9月17日计算机视觉基础学习笔记——认识机器学习

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前言

本文为9月17日计算机视觉基础学习笔记——认识机器学习，分为两个章节：

• Week 2 homework；
• 机器学习。

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一、Week 2 homework

生成10张图片，对应数字 0-9。对这10张图片提取特征x，用一个判别器f(x)来决策输出结果y。

import torch

def generate_data():
    '''
    本函数生成0-9，10个数字的图片矩阵
    '''
    
    image_data=[]
    num_0 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,1,0,0,1,0],
    [0,1,0,0,1,0],
    [0,1,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_0)
    num_1 = torch.tensor(
    [[0,0,0,1,0,0],
    [0,0,1,1,0,0],
    [0,0,0,1,0,0],
    [0,0,0,1,0,0],
    [0,0,1,1,1,0],
    [0,0,0,0,0,0]])
    image_data.append(num_1)
    num_2 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,1,0,0,1,0],
    [0,0,0,1,0,0],
    [0,0,1,0,0,0],
    [0,1,1,1,1,0],
    [0,0,0,0,0,0]])
    image_data.append(num_2)
    num_3 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,0,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_3)
    num_4 = torch.tensor(
    [
    [0,0,0,0,1,0],
    [0,0,0,1,1,0],
    [0,0,1,0,1,0],
    [0,1,1,1,1,1],
    [0,0,0,0,1,0],
    [0,0,0,0,0,0]])
    image_data.append(num_4)
    num_5 = torch.tensor(
    [
    [0,1,1,1,0,0],
    [0,1,0,0,0,0],
    [0,1,1,1,0,0],
    [0,0,0,0,1,0],
    [0,1,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_5)
    num_6 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,1,0,0,0,0],
    [0,1,1,1,0,0],
    [0,1,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_6)
    num_7 = torch.tensor(
    [
    [0,1,1,1,1,0],
    [0,0,0,0,1,0],
    [0,0,0,1,0,0],
    [0,0,0,1,0,0],
    [0,0,0,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_7)
    num_8 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,1,0,0,1,0],
    [0,0,1,1,0,0],
    [0,1,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_8)
    num_9 = torch.tensor(
    [[0,0,1,1,1,0],
    [0,1,0,0,1,0],
    [0,0,1,1,1,0],
    [0,0,0,0,1,0],
    [0,0,0,0,1,0],
    [0,0,0,0,0,0]])
    image_data.append(num_9)
    return image_data

import matplotlib.pyplot as plt
image_data = generate_data()
print(image_data[0])
>>> tensor([[0, 0, 1, 1, 0, 0],
            [0, 1, 0, 0, 1, 0],
            [0, 1, 0, 0, 1, 0],
            [0, 1, 0, 0, 1, 0],
            [0, 0, 1, 1, 0, 0],
            [0, 0, 0, 0, 0, 0]])

# 方法：提取特征
def get_feature(x):
    return torch.sum(x, 0)

print(get_feature(image_data[2]))
print(get_feature(image_data[8]))
>>> tensor([0, 2, 3, 3, 2, 0])
>>> tensor([0, 2, 3, 3, 2, 0])

i = 0
def model(x, image_data):
    y = -1
    
    for i in range(0, 10):
        diff_tmp = get_feature(x) - get_feature(image_data[i])
        
        if torch.sum(torch.abs(diff_tmp)) == 0:
            y = i
            break
    print("{} --识别为--> {}".format(x, y))
    return y

model(image_data[8], image_data)
>>> tensor([[0, 0, 1, 1, 0, 0],
            [0, 1, 0, 0, 1, 0],
            [0, 0, 1, 1, 0, 0],
            [0, 1, 0, 0, 1, 0],
            [0, 0, 1, 1, 0, 0],
            [0, 0, 0, 0, 0, 0]]) --识别为--> 2
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二、机器学习

1、线性回归模型

• 特征： $\text{V}:[v_1,v_2,\dots ,v_6{]}^T$；
• 判别式： $y=\text{W}\ast \text{V}=[{\omega }_1,{\omega }_2,\dots ,{\omega }_6][v_1,v_2,\dots ,v_6{]}^T$.
• Loss： ${\sum }_i({\hat{y}}_i-y_i{)}^2$.
• 求 loss 的最小值对应的 $\omega$： $\partial \frac{{\sum }_i({\hat{y}}_i-y_i{)}^2}{\partial w_j}=0$.

