\u672c\u8282\u5c06\u4ecb\u7ecd\u7ebf\u6027\u4ee3\u6570\u4e2d\u7684\u57fa\u672c\u6570\u5b66\u5bf9\u8c61\u3001\u7b97\u672f\u548c\u8fd0\u7b97，\u5e76\u7528\u6570\u5b66\u7b26\u53f7\u548c\u76f8\u5e94\u7684\u4ee3\u7801\u5b9e\u73b0\u6765\u8868\u793a\u5b83\u4eec\u3002<\/p>\n
.<\/p>\n
\u5b9a\u4e49：\u4ec5\u5305\u542b\u4e00\u4e2a\u6570\u503c\u7684\u91cf\u79f0\u4e3a\u6807\u91cf（\u96f6\u7ef4\u5f20\u91cf），\u4f8b\u5982\u6e29\u5ea6\u503c\u3002<\/p>\n
\u8868\u793a：\u6807\u91cf\u53d8\u91cf\u7528\u666e\u901a\u5c0f\u5199\u5b57\u6bcd\u8868\u793a（\u5982 x,y,z），\u5c5e\u4e8e\u5b9e\u6570\u7a7a\u95f4 R\u3002<\/p>\n
\u64cd\u4f5c：\u6807\u91cf\u652f\u6301\u52a0\u6cd5\u3001\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u6307\u6570\u8fd0\u7b97\u3002<\/p>\n
import torch
\nx = torch.tensor(3.0, dtype=torch.float32)
\ny = torch.tensor(2.0, dtype=torch.float32)
\nx + y, x * y, x \/ y, x**y<\/p>\n
# \u8f93\u51fa：
\n(tensor(5.), tensor(6.), tensor(1.5000), tensor(9.))<\/p>\n
.<\/p>\n
\u5b9a\u4e49：\u5411\u91cf\u662f\u6807\u91cf\u503c\u7ec4\u6210\u7684\u5217\u8868（\u4e00\u7ef4\u5f20\u91cf）\u3002<\/p>\n
\u8868\u793a：\u5411\u91cf\u7528\u7c97\u4f53\u3001\u5c0f\u5199\u5b57\u6bcd\u8868\u793a，\u901a\u5e38\u4e3a\u5217\u5411\u91cf（\u5982 <\/p>\n x <\/p>\n = <\/p>\n [ <\/p>\n x <\/p>\n 1 <\/p>\n , <\/p>\n x <\/p>\n 2 <\/p>\n , <\/p>\n \u2026 <\/p>\n , <\/p>\n x <\/p>\n n <\/p>\n ] <\/p>\n ) <\/p>\n \\\\mathbf{x} = [x_1, x_2, \\\\dots, x_n] ) <\/p>\n <\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>])<\/span><\/span><\/span><\/span><\/span>，\u7ef4\u5ea6\u4e3a\u5143\u7d20\u6570\u91cf <\/p>\n ( <\/p>\n x <\/p>\n \u2208 <\/p>\n R <\/p>\n n <\/p>\n ) <\/p>\n ( \\\\mathbf{x} \\\\in \\\\mathbb{R}^n ) <\/p>\n <\/span><\/span>(<\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>R<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n \u64cd\u4f5c：\u901a\u8fc7\u7d22\u5f15\u8bbf\u95ee\u5411\u91cf\u7684\u5143\u7d20\u3002<\/p>\n x = torch.arange(4, dtype=torch.float32) # \u521b\u5efa\u5411\u91cf：\u751f\u6210 [0., 1., 2., 3.]