WHY MATHEMATICS?

Why Mathematics? | Tensor-cell2cell Communication Contexts: Four-Way Tensors, Nonnegative CP Decomposition, Rank and Latent Factors

How Tensor-cell2cell organizes many communication networks into one context-aware array and extracts coordinated ligand-receptor, sender and receiver programmes without treating latent factors as causal mechanisms.

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

Construct one aligned context-by-interaction-by-sender-by-receiver tensor
Approximate communication with nonnegative rank-one components
Choose and stabilize the factor rank across random starts
Relate context loadings to phenotype at the sample level

Chapter map

1. Input communication scores
2. Context mode
3. Ligand-receptor mode
4. Sender mode
5. Receiver mode
6. Tensor entry
7. Rank-one outer product
8. CP/PARAFAC sum
9. Nonnegative constraints
10. Alternating optimization
11. Reconstruction error
12. Rank selection
13. Scale and permutation ambiguity
14. Factor interpretation
15. Phenotype association
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. Input communication scores

A small example makes the mechanism visible. A prior tool supplies ligand-receptor scores for sender-receiver pairs in every context.

A concrete situation

Ten donors each yield a network.

What the mathematics does not license

Upstream assumptions carry forward. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record method and version. 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. Context mode

This is where modelling choices begin to affect evidence. The first tensor axis represents samples, conditions, times or regions.

A concrete situation

Rows correspond to twelve patients.

What the mathematics does not license

Technical replicates are not independent patients. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Define the unit. 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. Ligand-receptor mode

The central mathematical idea is concrete. The second axis aligns interaction pairs across contexts.

A concrete situation

Five hundred pairs form columns.

What the mathematics does not license

Missing pairs need a declared rule. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep a common vocabulary. 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. Sender mode

Here the geometry becomes operational. The third axis indexes source cell types.

A concrete situation

Six sender groups are aligned.

What the mathematics does not license

Cluster labels can drift across samples. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Harmonize annotations. 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. Receiver mode

This step deserves a careful pause. The fourth axis indexes target cell types.

A concrete situation

Seven receiver groups are aligned.

What the mathematics does not license

Absent cell types create structural missingness. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Distinguish zero and missing. 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. Tensor entry

The useful question is not only how to calculate, but what the calculation assumes. One scalar has four coordinates.

A concrete situation

X patient, L-R pair, macrophage, T cell equals 0.8.

What the mathematics does not license

Score scale may differ by context. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Normalize transparently. 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. Rank-one outer product

A small example makes the mechanism visible. Four loading vectors multiply to create one component tensor.

A concrete situation

A context pattern combines with one signaling module.

What the mathematics does not license

Multiplicative structure is restrictive. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Inspect 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.

8. CP/PARAFAC sum

This is where modelling choices begin to affect evidence. Several rank-one tensors add to approximate the data.

A concrete situation

Four components reconstruct major programmes.

What the mathematics does not license

Factors can overlap. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report component correlations. 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. Nonnegative constraints

The central mathematical idea is concrete. All loadings remain at or above zero.

A concrete situation

Memberships add without sign cancellation.

What the mathematics does not license

Nonnegativity can raise required rank. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare reconstruction. 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. Alternating optimization

Here the geometry becomes operational. One factor matrix is updated while others are held fixed.

A concrete situation

Loss falls across iterations.

What the mathematics does not license

Local minima depend on initialization. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Run many seeds. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

11. Reconstruction error

This step deserves a careful pause. A norm measures the gap between observed and fitted tensors.

A concrete situation

Rank five reduces relative error.

What the mathematics does not license

Lower training error always favors more factors. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use stability and holdout 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.

12. Rank selection

The useful question is not only how to calculate, but what the calculation assumes. Elbows and factor stability guide component count.

A concrete situation

Ranks four to six are compared.

What the mathematics does not license

No elbow is a warning, not permission to guess. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Publish the 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.

13. Scale and permutation ambiguity

A small example makes the mechanism visible. Component order is arbitrary and scale can shift among modes.

