You had me at Kronecker delta... :-) +1
On Thu, Aug 14, 2014 at 3:07 PM, Pierre-Andre Noel < noel.pierre.an...@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 > _______________________________________________ > NumPy-Discussion mailing list > NumPy-Discussion@scipy.org > http://mail.scipy.org/mailman/listinfo/numpy-discussion >
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