Chapter map

1. 1. Why shared and specific programmes coexist
2. 2. The non-negative matrix idea
3. 3. Integrative NMF
4. 4. Shared metagenes
5. 5. Dataset-specific metagenes
6. 6. A small factorisation
7. 7. Choosing the factor rank
8. 8. The dataset-specific penalty
9. 9. Alternating non-negative optimisation
10. 10. Factor loading geometry
11. 11. Quantile normalisation
12. 12. Joint clustering and interpretation
13. 13. Cross-species and multi-omic mapping
14. 14. Rare states and real differences
15. 15. A student parts-based project
16. 16. Governance and careers

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

---

1. Why shared and specific programmes coexist

The central mathematical idea is concrete. Different datasets can contain the same cell identities and also platform, species or modality-specific signals. One model should not confuse sharing with sameness.

A concrete situation

RNA and chromatin measurements may express related cell types through different features.

What the mathematics does not license

A shared factor does not imply identical molecular mechanisms. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

List expected shared and dataset-specific structures. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

2. The non-negative matrix idea

Here the geometry becomes operational. NMF approximates a non-negative data matrix with non-negative cell loadings times non-negative feature factors. Components add rather than cancel.

A concrete situation

A cell can combine neuronal and activity programmes with positive weights.

What the mathematics does not license

NMF solutions can depend on scale, rank and initialisation. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Multiply a two-factor loading row by a factor matrix. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

3. Integrative NMF

This step deserves a careful pause. LIGER extends factorisation across datasets with shared metagenes and dataset-specific contributions. Cell loadings locate each cell in factor space.

A concrete situation

Two species can share a neuronal factor and retain species-specific expression.

What the mathematics does not license

The decomposition is fitted, not uniquely discovered. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Write dimensions for cells, genes, factors and datasets. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

4. Shared metagenes

The useful question is not only how to calculate, but what the calculation assumes. Shared factor weights identify feature groups contributing similarly across datasets under the model.

A concrete situation

A synaptic programme can recur across experiments.

What the mathematics does not license

Correlated genes can exchange weights across fits. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Bootstrap factor subspaces and feature groups. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

5. Dataset-specific metagenes

A small example makes the mechanism visible. Specific factors describe extra feature patterns needed by one dataset. A tuning penalty controls their influence relative to sharing.

A concrete situation

A sequencing chemistry may create a dataset-specific programme.

What the mathematics does not license

Specific does not mean artefact; it may be genuine biology. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Relate specific loadings to technical and biological metadata. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

6. A small factorisation

This is where modelling choices begin to affect evidence. Let two cells load on two factors with H=[[2,0],[1,1]] and simple non-negative metagenes. The product reconstructs additive feature profiles.

A concrete situation

The second cell combines both programmes.

What the mathematics does not license

Real algorithms minimise reconstruction with constraints and penalties. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Calculate the matrix product and residuals. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

7. Choosing the factor rank

The central mathematical idea is concrete. Too few factors merge distinct identities; too many split noise or duplicate programmes. Rank needs stability and held-out evidence.

A concrete situation

One neuronal type may split only at high rank.

What the mathematics does not license

Training reconstruction always improves with flexibility. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot held-out error and factor stability across ranks. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

8. The dataset-specific penalty

Here the geometry becomes operational. The penalty balances shared and specific structure. Strong sharing can overalign; weak sharing can leave datasets separated.

A concrete situation

A cross-species analysis may need more specific structure than two technical replicates.

What the mathematics does not license

There is no universally neutral penalty. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Map biological preservation against mixing across a grid. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

9. Alternating non-negative optimisation

This step deserves a careful pause. Factor blocks and cell loadings are updated iteratively under non-negativity. Convergence and several starts should be recorded.

A concrete situation

Different starts can exchange or split factors.

What the mathematics does not license

One converged solution is not proof of global optimality. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare objectives and aligned factor spaces across seeds. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

10. Factor loading geometry

The useful question is not only how to calculate, but what the calculation assumes. Cells close in the non-negative loading space share mixtures of programmes. Dominant factors support tentative joint clustering.

A concrete situation

Two datasets can place homologous neurons near one another.

What the mathematics does not license

Distance depends on normalisation and factor scaling. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Normalise three loading vectors and compare cosine similarity. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

11. Quantile normalisation

A small example makes the mechanism visible. LIGER aligns factor-loading distributions across datasets for comparable cells, helping shared types cluster together.

A concrete situation

Factor quantiles for one cell type can be matched across modalities.