A concrete situation

A sender vector doubles while another mode halves.

What the mathematics does not license

Raw loadings across runs are not aligned. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Normalize and match factors. 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. Factor interpretation

This is where modelling choices begin to affect evidence. High loadings identify contexts, pairs, senders and receivers acting together.

A concrete situation

One factor highlights interferon pairs.

What the mathematics does not license

A label can overstate mixed memberships. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Show full loadings. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

15. Phenotype association

The central mathematical idea is concrete. Context loadings are tested against sample metadata.

A concrete situation

A factor rises with disease severity.

What the mathematics does not license

Entry-level replication inflates significance. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Test at context level. 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

Here the geometry becomes operational. Version inputs, missingness, scaling, rank, seeds, solver and tests.

A concrete situation

A student factorizes a two-by-two-by-two cube.

What the mathematics does not license

A factor is an additive latent pattern. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Write a tensor audit 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

Tensor-cell2cell begins when communication scores exist for several biological contexts. Instead of comparing every context pair separately, it arranges the data as a four-way tensor: context by ligand-receptor pair by sender cell type by receiver cell type. Every entry has an address in all four modes. Missing cell types and zero scores must be handled consistently because they change the geometry of the tensor.

Nonnegative CP or PARAFAC decomposition approximates the tensor as a sum of rank-one components. Each component is an outer product of four nonnegative loading vectors. One factor may be high in severe disease contexts, emphasize a cytokine family, highlight myeloid senders and select lymphocyte receivers. The component is a coordinated mathematical pattern, not a newly discovered pathway by itself.

The chosen rank controls detail. Too few factors merge distinct programmes; too many split stable structure or fit noise. Reconstruction error, elbow plots, similarity across random initializations and biological interpretability all contribute to rank selection. Because CP factors are permutation-invariant and scale can move between modes, implementations normalize loadings and align factors before comparison.

Context loadings can be associated with phenotype variables, and ligand-receptor or cell loadings can be inspected as factor-specific modules. Those downstream tests require appropriate sample-level replication; thousands of tensor entries do not create thousands of independent patients. A responsible workflow preserves the original scores, reports preprocessing and sparsity, repeats factorizations across seeds and ranks, and validates factor associations in independent contexts.

Frequently asked questions

What is Tensor-cell2cell?

Tensor-cell2cell is an unsupervised framework that decomposes multi-context cell-communication scores into coordinated context, ligand-receptor, sender and receiver factors.

What are the four tensor modes?

Contexts, ligand-receptor pairs, sender cell types and receiver cell types.

What is CP decomposition?

It approximates a tensor as a sum of outer products; each component has one loading vector for every mode.

Why require nonnegative factors?

Communication scores and memberships are naturally nonnegative, making additive patterns easier to inspect without cancellation between positive and negative loadings.

How is the number of factors chosen?

Reconstruction error, elbows, factor similarity across seeds and usefulness of the resulting patterns are considered together.

Does a phenotype-associated factor prove mechanism?

No. It is an unsupervised pattern whose association must be tested at the independent-context level and validated in new samples or experiments.

Primary source and fact-check note

Erick Armingol, Hasmik M. Baghdassarian, Cameron Martino, Aurelien Perez-Lopez, Caitlin Aamodt, Rob Knight and Nathan E. Lewis, “Context-aware deconvolution of cell–cell communication with Tensor-cell2cell,” Nature Communications 13, 3665 (2022), DOI 10.1038/s41467-022-31369-2. 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

Alignment audit

Use consistent interactions and cell labels across contexts.

Missingness audit

Separate absent measurements from true zeros.

Scaling audit

Prevent one context or score range from dominating.

Rank audit

Publish error and stability across candidate ranks.

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

Align factors from many initializations.

Replication audit

Test phenotype links at the independent-context level.

Residual audit

Inspect structure the factors fail to explain.

Final claim

A latent factor is an additive pattern, not a causal pathway.

