Supervised Coupled Matrix-Tensor Factorization

• SCMTF is a low-rank framework that simultaneously decomposes temporal tensors and static matrices to identify interpretable clinical phenotypes.
• It integrates a supervised loss to directly guide latent factor extraction using clinical outcomes while handling missing data through explicit masking.
• Its application to ulcerative colitis demonstrates effective phenotype prediction with validated AUCs, highlighting potential for clinical stratification.

Supervised Coupled Matrix-Tensor Factorization (SCMTF) is a low-rank factorization framework designed to extract interpretable latent structures—termed phenotypes—from heterogeneous clinical data integrating temporal, static, and patient-reported variables. Its defining features are the simultaneous factorization (coupling) of a temporal tensor and a static matrix, explicit handling of missing data, and direct supervision by clinical outcomes, making it especially suited for phenotyping in domains with complex, multi-modal, and highly incomplete data such as patient-reported outcomes (PROs) in ulcerative colitis (Minoccheri et al., 24 Jun 2025).

1. Data Organization and Mathematical Formulation

SCMTF models datasets comprising $N$ patients, each described by both time-varying and static clinical features:

• $X \in \mathbb{R}^{N\times F \times T}$: Third-order tensor of temporal observations, combining $F$ temporal features (labs and PRO items) over $T$ time windows.
• $M \in \mathbb{R}^{N\times D}$: Matrix of $D$ static features per patient (e.g., demographics, medication history).
• $y \in \{0,1\}^N$: Binary outcome vector (e.g., medication persistence).

The unsupervised base is a coupled low-rank factorization:

• $X \approx [[w; A, B, C]]$, a rank-$r$ nonnegative CANDECOMP/PARAFAC (CP) decomposition.
• $M \approx [w; A, D]$, where the patient-mode factor $X \in \mathbb{R}^{N\times F \times T}$0 is shared.

Bias tensors $X \in \mathbb{R}^{N\times F \times T}$1 (size $X \in \mathbb{R}^{N\times F \times T}$2) and $X \in \mathbb{R}^{N\times F \times T}$3 (size $X \in \mathbb{R}^{N\times F \times T}$4) are introduced to account for feature- and patient-level offsets, ensuring that the factorized components model deviations—representing temporal phenotypes—rather than global shifts.

2. Supervised Extension

To direct the factor model towards predictive phenotypes, SCMTF integrates an explicit supervised loss. A two-layer neural classifier $X \in \mathbb{R}^{N\times F \times T}$5 maps each subject's phenotype vector $X \in \mathbb{R}^{N\times F \times T}$6 (row $X \in \mathbb{R}^{N\times F \times T}$7 of $X \in \mathbb{R}^{N\times F \times T}$8) to $X \in \mathbb{R}^{N\times F \times T}$9, an estimated probability for the clinical outcome.

The total loss optimized is:

$F$0

subject to $F$1.

Here,

• $F$2, $F$3: Binary masks for observed entries in $F$4, $F$5
• $F$6: $F$7 sparsity term
• $F$8: Cross-entropy (logistic) loss for outcome prediction
• $F$9: Balance between reconstruction and supervised loss
• $T$0: Sparsity penalty

3. Handling Missing Data

SCMTF is explicitly constructed to address missingness pervasive in PRO data:

• For the tensor $T$1, only observed entries (where $T$2) contribute to the squared error; unobserved data is ignored in both loss computation and gradient updates.
• Missing values in $T$3 are similarly handled via $T$4.
• This design allows for data imputation as a byproduct, as missing entries are completed by the fitted model.

A plausible implication is that this approach makes SCMTF applicable to high-missingness scenarios, such as longitudinal PRO data, without introducing bias from imputation preprocessors (Minoccheri et al., 24 Jun 2025).

