WHY MATHEMATICS?

Why Mathematics? | Tangram for Single-Cell Spatial Mapping: Mapping Matrices, Cosine Similarity, Density Priors and Entropy

How Tangram learns a nonnegative matrix that distributes single-cell profiles across spatial locations so their projected expression matches measured spatial patterns across genes and voxels.

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

Represent cell-to-location assignments as a probability matrix
Project single-cell expression into space with matrix multiplication
Balance cosine-similarity fit and a KL density prior
Validate mapping and imputed genes with genuine holdouts

Chapter map

1. Single-cell matrix S
2. Spatial matrix G
3. Training-gene intersection
4. Free mapping parameters
5. Row-wise softmax
6. Projected expression
7. Gene-wise cosine term
8. Voxel-wise cosine term
9. Density vector
10. KL density penalty
11. Entropy option
12. Gradient optimization
13. Worked two-cell map
14. Held-out-gene imputation
15. Mapping-mode comparison
16. Governance and learning

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

1. Single-cell matrix S

This step deserves a careful pause. Rows represent cells or cell-type averages and columns represent genes after declared preprocessing.

A concrete situation

S has 8,000 cells and 2,000 genes.

What the mathematics does not license

Dissociation and batch effects alter profiles. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Version normalization 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. Spatial matrix G

The useful question is not only how to calculate, but what the calculation assumes. Rows represent voxels, spots or segmented cells and columns contain measured genes.

A concrete situation

G has 1,500 voxels and 500 shared genes.

What the mathematics does not license

Resolution and sensitivity differ by assay. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record platform and units. 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. Training-gene intersection

A small example makes the mechanism visible. Optimization uses genes measured in both S and G, while other measured spatial genes can serve as tests.

A concrete situation

Four hundred genes train and one hundred are held out.

What the mathematics does not license

Choosing markers after viewing results leaks evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Freeze splits in advance. 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. Free mapping parameters

This is where modelling choices begin to affect evidence. An unconstrained matrix tilde-M is optimized before transformation.

A concrete situation

Each cell begins with scores for every voxel.

What the mathematics does not license

Initialization can influence a nonconvex objective. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use multiple starts. 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. Row-wise softmax

The central mathematical idea is concrete. Softmax turns each row into nonnegative mapping probabilities summing to one.

A concrete situation

Scores 2, 1 and 0 become a peaked three-location distribution.

What the mathematics does not license

A forced sum creates assignment even for unmatched cells. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Include novelty checks. 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. Projected expression

Here the geometry becomes operational. Matrix multiplication M-transpose S predicts a spatial expression matrix from mapped profiles.

A concrete situation

Cell contributions are summed within each voxel.

What the mathematics does not license

Summation assumes compatible scale. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit scaling choices. 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. Gene-wise cosine term

This step deserves a careful pause. For each training gene, cosine similarity compares its predicted and observed pattern across voxels.

A concrete situation

A cortical marker peaks in the same regions.

What the mathematics does not license

Cosine ignores absolute magnitude. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report scale errors 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.

8. Voxel-wise cosine term

The useful question is not only how to calculate, but what the calculation assumes. For each voxel, cosine similarity compares predicted and observed expression across genes.

A concrete situation

One tissue location has a similar multigene profile.

What the mathematics does not license

Dominant genes can shape direction. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Repeat with gene weighting. 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. Density vector

A small example makes the mechanism visible. Summing mapping mass over cells produces an implied spatial density across voxels.

A concrete situation

A large voxel receives more mapped cell mass.

What the mathematics does not license

Expected density may be uncertain. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Derive it from defensible evidence. 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. KL density penalty

This is where modelling choices begin to affect evidence. KL divergence discourages the implied density from departing from the expected distribution.

A concrete situation

A mapping that crowds one voxel incurs a larger penalty.

What the mathematics does not license

KL is asymmetric and punishes missing support strongly. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Test alternative density priors. 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. Entropy option

The central mathematical idea is concrete. Entropy of a cell row measures spread; a penalty can favour concentrated spatial assignments.

A concrete situation

Uniform mapping has higher entropy than a single-location peak.

What the mathematics does not license

Sharpness is not correctness. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare calibration and anatomy. 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. Gradient optimization

Here the geometry becomes operational. Automatic differentiation adjusts tilde-M to reduce the combined objective.

A concrete situation

Thousands of matrix entries update in one step.

