Arithmetic on tensor module elements, manifold objects: Always Return a Copy

[gh-mjungmath]

This question arose from ticket #30239, ​comment:36.

Should FiniteRankFreeModule and manifold objects always return a mutable copy, even for trivial operations? At least, this would be a consistent behavior.

As pointed out by Matthias, this already holds true for FreeModule:

sage: M = FreeModule(QQ, 3)
sage: v = M([1,2,3])
sage: w = v + 0
sage: w == v
True
sage: w is v
False

I feel quite torn about this, but slightly tend to the copy-version.

Addendum:

For FreeModule, we also have the following behavior:

sage: M = FreeModule(QQ, 3)
sage: M(0)
(0, 0, 0)
sage: M.zero()
(0, 0, 0)
sage: M.zero() is M(0)
False

I don't think that a parent should do that, especially when it already has a zero method. Should that be changed?

[mkoeppe]

Both M() and M(0) return a new mutable element. That's how one creates a new vector, whose components can then be modified. If you want the immutable 0 element, you can use M.zero().

The arithmetic operations should either always create an immutable element or always create a new mutable element. It would be very inconvenient if the result of an operation was sometimes immutable, sometimes a new mutable element, depending on the input. (And it would be highly problematic if sometimes it would return an existing mutable element.)

[egourgoulhon]

My concern here would be performance. For complicated symbolic expressions, creating a copy can have a significant CPU cost. The main concern is about scalar fields, because tensor fields arithmetics always end up to scalar fields arithmetics, the scalar fields being the tensor components in a given frame. For the time being, ScalarField._add_() starts with

        if self._is_zero:
            return other
        if other._is_zero:
            return self

If we change this to return a copy (I understand the arguments put forward by Matthias), then I would advocate for extensive benchmarks, with complicated tensor fields (those in the doctests are too simple), such as in those mentionned ​here.

[mkoeppe]

Symbolic expressions are immutable and therefore do not need to be copied.

The cost of making a copy should be dominated by copying the dictionaries in the Components objects.

[gh-mjungmath]

If making a copy is so costly, we should try to optimize it.

If I remember correctly, until Sage 9 or so, all elements emerged from arithmetics were created from scratch, even the trivial ones. Even if copying would slow the current code down a bit, it should be still faster than versions before Sage 9.

[egourgoulhon]

Replying to gh-mjungmath:

If making a copy is so costly, we should try to optimize it.

Copying a symbolic expression with hundreds of terms will always remain slower than returning self or other.

If I remember correctly, until Sage 9 or so, all elements emerged from arithmetics were created from scratch, even the trivial ones.

That's not true: already in Sage 7.4, we have the code snippet shown in comment:4.

[gh-mjungmath]

Replying to gh-mjungmath:

If I remember correctly, until Sage 9 or so, all elements emerged from arithmetics were created from scratch, even the trivial ones. Even if copying would slow the current code down a bit, it should be still faster than versions before Sage 9.

My memory was partially wrong. For scalar fields, the _is_zero variable was already implemented, for tensor fields it was not.

Furtermore, I've compared some computation times:

Current state:

sage: M = Manifold(2, 'M', structure='topological') # the 2-dimensional sphere S^2
sage: U = M.open_subset('U') # complement of the North pole
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: V = M.open_subset('V') # complement of the South pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: M.declare_union(U,V)   # S^2 is the union of U and V
sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
....:                                intersection_name='W',
....:                                restrictions1= x^2+y^2!=0,
....:                                restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), c_uv: (u^2+v^2)/(1+u^2+v^2)},
....:                    name='f')
sage: %timeit f+0
The slowest run took 177.32 times longer than the fastest. This could mean that an intermediate result is being cached.
100000 loops, best of 5: 4.22 µs per loop
sage: %timeit f*1
The slowest run took 50.53 times longer than the fastest. This could mean that an intermediate result is being cached.
100000 loops, best of 5: 11.8 µs per loop

Returning a copy:

sage: M = Manifold(2, 'M', structure='topological') # the 2-dimensional sphere S^2
sage: U = M.open_subset('U') # complement of the North pole
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: V = M.open_subset('V') # complement of the South pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: M.declare_union(U,V)   # S^2 is the union of U and V
sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
....:                                intersection_name='W',
....:                                restrictions1= x^2+y^2!=0,
....:                                restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), c_uv: (u^2+v^2)/(1+u^2+v^2)},
....:                    name='f')
sage: %timeit f+0
The slowest run took 23.78 times longer than the fastest. This could mean that an intermediate result is being cached.
10000 loops, best of 5: 60.6 µs per loop
sage: %timeit f*1
The slowest run took 24.43 times longer than the fastest. This could mean that an intermediate result is being cached.
10000 loops, best of 5: 33.4 µs per loop

That's a heavy loss of comutational time.

