PyTorch框架学习(二) — 张量操作与线性回归
1 张量的操作
1.1 拼接
torch.cat()
torch.cat(tensors, dim=0, *, out=None) → Tensor
【功能】:将张量按照 dim 维度进行拼接。
∙ \bullet ∙ tensors: 任何相同类型的python张量序列,其中的非空张量除了拼接维度上的形状可以不同,其他维度上的形状必须相同。注:python的序列数据只有 list
和 tuple
。
∙ \bullet ∙
dim:
要拼接的维度,其中,dim
∈ \in ∈
[0, len(tensor[0])) 。
【代码】:
import torch t = torch.randn(2, 3) t_0 = torch.cat([t, t, t], dim=0) t_1 = torch.cat([t, t, t], dim=1) print("t = {} shape = {} t_0 = {} shape = {} t_1 = {} shape = {}" .format(t, t.shape, t_0, t_0.shape, t_1, t_1.shape))
【结果】:
t = tensor([[-0.6014, -1.0122, -0.3023], [-1.2277, 0.9198, -0.3485]]) shape = torch.Size([2, 3]) t_0 = tensor([[-0.6014, -1.0122, -0.3023], [-1.2277, 0.9198, -0.3485], [-0.6014, -1.0122, -0.3023], [-1.2277, 0.9198, -0.3485], [-0.6014, -1.0122, -0.3023], [-1.2277, 0.9198, -0.3485]]) shape = torch.Size([6, 3]) t_1 = tensor([[-0.6014, -1.0122, -0.3023, -0.6014, -1.0122, -0.3023, -0.6014, -1.0122, -0.3023], [-1.2277, 0.9198, -0.3485, -1.2277, 0.9198, -0.3485, -1.2277, 0.9198, -0.3485]]) shape = torch.Size([2, 9])
torch.stack()
torch.stack(tensors, dim=0, *, out=None) → Tensor
【功能】:沿着一个新维度对输入张量序列进行连接。序列中所有的张量都应该为相同形状。可以理解为:把多个2维的张量凑成一个3维的张量;多个3维的凑成一个4维的张量…以此类推,也就是在增加新的维度进行堆叠。
∙ \bullet ∙ tensors: 待连接的张量序列。注:python的序列数据只有 list
和 tuple
。
∙ \bullet ∙ dim: 新的维度, 其中 dim ∈ \in ∈ [0, len(out)) 。注: len(out)
是生成数据的维度大小,也就是 out
的维度值。
【代码】:
import torch t1 = torch.tensor([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) t2 = torch.tensor([[10, 20, 30], [40, 50, 60], [70, 80, 90]]) t_stack_0 = torch.stack([t1, t2], dim=0) print("t_stack_0:{}".format(t_stack_0)) print("t_stack_0.shape:{} ".format(t_stack_0.shape)) t_stack_1 = torch.stack([t1, t2], dim=1) print("t_stack_1:{}".format(t_stack_1)) print("t_stack_1.shape:{} ".format(t_stack_1.shape)) t_stack_2 = torch.stack([t1, t2], dim=2) print("t_stack_2:{}".format(t_stack_2)) print("t_stack_2.shape:{} ".format(t_stack_2.shape)) t_stack_3 = torch.stack([t1, t2], dim=3) print("t_stack_3.shape:{}".format(t_stack_3.shape))
【结果】:
t_stack_0:tensor([[[ 1, 2, 3], [ 4, 5, 6], [ 7, 8, 9]], [[10, 20, 30], [40, 50, 60], [70, 80, 90]]]) t_stack_0.shape:torch.Size([2, 3, 3]) t_stack_1:tensor([[[ 1, 2, 3], [10, 20, 30]], [[ 4, 5, 6], [40, 50, 60]], [[ 7, 8, 9], [70, 80, 90]]]) t_stack_1.shape:torch.Size([3, 2, 3]) t_stack_2:tensor([[[ 1, 10], [ 2, 20], [ 3, 30]], [[ 4, 40], [ 5, 50], [ 6, 60]], [[ 7, 70], [ 8, 80], [ 9, 90]]]) t_stack_2.shape:torch.Size([3, 3, 2]) IndexError: Dimension out of range (expected to be in range of [-3, 2], but got 3)
【结果说明】:当dim=3时,此时dim= len(out)=3,所以溢出报错。
1.2 切分
torch.chunk()
torch.chunk(input, chunks, dim=0) → List of Tensors
【功能】:将张量按照维度 dim 进行平均切分。若不能整除,则最后一份张量小于其他张量。
∙ \bullet ∙ input: 待切分的张量。
∙ \bullet ∙ chunks: 要切分的份数。
∙ \bullet ∙ dim: 要切分的维度。
【代码】:
import torch a = torch.ones((2, 7)) print("a = {}".format(a)) list_of_tensors_1 = torch.chunk(a, chunks=3, dim=1) for idx, t in enumerate(list_of_tensors_1): print("第{}个张量: {},shape is {}".format(idx+1, t, t.shape)) print(" ") b = torch.arange(11) print("b = {}".format(b)) list_of_tensors_2 = b.chunk(6) for idx, t in enumerate(list_of_tensors_2): print("第{}个张量: {},shape is {}".format(idx+1, t, t.shape)) print(" ") c = torch.arange(12) print("c = {}".format(c)) list_of_tensors_3 = c.chunk(6) for idx, t in enumerate(list_of_tensors_3): print("第{}个张量: {},shape is {}".format(idx+1, t, t.shape))
【结果】:
a = tensor([[1., 1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1., 1.]]) 第1个张量: tensor([[1., 1., 1.], [1., 1., 1.]]),shape is torch.Size([2, 3]) 第2个张量: tensor([[1., 1., 1.], [1., 1., 1.]]),shape is torch.Size([2, 3]) 第3个张量: tensor([[1.], [1.]]),shape is torch.Size([2, 1]) b = tensor([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) 第1个张量: tensor([0, 1]),shape is torch.Size([2]) 第2个张量: tensor([2, 3]),shape is torch.Size([2]) 第3个张量: tensor([4, 5]),shape is torch.Size([2]) 第4个张量: tensor([6, 7]),shape is torch.Size([2]) 第5个张量: tensor([8, 9]),shape is torch.Size([2]) 第6个张量: tensor([10]),shape is torch.Size([1]) c = tensor([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]) 第1个张量: tensor([0, 1]),shape is torch.Size([2]) 第2个张量: tensor([2, 3]),shape is torch.Size([2]) 第3个张量: tensor([4, 5]),shape is torch.Size([2]) 第4个张量: tensor([6, 7]),shape is torch.Size([2]) 第5个张量: tensor([8, 9]),shape is torch.Size([2]) 第6个张量: tensor([10, 11]),shape is torch.Size([2])