代码如下：

import torch
from itertools import product

def generate_data():
    '''
    本函数生成0-9，10个数字的图片矩阵
    :return: image_data,image_label
    '''

    image_data=[]
    num_0 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,1,0,0,1,0],
    [0,1,0,0,1,0],
    [0,1,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_0)
    num_1 = torch.tensor(
    [[0,0,0,1,0,0],
    [0,0,1,1,0,0],
    [0,0,0,1,0,0],
    [0,0,0,1,0,0],
    [0,0,1,1,1,0],
    [0,0,0,0,0,0]])
    image_data.append(num_1)
    num_2 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,1,0,0,1,0],
    [0,0,0,1,0,0],
    [0,0,1,0,0,0],
    [0,1,1,1,1,0],
    [0,0,0,0,0,0]])
    image_data.append(num_2)
    num_3 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,0,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_3)
    num_4 = torch.tensor(
    [
    [0,0,0,0,1,0],
    [0,0,0,1,1,0],
    [0,0,1,0,1,0],
    [0,1,1,1,1,1],
    [0,0,0,0,1,0],
    [0,0,0,0,0,0]])
    image_data.append(num_4)
    num_5 = torch.tensor(
    [
    [0,1,1,1,0,0],
    [0,1,0,0,0,0],
    [0,1,1,1,0,0],
    [0,0,0,0,1,0],
    [0,1,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_5)
    num_6 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,1,0,0,0,0],
    [0,1,1,1,0,0],
    [0,1,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_6)
    num_7 = torch.tensor(
    [
    [0,1,1,1,1,0],
    [0,0,0,0,1,0],
    [0,0,0,1,0,0],
    [0,0,0,1,0,0],
    [0,0,0,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_7)
    num_8 = torch.tensor(
    [[0,0,1,1,0,0],
    [0,1,0,0,1,0],
    [0,0,1,1,0,0],
    [0,1,0,0,1,0],
    [0,0,1,1,0,0],
    [0,0,0,0,0,0]])
    image_data.append(num_8)
    num_9 = torch.tensor(
    [[0,0,1,1,1,0],
    [0,1,0,0,1,0],
    [0,0,1,1,1,0],
    [0,1,0,0,1,0],
    [0,0,0,0,1,0],
    [0,0,0,0,0,0]])
    image_data.append(num_9)
    image_label=[0,1,2,3,4,5,6,7,8,9]
    return image_data,image_label

# 方法：提取特征
def get_feature(x):
    feature = [0, 0, 0, 0]
    # 提取图像x的特征 feature 的代码
    def get_shadow(x, dim):
        print("提取特征，x 的尺寸：", x.size())
        feature = torch.sum(x, dim)
        feature = feature.float()
        print("提取特征，feature 的尺寸：", feature.size())

        # 归一化
        for i in range(feature.shape[0]):
            feature[i] = feature[i] / sum(feature)

        feature = feature.view(1, 6)
        return feature

    feature = get_shadow(x, 0)

    return feature

def model(feature, weights):
    y = -1
    '''
    # 下面添加对feature进行决策的代码，
    判定出feature 属于[0,1,2,3,...9]哪个类别
    '''
    feature = torch.cat((feature, torch.tensor(1.0).view(1, 1)), 1)
    y = feature.mm(weights)
    return y

def train_model(image_data, image_label, weights):
    for epoch in range(3000):
        loss = 0
        for i in range(len(image_data)):
            feature = get_feature(image_data[i])
            y = model(feature, weights)

            # 计算 yhat 和期望 iamge_label 的差距
            loss += 0.5 * (y.item() - image_label[i])**2  # .item()返回的是一个浮点型数据，精度更高

            # 更新权重 w
            # w = w - lr * (y - y1) * x
            feature = feature.view(6)
            lr = -0.05
            weights[0, 0] = weights[0, 0] + (y.item() - image_label[i]) * feature[0] * lr
            weights[1, 0] = weights[1, 0] + (y.item() - image_label[i]) * feature[1] * lr
            weights[2, 0] = weights[2, 0] + (y.item() - image_label[i]) * feature[2] * lr
            weights[3, 0] = weights[3, 0] + (y.item() - image_label[i]) * feature[3] * lr
            weights[4, 0] = weights[4, 0] + (y.item() - image_label[i]) * feature[4] * lr
            weights[5, 0] = weights[5, 0] + (y.item() - image_label[i]) * feature[5] * lr
            weights[6, 0] = weights[6, 0] + (y.item() - image_label[i]) * lr

        loss = 0

    return weights

if __name__ == '__main__':
    # 生成 w0
    weights = torch.rand(7, 1)
    image_data, image_label = generate_data()

    # 打印出 0 的图像
    print("数字 0 的图片是：", image_data[0])
    print("-" * 20)

    # 打印出8的图像
    print("数字 8 的图片是：", image_data[8])
    print("-" * 20)

    # 训练模型
    weights = train_model(image_data, image_label, weights)

    # 测试：识别每张图片
    print("对每张图片进行识别")

    for i in range(6):
        x = image_data[i]
        feature = get_feature(x)
        y = model(feature, weights)

        print("图像{}的分类结果是：{}，它的特征是：{}".format(i, y, feature))
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2、逻辑回归模型

• 非0即1；

• 线性模型取对数。

• Sigmoid 函数：
  $$\sigma (a)=\frac{1}{1+e^{-a}}$$

其导数为：$\frac{\partial \sigma (a)}{\partial a}=\sigma (a)(1-\sigma (a))$.

