User talk:⊂⍺m: Difference between revisions
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(The historical quote relating inner and outer products is misleading. But maybe my accompanying clarification should be moved elsewhere?) Tags: Mobile edit Mobile web edit |
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The Falkoff/Iverson quote, while providing illuminating historical context, | 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 <syntaxhighlight lang=apl inline>f/¨ (⊂[⍴⍴x]x)∘.g ⊂[1]y</syntaxhighlight> 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 <syntaxhighlight lang=apl inline>X⊂⍤(f/)⍤g⍤1⍤1 99⊢Y⍉⍨¯1⌽⍳≢⍴Y</syntaxhighlight>, the collapsing of dimensions is due more generally to <syntaxhighlight lang=apl inline>⊂⍤…⍤g⍤1</syntaxhighlight> mapping two vectors to a scalar. Clearly the same holds for <syntaxhighlight lang=apl inline>…⊂⍤g⍤1…</syntaxhighlight>, i.e., the same expression with the reduction removed. | Equivalently from a slightly different angle, if we use the model <syntaxhighlight lang=apl inline>X⊂⍤(f/)⍤g⍤1⍤1 99⊢Y⍉⍨¯1⌽⍳≢⍴Y</syntaxhighlight>, the collapsing of dimensions is due more generally to <syntaxhighlight lang=apl inline>⊂⍤…⍤g⍤1</syntaxhighlight> mapping two vectors to a scalar. Clearly the same holds for <syntaxhighlight lang=apl inline>…⊂⍤g⍤1…</syntaxhighlight>, i.e., the same expression with the reduction removed. | ||
[[User:⊂⍺m|⊂⍺m]] ([[User talk:⊂⍺m|talk]]) 22:45, 25 June 2023 (UTC) ⊂⍺m | |||