WHY MATHEMATICS?

Why Mathematics? | CellDART Domain Adaptation: Pseudo-Spots, KL Proportion Loss, Binary Cross-Entropy and Adversarial Training

How CellDART learns cell-type proportions from labelled synthetic mixtures while aligning them with unlabelled spatial spots through an adversarial objective.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Build pseudo-spots with known mixture proportions
Measure proportion error with KL divergence
Use binary cross-entropy to detect dataset domain
Balance prediction and domain confusion without hiding transfer risk

Chapter map

1. Annotated reference cells
2. Marker-gene feature set
3. Pseudo-spot sampling
4. Random mixture weights
5. Known target proportions
6. Synthetic expression
7. Shared feature extractor
8. Source proportion classifier
9. KL proportion loss
10. Domain classifier
11. Binary cross-entropy
12. Adversarial label inversion
13. Alternating optimization
14. Trade-off weight
15. Spatial proportion prediction
16. Transfer validation

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

1. Annotated reference cells

The useful question is not only how to calculate, but what the calculation assumes. Each single cell has expression and a cell-type label.

A concrete situation

Eight types contribute 12,000 cells.

What the mathematics does not license

Incorrect labels enter every synthetic target. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Audit labels and donors. 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. Marker-gene feature set

A small example makes the mechanism visible. Selected genes define the neural input.

A concrete situation

Five hundred markers remain.

What the mathematics does not license

Markers may encode one platform. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat with alternative feature sets. 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. Pseudo-spot sampling

This is where modelling choices begin to affect evidence. A fixed number of reference cells are drawn.

A concrete situation

Eight cells form one pseudo-spot.

What the mathematics does not license

Fixed size may not match tissue density. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Sweep mixture sizes. 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. Random mixture weights

The central mathematical idea is concrete. Nonnegative weights sum to one.

A concrete situation

Weights are [0.4,0.3,0.2,0.1].

What the mathematics does not license

The training distribution sets expected mixtures. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot the simulated simplex coverage. 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.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. Known target proportions

Here the geometry becomes operational. Weighted labels form the supervision vector.

A concrete situation

Type A target is 0.4.

What the mathematics does not license

RNA contribution may differ from cell fraction. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Name the target precisely. 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. Synthetic expression

This step deserves a careful pause. Weighted cell profiles produce the pseudo-spot.

A concrete situation

Eight vectors combine into x-p.

What the mathematics does not license

Noise and capture efficiency are simplified. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Add realistic perturbations. 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. Shared feature extractor

The useful question is not only how to calculate, but what the calculation assumes. One network maps both domains to latent features.

A concrete situation

f(x) has 64 components.

What the mathematics does not license

Shared weights can erase domain-specific biology. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Inspect retained class separation. 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. Source proportion classifier

A small example makes the mechanism visible. A head predicts K nonnegative proportions.

A concrete situation

Prediction is [0.38,0.31,0.21,0.10].

What the mathematics does not license

Simplex output forces known types to absorb all mass. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check unknown-state 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.

9. KL proportion loss

This is where modelling choices begin to affect evidence. Known and predicted distributions are compared.

A concrete situation

Sum y log(y over p) is 0.012.

What the mathematics does not license

Zeros need stable conventions. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Publish direction and epsilon. 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. Domain classifier

The central mathematical idea is concrete. A second head predicts pseudo versus real.

A concrete situation

Domain probability is 0.72.

What the mathematics does not license

Accuracy can reflect trivial batch cues. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Inspect feature attribution. 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. Binary cross-entropy

Here the geometry becomes operational. Domain labels score discrimination.

A concrete situation

Minus log 0.72 penalizes a correct pseudo label.

What the mathematics does not license

Confident errors dominate. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Calibrate domain probabilities. 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. Adversarial label inversion

This step deserves a careful pause. Feature updates try to fool the domain classifier.

A concrete situation

A real spot is trained toward the pseudo label.

What the mathematics does not license

Confusion can remove biological signal. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Monitor class structure during 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.

