Exact local subdivision
=======================

When to split a model
---------------------

Subdivision replaces one fitted cell by two local Bézier models whose images
together reproduce the parent. Use it to isolate a region for inspection,
evaluate pieces in separate downstream systems, or prepare a later local
refinement. It is useful when one global model spans regions with different
geometric scales and you want explicit charts for those regions.

Subdivision itself is an exact change of parameterization. It does not
collect observations, optimize coefficients, change polynomial degree, or
reduce prediction error. If a parent underfits a localized feature, its exact
children still underfit that feature. Error reduction requires an additional
refitting step, usually with new observations or independently adjustable
piecewise coefficients. Keep this distinction visible in before/after plots.

``subdivide`` acts on one selected cell. It returns a ``SubdivisionResult``
with two models and two parameter maps. It does not automatically update
neighboring cells of an assembled domain, rebuild shared topology, or create
a globally remeshed model archive. The maps retain the selected parent cell
ID so that evaluation can be traced back to the original model.

A complete exact split
----------------------

The example fits a scalar quadratic on a segment and splits the parameter
edge at fraction .4. Each child has its own full local barycentric domain.
The map converts those local points back to the appropriate parent interval.

.. testcode:: splitting

    import torch
    from torch_bsf import BezierSpace, Samples, fit
    from torch_bsf.sampling import grid
    from torch_bsf.splitting import EdgeSplit, LongestEdge, subdivide

    space = BezierSpace.simplex(1, degree=2)
    observations = grid(space, 8)
    x = observations.parameters[:, 1]
    model = fit(space, Samples(observations, x.square()[:, None])).require_success().model
    pieces = subdivide(model, cell=0, policy=EdgeSplit(0, 1, fraction=.4))
    for child, mapping in zip(pieces.models, pieces.maps, strict=True):
        child_points = grid(child.space, 40)
        parent_points = mapping.to_parent(child_points)
        assert torch.allclose(child(child_points), model(parent_points), atol=1e-12)
    assert len(pieces.models) == len(pieces.maps) == 2

The selected parent is unchanged. Child coefficient tensors retain its dtype
and device, while each child receives a single-cell space. The representation
is local even if the original model was assembled from many cells. Always
pair each child with its corresponding map; swapping maps produces an
incorrect comparison despite identical coordinate widths.

.. figure:: ../_static/guides/subdivision.png
   :width: 100%
   :alt: A parent curve and two child curves displayed in the same parent parameter coordinate.

   Before: one fitted cubic. After: two exact pieces at fraction .4, plotted
   against parent coordinates. The curves join without changing the represented
   response. The generator checks maximum child/parent discrepancies below
   1e-12; the figure does not claim a reduction in fitting error.

The figure uses the noisy-data cubic from
``examples/generate_guide_figures.py``, while the executable example above
uses an exactly representable quadratic. The same map invariant applies in
both cases. The figure's measured errors are stored in
:download:`metrics.json <../_static/guides/metrics.json>`.

Understand the split fraction
-----------------------------

For an edge joining factor vertices :math:`v_i` and :math:`v_j`, fraction
:math:`s` inserts a new vertex

.. math::

   v_{\mathrm{new}}=(1-s)v_i+s v_j,\qquad 0<s<1.

The first child replaces vertex :math:`j` by the new vertex, and the second
replaces vertex :math:`i`. With child coordinates :math:`u`, the first map is

.. math::

   t_i=u_i+(1-s)u_j,\qquad t_j=s u_j,

while the second map is

.. math::

   t_i=(1-s)u_i,\qquad t_j=s u_i+u_j.

Other coordinates in that factor are unchanged. Other factors of a product
are also unchanged. These formulas preserve nonnegativity and the per-factor
sum of one, so each full child simplex maps into its parent piece.

The first piece covers parent points satisfying
:math:`t_j\le s(t_i+t_j)` and the second covers
:math:`t_j\ge s(t_i+t_j)`. Points on the dividing face belong to both.
They must produce the same output under either exact representation.
The child coefficients are obtained by Bézier subdivision, not by fitting
outputs sampled at a few new locations.

Increasing ``fraction`` moves the new vertex toward vertex ``j``. It enlarges
the first child's portion of the selected edge and shrinks the second's.
``fraction=.5`` bisects it. Values near zero or one create a very small
piece, which may be useful for a localized region but can magnify coordinate
conversion errors. Fractions must be strictly between zero and one; endpoints
would create a degenerate piece.

``i`` and ``j`` are distinct vertex indices within the selected factor,
not global coefficient indices or output-coordinate indices. On a prism of
type ``(2, 1)``, ``factor=0`` selects the triangular factor with three vertices,
while ``factor=1`` selects the segment factor with two. Raising polynomial
degree does not add vertices to these factors. Check geometry rather than
control-net size when choosing an edge.

Choose a policy
---------------

.. list-table:: Policies and appropriate uses
   :header-rows: 1
   :widths: 24 38 38

   * - Policy
     - Selection
     - What to check
   * - ``EdgeSplit(i, j, fraction=.5, factor=0)``
     - Explicit edge and location.
     - Use when the region of interest is known in parameter coordinates.
   * - ``LongestEdge(fraction=.5)``
     - Longest chord between selected cell vertex images.
     - Output units affect which chord is longest.
   * - ``ResidualSplit(samples, resolution=10)``
     - Edge/fraction minimizing the sum of unweighted child MSEs over a grid.
     - Groups current errors; does not refit or guarantee lower error.

