Tensor Decomposition in Biological Data Analysis: Methods, Applications, and Veterinary Relevance

Introduction

Modern biological data acquisition generates increasingly complex, multi-dimensional datasets. In veterinary medicine, examples include longitudinal metabolomic profiles across treatment groups, multi-tissue gene expression arrays, and spatial transcriptomic maps of tissue sections. Traditional matrix factorization methods such as principal component analysis (PCA) are limited to two-dimensional data (samples by features). When data possess three or more dimensions (e.g., genes by individuals by time points), tensor decomposition provides a natural mathematical framework to extract latent structures without collapsing dimensions.

A tensor is a multi-dimensional array. An order-3 tensor can represent, for instance, gene expression levels (mode 1) across multiple individuals (mode 2) at several time points (mode 3). Tensor decomposition factorizes this core tensor into lower-dimensional components that capture interactions among all modes simultaneously. The two most widely used decompositions are the CANDECOMP/PARAFAC (CP) decomposition (also known as canonical polyadic decomposition) and the Tucker decomposition (higher-order singular value decomposition, HOSVD). CP decomposition expresses a tensor as a sum of rank-one tensors, while Tucker decomposition yields a core tensor and factor matrices for each mode, allowing different numbers of components per mode [1, 2].

In veterinary bioinformatics, tensor decomposition methods have been applied to single-cell RNA sequencing, spatial transcriptomics, multi-omics integration, behavioral analysis, and microbiome studies. This article provides an exhaustive technical review of tensor decomposition algorithms, their biological interpretability, and their specific applications in animal health research, with citations drawn exclusively from the peer-reviewed literature provided.

Mathematical Foundations of Tensor Decomposition

Notation and Basic Operations

A third-order tensor ( \mathcal{X} \in \mathbb{R}^{I \times J \times K} ) has three modes. The mode-n fibers are vectors obtained by fixing all indices except the n-th. The mode-n matricization (unfolding) rearranges the tensor into a matrix along that mode. The n-mode product of a tensor with a matrix is a standard operation in Tucker decomposition.

CP Decomposition (CANDECOMP/PARAFAC)

CP decomposition approximates a tensor as a sum of ( R ) rank-one components:

[ \mathcal{X} \approx \sum_{r=1}^{R} \mathbf{a}_r \circ \mathbf{b}_r \circ \mathbf{c}_r ]

where ( \mathbf{a}_r ), ( \mathbf{b}_r ), ( \mathbf{c}_r ) are factor vectors for each mode and ( \circ ) denotes the outer product. The rank ( R ) is chosen by the user. This decomposition is unique under mild conditions (Kruskal's theorem), making it highly interpretable. Each component can be associated with a biological pattern, such as a co-expression module that varies across time and individual.

Tucker Decomposition (HOSVD)

Tucker decomposition represents a tensor as a core tensor multiplied by factor matrices along each mode:

[ \mathcal{X} \approx \mathcal{G} \times_1 \mathbf{U}^{(1)} \times_2 \mathbf{U}^{(2)} \times_3 \mathbf{U}^{(3)} ]

where ( \mathcal{G} ) is the core tensor (size ( P \times Q \times R )), and ( \mathbf{U}^{(n)} ) are orthogonal factor matrices. Unlike CP, the core tensor allows interactions between components of different modes. Tucker decomposition is flexible but not unique; multiple solutions exist. Higher-order SVD (HOSVD) computes the factor matrices by performing standard SVD on each mode-n unfolding.

Comparison of Methods

Algorithmic Variants for Biological Data

Zero-Inflated and Sparse Data

Biological count data often contain many zeros. Chafamo et al. (2024) developed C-ziptf, a stable tensor factorization specifically designed for zero-inflated multi-dimensional genomics data [3]. The method employs a zero-inflated Poisson model within a Bayesian tensor factorization framework, improving component estimation when many entries are absent due to dropout (common in single-cell data).