13. Alternating optimization

The useful question is not only how to calculate, but what the calculation assumes. Feature and source weights alternate with domain weights.

A concrete situation

One step freezes D; the next freezes f and S.

What the mathematics does not license

Unstable games depend on seeds and learning rates. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot every loss curve. 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. Trade-off weight

A small example makes the mechanism visible. Alpha balances source and adversarial objectives.

A concrete situation

Total loss is Ls plus 0.1 times Ladv.

What the mathematics does not license

One number changes the transfer target. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Sweep alpha on held-out donors. 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. Spatial proportion prediction

This is where modelling choices begin to affect evidence. The trained source head processes real spots.

A concrete situation

Spot j predicts 60% A and 40% B.

What the mathematics does not license

Prediction is not a direct count. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare with imaging and histology. 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. Transfer validation

The central mathematical idea is concrete. Synthetic, donor and platform holdouts test different errors.

A concrete situation

A whole tissue section is withheld.

What the mathematics does not license

Pseudo accuracy can hide domain shift. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Report each validation level separately. 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

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

CellDART frames spatial deconvolution as a domain-adaptation problem. It builds labelled pseudo-spots by sampling annotated single cells, assigning random mixture weights and combining their expression. Each pseudo-spot therefore has a known cell-type proportion vector. A source network learns to predict that vector, while real spatial spots remain unlabelled and may follow a different measurement distribution.

The source classifier is trained with Kullback–Leibler divergence between the known and predicted proportion distributions. A domain classifier uses binary cross-entropy to tell pseudo-spots from real spots. Adversarial training alternates updates: the feature extractor and proportion predictor improve the biological task while trying to confuse the domain classifier, then the domain classifier improves its own discrimination with the other weights fixed.

A trade-off weight balances proportion accuracy against domain confusion. If it is too small, the model may fit synthetic mixtures yet transfer poorly; if too large, it may erase signals that distinguish real cell types. Fixed pseudo-spot cell counts, marker-gene selection, random mixture weights, normalization and reference coverage all define the training distribution before optimization begins.

The final proportions are conditional predictions, not direct cell counts. A softmax-like simplex allocates mass among known types, so an omitted population can be redistributed among familiar classes. Good validation separates pseudo-spot holdouts from real-tissue transfer, repeats random seeds and mixture designs, withholds whole donors or platforms and compares predictions with independent anatomy or imaging.

Frequently asked questions

What is CellDART?

CellDART is a neural domain-adaptation method that learns cell-type proportions from labelled single-cell pseudo-spots and transfers the predictor to spatial spots.

Why create pseudo-spots?

Combining annotated single cells produces synthetic mixtures whose proportion vectors are known and can supervise learning.

What does KL divergence measure?

It measures the mismatch between the known and predicted cell-type proportion distributions, with direction and zero handling stated explicitly.

Why use binary cross-entropy?

The domain classifier uses it to distinguish pseudo-spots from real spatial spots.

What makes training adversarial?

The feature extractor tries to support proportion prediction while making the two domains harder to distinguish; the domain classifier tries to distinguish them.

Does domain confusion prove biological alignment?

No. Technical domains can overlap while cell states remain mismatched, so real-tissue validation is still required.

Primary source and fact-check note

Sungwoo Bae, Kwon Joong Na, Jaemoon Koh, Dong Soo Lee, Hongyoon Choi and Young Tae Kim, “CellDART: cell type inference by domain adaptation of single-cell and spatial transcriptomic data,” Nucleic Acids Research 50(10), e57 (2022), DOI 10.1093/nar/gkac084. 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

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

Reference audit

Version labels, donors and feature genes.

Pseudo audit

Publish mixture sizes, weight distribution and random seeds.

KL audit

State divergence direction, epsilon and reduction.

Domain audit

Keep classifier accuracy and calibration.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Adversarial audit

Publish alternating updates, alpha and loss curves.

Transfer audit

Withhold whole donors, sections and platforms.

Unknown audit

Check residuals for omitted cell types.

