Advanced Topics

The common space, model and observation types connect regression, optimization and optional integrations. The guides follow a fitting workflow: prepare observations, inspect predictions, then adjust the model or acquire more data. Continue with domain construction when you need products or connected cells. Objective minimization has its own sequence for differentiable costs without regression targets, followed by optional integrations and larger-scale solves.

Each practical guide explains when the feature is needed, provides runnable examples, and describes how settings affect results. Before/after figures state what changed and retain reproduction scripts and numerical evidence.

Choose a guide by the question you need to answer

Table 8 Practical entry points

Question	Start here	What to inspect afterward
Only points on a curve or surface are known; parameter coordinates are unknown.	Fitting from values alone	Local convergence, inferred coordinates and reconstruction distances
Outputs have very different scales.	Output scaling	Errors in original units
A fit oscillates between measurements.	Cross-validation and degree search, Regularization and numerical evidence	Independent prediction errors
Endpoint or boundary values are exact.	Fixed controls	Full boundary coverage of the constraints
Reference evaluations are expensive.	Explicit ensembles and active learning	Errors on a fixed external validation set
A measurement budget must be split among vertices, edges and faces.	Optimal skeleton sampling	Rows per face and each face’s least-squares rank
A plot is coarse or misses a feature.	Visualization, Grid sampling	Numerical residuals and render resolution
Independent parameter groups vary together.	Single and assembled domains	Per-factor sums and coordinate order
Neighboring local regions must agree.	Single and assembled domains	Predictions along identified faces
Exact data on a mesh must not overshoot.	Shape-preserving interpolation on simplicial meshes	Prediction range in each cell
A large fit exhausts memory.	Solvers, performance and optional execution	Sparse termination and conditioning
There are costs but no target responses.	Explicit initial models, Objective minimization	Full-sample objective and gradient stopping

Guide chapters

Start by recording the domain, output units and measurement process. Use sampling and visualization to examine coverage before changing the model. Then choose capacity with held-out data. A larger degree can reduce bias, but increases controls and can worsen conditioning; regularization imposes an additional assumption rather than adding measurements.

Keep the evaluation set fixed while comparing settings. If you change both degree and training locations, a better result cannot be attributed to either change alone. Boundary constraints, coefficient penalties and output transforms should be recorded with every comparison because they change what problem is being solved.

FitResult.success describes numerical termination. Observation rank describes whether data identify the free controls, while held-out errors describe predictive performance. None replaces the others. An accurate interior fit can still miss an important boundary, and a regularized full-rank system can still have an underdetermined data matrix.

For regression, follow the first two groups from data preparation through validation and refinement. The remaining groups are independent paths: choose them when your domain, objective or execution environment requires it.

Prepare data and fit

• Output scaling
• Sampling and grids
• Optimal skeleton sampling
• Theory of the skeleton risk
• Further evaluations of the skeleton risk
• Quadratic codimension-two multiplicities
• Uniform construction for codimension two
• Boundary transfer and a codimension-three edge bound
• Positive diagonal formulas for nonvertex risk
• Tetrahedron vertex risk
• Vertex coefficients at every fixed vertex count
• The vertex leading constant at large vertex count
• Explicit vertex errors at every fixed vertex count
• Single-log vertex errors for every fixed M at least five
• Tetrahedron vertex degree recurrence
• A simultaneous vertex-count and degree regime
• Quantitative tetrahedron vertex asymptotics
• Fixed-degree risk comparisons in every dimension
• An all-degree sufficient region for large vertex counts
• An explicit winning degree tail at every fixed vertex count
• Tetrahedron risk comparison at every degree
• Five-vertex vertex risk
• Five-vertex vertex degree recurrence
• Quantitative five-vertex vertex asymptotics
• Five-vertex risk comparison at every degree
• Six-vertex risk comparison at every degree
• Risk on an assembled simploidal set
• Exact verification of the skeleton risk
• Fitting from values alone

Inspect and improve a fit

• Visualization
• Grid sampling
• Cross-validation and degree search
• Regularization and numerical evidence
• Fixed controls
• Explicit ensembles and active learning

Start with the geometric simploid guide before interpolation or subdivision. Exact subdivision changes charts while preserving the represented function; assess it differently from refitting. The category-theory chapter is an optional mathematical continuation about faces, degeneracies and gluing.

Build and refine domains

• Single and assembled domains
• Shape-preserving interpolation on simplicial meshes
• Exact local subdivision
• The Category of Simploidal Sets

Optimize differentiable objectives

• Explicit initial models
• Objective minimization

Integrations and performance

• Scikit-learn integration
• Solvers, performance and optional execution