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: Input communication scores

Start from this mechanism: A prior tool supplies ligand-receptor scores for sender-receiver pairs in every context. Recreate the small case: Ten donors each yield a network. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Upstream assumptions carry forward. Complete the transfer task: Record method and version. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 2: Context mode

Start from this mechanism: The first tensor axis represents samples, conditions, times or regions. Recreate the small case: Rows correspond to twelve patients. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Technical replicates are not independent patients. Complete the transfer task: Define the unit. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 3: Ligand-receptor mode

Start from this mechanism: The second axis aligns interaction pairs across contexts. Recreate the small case: Five hundred pairs form columns. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Missing pairs need a declared rule. Complete the transfer task: Keep a common vocabulary. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 4: Sender mode

Start from this mechanism: The third axis indexes source cell types. Recreate the small case: Six sender groups are aligned. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Cluster labels can drift across samples. Complete the transfer task: Harmonize annotations. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

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: Receiver mode

Start from this mechanism: The fourth axis indexes target cell types. Recreate the small case: Seven receiver groups are aligned. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Absent cell types create structural missingness. Complete the transfer task: Distinguish zero and missing. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 6: Tensor entry

Start from this mechanism: One scalar has four coordinates. Recreate the small case: X patient, L-R pair, macrophage, T cell equals 0.8. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Score scale may differ by context. Complete the transfer task: Normalize transparently. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 7: Rank-one outer product

Start from this mechanism: Four loading vectors multiply to create one component tensor. Recreate the small case: A context pattern combines with one signaling module. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Multiplicative structure is restrictive. Complete the transfer task: Inspect residuals. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 8: CP/PARAFAC sum

Start from this mechanism: Several rank-one tensors add to approximate the data. Recreate the small case: Four components reconstruct major programmes. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Factors can overlap. Complete the transfer task: Report component correlations. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

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: Nonnegative constraints

Start from this mechanism: All loadings remain at or above zero. Recreate the small case: Memberships add without sign cancellation. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Nonnegativity can raise required rank. Complete the transfer task: Compare reconstruction. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 10: Alternating optimization

Start from this mechanism: One factor matrix is updated while others are held fixed. Recreate the small case: Loss falls across iterations. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Local minima depend on initialization. Complete the transfer task: Run many seeds. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 11: Reconstruction error

Start from this mechanism: A norm measures the gap between observed and fitted tensors. Recreate the small case: Rank five reduces relative error. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Lower training error always favors more factors. Complete the transfer task: Use stability and holdout checks. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 12: Rank selection

Start from this mechanism: Elbows and factor stability guide component count. Recreate the small case: Ranks four to six are compared. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: No elbow is a warning, not permission to guess. Complete the transfer task: Publish the curve. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

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: Scale and permutation ambiguity

Start from this mechanism: Component order is arbitrary and scale can shift among modes. Recreate the small case: A sender vector doubles while another mode halves. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Raw loadings across runs are not aligned. Complete the transfer task: Normalize and match factors. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 14: Factor interpretation

Start from this mechanism: High loadings identify contexts, pairs, senders and receivers acting together. Recreate the small case: One factor highlights interferon pairs. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: A label can overstate mixed memberships. Complete the transfer task: Show full loadings. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 15: Phenotype association

Start from this mechanism: Context loadings are tested against sample metadata. Recreate the small case: A factor rises with disease severity. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: Entry-level replication inflates significance. Complete the transfer task: Test at context level. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.

Workshop 16: Governance and learning

Start from this mechanism: Version inputs, missingness, scaling, rank, seeds, solver and tests. Recreate the small case: A student factorizes a two-by-two-by-two cube. Write tensor indices, entry values, factor loadings, outer products, reconstruction residuals, ranks, aligned components and context-level tests with dimensions, units and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation directly: A factor is an additive latent pattern. Complete the transfer task: Write a tensor audit card. Change exactly one distance, marginal, regularization, atlas weight, background universe, propagation setting, tensor rank, missing-value rule or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, assumption, model output, statistical evidence, biological interpretation and practical decision on separate notebook lines. When the question concerns transfer, reserve complete donors, tissues, sections, contexts or marker genes outside tuning.