Pullbacks (preimages) of manifold subsets under continuous maps

mkoeppe commented 3 years ago

Similar to #31653, given a continuous map \Phi: N -> M and a manifold subset S of M, we define the pullback (preimage) of S as the subset of N of points p with \Phi(p) in S.

Given a real scalar field Phi: N -> R and a RealSet S, we define the pullback in the same way.

Also, we view a chart C as a continuous function Phi: C.domain() -> R^n and allow pulling back any subset of R^n (an object with a __contains__ method; for example polyhedra, lattices, linear subspaces, tensor modules) by it as well.

In all cases, because Phi is continuous, topological closures/interiors pull back.

An application is in #31981.

Depends on #31883 Depends on #31904 Depends on #31653 Depends on #31916 Depends on #31644 Depends on #31959 Depends on #31990 Depends on #21243

CC: @egourgoulhon @tscrim @mjungmath

Component: manifolds

Author: Matthias Koeppe

Branch/Commit: 4558e26

Reviewer: Eric Gourgoulhon

Issue created by migration from https://trac.sagemath.org/ticket/31688

mkoeppe commented 3 years ago

Changed dependencies from #31883, #31904, #31653, #31916 to #31883, #31904, #31653, #31916, #31959

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

`6bef52b`	`ConvexSet_base._test_convex_set: Run the testsuite of relint`
`6ab5677`	`RelativeInterior.is_universe: New`
`c085d30`	`Polyhedron_base.interior: Handle the empty polyhedron correctly`
`686d0af`	`Polyhedron_base.product: Add doctest for alias 'cartesian_product'`
`2b1d108`	`Merge #31919`
`7323b10`	`ConvexSet_base._test_contains: Only test extension to AA for exact base rings`
`e0d2eeb`	`Merge #31959`
`e6c3ed5`	`RelativeInterior.__hash__: New`
`4c7d57f`	`src/sage/geometry/relative_interior.py: Fix doctest output`
`9ea0cea`	`ManifoldSubsetPullback: Do not trip over relative interiors of cones`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from e802a21 to 9ea0cea

egourgoulhon commented 3 years ago

comment:34

As revealed by the patchbot, there are some doctest failures, as well as some coverage and pyflakes issues.

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from 9ea0cea to d591234

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

`94e6858`	`RelativeInterior.ambient, ambient_vector_space, is_universe: New`
`0c9bc94`	`ConvexSet_base: Add default implementations of ambient, ambient_dim; add doctests`
`d591234`	`Merge #31959`

mkoeppe commented 3 years ago

comment:36

Thanks for taking a look! Yes, I'll have to add more documentation and examples.

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

87f3e30 ManifoldSubsetPullback: Add documentation and tests

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from d591234 to 87f3e30

mkoeppe commented 3 years ago

Changed dependencies from #31883, #31904, #31653, #31916, #31959 to #31883, #31904, #31653, #31916, #31959, #31990

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

`7d3ae5c`	`Merge #31959`
`2b1040c`	`ConvexSet_base.{an_element,some_elements}, Polyhedron.some_elements, RelativeInterior.some_elements: New`
`a884517`	`PolyhedronFace.some_elements: New`
`f6cc8f7`	`Merge #31990`
`1a4805c`	`ManifoldSubsetPullback: For charts, use .point as the inverse`
`754e002`	`ManifoldSubsetPullback._an_element_, some_elements: Implement`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from 87f3e30 to 754e002

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from 754e002 to a8e607c

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

a8e607c ManifoldSubsetPullback._coord_def: Add documentation and examples

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

`ce91e44`	`src/sage/geometry/relative_interior.py: Fix doctest output`
`b19b3cc`	`Merge #31959`
`c49ab33`	`Merge #31990`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from a8e607c to c49ab33