<\/p>\n print(x) # \u8f93\u51fa： \u957f\u5ea6\u3001\u7ef4\u5ea6\u548c\u5f62\u72b6<\/p>\n \u957f\u5ea6：\u5411\u91cf\u7684\u5143\u7d20\u6570\u91cf\u3002<\/p>\n<\/li>\n \u7ef4\u5ea6：\u540c\u5411\u91cf\u7684\u957f\u5ea6，\u8868\u793a\u5411\u91cf\u7684\u5143\u7d20\u6570\u91cf\u3002<\/p>\n<\/li>\n \u5f62\u72b6：\u5f20\u91cf\u7684\u5f62\u72b6\u8868\u793a\u6bcf\u4e2a\u8f74\u7684\u957f\u5ea6\u3002<\/p>\n<\/li>\n<\/ul>\n print(len(x)) # len()\u51fd\u6570\u6765\u8bbf\u95ee\u5f20\u91cf\u7684\u957f\u5ea6 # \u8f93\u51fa： \u6ce8：\u7ef4\u5ea6\u5728\u4e0d\u540c\u4e0a\u4e0b\u6587\u4e2d\u6709\u4e0d\u540c\u542b\u4e49\u3002\u5411\u91cf\u7684\u7ef4\u5ea6\u6307\u5176\u957f\u5ea6，\u800c\u5f20\u91cf\u7684\u7ef4\u5ea6\u6307\u5176\u8f74\u7684\u6570\u91cf\u3002<\/p>\n .<\/p>\n \u5b9a\u4e49：\u77e9\u9635\u662f\u4e8c\u7ef4\u5f20\u91cf，\u5f62\u72b6\u7531 (\u884c\u6570, \u5217\u6570) \u5b9a\u4e49\u3002<\/p>\n \u8868\u793a：\u6570\u5b66\u4e2d\u8868\u793a\u4e3a\u7c97\u4f53\u5927\u5199\u5b57\u6bcd（\u5982 <\/p>\n A <\/p>\n \u2208 <\/p>\n R <\/p>\n m <\/p>\n \u00d7 <\/p>\n n <\/p>\n \\\\mathbf{A} \\\\in \\\\mathbb{R}^{m\u00d7n} <\/p>\n <\/span><\/span>A<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>R<\/span><\/span>m<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>），\u5f62\u72b6\u4e3a (m,n)；<\/p>\n \u8bbf\u95ee：\u901a\u8fc7\u7d22\u5f15 A[i][j] \u6216 A[i, j] \u83b7\u53d6\u7b2c i \u884c\u7b2c j \u5217\u7684\u5143\u7d20；<\/p>\n A = torch.arange(20).reshape(5, 4) # \u8f93\u51fa： \u77e9\u9635\u7684\u8f6c\u7f6e（transpose）：\u662f\u4ea4\u6362\u5176\u884c\u548c\u5217，\u8868\u793a\u4e3a <\/p>\n A <\/p>\n T <\/p>\n \\\\mathbf{A}^T <\/p>\n <\/span><\/span>A<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n print(A.T)<\/p>\n # \u8f93\u51fa： \u65b9\u9635（square matrix）：\u5f53\u77e9\u9635\u5177\u6709\u76f8\u540c\u6570\u91cf\u7684\u884c\u548c\u5217\u65f6，\u88ab\u79f0\u4e3a\u65b9\u9635；<\/p>\n \u5bf9\u79f0\u77e9\u9635（symmetric matrix）：\u5982\u679c\u77e9\u9635\u7b49\u4e8e\u5176\u8f6c\u7f6e，\u5219\u79f0\u4e3a\u5bf9\u79f0\u77e9\u9635\u3002<\/p>\n B = torch.tensor([[1, 2, 3], [2, 0, 4], [3, 4, 5]]) # \u8f93\u51fa\u7ed3\u679c： .