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Multi-view biclustering via non-negative matrix tri-factorisation

This paper proposes ResNMTF, a novel multi-view biclustering method based on non-negative matrix tri-factorisation that automatically discovers overlapping and non-exhaustive biclusters across diverse views without requiring prior knowledge of cluster numbers, while also introducing the bisilhouette score as a new intrinsic metric for evaluation and hyperparameter tuning.

Original authors: Ella S. C. Orme, Theodoulos Rodosthenous, Marina Evangelou

Published 2026-03-13

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CC BY 4.0

Original authors: Ella S. C. Orme, Theodoulos Rodosthenous, Marina Evangelou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a massive mystery involving thousands of suspects (rows) and thousands of clues (columns). But here's the twist: you aren't just looking at one list of clues. You have multiple notebooks (views) describing the same group of people, but each notebook was written by a different witness with a different perspective. One notebook might list their DNA, another their shopping habits, and a third their social media posts.

This is the world of Multi-View Data. The problem is that not every clue in every notebook is useful for every suspect. Some clues only matter for a specific group of people. Your goal is to find these hidden groups (clusters) and the specific clues that define them, all at the same time. This is called Biclustering.

The paper you shared introduces two main things to help detectives like us: a new, smarter detective tool called ResNMTF, and a new way to grade the detective's work called the Bisilhouette Score.

Here is the breakdown in simple terms:

1. The Problem: Too Much Noise, Too Many Perspectives

In the past, if you wanted to find groups in data, you might just mash all the notebooks together into one giant pile. But that's messy. If one notebook is full of scribbles (noise), it ruins the whole pile. Or, if you look at each notebook separately, you miss the big picture where the groups overlap.

Existing tools were like rigid rulebooks:

They often demanded you know exactly how many groups you were looking for before you started (like guessing the number of suspects before finding them).
They struggled if the groups overlapped (a suspect belonging to two gangs) or if some suspects didn't belong to any gang at all.
They couldn't easily handle the situation where Notebook A and Notebook B share the same people but different clues, while Notebook C shares different people but the same clues.

2. The Solution: ResNMTF (The Flexible Detective)

The authors created a new method called ResNMTF (Restrictive Non-negative Matrix Tri-Factorisation). Think of this as a super-smart, flexible detective agent.

The "Tri-Factor" Magic: Imagine you are trying to organize a messy room. Instead of just sorting items, you break the problem down into three parts:
1. Who is in the room (Rows/People).
2. What items are in the room (Columns/Clues).
3. How they fit together (The hidden pattern).
  ResNMTF uses math to separate these three layers simultaneously.
The "Restrictive" Superpower: This is the cool part. ResNMTF lets you tell the detective: "Hey, Notebook 1 and Notebook 2 are talking about the same people, so treat their 'Who' lists as similar." Or, "Notebook 1 and Notebook 3 are talking about the same clues, so treat their 'What' lists as similar."
It's like having a detective who can say, "Okay, I know these two witnesses saw the same car, so I'll link their notes, but I know this third witness saw a different car, so I'll keep those notes separate." It handles any mix of shared people or shared clues without getting confused.
No Crystal Ball Needed: Unlike older tools, ResNMTF doesn't need you to guess how many groups exist. It tries different numbers, checks which one makes the most sense, and even has a "garbage collector" feature. If it finds a group that looks like random noise (a false lead), it throws it out automatically.

3. The Grading System: The Bisilhouette Score

Once the detective finds the groups, how do you know if they did a good job?

The Old Way (Silhouette Score): Imagine grading a class. The old method asked, "How close are the students sitting together?" and "How far are they from the next group?" But this broke down if students sat in two groups at once (overlap) or if some students sat alone (non-exhaustive).
The New Way (Bisilhouette Score): The authors invented a new grading system specifically for this messy, overlapping world.
- It looks at a group of people only using the clues that define that specific group.
- It asks: "Are these people tightly packed together based on these specific clues, and are they clearly different from everyone else?"
- The Score: It gives a score between -1 and 1. A high score means the detective found tight, clear groups. A low score means the groups are messy or fake.

Why is this score a game-changer?
It acts as a compass. Since you don't know the "true" answer in real life (unsupervised learning), you can't use a standard answer key. The Bisilhouette Score tells you, "Hey, if you change this setting, the groups get messier. If you change it this way, they get tighter." It helps you tune the detective's settings to get the best result without needing to know the answer beforehand.

