Multi-modal Knowledge Decomposition (MKD)

• Multi-modal Knowledge Decomposition (MKD) is a design paradigm that explicitly factors complementary, modality-specific, and modality-general information instead of relying on monolithic fusion.
• MKD is implemented in diverse setups—including offline teacher–student distillation, online co-distillation, relation-aware expert decomposition, and information-theoretic interaction modeling.
• Its practical applications include enhancing uni-modal inference in clinical settings, improving biomarker prediction, and structuring multi-modal representations for robust performance.

Multi-modal Knowledge Decomposition (MKD) denotes a family of multimodal learning strategies that explicitly separate, transfer, or align information originating from different modalities instead of treating multimodal fusion as a single undifferentiated operation. In its earliest explicit formulation, KD-Net defines MKD as distilling the complementary information present in multiple MRI modalities from a trained multi-modal teacher into a mono-modal student through output-space soft targets, bottleneck-level latent alignment, and hard-label segmentation supervision (Hu et al., 2021). Later work extends the term to online distillation for paired genomics–pathology data, where modality-specific and modality-general features are jointly learned and regularized by covariance alignment, orthogonality, similarity preservation, and collaborative KL-based teaching (Zhang et al., 24 Aug 2025). Closely related decomposition-based paradigms in multi-modal knowledge graphs and information-theoretic interaction learning broaden the same organizing intuition: multimodal systems can benefit when shared, specialized, and synergistic factors are modeled explicitly rather than left implicit (Zhang et al., 2024, Yang et al., 10 Jun 2026).

1. Conceptual scope and major formulations

Across the cited literature, MKD is not a single fixed algorithm but a design pattern. The decomposition target varies by domain: complementary information can be distilled from a multi-modal teacher into a mono-modal student; modality-specific and modality-general factors can be separated during online training; relation-conditioned expert views can be disentangled within each modality; or predictive information can be factored into redundant, unique, and synergistic atoms. What remains constant is the attempt to make cross-modal structure explicit.

Work	Decomposition target	Principal mechanism
KD-Net (Hu et al., 2021)	Complementary MRI information	Soft targets, latent KL, segmentation loss
Online MKD (Zhang et al., 24 Aug 2025)	Modality-specific and modality-general features	CORAL, orthogonality, SKD, CLOD
MoMoK (Zhang et al., 2024)	Relation-conditioned modality perspectives	Mixture-of-experts, gating, CLUB
DMIL (Yang et al., 10 Jun 2026)	Redundant, unique, synergistic interactions	Variational decomposition, sample-specific gating

This heterogeneity matters methodologically. In segmentation and biomarker prediction, decomposition is used primarily to recover uni-modal inference quality when full multimodal acquisition is impractical. In knowledge graphs, decomposition is relation-aware and expert-centric. In information-theoretic interaction learning, decomposition becomes a formal account of what types of predictive information multimodal models should capture. A plausible implication is that MKD has evolved from a task-specific distillation procedure into a broader multimodal inductive bias.

2. KD-Net and the original explicit MKD formulation

KD-Net instantiates MKD in a generalized distillation setting for medical image segmentation, where a multi-modal teacher network $f_T$ is trained on all available MRI sequences and a mono-modal student network $f_S$ is trained on a single modality such as T1-CE (Hu et al., 2021). The teacher is written as

$f_T(X_i)=D_t(E_t(X_i)),$

where $E_t$ and $D_t$ are the encoder and decoder of a 3D U-Net, and the student predicts

$$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$

The student objective combines three terms: $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$ Here $L_{KD}$ is the temperature-scaled KL distillation loss between teacher and student soft predictions, $L_{CE}$ is the sum of a Dice-loss term and binary cross-entropy against the ground-truth segmentation, and $L_{KL}$ aligns normalized teacher and student bottleneck activations through $f_S$0.

The architectural and optimization choices are specified. Both teacher and student use a 3D U-Net as in nnUNet with five encoding blocks and five decoding blocks with skip-connections; each block has two $f_S$1 convolutions, ReLU, and batch-norm. Input volumes are cropped to $f_S$2 voxels with on-the-fly random flips, and only non-zero voxels are z-score normalized per modality. The teacher receives four channels—T1, T1-CE, T2, and FLAIR—whereas the student receives only T1-CE. Optimization uses Adam with initial learning rate $f_S$3, reduced by factor $f_S$4 if validation loss plateaus for 50 epochs, for up to 500 epochs. Grid search selects $f_S$5, $f_S$6, and $f_S$7.