``LongestEdge`` is a geometry heuristic based on vertex chords in output
space. It does not measure the arc length of a curved edge or locate the
largest curvature. A loop whose endpoint images coincide can have a short
chord despite a long curved path. Rescaling one output coordinate can change
which chord is selected, so keep units consistent when comparing policies.

``ResidualSplit`` requires equally weighted observations on the selected
cell. It uses current model residuals to assess how observations group into
two pieces. A larger ``resolution`` examines a finer split-fraction grid and
costs more evaluation work. It does not increase the model's degree or add
training points. Since the score sums two child means rather than recomputing
a single observation-weighted global MSE, it can favor partitions with very
different point counts. Inspect those counts and the selected location before
interpreting the policy as an accuracy improvement.

No policy changes the exactness guarantee: once an edge and fraction have
been selected, the two local models represent the same parent function.
Choose a policy according to the next operation, such as balanced local data
sets or a physically meaningful region boundary, rather than expecting one
policy to repair the parent's bias automatically.

Map existing observations into children
---------------------------------------

A child model expects child coordinates. Passing parent coordinates directly
may produce a plausible but unrelated response. Use the inverse map to obtain
both transformed coordinates and a membership mask:

.. testcode:: splitting

    first_child = pieces.models[0]
    first_map = pieces.maps[0]
    local_points, mask = first_map.from_parent(observations)
    assert len(local_points) == int(mask.sum())
    assert torch.allclose(first_child(local_points), model(observations)[mask], atol=1e-12)
    local_values = x.square()[:, None][mask]
    local_samples = Samples(local_points, local_values)
    local_fit = fit(first_child.space, local_samples).require_success()
    assert local_fit.metrics["data_mse"] < 1e-24

``from_parent`` ignores rows from other parent cells and returns a mask over
the original batch. Apply that mask to values and weights as well as parameters.
The returned child ``Points`` use the local single-cell ID. Do not keep the
parent's cell ID when feeding them to the child. To compare the child with
an assembled parent, call ``to_parent`` so the correct original ID is restored.

Points on the cut can appear in both child data sets. Decide whether that is
desired for subsequent fitting and evaluation, and avoid double-counting them
in a pooled global error. Exact children agree on the cut, but two independent
refits need not agree there. A shared continuity constraint or a newly assembled
domain is required if that equality must remain exact after coefficients move.

Refitting and recursive use
---------------------------

If the goal is a more accurate piecewise surrogate, first preserve the exact
split as a baseline. Map observations, inspect per-piece coverage, obtain
additional true responses where needed, and refit each admissible model.
Compare every candidate in parent coordinates on an independent evaluation
set. This makes it clear whether improvement comes from new information,
extra piecewise freedom, or merely a denser plot.

Refitting the same degree on smaller domains can resolve localized behavior,
but it increases total independent coefficients and may overfit a small child
with few observations. Check free-control count and data rank separately for
each child. Keep important boundary values fixed or compile a glued space if
the pieces must remain continuous. Smoothness penalties within separate child
models do not by themselves enforce agreement across their interface.

For recursive subdivision, each returned map relates one level of child
coordinates to that level's parent. Compose the maps in order to reach the
original chart, and retain the path of cell IDs. Repeatedly splitting does
not automatically compose maps or maintain a hierarchy for you. Save the
hierarchy in application metadata if downstream consumers need it.

A practical stopping rule considers local held-out error, minimum sample
count and smallest permitted region size. Stopping only when training error
is tiny can generate many unsupported pieces. A split at an extreme fraction
can also leave one child without enough informative measurements; check this
before launching a local degree search or additional experiment batch.

For adjacent cells in an assembled space, independently subdividing one patch
leaves neighboring topology unchanged. The returned models can be visualized
locally, but they are not a conforming replacement mesh. Rebuild and validate
that mesh explicitly when it is required. See :doc:`simploidal` for shared
coefficient construction, :doc:`partial` for exact boundary values, and
:doc:`auto_degree` for deciding whether additional local polynomial capacity
is supported by independent errors.

Keep comparisons in the same physical coordinates
-------------------------------------------------

A useful split report includes the parent cell, factor and edge, the chosen
fraction, child observation counts, and the maximum child-versus-parent
prediction difference before any refit. Then report independent errors after
refitting as a second comparison. This sequence distinguishes exact geometry
preservation from an actual improvement in the surrogate.

For a scalar graph, plot against the parent parameter or a known physical
coordinate, as the figure does. Plotting both children against their own local
zero-to-one coordinate stretches each piece to the same width and can make
them appear unrelated to the parent. For an output-space image, use the same
projection and common axis limits. A different control polygon after splitting
is expected and is not itself a change of the represented function.

Retain original observation IDs when applying membership masks so that later
error reports can identify repeated cut-boundary rows. In a recursive process,
store the complete mapping path beside each leaf model. A local model archive
alone cannot reconstruct which portion of the original experiment its local
coordinates describe.