Chapter map

1. 1. Spatial Euclidean distance
2. 2. Radius adjacency
3. 3. Cosine expression similarity
4. 4. Feature kNN graph
5. 5. Self-loop augmentation
6. 6. Degree normalization
7. 7. Spatial convolution
8. 8. Feature convolution
9. 9. Co-convolution
10. 10. View attention
11. 11. Fused embedding
12. 12. ZINB mean
13. 13. Zero-inflation probability
14. 14. ZINB negative log-likelihood
15. 15. Spatial contrast loss
16. 16. Weighted total loss

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

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1. Spatial Euclidean distance

The central mathematical idea is concrete. Coordinate differences define physical proximity.

A concrete situation

Two spots are 400 coordinate units apart.

What the mathematics does not license

Units differ across platforms. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Preserve and report 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.

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2. Radius adjacency

Here the geometry becomes operational. Edges appear when distance is no greater than r.

A concrete situation

The paper used r equal to 550 in examples.

What the mathematics does not license

A copied radius may be meaningless elsewhere. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Plot degree across r. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Cosine expression similarity

This step deserves a careful pause. Dot products are normalized by vector lengths.

A concrete situation

Parallel profiles have similarity near one.

What the mathematics does not license

Low-norm noisy vectors are unstable. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Filter and inspect norms. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Feature kNN graph

The useful question is not only how to calculate, but what the calculation assumes. Each spot connects to its most similar expression neighbours.

A concrete situation

k equals ten yields ten outgoing candidates.

What the mathematics does not license

Distant tissue can become connected. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Map long edges. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Self-loop augmentation

A small example makes the mechanism visible. Identity edges retain a spot own features.

A concrete situation

A-tilde equals A plus I.

What the mathematics does not license

Self-mass can dominate sparse graphs. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect diagonal influence. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Degree normalization

This is where modelling choices begin to affect evidence. D inverse square-root scales both ends of each edge.

A concrete situation

A hub message receives smaller weight.

What the mathematics does not license

Normalization is not biological calibration. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare degree-stratified error. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Spatial convolution

The central mathematical idea is concrete. Graph layers aggregate expression on the spatial graph.

A concrete situation

Hs updates through normalized As and ReLU.

What the mathematics does not license

Repeated layers can oversmooth. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Measure boundary contrast by depth. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Feature convolution

Here the geometry becomes operational. A parallel branch aggregates expression-similar spots.

A concrete situation

Hf updates through normalized Af.

What the mathematics does not license

Feature leakage may connect batches. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Hold out sections. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Co-convolution

This step deserves a careful pause. A shared branch captures dependencies across both views.

A concrete situation

Hc contains jointly learned features.

What the mathematics does not license

Shared structure can reflect shared artifact. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use single-view ablations. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. View attention

The useful question is not only how to calculate, but what the calculation assumes. Tanh scores and softmax yield weights for Hs, Hf and Hc.

A concrete situation

Weights 0.2, 0.3 and 0.5 sum to one.

What the mathematics does not license

Attention is objective-dependent. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

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

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11. Fused embedding

A small example makes the mechanism visible. A linear layer transforms the weighted view sum.

A concrete situation

H has thirty latent features per spot.

What the mathematics does not license

Width affects bottleneck strength. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Sweep latent dimension. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. ZINB mean

This is where modelling choices begin to affect evidence. The decoder predicts a positive expected count.

A concrete situation

Mean five with dispersion two is over-dispersed.

What the mathematics does not license

Normalization and raw-count use must match. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Document decoder input and target. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Zero-inflation probability

The central mathematical idea is concrete. A mixture component adds excess probability at zero.

A concrete situation

Pi equals 0.3 for a spot-gene pair.

What the mathematics does not license

NB alone may explain many zeros. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare NB and ZINB. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. ZINB negative log-likelihood

Here the geometry becomes operational. Observed counts score decoder parameters.

A concrete situation

A low probability count adds a large loss.