【结果说明】:由于 7 不能整除 3,7/3 再向上取整是 3,因此前两个维度是 [2, 3],所以最后一个切分的张量维度是 [2,1]。
torch.split()
torch.split(tensor, split_size_or_sections, dim=0)
【功能】:将张量按照维度 dim 进行平均切分。可以指定每一个分量的切分长度。
∙ \bullet ∙ tensor: 待切分的张量。
∙ \bullet ∙ split_size_or_sections: 为 int 时,表示每一份的长度,如果不能被整除,则最后一份张量小于其他张量;为 list 时,按照 list 元素作为每一个分量的长度切分。如果 list 元素之和不等于切分维度 (dim) 的值,就会报错。
∙ \bullet ∙ dim: 要切分的维度。
【代码】:
import torch a = torch.ones((2, 5)) print("a = {}".format(a)) list_of_tensors_1 = torch.split(a, [2, 1, 2], dim=1) for idx, t in enumerate(list_of_tensors_1): print("第{}个张量:{}, shape is {}".format(idx + 1, t, t.shape)) print(" ") b = torch.arange(10).reshape(5, 2) print("b = {}".format(b)) list_of_tensors_2 = torch.split(b, 2) for idx, t in enumerate(list_of_tensors_2): print("第{}个张量:{}, shape is {}".format(idx + 1, t, t.shape)) print(" ") c = torch.arange(10).reshape(5, 2) print("c = {}".format(c)) list_of_tensors_3 = torch.split(c, [1, 4]) for idx, t in enumerate(list_of_tensors_3): print("第{}个张量:{}, shape is {}".format(idx + 1, t, t.shape))
【结果】:
a = tensor([[1., 1., 1., 1., 1.], [1., 1., 1., 1., 1.]]) 第1个张量:tensor([[1., 1.], [1., 1.]]), shape is torch.Size([2, 2]) 第2个张量:tensor([[1.], [1.]]), shape is torch.Size([2, 1]) 第3个张量:tensor([[1., 1.], [1., 1.]]), shape is torch.Size([2, 2]) b = tensor([[0, 1], [2, 3], [4, 5], [6, 7], [8, 9]]) 第1个张量:tensor([[0, 1], [2, 3]]), shape is torch.Size([2, 2]) 第2个张量:tensor([[4, 5], [6, 7]]), shape is torch.Size([2, 2]) 第3个张量:tensor([[8, 9]]), shape is torch.Size([1, 2]) c = tensor([[0, 1], [2, 3], [4, 5], [6, 7], [8, 9]]) 第1个张量:tensor([[0, 1]]), shape is torch.Size([1, 2]) 第2个张量:tensor([[2, 3], [4, 5], [6, 7], [8, 9]]), shape is torch.Size([4, 2])
1.3 索引
torch.index_select()
torch.index_select(input, dim, index, *, out=None) → Tensor
【功能】:在维度 dim 上,按照 index 索引取出数据拼接为张量返回。
∙ \bullet ∙ input: 待索引的张量。
∙ \bullet ∙ dim: 要索引的维度。
∙ \bullet ∙ index(IntTensor or LongTensor): 要索引数据的序号,1D张量。
【代码】:
import torch t = torch.randint(0, 9, size=(3, 3)) # 创建均匀分布 idx = torch.tensor([0, 2], dtype=torch.long) # 注意 idx 的 dtype 不能指定为 torch.float print("idx: {}".format(idx)) t_select_1 = torch.index_select(t, dim=0, index=idx) # 取出第 0 行和第 2 行 print("t: {} t_select_1: {} ".format(t, t_select_1)) x = torch.randn(3, 4) indices = torch.tensor([0, 2]) t_select_2 = torch.index_select(x, 0, indices) # 取出第 0 行和第 2 行 print("x: {}".format(x)) print("t_select_2: {}".format(t_select_2)) t_select_3 = torch.index_select(x, 1, indices) # 取出第 0 列和第 2 列 print("t_select_3: {}".format(t_select_3))
【结果】:
idx: tensor([0, 2]) t: tensor([[5, 3, 6], [4, 6, 2], [8, 7, 4]]) t_select_1: tensor([[5, 3, 6], [8, 7, 4]]) x: tensor([[-0.7894, -0.9907, -1.9858, 0.5365], [ 1.7030, -2.0950, -0.9801, 0.2507], [-0.1537, 0.9861, 0.0340, -1.5576]]) t_select_2: tensor([[-0.7894, -0.9907, -1.9858, 0.5365], [-0.1537, 0.9861, 0.0340, -1.5576]]) t_select_3: tensor([[-0.7894, -1.9858], [ 1.7030, -0.9801], [-0.1537, 0.0340]])
torch.masked_select()
torch.masked_select(input, mask, *, out=None) → Tensor
【功能】:按照 mask 中的 True 进行索引拼接得到一维张量返回。
∙ \bullet ∙ input(Tensor): 待索引的张量。
∙ \bullet ∙ mask(BoolTensor): 与 input 同形状的布尔类型张量。
【代码】:
import torch t = torch.randint(0, 9, size=(3, 3)) print("t: {}".format(t)) mask = t.le(5) print("mask: {}".format(mask)) t_select = torch.masked_select(t, mask) # 取出小于等于 5 的数 print("t_select: {} ".format(t_select)) x = torch.randn(3, 4) print("x: {}".format(x)) mask = x.ge(0.5) print("mask: {}".format(mask)) x_select = torch.masked_select(x, mask) print("x_select: {} ".format(x_select)) # 取出大于等于 0.5 的数
【结果】:
t: tensor([[5, 3, 6], [4, 6, 2], [8, 7, 4]]) mask: tensor([[ True, True, False], [ True, False, True], [False, False, True]]) t_select: tensor([5, 3, 4, 2, 4]) x: tensor([[-0.7894, -0.9907, -1.9858, 0.5365], [ 1.7030, -2.0950, -0.9801, 0.2507], [-0.1537, 0.9861, 0.0340, -1.5576]]) mask: tensor([[False, False, False, True], [ True, False, False, False], [False, True, False, False]]) x_select: tensor([0.5365, 1.7030, 0.9861])
【注意】:最后返回的是一维张量。
1.4 变换
torch.reshape()
torch.reshape(input, shape) → Tensor
【功能】:变换张量的形状。当张量在内存中是连续时,返回的张量和原来的张量共享数据内存,改变一个变量时,另一个变量也会被改变。
∙ \bullet ∙ input(Tensor): 待变换的张量。
