einsum
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I wrote a new function, einsum, which implements Einstein summation notation, and I'd like comments/thoughts from people who might be interested in this kind of thing.
In testing it, it is also faster than many of NumPy's built-in functions, except for dot and inner. At the bottom of this email you can find the documentation blurb I wrote for it, and here are some timings:
In [1]: import numpy as np In [2]: a = np.arange(25).reshape(5,5) In [3]: timeit np.einsum('ii', a) 100000 loops, best of 3: 3.45 us per loop
In [4]: timeit np.trace(a) 100000 loops, best of 3: 9.8 us per loop In [5]: timeit np.einsum('ii->i', a) 1000000 loops, best of 3: 1.19 us per loop In [6]: timeit np.diag(a)
100000 loops, best of 3: 7 us per loop In [7]: b = np.arange(30).reshape(5,6) In [8]: timeit np.einsum('ij,jk', a, b) 10000 loops, best of 3: 11.4 us per loop
In [9]: timeit np.dot(a, b) 100000 loops, best of 3: 2.8 us per loop In [10]: a = np.arange(10000.) In [11]: timeit np.einsum('i->', a)
10000 loops, best of 3: 22.1 us per loop In [12]: timeit np.sum(a) 10000 loops, best of 3: 25.5 us per loop -Mark The documentation: einsum(subscripts, *operands, out=None, dtype=None, order='K', casting='safe') Evaluates the Einstein summation convention on the operands. Using the Einstein summation convention, many common multi-dimensional array operations can be represented in a simple fashion. This function provides a way compute such summations.
The best way to understand this function is to try the examples below, which show how many common NumPy functions can be implemented as calls to einsum. The subscripts string is a comma-separated list of subscript labels, where each label refers to a dimension of the corresponding operand. Repeated subscripts labels in one operand take the diagonal. For example,
``np.einsum('ii', a)`` is equivalent to ``np.trace(a)``. Whenever a label is repeated, it is summed, so ``np.einsum('i,i', a, b)`` is equivalent to ``np.inner(a,b)``. If a label appears only once,
it is not summed, so ``np.einsum('i', a)`` produces a view of ``a`` with no changes. The order of labels in the output is by default alphabetical. This
means that ``np.einsum('ij', a)`` doesn't affect a 2D array, while ``np.einsum('ji', a)`` takes its transpose. The output can be controlled by specifying output subscript labels
as well. This specifies the label order, and allows summing to be disallowed or forced when desired. The call ``np.einsum('i->', a)`` is equivalent to ``np.sum(a, axis=-1)``, and
``np.einsum('ii->i', a)`` is equivalent to ``np.diag(a)``. It is also possible to control how broadcasting occurs using an ellipsis. To take the trace along the first and last axes,
you can do ``np.einsum('i...i', a)``, or to do a matrix-matrix product with the left-most indices instead of rightmost, you can do ``np.einsum('ij...,jk...->ik...', a, b)``.
When there is only one operand, no axes are summed, and no output parameter is provided, a view into the operand is returned instead of a new array. Thus, taking the diagonal as ``np.einsum('ii->i', a)``
produces a view. Parameters ---------- subscripts : string Specifies the subscripts for summation. operands : list of array_like
These are the arrays for the operation. out : None or array If provided, the calculation is done into this array. dtype : None or data type If provided, forces the calculation to use the data type specified.
Note that you may have to also give a more liberal ``casting`` parameter to allow the conversions. order : 'C', 'F', 'A', or 'K' Controls the memory layout of the output. 'C' means it should
be Fortran contiguous. 'F' means it should be Fortran contiguous, 'A' means it should be 'F' if the inputs are all 'F', 'C' otherwise. 'K' means it should be as close to the layout as the inputs as
is possible, including arbitrarily permuted axes. casting : 'no', 'equiv', 'safe', 'same_kind', 'unsafe' Controls what kind of data casting may occur. Setting this to
'unsafe' is not recommended, as it can adversely affect accumulations. 'no' means the data types should not be cast at all. 'equiv' means only byte-order changes are allowed. 'safe' means only casts
which can preserve values are allowed. 'same_kind' means only safe casts or casts within a kind, like float64 to float32, are allowed. 'unsafe' means any data conversions may be done.