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from c49ab33 to 7ae21da

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

`f0e7c58`	`ambient_vector_space docstring: Fix bad blocks`
`200d967`	`ConvexSet_base.ambient doctest: Actually test the method`
`1c2bef5`	`Merge #31959`
`ed898e6`	ConvexSet_base.an_element, some_elements: Go through generator method `_some_elements_`, make some_elements return a list
`7ae21da`	`Merge #31990`

egourgoulhon commented 3 years ago

Reviewer: Eric Gourgoulhon

egourgoulhon commented 3 years ago

comment:44

Thanks for the last changes. Some comments regarding the documentation:

there is no entry for the class ManifoldSubsetPullback in the reference manual
in the files continuous_map.py and subsets/pullback.py, each instance of :class:`ManifoldSubset` should be replaced by :class:`~sage.manifolds.subset.ManifoldSubset`
the docstring of ScalarField.pullback should have an INPUT and OUTPUT sections similar to those of ContinuousMap.preimage. By the way, to be consistent with ContinuousMap.preimage, shouldn't ScalarField.pullback be renamed to ScalarField.preimage, with pullback defined as an alias to preimage?
the INPUT and OUTPUT sections of DiffMap.pullback should be updated to take into account the new arguments.

mkoeppe commented 3 years ago

Changed dependencies from #31883, #31904, #31653, #31916, #31959, #31990 to #31883, #31904, #31653, #31916, #31644, #31959, #31990

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from 7ae21da to 3877c5f

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

`8d3ad85`	`Merge #31764`
`60505b0`	`Merge #31798`
`b9909c0`	`ManifoldSubsetClosure: Add examples, fix docstring markup`
`16c6a72`	`ManifoldSubsetClosure, ManifoldSubset.closure: Improve documentation`
`2514586`	`Merge #31644`
`3ca139a`	`src/doc/en/reference/manifolds: Add pullback`
`1d430ee`	`src/sage/manifolds/continuous_map.py, src/sage/manifolds/subsets: Use qualified class name in docstrings`
`1266de9`	`src/sage/geometry/convex_set.py: Fix docstring markup`
`df64125`	`ScalarField.preimage: Make pullback an alias; expand documentation`
`3877c5f`	`ManifoldSubsetPullback.is_closed: Remove unfinished NNC_Polyhedron example`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

`0c549f0`	`Chart.pullback, preimage: New`
`7bb720e`	`ManifoldSubsetPullback.__contains__: Handle the case of charts, which return tuples of coordinates, not vectors`
`60ea044`	`ContinuousMap.preimage, ScalarField.preimage: Document that output can also be a manifold`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from 3877c5f to 60ea044

mkoeppe commented 3 years ago

Changed dependencies from #31883, #31904, #31653, #31916, #31644, #31959, #31990 to #31883, #31904, #31653, #31916, #31644, #31959, #31990, #21243

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

`5b8ceca`	InternalRealInterval, RealSet: Add `_latex_` methods
`69ca854`	`RealSet._repr_: Use unicode cup sign instead of +`
`dbdfc06`	`InternalRealInterval, RealSet: Remove extra whitespace in latex, add documentation`
`a0e3bf5`	`Merge #31880`
`181b128`	`RealSet.is_open, is_closed, closure, interior, boundary: New`
`7f56338`	`PiecewiseFunction: Adjust doctests for changed RealSet repr`
`8cd74b5`	`Merge #31880`
`8fdb104`	`RealSet.boundary: Add another doctest`
`9d41f0e`	`src/sage/sets/real_set.py: Remove unnecessary import`
`c36e43b`	`Merge #21243`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from 60ea044 to c36e43b

egourgoulhon commented 3 years ago

comment:50

In the docstring of ManifoldSubsetPullback:

in the INPUT section, there seems to be no entry for ConvexSet_base in the documentation
in the following part of the EXAMPLES block:

sage: r_squared = M.scalar_field(x^2+y^2)
sage: r_squared.set_immutable()
sage: cl_I = RealSet((1, 4)); cl_I
(1, 4)
sage: cl_O = ManifoldSubsetPullback(r_squared, None, I); cl_O

I guess I in the last line (which, at this stage, is the imaginary number i) should be replaced by cl_I and the subsequent True and False should become False and True respectively. But then, cl_I and cl_O are identical to I and O in the example that follows...

