Multimodal Representation Decomposition (MRD)

• MRD is a modeling strategy that explicitly separates multimodal signals into shared and modality-specific components to address redundancy and interference.
• It employs various decomposition templates—such as shared-specific splits, low-rank/sparse methods, and information-theoretic approaches—to enhance prediction and interpretability.
• Empirical findings in scene representation, medical prediction, and multimodal detection demonstrate improved robustness and clearer feature attribution.

Searching arXiv for papers on Multimodal Representation Decomposition and closely related formulations. Multimodal Representation Decomposition (MRD) denotes a class of modeling strategies that explicitly partition multimodal signals into structurally distinct components rather than treating fusion as a single undifferentiated operation. Across recent work, these components have included shared versus modality-specific factors, low-rank common structure versus sparse private structure, modality-invariant anatomical content versus modality appearance, independent modality subspaces for gradient routing, and target-conditioned unique, redundant, and synergistic information. The recurring motivation is that multimodal systems often suffer from redundancy, modality conflict, alignment failure, gradient interference, or missing-modality fragility when representations are fused too early or too rigidly (Gu et al., 15 Jul 2025, Kaushik et al., 27 Jan 2026, Liu et al., 7 Jul 2025, Chen et al., 18 May 2026, Ouyang et al., 2021, Ma et al., 26 Dec 2025, Tian et al., 8 Jun 2025, Wang et al., 24 Sep 2025, Shao et al., 19 Nov 2025).

1. Problem setting and motivating pathologies

Recent MRD formulations arise from several distinct but related failure modes. In multimodal scene representation, MMOne identifies property disparity and granularity disparity: different modalities may require different feature dimensionalities, obey different physical behavior, and prefer different numbers and sizes of primitives. RGB may prefer many small Gaussians near sharp edges, whereas thermal may prefer fewer larger Gaussians for smoother fields (Gu et al., 15 Jul 2025). In multimodal embedding decomposition, standard sparse autoencoders may learn split dictionaries, in which most features are effectively unimodal and paired embeddings share no common active neurons, while also producing many dead neurons and severe degradation on zero-shot cross-modal tasks (Kaushik et al., 27 Jan 2026).

Other MRD papers formulate the problem as one of redundancy and interference in downstream prediction. MurreNet argues that straightforward fusion of histopathology and genomic embeddings fails to capture both modality-specific and modality-common interactions, yielding limited understanding of multimodal correlations and suboptimal survival prediction (Liu et al., 7 Jul 2025). In multimodal recommendation, MRdIB attributes degradation to redundant and irrelevant information that is not filtered before fusion, while in multimodal object detection, RSC-MD gives a theoretical account of fusion degradation in terms of suppressed unimodal gradients and stronger suppression for weaker modalities (Wang et al., 24 Sep 2025, Shao et al., 19 Nov 2025).

A separate line of work emphasizes that apparent cross-modal consistency is not itself sufficient evidence of decomposition. In multi-modal brain MRI, cross-reconstruction-based methods can still leak information between latent factors, and adversarial penalties may regularize only one factor and remain unstable. That work therefore treats disentanglement as a problem of explicitly constraining relationships among representations across subjects and modalities (Ouyang et al., 2021). This suggests that MRD is best understood not as a single architecture, but as an explicit response to identifiable multimodal failure modes.

2. Canonical decomposition forms

Despite their diversity, MRD methods usually instantiate a small number of decomposition templates.

Decomposition form	Representative factorization	Example paper
Shared / modality-specific	$$h_p^o \mapsto (h_p^s,h_p^c),\; h_g^o \mapsto (h_g^s,h_g^c)$$	MurreNet (Liu et al., 7 Jul 2025)
Shared / specific with orthogonality	$$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$	CodeBind (Chen et al., 18 May 2026)
Low-rank / sparse	$$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$	Affective MRD (Tian et al., 8 Jun 2025)
Anatomy / modality appearance	$x_i^p \mapsto (s_i^p,z_i^p)$	Brain MRI MRD (Ouyang et al., 2021)
Unique / redundant / synergistic	$I(Y;Z_1,Z_2)=U_1+U_2+R+S$	PIDReg, MRdIB (Ma et al., 26 Dec 2025, Wang et al., 24 Sep 2025)
Shared geometry / modality-specific appearance	$z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$	MMOne reinterpretation (Gu et al., 15 Jul 2025)

The most common pattern is a shared-specific split. MurreNet decomposes pathology and genomics into four vectors, $h_p^s, h_p^c, h_g^s, h_g^c$, where the superscripts denote modality-specific and modality-common representations, respectively. CodeBind uses projection heads to form a modality-shared component that carries cross-modal semantic invariants and a modality-specific component that preserves modality-unique details, with an explicit orthogonality condition between the two (Liu et al., 7 Jul 2025, Chen et al., 18 May 2026).