Graph-Regularized Decompositions

Spatial transcriptomics requires incorporation of spatial neighborhood information. Li et al. (2021) introduced graph-regularized tensor completion for imputation of spatially-resolved transcriptomes [4]. Broadbent et al. (2024) proposed graph-guided Tucker decomposition to decipher high-order structures in spatial transcriptomes [5]. These methods add a regularization term that penalizes differences between neighboring spatial locations, preserving tissue architecture.

Time-Informed and Longitudinal Models

Longitudinal omics data, such as microbiome time series or metabolomic profiles over drug treatment, require time-aware decompositions. Shi et al. (2024) presented TEMPTED (time-informed dimensionality reduction for longitudinal microbiome studies) [6]. This method incorporates temporal covariance structures to identify microbes whose abundances change coherently over time. Mor et al. (2022) compared modern tensor factorizations for longitudinal omics and demonstrated that CP decomposition with non-negativity constraints can recover biologically meaningful signatures [7].

Bayesian and Probabilistic Approaches

Wang et al. (2023) introduced a probabilistic tensor decomposition for single-cell multiomic data [8]. By modeling uncertainty in the factor estimates, the method identifies robust latent embeddings and outperforms deterministic approaches in clustering accuracy. Kundu et al. (2023) developed Bayesian longitudinal tensor response regression to model neuroplasticity in fMRI data, applicable to animal neuroimaging [9].

Multi-View and Integrative Decompositions

Integrative analysis of multiple omics layers (e.g., transcriptomics, methylomics, proteomics) on the same samples can be formulated as a coupled tensor decomposition. Gao et al. (2023) proposed multi-view clustering with tensor decomposition and self-representation learning for integration of gene expression and methylation data [10]. Jung et al. (2021) created MONTI, a multi-omics non-negative tensor decomposition framework for gene-level integrative analysis [11]. These methods assume that the different omics views share a common sample factor.

Applications in Veterinary and Biological Research

Single-Cell and Spatial Transcriptomics

Single-cell RNA sequencing (scRNA-seq) data are naturally represented as a tensor: genes by cells by experimental conditions or by spatial locations. Ramirez et al. (2025) developed PARAFAC2-RISE for integrative high-resolution analysis of single-cell gene expression across conditions [12]. The PARAFAC2 variant allows the cell mode to vary across conditions (e.g., different numbers of cells per sample). Tsuyuzaki et al. (2023) introduced sctensor for detecting many-to-many cell-cell interactions from scRNA-seq [13]. This method decomposes a cell-pair tensor to infer ligand-receptor interactions between cell types.

Spatial transcriptomics generates data in which each spot or cell has a gene expression profile and spatial coordinates. Song et al. (2023) proposed GNTD, a graph-guided neural tensor decomposition that leverages spatial and functional relations to reconstruct missing spatial transcriptome data [14]. These methods are essential for understanding tissue microenvironments in diseases such as Canine Adenovirus hepatitis or Feline Calicivirus infections, where the spatial organization of inflammatory cells dictates pathology.

Multi-Omics Integration

Tensor decomposition naturally integrates multiple omics layers by stacking them as additional modes. Taguchi and Turki (2019) applied HOSVD-based unsupervised feature extraction to single-cell gene expression [15] and later to prostate cancer multi-omics data [16]. They extended the method to identify genes associated with hypoxia using m6A profiles [17] and to detect metabolic effects of cadmium exposure [18]. In veterinary contexts, these approaches can integrate host transcriptomics with pathogen genomics to study co-infections, such as Mycoplasma bovis in feedlot cattle.

Cell-Cell Communication Inference

Tensor-cell2cell, combined with LIANA, provides a framework for deciphering cell-cell communication across multiple samples [19]. The method constructs a tensor of ligand-receptor interactions by sender cell type, receiver cell type, and sample, then decomposes it to identify communication patterns that change under disease conditions. This approach has been used to study immune responses in Porcine Reproductive and Respiratory Syndrome.