∙ \bullet ∙ shape(tuple of python:ints): 新张量的形状。
【代码】:
import torch t = torch.randperm(8) # 生成 0 到 8 的随机排列 print("t: {}".format(t)) t_reshape = torch.reshape(t, (-1, 2, 2)) print("t_reshape: {}".format(t_reshape)) print("t_reshape_shape: {} ".format(t_reshape.shape)) a = torch.arange(4) print("a: {}".format(a)) a_reshape = torch.reshape(a, (2, 2)) print("a_reshape: {}".format(a_reshape)) print("a_reshape_shape: {} ".format(a_reshape.shape)) b = torch.tensor([[0, 1], [2, 3]]) print("b: {}".format(b)) b_reshape = torch.reshape(b, (-1, )) print("b_reshape: {}".format(b_reshape)) print("b_reshape_shape: {} ".format(b_reshape.shape)) # 修改原来的张量的一个元素,新张量也会被改变 t[0] = 1024 print("t: {}".format(t)) print("t_reshape: {}".format(t_reshape)) print("t.data内存地址: {}".format(id(t.data))) print("t_reshape.data内存地址: {}".format(id(t_reshape.data)))
【结果】:
t: tensor([1, 0, 2, 5, 4, 7, 3, 6]) t_reshape: tensor([[[1, 0], [2, 5]], [[4, 7], [3, 6]]]) t_reshape_shape: torch.Size([2, 2, 2]) a: tensor([0, 1, 2, 3]) a_reshape: tensor([[0, 1], [2, 3]]) a_reshape_shape: torch.Size([2, 2]) b: tensor([[0, 1], [2, 3]]) b_reshape: tensor([0, 1, 2, 3]) b_reshape_shape: torch.Size([4]) t: tensor([1024, 0, 2, 5, 4, 7, 3, 6]) t_reshape: tensor([[[1024, 0], [ 2, 5]], [[ 4, 7], [ 3, 6]]]) t.data内存地址: 140396336572672 t_reshape.data内存地址: 140396336572672
torch.transpose()
torch.transpose(input, dim0, dim1) → Tensor
【功能】:交换张量的两个维度。常用于图像的变换,比如把 c ∗ h ∗ w c\ast h\ast w 变换为 h ∗ w ∗ c h\ast w\ast c 。
∙ \bullet ∙ input(Tensor): 待交换的张量。
∙ \bullet ∙ dim0(int): 要交换的第一个维度。
∙ \bullet ∙ dim1(int): 要交换的第二个维度。
【代码】:
import torch t = torch.rand((2, 3, 4)) print("t: {}".format(t)) print("t shape: {} ".format(t.shape)) t_transpose = torch.transpose(t, dim0=1, dim1=2) print("t_transpose: {}".format(t_transpose)) print("t_transpose shape: {}".format(t_transpose.shape))
【结果】:
t: tensor([[[0.4581, 0.4829, 0.3125, 0.6150], [0.2139, 0.4118, 0.6938, 0.9693], [0.6178, 0.3304, 0.5479, 0.4440]], [[0.7041, 0.5573, 0.6959, 0.9849], [0.2924, 0.4823, 0.6150, 0.4967], [0.4521, 0.0575, 0.0687, 0.0501]]]) t shape: torch.Size([2, 3, 4]) t_transpose: tensor([[[0.4581, 0.2139, 0.6178], [0.4829, 0.4118, 0.3304], [0.3125, 0.6938, 0.5479], [0.6150, 0.9693, 0.4440]], [[0.7041, 0.2924, 0.4521], [0.5573, 0.4823, 0.0575], [0.6959, 0.6150, 0.0687], [0.9849, 0.4967, 0.0501]]]) t_transpose shape: torch.Size([2, 4, 3])
torch.t()
torch.t(input) → Tensor
【功能】:2 维张量转置,对于 2 维矩阵而言,等价于 torch.transpose(input, 0, 1)
。
∙ \bullet ∙ input(Tensor): 待转置的张量。
【代码】:
import torch x1 = torch.randn(()) print("x1: {}".format(x1)) x1_t = torch.t(x1) print("x1_t: {} ".format(x1_t)) x2 = torch.randn(3) print("x2: {}".format(x2)) x2_t = torch.t(x2) print("x2_t: {} ".format(x2_t)) x3 = torch.randn(2, 3) print("x3: {}".format(x3)) x3_t = torch.t(x3) print("x3_t: {}".format(x3_t))
【结果】:
x1: -0.6013928055763245 x1_t: -0.6013928055763245 x2: tensor([-1.0122, -0.3023, -1.2277]) x2_t: tensor([-1.0122, -0.3023, -1.2277]) x3: tensor([[ 0.9198, -0.3485, -0.8692], [-0.9582, -1.1920, 1.9050]]) x3_t: tensor([[ 0.9198, -0.9582], [-0.3485, -1.1920], [-0.8692, 1.9050]])
【结果说明】:对于0-D和1-D张量,输出结果仍然为输入本身。
torch.squeeze()
torch.squeeze(input, dim=None, *, out=None) → Tensor
【功能】:压缩长度为 1 的维度。
∙ \bullet ∙ input(Tensor): 待压缩的张量。
∙ \bullet ∙ dim(int, optional): 若为 None,则移除所有长度为 1 的维度;若指定维度,则当且仅当该维度长度为 1 时可以移除。
【代码】:
import torch t = torch.rand((1, 2, 3, 1)) # 维度 0 和 3 的长度是 1 t_sq = torch.squeeze(t) # 可以移除维度 0 和 3 t_0 = torch.squeeze(t, dim=0) # 可以移除维度 0 t_1 = torch.squeeze(t, dim=1) # 不能移除维度 1 print("t.shape: {}".format(t.shape)) print("t_sq.shape: {}".format(t_sq.shape)) print("t_0.shape: {}".format(t_0.shape)) print("t_1.shape: {}".format(t_1.shape))
【结果】:
t.shape: torch.Size([1, 2, 3, 1]) t_sq.shape: torch.Size([2, 3]) t_0.shape: torch.Size([2, 3, 1]) t_1.shape: torch.Size([1, 2, 3, 1])
torch.unsqueeze()
torch.unsqueeze(input, dim) → Tensor
【功能】:根据 dim 扩展维度,长度为 1。
∙ \bullet ∙ input(Tensor): 待扩展的张量。
∙ \bullet ∙ dim(int): 指定扩展的维度。
【代码】:
import torch t = torch.tensor([1, 2, 3, 4]) t_0 = torch.unsqueeze(t, 0) # 在维度 0 扩展 t_1 = torch.unsqueeze(t, 1) # 在维度 1 扩展 print("t.shape: {}".format(t.shape)) print("t_0.shape: {}".format(t_0.shape)) print("t_1.shape: {}".format(t_1.shape))
【结果】:
t.shape: torch.Size([4]) t_0.shape: torch.Size([1, 4]) t_1.shape: torch.Size([4, 1])
2 张量的数学运算
2.1 加法运算
torch.add()
torch.add(input, other, *, alpha=1, out=None) → Tensor
【功能】:逐元素相加,计算公式为: o u t i = i n p u t i + a l p h a × o t h e r i out_{i} = input_{i} + alpha \times other_{i} o u t i = i n p u t i + a l p h a × o t h e r i 。
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ other(Tensor or Number): 与input相加的张量或数值。
∙ \bullet ∙ alpha(Number): 乘项因子。
【代码】:
import torch a = torch.randn(4) print("a: {}".format(a)) a_add = torch.add(a, 20) print("a_add: {} ".format(a_add)) b = torch.randn(4) print("b: {}".format(b)) c = torch.randn(4, 1) print("c: {}".format(c)) b_add_c = torch.add(b, c, alpha=10) print("b_add_c: {} ".format(b_add_c)) d = b + 10 * c print("d: {} ".format(d))
【结果】:
a: tensor([-0.6014, -1.0122, -0.3023, -1.2277]) a_add: tensor([19.3986, 18.9878, 19.6977, 18.7723]) b: tensor([ 0.9198, -0.3485, -0.8692, -0.9582]) c: tensor([[-1.1920], [ 1.9050], [-0.9373], [-0.8465]]) b_add_c: tensor([[-11.0006, -12.2689, -12.7896, -12.8786], [ 19.9698, 18.7015, 18.1808, 18.0918], [ -8.4535, -9.7218, -10.2425, -10.3315], [ -7.5448, -8.8131, -9.3339, -9.4228]]) d: tensor([[-11.0006, -12.2689, -12.7896, -12.8786], [ 19.9698, 18.7015, 18.1808, 18.0918], [ -8.4535, -9.7218, -10.2425, -10.3315], [ -7.5448, -8.8131, -9.3339, -9.4228]])
【结果说明】: torch.add(b, c, alpha=10)
等价于 b + 10 * c
。
2.2 减法运算
torch.sub()
torch.sub(input, other, *, alpha=1, out=None) → Tensor
【功能】:逐元素相减,计算公式为: o u t i = i n p u t i − a l p h a × o t h e r i out_{i} = input_{i} – alpha \times other_{i} o u t i = i n p u t i − a l p h a × o t h e r i 。
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ other(Tensor or Number): 与input相减的张量或数值。
∙ \bullet ∙ alpha(Number): 乘项因子。
【代码】:
import torch a = torch.tensor((1, 2)) print("a: {}".format(a)) b = torch.tensor((0, 1)) print("b: {}".format(b)) a_sub_b = torch.sub(a, b, alpha=2) print("a_sub_b: {} ".format(a_sub_b)) c = a - 2 * b print("c: {}".format(c))
【结果】:
a: tensor([1, 2]) b: tensor([0, 1]) a_sub_b: tensor([1, 0]) c: tensor([1, 0])
【结果说明】: torch.sub(a, b, alpha=2)
等价于 a - 2 * b
。
2.3 哈达玛积运算(element wise,对应元素相乘)
torch.mul()
torch.mul(input, other, *, out=None) → Tensor
【功能】:逐元素相乘,计算公式为: o u t i = i n p u t i × o t h e r i out_{i} = input_{i} \times other_{i} o u t i = i n p u t i × o t h e r i 。
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ other(Tensor or Number): 与input相乘的张量或数值。
【代码】:
import torch a = torch.randn(3) print("a: {}".format(a)) a_mul1 = torch.mul(a, 100) a_mul2 = a * 100 print("a_mul1: {}".format(a_mul1)) print("a_mul2: {} ".format(a_mul2)) b = torch.randn(4, 1) c = torch.randn(1, 4) print("b: {}".format(b)) print("c: {}".format(c)) b_mul_c_1 = torch.mul(b, c) b_mul_c_2 = b * c print("b_mul_c_1: {}".format(b_mul_c_1)) print("b_mul_c_2: {}".format(b_mul_c_2))
【结果】:
a: tensor([-0.6014, -1.0122, -0.3023]) a_mul1: tensor([ -60.1393, -101.2210, -30.2269]) a_mul2: tensor([ -60.1393, -101.2210, -30.2269]) b: tensor([[-1.2277], [ 0.9198], [-0.3485], [-0.8692]]) b: tensor([[-0.9582, -1.1920, 1.9050, -0.9373]]) b_mul_c_1: tensor([[ 1.1763, 1.4635, -2.3387, 1.1508], [-0.8814, -1.0965, 1.7523, -0.8622], [ 0.3339, 0.4154, -0.6638, 0.3266], [ 0.8328, 1.0361, -1.6558, 0.8147]]) b_mul_c_2: tensor([[ 1.1763, 1.4635, -2.3387, 1.1508], [-0.8814, -1.0965, 1.7523, -0.8622], [ 0.3339, 0.4154, -0.6638, 0.3266], [ 0.8328, 1.0361, -1.6558, 0.8147]])
【结果说明】: torch.mul(b, c)
等价于 b * c
。
2.4 除法运算
torch.div()
torch.div(input, other, *, rounding_mode=None, out=None) → Tensor
【功能】:逐元素相除,计算公式为: o u t i = i n p u t i o t h e r i out_{i} = \frac{input_{i}}{other_{i}} o u t i = o t h e r i i n p u t i 。
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ other(Tensor or Number): 与input相除的张量或数值。
∙ \bullet ∙ rounding_mode(str, optional): “None”– 不执行舍入,等价于Python中的 /
运算,或者 np.true_divide
;“trunc”– 将除法结果四舍五入到零,等价于C风格的整除运算;“floor”– 向下舍入除法的结果,等价于Python中的 //
运算,或者 np.floor_divide
;
【代码】:
import torch x = torch.tensor([0.3810, 1.2774, -0.2972, -0.3719, 0.4637]) print("x: {}".format(x)) x_div = torch.div(x, 0.5) print("x_div: {}".format(x_div)) a = torch.tensor([[-0.3711, -1.9353, -0.4605, -0.2917], [0.1815, -1.0111, 0.9805, -1.5923], [0.1062, 1.4581, 0.7759, -1.2344], [-0.1830, -0.0313, 1.1908, -1.4757]]) b = torch.tensor([0.8032, 0.2930, -0.8113, -0.2308]) a_div_b1 = torch.div(a, b) print("a_div_b1: {}".format(a_div_b1)) a_div_b2 = a / b print("a_div_b2: {}".format(a_div_b2))
