Chapter map

1. 1. Quality filtering
2. 2. Log normalization
3. 3. Principal components
4. 4. Z-score scaling
5. 5. Coordinate-to-grid map
6. 6. Batch normalization
7. 7. Convolutional block
8. 8. Refinement module
9. 9. MSE pre-training
10. 10. Pseudo-label assignment
11. 11. Cross-entropy loss
12. 12. L2 regularization
13. 13. Edge-strength loss
14. 14. Joint objective
15. 15. Multi-section alignment
16. 16. Domain-derived analysis

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. Quality filtering

Here the geometry becomes operational. Low-quality spots and genes are removed before modelling.

A concrete situation

A gene detected in too few spots is excluded.

What the mathematics does not license

Filtering can erase rare domains. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Publish thresholds and retained counts. 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. Log normalization

This step deserves a careful pause. Counts are scaled and transformed before PCA.

A concrete situation

A library-size-adjusted value becomes log one plus count.

What the mathematics does not license

Transformation changes variance. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Compare robust preprocessing. 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. Principal components

The useful question is not only how to calculate, but what the calculation assumes. Linear directions compress the expression matrix.

A concrete situation

Thirty PCs become tensor channels.

What the mathematics does not license

A low rank can omit weak anatomy. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Vary retained dimensions. 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. Z-score scaling

A small example makes the mechanism visible. Each channel is centred and scaled.

A concrete situation

One PC has mean zero and variance one.

What the mathematics does not license

Outliers affect mean and standard deviation. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect robust scaling. 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. Coordinate-to-grid map

This is where modelling choices begin to affect evidence. Every spot is assigned to a tensor position.

A concrete situation

A hexagonal spot becomes one occupied grid cell.

What the mathematics does not license

Irregular geometry creates empty positions. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot the occupancy mask. 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. Batch normalization

The central mathematical idea is concrete. Mini-batch channel statistics stabilize optimization.

A concrete situation

Activations are rescaled before convolution.

What the mathematics does not license

Small batches make estimates noisy. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare evaluation statistics. 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. Convolutional block

Here the geometry becomes operational. Local filters combine neighbouring tensor features.

A concrete situation

A three-by-three window mixes nearby spots.

What the mathematics does not license

A rectangular kernel approximates assay geometry. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Test receptive fields. 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. Refinement module

This step deserves a careful pause. Later layers sharpen the latent segmentation representation.

A concrete situation

yn contains d output features.

What the mathematics does not license

Refinement can amplify early mistakes. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Run ablations and 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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9. MSE pre-training

The useful question is not only how to calculate, but what the calculation assumes. Squared differences initialize output near input features.

A concrete situation

Residuals one and two contribute one and four.

What the mathematics does not license

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

Try it, check it, transfer it

Inspect channel-wise 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.

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10. Pseudo-label assignment

A small example makes the mechanism visible. Current output features create temporary classes.

A concrete situation

Argmax chooses one label for a spot.

What the mathematics does not license

Hard labels hide uncertainty. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Keep margins and entropy. 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. Cross-entropy loss

This is where modelling choices begin to affect evidence. Predicted class probabilities are scored against pseudo-labels.

A concrete situation

Minus log 0.8 is about 0.223.

What the mathematics does not license

The target is model-generated. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Delay and refresh labels carefully. 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. L2 regularization

The central mathematical idea is concrete. Large weights receive a quadratic penalty.

A concrete situation

Doubling a weight quadruples its penalty.

What the mathematics does not license

Too much shrinkage underfits. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Sweep the coefficient. 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. Edge-strength loss

Here the geometry becomes operational. Adjacent output differences are penalized.

A concrete situation

A sharp latent jump adds cost.

What the mathematics does not license

Real boundaries also have jumps. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Validate thin structures. 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. Joint objective

This step deserves a careful pause. Segmentation and edge losses are weighted together.

A concrete situation

Lambda controls spatial coherence.