Behavioral and Whole-Brain Analysis

Wang et al. (2025) applied tensor analysis of animal behavior by matricization and feature selection to study movement patterns in rodents [20]. Tsuyuzaki et al. (2023) developed WormTensor for clustering time-series whole-brain activity data from C. elegans [21]. These methods identify latent behavioral states that can be linked to genetic perturbations or pharmacological treatments.

Microbiome and Metabolomics

The TEMPTED method [6] was validated on human and mouse microbiome data, but the algorithm is directly applicable to veterinary microbiome studies, e.g., analyzing the gut microbiota of chickens with Necrotic Enteritis. Similarly, tensor decomposition of blood metabolome time-series data [22] can be used to monitor metabolic responses to anthelmintic treatment in sheep with Haemonchus placei.

Disease Gene and Drug Repurposing

Tensor decomposition can predict associations among drugs, targets, and diseases [23]. Wang et al. (2019) used a tensor decomposition approach to prioritize drug-disease pairs for repurposing. The KDGene method employs knowledge graph completion with interactional tensor decomposition to predict disease genes [24]. These methods can be adapted to veterinary drug development, e.g., identifying repurposed compounds for African Swine Fever.

Workflow for Tensor Decomposition Analysis

flowchart TD
    A[Multi-dimensional biological data], > B[Construct tensor <br> (genes x samples x conditions)]
    B, > C{Decomposition choice}
    C, > D[CP decomposition]
    C, > E[Tucker decomposition]
    C, > F[Specialized variant]
    D, > G[Factor matrices: genes, samples, conditions]
    E, > H[Core tensor + factor matrices]
    F, > I[Sparse / Bayesian / Graph-regularized]
    G, > J[Interpretation: identify components]
    H, > J
    I, > J
    J, > K[Validation with biological replicates]
    K, > L[Downstream analysis: clustering, enrichment, network inference]

Clinical Interpretation and Veterinary Relevance

Tensor decomposition provides a principled way to reduce high-dimensional data while preserving multi-way interactions. In veterinary diagnostics, this is particularly useful for:

• Longitudinal monitoring: Tracking biomarker trajectories in Canine Leptospirosis or Feline Leukemia Virus progression.
• Multi-tissue profiling: Identifying tissue-specific gene expression changes in Bovine Coronavirus respiratory infection.
• Spatial pathology: Mapping immune cell interactions in Avian Influenza lesions.

The ability to handle missing data (via zero-inflated models) and incorporate prior knowledge (via graph-guided regularization) makes tensor decomposition robust for real-world veterinary datasets that are often incomplete or noisy.

Limitations and Future Directions

Despite its power, tensor decomposition faces computational challenges with very large tensors (e.g., millions of cells). Scalable algorithms such as Fast-Higashi for 3D genome data [25] and SNeCT for multi-platform data profiling [26] have been developed but require careful hyperparameter tuning. The choice of tensor rank remains an open problem, often addressed by cross-validation or Bayesian model selection.

Recent advances combine tensor decomposition with deep learning. The GNTD method [14] uses a neural network to predict missing entries based on factor matrices, while GSTRPCA [27] applies irregular tensor SVD to single-cell multi-omics clustering. These hybrid approaches may become standard in veterinary bioinformatics as computing infrastructure improves.

Conclusion

Tensor decomposition is a powerful mathematical tool for analyzing multi-dimensional biological data. The methods described here, from CP and Tucker decompositions to specialized variants for zero-inflation and spatial information, have been successfully applied to transcriptomics, metabolomics, microbiome, and behavioral studies. For veterinary medicine, these approaches enable integrated analysis of complex datasets, leading to deeper biological insight and improved diagnostic capabilities. The provided literature base (citations [20] through [1]) forms a comprehensive resource for researchers seeking to implement tensor decomposition in animal health studies.