What the mathematics does not license

Likelihood fit is not domain truth. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect held-out genes. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Spatial contrast loss

This step deserves a careful pause. Cosine scores distinguish neighbours from non-neighbours.

A concrete situation

Sigmoid maps a similarity to a pair score.

What the mathematics does not license

False negatives can split related regions. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Try distance-banded negatives. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Weighted total loss

The useful question is not only how to calculate, but what the calculation assumes. Alpha, beta and gamma balance consistency, count fit and space.

A concrete situation

Gamma doubles the neighbour penalty.

What the mathematics does not license

Scales are not directly comparable. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Publish raw component 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.

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A seven-day practice lab

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

Questions for a student, parent or mentor

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

A method-specific audit before deployment

Spatial-MGCN represents the same spots through two graphs. A radius graph connects physically close spots, while a feature k-nearest-neighbour graph connects similar expression vectors using cosine similarity. Normalized graph convolution is applied separately to each view. A co-convolution branch combines both graphs, allowing the model to learn information unique to spatial structure, unique to expression similarity and shared across the two.

An attention module calculates trainable importance weights for the spatial, feature and co-convolution embeddings. A softmax makes the three weights comparable, and a final linear layer produces the fused representation. Attention answers which view helped the trained objective under the chosen graphs. It does not tell us which biological process caused a tissue boundary.

The decoder models raw expression through a zero-inflated negative binomial distribution. Its mean, dispersion and excess-zero probability address count discreteness, over-dispersion and sparsity. The reconstruction loss is the negative log-likelihood. A spatial regularization term uses cosine similarities of learned embeddings to pull declared spatial neighbours together and push non-neighbours apart. The total objective balances consistency, count reconstruction and spatial regularization with three weights.

Every component can help or mislead. The reported radius of 550 belongs to the paper’s coordinate systems, not every assay. A feature neighbour may join distant but related tissue, while a spatial edge may cross a sharp boundary. The zero-inflated assumption may fit some preprocessing pipelines better than others. Responsible use tests graph rules, loss weights, ablations, seeds, clustering parameters and transfer to complete held-out sections.

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Frequently asked questions

What is Spatial-MGCN?

Spatial-MGCN is a multi-view graph convolutional network that learns from spatial and expression-similarity graphs and fuses their embeddings with attention.

How are the two graphs different?

The spatial graph uses a coordinate radius; the feature graph uses k-nearest neighbours under expression cosine similarity.

What is co-convolution?

It is a branch designed to learn information shared by the spatial and feature graph views.

Why use a ZINB decoder?

A zero-inflated negative binomial likelihood can represent discrete, over-dispersed and sparse expression counts.

What does spatial regularization do?

It rewards high embedding similarity for declared spatial neighbours and low similarity for spatial non-neighbours.

Do attention weights explain biology?

No. They quantify view importance for the trained objective under chosen graphs and loss weights.

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Primary source and fact-check note

Bo Wang, Jiawei Luo, Ying Liu et al., “Spatial-MGCN: a novel multi-view graph convolutional network for identifying spatial domains with attention mechanism,” Briefings in Bioinformatics 24(5), bbad262 (2023), DOI 10.1093/bib/bbad262. 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

• STAGATE graph attention autoencoding
• CCST contrastive graph clustering
• SpaceFlow distance regularization
• How Mathematics Works

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

Extended laboratory: topic-specific validation and responsible use

Spatial graph audit

Preserve coordinate units, radius and degree.

Feature graph audit

Map cosine neighbours and long edges.

View audit

Compare spatial, feature and co-convolution ablations.

Attention audit

Report view weights across seeds.

Count audit

Compare NB and ZINB residuals.

Loss audit

Publish every raw and weighted component.

Transfer audit

Hold out whole sections and platforms.