A second pattern is structural decomposition of aligned features. In affective computing, aligned visual and textual features are jointly decomposed into a shared low-rank matrix $\mathbf L$ and sparse private matrices $\mathbf S_I,\mathbf S_T$, so that common emotional content is captured by the low-rank part and unique visual or textual cues remain sparse residuals (Tian et al., 8 Jun 2025). In MMOne, a related structural split occurs at the level of 3D Gaussians: geometry is treated as shared, whereas appearance is modality-dependent, and only the shared code plus the active modality code participate in rendering (Gu et al., 15 Jul 2025).

A third pattern is information-theoretic decomposition with respect to a target. PIDReg and MRdIB decompose predictive information into unique, redundant, and synergistic terms. In this view, the central object is not merely whether a latent is shared across modalities, but whether it contributes information about $Y$ uniquely, redundantly, or only jointly with another modality (Ma et al., 26 Dec 2025, Wang et al., 24 Sep 2025). This broadens MRD from geometric or architectural factorization to target-conditioned information accounting.

3. Architectural mechanisms and optimization strategies

MRD implementations differ sharply in the operators used to realize decomposition. MMOne uses a modality modeling module with learned indicators $$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$0 and modality-specific feature vectors $$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$1, followed by soft prune and decomposition steps. If $$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$2, a Gaussian can be turned off for a modality or deleted entirely; if per-modality gradients on the same Gaussian disagree enough, measured by $$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$3, the Gaussian is cloned so that different modalities can specialize on different children (Gu et al., 15 Jul 2025).

In multimodal embedding spaces, the group-sparse SAE approach implements MRD through a shared encoder-decoder with TopK sparsification,

$$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$4

shared random masking before TopK, and a mixed $$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$5 penalty

$$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$6

The masking forces both modalities to contend with the same temporarily unavailable atoms, while the group penalty encourages paired codes to co-activate on the same dictionary coordinates (Kaushik et al., 27 Jan 2026).

MurreNet realizes MRD through separate MLPs for pathology-specific and genomics-specific features, combined with a co-attention-derived common encoder. Its shared vectors are computed as

$$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$7

where $$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$8. Three regularizers are then imposed: a similarity loss on common embeddings, a KL-based difference loss between common and specific embeddings, and a reconstruction loss from concatenated shared and specific parts (Liu et al., 7 Jul 2025).

The neuroimaging formulation uses a different route. Instead of separate encoders per modality, it employs conditional convolution, in which each convolutional layer is a mixture of expert kernels selected by modality-dependent routing weights: $$z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}, \langle z^{\mathcal M}_{\mathrm{shared}}, z^{\mathcal M}_{\mathrm{spec}}\rangle=0$$9 A hinge-style similarity loss then enforces that anatomical codes for the same subject across modalities are closer than anatomical codes of different subjects within the same modality, while modality codes obey the opposite pattern (Ouyang et al., 2021).

CodeBind combines shared-specific projection heads with unified compositional vector quantization. A single shared codebook stores cross-modal semantics, while each modality has its own specific codebook. Shared embeddings are aligned with InfoNCE, specific embeddings are regularized with an orthogonality loss and a uniformity loss, and code usage is stabilized by VQ commitment, cross-modal code matching, and codevector regularization (Chen et al., 18 May 2026).

RSC-MD is distinctive in that its decomposition is primarily enforced in the backward pass. Auxiliary unimodal heads amplify each backbone’s own gradient, while the Modality Decoupling module imposes a Jacobian mask that zeroes cross-branch gradient flow. Formally, $$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$0 for $$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$1, so the branches no longer fight each other through shared gradients (Shao et al., 19 Nov 2025). This is still MRD in the sense that the learned representation spaces are driven toward independent modality-specific subspaces.

4. Statistical and theoretical interpretations

Several MRD papers make explicit claims about what decomposition does and does not guarantee. The MMOne paper itself does not introduce a black-box VAE or explicit ELBO, but its mechanism has been reinterpreted in latent-variable language by assigning each Gaussian a shared latent $$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$2 for geometry and modality-specific latents $$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$3 for appearance, with factorized Gaussian priors

$$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$4

Under this reinterpretation, shared and specific factors are encouraged by prior factorization, while activity is implemented by learned modality indicators $$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$5 (Gu et al., 15 Jul 2025). The summary explicitly notes that a factorized mutual-information penalty would be natural in this view, but is not part of MMOne itself.