Returns ------- output : ndarray The calculation based on the Einstein summation convention. See Also --------
dot, inner, outer, tensordot Examples -------- >>> a = np.arange(25).reshape(5,5) >>> b = np.arange(5)
>>> c = np.arange(6).reshape(2,3) >>> np.einsum('ii', a) 60 >>> np.trace(a) 60 >>> np.einsum('ii->i', a)
array([ 0, 6, 12, 18, 24]) >>> np.diag(a) array([ 0, 6, 12, 18, 24]) >>> np.einsum('ij,j', a, b) array([ 30, 80, 130, 180, 230])
>>> np.dot(a, b) array([ 30, 80, 130, 180, 230]) >>> np.einsum('ji', c) array([[0, 3], [1, 4], [2, 5]])
>>> c.T array([[0, 3], [1, 4], [2, 5]]) >>> np.einsum(',', 3, c) array([[ 0, 3, 6],
[ 9, 12, 15]]) >>> np.multiply(3, c) array([[ 0, 3, 6], [ 9, 12, 15]]) >>> np.einsum('i,i', b, b)
30 >>> np.inner(b,b) 30 >>> np.einsum('i,j', np.arange(2)+1, b) array([[0, 1, 2, 3, 4], [0, 2, 4, 6, 8]])
>>> np.outer(np.arange(2)+1, b) array([[0, 1, 2, 3, 4], [0, 2, 4, 6, 8]]) >>> np.einsum('i...->', a) array([50, 55, 60, 65, 70])
>>> np.sum(a, axis=0) array([50, 55, 60, 65, 70]) >>> a = np.arange(60.).reshape(3,4,5) >>> b = np.arange(24.).reshape(4,3,2)
>>> np.einsum('ijk,jil->kl', a, b) array([[ 4400., 4730.], [ 4532., 4874.], [ 4664., 5018.], [ 4796., 5162.],
[ 4928., 5306.]]) >>> np.tensordot(a,b, axes=([1,0],[0,1])) array([[ 4400., 4730.], [ 4532., 4874.], [ 4664., 5018.],
[ 4796., 5162.], [ 4928., 5306.]]) _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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On Wed, Jan 26, 2011 at 11:27 AM, Mark Wiebe <[hidden email]> wrote:
> I wrote a new function, einsum, which implements Einstein summation > notation, and I'd like comments/thoughts from people who might be interested > in this kind of thing. This sounds really cool! I've definitely considered doing something like this previously, but never really got around to seriously figuring out any sensible API. Do you have the source up somewhere? I'd love to try it out myself. --Josh _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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On Wed, Jan 26, 2011 at 1:36 PM, Joshua Holbrook <[hidden email]> wrote:
You can check out the new_iterator branch from here: $ git clone https://github.com/m-paradox/numpy.git Cloning into numpy... -Mark _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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On Wed, Jan 26, 2011 at 12:48 PM, Mark Wiebe <[hidden email]> wrote:
> On Wed, Jan 26, 2011 at 1:36 PM, Joshua Holbrook <[hidden email]> > wrote: >> >> On Wed, Jan 26, 2011 at 11:27 AM, Mark Wiebe <[hidden email]> wrote: >> > I wrote a new function, einsum, which implements Einstein summation >> > notation, and I'd like comments/thoughts from people who might be >> > interested >> > in this kind of thing. >> >> This sounds really cool! I've definitely considered doing something >> like this previously, but never really got around to seriously >> figuring out any sensible API. >> >> Do you have the source up somewhere? I'd love to try it out myself. > > You can check out the new_iterator branch from here: > https://github.com/m-paradox/numpy > $ git clone https://github.com/m-paradox/numpy.git > Cloning into numpy... > -Mark > Thanks for the link! How closely coupled is this new code with numpy's internals? That is, could you factor it out into its own package? If so, then people could have immediate use out of it without having to integrate it into numpy proper. --Josh _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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On Wed, Jan 26, 2011 at 2:01 PM, Joshua Holbrook <[hidden email]> wrote:
The code depends heavily on the iterator I wrote, and I think the idea itself depends on having a good dynamic multi-dimensional array library. When the numpy-refactor branch is complete, this would be part of libndarray, and could be used directly from C without depending on Python.