4. How It Works in Real Life

The authors tested this on two types of cases:

Fake Cases (Synthetic Data): They created fake data where they knew the answer. ResNMTF found the groups perfectly, even when the data was noisy or the groups overlapped. It beat other famous detective tools (like GFA and iSSVD) consistently.
Real Cases:
- News Articles: Grouping articles from BBC, Reuters, and The Guardian. ResNMTF found that articles about "Sports" were grouped together across all three newspapers, even though the words used were slightly different.
- Medical Data (Cancer & Cells): They looked at patient data (genes, proteins, etc.). ResNMTF successfully identified different types of cancer cells and the specific genes that defined them, even when some patients had mixed signals or missing data.

The Bottom Line

This paper gives us a new, flexible tool (ResNMTF) to find hidden patterns in complex, multi-source data, and a new ruler (Bisilhouette Score) to measure how good those patterns are.

The Analogy:
If finding patterns in data is like organizing a chaotic library where books are written in different languages and about different topics:

Old tools tried to force all books into one shelf or asked you to guess how many shelves you needed.
ResNMTF is a librarian who can sort books by author and topic simultaneously, realizing that some authors write for different sections, and some topics appear in different languages.
The Bisilhouette Score is the librarian's checklist to ensure the shelves are neat and the books actually belong there, without needing a manager to tell them what the final arrangement should look like.

This makes it much easier for scientists and data analysts to discover hidden truths in complex data without needing a crystal ball.

1. Problem Statement

The paper addresses the challenges of multi-view biclustering (also known as co-clustering), which involves simultaneously clustering rows and columns of data matrices across multiple data sources (views).

Context: Modern datasets often consist of multiple views describing the same objects (e.g., genomics, proteomics, and methylomics data for the same patients).
Challenges:
- Non-exhaustivity and Non-exclusivity: Rows and columns may not belong to any cluster or may belong to multiple clusters (overlapping).
- View Heterogeneity: Different views may share different subsets of rows (samples) or columns (features). Some views may share rows but have unique features, while others share features but have unique samples.
- Lack of Ground Truth: In unsupervised settings, the true number of biclusters ( $K$ ) is unknown.
- Evaluation Gap: There is a lack of suitable intrinsic measures (internal validation metrics) specifically designed for biclusters that account for their structural complexity (compactness and separation) while handling overlap and non-exhaustivity. Existing methods often rely on extrinsic measures (requiring ground truth) or treat row/column clustering separately, which fails to capture the joint structure.

2. Methodology

The authors propose two main innovations: a new algorithm (ResNMTF) and a new evaluation metric (Bisilhouette Score).

A. ResNMTF (Restrictive Non-negative Matrix Tri-Factorisation)

ResNMTF is a multi-view biclustering algorithm based on Non-negative Matrix Tri-Factorisation (NMTF).

Mathematical Formulation:
For $n_v$ $n_{v}$ views, the data matrix $X^{(v)}$ $X^{(v)}$ is approximated as $X^{(v)} \approx F^{(v)} S^{(v)} (G^{(v)})^T$ $X^{(v)} \approx F^{(v)} S^{(v)} (G^{(v)})^{T}$ .
- $F^{(v)}$ : Row cluster indicators.
- $G^{(v)}$ : Column cluster indicators.
- $S^{(v)}$ : Interaction matrix between row and column clusters.
Objective Function:
The method minimizes the reconstruction error across all views while enforcing regularization between views to share information:
$\min \sum_{v} \| X^{(v)} - F^{(v)} S^{(v)} (G^{(v)})^T \|^2 + \sum_{v<w} (\phi_{vw} \|F^{(v)} - F^{(w)}\|^2 + \xi_{vw} \|S^{(v)} - S^{(w)}\|^2 + \psi_{vw} \|G^{(v)} - G^{(w)}\|^2)$
- Flexibility: The tuning parameters ( $\phi, \xi, \psi$ ) allow the user to specify exactly which dimensions (rows, columns, or both) should be shared across views. This handles complex scenarios where View 1 and 2 share rows, but View 1 and 3 share columns.
Optimization:
- Uses multiplicative update rules (similar to Lee & Seung) to solve the non-convex problem.
- Includes a specific initialization strategy based on Singular Value Decomposition (SVD) to ensure non-negativity and proper scaling.
Post-Processing:
- Spurious Bicluster Removal: Uses a resampling approach with Jensen-Shannon Divergence (JSD) to distinguish true signal from noise. If a bicluster's factor distribution is indistinguishable from shuffled (noise) data, it is removed.
- Stability Analysis: Uses sub-sampling to assess the stability of biclusters. Unstable biclusters (those that do not persist across subsamples) are removed.

