KNN改进

转自http://ben1024.blogbus.com/logs/41046442.html

近邻的非正式描述，就是给定一个样本集exset，样本数为M，每个样本点是N维向量，对于给定目标点d，d也为N维向量，要从exset中找出与d距离最近的k个点(k<=N)，当k=1时，knn问题就变成了最近邻问题。最naive的方法就是求出exset中所有样本与d的距离，进行按出小到大排序，取前k个即为所求，但这样的复杂度为O(N)，当样本数大时，效率非常低下. 我实现了层次knn(HKNN)和kdtree knn，它们都是通过对树进行剪枝达到提高搜索效率的目的，hknn的剪枝原理是(以最近邻问题为例)，如果目标点d与当前最近邻点x的距离，小于d与某结点Kp中心的距离加上Kp的半径，那么结点Kp中的任何一点到目标点的距离都会大于d与当前最近邻点的距离，从而它们不可能是最近邻点(K近邻问题类似于它)，这个结点可以被排除掉。 kdtree对样本集所在超平面进行划分成子超平面，剪枝原理是， 如果某个子超平面与目标点的最近距离大于d与当前最近点x的距离，则该超平面上的点到d的距离都大于当前最近邻点，从而被剪掉。

matlab下实现：

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VecDist.m

 

function y = VecDist(a, b)

%%返回两向量距离的平方

assert(length(a) == length(b));

y = sum((a-b).^2);

end

 

下面是HKNN的代码

 

Node.m

 

classdef Node < handle

    %UNTITLED2 Summary of this class goes here

    %   Detailed explanation goes here

    % Node 层次树中的一个结点，对应一个样本子集Kp   

    properties

        Np; %Kp的样本数

        Mp; %Kp的样本均值，即中心

        Rp; %Kp中样本到Mp的最大距离

        Leafs; %生成的子节点的叶子,C * k矩阵,C为中心数量，k是样本维数。如果不是叶结点，则为空

        SubNode; %子节点, 行向量

    end  

    methods

        function obj = Node(samples, maxLeaf)

            global SAMPLES

            %samples是个列向量，它里面的元素是SAMPLES的行的下标,而不是SAMPLES行向量，使用全局变量是出于效率上的考虑

            obj.Np = length(samples);

            if (obj.Np <= maxLeaf)

                obj.Leafs = samples;

            else

%                 opts = statset('MaxIter',100);

%                 [IDX] = kmeans(SAMPLES(samples, :), maxLeaf, 'EmptyAction','singleton','Options',opts);

                [IDX] = kmeans(SAMPLES(samples, :), maxLeaf, 'EmptyAction','singleton');

                for k = 1:maxLeaf

                    idxs = (IDX == k);

                    samp = samples(idxs);

                    newObj = Node(samp, maxLeaf);

                    obj.SubNode = [obj.SubNode newObj];%SubNode为空说明当层的Centers是叶结点

                end

            end

            obj.Mp = mean(SAMPLES(samples, :), 1);

            dist = zeros(1, obj.Np);

            for t = 1:obj.Np

                dist(t) = VecDist(SAMPLES(samples(t), :), obj.Mp);

            end

            obj.Rp = max(dist);

        end

    end

end

 

SearchKNN.m

 

function SearchKnn(Node)

global KNNVec KNNDist B DEST SAMPLES

m = length(Node.Leafs);

if m ~= 0

    %叶结点

    %是叶结点

    for k = 1:m

        D_X_Xi = VecDist(DEST, SAMPLES(Node.Leafs(k), :));

        if (D_X_Xi < B)

            [Dmax, I] = max(KNNDist);

            KNNDist(I) = D_X_Xi;

            KNNVec(I) = Node.Leafs(k);

            B = max(KNNDist);

        end

    end

else

    %非叶结点

    tab = Node.SubNode;

    D = zeros(size(tab));

    delMark = zeros(size(tab));

    for k = 1:length(tab)

        D(k) = VecDist(DEST, tab(k).Mp);

        if (D(k) > B + tab(k).Rp)

            delMark(k) = 1;

        end

    end

    tab(delMark == 1) = [];

    for k = 1:length(tab)

        SearchKnn(tab(k));

    end   

end

 

下面是kdtree的代码

 

KDTree.m

 

classdef KDTree < handle

    %UNTITLED2 Summary of this class goes here

    %   Detailed explanation goes here

     

    properties

        dom_elt; %A point from Kd_d space, point associated with the current node

        split_pos;%分割位置，比如对于K维向量，这个位置可以是从1到k

        left;%左子树

        right;%右子树

        bNULL;%标识这个结点是否是NULL

    end

     

    methods (Static)

        function [sample, index, split] = ChoosePivot1(samples)

            global SAMPLES

            dimVar = var(SAMPLES(samples, :));

            [maxVar, split] = max(dimVar);%分界点的维，即从第多少维处分

            [sorted, IDX] = sort(SAMPLES(samples, split));

            n = length(IDX);

            index = IDX(round(n/2));

            sample = samples(index);

        end

        function [sample, index, split] = ChoosePivot2(samples)

            %第二种pivot选择策略,选择范围最长的那维作为pivot

            %注意：这个选择策略是以树的不平衡性换取剪枝时的效果，对于有些数据分布，性能可能反而下降

            global SAMPLES

            [upper, I] = max(SAMPLES(samples, :), [], 1);%按列取最大值

            [bottom, I] = min(SAMPLES(samples, :), [], 1);%

            range = upper-bottom;%行向量

            [maxRange, split] = max(range);%分界点的维，即从第多少维处分

            [sorted, IDX] = sort(SAMPLES(samples, split));

            n = length(IDX);

            index = IDX(round(n/2));

            sample = samples(index);         

        end

        function [exleft, exright] = SplitExset(exset, ex, pivot)

            global SAMPLES

            vec = SAMPLES(exset, pivot);%列向量

            flag = (vec <= SAMPLES(ex, pivot));

            exleft = exset(flag);

            flag = ~flag;

            exright = exset(flag);

        end

    end

     

    methods

        function obj = KDTree(exset)

