Tensor-Based Fusion in Multimodal Data

• Tensor-based fusion is a mathematical framework that preserves higher-order relationships by using multiway tensor representations for integrating heterogeneous data.
• It employs methodologies like coupled matrix-tensor factorization, tensor product fusion layers, and low-rank approximations to optimize multimodal data fusion.
• Its applications span neuroimaging, hyperspectral imaging, and machine learning, offering improved accuracy, interpretability, and computational efficiency.

Tensor-based fusion refers to a collection of mathematical frameworks and algorithmic paradigms that employ the multi-way structure of tensors to integrate, jointly analyze, or efficiently compute over heterogeneous, multivariate, or multimodal data. Central to these approaches is the retention and exploitation of the higher-order relationships inherent in the data, as opposed to traditional methods which often vectorize, flatten, or otherwise disrupt modal dependencies. Tensor-based fusion has applications across neuroimaging, hyperspectral imaging, multimodal machine learning, deep neural network compilation, distributed training, hardware-aware operator fusion, and more. This article canvasses the foundational models, algorithmic strategies, and key disciplinary applications of tensor-based fusion.

1. Mathematical Principles and Canonical Models

At its core, tensor-based fusion involves the construction and utilization of multi-way data representations—tensors—preserving the relationships among multiple axes such as time, space, frequency, modality, or subject. For a third-order tensor $X \in \mathbb{R}^{I \times J \times K}$, each element $X_{i,j,k}$ encodes data indexed by three interacting axes (e.g., subject, time, electrode for EEG).

Common fusion architectures include:

• Coupled Matrix-Tensor Factorization (CMTF): Simultaneously factorizes an $n$-way tensor $X$ and a matrix $Y$, sharing factors along specified modes. In the context of multimodal neuroimaging (EEG–fMRI fusion), factors on the subject mode are shared. The optimization objective tightly couples the decompositions via shared latent structures and sparsity-inducing penalties on component importance (Acar et al., 2016).
• Tensor Product Fusion Layers: Used extensively in neural networks for multimodal data, these layers compute the outer product of modality embeddings, explicitly modeling all unimodal, bimodal, ..., multimodal interactions. Augmentations via constants enable lower-order slices to capture main effects, while the full tensor captures higher-order interactions (Zadeh et al., 2017, Xiang et al., 2024, Varshneya et al., 2024).
• Sum and Combination of Tensor Networks: When operating in the tensor network formalism (e.g., Tucker, CP, TT), fusion can involve the exact algebraic sum of isomorphic tensor networks, resulting in block-tensors or concatenated factors that maintain full feature traceability (Calvi et al., 2017).
• Low-Rank and Bayesian Factorizations: Fusion modules may include explicit low-rank constraints (CP/Tucker decompositions) or hierarchical Bayesian priors (FCTN with Gamma sparsity) to increase interpretability, robustness, or computational efficiency (Shan et al., 21 Oct 2025, Zhao et al., 2023).
• Coupled Inverse Problems: In image fusion, the high-resolution target tensor is reconstructed by inverting a set of degradation operators (blur, downsampling, spectral mixing), all while enforcing structural or physics-informed constraints (Gao et al., 12 Mar 2026).

2. Algorithmic Strategies for Tensor-Based Fusion

Tensor-based fusion algorithms share several design patterns which are domain- and modality-agnostic:

• Alternating Least Squares (ALS) and Block-Coordinate Descent: Used for optimizing complex coupled objectives in CMTF, MPLS, and Tucker-based joint decompositions (Karahan et al., 2015, Acar et al., 2016).
• Proximal and Penalty Methods: For problems involving constraints (e.g., nonnegativity, simplex constraints, nuclear norms), ADMM (Alternating Direction Method of Multipliers) and Moreau envelope smoothing are implemented, crucial for solving coupled tensor inversions (Tenfuse) (Gao et al., 12 Mar 2026).
• Nonlinear Conjugate Gradient and Custom Solvers: Applied in CMTF and related tensor-matrix coupled systems for simultaneous factor updates and sparsity-regularized objective minimization (Acar et al., 2016).
• Variational Bayesian Inference and EM: Fully Bayesian tensor fusion models employ mean-field variational approximations and EM for estimating hierarchical sparsity and noise parameters, supporting automatic rank and regularization adaptation (Shan et al., 21 Oct 2025).
• Block-Diagonal and Core Concatenation: Exact tensor network fusion is algorithmically realized by constructing superdiagonal-block core tensors and stacking corresponding factor matrices, enabling non-destructive feature-level fusion (Calvi et al., 2017).
• Greedy and Search-Based Fusion Planners: In distributed compilation (DisCo), operator and tensor fusion is jointly optimized via backtracking search supported by GNN-based simulators. Hardware-aware compilers (FusionStitching, Neptune, Mambalaya) use schedule planning, dependency analysis, and algebraic rewrites to fuse operator chains and minimize DRAM traffic (Zhao et al., 9 Oct 2025, Yi et al., 2022, Long et al., 2018, Odemuyiwa et al., 4 Apr 2026).

