References ========== The bibliography supports the mathematical background and the application formulations discussed in this documentation. Use the links from each chapter to identify the result relevant to its assumptions. A theorem about the reference optimization problem does not by itself give an error bound for a finite-data Bézier fit. Fitting, approximation and statistical behavior ----------------------------------------------- :cite:t:`kobayashi2019bezier` introduces Bézier simplex fitting for Pareto fronts and its approximation perspective. :cite:t:`tanaka2020asymptotic` studies statistical risk. :doc:`advanced/skeleton_sampling` extends its optimal sample ratio to every dimension and degree, and :doc:`advanced/skeleton_risk_theory` proves the result, using the polynomial small-ball inequality of :cite:t:`carbery2001distributional`. Read these alongside :doc:`whatis` and :doc:`advanced/model_selection`: expressiveness at a sufficiently large degree and accuracy from a particular finite sample are distinct questions. :cite:t:`hikima2025stochastic` concerns objective-based stochastic optimization of a Bézier representation. Its setting is discussed in :doc:`advanced/objective_minimization`; it should be distinguished from linear least-squares regression on known parameter-response pairs. Topology and domain construction -------------------------------- :cite:t:`mizota2021unconstrained` provides the unconstrained strongly convex results stated in :doc:`whatis`. Check smoothness, feasibility and objective assumptions before transferring a statement to a constrained application. Weak simpliciality permits a many-to-one parameterization; it is stronger to claim that the image is homeomorphic to a simplex. :cite:t:`hamada2018data,hamada2020test` discuss topology assessment from data under their stated assumptions. A sampled picture or failure to detect a hole is not independently a proof of simplex topology. For product cells and their faces, see :cite:t:`peltier2009simploidal,peltier2018simploidal` and the geometric introduction in :doc:`advanced/simploidal`. The mathematical continuation in :doc:`advanced/simploidal_category` uses the category-theory background of :cite:t:`maclane1998categories,maclane1992sheaves`. Shape-preserving interpolation on simplicial meshes (:doc:`advanced/interpolation`) generalizes the monotone piecewise cubic interpolation of :cite:t:`fritsch1980monotone`. Its face controls use the degree elevation of Bézier triangles described by :cite:t:`farin2002curves`. Application sources and reproducible examples --------------------------------------------- Each :doc:`application ` links its formulation to relevant literature and names the script used for its figures. These scripts are small numerical illustrations of specified targets. Consult the page's parameter range, validation set and feasibility checks when interpreting its reported error; a reference to a broader application does not make the toy experiment an implementation of that complete system. Bibliography ------------ .. bibliography::