Final claim

Attention and imputation remain model outputs.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Spatial Euclidean distance

Begin with this mechanism: Coordinate differences define physical proximity. Recreate the concrete case: Two spots are 400 coordinate units apart. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Units differ across platforms. Complete the transfer step: Preserve and report units. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 2: Radius adjacency

Begin with this mechanism: Edges appear when distance is no greater than r. Recreate the concrete case: The paper used r equal to 550 in examples. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A copied radius may be meaningless elsewhere. Complete the transfer step: Plot degree across r. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 3: Cosine expression similarity

Begin with this mechanism: Dot products are normalized by vector lengths. Recreate the concrete case: Parallel profiles have similarity near one. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Low-norm noisy vectors are unstable. Complete the transfer step: Filter and inspect norms. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 4: Feature kNN graph

Begin with this mechanism: Each spot connects to its most similar expression neighbours. Recreate the concrete case: k equals ten yields ten outgoing candidates. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Distant tissue can become connected. Complete the transfer step: Map long edges. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 5: Self-loop augmentation

Begin with this mechanism: Identity edges retain a spot own features. Recreate the concrete case: A-tilde equals A plus I. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Self-mass can dominate sparse graphs. Complete the transfer step: Inspect diagonal influence. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 6: Degree normalization

Begin with this mechanism: D inverse square-root scales both ends of each edge. Recreate the concrete case: A hub message receives smaller weight. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Normalization is not biological calibration. Complete the transfer step: Compare degree-stratified error. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 7: Spatial convolution

Begin with this mechanism: Graph layers aggregate expression on the spatial graph. Recreate the concrete case: Hs updates through normalized As and ReLU. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Repeated layers can oversmooth. Complete the transfer step: Measure boundary contrast by depth. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 8: Feature convolution

Begin with this mechanism: A parallel branch aggregates expression-similar spots. Recreate the concrete case: Hf updates through normalized Af. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Feature leakage may connect batches. Complete the transfer step: Hold out sections. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 9: Co-convolution

Begin with this mechanism: A shared branch captures dependencies across both views. Recreate the concrete case: Hc contains jointly learned features. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Shared structure can reflect shared artifact. Complete the transfer step: Use single-view ablations. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 10: View attention

Begin with this mechanism: Tanh scores and softmax yield weights for Hs, Hf and Hc. Recreate the concrete case: Weights 0.2, 0.3 and 0.5 sum to one. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Attention is objective-dependent. Complete the transfer step: Report weights across seeds. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 11: Fused embedding

Begin with this mechanism: A linear layer transforms the weighted view sum. Recreate the concrete case: H has thirty latent features per spot. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Width affects bottleneck strength. Complete the transfer step: Sweep latent dimension. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 12: ZINB mean

Begin with this mechanism: The decoder predicts a positive expected count. Recreate the concrete case: Mean five with dispersion two is over-dispersed. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Normalization and raw-count use must match. Complete the transfer step: Document decoder input and target. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 13: Zero-inflation probability

Begin with this mechanism: A mixture component adds excess probability at zero. Recreate the concrete case: Pi equals 0.3 for a spot-gene pair. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: NB alone may explain many zeros. Complete the transfer step: Compare NB and ZINB. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 14: ZINB negative log-likelihood

Begin with this mechanism: Observed counts score decoder parameters. Recreate the concrete case: A low probability count adds a large loss. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Likelihood fit is not domain truth. Complete the transfer step: Inspect held-out genes. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 15: Spatial contrast loss

Begin with this mechanism: Cosine scores distinguish neighbours from non-neighbours. Recreate the concrete case: Sigmoid maps a similarity to a pair score. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: False negatives can split related regions. Complete the transfer step: Try distance-banded negatives. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 16: Weighted total loss

Begin with this mechanism: Alpha, beta and gamma balance consistency, count fit and space. Recreate the concrete case: Gamma doubles the neighbour penalty. Write coordinate distances, radius edges, cosine similarities, k-neighbour sets, normalized adjacency entries, graph messages, view embeddings, attention weights, count means, dispersions, zero probabilities, likelihood terms and contrast scores with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Scales are not directly comparable. Complete the transfer step: Publish raw component curves. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.