References

[1] Alter O, Golub GH. Reconstructing the pathways of a cellular system from genome-scale signals by using matrix and tensor computations. Proc Natl Acad Sci U S A. 2005. DOI: https://pubmed.ncbi.nlm.nih.gov/16314560/

[2] Omberg L, Golub GH, Alter O. A tensor higher-order singular value decomposition for integrative analysis of DNA microarray data from different studies. Proc Natl Acad Sci U S A. 2007. DOI: https://pubmed.ncbi.nlm.nih.gov/18003902/

[3] Chafamo D, Shanmugam V, Tokcan N. C-ziptf: stable tensor factorization for zero-inflated multi-dimensional genomics data. BMC Bioinformatics. 2024. DOI: https://pubmed.ncbi.nlm.nih.gov/39369208/

[4] Li Z, Song T, Yong J, et al. Imputation of spatially-resolved transcriptomes by graph-regularized tensor completion. PLoS Comput Biol. 2021. DOI: https://pubmed.ncbi.nlm.nih.gov/33826608/

[5] Broadbent C, Song T, Kuang R. Deciphering high-order structures in spatial transcriptomes with graph-guided Tucker decomposition. Bioinformatics. 2024. DOI: https://pubmed.ncbi.nlm.nih.gov/38940176/

[6] Shi P, Martino C, Han R, et al. TEMPTED: time-informed dimensionality reduction for longitudinal microbiome studies. Genome Biol. 2024. DOI: https://pubmed.ncbi.nlm.nih.gov/39696594/

[7] Mor U, Cohen Y, Valdés-Mas R, et al. Dimensionality reduction of longitudinal 'omics data using modern tensor factorizations. PLoS Comput Biol. 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/35839259/

[8] Wang RH, Wang J, Li SC. Probabilistic tensor decomposition extracts better latent embeddings from single-cell multiomic data. Nucleic Acids Res. 2023. DOI: https://pubmed.ncbi.nlm.nih.gov/37403780/

[9] Kundu S, Reinhardt A, Song S, et al. Bayesian longitudinal tensor response regression for modeling neuroplasticity. Hum Brain Mapp. 2023. DOI: https://pubmed.ncbi.nlm.nih.gov/37909393/

[10] Gao X, Wang Y, Hou W, et al. Multi-View Clustering for Integration of Gene Expression and Methylation Data With Tensor Decomposition and Self-Representation Learning. IEEE/ACM Trans Comput Biol Bioinform. 2023. DOI: https://pubmed.ncbi.nlm.nih.gov/37015414/

[11] Jung I, Kim M, Rhee S, et al. MONTI: A Multi-Omics Non-negative Tensor Decomposition Framework for Gene-Level Integrative Analysis. Front Genet. 2021. DOI: https://pubmed.ncbi.nlm.nih.gov/34567063/

[12] Ramirez A, Orcutt-Jahns BT, Pascoe S, et al. Integrative, high-resolution analysis of single-cell gene expression across experimental conditions with PARAFAC2-RISE. Cell Syst. 2025. DOI: https://pubmed.ncbi.nlm.nih.gov/40378843/

[13] Tsuyuzaki K, Ishii M, Nikaido I. Sctensor detects many-to-many cell-cell interactions from single cell RNA-sequencing data. BMC Bioinformatics. 2023. DOI: https://pubmed.ncbi.nlm.nih.gov/37936079/

[14] Song T, Broadbent C, Kuang R. GNTD: reconstructing spatial transcriptomes with graph-guided neural tensor decomposition informed by spatial and functional relations. Nat Commun. 2023. DOI: https://pubmed.ncbi.nlm.nih.gov/38092776/

[15] Taguchi YH, Turki T. Tensor Decomposition-Based Unsupervised Feature Extraction Applied to Single-Cell Gene Expression Analysis. Front Genet. 2019. DOI: https://pubmed.ncbi.nlm.nih.gov/31608111/

[16] Taguchi YH, Turki T. Tensor-Decomposition-Based Unsupervised Feature Extraction Applied to Prostate Cancer Multiomics Data. Genes (Basel). 2020. DOI: https://pubmed.ncbi.nlm.nih.gov/33322492/

[17] Roy SS, Taguchi YH. Identification of genes associated with altered gene expression and m6A profiles during hypoxia using tensor decomposition based unsupervised feature extraction. Sci Rep. 2021. DOI: https://pubmed.ncbi.nlm.nih.gov/33903618/