B. Bisilhouette Score

To address the lack of intrinsic measures, the authors extend the standard Silhouette Score to the bicluster context.

Concept: Instead of calculating the silhouette for all data points in a global cluster, the score is calculated for a specific bicluster $(R_k, C_k)$ $(R_{k}, C_{k})$ by:
1. Subsetting the data matrix to only include columns $C_k$ .
2. Calculating the silhouette coefficient for rows in $R_k$ based on this subset.
3. Averaging these coefficients to get a score for the bicluster.
Properties:
- Handles overlap and non-exhaustivity (rows/columns can belong to multiple or no clusters).
- Evaluates both compactness (within-cluster distance) and separation (distance to nearest other cluster).
- Can be used to determine the optimal number of biclusters ( $K$ ) by maximizing the score.

3. Key Contributions

ResNMTF Algorithm: A novel multi-view biclustering method that offers unprecedented flexibility in defining relationships between views (shared rows, shared columns, or both) without requiring pre-specified cluster counts.
Bisilhouette Score: The first intrinsic measure specifically designed for biclustering that accounts for the joint structure of rows and columns, enabling unsupervised model selection and hyperparameter tuning.
Stability Framework: A robust procedure combining spurious bicluster removal (via JSD) and stability analysis (via sub-sampling) to ensure only reliable, signal-driven biclusters are returned.
Comprehensive Evaluation: Extensive testing on synthetic and real-world multi-omics and text datasets, demonstrating superiority over state-of-the-art methods like iSSVD and GFA.

4. Results

The performance was evaluated on synthetic data and four real-world datasets: 3Sources (news articles), BBCSport (sports news), A549 (single-cell multi-omics), and TCGA (cancer genomics).

Comparison with Competitors:
- ResNMTF consistently outperformed iSSVD and GFA (Group Factor Analysis) in terms of F-score and Relevance on most datasets.
- ResNMTF generally outperformed the single-view NMTF (unrestricted), proving the benefit of multi-view integration.
- iSSVD struggled with default parameters and required significant signal strength to detect biclusters. GFA was robust to noise but less sensitive to varying cluster numbers.
Handling Complexity:
- ResNMTF successfully identified overlapping and non-exhaustive biclusters.
- Performance remained stable even with varying noise levels and numbers of views.
Bisilhouette Score Efficacy:
- The Bisilhouette score showed a very high correlation (Pearson $r \approx 0.94$ ) with the F-score (extrinsic measure) in synthetic experiments, correctly identifying the best performing method in 100% of scenarios.
- It proved effective for tuning hyperparameters (specifically the restriction parameters) in unsupervised settings, though it tended to favor removing weaker biclusters to maximize the average score.
- It served as a useful visualization tool to inspect the quality of individual biclusters.

5. Significance

Bridging the Evaluation Gap: The introduction of the Bisilhouette Score fills a critical void in the literature, providing a standard internal metric for biclustering that does not rely on ground truth labels. This is essential for applying biclustering to real-world biological and social science data where labels are often unavailable.
Flexible Multi-View Integration: Unlike previous methods that enforce strict consensus (all views share the same structure) or treat views independently, ResNMTF allows for partial sharing of structure. This is crucial for realistic scenarios (e.g., multi-omics studies where different assays measure different features on the same patients).
Robustness: The combination of spurious removal and stability analysis ensures that the results are not artifacts of noise, making the method reliable for exploratory data analysis.
Practical Utility: The authors provide open-source R packages (resnmtf and bisilhouette), facilitating the adoption of these methods by the broader research community.

In conclusion, the paper presents a mathematically rigorous and practically flexible framework for multi-view biclustering, solving key issues related to model selection, view integration, and result validation in unsupervised learning.