What the mathematics does not license

Different composition can make quantiles incomparable. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect factor distributions before and after alignment. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

12. Joint clustering and interpretation

This is where modelling choices begin to affect evidence. Clusters are built in aligned factor space, while shared and specific metagenes explain their programmes.

A concrete situation

A joint cluster can include cells from RNA and chromatin assays.

What the mathematics does not license

Cluster agreement does not establish direct one-to-one molecular equivalence. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Validate clusters with independent markers and donor support. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

13. Cross-species and multi-omic mapping

The central mathematical idea is concrete. Linked features can connect species or modalities, but mapping rules determine the scientific meaning.

A concrete situation

Orthologs can connect mouse and human genes.

What the mathematics does not license

One-to-many mappings can distort weights. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit mapping coverage and ambiguous features. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

14. Rare states and real differences

Here the geometry becomes operational. Rare or dataset-specific cells may have weak shared-factor support. Preserve them with uncertainty.

A concrete situation

A disease-only state should not be forced into a healthy cluster.

What the mathematics does not license

Overnormalisation can remove the very difference under study. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Review cells with large specific-factor loadings. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

15. A student parts-based project

This step deserves a careful pause. Factorise two tiny non-negative matrices with one shared pattern and one local pattern.

A concrete situation

Known factors make reconstruction and ambiguity visible.

What the mathematics does not license

Small examples omit count noise and high dimensionality. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Change rank and penalty, then explain the trade-off. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

16. Governance and careers

The useful question is not only how to calculate, but what the calculation assumes. Archive mappings, normalisation, rank, penalty, seeds, factors, alignment and clusters. Integrative factorisation combines algebra, optimisation and domain science.

A concrete situation

Students can begin with multiplication and residuals.

What the mathematics does not license

Factors guarantee no cell identity, mechanism or career outcome. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Create a factor ledger with evidence and limits. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

A LIGER audit starts with several single-cell datasets whose cells need not be the same, but whose features can be linked. Integrative non-negative matrix factorisation represents each dataset through non-negative cell loadings, shared metagenes and dataset-specific metagenes. The decomposition is designed to expose common cell identities while preserving meaningful differences.

Save feature mappings, normalisation, factor count, dataset-specific penalty, initialisations, convergence, factor matrices and quantile-normalisation choices. Non-negativity makes parts-based interpretation easier, but factors can remain non-unique and correlated. Stability across starts and held-out donors matters more than a single clean factor plot.

Evaluate shared cell types, dataset-specific biology, neighbourhoods, markers and rare populations. A shared factor is an association pattern, not a causal pathway. Quantile normalisation can align factor-load distributions across datasets, so it must be checked for overcorrection when composition or biology truly differs.

---

Frequently asked questions

What does LIGER factorise?

It factorises each dataset into non-negative cell loadings and metagene factors, with shared and dataset-specific feature patterns.

Why non-negative factors?

Non-negativity creates additive, parts-based components that are often easier to interpret in count-like biological data.

What is a shared metagene?

It is a non-negative feature-weight pattern used across datasets to represent a recurring programme.

What is a dataset-specific factor?

It captures feature structure associated with one dataset beyond the shared metagene pattern.

Why use quantile normalisation?

It aligns factor-loading distributions for comparable cell populations across datasets before joint clustering.

How should LIGER be validated?

Use several starts, factor sensitivity, donor holdouts, marker and neighbourhood checks, preserved dataset-specific biology and rare-state review.

---

Primary source and fact-check note

Joshua D. Welch, Velina Kozareva, Ashley Ferreira et al., “Single-Cell Multi-omic Integration Compares and Contrasts Features of Brain Cell Identity,” Cell 177(7), 1873–1887.e17 (2019), DOI 10.1016/j.cell.2019.05.006. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• MOFA multi-omics factor analysis
• JIVE multiblock integration
• Harmony single-cell integration

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: validation, transfer and responsible use

Factor ledger

Save rank, penalty, start, objective, shared metagenes, specific metagenes and cell loadings. Align factors across starts before comparing them.

Non-negativity check

Confirm all constrained matrices are non-negative within tolerance. Negative values can indicate a preprocessing or implementation mismatch.

Scale audit

Different dataset sizes and count scales can dominate reconstruction. Record normalisation and dataset weights. Repeat with balanced contributions where scientifically justified.

Rank stability

Compare reconstruction, neighbourhoods and metagene subspaces across ranks. A higher rank that splits one stable factor into copies may not add biology.

Penalty frontier

Plot shared-cell mixing against preservation of dataset-specific markers over a penalty grid. Choose a defensible region, not the prettiest embedding.

