Multimodal Factorization Model (MFM)

Updated 31 October 2025

Multimodal factorization models are frameworks that decompose heterogeneous data into shared and modality-specific latent factors, enabling efficient fusion and robust predictions.
They employ diverse architectures—such as latent variable models, tensor/matrix decompositions, and transformer-based methods—to capture complex intra- and cross-modal interactions.
Empirical studies demonstrate that MFMs improve interpretability, scalability, and accuracy in tasks like recommendation, sentiment analysis, and cross-modal retrieval.

A multimodal factorization model (MFM) is a structured framework for extracting, fusing, and disentangling information from heterogeneous data sources by decomposing multimodal data into shared and modality-specific representations. MFMs have emerged as a central methodology for multimodal learning, enabling explicit modeling of complex intra-modal and cross-modal interactions, facilitating tasks from recommendation to sentiment analysis, and providing interpretability, efficiency, and robustness to missing data.

1. Conceptual Foundations and Rationale

The primary objective of MFMs is to address the twofold challenge of (1) capturing both intra-modal and complex cross-modal interactions relevant for prediction and (2) providing robustness to missing, noisy, or redundant modalities. The key paradigm is to explicitly factorize latent spaces into:

Shared (multimodal) latent factors: Capturing information common and necessary across modalities (typically for discriminative or joint tasks).
Modality-specific latent factors: Capturing information unique to individual modalities (typically for generative modeling, reconstruction, or regularization).

This structured factorization establishes a foundation for disentangling and utilizing the unique and shared information inherent in multimodal sources, facilitating interpretability and selective fusion.

MFMs have demonstrated efficacy across diverse problem domains, including but not limited to recommendation systems (Geng et al., 2023), sentiment/emotion recognition (Tsai et al., 2018, Barezi et al., 2018, Sahay et al., 2020), cross-modal retrieval (Matsuishi et al., 29 May 2025), biomedicine (Liu et al., 14 Jul 2025), and topic modeling (Virtanen et al., 2012).

2. Representative Model Architectures

MFMs have been instantiated using various architectural strategies, each reflecting different computational and application constraints:

Factorized Latent Variable Models: These models introduce separate latent variables for shared (discriminative, multimodal) and private (generative, modality-specific) components (Tsai et al., 2018). Architectures consist of encoders mapping data (and possibly labels) to latent spaces and decoders for reconstruction or prediction.
Tensor and Matrix Factorization Approaches: Classical methods leverage outer-product fusion (Tensor Fusion Networks [Zadeh et al., 2017]), low-rank CP/Tucker decompositions (Barezi et al., 2018, Sahay et al., 2020), and matrix decomposition for compact joint representations (e.g., BoNMF (Xiang et al., 12 Jul 2024)).
Prompt-Based Foundation Architectures: Recent MFM implementations for recommendations (VIP5 (Geng et al., 2023)) employ a token-based unification strategy over a backbone foundation model, aggregating multimodal representations via multimodal personalized prompts, and perform parameter-efficient training (e.g., adapters).
Transformer-Based Modal Interaction Factorization: Explicit modeling of all intra- and inter-modal interactions at the attention module level (e.g., Factorized Multimodal Transformer (Zadeh et al., 2019)) enables explicit spatio-temporal factorization and global context aggregation.

Architecture	Key Factorization Strategy	Target Domains
Latent variable	Discriminative vs. generative latent split	Sentiment, emotion, missing modality
Tensor/matrix	Low-rank/shared-specific decomposition	Multimodal fusion, recommendation
Prompt-based	Token-based prompt unification	Recommender systems
Attention-based	Inter-modal factor-specific self-attention	Sequential, spatio-temporal tasks

3. Mathematical Formulation of Multimodal Factorization

A unifying property is the explicit probabilistic or algebraic factorization of the multimodal data distribution:

Let $\mathbf{X}_{1:M}$ denote $M$ modalities, and $\mathbf{Y}$ the target label.

Latent variable decomposition:
- Joint discriminative: $\mathbf{Z}_y$ (shared across all)
- Private generative: $\mathbf{Z}_{a_i}$ for each modality $i$
Overall generative model:

$P(\hat{\mathbf{X}}_{1:M}, \hat{\mathbf{Y}}) = \int P(\hat{\mathbf{Y}} | \mathbf{F}_y) \prod_{i=1}^M P(\hat{\mathbf{X}}_i | \mathbf{F}_{a_i}, \mathbf{F}_y) P(\mathbf{F}_y | \mathbf{Z}_y) \prod_{i=1}^M P(\mathbf{F}_{a_i} | \mathbf{Z}_{a_i}) P(\mathbf{Z}_y) \prod_{i=1}^M P(\mathbf{Z}_{a_i}) d\mathbf{F}d\mathbf{Z}$

Given features from $M$ modalities $x_1,\ldots,x_M$ ,

Full tensor fusion: Outer product produces prohibitive $O(\prod_i d_i)$ size.
Low-rank CP/Tucker factorization: Decomposes the fusion tensor with per-modality ranks or factors, e.g.

$h = \sum_{k=1}^r \prod_{m=1}^M \mathbf{w}_m^{(k)T} [x_m; 1]$

$\mathcal{W} = \mathcal{R} \times_1 W_1 \cdots \times_{M+1} W_{M+1}$

with $\mathcal{R}$ as core, $W_i$ per-modality factor matrices.

