通过嵌套循环构造Matlab数组

问题描述：

假设我有三个维度为 mxn 的三个矩阵 A_1 ， A_2 ， A_3 m 和 n 很大.这些矩阵严格包含正数.

Suppose I have three matrices A_1, A_2, A_3 each of dimension mxn, where m and n are large. These matrices contain strictly positive numbers.

我想构造一个尺寸为 mx1 的矩阵 check ，使得对于每个 i = 1，...，m :

I want to construct a matrix check of dimension mx1 such that, for each i=1,...,m:

检查(i)= 1 是否存在 j，k 这样

A_1(i，1)+ A_2(j，1)+ A_3(k，1)

在我的情况下， m 大( m = 10 ^ 5 )而 n = 500 .因此，我希望您能帮助您找到一种有效的方法.

In my case m is large (m=10^5) and n=500. Therefore, I would like your help to find an efficient way to do this.

下面，我重现一个示例.我强加了 m 小于实际值，并报告了构造 check 的不完整且可能无效的尝试.

Below I reproduce an example. I impose m smaller than in reality and report my incomplete and probably inefficient attempt to construct check.

clear
rng default
m=4;
n=500;
A_1=betarnd(1,2,m,n);
A_2=betarnd(1,2,m,n);
A_3=betarnd(1,2,m,n);
check=zeros(m,1);
for i=1:m
    for j=1:m
        for k=1:m
            if A_1(i,1)+A_2(j,1)+A_3(k,1)<=quantile(A_1(i,2:end)+A_2(j,2:end)+A_3(k,2:end), 0.95)
              check(i)=1;
              STOP TO LOOP OVER j AND k, MOVE TO THE NEXT i (INCOMPLETE!)              
           else
            KEEP SEARCHING FOR j,k SUCH THAT THE CONDITION IS SATISFIED (INCOMPLETE!) 
           end
       end
    end
end

答

给出标量 x 和向量 v ，表达式 x< = quantile(v，.95)可以写为 sum(x> v)<Q ，其中 Q = .95 * numel(v) *.

Given a scalar x and a vector v the expression x <=quantile (v, .95) can be written as sum( x > v) < Q where Q = .95 * numel(v) *.

A_1 也可以在循环之前进行拆分，以避免额外的索引编制.此外，可以移除最内层的循环，从而有利于向量化.

Also A_1 can be splitted before the loop to avoid extra indexing. Moreover the most inner loop can be removed in favor of vectorization.

Af_1 = A_1(:,1);
Af_2 = A_2(:,1);
Af_3 = A_3(:,1);
As_1 = A_1(:,2:end);
As_2 = A_2(:,2:end);
As_3 = A_3(:,2:end);
Q = .95 * (n -1);
for i=1:m
    for j=1:m
        if any (sum (Af_1(i) + Af_2(j) + Af_3 > As_1(i,:) + As_2(j,:) + As_3, 2) < Q)
            check(i) = 1;
            break; 
        end             
    end
end

通过重新排列不等式和预计算中涉及的表达式，可以实现更多优化:

More optimization can be achieved by rearranging the expressions involved in the inequality and pre-computation:

lhs = A_3(:,1) - A_3(:,2:end);
lhsi = A_1(:,1) - A_1(:,2:end);
rhsj = A_2(:,2:end) - A_2(:,1);
Q = .95 * (n - 1);
for i=1:m
    LHS = lhs + lhsi(i,:);
    for j=1:m
        if any (sum (LHS > rhsj(j,:), 2) < Q)
            check(i) = 1;
            break; 
        end             
    end
end

请注意，由于 quantile 您可能会得到略有不同的结果.

Note that because of the method that is used in the computation of quantile you may get a slightly different result.

通过嵌套循环构造Matlab数组

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