Final claim

Domain confusion does not prove biological equivalence.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Annotated reference cells

Begin with this mechanism: Each single cell has expression and a cell-type label. Recreate the concrete case: Eight types contribute 12,000 cells. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Incorrect labels enter every synthetic target. Complete the transfer step: Audit labels and donors. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 2: Marker-gene feature set

Begin with this mechanism: Selected genes define the neural input. Recreate the concrete case: Five hundred markers remain. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Markers may encode one platform. Complete the transfer step: Repeat with alternative feature sets. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 3: Pseudo-spot sampling

Begin with this mechanism: A fixed number of reference cells are drawn. Recreate the concrete case: Eight cells form one pseudo-spot. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Fixed size may not match tissue density. Complete the transfer step: Sweep mixture sizes. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 4: Random mixture weights

Begin with this mechanism: Nonnegative weights sum to one. Recreate the concrete case: Weights are [0.4,0.3,0.2,0.1]. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: The training distribution sets expected mixtures. Complete the transfer step: Plot the simulated simplex coverage. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: Known target proportions

Begin with this mechanism: Weighted labels form the supervision vector. Recreate the concrete case: Type A target is 0.4. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: RNA contribution may differ from cell fraction. Complete the transfer step: Name the target precisely. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 6: Synthetic expression

Begin with this mechanism: Weighted cell profiles produce the pseudo-spot. Recreate the concrete case: Eight vectors combine into x-p. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Noise and capture efficiency are simplified. Complete the transfer step: Add realistic perturbations. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 7: Shared feature extractor

Begin with this mechanism: One network maps both domains to latent features. Recreate the concrete case: f(x) has 64 components. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Shared weights can erase domain-specific biology. Complete the transfer step: Inspect retained class separation. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 8: Source proportion classifier

Begin with this mechanism: A head predicts K nonnegative proportions. Recreate the concrete case: Prediction is [0.38,0.31,0.21,0.10]. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Simplex output forces known types to absorb all mass. Complete the transfer step: Check unknown-state residuals. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: KL proportion loss

Begin with this mechanism: Known and predicted distributions are compared. Recreate the concrete case: Sum y log(y over p) is 0.012. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Zeros need stable conventions. Complete the transfer step: Publish direction and epsilon. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 10: Domain classifier

Begin with this mechanism: A second head predicts pseudo versus real. Recreate the concrete case: Domain probability is 0.72. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Accuracy can reflect trivial batch cues. Complete the transfer step: Inspect feature attribution. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 11: Binary cross-entropy

Begin with this mechanism: Domain labels score discrimination. Recreate the concrete case: Minus log 0.72 penalizes a correct pseudo label. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Confident errors dominate. Complete the transfer step: Calibrate domain probabilities. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 12: Adversarial label inversion

Begin with this mechanism: Feature updates try to fool the domain classifier. Recreate the concrete case: A real spot is trained toward the pseudo label. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Confusion can remove biological signal. Complete the transfer step: Monitor class structure during alignment. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: Alternating optimization

Begin with this mechanism: Feature and source weights alternate with domain weights. Recreate the concrete case: One step freezes D; the next freezes f and S. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Unstable games depend on seeds and learning rates. Complete the transfer step: Plot every loss curve. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 14: Trade-off weight

Begin with this mechanism: Alpha balances source and adversarial objectives. Recreate the concrete case: Total loss is Ls plus 0.1 times Ladv. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: One number changes the transfer target. Complete the transfer step: Sweep alpha on held-out donors. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 15: Spatial proportion prediction

Begin with this mechanism: The trained source head processes real spots. Recreate the concrete case: Spot j predicts 60% A and 40% B. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Prediction is not a direct count. Complete the transfer step: Compare with imaging and histology. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 16: Transfer validation

Begin with this mechanism: Synthetic, donor and platform holdouts test different errors. Recreate the concrete case: A whole tissue section is withheld. Write cell labels, mixture weights, target proportions, latent features, KL terms, domain probabilities, cross-entropy terms, trade-off weights, loss curves and predicted fractions with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Pseudo accuracy can hide domain shift. Complete the transfer step: Report each validation level separately. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.