Chapter map

1. 1. Why learn the comparison itself
2. 2. Reference quality control
3. 3. Scaling and log transformation
4. 4. Rare-class exclusion
5. 5. M3Drop feature selection
6. 6. Similarity constraints
7. 7. Transformation matrix
8. 8. Discriminative component analysis
9. 9. Bootstrap stability
10. 10. Transformed centroids
11. 11. Pearson similarity after transformation
12. 12. Class-specific last-one-percent threshold
13. 13. Rejecting unknown cells
14. 14. Ambiguity rule
15. 15. Multilabel dependence maximisation
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.

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1. Why learn the comparison itself

This step deserves a careful pause. A fixed Pearson or cosine similarity may fit one atlas but fail under different annotation depth and heterogeneity. Metric learning adapts geometry to labelled relationships.

A concrete situation

Fine-grained subtypes require tighter separation than broad lineages.

What the mathematics does not license

Learned geometry can encode reference bias. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Hold out whole studies. 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.

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2. Reference quality control

The useful question is not only how to calculate, but what the calculation assumes. The paper’s defaults retained cells with more than 500 genes, more than 1,500 UMIs and under 10% mitochondrial genes.

A concrete situation

A low-quality cell fails two criteria.

What the mathematics does not license

Thresholds depend on protocol and should not be universalised. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State and justify current criteria. 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.

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3. Scaling and log transformation

A small example makes the mechanism visible. Counts are scaled to 10,000 per cell and transformed with log(counts + 1) before learning.

A concrete situation

A count of zero remains zero.

What the mathematics does not license

Library normalisation does not remove every composition effect. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare distributions by donor and class. 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.

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4. Rare-class exclusion

This is where modelling choices begin to affect evidence. The study excluded reference cell types with fewer than 10 cells because they supplied too little information for reliable learning.

A concrete situation

A seven-cell type is removed.

What the mathematics does not license

Exclusion protects estimation but loses rare biology. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Maintain a separate rare-cell review. 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.

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5. M3Drop feature selection

The central mathematical idea is concrete. M3Drop uses the relationship between mean expression and dropout to identify informative genes for the reference.

A concrete situation

A gene is detected less often than its mean predicts.

What the mathematics does not license

Dropout models depend on technology and preprocessing. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Resample selected 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.

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6. Similarity constraints

Here the geometry becomes operational. Training pairs or groups encode which reference cells should be close and which should be separated.

A concrete situation

Two cells with the same label form similar evidence.

What the mathematics does not license

Labels may merge real substructure. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect within-label heterogeneity. 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.

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7. Transformation matrix

This step deserves a careful pause. Multiplying expression by the learned matrix produces coordinates where labelled relationships are more discriminative.

A concrete situation

A 2D toy matrix rotates and rescales two genes.

What the mathematics does not license

Unconstrained matrices can amplify noise. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect singular values and condition. 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.

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8. Discriminative component analysis

The useful question is not only how to calculate, but what the calculation assumes. DCA chooses the transformation from similarity and dissimilarity information for single-label assignment.

A concrete situation

Between-class separation rises relative to within-class spread.

What the mathematics does not license

Optimised training separation is not test transfer. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use grouped external folds. 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.

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9. Bootstrap stability

A small example makes the mechanism visible. The published single-label procedure bootstraps ten times to reduce sampling imbalance and stabilise the learned measurement.

A concrete situation

Rare and common classes contribute across resamples.

What the mathematics does not license

Ten is a workflow choice, not a guarantee. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report matrix and prediction variability. 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.

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10. Transformed centroids

This is where modelling choices begin to affect evidence. Each reference class is summarised by its centroid in the learned space, and query cells are compared with these centroids.

A concrete situation

A query lies close to one immune-class mean.

What the mathematics does not license

A centroid hides multimodal classes. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Plot within-class distance structure. 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.

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11. Pearson similarity after transformation

The central mathematical idea is concrete. The study used Pearson correlation in the transformed space while noting that other similarities were tested.

A concrete situation

A query and centroid have aligned centred profiles.

What the mathematics does not license

Correlation ignores overall magnitude. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare cosine and Spearman in validation. 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.

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12. Class-specific last-one-percent threshold

Here the geometry becomes operational. For each class, the similarity distribution of its own reference cells to the centroid yields a lower boundary at the last one percent.

A concrete situation

Ninety-nine percent of training-class similarities exceed the threshold.

What the mathematics does not license

In-sample quantiles can be optimistic. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Estimate thresholds inside training folds. 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.

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13. Rejecting unknown cells

This step deserves a careful pause. If no query-to-centroid similarity exceeds its matching threshold, the query is unassigned.

A concrete situation

A withheld lineage clears none.

What the mathematics does not license

A shifted known class may also be rejected. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Separate unknown detection from domain failure. 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.