<\/p>\n \u5f20\u91cf\u662f\u591a\u7ef4\u6570\u7ec4\u7684\u901a\u7528\u8868\u793a：<\/p>\n \u6807\u91cf：0\u9636\u5f20\u91cf（\u5355\u4e2a\u6570\u503c）<\/p>\n<\/li>\n \u5411\u91cf：1\u9636\u5f20\u91cf（\u4e00\u7ef4\u6570\u7ec4）<\/p>\n<\/li>\n \u77e9\u9635：2\u9636\u5f20\u91cf（\u4e8c\u7ef4\u6570\u7ec4）<\/p>\n<\/li>\n \u9ad8\u9636\u5f20\u91cf：\u59823\u9636\u5f20\u91cf\u53ef\u8868\u793a\u56fe\u50cf（\u9ad8\u5ea6\u00d7\u5bbd\u5ea6\u00d7\u989c\u8272\u901a\u9053）<\/p>\n<\/li>\n<\/ul>\n \u793a\u4f8b：\u521b\u5efa3\u9636\u5f20\u91cf（2\u00d73\u00d74）<\/p>\n import torch .<\/p>\n \u5f62\u72b6\u4e0d\u53d8\u6027<\/p>\n \u4e00\u5143\u8fd0\u7b97（\u5982\u53d6\u7edd\u5bf9\u503c）\u4e0d\u6539\u53d8\u5f20\u91cf\u5f62\u72b6；<\/p>\n<\/li>\n \u4e8c\u5143\u8fd0\u7b97（\u5982\u52a0\u6cd5）\u8981\u6c42\u8f93\u5165\u5f62\u72b6\u76f8\u540c，\u8f93\u51fa\u5f62\u72b6\u4e0d\u53d8；<\/p>\n<\/li>\n<\/ul>\n \u4f8b\u5982，\u5c06\u4e24\u4e2a\u76f8\u540c\u5f62\u72b6\u7684\u77e9\u9635\u76f8\u52a0，\u4f1a\u5728\u8fd9\u4e24\u4e2a\u77e9\u9635\u4e0a\u6267\u884c\u5143\u7d20\u52a0\u6cd5\u3002<\/p>\n A = torch.arange(20, dtype=torch.float32).reshape(5, 4) # \u8f93\u51fa： Hadamard\u79ef（\u5143\u7d20\u4e58\u6cd5）<\/p>\n \u2299 <\/p>\n \\\\odot <\/p>\n <\/span><\/span>\u2299<\/span><\/span><\/span><\/span><\/span>，\u6570\u5b66\u5b9a\u4e49： <\/p>\n ( <\/p>\n A <\/p>\n \u2299 <\/p>\n B <\/p>\n ) <\/p>\n i <\/p>\n j <\/p>\n = <\/p>\n A <\/p>\n i <\/p>\n j <\/p>\n \u00d7 <\/p>\n B <\/p>\n i <\/p>\n j <\/p>\n (A \\\\odot B)_{ij} = A_{ij} \\\\times B_{ij} <\/p>\n <\/span><\/span>(<\/span>A<\/span><\/span>\u2299<\/span><\/span><\/span><\/span>B<\/span>)<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>B<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n A = torch.arange(20, dtype=torch.float32).reshape(5,4) # \u8f93\u51fa： \u5f20\u91cf\u4e0e\u6807\u91cf\u7684\u8fd0\u7b97<\/p>\n a = 2 # \u8f93\u51fa： [[14, 15, 16, 17], .<\/p>\n \u5f20\u91cf\u964d\u7ef4\u64cd\u4f5c\u53ef\u4ee5\u51cf\u5c11\u5f20\u91cf\u7684\u7ef4\u5ea6，\u5e38\u7528\u7684\u964d\u7ef4\u65b9\u6cd5\u5305\u62ec\u6c42\u548c\u548c\u6c42\u5e73\u5747\u503c\u3002<\/p>\n （1）\u6c42\u548c\u64cd\u4f5c<\/p>\n # \u5411\u91cf\u6c42\u548c # \u77e9\u9635\u6c42\u548c # axis=0：\u5bf9\u77e9\u9635\u7684\u884c\u6c42\u548c # axis=1：\u5bf9\u77e9\u9635\u7684\u5217\u6c42\u548c # \u591a\u8f74\u6c42\u548c：\u540c\u65f6\u6cbf\u591a\u4e2a\u8f74\u6c42\u548c，\u7b49\u4ef7\u4e8e\u5bf9\u6240\u6709\u5143\u7d20\u6c42\u548c\u3002 （2）\u6c42\u5e73\u5747\u503c\u64cd\u4f5c<\/p>\n print(A.mean()) # \u8f93\u51fa：tensor(9.5000) print(A.mean(axis=0)) # \u8f93\u51fa：tensor([ 8., 9., 10., 11.]) （3）\u975e\u964d\u7ef4\u6c42\u548c<\/p>\n sum_A = A.sum(axis=1, keepdims=True) # \u5229\u7528\u5e7f\u64ad\u5b9e\u73b0\u9010\u884c\u5f52\u4e00\u5316 A.cumsum(axis=0) .<\/p>\n \u70b9\u79ef（dot product）： \u662f\u76f8\u540c\u4f4d\u7f6e\u7684\u6309\u5143\u7d20\u4e58\u79ef\u7684\u548c\u3002<\/p>\n \u8868\u793a：\u4e3a <\/p>\n x <\/p>\n \u22a4 <\/p>\n y <\/p>\n ( <\/p>\n \u6216 <\/p>\n \u27e8 <\/p>\n x <\/p>\n , <\/p>\n y <\/p>\n \u27e9 <\/p>\n ) <\/p>\n \\\\mathbf{x}^\\\\top \\\\mathbf{y} (\u6216\\\\langle \\\\mathbf{x}, \\\\mathbf{y} \\\\rangle) <\/p>\n <\/span><\/span>x<\/span>
\nprint(x[3]) # \u8bbf\u95ee\u5143\u7d20：\u83b7\u53d6\u7d22\u5f15\u4e3a3\u7684\u5143\u7d20（\u7b2c4\u4e2a\u5143\u7d20）<\/p>\n
\ntensor([0., 1., 2., 3.])
\ntensor(3.)<\/p>\n\n
\nprint(x.shape) # .shape\u5c5e\u6027\u8bbf\u95ee\u5411\u91cf\u7684\u957f\u5ea6<\/p>\n
\n4
\ntorch.Size([4])<\/p>\n3）\u77e9\u9635<\/h4>\n
\nprint(A)<\/p>\n
\ntensor([[ 0, 1, 2, 3],
\n [ 4, 5, 6, 7],
\n [ 8, 9, 10, 11],
\n [12, 13, 14, 15],
\n [16, 17, 18, 19]])<\/p>\n
\ntensor([[ 0, 4, 8, 12, 16],
\n [ 1, 5, 9, 13, 17],
\n [ 2, 6, 10, 14, 18],
\n [ 3, 7, 11, 15, 19]])<\/p>\n
\nprint(B == B.T)<\/p>\n
\ntensor([[True, True, True],
\n [True, True, True],
\n [True, True, True]])<\/p>\n4）\u5f20\u91cf<\/h4>\n
\n
\nX = torch.arange(24).reshape(2, 3, 4)
\n# \u8f93\u51fa\u4e24\u4e2a 3\u00d74 \u77e9\u9635\u7684\u5806\u53e0
\ntensor([[[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]],
\n [[12, 13, 14, 15], [16, 17, 18, 19], [20, 21, 22, 23]]])<\/p>\n5）\u5f20\u91cf\u7684\u57fa\u672c\u6027\u8d28<\/h4>\n
\n
\nB = A.clone() # \u901a\u8fc7\u5206\u914d\u65b0\u5185\u5b58，\u5c06A\u7684\u4e00\u4e2a\u526f\u672c\u5206\u914d\u7ed9B
\nA, A + B<\/p>\n
\n(tensor([[ 0., 1., 2., 3.],
\n [ 4., 5., 6., 7.],
\n [ 8., 9., 10., 11.],
\n [12., 13., 14., 15.],
