You are doing an awful lot of completely unnecessary copying of arrays.

Try this clean-compiling (unlike your posted code), but untested version, It should be substantially faster:

#! perl -slw
use strict;

## Variables and Data Structures
my $count = 1;
my @probesArray;
my %probes;
my $size = 100;

## Reading the file
open my $FILE, "data.txt" or die "ERROR: cannot read file $!\n";
while (my $line = <$FILE>){
    chomp $line;
    my @line = split '\t', $line;
    $probes{ $line[0] } = [ @line[ 1 .. $#line ] ]; ## value of the ha
+sh as an array
    ## correlation between 1-2 or 2-1 will be same so using calculatin
+g only once can be done thru this array
    $probesArray[ $count ] = $line[ 0 ];
    $count++;
}
## Frankly speaking reading of the file takes less than 3 sec with 50,
+000 categories each having 100 values
close $FILE;

## Correlation Calculation
for( my $i = 0; $i <= $count-1; $i++ ){
    for( my $j = $i+1; $j <= $count; $j++ ){
        my $cor = correlation( $probes{ $probesArray[$i] }, $probes{ $
+probesArray[$j] }, $size ); ## correlation is the subroutine
        $probes{ $probesArray[$i] . "-" . $probesArray[$j] } = $cor;
#       print $count,"\t",$probesArray[$i]."-".$probesArray[$j],"\t",$
+cor,"\n";
        $count++;
    }
}

## Subroutines
## Pearson Correlation Coefficient
sub correlation {
    my( $arr1, $arr2, $size ) = @_;
    my( $mean_x, $mean_y ) = mean( $arr1, $arr2, $size );
    my( $ssxx, $ssxy, $ssyy ) = ss( $arr1, $arr2, $mean_x, $mean_y );
    my $cor = $ssxy / sqrt( $ssxx * $ssyy );
    return $cor ;
}

sub mean {
    my( $arr1, $arr2, $size ) = @_;
    my $mu_x = sum( @$arr1 ) / $size;
    my $mu_y = sum( @$arr2 ) / $size;
    return($mu_x,$mu_y);
}

## Sum of Squared Deviations to the mean
sub ss {
    my( $arr1, $arr2, $mean_x, $mean_y ) = @_;
    my( $ssxx, $ssxy, $ssyy ) = (0) x 3;

    ## looping over all the samples
    for( my $i = 0; $i < @$arr1; $i++ ){
        $ssxx = $ssxx + ( $arr1->[$i] - $mean_x )**2;
        $ssxy = $ssxy + ( $arr1->[$i] - $mean_x ) * ( $arr2->[$i] - $m
+ean_y );
        $ssyy = $ssyy + ( $arr2->[$i] - $mean_y )**2;
    }
    return ($ssxx, $ssxy, $ssyy);
}
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