Question

我使用Matlab执行大量迭代，使用偏最小二乘法（PLS）作为每次迭代中处理数据的算法。它运行得很慢。使用profile来监控每行代码所用的时间，显示行X = X-t*p'，其中X是一个大小为r的矩阵c和t和p是大小为r和c的列向量，大部分时间都是成本。我认为原因可能是t*p'的内存扩展，它从两个向量生成一个矩阵，这是一个众所周知的原因，它使Matlab中的计算速度变慢。有没有办法加快这个？

PLS代码是：

for iter = 1:maxiter % maxiter is defaultedly set as 10,000
    % preparing data....
    ......
    % perform PLS using NIPALS algorithm
    for a = 1:A
        x = X'*y;
        w = x/norm(x);
        t = X*w;
        tt = sum(t.^2);
        p = X'*(t/tt);
        q = sum(y.*t)/tt;
        X = X-t*p';     % updata X, this line costs most of the time
        y = y-t*q;        % updata y
    end

    % output w, P, q
    .......
end

Answer 1

为避免X与X = X-t*p';之间的通缩，有三种方法，如"Computational performance and cross-validation error precision of five PLS algorithms using designed and real data sets"中所述。我使用了以下算法：

SIMPLS

for iter = 1:maxtest
    % preparing data...
    ... ...
    for a = 1:A
        % for a vector, eigenvector of S'*S is always 1
        ri = s;
        t = X*ri;
        normt = sqrt(sum(t.^2));
        t = t/normt;
        ri = ri/normt;
        % X loadings
        pi = X'*t;
        q = y'*t;
        %u = y*q;
        v = pi;
        if a > 1
            v = v-V(:,1:a-1)*(V(:,1:a-1)'*pi);
            %u = u-T(:,1:a-1)*(T(:,1:a-1)'*u);
        end
        v = v/sqrt(sum(v.^2));
        s = s-v*(v'*s);

        V(:,a) = v;
        R(:,a) = ri;
        Q(a) = q;
    end
    % PLS coefficient...
    b = R*Q;
    ...
end

NIPALS-y

for iter = 1:maxtest
    % preparint data...
    ... ...
    % NIPALS
    for a = 1:A
        x = X'*y;
        rx = x/norm(x);
        w = rx;
        if a > 1
            rx = rx-R(:,1:a-1)*(P(:,1:a-1)'*w);
        end
        t = X*rx;
        tt = sum(t.^2);
        P(:,a) = X'*t/tt;
        q(a) = y'*t/tt;
        y = y-t*q(a);
        % W(:, a) = w;
        R(:, a) = rx;
    end

    % PLS coefficient...
    b = R*q;
    ... ...
end

双对角化

for iter = 1:maxtest
    v = X'*y/norm(X'*y);
    u = X*v;
    normu = sqrt(sum(u.^2));
    u = u/normu;
    R(1) = normu;
    V(:,1) = v;
    U(:,1) = u;
    for a = 2:A
        v = X'*u-normu*v;
        normv = sqrt(sum(v.^2));
        v = v/normv;
        u = X*v-normv*u;
        normu = sqrt(sum(u.^2));
        u = u/normu;
        U(:,a) = u;
        V(:,a) = v;
        R(a-1,a) = normv;
        R(a,a) = normu;
    end
    % PLS coefficients...
    b = V*(R\(U'*y));
end

使用大小为43 by 195的测试数据和10倍交叉验证，双向对角化方法是最快的，与论文结果一致。

在PLS中将列向量与行向量快速元素相乘

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