Welcome to PyTorch-BSF!

Fit smooth manifolds with explicit geometry and numerical evidence.

PyTorch-BSF brings Bézier Simploid Fitting (BSF) to PyTorch. A Bézier simplex is a high-dimensional generalization of the Bézier curve: it can model an arbitrarily complex point cloud as a smooth parametric hyper-surface in any number of dimensions (see Bézier simplex fitting). Bézier simploids go further: they are tensor products of Bézier simplices defined on products of simplices such as prisms and cubes, and simploidal sets glue such cells along common faces. This makes them a natural tool for representing Pareto fronts in multi-objective optimization, interpolating scattered observations, and fitting geometric structures in high-dimensional spaces.

The project is on GitHub.

Note

BSF formerly stood for Bézier Simplex Fitting, which remains the special case where the parameter domain is a single simplex.

Practical guides include reproducible comparisons, parameter tuning and interpretation of numerical evidence for the unified API.

User guide

• What is Bézier simplex fitting?
  • Bézier Simplex
  • Bézier Simplex Fitting
  • Approximation Theorem
  • Beyond Simplices: Bézier Simploids
  • Relation to Multi-Objective Optimization
  • Statistical Test for Simplicial Topology
• Installation
  • Choose dependencies for your workflow
  • Check the environment you actually run
  • Upgrading from the earlier API
• Quickstart
  • Verified Python workflow
  • Prepare your own observations
  • Read results before choosing a more complicated model
  • Next steps and reproducibility
• Migrating to the unified v0 API
  • Separate geometry, observations and fitted state
  • Translate penalties and constraints deliberately
  • Replace implicit preprocessing and model storage
  • Validation and integration changes
• Advanced Topics
  • Choose a guide by the question you need to answer
  • Follow a fit through the guides
  • Guide chapters
• Applications
  • What is fitted in each application?
  • A common reproduction and validation workflow
  • Choosing a domain and a next guide
• Frequently asked questions
  • Do I need Conda, Lightning or MLflow?
  • How do I predict?
  • Does solver success imply identifiable coefficients?
  • How do I choose a degree?
  • Where did the old APIs go?
  • What are the objects I need for an ordinary fit?
  • Why does a one-column input produce a constant model?
  • How do I choose between simplex, product and assembled spaces?
  • Why can a successful model still be poor?
  • What should I adjust when the fit oscillates?
  • When should I normalize outputs?
  • Are control points predicted values on a grid?
  • How can I preserve a known endpoint exactly?
  • Will splitting reduce my prediction error?
  • Does active learning guarantee better accuracy each round?
  • Should I switch to the sparse solver or a GPU?
  • Why does optimization report failure after using every iteration?
  • What happens after a numerical solve fails?
  • How should I save a model and reproduce its figures?
  • Can I compare before/after runs with different evaluation points?
• References
  • Fitting, approximation and statistical behavior
  • Topology and domain construction
  • Application sources and reproducible examples
  • Bibliography

API reference

• API reference guide
  • Python API

Indices and tables

• Index

• Module Index

• Search Page