【结果】:
x: tensor([ 0.3810, 1.2774, -0.2972, -0.3719, 0.4637]) x_div: tensor([ 0.7620, 2.5548, -0.5944, -0.7438, 0.9274]) a_div_b1: tensor([[-0.4620, -6.6051, 0.5676, 1.2639], [ 0.2260, -3.4509, -1.2086, 6.8990], [ 0.1322, 4.9764, -0.9564, 5.3484], [-0.2278, -0.1068, -1.4678, 6.3938]]) a_div_b2: tensor([[-0.4620, -6.6051, 0.5676, 1.2639], [ 0.2260, -3.4509, -1.2086, 6.8990], [ 0.1322, 4.9764, -0.9564, 5.3484], [-0.2278, -0.1068, -1.4678, 6.3938]])
【结果说明】: torch.div(a, b)
等价于 a / b
。
2.5 特殊运算 torch.addcdiv
torch.addcdiv()
torch.addcdiv(input, tensor1, tensor2, *, value=1, out=None) → Tensor
【功能】:计算公式为: o u t i = i n p u t i + v a l u e × t e n s o r 1 i t e n s o r 2 i out_{i} = input_{i} + value\times \frac{tensor1_{i}}{tensor2_{i}} o u t i = i n p u t i + v a l u e × t e n s o r 2 i t e n s o r 1 i
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ tensor1(Tensor): 分子张量。
∙ \bullet ∙ tensor2(Tensor): 分母张量。
∙ \bullet ∙ value(Number, optional): 乘积因子。
【代码】:
import torch t = torch.randn(1, 3) print("t: {}".format(t)) t1 = torch.randn(3, 1) print("t1: {}".format(t1)) t2 = torch.randn(1, 3) print("t2: {}".format(t2)) result = torch.addcdiv(t, t1, t2, value=0.5) print("result: {}".format(result))
【结果】:
t: tensor([[-0.6014, -1.0122, -0.3023]]) t1: tensor([[-1.2277], [ 0.9198], [-0.3485]]) t2: tensor([[-0.8692, -0.9582, -1.1920]]) result: tensor([[ 0.1048, -0.3716, 0.2127], [-1.1305, -1.4922, -0.6881], [-0.4009, -0.8304, -0.1561]])
2.6 特殊运算 torch.addcmul
torch.addcmul()
torch.addcmul(input, tensor1, tensor2, *, value=1, out=None) → Tensor
【功能】:计算公式为: o u t i = i n p u t i + v a l u e × t e n s o r 1 i × t e n s o r 2 i out_{i} = input_{i} + value\times tensor1_{i}\times tensor2_{i} o u t i = i n p u t i + v a l u e × t e n s o r 1 i × t e n s o r 2 i
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ tensor1(Tensor): 张量,乘子1。
∙ \bullet ∙ tensor2(Tensor): 张量,乘子2。
∙ \bullet ∙ value(Number, optional): 乘积因子。
【代码】:
import torch t = torch.randn(1, 3) print("t: {}".format(t)) t1 = torch.randn(3, 1) print("t1: {}".format(t1)) t2 = torch.randn(1, 3) print("t2: {}".format(t2)) result = torch.addcmul(t, t1, t2, value=0.1) print("result: {}".format(result))
【结果】:
t: tensor([[-0.6014, -1.0122, -0.3023]]) t1: tensor([[-1.2277], [ 0.9198], [-0.3485]]) t2: tensor([[-0.8692, -0.9582, -1.1920]]) result: tensor([[-0.4947, -0.8946, -0.1559], [-0.6813, -1.1003, -0.4119], [-0.5711, -0.9788, -0.2607]])
2.7 幂函数
torch.pow()
torch.pow(input, exponent, *, out=None) → Tensor
【功能】:当 exponent
为标量时,计算公式为: o u t i = i n p u t i e x p o n e n t out_{i} = input_{i}^{exponent} o u t i = i n p u t i e x p o n e n t ;当 exponent
为张量时,计算公式为: o u t i = i n p u t i e x p o n e n t i out_{i} = input_{i}^{exponent_{i} } o u t i = i n p u t i e x p o n e n t i 。
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ exponent(float or tensor): 指数值。
【代码】:
import torch t = torch.tensor([1, 2, 3, 4]) print("t: {}".format(t)) t1 = torch.pow(t, 2) print("t1: {}".format(t1)) exp = torch.arange(1, 5) print("exp: {}".format(exp)) t2 = torch.pow(t, exp) print("t2: {}".format(t2))
【结果】:
t: tensor([1, 2, 3, 4]) t1: tensor([ 1, 4, 9, 16]) exp: tensor([1, 2, 3, 4]) t2: tensor([ 1, 4, 27, 256])
torch.pow()
torch.pow(self, exponent, *, out=None) → Tensor
【功能】:计算公式为: o u t i = s e l f e x p o n e n t i out_{i} = self^{exponent_{i} } o u t i = s e l f e x p o n e n t i 。
∙ \bullet ∙ self(float): 幂运算的标量基值。
∙ \bullet ∙ exponent(tensor): 指数张量。
【代码】:
import torch exp = torch.arange(1, 5) print("exp: {}".format(exp)) base = 2 result = torch.pow(base, exp) print("result: {}".format(result))
【结果】:
exp: tensor([1, 2, 3, 4]) result: tensor([ 2, 4, 8, 16])
2.7 指数函数
torch.exp()
torch.exp(input, *, out=None) → Tensor
【功能】:计算公式为: o u t i = e i n p u t i out_{i} = e^{input_{i} } o u t i = e i n p u t i 。
∙ \bullet ∙ input(Tensor): 输入的张量。
【代码】:
import torch import math t = torch.tensor([0, math.log(2.)]) print("t: {}".format(t)) result = torch.exp(t) print("result: {}".format(result))
【结果】:
t: tensor([0.0000, 0.6931]) result: tensor([1., 2.])
