in reply to Re^2: Fast sliding submatrix sums with PDL (inspired by PWC 248 task 2)
in thread Fast sliding submatrix sums with PDL (inspired by PWC 248 task 2)

For large submatrices one can use a Fourier transforms to perform the convolution, as in

use v5.36;
use PDL;
use PDL::FFT;
my $matrix=pdl(shift); # Format "[[m11,m21...],[m21...]...]"
my $width=my $w=shift;
my $height=my $h=shift;
my $small=ones($w%2?$w:$w+1, $h%2?$h:$h+1);
$small->slice(-1).=0, ++$w unless $w%2; # zero row and/or column for e
+ven kernels
$small->slice([],-1).=0, ++$h unless $h%2;
my $kernel=kernctr($matrix, $small); #full kernel
my $result=$matrix->copy;
$result->fftconvolve($kernel);
say "$matrix $width $height -> ",
    $result->slice([floor(($width-1)/2),floor(-($width+1)/2)],
                   [floor(($height-1)/2),floor(-($height+1)/2)]);

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It would be interesting to also compare it.

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Re^4: Fast sliding submatrix sums with PDL (inspired by PWC 248 task 2)
by Anonymous Monk on Dec 28, 2023 at 11:35 UTC

    Interesting. FFT for this task, who would have thought a few days ago. Looks like it runs in almost constant time, regardless of kernel size; and of course slower than special solutions. I had to add a couple of checks to accommodate for (unimportant) corner test case of submatrix with dims (4,3) and example matrix.

    sub sms_WxH_PDL_fftconvolve ( $m, $w, $h ) {
    my ( $W, $H ) = $m-> dims;

    my $small = ones( $w%2?$w:$w+1, $h%2?$h:$h+1 );
    $small-> slice( -1 )    .= 0 unless $w%2; # zero row and/or column
+ for even kernels
    $small-> slice( [],-1 ) .= 0 unless $h%2;

    my $result = $m-> copy;
    $result = append( $result, 0 ) if $W == $w and not $W % 2;
    $result = append( $result-> transpose, 0 )-> transpose 
                                   if $H == $h and not $H % 2;

    my $kernel = kernctr( $result, $small ); #full kernel
    $result-> fftconvolve( $kernel );
    
    $result = $result-> slice( '0:-2','' ) if $W == $w and not $W % 2;
    $result = $result-> slice( '','0:-2' ) if $H == $h and not $H % 2;

    return $result-> slice(
        [floor(($w-1)/2),floor(-($w+1)/2)],
        [floor(($h-1)/2),floor(-($h+1)/2)]
    )
}

__END__

    Time (s) vs. N (NxN submatrix, PDL: Double D [1500,1500] matrix)

      +-----------------------------------------------------------+
      |    +      +      +      +       +      +      +      +    |
      |     A                                                     |
      |                                                           |
      |                                                           |
    2 |-+                                                       +-|
      |    B   B                        B  B      B               |
      |    BB         B      B      B          B         B   B    |
      |     B     B      B      B                     B           |
      |                         C   C                             |
      |                      C          C                         |
  1.5 |-+   A                              C                    +-|
      |                  C                                        |
      |                                        C                  |
      |                                                           |
      |               C                                           |
    1 |-+                                         C             +-|
      |     A                                                     |
      |           C                                   C           |
      |                                                           |
      |                                                           |
      |                                                  C        |
  0.5 |-+  A                                                    +-|
      |        C                                                  |
      |                                                      C    |
      |    A                                                      |
      |     D  D                                                  |
      |    CC     D   D  D   D  D   D   D  D   D  D   D  D        |
    0 |-+                                                    D  +-|
      |    +      +      +      +       +      +      +      +    |
      +-----------------------------------------------------------+
           0     200    400    600     800    1000   1200   1400
                         sms_WxH_PDL_conv2d    A
                    sms_WxH_PDL_fftconvolve    B
                           sms_WxH_PDL_lags    C
                        sms_WxH_PDL_sliding    D


  +------+-------+-------+-------+-------+
  | N    | A     | B     | C     | D     |
  +------+-------+-------+-------+-------+
  | 2    | 0.023 | 1.792 | 0.021 |       |
  | 4    | 0.065 | 1.854 | 0.036 |       |
  | 8    | 0.237 | 1.896 | 0.031 |       |
  | 12   | 0.534 | 1.865 | 0.049 |       |
  | 16   | 0.930 | 1.740 | 0.039 |       |
  | 20   | 1.432 | 1.812 | 0.057 |       |
  | 25   | 2.224 | 1.716 | 0.081 | 0.099 |
  | 100  |       | 1.899 | 0.380 | 0.107 |
  | 200  |       | 1.779 | 0.826 | 0.075 |
  | 300  |       | 1.836 | 1.143 | 0.089 |
  | 400  |       | 1.760 | 1.393 | 0.065 |
  | 500  |       | 1.826 | 1.560 | 0.047 |
  | 600  |       | 1.740 | 1.659 | 0.036 |
  | 700  |       | 1.818 | 1.620 | 0.049 |
  | 800  |       | 1.919 | 1.589 | 0.031 |
  | 900  |       | 1.917 | 1.484 | 0.026 |
  | 1000 |       | 1.789 | 1.260 | 0.008 |
  | 1100 |       | 1.880 | 1.068 | 0.010 |
  | 1200 |       | 1.745 | 0.823 | 0.010 |
  | 1300 |       | 1.841 | 0.562 | 0.008 |
  | 1400 |       | 1.841 | 0.273 | 0.003 |
  +------+-------+-------+-------+-------+

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