SVM

回顾一下之前的SVM，找到一个间隔最大的函数，使得正负样本离该函数是最远的，是否最远不是看哪个点离函数最远，而是找到一个离函数最近的点看他是不是和该分割函数离的最近的。

使用large margin来regularization。
之前讲SVM的算法：https://www.jianshu.com/p/8fd28df734a0

线性分类

线性SVM就是一种线性分类的方法。输入，输出，每一个样本的权重是，偏置项bias是。得分函数
算出这么多个类别，哪一个类别的分数高，那就是哪个类别。比如要做的图像识别有三个类别，假设这张图片有4个像素，拉伸成单列：

得到的结果很明显是dog分数最大，cat的分数最低，但是图片很明显是猫，什么分类器是错误的。
一般来说习惯会把w和b合并了，x加上一个全为1的列，于是有

损失函数

之前的SVM是把正负样本离分割函数有足够的空间，虽然正确的是猫，但是猫的得分是最低的，常规方法是将猫的分数提高，这样才可以提高猫的正确率。但是SVM里面是要求一个间隔最大化，提到这里来说，其实就是cat score不仅仅是要大于其他的分数，而且是要有一个最低阈值，cat score不能低于这个分数。
所以正确的分类score应该是要大于其他的分类score一个阈值：
就是正确分类的分数，就是其他分类的分数。所以，这个损失函数就是：只有正确的分数比其他的都大于一个阈值才为0，否则都是有损失的。

只有

首先要读入每一个文件的数据，先用load_pickle把文件读成字典形式，取出来。因为常规的图片都是(数量，高，宽，RGB颜色)，在loadCIFAR_batch要用transpose来把维度调换一下。最后把每一个文件的数据都集合起来。
之后就是数据的格式调整了：

def data_validation(x_train, y_train, x_test, y_test):
    num_training = 49000
    num_validation = 1000
    num_test = 1000
    num_dev = 500
    mean_image = np.mean(x_train, axis=0)
    x_train -= mean_image
    mask = range(num_training, num_training + num_validation)
    X_val = x_train[mask]
    Y_val = y_train[mask]
    mask = range(num_training)
    X_train = x_train[mask]
    Y_train = y_train[mask]
    mask = np.random.choice(num_training, num_dev, replace=False)
    X_dev = x_train[mask]
    Y_dev = y_train[mask]
    mask = range(num_test)
    X_test = x_test[mask]
    Y_test = y_test[mask]
    X_train = np.reshape(X_train, (X_train.shape[0], -1))
    X_val = np.reshape(X_val, (X_val.shape[0], -1))
    X_test = np.reshape(X_test, (X_test.shape[0], -1))
    X_dev = np.reshape(X_dev, (X_dev.shape[0], -1))
    X_train = np.hstack([X_train, np.ones((X_train.shape[0], 1))])
    X_val = np.hstack([X_val, np.ones((X_val.shape[0], 1))])
    X_test = np.hstack([X_test, np.ones((X_test.shape[0], 1))])
    X_dev = np.hstack([X_dev, np.ones((X_dev.shape[0], 1))])
    return X_val, Y_val, X_train, Y_train, X_dev, Y_dev, X_test, Y_test
    pass

数据要变成一个长条。
先看看数据长啥样：

def showPicture(x_train, y_train):
    classes = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
    num_classes = len(classes)
    samples_per_classes = 7
    for y, cls in enumerate(classes):
        idxs = np.flatnonzero(y_train == y)
        idxs = np.random.choice(idxs, samples_per_classes, replace=False)
        for i, idx in enumerate(idxs):
            plt_index = i*num_classes +y + 1
            plt.subplot(samples_per_classes, num_classes, plt_index)
            plt.imshow(x_train[idx].astype('uint8'))
            plt.axis('off')
            if i == 0:
                plt.title(cls)
    plt.show()

然后就是使用谷歌的公式了：

    def loss(self, x, y, reg):
        loss = 0.0
        dw = np.zeros(self.W.shape)
        num_train = x.shape[0]
        scores = x.dot(self.W)
        correct_class_score = scores[range(num_train), list(y)].reshape(-1, 1)
        margin = np.maximum(0, scores - correct_class_score + 1)
        margin[range(num_train), list(y)] = 0
        loss = np.sum(margin)/num_train + 0.5 * reg * np.sum(self.W*self.W)

        num_classes = self.W.shape[1]
        inter_mat = np.zeros((num_train, num_classes))
        inter_mat[margin > 0] = 1
        inter_mat[range(num_train), list(y)] = 0
        inter_mat[range(num_train), list(y)] = -np.sum(inter_mat, axis=1)

        dW = (x.T).dot(inter_mat)
        dW = dW/num_train + reg*self.W
        return loss, dW
        pass

操作都是常规操作，算出score然后求loss最后SGD求梯度更新W。

    def train(self, X, y, learning_rate=1e-3, reg=1e-5, num_iters=100,batch_size=200, verbose=False):
        num_train, dim = X.shape
        num_classes = np.max(y) + 1
        if self.W is None:
            self.W = 0.001 * np.random.randn(dim, num_classes)
        # Run stochastic gradient descent to optimize W
        loss_history = []
        for it in range(num_iters):
            X_batch = None
            y_batch = None
            idx_batch = np.random.choice(num_train, batch_size, replace = True)
            X_batch = X[idx_batch]
            y_batch = y[idx_batch]
            # evaluate loss and gradient
            loss, grad = self.loss(X_batch, y_batch, reg)
            loss_history.append(loss)
            self.W -=  learning_rate * grad
            if verbose and it % 100 == 0:
                print('iteration %d / %d: loss %f' % (it, num_iters, loss))
        return loss_history
        pass

预测：

    def predict(self, X):
        y_pred = np.zeros(X.shape[0])
        scores = X.dot(self.W)
        y_pred = np.argmax(scores, axis = 1)
        return y_pred

最后运行函数：

 svm = LinearSVM()
    tic = time.time()
    cifar10_name = '../Data/cifar-10-batches-py'
    x_train, y_train, x_test, y_test = loadCIFAR10(cifar10_name)
    X_val, Y_val, X_train, Y_train, X_dev, Y_dev, X_test, Y_test = data_validation(x_train, y_train, x_test, y_test)
    loss_hist = svm.train(X_train, Y_train, learning_rate=1e-7, reg=2.5e4,
                          num_iters=3000, verbose=True)
    toc = time.time()
    print('That took %fs' % (toc - tic))
    plt.plot(loss_hist)
    plt.xlabel('Iteration number')
    plt.ylabel('Loss value')
    plt.show()
    y_test_pred = svm.predict(X_test)
    test_accuracy = np.mean(Y_test == y_test_pred)
    print('accuracy: %f' % test_accuracy)
    w = svm.W[:-1, :]  # strip out the bias
    w = w.reshape(32, 32, 3, 10)
    w_min, w_max = np.min(w), np.max(w)
    classes = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
    for i in range(10):
        plt.subplot(2, 5, i + 1)
        wimg = 255.0 * (w[:, :, :, i].squeeze() - w_min) / (w_max - w_min)
        plt.imshow(wimg.astype('uint8'))
        plt.axis('off')
        plt.title(classes[i])
    plt.show()

首先是画出整个loss函数趋势：

最后再可视化一下w权值，看看每一个种类提取处理的特征是什么样子的：