Visual/textual tokens unified: Features $x$ from an image $\mathbf{I}$ are projected,

$[p_1, ..., p_k] = f(x)$

and concatenated as prompt tokens. All modalities are processed as sequences in a shared Transformer backbone.

4. Parameter Efficiency and Scalability

A central claim in recent MFM designs is that correct factorization enables substantial efficiency and scalability:

Adapter- or Low-rank-based Tuning: Rather than full backbone fine-tuning, parameter-efficient strategies (e.g., adapters, LoRA (Geng et al., 2023, Jiang et al., 2 Mar 2025)) permit adaptation with less than 4% of parameters, conferring faster convergence, reduced computation and memory, and improved generalization.
Pairwise Inter-View Factorization: Information-theoretic designs such as CooKIE (Namgung et al., 15 May 2025) only model pairwise shared and unique information, avoiding combinatorial explosion ( $O(2^m)$ objectives for $m$ modalities) seen in classic high-order factorization, while maintaining state-of-the-art predictive performance.
Advantage of Low-rank Tensor Models: Compared to explicit high-dimensional fusion, low-rank constructions reduce parameter and computation needs by up to 45% without sacrificing accuracy (Sahay et al., 2020, Barezi et al., 2018).

These architectural considerations directly enable scaling MFMs to highly multimodal, high-dimensional settings (e.g., biomedical data integration (Liu et al., 14 Jul 2025), large-scale recommendation (Geng et al., 2023), or geospatial analysis (Namgung et al., 15 May 2025)).

5. Interpretability, Regularization, and Handling Redundancy

Factorization models provide mechanisms for interpretability and regularization:

Modality Contribution Analysis: Varying rank (Tucker factorization) or compressing specific modalities quantifies the unique vs. redundant contribution and identifies where cross-modal information is duplicative (Barezi et al., 2018, Tsai et al., 2018).
Regularization: Reduced parameter space and explicit separation of shared/redundant information acts as a learned regularizer, mitigating overfitting, especially in small-sample or high-dimensional contexts.
Robustness to Missing Modalities: Surrogate inference networks (Tsai et al., 2018) or generative surrogates allow conditional reconstruction and predictive robustness even when arbitrary subsets of modalities are missing at inference.

Interpretability Mechanism	Explanation
Per-modality compression	Quantifies unique information and redundancies
Mutual information ratios	Measures influence of shared vs. specific factors (e.g., $r_i$ score)
Gradient- or attention-based	Identifies salient features influencing interaction/fusion

6. Empirical Validation and Representative Results

Empirical studies across tasks and domains substantiate the advantages of MFM approaches:

Recommendation Systems: Models such as VIP5 (Geng et al., 2023) establish state-of-the-art performance on multi-group recommendation and explanation generation, outperforming text-only and task-specific baselines with parameter-efficient adapter tuning.
Sentiment and Emotion Recognition: Low-rank and Tucker-decomposed factorization models achieve 1–4% gains over prior state-of-the-art in F1/accuracy and drastically lower parameter counts (Sahay et al., 2020, Barezi et al., 2018). Factorized representation models (Tsai et al., 2018) achieve competitive or superior performance in both full- and missing-modality scenarios.
Cross-modal Retrieval/Activity Recognition: Contrastive foundation models aggregate four or more modalities (video, IMU, MoCap, text) for retrieval and activity recognition, enabling superior zero-shot and transfer generalization (Matsuishi et al., 29 May 2025).
Topic Modeling: Factorized multi-modal topic models (Virtanen et al., 2012) achieve lower conditional perplexity in cross-modal prediction by learning both shared and private topics, overcoming limitations of classic topic models and nonparametric extensions.

Task/Domain	Model	Reported Improvement
RecSys	VIP5	SOTA HR@5/NDCG@5/BLEU/ROUGE (Geng et al., 2023)
Sentiment, Emotion	MRRF, LMF	1–4% F1/Acc gains, up to 45% faster training (Barezi et al., 2018, Sahay et al., 2020)
Cross-modal Retrieval	AURA-MFM	F1: 0.62 vs. 0.07 baseline zero-shot (Matsuishi et al., 29 May 2025)
Multistudy Bio	MMGFM	Significantly better factor recovery/efficiency (Liu et al., 14 Jul 2025)

7. Theoretical and Practical Impact

MFMs advance multimodal learning by providing frameworks for:

Principled integration of diverse modalities at scale, substantiated by theoretical guarantees (e.g., identifiability, consistency (Liu et al., 14 Jul 2025)).
Operational scalability, enabling factorization in high modality count contexts without combinatorial growth of objectives (Namgung et al., 15 May 2025).
Interpretability and pruning, supporting model selection and resource allocation based on modality utility (Barezi et al., 2018).
Unified architectures for emergent tasks in recommendation, retrieval, and analysis, particularly with large foundation models (Geng et al., 2023, Matsuishi et al., 29 May 2025).

A plausible implication is that future directions will increasingly focus on further scaling (across data types, domains, and samples), automated discovery of optimal factorized structure, and integrating MFM frameworks with robust missing data imputation and transfer learning protocols.

In summary, multimodal factorization models offer a comprehensive and rigorous approach to unifying, disentangling, and leveraging information in multimodal machine learning, with architectures and methodologies that balance expressiveness, efficiency, and practical robustness for a wide range of ML applications.