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14. Ambiguity rule

The useful question is not only how to calculate, but what the calculation assumes. When several thresholds are passed and the two largest similarities differ by at most 0.05, the paper labels the query unassigned as ambiguous.

A concrete situation

Two scores 0.86 and 0.83 are too close.

What the mathematics does not license

The fixed gap is context-specific. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Plot error and coverage across gap values. 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.

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15. Multilabel dependence maximisation

A small example makes the mechanism visible. MDDM extends the learned transformation to preserve dependence with several labels, such as identity and time.

A concrete situation

One cell receives lineage and developmental-stage labels.

What the mathematics does not license

Correlated labels can inflate apparent success. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use joint and per-label metrics. 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.

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16. Governance and careers

This is where modelling choices begin to affect evidence. Version quality rules, features, constraints, matrices, centroids, thresholds, ambiguity and folds. This connects linear algebra, metric learning and decision theory.

A concrete situation

Students can begin with matrix transformations and correlations.

What the mathematics does not license

An assignment is conditional on the reference. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Write a learned-metric model card. 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.

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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 scLearn audit begins with quality-controlled reference cells, M3Drop-selected genes and reliable labels. It learns a transformation matrix through discriminative component analysis for one label, or multilabel dimension reduction via dependence maximisation for several labels, then compares transformed query cells with transformed class centroids.

Record quality-control thresholds, removed rare classes, selected features, similarity and dissimilarity constraints, bootstrap samples, transformation matrices, centroid calculations, within-class similarity distributions, the class-specific last-one-percent thresholds and the 0.05 ambiguity rule described in the paper. Every threshold is conditional on the reference distribution and annotation resolution.

Validate positive-control and rejection tasks separately, with complete studies and cell types held out. Report known-class accuracy, unassigned sensitivity and specificity, ambiguity rates, bootstrap stability, micro-F1 for multilabel tasks and sensitivity to rare classes. Learned geometry improves adaptation only when reference labels and sampling support the intended query.

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Frequently asked questions

What is scLearn?

scLearn is a metric-learning framework that learns a transformation for cell comparison and class-specific thresholds for assigning or rejecting query cells.

What preprocessing did the paper describe?

The reference workflow used gene, UMI and mitochondrial-percentage quality checks, scaled counts to 10,000, applied log(counts + 1), removed classes with fewer than 10 cells and selected genes with M3Drop.

What does discriminative component analysis learn?

It learns a matrix that brings samples constrained as similar closer while separating samples constrained as dissimilar, creating a reference-specific comparison space.

How are thresholds learned?

For each transformed reference class, scLearn calculates similarities to its centroid and uses the last one percent of that class distribution as the class-specific threshold described in the paper.

When is a query unassigned?

A query is unassigned when no class similarity clears its threshold; it is also treated as ambiguous when several clear thresholds and the largest two similarities differ by no more than 0.05.

How does multilabel scLearn differ?

The paper used multilabel dimension reduction via dependence maximisation to learn a transformation that can support several labels, such as cell type and developmental time, and evaluated results with micro-F1.

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Primary source and fact-check note

Bin Duan, Chenyu Zhu, Guohui Chuai et al., “Learning for single-cell assignment,” Science Advances 6, eabd0855 (2020), DOI 10.1126/sciadv.abd0855. 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

• scPred dimensional reduction and SVMs
• CaSTLe transfer learning
• scHPL progressive hierarchies
• How Mathematics Works

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: topic-specific validation and responsible use

Quality-control ledger

Record every cell removed by gene, UMI, mitochondrial and rare-class criteria. Protocol-specific thresholds need justification.

Feature-stability audit

Repeat M3Drop selection across donor and bootstrap samples. Store selection frequencies and lost known markers.

Transformation audit

Archive matrices, singular values and transformed coordinates. Extreme scaling can make apparent separation numerically fragile.

Bootstrap comparison

Compare the ten learned measurements and their predictions. Report disagreement rather than only an aggregate model.

Threshold calibration

Estimate class thresholds inside training folds and validate coverage on new donors. In-sample one-percent tails can be optimistic.

Ambiguity-curve audit

Vary the 0.05 top-score gap and plot exact assignments, errors and unassigned coverage.

Rare-and-unseen test

Withhold one supported class and separately challenge a class with fewer than ten examples. Rejection and rarity are different problems.