2.8 对数函数
torch.log()
torch.log(input, *, out=None) → Tensor
【功能】:计算公式为: o u t i = l o g e ( i n p u t i ) out_{i} = log_{e}^{(input_{i}) } o u t i = l o g e ( i n p u t i ) 。
∙ \bullet ∙ input(Tensor): 输入的张量。
【代码】:
import torch import math t = torch.tensor([math.e, math.exp(2), math.exp(3)]) print("t: {}".format(t)) result = torch.log(t) print("result: {}".format(result))
【结果】:
t: tensor([ 2.7183, 7.3891, 20.0855]) result: tensor([1., 2., 3.])
torch.log2()
torch.log2(input, *, out=None) → Tensor
【功能】:计算公式为: o u t i = l o g 2 ( i n p u t i ) out_{i} = log_{2}^{(input_{i}) } o u t i = l o g 2 ( i n p u t i ) 。
∙ \bullet ∙ input(Tensor): 输入的张量。
【代码】:
import torch t = torch.tensor([2., 4., 8.]) print("t: {}".format(t)) result = torch.log2(t) print("result: {}".format(result))
【结果】:
t: tensor([2., 4., 8.]) result: tensor([1., 2., 3.])
torch.log10()
torch.log10(input, *, out=None) → Tensor
【功能】:计算公式为: o u t i = l o g 10 ( i n p u t i ) out_{i} = log_{10}^{(input_{i}) } o u t i = l o g 1 0 ( i n p u t i ) 。
∙ \bullet ∙ input(Tensor): 输入的张量。
【代码】:
import torch t = torch.tensor([10., 100., 1000.]) print("t: {}".format(t)) result = torch.log10(t) print("result: {}".format(result))
【结果】:
t: tensor([ 10., 100., 1000.]) result: tensor([1., 2., 3.])
2.9 三角函数
torch.sin()
torch.sin(input, *, out=None) → Tensor
【功能】:计算公式为: o u t i = s i n ( i n p u t i ) out_{i} = sin\left ( input_{i} \right ) o u t i = s i n ( i n p u t i ) 。
∙ \bullet ∙ input(Tensor): 输入的张量。
【代码】:
import torch import math t = torch.tensor([0., 1 / 6 * math.pi, 1 / 3 * math.pi, 1 / 2 * math.pi]) print("t: {}".format(t)) result = torch.sin(t) print("result: {}".format(result))
【结果】:
t: tensor([0.0000, 0.5236, 1.0472, 1.5708]) result: tensor([0.0000, 0.5000, 0.8660, 1.0000])
同理可以得到其他的三角函数如下所示:
torch.cos(input, *, out=None) → Tensor
torch.tan(input, *, out=None) → Tensor
torch.asin(input, *, out=None) → Tensor # 反正弦函数
torch.acos(input, *, out=None) → Tensor # 反余弦函数
torch.atan(input, *, out=None) → Tensor # 反正切函数
torch.atan2(input, other, *, out=None) → Tensor # input/other对应元素的反正切函数
torch.sinh(input, *, out=None) → Tensor # 双曲正弦
torch.cosh(input, *, out=None) → Tensor # 双曲余弦
torch.tanh(input, *, out=None) → Tensor # 双曲正切
2.10 矩阵乘法
torch.mm()
torch.mm(input, mat2, *, out=None) → Tensor
【功能】:如果 input 为 ( n × m ) \left ( n\times m \right ) ( n × m ) 张量,mat2 为 ( m × p ) \left ( m\times p \right ) ( m × p ) 张量,则输出一个 ( n × p ) \left ( n\times p \right ) ( n × p ) 张量。
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ mat2(Tensor): 与 input 执行相乘运算的张量。
【代码】:
import torch mat1 = torch.tensor([[1, 2, 3], [0, 1, 4]]) print("mat1: {}".format(mat1)) mat2 = torch.tensor([[3, 2, 1], [2, 1, 4], [0, 0, 1]]) print("mat2: {}".format(mat2)) result = torch.mm(mat1, mat2) print("result: {}".format(result))
【结果】:
mat1: tensor([[1, 2, 3], [0, 1, 4]]) mat2: tensor([[3, 2, 1], [2, 1, 4], [0, 0, 1]]) result: tensor([[ 7, 4, 12], [ 2, 1, 8]])
【注意】: t o r c h . m m {\color{Red} torch.mm} t o r c h . m m 函数不支持 Python 的广播机制,所以必须保证维度能够相乘。
torch.bmm()
torch.bmm(input, mat2, *, out=None) → Tensor
【功能】:实现矩阵乘法的批量运算。input 和 mat2 必须为 3D 张量,并且必须包含相同数量的矩阵。如果 input 为 ( b × n × m ) \left ( b \times n \times m \right ) ( b × n × m ) 张量,mat2 为 ( b × m × p ) \left ( b \times m\times p \right ) ( b × m × p ) 张量,则输出一个 ( b × n × p ) \left ( b \times n\times p \right ) ( b × n × p ) 张量。
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ mat2(Tensor): 与 input 执行相乘运算的张量。
【代码】:
import torch tensor1 = torch.rand(2, 3, 4) tensor2 = torch.rand(2, 4, 3) print("tensor1: {} tensor1 size: {}".format(tensor1, tensor1.shape)) print("tensor2: {} tensor2 size: {}".format(tensor2, tensor2.shape)) result = torch.bmm(tensor1, tensor2) print("result: {} result size: {}".format(result, result.shape))
【结果】:
tensor1: tensor([[[0.4581, 0.4829, 0.3125, 0.6150], [0.2139, 0.4118, 0.6938, 0.9693], [0.6178, 0.3304, 0.5479, 0.4440]], [[0.7041, 0.5573, 0.6959, 0.9849], [0.2924, 0.4823, 0.6150, 0.4967], [0.4521, 0.0575, 0.0687, 0.0501]]]) tensor1 size: torch.Size([2, 3, 4]) tensor2: tensor([[[0.0108, 0.0343, 0.1212], [0.0490, 0.0310, 0.7192], [0.8067, 0.8379, 0.7694], [0.6694, 0.7203, 0.2235]], [[0.9502, 0.4655, 0.9314], [0.6533, 0.8914, 0.8988], [0.3955, 0.3546, 0.5752], [0.4787, 0.5782, 0.7536]]]) tensor2 size: torch.Size([2, 4, 3]) result: tensor([[[0.6924, 0.7355, 0.7807], [1.2310, 1.2996, 1.0725], [0.7621, 0.8103, 0.8333]], [[1.7798, 1.6407, 2.2993], [1.0740, 1.0713, 1.4340], [0.5183, 0.3150, 0.5500]]]) result size: torch.Size([2, 3, 3])
torch.matmul()
torch.matmul(input, other, *, out=None) → Tensor
【功能】:矩阵的乘法运算,输入可以是高维的。
∙ \bullet ∙ input(Tensor): 输入的张量。
∙ \bullet ∙ mat2(Tensor): 与 input 执行相乘运算的张量。
(1)若两个都是1D的张量,则返回两个张量的点积,结果为标量。
【代码】:
import torch tensor1 = torch.tensor([2]) tensor2 = torch.tensor([3]) result = torch.matmul(tensor1, tensor2) print("result: {}".format(result))
【结果】:
result: 6
【注意】:对于两个输入都是1D的张量, t o r c h . m a t m u l {\color{Red} torch.matmul} t o r c h . m a t m u l 函数不包含 o u t {\color{Red} out} o u t 参数。
(2)若两个都是2D的张量,则按照矩阵相乘规则返回2D张量,此时和 t e n s o r . m m {\color{Red} tensor.mm} t e n s o r . m m 函数用法相同。
【代码】:
import torch tensor1 = torch.tensor([[1, 2, 3], [0, 1, 4]]) tensor2 = torch.tensor([[3, 2, 1], [2, 1, 4], [0, 0, 1]]) result = torch.matmul(tensor1, tensor2) print("result: {}".format(result))
【结果】:
result: tensor([[ 7, 4, 12], [ 2, 1, 8]])
(3)若input维度1D,other维度2D,则先将1D的维度扩充到2D(1D的维数前面+1),然后得到结果后再将此维度去掉,得到的输出与input的维度相同。即使作广播处理,input的维度也要和other维度做对应关系。
【代码】:
import torch tensor1 = torch.rand(4) tensor2 = torch.rand(4, 3) print("tensor1: {} tensor1 size: {}".format(tensor1, tensor1.shape)) print("tensor2: {} tensor2 size: {}".format(tensor2, tensor2.shape)) result = torch.matmul(tensor1, tensor2) print("result: {} result size: {}".format(result, result.shape))
【结果】:
tensor1: tensor([0.4581, 0.4829, 0.3125, 0.6150]) tensor1 size: torch.Size([4]) tensor2: tensor([[0.2139, 0.4118, 0.6938], [0.9693, 0.6178, 0.3304], [0.5479, 0.4440, 0.7041], [0.5573, 0.6959, 0.9849]]) tensor2 size: torch.Size([4, 3]) result: tensor([1.0800, 1.0537, 1.3031]) result size: torch.Size([3])
(4)若input维度2D,other维度1D,则返回两者的点积结果。
【代码】:
import torch tensor1 = torch.rand(3, 4) tensor2 = torch.rand(4) print("tensor1: {} tensor1 size: {}".format(tensor1, tensor1.shape)) print("tensor2: {} tensor2 size: {}".format(tensor2, tensor2.shape)) result = torch.matmul(tensor1, tensor2) print("result: {} result size: {}".format(result, result.shape))
【结果】:
tensor1: tensor([[0.4581, 0.4829, 0.3125, 0.6150], [0.2139, 0.4118, 0.6938, 0.9693], [0.6178, 0.3304, 0.5479, 0.4440]]) tensor1 size: torch.Size([3, 4]) tensor2: tensor([0.7041, 0.5573, 0.6959, 0.9849]) tensor2 size: torch.Size([4]) result: tensor([1.4148, 1.8176, 1.4376]) result size: torch.Size([3])
(5)如果一个维度至少是1D,另外一个大于2D,则返回的是一个批矩阵乘法( a batched matrix multiply)
∙ \bullet ∙ 若input是1D,other是大于2D的,则类似于规则(3)。
【代码】:
import torch tensor1 = torch.rand(3) tensor2 = torch.rand(2, 3, 4) print("tensor1: {} tensor1 size: {}".format(tensor1, tensor1.shape)) print("tensor2: {} tensor2 size: {}".format(tensor2, tensor2.shape)) result = torch.matmul(tensor1, tensor2) print("result: {} result size: {}".format(result, result.shape))
【结果】:
tensor1: tensor([0.4581, 0.4829, 0.3125]) tensor1 size: torch.Size([3]) tensor2: tensor([[[0.6150, 0.2139, 0.4118, 0.6938], [0.9693, 0.6178, 0.3304, 0.5479], [0.4440, 0.7041, 0.5573, 0.6959]], [[0.9849, 0.2924, 0.4823, 0.6150], [0.4967, 0.4521, 0.0575, 0.0687], [0.0501, 0.0108, 0.0343, 0.1212]]]) tensor2 size: torch.Size([2, 3, 4]) result: tensor([[0.8885, 0.6163, 0.5223, 0.7998], [0.7067, 0.3556, 0.2594, 0.3528]]) result size: torch.Size([2, 4])
∙ \bullet ∙ 若other是1D,input是大于2D的,则类似于规则(4)。
【代码】:
import torch tensor1 = torch.rand(2, 3, 4) tensor2 = torch.rand(4) print("tensor1: {} tensor1 size: {}".format(tensor1, tensor1.shape)) print("tensor2: {} tensor2 size: {}".format(tensor2, tensor2.shape)) result = torch.matmul(tensor1, tensor2) print("result: {} result size: {}".format(result, result.shape))
【结果】:
tensor1: tensor([[[0.4581, 0.4829, 0.3125, 0.6150], [0.2139, 0.4118, 0.6938, 0.9693], [0.6178, 0.3304, 0.5479, 0.4440]], [[0.7041, 0.5573, 0.6959, 0.9849], [0.2924, 0.4823, 0.6150, 0.4967], [0.4521, 0.0575, 0.0687, 0.0501]]]) tensor1 size: torch.Size([2, 3, 4]) tensor2: tensor([0.0108, 0.0343, 0.1212, 0.0490]) tensor2 size: torch.Size([4]) result: tensor([[0.0895, 0.1481, 0.1062], [0.1593, 0.1186, 0.0176]]) result size: torch.Size([2, 3])
∙ \bullet ∙ 若input和other都是3D的,则与 t o r c h . b m m {\color{Red} torch.bmm} t o r c h . b m m 函数功能一样。
∙ \bullet ∙ 如果input中某一维度满足可以广播,那幺也是可以进行相乘操作的。如果 input 为 ( j × 1 × n × m ) \left ( j \times 1 \times n \times m \right ) ( j × 1 × n × m ) 张量,other 为 ( k × m × p ) \left ( k \times m\times p \right ) ( k × m × p ) 张量,则输出一个 ( j × k × n × p ) \left ( j \times k \times n\times p \right ) ( j × k × n × p ) 张量。
【代码】:
import torch tensor1 = torch.rand(2, 1, 2, 4) tensor2 = torch.rand(2, 4, 3) print("tensor1: {} tensor1 size: {}".format(tensor1, tensor1.shape)) print("tensor2: {} tensor2 size: {}".format(tensor2, tensor2.shape)) result = torch.matmul(tensor1, tensor2) print("result: {} result size: {}".format(result, result.shape))
【结果】:
tensor1: tensor([[[[0.4581, 0.4829, 0.3125, 0.6150], [0.2139, 0.4118, 0.6938, 0.9693]]], [[[0.6178, 0.3304, 0.5479, 0.4440], [0.7041, 0.5573, 0.6959, 0.9849]]]]) tensor1 size: torch.Size([2, 1, 2, 4]) tensor2: tensor([[[0.2924, 0.4823, 0.6150], [0.4967, 0.4521, 0.0575], [0.0687, 0.0501, 0.0108], [0.0343, 0.1212, 0.0490]], [[0.0310, 0.7192, 0.8067], [0.8379, 0.7694, 0.6694], [0.7203, 0.2235, 0.9502], [0.4655, 0.9314, 0.6533]]]) tensor2 size: torch.Size([2, 4, 3]) result: tensor([[[[0.4164, 0.5295, 0.3430], [0.3481, 0.4416, 0.2103]], [[0.9301, 1.3436, 1.3915], [1.3026, 1.5286, 1.7407]]], [[[0.3976, 0.5286, 0.4266], [0.5643, 0.7458, 0.5208]], [[0.8973, 1.2344, 1.5302], [1.4484, 2.0080, 2.2456]]]]) result size: torch.Size([2, 2, 2, 3])
3 线性回归
最后,利用上面介绍的张量的基本操作来实现一个简单的线性回归分析,因为线性回归是深度学习的基础,所以有必要先来学习一下如何利用pytorch来实现一个简单的线性回归。
线性回归是分析一个变量 ( y y y ) 与另外一 (多) 个变量 ( x x x ) 之间的关系的方法。一般可以写成 y = w x + b y=wx+b w x + b 。线性回归的目的就是求解参数 w , b w, b w , b 。
线性回归的求解可以分为 3 步:
(1)确定模型: y = w x + b y=wx+b w x + b
(2)选择损失函数,一般使用均方误差 MSE: 1 m ∑ i = 1 m ( y i − y ^ i ) 2 \frac{1}{m}\sum_{i=1}^{m} \left ( y_{i}-\hat{y} _{i} \right ) ^{2} ∑ i = 1 m ( y i − y ^ i ) 2 。其中 y ^ i \hat{y} _{i} y ^ i 是预测值, y i y_{i} y i 是真实值。
(3)使用梯度下降法求解梯度 (其中 α \alpha α 是学习率),并更新参数: w = w − α ∗ w . g r a d w = w – \alpha * w.grad w . g r a d b = b − α ∗ b . g r a d b = b -\alpha * b.grad b . g r a d
【代码】:
import torch import numpy as np import matplotlib.pyplot as plt torch.manual_seed(10) r = 0.05 # 学习率 x = torch.rand(20, 1) * 10 # 创建训练数据 y = 2 * x + (5 + torch.randn(20, 1)) # torch.randn(20, 1) 用于添加噪声 # 构建线性回归参数 w = torch.randn(1, requires_grad=True) b = torch.zeros(1, requires_grad=True) # 迭代训练 1000 次 for iteration in range(2000): # 前向传播,计算预测值 wx = torch.mul(w, x) y_hat = torch.add(wx, b) # 计算 MSE loss loss = (1 / 20 * (y - y_hat) ** 2).mean() # 反向传播 loss.backward() # 更新参数 b.data.sub_(r * b.grad) w.data.sub_(r * w.grad) # 每次更新参数之后,都要清零张量的梯度 w.grad.zero_() b.grad.zero_() # 绘图,每隔 20 次重新绘制直线 if iteration % 20 == 0: plt.scatter(x.data.numpy(), y.data.numpy()) plt.plot(x.data.numpy(), y_hat.data.numpy(), 'r-', lw=3) plt.xlim(1.5, 10) plt.ylim(8, 28) plt.title("Iteration: {} w: {} b: {} loss: {}" .format(iteration, w.data.numpy(), b.data.numpy(), loss.data.numpy())) plt.pause(0.5)
【结果】:
迭代0次的结果如下:
迭代2000次的结果如下:
迭代20000次的结果如下:
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