[Numpy-discussion] Proposed new feature for numpy.einsum: repeated output subscripts as diagonal
Stephan Hoyer
shoyer at gmail.com
Thu Aug 14 15:42:51 EDT 2014
I think this would be very nice addition.
On Thu, Aug 14, 2014 at 12:21 PM, Benjamin Root <ben.root at ou.edu> wrote:
> You had me at Kronecker delta... :-) +1
>
>
> On Thu, Aug 14, 2014 at 3:07 PM, Pierre-Andre Noel <
> noel.pierre.andre at gmail.com> wrote:
>
>> (I created issue 4965 earlier today on this topic, and I have been
>> advised to email to this mailing list to discuss whether it is a good
>> idea or not. I include my original post as-is, followed by additional
>> comments.)
>>
>> I think that the following new feature would make `numpy.einsum` even
>> more powerful/useful/awesome than it already is. Moreover, the change
>> should not interfere with existing code, it would preserve the
>> "minimalistic" spirit of `numpy.einsum`, and the new functionality would
>> integrate in a seamless/intuitive manner for the users.
>>
>> In short, the new feature would allow for repeated subscripts to appear
>> in the "output" part of the `subscripts` parameter (i.e., on the
>> right-hand side of `->`). The corresponding dimensions in the resulting
>> `ndarray` would only be filled along their diagonal, leaving the off
>> diagonal entries to the default value for this `dtype` (typically zero).
>> Note that the current behavior is to raise an exception when repeated
>> output subscripts are being used.
>>
>> This is simplest to describe with an example involving the dual behavior
>> of `numpy.diag`.
>>
>> ```python
>> # Extracting the diagonal of a 2-D array.
>> A = arange(16).reshape(4,4)
>> print(diag(A)) # Output: [ 0 5 10 15 ]
>> print(einsum('ii->i', A)) # Same as previous line (current behavior).
>>
>> # Constructing a diagonal 2-D array.
>> v = arange(4)
>> print(diag(v)) # Output: [[0 0 0 0] [0 1 0 0] [0 0 2 0] [0 0 0 3]]
>> print(einsum('i->ii', v)) # New behavior would be same as previous line.
>> # The current behavior of the previous line is to raise an exception.
>> ```
>>
>> By opposition to `numpy.diag`, the approach generalizes to higher
>> dimensions: `einsum('iii->i', A)` extracts the diagonal of a 3-D array,
>> and `einsum('i->iii', v)` would build a diagonal 3-D array.
>>
>> The proposed behavior really starts to shine in more intricate cases.
>>
>> ```python
>> # Dummy values, these should be probabilities to make sense below.
>> P_w_ab = arange(24).reshape(3,2,4)
>> P_y_wxab = arange(144).reshape(3,3,2,2,4)
>>
>> # With the proposed behavior, the following two lines should be
>> equivalent.
>> P_xyz_ab = einsum('wab,xa,ywxab,zy->xyzab', P_w_ab, eye(2), P_y_wxab,
>> eye(3))
>> also_P_xyz_ab = einsum('wab,ywaab->ayyab', P_w_ab, P_y_wxab)
>> ```
>>
>> If this is not convincing enough, replace `eye(2)` by
>> `eye(P_w_ab.shape[1])` and replace `eye(3)` by `eye(P_y_wxab.shape[0])`,
>> then imagine more dimensions and repeated indices... The new notation
>> would allow for crisper codes and reduce the opportunities for dumb
>> mistakes.
>>
>> For those who wonder, the above computation amounts to
>> $P(X=x,Y=y,Z=z|A=a,B=b) = \sum_w P(W=w|A=a,B=b) P(X=x|A=a)
>> P(Y=y|W=w,X=x,A=a,B=b) P(Z=z|Y=y)$ with $P(X=x|A=a)=\delta_{xa}$ and
>> $P(Z=z|Y=y)=\delta_{zy}$ (using LaTeX notation, and $\delta_{ij}$ is
>> [Kronecker's delta](http://en.wikipedia.org/wiki/Kronecker_delta)).
>>
>> (End of original post.)
>>
>> I have been told by @jaimefrio that "The best way of getting a new
>> feature into numpy is putting it in yourself." Hence, if discussions
>> here do reveal that this is a good idea, then I may give a try at coding
>> it myself. However, I currently know nothing of the inner workings of
>> numpy/ndarray/einsum, and I have higher priorities right now. This means
>> that it could take a long while before I contribute any code, if I ever
>> do. Hence, if anyone feels like doing it, feel free to do so!
>>
>> Also, I am aware that storing a lot of zeros in an `ndarray` may not, a
>> priori, be a desirable avenue. However, there are times where you have
>> to do it: think of `numpy.eye` as an example. In my case of application,
>> I use such diagonal structures in the initialization of an `ndarray`
>> which is later updated through an iterative process. After these
>> iterations, most of the zeros will be gone. Do other people see a use
>> for such capabilities?
>>
>> Thank you for your time and have a nice day.
>>
>> Sincerely,
>>
>> Pierre-André Noël
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>> NumPy-Discussion at scipy.org
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>>
>
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