4. Optimization and Implementation

SCMTF is optimized end-to-end in a deep learning environment (TensorLy + PyTorch):

• Variables $T$5: Updated via Adam with $T$6-proximal steps, enforcing nonnegativity by projection after each gradient step.
• Classifier parameters $T$7: Trained with stochastic gradient descent (SGD) with momentum.
• Hyperparameters (rank $T$8, loss balance $T$9, learning rate, and $M \in \mathbb{R}^{N\times D}$0 penalty $M \in \mathbb{R}^{N\times D}$1) are determined via grid search. Optimal settings in the reported work were $M \in \mathbb{R}^{N\times D}$2, $M \in \mathbb{R}^{N\times D}$3, learning rate $M \in \mathbb{R}^{N\times D}$4, and $M \in \mathbb{R}^{N\times D}$5.

5. Application to Ulcerative Colitis Phenotyping

In the published UC application, data included $M \in \mathbb{R}^{N\times D}$6 patients with:

• $M \in \mathbb{R}^{N\times D}$7 ($M \in \mathbb{R}^{N\times D}$8): Static features (age, sex, disease duration/location, endoscopic severity, medication status).
• $M \in \mathbb{R}^{N\times D}$9 ($D$0): Temporal features, such as 4-month binned labs (CRP, calprotectin, Hgb, WBC) and patient-reported Likert-scale surveys across 7 windows centered on medication change.

Phenotypes are defined by the columns of factor matrices $D$1:

• $D$2: Patient $D$3’s membership in phenotype $D$4.
• $D$5: Loading of temporal feature $D$6 for $D$7.
• $D$8: Time profile across the 7 windows for $D$9.
• $y \in \{0,1\}^N$0: Static-feature contribution to $y \in \{0,1\}^N$1.

Downstream prediction of medication persistence used the $y \in \{0,1\}^N$2 factors as inputs:

• Neural network classifier inside SCMTF achieved test set AUC $y \in \{0,1\}^N$3.
• Random Forest on $y \in \{0,1\}^N$4 achieved AUC $y \in \{0,1\}^N$5 at $y \in \{0,1\}^N$6 months, $y \in \{0,1\}^N$7 at $y \in \{0,1\}^N$8 months.

Three phenotypes—indexed $y \in \{0,1\}^N$9, $X \approx [[w; A, B, C]]$0, and $X \approx [[w; A, B, C]]$1—were most predictive:

• 24 (“chronic arthritis-pain-lab”): Persistently high pain, bloating, arthritis, elevated albumin.
• 16 (“episodic emotional-and-arthritis symptoms”): Fluctuating arthritis, anger, worry, CRP/WBC—relevant for $X \approx [[w; A, B, C]]$2-month prediction.
• 27 (“hemoglobin-driven”): Variable Hgb and arthritis—most relevant at $X \approx [[w; A, B, C]]$3 months.

This suggests that PRO-based and mixed temporal phenotypes, previously excluded due to data challenges, can be leveraged for clinically-relevant stratification (Minoccheri et al., 24 Jun 2025).

6. Interpretability and Domain Significance

SCMTF produces sparse, nonnegative, and interpretable phenotypes by construction:

• Each phenotype connects static and temporal clinical features, with explicit temporal profiles.
• Phenotype-specific patterns (e.g., symptom trajectories, biomarker dynamics) are recoverable and clinically characterizable.
• The use of nonnegativity and sparsity encourages each phenotype to capture distinct clinical presentations.

A plausible implication is that such structure facilitates downstream clinical interpretation and discovery of actionable patient subgroups, especially where symptom data has previously been omitted (Minoccheri et al., 24 Jun 2025).

7. Distinguishing Features and Research Impact

SCMTF is the first tensor-based approach to be both supervised and coupled, with the inaugural application to ulcerative colitis and patient-reported outcomes. By jointly factorizing static and temporal modalities, effectively handling missing values, and incorporating supervision via a flexible neural classifier, SCMTF advances computational phenotyping in settings with heterogeneous and incomplete clinical data. Its empirical performance in medication-persistence prediction and clinical interpretability demonstrates the viability of low-rank matrix and tensor factorization in domains and data regimes previously considered inaccessible (Minoccheri et al., 24 Jun 2025).