Final claim

Name features, constraints, transformation, similarity, thresholds and folds. Learned geometry does not remove reference dependence.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Why learn the comparison itself

Start with the mechanism: A fixed Pearson or cosine similarity may fit one atlas but fail under different annotation depth and heterogeneity. Metric learning adapts geometry to labelled relationships. The worked case is concrete: Fine-grained subtypes require tighter separation than broad lineages. Expose the boundary: Learned geometry can encode reference bias. Complete this evidence step: Hold out whole studies. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 2: Reference quality control

Start with the mechanism: The paper’s defaults retained cells with more than 500 genes, more than 1,500 UMIs and under 10% mitochondrial genes. The worked case is concrete: A low-quality cell fails two criteria. Expose the boundary: Thresholds depend on protocol and should not be universalised. Complete this evidence step: State and justify current criteria. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 3: Scaling and log transformation

Start with the mechanism: Counts are scaled to 10,000 per cell and transformed with log(counts + 1) before learning. The worked case is concrete: A count of zero remains zero. Expose the boundary: Library normalisation does not remove every composition effect. Complete this evidence step: Compare distributions by donor and class. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 4: Rare-class exclusion

Start with the mechanism: The study excluded reference cell types with fewer than 10 cells because they supplied too little information for reliable learning. The worked case is concrete: A seven-cell type is removed. Expose the boundary: Exclusion protects estimation but loses rare biology. Complete this evidence step: Maintain a separate rare-cell review. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 5: M3Drop feature selection

Start with the mechanism: M3Drop uses the relationship between mean expression and dropout to identify informative genes for the reference. The worked case is concrete: A gene is detected less often than its mean predicts. Expose the boundary: Dropout models depend on technology and preprocessing. Complete this evidence step: Resample selected features. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 6: Similarity constraints

Start with the mechanism: Training pairs or groups encode which reference cells should be close and which should be separated. The worked case is concrete: Two cells with the same label form similar evidence. Expose the boundary: Labels may merge real substructure. Complete this evidence step: Inspect within-label heterogeneity. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 7: Transformation matrix

Start with the mechanism: Multiplying expression by the learned matrix produces coordinates where labelled relationships are more discriminative. The worked case is concrete: A 2D toy matrix rotates and rescales two genes. Expose the boundary: Unconstrained matrices can amplify noise. Complete this evidence step: Inspect singular values and condition. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 8: Discriminative component analysis

Start with the mechanism: DCA chooses the transformation from similarity and dissimilarity information for single-label assignment. The worked case is concrete: Between-class separation rises relative to within-class spread. Expose the boundary: Optimised training separation is not test transfer. Complete this evidence step: Use grouped external folds. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 9: Bootstrap stability

Start with the mechanism: The published single-label procedure bootstraps ten times to reduce sampling imbalance and stabilise the learned measurement. The worked case is concrete: Rare and common classes contribute across resamples. Expose the boundary: Ten is a workflow choice, not a guarantee. Complete this evidence step: Report matrix and prediction variability. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 10: Transformed centroids

Start with the mechanism: Each reference class is summarised by its centroid in the learned space, and query cells are compared with these centroids. The worked case is concrete: A query lies close to one immune-class mean. Expose the boundary: A centroid hides multimodal classes. Complete this evidence step: Plot within-class distance structure. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 11: Pearson similarity after transformation

Start with the mechanism: The study used Pearson correlation in the transformed space while noting that other similarities were tested. The worked case is concrete: A query and centroid have aligned centred profiles. Expose the boundary: Correlation ignores overall magnitude. Complete this evidence step: Compare cosine and Spearman in validation. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 12: Class-specific last-one-percent threshold

Start with the mechanism: For each class, the similarity distribution of its own reference cells to the centroid yields a lower boundary at the last one percent. The worked case is concrete: Ninety-nine percent of training-class similarities exceed the threshold. Expose the boundary: In-sample quantiles can be optimistic. Complete this evidence step: Estimate thresholds inside training folds. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 13: Rejecting unknown cells

Start with the mechanism: If no query-to-centroid similarity exceeds its matching threshold, the query is unassigned. The worked case is concrete: A withheld lineage clears none. Expose the boundary: A shifted known class may also be rejected. Complete this evidence step: Separate unknown detection from domain failure. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 14: Ambiguity rule

Start with the mechanism: When several thresholds are passed and the two largest similarities differ by at most 0.05, the paper labels the query unassigned as ambiguous. The worked case is concrete: Two scores 0.86 and 0.83 are too close. Expose the boundary: The fixed gap is context-specific. Complete this evidence step: Plot error and coverage across gap values. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 15: Multilabel dependence maximisation

Start with the mechanism: MDDM extends the learned transformation to preserve dependence with several labels, such as identity and time. The worked case is concrete: One cell receives lineage and developmental-stage labels. Expose the boundary: Correlated labels can inflate apparent success. Complete this evidence step: Use joint and per-label metrics. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 16: Governance and careers

Start with the mechanism: Version quality rules, features, constraints, matrices, centroids, thresholds, ambiguity and folds. This connects linear algebra, metric learning and decision theory. The worked case is concrete: Students can begin with matrix transformations and correlations. Expose the boundary: An assignment is conditional on the reference. Complete this evidence step: Write a learned-metric model card. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.