The group-sparse SAE work provides a complementary theorem-level perspective. It argues that even if paired embeddings satisfy $$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$6, a sparse decomposition can still be split, with disjoint supports for paired codes. Its Theorem 1 then shows that the existence of a split dictionary on an aligned embedding space implies the existence of a non-split dictionary with improved modality alignment (Kaushik et al., 27 Jan 2026). The immediate implication is that dense-space alignment does not automatically transfer to sparse latent alignment.

The neuroimaging work directly challenges the widespread assumption that cross-reconstruction suffices for disentanglement. Its argument is that both latent factors can still copy the input, and adversarial penalties only partially address leakage. It therefore replaces indirect reconstruction logic with an explicit margin-based similarity regularizer on representation relations across subjects and modalities (Ouyang et al., 2021). This suggests a broader lesson: MRD usually requires constraints on who should resemble whom, not only on what should reconstruct what.

Information-theoretic MRD exposes a different theoretical issue: decomposition may be conceptually well motivated but mathematically underdetermined. PIDReg notes that

$$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$7

provides only three equations for four unknowns. It resolves this by imposing a Gaussianity assumption on $$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$8 and using the union-information constraint, which makes the PID terms analytically computable from covariance structure (Ma et al., 26 Dec 2025). MRdIB adopts a related PID vocabulary, but operationalizes it through a multimodal information bottleneck, a MINE-based redundancy estimator, and separate unique-information objectives rather than through the same Gaussian closed form (Wang et al., 24 Sep 2025).

5. Empirical performance across domains

The reported empirical gains from MRD span 3D representation, embedding interpretability, medical prediction, affective computing, detection, recommendation, and multimodal alignment.

Setting	Reported effect	Citation
RGB + Thermal scene representation	RGB PSNR $$\mathbf I=\mathbf L+\mathbf S_I,\; \mathbf T=\mathbf L+\mathbf S_T$$9 dB, Thermal PSNR $x_i^p \mapsto (s_i^p,z_i^p)$0 dB, Gaussian count $x_i^p \mapsto (s_i^p,z_i^p)$1	(Gu et al., 15 Jul 2025)
Image/Text sparse decomposition	CLIP CIFAR-10 $x_i^p \mapsto (s_i^p,z_i^p)$2 with SAE, recovered to $x_i^p \mapsto (s_i^p,z_i^p)$3 with MGSAE	(Kaushik et al., 27 Jan 2026)
Histopathology + genomics survival prediction	MurreNet exceeds previous SOTA by $x_i^p \mapsto (s_i^p,z_i^p)$4–$x_i^p \mapsto (s_i^p,z_i^p)$5 in Concordance index on six TCGA cohorts	(Liu et al., 7 Jul 2025)
Nine-modality alignment	LLVIP $x_i^p \mapsto (s_i^p,z_i^p)$6, FLIR_v2 $x_i^p \mapsto (s_i^p,z_i^p)$7, AudioSet mAP $x_i^p \mapsto (s_i^p,z_i^p)$8	(Chen et al., 18 May 2026)
Missing-modality brain MRI	Missing T1 in BraTS: Standard+Zero $x_i^p \mapsto (s_i^p,z_i^p)$9 Dice, MRD $I(Y;Z_1,Z_2)=U_1+U_2+R+S$0	(Ouyang et al., 2021)
Multimodal regression	CT Slices RMSE $I(Y;Z_1,Z_2)=U_1+U_2+R+S$1 vs. $I(Y;Z_1,Z_2)=U_1+U_2+R+S$2; brain-age MAE $I(Y;Z_1,Z_2)=U_1+U_2+R+S$3 vs. $I(Y;Z_1,Z_2)=U_1+U_2+R+S$4	(Ma et al., 26 Dec 2025)
Multimodal detection	FLIR mean AP$I(Y;Z_1,Z_2)=U_1+U_2+R+S$5 $I(Y;Z_1,Z_2)=U_1+U_2+R+S$6; LLVIP mean AP$I(Y;Z_1,Z_2)=U_1+U_2+R+S$7 $I(Y;Z_1,Z_2)=U_1+U_2+R+S$8	(Shao et al., 19 Nov 2025)
Affective computing with LLM prompting	Twitter-15 $I(Y;Z_1,Z_2)=U_1+U_2+R+S$9 in Acc/macro-F1	(Tian et al., 8 Jun 2025)
Recommendation	Average improvement of $z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$0 in Recall@5 and NDCG@5; VBPR gains up to $z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$1 recall	(Wang et al., 24 Sep 2025)

Several empirical patterns recur. First, decomposition often improves accuracy and compactness simultaneously. MMOne reduces Gaussian count to approximately one third on RGB+Thermal and to approximately $z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$2 of LangSplat on RGB+Language, while preserving or improving quality metrics (Gu et al., 15 Jul 2025). CodeBind similarly reports nearly $z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$3 codebook usage for both shared and specific codebooks under its compositional design (Chen et al., 18 May 2026).

