14–15
• The reproduced baselines train each modality independently or train all modalities together under a single supervised task objective.Source: §A.2 · p. 15

A.3 HYPERPARAMETER TUNING

• The final comparison uses validation-selected hyperparameters, with mask ratio and batch size for MASK and InfoNCE temperature and batch size for CONTRAST.Table 7 summarizes the selected settings for each dataset; Table 8 reports validation sensitivity.Source: §A.3 · p. 15, Table 7 · p. 15

A.4 LOSS ABLATION

• A one-factor-at-a-time validation procedure examines loss-coefficient contributions, with results reported in the loss-ablation tables.The supplied table captions identify hyperparameter sensitivity and loss-coefficient ablations on validation datasets.Source: §A.4 · p. 15, Table 9 · p. 16, Table 8 · p. 16

B CONNECTION TO PARTIAL INFORMATION DECOMPOSITION (PID)

• PID separates each view’s unique information, shared redundancy, and joint-only synergy, providing a framework for analyzing predictive interactions between modalities.The section applies this decomposition to the latent model’s shared nuisance and task-relevant factors.Source: B CONNECTION TO PARTIAL INFORMATION… · pp. 16–17

B.1 THE ADVERSARIAL LATENT FRAMEWORK

• The latent framework assumes mutually orthogonal Gaussian factors: shared predictive, shared nuisance, view-specific predictive, and independent private noise.The nuisance loadings are orthogonal to the combined task-relevant loadings, which are assumed to have full column rank.Source: §B.1 · p. 17
• Adversarial distributions preserve source-target marginals and vary cross-source covariance; in the Gaussian model, this is represented as noise injected between independent private noises.The resulting covariance must remain positive semidefinite to define a valid adversarial distribution.Source: §B.1 · p. 17, B CONNECTION TO PARTIAL INFORMATION… · p. 17

B.2 THE CONDITIONAL INDEPENDENCE ADVERSARY (q∗CE)

• Maximizing conditional entropy defines the least-informative coupling and seeks conditional independence between views given Y.For jointly Gaussian variables, zero conditional cross-covariance implies conditional independence.Source: §B.2 · p. 18
• The chosen conditional cross-covariance makes the full joint covariance positive semidefinite, so q∗CE is feasible in the constraint space.The argument uses the Schur complement and the fixed positive target variance.Source: §B.2 · p. 18
• For shared nuisance zc, q∗CE injects negative cross-covariance to decouple the views and prevent cooperative subtraction of the nuisance.In the example U1 = Y + zc and U2 = zc, joint regression can subtract U2 to recover Y.Source: §B.2 · p. 19
• For shared task signal z12, q∗CE cancels cross-view signal correlation, removing the redundancy that would let joint averaging suppress independent noise.The resulting baseline eliminates the averaging mechanism for cleaner prediction.Source: §B.2 · p. 19
• For independent task factors that jointly cause Y, q∗CE cancels the induced explaining-away correlation and removes V-structure exploitation.The section identifies suppression, averaging, and V-structure exploitation as three sources of synergy eliminated by the baseline.Source: §B.2 · pp. 18–19

B.3 THE MINIMUM MUTUAL INFORMATION ADVERSARY (q∗MI)

• Because source-target marginals are fixed, minimizing mutual information is equivalent to maximizing residual generalized variance through the cross-source covariance.The determinant identity connects residual target variance to the joint covariance determinant.Source: §B.3 · p. 19
• Unlike q∗CE, q∗MI sacrifices conditional independence, injecting task-subspace correlation that hinders regression while retaining conditional dependence between views.The q∗CE objective instead decouples task variables to achieve zero conditional mutual information.Source: §B.3 · pp. 20–21
• In the scalar model, the optimal adversarial correlation is the unique root satisfying the positive-semidefinite covariance constraint, and it matches the signal-correlation sign.The stated candidate roots are c = λ1v2/λ2 and c = λ2v1/λ1.Source: §B.3 · p. 20
• Signal-aligned injected noise makes any linear combination that boosts the signal also amplify adversarial noise, preventing synergistic noise cancellation.For positively aligned signal loadings, the example contrasts this with averaging views to suppress independent noise.Source: §B.3 · pp. 20–21

B.4 SHARED PROPERTIES AND ANALYTIC BOUNDS

• Both adversaries preserve each view’s marginal predictive relationship with Y, so minimizing joint information reduces to minimizing conditional mutual information.The q∗CE adversary is a sub-optimal minimizer of joint information, making its conditional mutual information an upper bound on true unique information.Source: §B.4 · p. 21
• The q∗CE proxy underestimates synergy and redundancy while overestimating unique information relative to the corresponding q∗MI quantities.These are analytic bounds obtained by substituting q∗CE into the PID equations.Source: §B.4 · p. 21
• Conditional mutual information is not identical to unique information: it equals unique information plus synergy, which the adversaries remove.Source: §B.4 · p. 21

B.5 BRIDGE TO MULTI-VIEW LEARNING

• Conditional independence does not imply redundancy, and a large SCE indicates synergy from cooperative noise cancellation that can make conditional-independence algorithms suboptimal to joint models.SCE approximates true synergy and quantifies violation of the conditional-independence assumption.Source: §B.5 · pp. 21–22