3. Applications in Multimodal Data and Remote Sensing

Tensor-based fusion is extensively adopted in domains with inherently multiway data:

Neuroimaging Fusion

• EEG–fMRI Fusion with CMTF: EEG data, modeled as a three-way tensor (subject × time × electrode), is fused with fMRI matrices (subject × voxel) by sharing the subject-mode factors. Sparsity on component weights reveals shared vs. modality-specific patterns, enabling extraction of neural correlates of schizophrenia, such as N2–P3 temporal components and DMN spatial signatures. Electrode selection strongly affects interpretability and statistical significance (Acar et al., 2016).
• Tensor Regression for Multimodal Causality: Multiway partial least squares (MPLS) and CMTF are unified under graphical (Markov–Penrose) frameworks for multimodal fusion, and Granger causality is formulated as tensor regression with atomic PARAFAC decomposition, extracting interpretable neurobiological “atoms” (Karahan et al., 2015).

Hyperspectral–Multispectral Image Fusion

• Coupled Tensor Decomposition: HR-HSI and HR-MSI fusion leverages joint Tucker models, with algorithms (CT-/CB-STAR) guaranteeing recovery of the high-resolution image even under spatial/spectral variability, outperforming previous matrix-based or unstructured methods (Borsoi et al., 2020).
• Blind Tensor Fusion Framework: Tenfuse jointly reconstructs the HR-HSI tensor and unknown degradation operators (PSFs and spectral responses), employing ADMM with TTNN regularization and simplex constraints. Real-time, training-free fusion is achieved for diverse image pairs, with empirical PSNR up to 49.3 dB (Gao et al., 12 Mar 2026).
• Bayesian FCTN: The Bayesian Fully-Connected Tensor Network incorporates hierarchical sparsity and Gaussian noise modeling, attaining state-of-the-art robustness and quantitative performance (e.g., CAVE, Harvard, Pavia) across SNR and downsampling regimes (Shan et al., 21 Oct 2025).

Computer Vision and Multimodal Machine Learning

• Multimodal Sentiment/Education Analytics: Outer-product-based Tensor Fusion Networks (TFN) explicitly encode unimodal, bimodal, and trimodal interactions for sentiment and emotion analysis, outperforming simple concatenation and dot-product fusion. Trimodal blocks account for substantial accuracy gain (e.g., 93.65% for emotion recognition in BERT-ViT TFN models) (Zadeh et al., 2017, Xiang et al., 2024).
• Interpretability: The InTense model imposes a block-norm (L_p) penalty on tensor-product fusion weights, producing normalized relevance scores for each modality and interaction term with theoretical guarantees (unbiasedness, no spurious higher-order effects). These scores are consistent with empirical unimodal/bimodal performance (Varshneya et al., 2024).
• Tensorized Hardware Acceleration: In TOMFN, low-rank CP decompositions and TT (tensor train) factorization drastically compress the fusion and self-attention modules, enabling implementation in photonic hardware with a 51.3× reduction in device count and energy efficiency of $3.7\times10^{13}$ MAC/J (Zhao et al., 2023).
• Person Re-Identification (PRe-ID): High-Dimensional Feature Fusion (HDFF) and TXQDA combine CNN, LOMO, and GOG descriptors into third-order tensors, with multilinear subspace learning optimizing per-mode scatter ratios. Empirical gains in Rank-1 matching demonstrate the effectiveness of the multilinear tensor-based fusion, notably outperforming vectorized or single-feature baselines (Chouchane et al., 9 May 2025, Gharbi et al., 2023).

4. Tensor Fusion in Deep Learning, Compilation, and Distributed Systems

Tensor-based fusion extends into optimization of computational graphs, especially in neural network training and inference:

• Operator Fusion and Scheduling: Deep learning compilers (e.g., Neptune, FusionStitching, Mambalaya) perform end-to-end kernel fusion by restructuring chains of tensor operations. Advanced schemes involve breaking cross-iteration reduction dependencies (softmax, attention) into algebraically correct, batch-wise fused kernels, achieving up to 1.35× speedup over best-in-class alternatives (Zhao et al., 9 Oct 2025, Long et al., 2018, Odemuyiwa et al., 4 Apr 2026).
• Distributed Training: DisCo maximizes overlap between computation and communication through joint operator and tensor fusion (gradient AllReduce). Both forms of fusion are optimized by search-guided simulators leveraging GNN-based runtime prediction, with observed per-iteration speedup up to 26.7% (Yi et al., 2022).
• Hardware/Architecture-Aware Fusion: The Mambalaya accelerator fuses extended Einstein summation (Einsum) operator cascades by classifying iteration space relationships (rank-isomorphic, subset, superset, disjoint), generating output/input-stationary schedules that exploit on-chip buffer hierarchies and configurable PE arrays (Odemuyiwa et al., 4 Apr 2026).