• 特征：$\varphi (x)$，那么逻辑回归将这个例子属于第1类的概率建模为:
  $$p(t=1|\text{x};\text{w})=\sigma ({\text{w}}^T\varphi (\text{x}))$$
  并将属于0类的例子的概率定义为:

$$p(t=0|\text{x};\text{w})=1-p(t=1|\text{x};\text{w})=1-\sigma ({\text{w}}^T\varphi (\text{x}))$$

• 极大似然估计（Maximum Likelihood Estimate）： 当前事件发生概率最大时，对应的概率密度函数中的参数：
  $$\begin{gathered} L(\text{w})=\prod _{i=1}^{N}p(t^{(i)}|{\text{x}}^{(i)};\text{w}) \\ p(t^{(i)}|{\text{x}}^{(i)};\text{w})=p(t=1|{\text{x}}^{(i)};\text{w}{)}^{t^{(i)}}p(t=0|{\text{x}}^{(i)};\text{w}{)}^{1-t^{(i)}} \end{gathered}$$

试图找到最大化似然 $L(\text{w})$的 $\text{w}$，即找到最大化对数似然 $l(\text{w})=logL(\text{w})$ 的 $\text{w}$，为简化推导，设 $\varphi (\text{x})=\text{x}$：

$$\begin{gathered} argmaxl(\text{w})=argmaxlog\prod _{i=1}^{N}p(t^{(i)}|{\text{x}}^{(i)};\text{w}) \\ =argmax\sum _{i=1}^{N}logp(t^{(i)}|{\text{x}}^{(i)};\text{w}) \\ =argmax\sum _{i=1}^{N}log[p(t=1|{\text{x}}^{(i)};\text{w}{)}^{t^{(i)}}p(t=0|{\text{x}}^{(i)};\text{w}{)}^{1-t^{(i)}}] \\ =argmax\sum _{i=1}^{N}log[p(t=1|{\text{x}}^{(i)};\text{w}{)}^{t^{(i)}}]+log[p(t=0|{\text{x}}^{(i)};\text{w}{)}^{1-t^{(i)}}]] \\ =argmax\sum _{i=1}^N{t}^{(i)}log[p(t=1|{\text{x}}^{(i)};\text{w})]+(1-t^{(i)})log[p(t=0|{\text{x}}^{(i)};\text{w})] \\ =argmax\sum _{i=1}^N{t}^{(i)}log[\sigma ({\text{w}}^T{\text{x}}^{(i)})]+(1-t^{(i)})log[1-\sigma ({\text{w}}^T{\text{x}}^{(i)})] \end{gathered}$$

为找到使该表达式最大的 $\text{w}$，求其对 $\text{w}$ 的导数：

$$\begin{gathered} {▽}_{\text{w}}l(\text{w})={▽}_{\text{w}}\sum _{i=1}^N{t}^{(i)}log[\sigma ({\text{w}}^T{\text{x}}^i)]+(1-t^{(i)})log[1-\sigma ({\text{w}}^T{\text{x}}^{(i)})] \\ =\sum _{i=1}^N{t}^{(i)}(\frac{1}{\sigma ({\text{w}}^T{\text{x}}^i)})\times (\sigma ({\text{w}}^T{\text{x}}^i)(1-\sigma ({\text{w}}^T{\text{x}}^i)))\times {\text{x}}^{(i)}+(1-t^{(i)})(\frac{1}{1-\sigma ({\text{w}}^T{\text{x}}^i)})\times (-1)\times (\sigma ({\text{w}}^T{\text{x}}^i)(1-\sigma ({\text{w}}^T{\text{x}}^i)))\times {\text{x}}^i \\ =\sum _{i=1}^N{t}^{(i)}(1-\sigma ({\text{w}}^T{\text{x}}^{(i)}))-(1-t^{(i)})(-1)(\sigma ({\text{w}}^T{\text{x}}^{(i)}){\text{x}}^{(i)} \\ =\sum _{i=1}^N{t}^{(i)}{\text{x}}^{(i)}-\sigma ({\text{w}}^T{\text{x}}^{(i)}){\text{x}}^{(i)} \end{gathered}$$

得到：

$${▽}_{\text{w}}l(\text{w})=\sum _{i=1}^N(t^{(i)}-\sigma ({\text{w}}^T{\text{x}}^{(i)})){\text{x}}^{(i)}$$

可见，无法使 ${▽}_{\text{w}}l(\text{w})=0$ 并求得 $\text{w}$，因为 $t^{(i)}$ 为 0 或 1，而 sigmoid 函数不可能为 0 或 1。因此只能用对似然函数进行梯度上升计算 $\text{w}$。

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