Training proceeds in two stages. First, the teacher is trained on full multi-modal data with $f_S$8 until convergence, after which its weights are frozen. Second, the student is trained on single-modality inputs with the combined loss, while gradients flow only into $f_S$9. This separation makes the decomposition asymmetric: the teacher is the source of multi-modal semantics, and the student is the recipient that must internalize them without ever observing the missing modalities at inference.

On BraTS 2018, the setup uses 285 training cases, four MRI modalities, three tumor sub-regions—enhancing tumor (ET), tumor core (TC), and whole tumor (WT)—and three-fold cross-validation. The baseline mono-modal T1-CE network achieves $f_T(X_i)=D_t(E_t(X_i)),$0 Dice on ET, $f_T(X_i)=D_t(E_t(X_i)),$1 on TC, and $f_T(X_i)=D_t(E_t(X_i)),$2 on WT. The four-modality teacher achieves $f_T(X_i)=D_t(E_t(X_i)),$3, $f_T(X_i)=D_t(E_t(X_i)),$4, and $f_T(X_i)=D_t(E_t(X_i)),$5. HeMIS with mono T1-CE reports 60.8, 58.5, and 58.5; U-HVED with mono T1-CE reports 65.5, 66.7, and 62.4. KD-Net with mono T1-CE reports $f_T(X_i)=D_t(E_t(X_i)),$6 on ET, $f_T(X_i)=D_t(E_t(X_i)),$7 on TC, and $f_T(X_i)=D_t(E_t(X_i)),$8 on WT, corresponding to $f_T(X_i)=D_t(E_t(X_i)),$9 on ET, $E_t$0 on TC, and approximately equal WT relative to the mono-modal baseline. The ablation study shows that removing either the soft-target distillation term $E_t$1 or the latent-alignment term $E_t$2 degrades performance, confirming that the two transfer channels are complementary.

Within this formulation, MKD has a precise meaning: output-space relations are transferred through soft targets, semantic latent features are transferred through bottleneck alignment, and low-level appearance adaptation remains student-specific. The practical significance follows directly from the medical setting described in the paper: most of the time, only one modality is acquired, so preserving high-level multi-modal semantics in a single-sequence model addresses a clinical acquisition constraint rather than merely improving benchmark accuracy.

3. Online MKD for biomarker prediction in breast cancer histopathology

The 2025 histopathology framework generalizes MKD from offline teacher-to-student transfer to online, jointly optimized distillation over three peer networks (Zhang et al., 24 Aug 2025). Teacher $E_t$3 processes genomic data $E_t$4 and outputs a genomics-specific feature $E_t$5 and logits $E_t$6. Teacher $E_t$7 fuses genomics and pathology via a Kronecker-product aggregator to produce a modality-general feature $E_t$8 and logits $E_t$9. Student $D_t$0 processes pathology embeddings $D_t$1 and yields a pathology-specific feature $D_t$2 and logits $D_t$3. The stated role allocation is explicit: $D_t$4 captures genomics-unique information, $D_t$5 captures features shared across genomics and pathology, and $D_t$6 learns to mimic both the unique pathology cues and the modality-general cues so that it can operate on pathology slides alone at test time.

The pathology branch uses Attention-based MIL: $D_t$7 with attention weights

$D_t$8

An analogous pipeline yields $D_t$9, and $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$0 fuses $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$1 and $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$2 through a Kronecker product and ABMIL to obtain $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$3.

The MKD loss itself is defined by two alignment terms. The CORAL term aligns covariance structure across the three decomposed features: $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$4 and the orthogonality term encourages the features to focus on distinct aspects: $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$5 The combined MKD objective is

$$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$6

with $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$7 set to $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$8 in experiments.