-Mark _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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On Wed, Jan 26, 2011 at 16:43, Mark Wiebe <[hidden email]> wrote:
> On Wed, Jan 26, 2011 at 2:01 PM, Joshua Holbrook <[hidden email]> > wrote: >> >> <snip> >> How closely coupled is this new code with numpy's internals? That is, >> could you factor it out into its own package? If so, then people could >> have immediate use out of it without having to integrate it into numpy >> proper. > > The code depends heavily on the iterator I wrote, and I think the idea > itself depends on having a good dynamic multi-dimensional array library. > When the numpy-refactor branch is complete, this would be part of > libndarray, and could be used directly from C without depending on Python. It think his real question is whether einsum() and the iterator stuff can live in a separate module that *uses* a released version of numpy rather than a development branch. -- Robert Kern "I have come to believe that the whole world is an enigma, a harmless enigma that is made terrible by our own mad attempt to interpret it as though it had an underlying truth." -- Umberto Eco _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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>
> It think his real question is whether einsum() and the iterator stuff > can live in a separate module that *uses* a released version of numpy > rather than a development branch. > > -- > Robert Kern > Indeed, I would like to be able to install and use einsum() without having to install another version of numpy. Even if it depends on features of a new numpy, it'd be nice to have it be a separate module. --Josh _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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In reply to this post by Mark Wiebe
Mark, interesting idea. Given the fact that in 2-d euclidean metric, the Einstein summation conventions are only a way to write out conventional matrix multiplications, do you consider at some point to include a non-euclidean metric in this thing? (As you have in special relativity, for example) Something along the lines: eta = np.diag(-1,1,1,1) a = np.array(1,2,3,4) b = np.array(1,1,1,1) such that einsum('i,i', a,b, metric=eta) = -1 + 2 + 3 + 4 I don't know how useful it would be, just a thought, Hanno Am 26.01.2011 um 21:27 schrieb Mark Wiebe: > I wrote a new function, einsum, which implements Einstein summation > notation, and I'd like comments/thoughts from people who might be > interested in this kind of thing. _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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On Thu, Jan 27, 2011 at 12:18:30AM +0100, Hanno Klemm wrote:
> interesting idea. Given the fact that in 2-d euclidean metric, the > Einstein summation conventions are only a way to write out > conventional matrix multiplications, do you consider at some point to > include a non-euclidean metric in this thing? (As you have in special > relativity, for example) In my experience, Einstein summation conventions are quite incomprehensible for people who haven't studies relativity (they aren't used much outside some narrow fields of physics). If you start adding metrics, you'll make it even harder for people to follow. My 2 cents, Gaël _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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In reply to this post by Joshua Holbrook
On Wed, Jan 26, 2011 at 3:05 PM, Joshua Holbrook <[hidden email]> wrote:
Ah, sorry for misunderstanding. That would actually be very difficult, as the iterator required a fair bit of fixes and adjustments to the core. The new_iterator branch should be 1.5 ABI compatible, if that helps.