What the mathematics does not license

One scale may not suit every tissue. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Publish a sensitivity grid. 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. Multi-section alignment

The useful question is not only how to calculate, but what the calculation assumes. Comparable learned domains support adjacent-section integration.

A concrete situation

Two brain slices align cortical layers.

What the mathematics does not license

Shared labels do not prove cell identity. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Hold out a full section. 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. Domain-derived analysis

A small example makes the mechanism visible. SVG and colocalization tests reuse domain assignments.

A concrete situation

A gene is enriched in one segment.

What the mathematics does not license

Upstream errors propagate. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Repeat downstream tests across segmentations. 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

SpaSEG turns a spatial transcriptomics experiment into an image-like tensor. Each measured spot becomes a pixel position and its principal-component feature vector becomes a set of channels. This change of representation lets ordinary convolutional operations learn from nearby locations without pretending that tissue is literally a photograph. Empty grid positions, irregular assay geometry, coordinate scaling and the mapping from spots to tensor cells remain modelling choices that must be recorded.

Training begins with a mean-squared reconstruction objective between the input feature vector and the network output. The method then assigns pseudo-labels from the current representation and optimizes a regularized cross-entropy segmentation loss together with an edge-strength loss. Cross-entropy makes the network predict its current classes more consistently; edge regularization discourages abrupt feature changes between adjacent tensor positions. The two terms can reduce fragmented domains, yet too much smoothing can erase a narrow true boundary.

The published architecture uses batch normalization, two stacked convolutional blocks and a refinement module. The final domain map supports adjacent-section integration, domain-specific spatially variable gene analysis and cell-colocalization summaries. These later tasks inherit every upstream choice. A domain-specific gene is conditional on the segmentation; an interaction map based on colocalization remains spatial association rather than proof of signalling.

SpaSEG reported tests across sequencing- and imaging-based technologies and large tissue maps. Performance comparisons are useful evidence under those datasets and metrics, not a universal guarantee. A strong analysis repeats pseudo-label initialization, loss weights, tensor resolution and domain count; preserves complete sections for testing; examines thin structures and missing tissue; and validates boundaries with independent anatomy or markers.

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Frequently asked questions

What is SpaSEG?

SpaSEG is an unsupervised convolutional method that represents spatial transcriptomics as an image-like tensor and learns coherent tissue domains.

What becomes a tensor channel?

After filtering, normalization, PCA and scaling, each spot has a low-dimensional feature vector whose components serve as channels at that spatial position.

Where do labels come from?

They are pseudo-labels derived iteratively from the current learned representation, not externally supplied ground truth.

Why use an edge-strength loss?

It discourages abrupt changes between adjacent tensor positions and therefore reduces fragmented domains.

Can edge smoothing remove biology?

Yes. A strong spatial penalty can blur narrow layers or sharp interfaces, so boundaries and loss weights need sensitivity checks.

Do colocalized cells prove communication?

No. Colocalization and domain co-membership describe spatial association; signalling requires molecular and experimental evidence.

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Primary source and fact-check note

Yong Bai, Xiangyu Guo, Keyin Liu et al., “SpaSEG: unsupervised deep learning for multi-task analysis of spatially resolved transcriptomics,” Genome Biology 26, 230 (2025), DOI 10.1186/s13059-025-03697-1. 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

• SpaCell multimodal tissue learning
• Giotto HMRF spatial domains
• SpaceFlow graph-based domains
• 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

Tensor audit

Plot occupied and empty grid cells.

Feature audit

Version filtering, normalization, PCA and scaling.

Pre-training audit

Inspect channel-wise reconstruction error.

Pseudo-label audit

Keep margins, entropy and update schedule.

Edge audit

Test thin structures across smoothness weights.

Section audit

Reserve complete tissues for transfer.

Downstream audit

Repeat SVG and colocalization tests across segmentations.

