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：
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
这种squared hinge loss SVM与linear hinge loss SVM相比较，特点是对违背间隔阈值要求的点加重惩罚，违背的越大，惩罚越大。某些实际应用中，squared hinge loss SVM的效果更好一些。具体使用哪个，可以根据实际问题，进行交叉验证再确定。
对于
最后还要增加的就是过拟合，regularization的限制了。L2正则化：
加上正则化之后就是：
N是训练样本的个数，取平均损失函数，
代码实现
首先是对CIFAR10的数据读取：
def load_pickle(f):
version = platform.python_version_tuple()
if version[0] == '2':
return pickle.load(f)
elif version[0] == '3':
return pickle.load(f, encoding='latin1')
raise ValueError("invalid python version: {}".format(version))
def loadCIFAR_batch(filename):
with open(filename, 'rb') as f:
datadict = load_pickle(f)
x = datadict['data']
y = datadict['labels']
x = x.reshape(10000, 3, 32, 32).transpose(0, 3, 2, 1).astype('float')
y = np.array(y)
return x, y
def loadCIFAR10(root):
xs = []
ys = []
for b in range(1, 6):
f = os.path.join(root, 'data_batch_%d' % (b, ))
x, y = loadCIFAR_batch(f)
xs.append(x)
ys.append(y)
X = np.concatenate(xs)
Y = np.concatenate(ys)
x_test, y_test = loadCIFAR_batch(os.path.join(root, 'test_batch'))
return X, Y, x_test, y_test
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：