            %输入向量集,SAMPLES的下标

            if isempty(exset)

                obj.bNULL = true;

            else

                obj.bNULL = false;

                [ex, index, split] = KDTree.ChoosePivot1(exset);

                %[ex, index, split] = KDTree.ChoosePivot2(exset);

                obj.dom_elt = ex;

                obj.split_pos = split;

                 

                exset_ = exset;%exset除去先作分割点的那个点

                exset_(exset == ex) = [];

                 

                %将exset_分成左右两个样本集

                [exsetLeft, exsetRight] = KDTree.SplitExset(exset_, ex, split);      

                %递归构造左右子树

                obj.left = KDTree(exsetLeft);

                obj.right = KDTree(exsetRight);

            end

        end

    end   

end

 

SearchKnn.m

 

function SearchKNN(kd, hr)

 

%SearchKNN Summary of this function goes here

%   Detailed explanation goes here

% kd 是 kdtree

% hr是输入超平面图,它是由两个点决定，类比平面和二维点，所有二维点都在平面上，

% 而平面上的一个矩形区域，可以由平面上的两个点决定

% 首次迭代，输入超平面为一个能覆盖所有点的超平面。对于二维，可以想像p1 = (-infinite, -infinite)

% 到p2 = （infinite, infinite)的平面可以覆盖二维平面所有点。可以推测一个可以覆盖K维空间所有点的的超平面图

% 应该是(-inf, -inf....-inf),k维,到正的相应无穷点

    global SAMPLES DEST MAX_DIST_SQD %global in

    %DIST_SQD, SQD是指距离的平方

    global KNNVec KNNDist %global out

    if kd.bNULL

        %kd是空的      

        return;

    end

    %kd不为空

    pivot = kd.dom_elt;%下标

    s = kd.split_pos;   

    %分割输入超平面

    %分割面是经过pivot并且cui直于第s维

    %还原是以二维情况联想，可以得到分割后的两个超平面图

    left_hr_right_point = hr(2,:);

    left_hr_right_point(s) = SAMPLES(pivot,s);

    left_hr = [hr(1,:);left_hr_right_point];%得到分割后的left 超平面

    right_hr_left_point = hr(1,:);

    right_hr_left_point(s) = SAMPLES(pivot, s);

    right_hr = [right_hr_left_point;hr(2,:)];%得到right 超平面

 

    % 判断目标点在哪个超平面上

    % 始终以二维情况来理解，不然比较抽象

    bTarget_in_left = (DEST(s) <= SAMPLES(pivot, s));

    nearer_kd = [];

    nearer_hr = [];

    further_kd = [];

    further_hr = [];

    if bTarget_in_left

        %如果在左边超平面上

        %那么最近点在kd的左孩子上

        nearer_kd = kd.left;

        nearer_hr = left_hr;

        further_kd = kd.right;

        further_hr = right_hr;

    else

        %在右孩子上

        nearer_kd = kd.right;

        nearer_hr = right_hr;

        further_kd = kd.left;

        further_hr = left_hr;

    end

    SearchKNN(nearer_kd, nearer_hr);

    % A nearer point could only lie in further_kd if there were some

    % part of further_hr within distance sqrt(MAX_DIST_SQD) of target

    sqrt_Maxdist = sqrt(MAX_DIST_SQD);

%     剪枝就在这里

    bIntersect = CheckInterSect(further_hr, sqrt_Maxdist, DEST);

    if ~bIntersect

        %如果不相交,没有必要继续搜索了

        return;

    end

    %如果超平面与超球有相交部分

    d = VecDist(SAMPLES(pivot, :), DEST);

    if d < MAX_DIST_SQD

        [Dmax, I] = max(KNNDist);

        KNNVec(I) = pivot;

        KNNDist(I) = d;

        MAX_DIST_SQD = max(KNNDist);

    end   

    SearchKNN(further_kd, further_hr);

end

     

function bIntersect = CheckInterSect(hr, radius, t)

%检查以点t为中心，radius为半径的圆，与超平面hr是否相交,为方便

%在超平面上找到一个距t最近的点，如果这个距离小于等于radius，则相交

%如何确定超平面上到t最近的点p：

%假设超平面hr在第i维的上限和下限分别是hri_max, hri_min,则有

%       hri_min, if ti <= hri_min

% pi = ti, if hri_min < ti < hri_max

%       hri_max, if ti >= hri_max

 

    p = zeros(size(t));%超平面上与t最近的点,待求

    minHr = hr(1,:);maxHr = hr(2,:);

%     for k = 1:length(t)

%         if (t(k) <= minHr(k))

%             p(k) = minHr(k);

%         elseif (t(k) >= maxHr(k))

%             p(k) = maxHr(k);

%         else

%             p(k) = t(k);

%         end

%     end

    flag1 = (t <= minHr);p(flag1) = minHr(flag1);

    flag2 = (t >= maxHr);p(flag2) = maxHr(flag2);

    flag3 = ~(flag1 | flag2);p(flag3) = t(flag3);   

     

    if (VecDist(p, t) >radius^2)

        bIntersect = false;

    else

        bIntersect = true;

    end

end