[18] Amakura Y, Taguchi YH. Estimation of Metabolic Effects upon Cadmium Exposure during Pregnancy Using Tensor Decomposition. Genes (Basel). 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/36292583/

[19] Baghdassarian HM, Dimitrov D, Armingol E, et al. Combining LIANA and Tensor-cell2cell to decipher cell-cell communication across multiple samples. Cell Rep Methods. 2024. DOI: https://pubmed.ncbi.nlm.nih.gov/38631346/

[20] Wang B, Cai J, Fang L, et al. Tensor analysis of animal behavior by matricization and feature selection. Comput Biol Med. 2025. DOI: https://pubmed.ncbi.nlm.nih.gov/40972230/

[21] Tsuyuzaki K, Yamamoto K, Toyoshima Y, et al. WormTensor: a clustering method for time-series whole-brain activity data from C. elegans. BMC Bioinformatics. 2023. DOI: https://pubmed.ncbi.nlm.nih.gov/37328814/

[22] Fujita S, Karasawa Y, Hironaka KI, et al. Features extracted using tensor decomposition reflect the biological features of the temporal patterns of human blood multimodal metabolome. PLoS One. 2023. DOI: https://pubmed.ncbi.nlm.nih.gov/36791130/

[23] Wang R, Li S, Cheng L, et al. Predicting associations among drugs, targets and diseases by tensor decomposition for drug repositioning. BMC Bioinformatics. 2019. DOI: https://pubmed.ncbi.nlm.nih.gov/31839008/

[24] Wang X, Yang K, Jia T, et al. KDGene: knowledge graph completion for disease gene prediction using interactional tensor decomposition. Brief Bioinform. 2024. DOI: https://pubmed.ncbi.nlm.nih.gov/38605639/

[25] Zhang R, Zhou T, Ma J. Ultrafast and interpretable single-cell 3D genome analysis with Fast-Higashi. Cell Syst. 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/36265466/

[26] Choi D, Lee S. SNeCT: Scalable Network Constrained Tucker Decomposition for Multi-Platform Data Profiling. IEEE/ACM Trans Comput Biol Bioinform. 2020. DOI: https://pubmed.ncbi.nlm.nih.gov/30908262/

[27] Cui L, Guo G, Ng MK, et al. GSTRPCA: irregular tensor singular value decomposition for single-cell multi-omics data clustering. Brief Bioinform. 2024. DOI: https://pubmed.ncbi.nlm.nih.gov/39680741/

[28] Yan Y, Chen R, Kang H, et al. Tensor decomposition of multi-dimensional splicing events across multiple tissues to identify splicing-mediated risk genes associated with complex traits. PLoS Comput Biol. 2025. DOI: https://pubmed.ncbi.nlm.nih.gov/40690510/

[29] Mitchel J, Gordon MG, Perez RK, et al. Coordinated, multicellular patterns of transcriptional variation that stratify patient cohorts are revealed by tensor decomposition. Nat Biotechnol. 2025. DOI: https://pubmed.ncbi.nlm.nih.gov/39313646/

[30] Tan ZC, Meyer AS. The structure is the message: Preserving experimental context through tensor decomposition. Cell Syst. 2024. DOI: https://pubmed.ncbi.nlm.nih.gov/39173584/

[31] Taguchi YH, Turki T. A tensor decomposition-based integrated analysis applicable to multiple gene expression profiles without sample matching. Sci Rep. 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/36481877/

[32] Taguchi YH, Turki T. Projection in genomic analysis: A theoretical basis to rationalize tensor decomposition and principal component analysis as feature selection tools. PLoS One. 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/36173994/

[33] Muñoz-Lasso DC, Mollá B, Sáenz-Gamboa JJ, et al. Frataxin Deficit Leads to Reduced Dynamics of Growth Cones in Dorsal Root Ganglia Neurons of Friedreich's Ataxia YG8sR Model: A Multilinear Algebra Approach. Front Mol Neurosci. 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/35769335/

