Rank (operator)

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This article is about the operator. See Rank for the number associated with every array. For numbers associated with a function specifying its argument rank, see function rank.
⍤

Rank (⍤) is a primitive dyadic operator which applies its left operand function to cells of its arguments specified by its right operand array.

Rank specification

The right operand specifies the rank of subarrays to which the left operand function is applied as follows: For left argument ⍺ and right argument ⍵, 

   ⍤    c   ⍝ Rank-c cells of ⍵ (monadic) or both arguments (dyadic)
   ⍤  b c   ⍝ Rank-b cells of ⍺ and rank-c cells of ⍵ (dyadic)
   ⍤a b c   ⍝ Rank-a cells of ⍵ (monadic), b-cells of ⍺ and c-cells of ⍵ (dyadic)

A non-negative right operand specifies the number of final axes to which the function applies. A negative right operand specifies complementary rank, i.e. the number of leading axes to be excluded. Negative rank can also be thought of as rank specification relative to the overall rank of the argument array.

Since a rank specification greater than the rank of the argument array means to apply the function to the whole array, 99, (⌊/⍬) or ∞, depending on the implementation, is "rank infinity" and always specifies the whole argument array.

Examples

Reverse order of rows in matrices of a 3D array: 

      ⊖⍤2⊢3 2 4⍴⎕A
EFGH
ABCD
    
MNOP
IJKL
    
UVWX
QRST

Laminate scalars from arrays of differing ranks: 

      'ABCD',⍤0⍤1⊢2 4⍴⍳8
A 1
B 2
C 3
D 4
   
A 5
B 6
C 7
D 8

Flat outer product: 

      -⍤1⍤1 99⍨3 2⍴6 7 1 1 2 4   ⍝ ↑∘.-⍨↓
 0  0
 5  6
 4  3
     
¯5 ¯6
 0  0
¯1 ¯3
     
¯4 ¯3
 1  3
 0  0

History

The rank operator was invented by Arthur Whitney in 1982 and first implemented in SHARP APL in 1983. It has been described as "a microcosm of APL history"[1], its evolution a progression from scalar extension, which has been in APL since its inception, through leading axis theory to a construct which is a generalisation of scalar extension, inner (matrix) product, outer product, maplist in LISP, map in modern functional programming languages and the broadcast facility in NumPy.

Rank vs Axis

Due to its ability to apply functions to specified subarrays, rank is frequently contrasted with bracket-axis. It provides nearly all of the functionality of the anomalous axis operator (f[a]) without its draw-backs.[2]

One of these draw-backs is that bracket-axis is specified ad hoc for each of the specific primitives on which it applies. Rank benefits from consistent behaviour when applied to any function, including user-defined functions. The ad hoc nature of bracket-axis definitions means that a generalised axis operator which works on any function, but behaves just as bracket-axis on those particular primitives, is impossible to formulate.

Here we show some bracket-axis constructs and their equivalent expressions which use rank but do not use bracket-axis.

Sum along axis

Rank k-cells are defined for k trailing axes, whereas axes are numbered from most major (first axis i.e. axis number 1) to least major (last axis). This leads to a simple and symmetrical relationship. 

+/[1+⍺-⍨≢⍴⍵] ≡ +⌿⍤⍺            ⍝ For scalar ⍺
+/[⍺]        ≡ +⌿⍤(1+⍺-⍨≢⍴⍵)   ⍝ For scalar ⍺

Enclose axes

Enclose-with-axis is equivalent to transposing desired axes to the end of the array's shape and enclosing subarrays of a rank matching the number of axes. 

EncloseAxes←{
    axes←⍳≢⍴⍵
    move←⍋(axes~⍺),⍺
    ⊂⍤(≢⍺)⊢move⍉⍺
}
⊂[⍺] ≡ ⍺∘EncloseAxes

Merge axes

Ravel-with-axis allows data to be merged along specified consecutive axes. The requirement that axes be consecutive is so that the data can remain in its original relative order.

Merging trailing axes is trivial. 

,[(-⍺)↑⍴⍵] ≡ ,⍤⍺

Merging leading axes is more involved, but can be expressed in one line. 

,[⍳⍺] ←→ {(1⌽⍳≢⍴z)⍉z←,⍤⍺((-⍺)⌽⍳≢⍴⍵)⍉⍵}

The general treatment benefits from being expanded. 

MergeAxes←{
    axes←⍳≢⍴⍵
    move←⍋(axes~⍺),⍺
    merged←,⍤(≢⍺)⊢move⍉⍵
    restore←((⍳≢⍴merged)~⊃⍺),⊃⍺
    restored⍉merged
}

,[⍺] ←→ ⍺∘MergeAxes   ⍝ for ∧/1=¯2-/⍺

Difference among dialects

Uniquely to NARS2000, negative rank k specifies taking the ∣k-cells from the array that results from permuting the k leading axes to the trailing end of the shape; the shape of the cells of argument a specified by rank k is (-k)↑⍴a whether k is negative or positive. It also allows axis specification for the rank operator, denoting what axis the results will go to in the frame; x f⍤[ax] l r⊢y is equivalent to ⊃[ax] x f⍤l r⊢y (noting that ⊃ in NARS2000 is Mix).

External links

Documentation

Publications

References

  1. ↑ Roger Hui and Morten Kromberg. APL since 1978. ACM HOPL IV. 2020-06.
  2. ↑ Robert Bernecky. An introduction to function rank at APL88. ACM SIGAPL APL Quote Quad, 18(2), pp.39-43. 1987.
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