-Mark
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In reply to this post by Hanno Klemm
On Wed, Jan 26, 2011 at 3:18 PM, Hanno Klemm <[hidden email]> wrote:
This particular example is already doable as follows: >>> eta = np.diag([-1,1,1,1])
>>> eta array([[-1, 0, 0, 0], [ 0, 1, 0, 0], [ 0, 0, 1, 0], [ 0, 0, 0, 1]]) >>> a = np.array([1,2,3,4]) >>> b = np.array([1,1,1,1])
>>> np.einsum('i,j,ij', a, b, eta) 8 I think that's right, did I understand you correctly? Cheers, Mark _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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Am 27.01.2011 um 00:29 schrieb Mark Wiebe:
Yes, that's what I had in mind. Thanks. _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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In reply to this post by Gael Varoquaux
On Wednesday, January 26, 2011, Gael Varoquaux
<[hidden email]> wrote: > On Thu, Jan 27, 2011 at 12:18:30AM +0100, Hanno Klemm wrote: >> interesting idea. Given the fact that in 2-d euclidean metric, the >> Einstein summation conventions are only a way to write out >> conventional matrix multiplications, do you consider at some point to >> include a non-euclidean metric in this thing? (As you have in special >> relativity, for example) > > In my experience, Einstein summation conventions are quite > incomprehensible for people who haven't studies relativity (they aren't > used much outside some narrow fields of physics). If you start adding > metrics, you'll make it even harder for people to follow. > > My 2 cents, > > Gaël > Just to dispel the notion that Einstein notation is only used in the study of relativity, I can personally attest that Einstein notation is used in the field of fluid dynamics and some aspects of meteorology. This is really a neat idea and I support the idea of packaging it as a separate module. Ben Root _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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> Ah, sorry for misunderstanding. That would actually be very difficult,
> as the iterator required a fair bit of fixes and adjustments to the core. > The new_iterator branch should be 1.5 ABI compatible, if that helps. I see. Perhaps the fixes and adjustments can/should be included with numpy standard, even if the Einstein notation package is made a separate module. > Just to dispel the notion that Einstein notation is only used in the > study of relativity, I can personally attest that Einstein notation is > used in the field of fluid dynamics and some aspects of meteorology. Einstein notation is also used in solid mechanics. --Josh _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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On Wed, Jan 26, 2011 at 5:02 PM, Joshua Holbrook
<[hidden email]> wrote: >> Ah, sorry for misunderstanding. That would actually be very difficult, >> as the iterator required a fair bit of fixes and adjustments to the core. >> The new_iterator branch should be 1.5 ABI compatible, if that helps. > > I see. Perhaps the fixes and adjustments can/should be included with > numpy standard, even if the Einstein notation package is made a > separate module. > <snip> > Indeed, I would like to be able to install and use einsum() without > having to install another version of numpy. Even if it depends on > features of a new numpy, it'd be nice to have it be a separate module. I don't really understand the desire to have this single function exist in a separate package. If it requires the new version of NumPy, then you'll have to install/upgrade either way...and if it comes as part of that new NumPy, then you are already set. Doesn't a separate package complicate things unnecessarily? It make sense to me if einsum consisted of many functions (such as Bottleneck). _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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In reply to this post by Benjamin Root-2
On Wed, Jan 26, 2011 at 7:35 PM, Benjamin Root <[hidden email]> wrote:
> On Wednesday, January 26, 2011, Gael Varoquaux > <[hidden email]> wrote: >> On Thu, Jan 27, 2011 at 12:18:30AM +0100, Hanno Klemm wrote: >>> interesting idea. Given the fact that in 2-d euclidean metric, the >>> Einstein summation conventions are only a way to write out >>> conventional matrix multiplications, do you consider at some point to >>> include a non-euclidean metric in this thing? (As you have in special >>> relativity, for example) >> >> In my experience, Einstein summation conventions are quite >> incomprehensible for people who haven't studies relativity (they aren't >> used much outside some narrow fields of physics). If you start adding >> metrics, you'll make it even harder for people to follow. >> >> My 2 cents, >> >> Gaël >> > > Just to dispel the notion that Einstein notation is only used in the > study of relativity, I can personally attest that Einstein notation is > used in the field of fluid dynamics and some aspects of meteorology. > This is really a neat idea and I support the idea of packaging it as a > separate module. So, if I read the examples correctly we finally get dot along an axis np.einsum('ijk,ji->', a, b) np.einsum('ijk,jik->k', a, b) or something like this. the notation might require getting used to but it doesn't look worse than figuring out what tensordot does. The only disadvantage I see, is that choosing the axes to operate on in a program or function requires string manipulation. Josef > > Ben Root > _______________________________________________ > NumPy-Discussion mailing list > [hidden email] > http://mail.scipy.org/mailman/listinfo/numpy-discussion > _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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In reply to this post by Mark Wiebe
Nice function, and wonderful that it speeds some tasks up.