Two further mechanisms extend the decomposition. Similarity-preserving Knowledge Distillation (SKD) aligns the pairwise similarity structure of student and teacher features within each mini-batch: $$\hat Y_i^S=f_S(\hat X_i),\qquad \hat X_i=X_i^k\in\mathbb R^{1\times H\times W\times D}.$$9 where, for example,

$$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$0

Collaborative Learning for Online Distillation (CLOD) replaces a fixed teacher hierarchy with symmetric mutual teaching: $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$1 The full loss is

$$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$2

The experimental protocol uses TCGA-BRCA with paired H&E-slide embeddings and RNA-Seq profiles under 5-fold cross-validation, and QHSU with H&E slides only as an external pathology-only test. Uni-modal pathology-only performance on TCGA-BRCA averages $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$3 AUC, $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$4 ACC, and $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$5 F1 for ER; $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$6 AUC, $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$7 ACC, and $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$8 F1 for PR; and $$L(\theta_S)=\lambda L_{KD}+(1-\lambda)L_{CE}+\alpha L_{KL}.$$9 AUC, $L_{KD}$0 ACC, and $L_{KD}$1 F1 for HER2. These results exceed all compared weakly-supervised MIL baselines by approximately $L_{KD}$2 AUC on ER and PR and are reported as new state-of-the-art in uni-modal inference.

Relative to KD-Net, this formulation changes the ontology of decomposition. The student no longer absorbs knowledge from a single frozen multi-modal expert. Instead, modality-specific and modality-general representations are co-evolved, structural relations between samples are preserved through SKD, and prediction agreement is enforced online by CLOD. This makes MKD a joint training principle rather than only a transfer procedure.

4. Expert-based decomposition in multi-modal knowledge graphs

MoMoK extends decomposition into the representation-learning regime of multi-modal knowledge graph completion and explicitly describes its mechanism as performing multi-modal knowledge decomposition (Zhang et al., 2024). Its architecture contains three interacting sub-modules. Relation-guided Modality Knowledge Experts (ReMoKE) decompose each modality $L_{KD}$3 into $L_{KD}$4 expert views: $L_{KD}$5 where $L_{KD}$6 may be a pre-trained BERT feature for text, a VGG feature for image, or a learned embedding for structure. A gating network conditioned on relation $L_{KD}$7 computes noisy, temperature-controlled attention weights: $L_{KD}$8 which yield the relation-aware modality embedding

$L_{KD}$9

Multi-modal Joint Decision (MuJoD) then recomposes across modalities. Each $L_{CE}$0 is projected to $L_{CE}$1, modality attention weights $L_{CE}$2 are computed through a shared soft-attention mechanism, and a joint embedding is formed as

$L_{CE}$3

Every modality, including the joint modality $L_{CE}$4, receives its own Tucker-style score function $L_{CE}$5, and inference sums these scores across $L_{CE}$6.

To prevent expert collapse, Expert Information Disentanglement (ExID) minimizes a CLUB-based upper bound on mutual information between pairs of expert embeddings within the same modality: $L_{CE}$7 with $L_{CE}$8 tuned, for example, to $L_{CE}$9. The CLUB model $L_{KL}$0 is updated alternately under $L_{KL}$1 to tighten the bound.

Empirically, MoMoK sets new state-of-the-art on four public MMKG benchmarks—MKG-W, MKG-Y, DB15K, and KVC16K. On DB15K, the reported example is MRR $L_{KL}$2 versus a previous best of approximately $L_{KL}$3, and Hit@1 $L_{KL}$4 versus $L_{KL}$5. Under synthetic Gaussian corruption of modality inputs, the model degrades more gracefully than baselines. Ablations show that removing the relation-aware temperature $L_{KL}$6 or the MoE noise $L_{KL}$7 hurts by approximately $L_{KL}$8–$L_{KL}$9 MRR, dropping joint fusion or joint training costs approximately $f_S$00–$f_S$01 MRR, and turning off ExID costs approximately $f_S$02 MRR. The best setting uses $f_S$03 experts per modality.

This version of decomposition differs from the distillation-centric meaning of MKD in KD-Net and the online teacher–student meaning in histopathology. Here the decomposition unit is not the whole modality but a set of relation-conditioned perspectives inside each modality, followed by learned recomposition both within and across modalities. A plausible implication is that MKD can serve as a representation-structuring principle even when the end task is neither uni-modal inference nor explicit teacher–student compression.

5. Information-theoretic decomposition and sample-specific interaction learning

A related but broader decomposition formalism appears in Decomposition-based Multimodal Interaction Learning (DMIL), which does not name its method MKD but provides an information-theoretic account of why multimodal predictors should isolate different interaction types (Yang et al., 10 Jun 2026). For two modalities $f_S$04 and $f_S$05 and target $f_S$06, DMIL decomposes total predictive mutual information into four nonnegative atoms: $f_S$07 with

$f_S$08

The paper’s central claim is that redundant, unique, and synergistic components vary per sample and that effective multimodal learning must both capture each component and adaptively emphasize the correct one on a per-sample basis.

