Talk:Outer Product

Context for the Falkoff/Iverson quote

The Falkoff/Iverson quote, while providing illuminating historical context, is a bit misleading—the reduction in inner product does not collapse any axes relative to the result shape, as it is applied only within each enclosure/box of the outer product. Rather, in ‪f/¨ (⊂[⍴⍴x]x)∘.g ⊂[1]y‬ the enclosure of one axis of each argument, prior to the outer product operation, is responsible for the collapsing of dimensions.

Equivalently from a slightly different angle, if we use the model X⊂⍤(f/)⍤g⍤1⍤1 99⊢Y⍉⍨¯1⌽⍳≢⍴Y, the collapsing of dimensions is due more generally to ⊂⍤…⍤g⍤1 mapping two vectors to a scalar. Clearly the same holds for ⊂⍤g⍤1, i.e., the same expression with the reduction removed.

⊂⍺m (talk) 22:45, 25 June 2023 (UTC)

Iverson's model may have been more like f⌿⍤(1-≢⍴X)⊢ X g⍤¯1⍤1 99⊢ Y, so that the flat intermediate result does have this extra axis. Maybe with a transpose like your second example, who knows. In a nested APL, g¨ should be used instead of g, but I think with this modification it correctly describes the semantics other than maybe evaluation order. I would say even in your examples there is an extra axis, but it's hidden away by a layer of nesting; forcing everything to be flat makes it literally an array axis at some point. --Marshall (talk) 01:05, 26 June 2023 (UTC)

You’re right, he may have described it in some different way conceptually—after all, the notation for it preceded any implementation. I was trying to save other APL novices the bit of confusion this had caused me, but upon reflection, the description is pretty vague anyway.

⊂⍺m (talk) 03:28, 26 June 2023 (UTC)

Yep, I can see how you'd get to that point. Thanks for taking the time to edit, I see lots of improvements! --Marshall (talk) 19:11, 27 June 2023 (UTC)