Initialisation robustness

Run several starts. Compare objectives, cell neighbourhoods and factor subspaces. Archive competing solutions when they are nearly tied.

Quantile audit

Inspect distributions and cell composition before and after quantile normalisation. Do not align quantiles for populations that lack a valid counterpart.

Specific-factor interpretation

Relate dataset-specific metagenes to technical and biological metadata. Preserve neutral labels until direct evidence distinguishes them.

Cross-modality mapping

Document gene-activity or feature-linking rules. Shared factors are only as meaningful as the cross-modality map.

Rare-state guard

Report cells dominated by specific factors or weakly assigned to shared clusters. They may represent discoveries, artefacts or unsupported regions.

Reconstruction audit

Rebuild one dataset from archived H, shared and specific factors. Compare residuals by gene and cell type.

Reader checkpoint

A trustworthy LIGER claim states feature mapping, rank, penalty, starts, factors, quantile alignment, validation and preserved specific biology.

Deep dive: how to stress-test the mathematical claim

Deep dive 1

Compare shared-factor reconstruction with dataset-specific reconstruction. A model that calls everything shared may have an overly strong penalty.

Deep dive 2

Rotate or rescale equivalent factorisations where possible to understand identifiability. Interpret stable groups, not arbitrary column order.

Deep dive 3

Inspect cells with high dataset-specific loadings. They define where integration is weakest or where unique biology is strongest.

Deep dive 4

Use donor-held-out clustering and factor scores. Random cell splits leave near-duplicate donor biology in training.

Deep dive 5

Test feature mappings by removing ambiguous orthologs or activity links. Shared factors should not depend entirely on uncertain links.

Deep dive 6

Measure exact-variable and pathway stability separately. Correlated variables can substitute without changing the programme.

Deep dive 7

Compare joint clusters before and after quantile normalisation. Large changes need biological explanation.

Deep dive 8

Preserve raw modality-specific measurements for downstream inference. Factor alignment is not measurement conversion.

Deep dive 9

Include a no-integration baseline. Sometimes dataset-specific analysis followed by comparison is more honest.

Deep dive 10

Distinguish shared covariance, cell identity, regulation and causation in the final language.

Worked evidence notebook

Matrix dimensions

For cells by genes X_i, write cell-load H_i and factor-by-gene shared and specific matrices so H_i(W+V_i) matches X_i.

Non-negative product

Multiply a small non-negative H by two metagene rows and inspect additive reconstruction.

Residual check

Subtract reconstructed from observed values and summarise residuals by dataset.

Rank table

Record rank, objective, held-out reconstruction and factor stability.

Quantile table

Compare factor quantiles for validated shared populations before and after alignment.

Factor evidence

For each factor, list shared genes, specific genes, markers, donor support and unresolved alternatives.

Practice, misconception and transfer

A parent-friendly starting point

Use non-negative multiplication to show how a cell combines additive programmes. Colours mixed from non-negative amounts provide an intuitive analogy.

A key misconception

Shared factors do not mean every dataset is identical. Specific factors are part of the model and can carry important biology.

When to pause

Pause when feature links across modalities or species are uncertain. Factorisation cannot repair a false mapping.

A useful extension

Compare two ranks and show how a stable factor can split. This teaches that model complexity changes the story.

Career transfer

NMF supports topics from image parts to document themes and molecular programmes, always with domain-specific validation.

Final learning move

Have a second person reconstruct one cell from archived factors and explain the residual.

A final worked scenario

A final factor scenario

RNA and chromatin datasets are linked through validated genes. LIGER finds shared neuronal factors and modality-specific factors. Several starts recover the same shared subspace, while one rare population is dominated by a specific factor and remains separate. Quantile alignment improves joint clustering for common types without erasing the rare state. The report includes rank and penalty sensitivity, raw markers and reconstruction. The mathematics is useful because it separates additive programmes that recur from programmes that belong to one modality, while leaving room for uncertainty and experiment.

Mathematical workshop: mechanisms, evidence and limits

Interpreting residuals rather than hiding them

Residuals are not merely leftover error. They show where the chosen shared and specific factors fail to reconstruct measurements. Summarise residual magnitude by gene, cell state, donor and modality, then inspect coherent patterns. A technical group of genes may suggest missing normalisation, while a stable biological programme may suggest insufficient rank or an omitted dataset-specific factor. Compare residuals on held-out donors so that flexibility is not mistaken for generalisation. When residual structure remains, report it as a boundary of the model and preserve it for follow-up rather than smoothing it away.