What the mathematics does not license

Loss weights trade objectives. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Publish weights and curves. 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. Worked two-cell map

This step deserves a careful pause. Map two three-gene cell profiles into two voxels, compute M-transpose S and compare cosine similarities.

A concrete situation

A nearly diagonal M reproduces two distinct regions.

What the mathematics does not license

Toy profiles are perfectly matched. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Add noise and an unmatched cell. 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. Held-out-gene imputation

The useful question is not only how to calculate, but what the calculation assumes. After learning M from training genes, project untrained single-cell genes and compare with measured spatial values.

A concrete situation

A held-out receptor recovers its regional pattern.

What the mathematics does not license

Correlation alone can hide bias. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report error and calibration. 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. Mapping-mode comparison

A small example makes the mechanism visible. Cell, cluster and constrained modes vary the units and number of profiles assigned to space.

A concrete situation

Cluster mode maps type averages while constrained mode selects cells.

What the mathematics does not license

They answer different biological questions. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Choose before analysis. 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 learning

This is where modelling choices begin to affect evidence. Version matrices, genes, density, objective weights, mode, seeds and validation. This joins linear algebra, geometry and optimization.

A concrete situation

Students can begin with matrix multiplication and cosine.

What the mathematics does not license

A mapping weight is model evidence. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Write a Tangram map 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.

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

A Tangram audit begins with a single-cell expression matrix S, a spatial expression matrix G and a shared training-gene set. A softmax transforms free parameters into a nonnegative mapping matrix M whose rows distribute cells or cell types across spatial locations. Gradient optimization seeks a mapping whose projected expression M-transpose times S resembles G across genes and locations while respecting an expected density distribution.

Record gene identifiers, cells or clusters, spatial voxels, normalization, training genes, held-out test genes, mapping mode, density prior, cosine-similarity terms, Kullback-Leibler density term, entropy option, optimizer, random seeds and convergence. Mapping cells, mapping clusters and constrained mapping answer different questions and must not be reported as interchangeable.

Validate on held-out genes, known anatomical markers, simulated placements, alternative density priors, multiple random starts, donor or section holdouts and mappings built from deliberately mismatched references. A large mapping probability means a profile helps reproduce the spatial measurements under the objective; it is not proof that the sequenced cell physically occupied that coordinate.

Frequently asked questions

What is Tangram?

Tangram is an optimization method that aligns single-cell or single-nucleus expression profiles with spatial measurements from the same tissue region by learning a probabilistic mapping matrix.

What does the mapping matrix contain?

Each row assigns one cell or cell type across spatial voxels with nonnegative values that sum to one after softmax. Entries are optimization weights, not directly observed locations.

Why use cosine similarity?

Cosine similarity compares expression-pattern direction while reducing sensitivity to total magnitude. Tangram uses gene-wise and voxel-wise similarity terms.

What is the density term?

A Kullback–Leibler divergence penalizes disagreement between the mapping-implied spatial density and an expected density distribution.

What does entropy regularization do?

An optional entropy term can encourage each cell mapping to concentrate on fewer locations. It changes sharpness and should be reported and sensitivity-tested.

Does a high mapping probability prove a cell location?

No. It shows that the profile helps reproduce the spatial data under the objective and constraints. Histology, measured markers and independent spatial assays provide stronger location evidence.

Primary source and fact-check note

Tommaso Biancalani, Gabriele Scalia, Lorenzo Buffoni et al., “Deep learning and alignment of spatially resolved single-cell transcriptomes with Tangram,” Nature Methods 18, 1352–1362 (2021), DOI 10.1038/s41592-021-01264-7. 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

Matrix audit

Version cells, voxels, genes, normalization and all identifier mappings.

Split audit

Freeze training and held-out genes before optimization.

Objective audit

Report both cosine terms, density KL weight and entropy setting.

Mode audit

State cell, cluster or constrained mode and the biological question it answers.

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.

Seed audit

Repeat nonconvex optimization from several initializations.

Density challenge

Vary expected density and measure changes in assignments.

Anatomy check

Compare with independent markers, histology and another spatial technology.

Final claim

Name objective and constraints. A mapping probability is not observed physical location.