[gh-mjungmath]

Replying to egourgoulhon:

Replying to gh-mjungmath:

If making a copy is so costly, we should try to optimize it.

Copying a symbolic expression with hundreds of terms will always remain slower than returning self or other.

If I remember correctly, until Sage 9 or so, all elements emerged from arithmetics were created from scratch, even the trivial ones.

That's not true: already in Sage 7.4, we have the code snippet shown in comment:4.

Yes sorry, I was mistaken. See above.

[mkoeppe]

Replying to gh-mjungmath:

Returning a copy:

... could you share the changes that you made for benchmarking this?

[egourgoulhon]

Replying to mkoeppe:

Symbolic expressions are immutable and therefore do not need to be copied.

You are right. Actually they are not copied when copying chart functions (and hence would not be copied when copying scalar fields): the code of ChartFunction.copy() is indeed:

        resu = type(self)(self.parent())
        for kk, vv in self._express.items():
            resu._express[kk] = vv
        resu._expansion_symbol = self._expansion_symbol
        resu._order = self._order
        return resu

Here if kk is 'SR', vv is a symbolic expression.

[mkoeppe]

Another implementation strategy if you consider Components objects an implementation detail that is not exposed to the user: Change them to "copy-on-write" semantics

---

New commits:

​4f3b69f	Trac 30291: simple checks for trivial cases in _mul_, _add_ and _div_
​b7b0b3f	return copies

[gh-mjungmath]

I've pushed my changes.

[gh-mjungmath]

Here's the output of

sage: %prun %timeit f+0

   Ordered by: internal time

   ncalls  tottime  percall  cumtime  percall filename:lineno(function)
   366666    1.064    0.000    1.584    0.000 chart_func.py:318(__init__)
   366666    1.033    0.000    2.745    0.000 chart_func.py:879(copy)
   122222    0.716    0.000    3.990    0.000 scalarfield.py:1480(copy)
       10    0.505    0.051    5.468    0.547 <magic-timeit>:1(inner)
   366666    0.395    0.000    0.463    0.000 chart.py:457(__getitem__)
   122222    0.391    0.000    0.483    0.000 scalarfield.py:1060(__init__)
   122222    0.176    0.000    0.543    0.000 scalarfield.py:1154(is_trivial_zero)
    61111    0.150    0.000    2.201    0.000 scalarfield.py:2423(_add_)
    61111    0.145    0.000    2.659    0.000 scalarfield.py:2632(_lmul_)
    61111    0.111    0.000    0.230    0.000 chart_func.py:810(is_trivial_zero)
    61111    0.103    0.000    2.762    0.000 unital_algebras.py:54(from_base_ring)
   549999    0.099    0.000    0.099    0.000 {method 'parent' of 'sage.structure.element.Element' objects}
   488888    0.087    0.000    0.087    0.000 {method 'items' of 'dict' objects}
   122222    0.084    0.000    0.313    0.000 scalarfield.py:1212(<genexpr>)
   366885    0.068    0.000    0.068    0.000 {built-in method builtins.isinstance}
   366671    0.058    0.000    0.058    0.000 {built-in method builtins.len}
    61111    0.052    0.000    0.073    0.000 calculus_method.py:278(is_trivial_zero)
   122222    0.047    0.000    0.047    0.000 scalarfield.py:1327(_init_derived)
    61111    0.046    0.000    0.046    0.000 chart_func.py:460(expr)
   122222    0.045    0.000    0.045    0.000 subset.py:483(manifold)
   122222    0.041    0.000    0.041    0.000 {method 'is_trivial_zero' of 'sage.symbolic.expression.Expression' objects}
    61112    0.038    0.000    0.326    0.000 {built-in method builtins.all}
    61111    0.015    0.000    0.015    0.000 {method 'values' of 'dict' objects}
    ...

with my pushed changes.