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from c36e43b to c540d35

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

c540d35 ManifoldSubsetPullback: Fix up some doctests with closed intervals

mkoeppe commented 3 years ago

comment:52

Thanks for spotting this - I have fixed this and a similar example

mkoeppe commented 3 years ago

comment:53

Replying to @egourgoulhon:

In the docstring of ManifoldSubsetPullback:

in the INPUT section, there seems to be no entry for ConvexSet_base in the documentation

You may need to run ./sage -docbuild reference/discrete_geometry html

egourgoulhon commented 3 years ago

comment:54

Replying to @mkoeppe:

Replying to @egourgoulhon:

In the docstring of ManifoldSubsetPullback:

in the INPUT section, there seems to be no entry for ConvexSet_base in the documentation

You may need to run ./sage -docbuild reference/discrete_geometry html

Yes indeed!

egourgoulhon commented 3 years ago

comment:55

Replying to @mkoeppe:

Thanks for spotting this - I have fixed this and a similar example

Thanks.

At the moment, we have

sage: M = Manifold(2, 'M')                                                                          
sage: X.<x,y> = M.chart()                                                                           
sage: M.identity_map().preimage(M)                                                                  
Subset Id_M_inv_M of the 2-dimensional differentiable manifold M
sage: M.identity_map().preimage(M) is M                                                             
False
sage: M.zero_scalar_field().preimage(RealSet.point(0))                                              
Subset zero_inv_{0} of the 2-dimensional differentiable manifold M
sage: M.zero_scalar_field().preimage(RealSet.point(0)) is M                                         
False

Do you think it would be easy to have both answers to be True? If no, this should not hamper the current ticket, but be differed to another ticket.

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from c540d35 to facad97

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

facad97 ContinuousMap.preimage: Handle identity_map specially

mkoeppe commented 3 years ago

comment:57

Sure, that's easy to do and certainly a good idea. Here it is for ContinuousMap.preimage already, I'll look into the case of the scalar fields

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

c2ecf3e ScalarField.preimage: Handle the case of the zero scalar field

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from facad97 to c2ecf3e

mkoeppe commented 3 years ago

comment:59

More general things such as detecting that the preimage of [0,oo) under squaring is the full domain will have to wait until a follow-up ticket

egourgoulhon commented 3 years ago

comment:60

Replying to @mkoeppe:

Sure, that's easy to do and certainly a good idea. Here it is for ContinuousMap.preimage already,

Thanks. Maybe one can add

if self._is_isomorphism and codomain_subset == self._codomain:
    return self._domain

egourgoulhon commented 3 years ago

comment:61

The patchbot reports a doctest error in src/sage/geometry/polyhedron/base.py as well as pyflakes errors in src/sage/manifolds/subsets/pullback.py.

mkoeppe commented 3 years ago

comment:62

Replying to @egourgoulhon:

Maybe one can add

if self._is_isomorphism and codomain_subset == self._codomain:
    return self._domain

I think I can do more generally

  if self._codomain.is_subset(codomain_subset):
      return self._domain

mkoeppe commented 3 years ago

comment:63

Replying to @egourgoulhon:

The patchbot reports a doctest error in src/sage/geometry/polyhedron/base.py

I've seen this one before locally, I don't think it's coming from this ticket, but I have opened #32030 for it

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from c2ecf3e to 3b1c428

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

3b1c428 ManifoldSubsetPullback: Make codomain_subset required 2nd init arg; fix pycodestyle/pyflakes warnings

egourgoulhon commented 3 years ago

comment:65

Replying to @mkoeppe:

Replying to @egourgoulhon:
Maybe one can add
if self._is_isomorphism and codomain_subset == self._codomain:
    return self._domain
I think I can do more generally
  if self._codomain.is_subset(codomain_subset):
      return self._domain

Yes, indeed! Being an isomorphism is not necessary in that case.

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Changed commit from 3b1c428 to d321b93

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 3 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

d321b93 ContinuousMap: Return domain if the map's codomain is contained in the given subset

sagemath / sage

Pullbacks (preimages) of manifold subsets under continuous maps #31688