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: Single-cell matrix S

Begin with this mechanism: Rows represent cells or cell-type averages and columns represent genes after declared preprocessing. Work through the concrete case: S has 8,000 cells and 2,000 genes. Make the limitation visible: Dissociation and batch effects alter profiles. Complete the evidence step: Version normalization and donors. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 2: Spatial matrix G

Begin with this mechanism: Rows represent voxels, spots or segmented cells and columns contain measured genes. Work through the concrete case: G has 1,500 voxels and 500 shared genes. Make the limitation visible: Resolution and sensitivity differ by assay. Complete the evidence step: Record platform and units. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 3: Training-gene intersection

Begin with this mechanism: Optimization uses genes measured in both S and G, while other measured spatial genes can serve as tests. Work through the concrete case: Four hundred genes train and one hundred are held out. Make the limitation visible: Choosing markers after viewing results leaks evidence. Complete the evidence step: Freeze splits in advance. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 4: Free mapping parameters

Begin with this mechanism: An unconstrained matrix tilde-M is optimized before transformation. Work through the concrete case: Each cell begins with scores for every voxel. Make the limitation visible: Initialization can influence a nonconvex objective. Complete the evidence step: Use multiple starts. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

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: Row-wise softmax

Begin with this mechanism: Softmax turns each row into nonnegative mapping probabilities summing to one. Work through the concrete case: Scores 2, 1 and 0 become a peaked three-location distribution. Make the limitation visible: A forced sum creates assignment even for unmatched cells. Complete the evidence step: Include novelty checks. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 6: Projected expression

Begin with this mechanism: Matrix multiplication M-transpose S predicts a spatial expression matrix from mapped profiles. Work through the concrete case: Cell contributions are summed within each voxel. Make the limitation visible: Summation assumes compatible scale. Complete the evidence step: Audit scaling choices. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 7: Gene-wise cosine term

Begin with this mechanism: For each training gene, cosine similarity compares its predicted and observed pattern across voxels. Work through the concrete case: A cortical marker peaks in the same regions. Make the limitation visible: Cosine ignores absolute magnitude. Complete the evidence step: Report scale errors separately. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 8: Voxel-wise cosine term

Begin with this mechanism: For each voxel, cosine similarity compares predicted and observed expression across genes. Work through the concrete case: One tissue location has a similar multigene profile. Make the limitation visible: Dominant genes can shape direction. Complete the evidence step: Repeat with gene weighting. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

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: Density vector

Begin with this mechanism: Summing mapping mass over cells produces an implied spatial density across voxels. Work through the concrete case: A large voxel receives more mapped cell mass. Make the limitation visible: Expected density may be uncertain. Complete the evidence step: Derive it from defensible evidence. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 10: KL density penalty

Begin with this mechanism: KL divergence discourages the implied density from departing from the expected distribution. Work through the concrete case: A mapping that crowds one voxel incurs a larger penalty. Make the limitation visible: KL is asymmetric and punishes missing support strongly. Complete the evidence step: Test alternative density priors. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 11: Entropy option

Begin with this mechanism: Entropy of a cell row measures spread; a penalty can favour concentrated spatial assignments. Work through the concrete case: Uniform mapping has higher entropy than a single-location peak. Make the limitation visible: Sharpness is not correctness. Complete the evidence step: Compare calibration and anatomy. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 12: Gradient optimization

Begin with this mechanism: Automatic differentiation adjusts tilde-M to reduce the combined objective. Work through the concrete case: Thousands of matrix entries update in one step. Make the limitation visible: Loss weights trade objectives. Complete the evidence step: Publish weights and curves. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

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: Worked two-cell map

Begin with this mechanism: Map two three-gene cell profiles into two voxels, compute M-transpose S and compare cosine similarities. Work through the concrete case: A nearly diagonal M reproduces two distinct regions. Make the limitation visible: Toy profiles are perfectly matched. Complete the evidence step: Add noise and an unmatched cell. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 14: Held-out-gene imputation

Begin with this mechanism: After learning M from training genes, project untrained single-cell genes and compare with measured spatial values. Work through the concrete case: A held-out receptor recovers its regional pattern. Make the limitation visible: Correlation alone can hide bias. Complete the evidence step: Report error and calibration. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 15: Mapping-mode comparison

Begin with this mechanism: Cell, cluster and constrained modes vary the units and number of profiles assigned to space. Work through the concrete case: Cluster mode maps type averages while constrained mode selects cells. Make the limitation visible: They answer different biological questions. Complete the evidence step: Choose before analysis. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 16: Governance and learning

Begin with this mechanism: Version matrices, genes, density, objective weights, mode, seeds and validation. This joins linear algebra, geometry and optimization. Work through the concrete case: Students can begin with matrix multiplication and cosine. Make the limitation visible: A mapping weight is model evidence. Complete the evidence step: Write a Tangram map card. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.