The additive programme idea

LIGER begins with a useful picture: each cell is built from non-negative amounts of latent programmes. A loading of zero means the programme contributes nothing; a larger loading means a larger additive contribution. Shared metagenes describe programmes available across datasets, while dataset-specific metagenes allow one technology or condition to express a different pattern. The model is therefore neither a demand for sameness nor a separate analysis of every dataset. It creates a controlled compromise. Students can read the matrices as a ledger of which programmes exist and how strongly each cell uses them.

A small reconstruction by hand

Take one cell with factor loadings H=(2,1). Let the shared metagenes for two genes be rows (1,0.5) and (0.2,1), and let the dataset-specific additions be (0.1,0) and (0,0.3). The reconstructed cell is 2(1.1,0.5)+1(0.2,1.3)=(2.4,2.3). Comparing this with observed values shows a residual for every gene. Repeating the calculation for several cells reveals whether error concentrates in one dataset or population. This hand calculation turns factorisation into an auditable prediction rather than an abstract decomposition.

Rank is a scientific choice

Rank controls how many programmes the model may use. Too small a rank merges distinct states and pushes their differences into residuals. Too large a rank can split one coherent programme into several unstable factors or fit noise. Reconstruction error normally improves as rank rises, so error alone cannot decide. A stronger selection table includes donor-held-out reconstruction, neighbourhood stability, metagene reproducibility across starts and interpretability against independent markers. The chosen rank should sit in a region where the scientific conclusion remains stable, not at a single mathematically convenient point.

Penalty and the shared-specific balance

The regularisation penalty influences how strongly the datasets are encouraged to use shared structure. If it is too weak, matching biology may remain separated. If it is too strong, real modality-specific or condition-specific programmes can be absorbed into an artificial consensus. A penalty sweep should therefore track two goals at once: mixing for known shared populations and preservation of known specific markers. Plotting these quantities exposes a frontier rather than a magical best number. The final choice can then be justified by the intended question and by sensitivity of the substantive conclusion.

Quantile normalisation needs matched populations

Quantile normalisation aligns distributions of factor loadings across datasets, but equal quantiles are meaningful only for populations that should correspond. Aligning a disease-only state with the nearest healthy state can erase the very difference under study. A careful workflow first uses factor and neighbourhood evidence to define supported shared groups, then aligns within those groups and leaves unsupported cells visible. Reports should show loading distributions before and after alignment, the cells included in each group and how conclusions change without normalisation. Alignment is a conditional operation, not a universal cleanup step.

Why starts and subspaces matter

Non-negative matrix factorisation is usually solved by iterative optimisation and can reach different local solutions from different starts. Factor columns may also swap order, so comparing column one with column one is unreliable. Align factors by their metagene similarity or compare the subspaces and cell neighbourhoods they span. When several starts have nearly equal objectives but different biological stories, archive them and narrow the conclusion. Stability across starts, donors and nearby ranks is more persuasive than the lowest objective from one lucky run.

A cross-modality caution

In RNA and chromatin integration, the model still needs a defensible way to connect features, such as gene activity linked to genes. That mapping contains biological assumptions about distance, regulation and measurement. A shared factor cannot rescue an incorrect link. Sensitivity tests can remove ambiguous links, compare alternative activity definitions and ask whether the shared programme persists. Downstream interpretation should distinguish a factor that predicts co-variation from a claim about regulatory mechanism. The latter needs targeted experiments or stronger causal evidence.

A useful final report

A strong LIGER report states dataset sizes, donor structure, feature map, normalisation, rank, penalty, starts, convergence, shared factors, specific factors and quantile groups. It reconstructs held-out data, checks rare states and preserves raw measurements for later inference. A careful conclusion might say that a shared neuronal programme recurred across modalities while an immune programme remained modality-specific and robust to nearby settings. This language makes the mathematical result precise without claiming unique factor identity, automatic mechanism or universal transfer to new datasets.

Method checklist and conclusion boundary

Reproducibility checklist

Freeze mappings, normalisation, rank, penalty, starts, shared and specific factors, quantile alignment and clusters. Compare subspaces across starts and donors; reconstruct one dataset from archived matrices.

Careful conclusion

A careful conclusion says that factor programmes recurred across named datasets while specific programmes remained. It does not claim unique factorisation, identical biology or causation.

Final evidence checks

Final evidence checks 1

Preserve shared and specific factors.

Final evidence checks 2

Report rank and penalty sensitivity.

Final evidence checks 3

Align factors before comparing starts.

Final evidence checks 4

Audit quantile normalisation by population.

Final evidence checks 5

Keep raw modalities for inference.

Final evidence checks 6

Require one reconstruction by a second analyst.