Second, MRD frequently improves robustness under modality imbalance or missing modalities. In brain MRI, the fused anatomical representation remains effective when one input modality is removed, whereas conventional zero-filling or mean replacement collapses performance (Ouyang et al., 2021). In object detection, gradient norms at an intermediate visible-branch SPPF layer are amplified by $z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$4–$z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$5 under RSC-MD, and unimodal branch AP lost under naïve joint training is substantially recovered (Shao et al., 19 Nov 2025).

Third, MRD often improves interpretability metrics directly rather than only downstream task performance. MGSAE increases “both-modal” atoms by $z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$6–$z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$7, reduces “neither” by $z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$8, shifts multimodal monosemanticity scores to the right, and improves semantic coherence of learned features (Kaushik et al., 27 Jan 2026). PIDReg reports decomposition values that reveal whether a dataset is redundancy-dominated or synergy-dominated, and uses those values to motivate modality selection decisions such as “sMRI-only” versus full multimodal inference (Ma et al., 26 Dec 2025).

6. Applications, misconceptions, and open directions

MRD has been used for at least four distinct purposes. One is representation quality under heterogeneous physics, as in MMOne’s scene modeling over RGB, thermal, and language (Gu et al., 15 Jul 2025). A second is interpretability and controllability, as in group-sparse SAE decomposition of CLIP and CLAP spaces, where concept naming and audio-text retrieval steering become more coherent when the dictionary is more multimodal (Kaushik et al., 27 Jan 2026). A third is robust multimodal prediction, exemplified by survival prediction, brain-age regression, and recommendation, where decomposition is tied directly to prediction targets rather than treated as an auxiliary property (Liu et al., 7 Jul 2025, Ma et al., 26 Dec 2025, Wang et al., 24 Sep 2025). A fourth is training stabilization, seen most clearly in RSC-MD, where MRD acts on optimization geometry rather than only latent semantics (Shao et al., 19 Nov 2025).

A common misconception is that any explicit shared/private split guarantees disentanglement. The brain MRI study explicitly argues otherwise, showing that cross-reconstruction and adversarial regularization do not naturally guarantee the desired factorization (Ouyang et al., 2021). Another misconception is that alignment in the original dense embedding space will necessarily persist after sparse decomposition; the group-sparse SAE analysis rejects this, showing that split sparse codes can exist even when paired dense embeddings are well aligned (Kaushik et al., 27 Jan 2026). A third misconception is that “MRD” names a single canonical algorithm. The literature instead uses the term for scene-Gaussian decomposition, SAE dictionary alignment, co-attention factorization, codebook factorization, low-rank/sparse factorization, gradient-space decoupling, and PID-based information decomposition (Gu et al., 15 Jul 2025, Liu et al., 7 Jul 2025, Chen et al., 18 May 2026, Tian et al., 8 Jun 2025, Shao et al., 19 Nov 2025, Ma et al., 26 Dec 2025).

Several directions emerge from the current formulations. MMOne and CodeBind both emphasize scalability to additional modalities, with MMOne extending to additional scene modalities and CodeBind validating on nine modalities while avoiding fully paired training through bridging modalities (Gu et al., 15 Jul 2025, Chen et al., 18 May 2026). The group-sparse SAE work states that extensions to $z_i=(z_i^{shared},z_i^1,\dots,z_i^M)$9 modalities can replace paired groups with an $h_p^s, h_p^c, h_g^s, h_g^c$0 penalty over $h_p^s, h_p^c, h_g^s, h_g^c$1 corresponding coordinates, while the same paper also proposes semi-paired training and hierarchical concepts as extensions (Kaushik et al., 27 Jan 2026). PID-based methods suggest a further shift from decomposition as structural disentanglement toward decomposition as decision-relevant information accounting, which may be especially important when modalities are complementary only on a subset of examples (Ma et al., 26 Dec 2025, Wang et al., 24 Sep 2025).

Taken together, the literature suggests that MRD is not a narrow subroutine but a general design principle: identify the source of multimodal interference, specify the factorization appropriate to that source, and enforce it through architectural, geometric, probabilistic, or information-theoretic constraints.