Final claim

A coherent segment remains a model-derived domain.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Quality filtering

Begin with this mechanism: Low-quality spots and genes are removed before modelling. Recreate the concrete case: A gene detected in too few spots is excluded. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Filtering can erase rare domains. Complete the transfer step: Publish thresholds and retained counts. 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: Log normalization

Begin with this mechanism: Counts are scaled and transformed before PCA. Recreate the concrete case: A library-size-adjusted value becomes log one plus count. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Transformation changes variance. Complete the transfer step: Compare robust preprocessing. 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: Principal components

Begin with this mechanism: Linear directions compress the expression matrix. Recreate the concrete case: Thirty PCs become tensor channels. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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 low rank can omit weak anatomy. Complete the transfer step: Vary retained dimensions. 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: Z-score scaling

Begin with this mechanism: Each channel is centred and scaled. Recreate the concrete case: One PC has mean zero and variance one. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Outliers affect mean and standard deviation. Complete the transfer step: Inspect robust scaling. 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: Coordinate-to-grid map

Begin with this mechanism: Every spot is assigned to a tensor position. Recreate the concrete case: A hexagonal spot becomes one occupied grid cell. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Irregular geometry creates empty positions. Complete the transfer step: Plot the occupancy mask. 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: Batch normalization

Begin with this mechanism: Mini-batch channel statistics stabilize optimization. Recreate the concrete case: Activations are rescaled before convolution. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Small batches make estimates noisy. Complete the transfer step: Compare evaluation statistics. 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: Convolutional block

Begin with this mechanism: Local filters combine neighbouring tensor features. Recreate the concrete case: A three-by-three window mixes nearby spots. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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 rectangular kernel approximates assay geometry. Complete the transfer step: Test receptive fields. 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: Refinement module

Begin with this mechanism: Later layers sharpen the latent segmentation representation. Recreate the concrete case: yn contains d output features. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Refinement can amplify early mistakes. Complete the transfer step: Run ablations and 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 9: MSE pre-training

Begin with this mechanism: Squared differences initialize output near input features. Recreate the concrete case: Residuals one and two contribute one and four. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Large errors dominate. Complete the transfer step: Inspect channel-wise residuals. 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: Pseudo-label assignment

Begin with this mechanism: Current output features create temporary classes. Recreate the concrete case: Argmax chooses one label for a spot. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Hard labels hide uncertainty. Complete the transfer step: Keep margins and entropy. 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: Cross-entropy loss

Begin with this mechanism: Predicted class probabilities are scored against pseudo-labels. Recreate the concrete case: Minus log 0.8 is about 0.223. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: The target is model-generated. Complete the transfer step: Delay and refresh labels carefully. 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: L2 regularization

Begin with this mechanism: Large weights receive a quadratic penalty. Recreate the concrete case: Doubling a weight quadruples its penalty. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Too much shrinkage underfits. Complete the transfer step: Sweep the coefficient. 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: Edge-strength loss

Begin with this mechanism: Adjacent output differences are penalized. Recreate the concrete case: A sharp latent jump adds cost. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Real boundaries also have jumps. Complete the transfer step: Validate thin structures. 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: Joint objective

Begin with this mechanism: Segmentation and edge losses are weighted together. Recreate the concrete case: Lambda controls spatial coherence. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: One scale may not suit every tissue. Complete the transfer step: Publish a sensitivity grid. 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: Multi-section alignment

Begin with this mechanism: Comparable learned domains support adjacent-section integration. Recreate the concrete case: Two brain slices align cortical layers. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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 labels do not prove cell identity. Complete the transfer step: Hold out a full section. 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: Domain-derived analysis

Begin with this mechanism: SVG and colocalization tests reuse domain assignments. Recreate the concrete case: A gene is enriched in one segment. Write quality indicators, normalized counts, PC scores, grid masks, convolution windows, latent channels, squared residuals, pseudo-label probabilities, cross-entropies, edge differences, loss weights, boundary errors and section-level 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: Upstream errors propagate. Complete the transfer step: Repeat downstream tests across segmentations. 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.