\n [16., 17., 18., 19.]]),
\n tensor([[ 0., 2., 4., 6.],
\n [ 8., 10., 12., 14.],
\n [16., 18., 20., 22.],
\n [24., 26., 28., 30.],
\n [32., 34., 36., 38.]]))<\/p>\n\n
\nB = A.clone() # \u590d\u5236A\u5230B
\nA * B # \u8f93\u51faHadamard\u79ef\u7ed3\u679c<\/p>\n
\ntensor([[ 0., 1., 4., 9.],
\n [ 16., 25., 36., 49.],
\n [ 64., 81., 100., 121.],
\n [144., 169., 196., 225.],
\n [256., 289., 324., 361.]])<\/p>\n\n
\nX = torch.arange(24).reshape(2, 3, 4)
\na + X, (a * X).shape<\/p>\n
\n(tensor([[[ 2, 3, 4, 5],
\n [ 6, 7, 8, 9],
\n [10, 11, 12, 13]],<\/p>\n
\n [18, 19, 20, 21],
\n [22, 23, 24, 25]]]),
\n torch.Size([2, 3, 4]))<\/p>\n6）\u964d\u7ef4<\/h4>\n
\n
\nx = torch.arange(4, dtype=torch.float32)
\nprint(x.sum()) # \u8f93\u51fa：tensor(6.)<\/p>\n
\nA = torch.arange(20, dtype=torch.float32).reshape(5,4)
\nprint(A.sum()) # \u8f93\u51fa：tensor(190.)<\/p>\n\n
\nA_sum_axis0 = A.sum(axis=0)
\nprint(A_sum_axis0, A_sum_axis0.shape) # \u8f93\u51fa：tensor([40., 45., 50., 55.]) torch.Size([4])<\/p>\n
\nA_sum_axis1 = A.sum(axis=1)
\nprint(A_sum_axis1, A_sum_axis1.shape) # \u8f93\u51fa：tensor([ 6., 22., 38., 54., 70.]) torch.Size([5])<\/p>\n
\nprint(A.sum(axis=[0,1])) # \u8f93\u51fa：tensor(190.)<\/p>\n\n
\nprint(A.sum() \/ A.numel()) # \u8f93\u51fa：tensor(9.5000)<\/p>\n\n
\nprint(A.sum(axis=0) \/ A.shape[0]) # \u8f93\u51fa：tensor([ 8., 9., 10., 11.])<\/p>\n\n
\n# \u8f93\u51fa\u5f62\u72b6\u53d8\u4e3a5\u00d71（\u4fdd\u7559\u4e8c\u7ef4\u7ed3\u6784）
\ntensor([[ 6.],
\n [22.],
\n [38.],
\n [54.],
\n [70.]])<\/p>\n
\nA \/ sum_A # \u6bcf\u884c\u5143\u7d20\u9664\u4ee5\u5bf9\u5e94\u884c\u548c
\ntensor([[0.0000, 0.1667, 0.3333, 0.5000],
\n [0.1818, 0.2273, 0.2727, 0.3182],
\n [0.2105, 0.2368, 0.2632, 0.2895],
\n [0.2222, 0.2407, 0.2593, 0.2778],
\n [0.2286, 0.2429, 0.2571, 0.2714]])<\/p>\n\n
\n# \u7ed3\u679c\u77e9\u9635\u663e\u793a\u9010\u6b65\u884c\u7d2f\u52a0\u8fc7\u7a0b：
\ntensor([[ 0, 1, 2, 3],
\n [ 4, 6, 8, 10], # 0\u884c+1\u884c
\n [12, 15, 18, 21], # \u524d\u4e24\u884c+2\u884c
\n [24, 28, 32, 36], # \u524d\u4e09\u884c+3\u884c
\n [40, 45, 50, 55]]) # \u5168\u90e8\u884c\u7d2f\u52a0<\/p>\n7）\u70b9\u79ef<\/h4>\n