some feedback: the following notation is a little counter intuitive to me: >>> np.einsum('i...->', a) array([50, 55, 60, 65, 70])
>>> np.sum(a, axis=0)
array([50, 55, 60, 65, 70])
Since there is nothing after the ->, I expected a scalar not a vector. I might suggest 'i...->...'Just noticed also a typo in the doc: order : 'C', 'F', 'A', or 'K' Controls the memory layout of the output. 'C' means it should
be Fortran contiguous. 'F' means it should be Fortran contiguous,should be changed to order : 'C', 'F', 'A', or 'K' Controls the memory layout of the output. 'C' means it should
be C contiguous. 'F' means it should be Fortran contiguous,
Hope this helps, Jonathan On Wed, Jan 26, 2011 at 2:27 PM, Mark Wiebe <[hidden email]> wrote: I wrote a new function, einsum, which implements Einstein summation notation, and I'd like comments/thoughts from people who might be interested in this kind of thing. -- Jonathan Rocher, Enthought, Inc. [hidden email] 1-512-536-1057 http://www.enthought.com _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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On Wed, Jan 26, 2011 at 6:41 PM, Jonathan Rocher <[hidden email]> wrote:
Nice function, and wonderful that it speeds some tasks up. Hmm, the dimension that's left is a a broadcast dimension, and the dimension labeled 'i' did go away. I suppose disallowing the empty output string and forcing a '...' is reasonable. Would disallowing broadcasting by default be a good approach? Then,
einsum('ii->i', a) would only except two dimensional inputs, and you would have to specify einsum('...ii->...i', a) to get the current default behavior for it.
Thanks, Mark _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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In reply to this post by josef.pktd
On Wed, Jan 26, 2011 at 5:23 PM, <[hidden email]> wrote:
I thought of various extensions to the notation, but the idea is tricky enough as is I think. Decoding a regex-like syntax probably wouldn't help.
The only disadvantage I see, is that choosing the axes to operate on One possibility would be for the Python exposure to accept lists or tuples of integers. The subscript 'ii' could be [(0,0)], and 'ij,jk->ik' could be [(0,1), (1,2), (0,2)]. Internally it would convert this directly to a C-string to pass to the API function.
-Mark _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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>>
>> The only disadvantage I see, is that choosing the axes to operate on >> in a program or function requires string manipulation. > > > One possibility would be for the Python exposure to accept lists or tuples > of integers. The subscript 'ii' could be [(0,0)], and 'ij,jk->ik' could be > [(0,1), (1,2), (0,2)]. Internally it would convert this directly to a > C-string to pass to the API function. > -Mark > What if you made objects i, j, etc. such that i*j = (0, 1) and etcetera? Maybe you could generate them with something like (i, j, k) = einstein((1, 2, 3)) . Feel free to disregard me since I haven't really thought too hard about things and might not even really understand what the problem is *anyway*. I'm just trying to help brainstorm. :) --Josh _______________________________________________ NumPy-Discussion mailing list [hidden email] http://mail.scipy.org/mailman/listinfo/numpy-discussion |
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