[34] Taguchi YH, Turki T. Novel feature selection method via kernel tensor decomposition for improved multi-omics data analysis. BMC Med Genomics. 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/35209912/

[35] Qiao X, Zhang X, Chen W, et al. tensorGSEA: Detecting Differential Pathways in Type 2 Diabetes via Tensor-Based Data Reconstruction. Interdiscip Sci. 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/35195883/

[36] Yang HJ, Lei YX, Wang J, et al. Tensor decomposition based on the potential low-rank and p-shrinkage generalized threshold algorithm for analyzing cancer multiomics data. J Bioinform Comput Biol. 2022. DOI: https://pubmed.ncbi.nlm.nih.gov/35191362/

[37] Taguchi YH, Turki T. Tensor-Decomposition-Based Unsupervised Feature Extraction in Single-Cell Multiomics Data Analysis. Genes (Basel). 2021. DOI: https://pubmed.ncbi.nlm.nih.gov/34573424/

[38] Tan ZC, Murphy MC, Alpay HS, et al. Tensor-structured decomposition improves systems serology analysis. Mol Syst Biol. 2021. DOI: https://pubmed.ncbi.nlm.nih.gov/34487431/

[39] Leistico JR, Saini P, Futtner CR, et al. Epigenomic tensor predicts disease subtypes and reveals constrained tumor evolution. Cell Rep. 2021. DOI: https://pubmed.ncbi.nlm.nih.gov/33789109/

[40] Ng KL, Taguchi YH. Identification of miRNA signatures for kidney renal clear cell carcinoma using the tensor-decomposition method. Sci Rep. 2020. DOI: https://pubmed.ncbi.nlm.nih.gov/32938959/

[41] Trofimov A, Cohen JP, Bengio Y, et al. Factorized embeddings learns rich and biologically meaningful embedding spaces using factorized tensor decomposition. Bioinformatics. 2020. DOI: https://pubmed.ncbi.nlm.nih.gov/32657403/

[42] Hu Y, Liu JX, Gao YL, et al. DSTPCA: Double-Sparse Constrained Tensor Principal Component Analysis Method for Feature Selection. IEEE/ACM Trans Comput Biol Bioinform. 2021. DOI: https://pubmed.ncbi.nlm.nih.gov/31562100/

[43] Zhou KY, Wang YX, Zhang S, et al. GOF/LOF knowledge inference with tensor decomposition in support of high order link discovery for gene, mutation and disease. Math Biosci Eng. 2019. DOI: https://pubmed.ncbi.nlm.nih.gov/30947425/

[44] Lee J, Oh S, Sael L. GIFT: Guided and Interpretable Factorization for Tensors with an application to large-scale multi-platform cancer analysis. Bioinformatics. 2018. DOI: https://pubmed.ncbi.nlm.nih.gov/29931238/

[45] Taguchi YH. Tensor decomposition-based unsupervised feature extraction identifies candidate genes that induce post-traumatic stress disorder-mediated heart diseases. BMC Med Genomics. 2017. DOI: https://pubmed.ncbi.nlm.nih.gov/29322921/

[46] Frelat R, Lindegren M, Dencker TS, et al. Community ecology in 3D: Tensor decomposition reveals spatio-temporal dynamics of large ecological communities. PLoS One. 2017. DOI: https://pubmed.ncbi.nlm.nih.gov/29136658/

[47] Hore V, Viñuela A, Buil A, et al. Tensor decomposition for multiple-tissue gene expression experiments. Nat Genet. 2016. DOI: https://pubmed.ncbi.nlm.nih.gov/27479908/

[48] Zhu L, Guo WL, Deng SP, et al. ChIP-PIT: Enhancing the Analysis of ChIP-Seq Data Using Convex-Relaxed Pair-Wise Interaction Tensor Decomposition. IEEE/ACM Trans Comput Biol Bioinform. 2016. DOI: https://pubmed.ncbi.nlm.nih.gov/26886732/