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  • hdu 5734 Acperience

     Acperience

    Time Limit: 4000/2000 MS (Java/Others)    Memory Limit: 65536/65536 K (Java/Others)
    Total Submission(s): 617    Accepted Submission(s): 329


    Problem Description
    Deep neural networks (DNN) have shown significant improvements in several application domains including computer vision and speech recognition. In computer vision, a particular type of DNN, known as Convolutional Neural Networks (CNN), have demonstrated state-of-the-art results in object recognition and detection.

    Convolutional neural networks show reliable results on object recognition and detection that are useful in real world applications. Concurrent to the recent progress in recognition, interesting advancements have been happening in virtual reality (VR by Oculus), augmented reality (AR by HoloLens), and smart wearable devices. Putting these two pieces together, we argue that it is the right time to equip smart portable devices with the power of state-of-the-art recognition systems. However, CNN-based recognition systems need large amounts of memory and computational power. While they perform well on expensive, GPU-based machines, they are often unsuitable for smaller devices like cell phones and embedded electronics.

    In order to simplify the networks, Professor Zhang tries to introduce simple, efficient, and accurate approximations to CNNs by binarizing the weights. Professor Zhang needs your help.

    More specifically, you are given a weighted vector W=(w1,w2,...,wn). Professor Zhang would like to find a binary vector B=(b1,b2,...,bn) (bi{+1,1}) and a scaling factor α0 in such a manner that WαB2 is minimum.

    Note that  denotes the Euclidean norm (i.e. X=x21++x2n−−−−−−−−−−−√, where X=(x1,x2,...,xn)).
     
    Input
    There are multiple test cases. The first line of input contains an integer T, indicating the number of test cases. For each test case:

    The first line contains an integers n (1n100000) -- the length of the vector. The next line contains n integers: w1,w2,...,wn (10000wi10000).
     
    Output
    For each test case, output the minimum value of WαB2 as an irreducible fraction "p/q" where pq are integers, q>0.
     
    Sample Input
    3 4 1 2 3 4 4 2 2 2 2 5 5 6 2 3 4
     
    Sample Output
    5/1 0/1 10/1
     
    Author
    zimpha
     
    Source

    Acperience

    展开式子, left| W-alpha B ight|^2=displaystylealpha^2sum_{i=1}^{n}b_i^2-2alphasum_{i=1}^n{w_ib_i}+sum_{i=1}^{n}w_i^2WαB2=α2i=1nbi22αi=1nwibi+i=1nwi2.

    由于b_iin {+1,-1}bi{+1,1}, 那么displaystylesum_{i=1}^{n}b_i^2=ni=1nbi2=n, 显然c=displaystylesum_{i=1}^{n}w_i^2c=i=1nwi2也是常数. 转化成求displaystylealpha^2n-2alphasum_{i=1}^n{w_ib_i}+cα2n2αi=1nwibi+c的最小值. 对于固定的alpha>0α>0, 只要displaystylesum_{i=1}^n{w_ib_i}i=1nwibi最大就好了. 显然b_i=sign(w_i)bi=sign(wi)的时候, displaystylesum_{i=1}^n{w_ib_i}=sum_{i=1}^{n}|w_i|i=1nwibi=i=1nwi最大. 进一步的, 上面显然是一个关于alphaα的二次方程, 于是当alpha=frac{1}{n}displaystylesum_{i=1}^n{w_ib_i}=frac{1}{n}displaystylesum_{i=1}^{n}{|w_i|}α=n1i=1nwibi=n1i=1nwi时, 取到最大值.

    化简下, 可以得到最小值是sum_{i=1}^n{w_i^2}-frac{1}{n}(displaystylesum_{i=1}^{n}|w_i|)^2i=1nwi2n1(i=1nwi)2    

    #include<iostream>
    #include<stdio.h>
    #include<math.h>
    #include<cmath>
    using namespace std;
    unsigned long long gcd(unsigned long long x,unsigned long long y)
    {
        unsigned long long a;
        if(x<y) a=x,x=y,y=a;
        while(y!=0){
            a=x%y;
            x=y;
            y=a;
        }
        return x;
        
    }
    int main(){
        unsigned long long sum1=0;
        unsigned long long sum2=0;
        int t;
        scanf("%d",&t);
        while(t--){
            int n;
            sum1=0;sum2=0;
            scanf("%d",&n);
            for(int i=0;i<n;i++){
                int tmp;
                scanf("%d",&tmp);
                sum1+=(tmp*tmp);
                sum2+=abs(tmp);
            }
            sum2*=sum2;
            sum1*=n;
            unsigned long long div=(gcd( sum1-sum2,n));
            printf("%lld/%lld
    ",(sum1-sum2)/div,n/div);
        }
        return 0;
    }

    left| W-alpha B ight|^2=displaystylealpha^2sum_{i=1}^{n}b_i^2-2alphasum_{i=1}^n{w_ib_i}+sum_{i=1}^{n}w_i^2

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  • 原文地址:https://www.cnblogs.com/superxuezhazha/p/5695551.html
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