Representing vector- and tensor-valued functions

[kohr-h]

When #861 is in, we'll have to deal with the issue of how to represent vector- and tensor-valued functions properly. Questions are:

• How to effectively deal with continuous functions with vector or tensor range? In particular vectorization?
• In DiscreteLp: Where to put the additional axes for the function range?
• Do we need properties for the two different parts of shape and ndim? If yes, how to name them?
• How do we, in the long run, deal with maintaining compatibility with ProductSpace based implementations of the same thing?
• More that I've forgotten now

Answer to 1: Clearly first, to keep compatibility with ProductSpace based versions of vector-valued functions.

Otherwise: ?

[adler-j]

How to effectively deal with continuous functions with vector or tensor range? In particular vectorization?

My suggestion is to use np.dtype(('float', (3,))) to represent vector valued functions.
with this syntax we would have:

>>> spc = odl.uniform_discr(0, 1, 10, dtype=(float, (3,)))
>>> spc.shape
(3, 10)
>>> spc.dspace.shape
(3, 10)
>>> spc.dtype
dtype(('<f8', (3,)))
>>> spc.dtype.shape
(3,)

pros: No need for another parameter, no redundancy
cons: Confusing to users not intimately familiar with numpy. Unsure if it works with pygpuarray

In DiscreteLp: Where to put the additional axes for the function range?

First is i.m.o. the only option, sadly. This needs to match productspace

Do we need properties for the two different parts of shape and ndim? If yes, how to name them?

Certainly, but I'm not sure what. Perhaps call the "full" shape and "full" ndim as now, to match numpy. Then perhaps value_shape/domain_shape etc?

How do we, in the long run, deal with maintaining compatibility with ProductSpace based implementations of the same thing?

No ideal solution here, but overall I'd say not to add too much to productspace, e.g. do not port everything but add as needed. Also try to make sure most things are transparent. We may want to reopen and solve #157 (Multidimensional product spaces) in order to make the syntax more similar for the basic cases.

More that I've forgotten now

I have trust in you!

[kohr-h]

How to effectively deal with continuous functions with vector or tensor range? In particular vectorization?

My suggestion is to use np.dtype(('float', (3,))) to represent vector valued functions.
pros: No need for another parameter, no redundancy
cons: Confusing to users not intimately familiar with numpy. Unsure if it works with pygpuarray

That's actually a very nice idea, makes total sense. It will also be a perfect match for continuous function spaces with out_dtype = np.dtype(float, (3,)).
I agree it's not obvious even to Numpy users (I'd guess the number of Numpy users that know this is around 20 % max) but it makes sense in every respect, and the value of not having to add a parameter is actually quite significant.
I'd also not worry too much about gpuarray, if there's no native support there we can do the conversion ourselves.

Do we need properties for the two different parts of shape and ndim? If yes, how to name them?

Certainly, but I'm not sure what. Perhaps call the "full" shape and "full" ndim as now, to match numpy. Then perhaps value_shape/domain_shape etc?

I had also thought of domain_shape so let's check that in. Actually value_shape is not bad either, it describes very closely the actual construction.

How do we, in the long run, deal with maintaining compatibility with ProductSpace based implementations of the same thing?

No ideal solution here, but overall I'd say not to add too much to productspace, e.g. do not port everything but add as needed. Also try to make sure most things are transparent. We may want to reopen and solve #157 (Multidimensional product spaces) in order to make the syntax more similar for the basic cases.

In some sense, the space ** value_shape implementation already does this, since that's what corresponds to the thing described above. So at least for this purpose, #157 is solved (although not multi-dim. indexing and such, but that gets messy).

[kohr-h]

This is now largely done in #861, at least on the level of FunctionSpace, but that's where the hard work goes. I'm quite happy with the syntax that it enables:

>>> intv = odl.IntervalProd(0, 1)
>>> scal_spc = odl.FunctionSpace(intv)
>>> elem = scal_spc.element(lambda x: x + 1)
>>> elem(0)
1.0
>>> elem([0, 1])
array([ 1.,  2.])
>>>
>>> vec_spc = odl.FunctionSpace(intv, out_dtype=(float, (2,)))  # 2 components
>>> # Possibility 1: sequence of functions
>>> elem1 = vec_spc.element([lambda x: x + 1, lambda x: x - 1])
>>> elem1(0)
array([ 1., -1.])
>>> elem1([0, 1])
array([[ 1.,  2.],
       [-1.,  0.]])
>>>
>>> # Possibility 2: function returning a sequence
>>> elem2 = vec_spc.element(lambda x: (x + 1, x - 1))
>>> elem2(0)
array([ 1., -1.])
>>> elem1([0, 1])
array([[ 1.,  2.],
       [-1.,  0.]])

Of course, things also work with higher order tensors, and you can use constants in sequences:

>>> tens_spc = odl.FunctionSpace(intv, out_dtype=(float, (2, 3)))
>>> elem = tens_spc.element([[lambda x: x, np.abs, np.negative],
                             [lambda x: x + 1, 0, 0]])
>>> elem1(0)
array([[ 0.,  0.,  0.],
       [ 1.,  0.,  0.]])
>>> elem([0, 1])
array([[[ 0.,  1.],
        [ 0.,  1.],
        [ 0., -1.]],

       [[ 1.,  2.],
        [ 0.,  0.],
        [ 0.,  0.]]])

Support for out is still patchy, and the linear space methods may not work. Of course, no extra tests yet, but we're getting there.

[adler-j]

Looking great to me!

[adler-j]

This is very related to #342.

[kohr-h]

I'll keep this issue open after the tensor branch is in, for the work on the discretized version of vector- and tensor-valued functions. That will remain as TODO.

[adler-j]

I agree, but that should be done ASAP once the branch is in.

[adler-j]

I guess this is on the table now

[kohr-h]

I'm working on it, have a WIP branch.