5. Interpretability, Identifiability, and Theoretical Guarantees

Tensor-based fusion frameworks offer notable properties:

• Identifiability: Coupled tensor models with structured low-rank constraints admit uniqueness guarantees (up to nullspaces of degradation operators). Theories establish necessary and sufficient conditions for exact HR image recovery, even under spatial/spectral variability (Borsoi et al., 2020).
• Interpretability: Block-wise parameterization (as in InTense or TFN) disentangles main effects and interactions, and yields explicit relevance scores for all modal subsets. Centering procedures prevent high-order blocks from mimicking lower-order effects (Varshneya et al., 2024).
• Separation of Shared/Specific Components: Sparsity or structurally restricted penalties on fusion weights (λ, σ in CMTF) ensure that only components with joint evidence across modalities appear as shared, supporting neuroscientific interpretation (Acar et al., 2016).

Table: Example Tensor Fusion Models, Objectives, and Properties

Model (Paper)	Fusion Objective / Method	Notable Guarantees / Interpretability
CMTF (Acar et al., 2016)	Minimize coupled Frobenius losses + 1-norm sparsity on λ, σ	Identifies shared vs. unshared factors; p-value significance per component
Tenfuse (Gao et al., 12 Mar 2026)	Blind inverse (ADMM, TTNN, simplex)	Real-time, training-free, spatial/spectral unmixing; convergence theory
TFN (Zadeh et al., 2017, Xiang et al., 2024)	Outer product augmented embeddings	Explicit all-order marginals; interpretable interaction blocks
InTense (Varshneya et al., 2024)	Block L_p-norm penalty on fusion layers	Provably unbiased relevance scores for each modal subset
Sum-of-TNs (Calvi et al., 2017)	Core block-diagonal sum of TNs	Exact feature-level sum, modular, rank additive
BFCTN (Shan et al., 21 Oct 2025)	Bayesian FCTN, variational EM	Automatic rank-sparsity adaptation, robustness to noise
HDFF+TXQDA (Chouchane et al., 9 May 2025)	Tensor concatenation + multilinear QDA	Substantial CMC Rank-1/20 performance gains

6. Limitations and Challenges

• Curse of Dimensionality: Naive tensor outer products or full fusion tensors quickly become infeasible for high-dimensional modalities; low-rank decompositions or block-sparse models are essential (Zadeh et al., 2017, Zhao et al., 2023).
• Model Selection and Hyperparameters: Performance is sensitive to choices of tensor ranks, regularization weights, and the number of interaction terms retained. Bayesian models help mitigate, but do not eliminate, these dependencies (Shan et al., 21 Oct 2025, Varshneya et al., 2024).
• Computational Complexity: Advanced fusion algorithms (e.g., variational Bayesian FCTN, ADMM for coupled inverse problems) remain more demanding than vectorized or singly-factored approaches; trade-offs between robustness and wallclock time must be addressed (Shan et al., 21 Oct 2025, Gao et al., 12 Mar 2026).
• Topology Matching in TN Fusion: Block-diagonal sum fusion of tensor networks requires isomorphic network topologies; fusion of arbitrary or mismatched topologies is not directly supported (Calvi et al., 2017).
• Noise and Missing Data: While probabilistic frameworks are more robust, handling strongly structured noise or outliers, and missing data, remains a challenge.

7. Summary and Future Directions

Tensor-based fusion subsumes a broad class of techniques that maintain, exploit, and efficiently compute over multi-way data structures for the purposes of multimodal integration, joint modeling, or computational optimization. It underpins advances in neuroimaging analysis, hyperspectral imaging, multimodal machine learning, and deep neural network execution, offering rigorous theoretical foundations (identifiability, interpretability), substantial empirical gains (accuracy, robustness, speed), and natural physical mappings (e.g., photonic hardware).

Open avenues include: fully adaptive rank determination (especially in Bayesian and hardware-constrained systems), scalable tensor factorization for extreme multimodal settings, nonlinear and kernelized tensor fusion extensions, and unified frameworks linking probabilistic graphical models with algebraic tensor representations (Varshneya et al., 2024, Shan et al., 21 Oct 2025, Gao et al., 12 Mar 2026). Existing toolboxes (e.g., ACMTF-OPT, CMTF-Toolbox) and reproducible algorithmic pipelines support further applied research and cross-domain innovation.