[gh-mjungmath]

        resu = type(self)(self.parent())
-        for kk, vv in self._express.items():
-            resu._express[kk] = vv
+       resu._express = self._express.copy()
        resu._expansion_symbol = self._expansion_symbol

is slightly faster btw:

sage: %timeit f+0
The slowest run took 5.64 times longer than the fastest. This could mean that an intermediate result is being cached.
10000 loops, best of 5: 59.3 µs per loop

And changing back to _is_zero instead of is_trivial_zero() yields a further minimal improvement:

sage: %timeit f+0
The slowest run took 17.22 times longer than the fastest. This could mean that an intermediate result is being cached.
10000 loops, best of 5: 55.5 µs per loop

[gh-mjungmath]

However, regarding my prun test, I think that

        resu = type(self)(self.parent())

is the costly line. And that one shouldn't be an issue for more complicated expressions.

[gh-mjungmath]

Replying to mkoeppe:

Another implementation strategy if you consider Components objects an implementation detail that is not exposed to the user: Change them to "copy-on-write" semantics

Something like this ​https://pypi.org/project/cowdict/ ?

[gh-mjungmath]

I wonder, is copying the chart function even necessary? If new expressions are set/added, a new chart function is created anyway,isn't it?

If the user wants the chart function directly, one can create a copy there.

[mkoeppe]

Replying to gh-mjungmath:

Replying to mkoeppe:

Another implementation strategy if you consider Components objects an implementation detail that is not exposed to the user: Change them to "copy-on-write" semantics

Something like this ​https://pypi.org/project/cowdict/ ?

Yes, same idea but with a different granularity. Add a flag to Component instances that keeps track of whether it belongs to a unique FiniteRankFreeModule instance. In FiniteRankFreeModule, don't copy the component, just mark it non-unique; instead, each time that you want to mutate a component, check first whether it's unique and if not, first copy it, then mutate. But complex code like this should only be written if absolutely necessary.

[gh-mjungmath]

Replying to mkoeppe:

Replying to gh-mjungmath:

Replying to mkoeppe:

Another implementation strategy if you consider Components objects an implementation detail that is not exposed to the user: Change them to "copy-on-write" semantics

Something like this ​https://pypi.org/project/cowdict/ ?

Yes, same idea but with a different granularity. Add a flag to Component instances that keeps track of whether it belongs to a unique FiniteRankFreeModule instance. In FiniteRankFreeModule, don't copy the component, just mark it non-unique; instead, each time that you want to mutate a component, check first whether it's unique and if not, first copy it, then mutate. But complex code like this should only be written if absolutely necessary.

That sounds like something we should attack.

However, the ingredient of scalar fields is ChartFunction, not Component. What do you think about my proposal in comment:19?

Or, alternatively, we apply the very same idea you proposed for Component to ChartFunction.

Edit:

But complex code like this should only be written if absolutely necessary.

I just overread this tiny but important part.

What about simply making those Components immutable which are bound to tensors? Similar as my proposal in comment:19. Each time the components of a tensor are changed via set_comp or add_comp, a whole new Component is created from scratch (self._new_comp is invoked). Making those components immutable which are used for tensors, wouldn't change much. Then we don't have to copy whole Components anymore, but only the dictionaries.

[mkoeppe]

Replying to gh-mjungmath:

Replying to mkoeppe:

Replying to gh-mjungmath:

Replying to mkoeppe:

Another implementation strategy if you consider Components objects an implementation detail that is not exposed to the user: Change them to "copy-on-write" semantics

[...]

Add a flag to Component instances that keeps track of whether it belongs to a unique FiniteRankFreeModule instance. [...]

That sounds like something we should attack.

I think they're lower-hanging fruit for improving efficiency. For example, #30307 could make copying components faster.

However, the ingredient of scalar fields is ChartFunction, not Component. What do you think about my proposal in comment:19?

Sorry, I'm not up to speed on chart functions yet.

[gh-mjungmath]

Watch my edit. Sorry for the chaos.

[tscrim]

Another useful thing for benchmarking is %lprun; see ​this tutorial.

[mkoeppe]

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.