This position is formalized by Theorem 1. If a latent interaction-composition variable $f_S$09 indicates whether a sample is redundancy-dominated, unique-dominated, or synergy-dominated, then any representation $f_S$10 satisfies

$f_S$11

The bound implies two simultaneous objectives: learning well on every interaction type and encoding enough information in $f_S$12 to identify the interaction type.

DMIL implements this through a variational architecture. In intra-modality decomposition, each modality representation $f_S$13 is split into a modality-predictive part $f_S$14 and a residual part $f_S$15, using a regularized-information objective and a VAE-style ELBO: $f_S$16 In consistency decomposition, the predictive factors $f_S$17 and $f_S$18 are further factored into a redundant component $f_S$19 and unique components $f_S$20 and $f_S$21, maximizing

$f_S$22

Synergy is constructed from residual factors: $f_S$23 where the elementwise product is introduced to capture classic nonlinear synergy such as XOR.

The method uses three training stages: pretraining encoders and $f_S$24 VAEs by minimizing $f_S$25; freezing Stage 1 and training the consistency VAE plus synergy construction with $f_S$26; and unfreezing all components for joint training with a gating network under $f_S$27. In the final stage, predictions from $f_S$28, $f_S$29, $f_S$30, and $f_S$31 are adaptively combined: $f_S$32 Because $f_S$33 is sample-dependent, the model can emphasize redundancy-dominated, unique-dominated, or synergy-dominated cases.

The empirical evidence spans real-world, synthetic, and trimodal settings. On CREMA-D, DMIL reports $f_S$34 ACC versus $f_S$35 for joint learning and $f_S$36 for ensemble; on Kinetic-Sounds, $f_S$37 versus $f_S$38 and $f_S$39; on UCF101, $f_S$40 versus $f_S$41 and $f_S$42; and on CMU-MOSEI, $f_S$43 versus $f_S$44 and $f_S$45. On VQAv2, the synergy metric is $f_S$46 for DMIL versus $f_S$47 for joint and $f_S$48 for ensemble. Removing the variational layers or one of the decomposition modules slows training and hurts final ACC by $f_S$49–$f_S$50, and on OOD splits the reported KS accuracy rises from $f_S$51 to $f_S$52.

Although DMIL is not presented under the MKD label, it supplies a principled decomposition vocabulary—redundancy, uniqueness, and synergy—that clarifies what several MKD-style methods are trying to operationalize. This suggests an information-theoretic generalization of MKD in which the central question is not merely how to fuse modalities, but which interaction component should be isolated or transferred for a given sample.

6. Recurring themes, misconceptions, and current interpretation

Several misconceptions follow from collapsing these methods into generic multimodal fusion. First, MKD is not equivalent to simple fusion. In KD-Net, the defining operation is the transfer of complementary multi-modal information into a student that only ever sees one MRI sequence at inference, with both output-space and latent-space supervision contributing to the transfer (Hu et al., 2021). Second, MKD is not restricted to an offline teacher–student hierarchy. The histopathology formulation trains three peer networks jointly, and CLOD explicitly makes the distillation symmetric rather than fixed-directional (Zhang et al., 24 Aug 2025). Third, decomposition is not confined to shared information alone. MoMoK decomposes each modality into multiple relation-conditioned expert perspectives and enforces diversity through mutual-information minimization, while DMIL separates redundant, unique, and synergistic interaction components and adaptively reweights them per sample (Zhang et al., 2024, Yang et al., 10 Jun 2026).

The practical motivation is equally consistent across otherwise different formulations. In medical imaging, acquiring multiple modalities is often not possible in a clinical setting because of limited physicians and scanners and the need to limit costs and scan time; most of the time, only one modality is acquired. In biomarker prediction, simultaneous acquisition of genomic and pathological information is often challenging because of cost or technical limitations (Hu et al., 2021, Zhang et al., 24 Aug 2025). These constraints explain why uni-modal inference after multi-modal training is such a prominent use case for MKD.

A plausible implication of the surveyed literature is that there is not yet a single canonical definition of MKD. The term presently spans at least four regimes: offline distillation from a multi-modal teacher to a mono-modal student, online co-distillation with modality-specific and modality-general features, relation-aware expert decomposition in MMKG representation learning, and explicit interaction decomposition into redundant, unique, and synergistic atoms. What unifies them is the rejection of monolithic fusion in favor of structured multimodal factorization. For research practice, that unifying perspective is more stable than any one implementation.