题目1
题目描述
结合Matlab环境的contour函数研究二元函数的等值线变化规律:
平面上任取四个点 $x_1$ , $x_2$ , $x_3$ , $x_4$ ,研究含有参数的二元函数 $f(x)=\sum^{4}_ {i=1}{\theta_ {i}e^{-\gamma||x-x_i||^2}}$ 等值线的变化规律，其中 $\theta_i$ 为实数，$\gamma$ 为正实数
- 至少要考虑相邻两点对应的 $\theta_i$ 同号和异号的情况
- 各个参数变化等值线的变化规律
- 突出0等值线
- 总结出一般规律推广到多于4个点
题解
初值:
p = [1 1; 1 -1; -1 -1; -1 1]; % 按四个象限的顺序编号
theta = [1 1 -1 -1];
gamma = 0.5;
X = -2:0.01:2;
Y = X;
- theta变化
左右相邻点同号 theta = [1 1 -1 -1]
上下相邻点同号 theta = [1 1 -1 -1]
相邻点异号 theta = [1 -1 1 -1]
- gamma变化
gamma = 5
gamma = 0.1
- 多于四个点
取第五个点 $(0, 0)$ 与 $\theta_5=1$
- 总结：该二元函数可以用于按 $\theta$ 的正负分类平面上的点集
代码
clc, clear
p = [1 1; 1 -1; -1 -1; -1 1; 0 0];
theta = [1 -1 1 -1 1];
gamma = 0.5;
X = -2:0.01:2;
Y = X;
n = length(X);
global m
m = length(theta);
Z = zeros(n, n);
for i = 1:n
for j = 1:n
Z(i, j) = f(X(i), Y(j), theta, gamma, p);
end
end
[X, Y] = meshgrid(X, Y);
contour(X, Y, Z, 'ShowText', 'on') % ShowText显示每条等值线的数值
colorbar % 将数值与色度图对应图例显示在右侧
hold on
for i = 1:m
plot(p(i, 1), p(i, 2), '*')
end
hold off
function z = f(x, y, theta, gamma, p)
z = 0;
global m
for i = 1:m
z = z + theta(i) * exp(-gamma .* norm([x, y] - p(i, :)));
end
return ;
end
题目2
题目描述
使用 $f(x)=\sum_ {i}{\theta_ {i}e^{-\gamma||x-x_ i||^2}}$ 作为分类器，用最小二乘法确定参数，分开不同颜色的数据
题解
- 数据生成
clc, clear
n = 200;
p1 = rand(50, 2);
p1 = [p1; rand(50, 2) - ones(50, 2)];
p2 = rand(50, 2) - [zeros(50, 1), ones(50, 1)];
p2 = [p2; rand(50, 2) - [ones(50, 1), zeros(50, 1)]];
hold on
plot(p1(:, 1), p1(:, 2), 'ob')
p1(:, 3) = -ones(100 ,1);
plot(p2(:, 1), p2(:, 2), 'or')
p2(:, 3) = ones(100, 1);
p = [p1 ,p2];
hold off
- 分类
% 非线性最小二乘法
clc
fun = @(x)f_lsq(x, p);
global m
m = length(p);
x0 = rand(n+1, 1);
x = lsqnonlin(fun, x0);
gamma = x(end);
theta = x(1:end-1);
X = -2:0.01:2;
Y = X;
Z = zeros(length(X),length(Y));
for i = 1:length(X)
for j = 1:length(Y)
Z(i,j) = f(X(i) , Y(j) , theta , gamma, p(:, 1:2) );
end
end
[X,Y] = meshgrid(X,Y);
contour(X, Y, Z, 'ShowText', 'on')
colorbar
hold on
for i = 1:m
if (p(i, 3)>0)
opt = "ob";
else
opt = "or";
end
plot(p(i, 1), p(i, 2), opt)
end
hold off
function z = f(x, y, theta, gamma, p)
z = 0;
global m
for i = 1:m
z = z + theta(i) * exp(-gamma .* norm([x, y] - p(i, :)));
end
return ;
end
function z = f_lsq(x, p)
gamma = x(end);
theta = x(1:end-1);
global m
z = zeros(1, m);
for i = 1:m
z(i) = f(p(i,1), p(i,2) , theta, gamma, p(:, 1:2) ) - p(i, 3);
end
end
题目3
题目描述
矩阵完备化的最小二乘法：
用随机方法生成一个较大的低秩矩阵，然后随机去掉里面大部分元素，再用最小二乘法恢复，比较前后异同
选择一张灰度图像，加入一些噪声，使用最小二乘法尝试恢复原图片
题解
- 生成低秩矩阵
A =
47 51 89 57 49 75 74 79 73 67
27 36 69 39 42 48 72 57 54 51
54 57 98 64 53 85 78 88 81 74
58 54 86 62 42 86 52 82 74 66
33 39 71 43 41 55 66 61 57 53
66 63 102 72 51 99 66 96 87 78
52 66 124 72 74 90 124 104 98 92
52 51 84 58 43 79 58 78 71 64
11 18 37 19 24 22 44 29 28 27
74 72 118 82 60 112 80 110 100 90
% 生成30个噪点
a =
Inf 51 89 57 49 75 74 79 73 67
27 Inf Inf 39 42 48 Inf Inf Inf 51
54 57 98 64 53 85 78 88 81 Inf
Inf Inf 86 Inf 42 86 52 82 74 Inf
33 39 71 Inf 41 Inf 66 Inf 57 Inf
66 63 102 72 Inf Inf 66 96 87 78
52 66 Inf Inf 74 90 124 104 Inf Inf
52 51 Inf 58 43 79 58 Inf Inf Inf
Inf Inf 37 Inf 24 22 Inf 29 28 27
74 72 118 82 60 112 80 110 Inf 90
- 复原
a =
47 51 89 57 49 75 74 79 73 67
27 35 66 39 42 48 65 55 53 51
54 57 98 64 53 85 78 88 81 75
58 54 86 60 42 86 52 82 74 66
33 39 71 45 41 55 66 61 57 54
66 63 102 72 52 99 66 96 87 78
52 66 124 78 74 90 124 104 101 94
52 51 84 58 43 79 58 78 71 64
12 18 37 22 24 22 40 29 28 27
74 72 118 82 60 112 80 110 99 90
残差平方和: 151
- 灰度图加噪声
代码
%% 低秩矩阵生成
clc, clear
n = 10;
B = randi(10, n, 2);
C = randi(10, 2, n);
A = B * C;
a = A;
m = 30;
id = randperm(n * n);
a(id(1:m)) = inf;
%% 图片加噪音
rgb = imread('Lenna.png');
rgb = imresize(rgb, 0.5, 'nearest');
A = rgb2gray(rgb);
disp(['原图秩：', num2str(rank(double(A)))])
[n , m] = size(A);
p = 5000; % 噪点数量
id = randperm(n * m);
a = A;
a(id(1:p)) = inf;
a = double(a);
imshow(a, [0,255]);
%%
clc
% A为nxm，B为nxr, C为rxm
r = 5; % 假设秩为5
a = double(a);
A = double(A);
x0 = randi(16, n+m, r);
% 非线性最小二乘法
fun = @(x)fmatrix(x, n, m, A);
% x = lsqnonlin(fun, x0);
A_Recover = round(x(1:n, :) * x(n+1:n+m, :)');
a(id(1:p)) = A_Recover(id(1:p));
imshow(a, [0,255]);
function y = fmatrix(x, n, m, a)
y = a - x(1:n, :) * x(n+1:n+m, :)';
y(y == inf) = 0;
end
