はじめに

Q＆A形式でのログです。

参考URL

[1] PyTorch公式サイト:torch.cdist — PyTorch 2.0 documentation
https://pytorch.org/docs/stable/generated/torch.cdist.html

[2] Twitter Yuta Suzuki:距離行列計算を自前からtorch.cdistに切り替えた話
https://twitter.com/resnant/status/1270998333671133185

[3] PyTorchのtorch.cdist関数は、2つの行列の全対ユークリッド(または任意のp-norm)距離を計算するのに便利なツールです。
https://runebook.dev/ja/docs/pytorch/generated/torch.cdist

[2][3]から外観を得て、[1]に沿って距離行列を作成した。

ソースコードは現在2か所からdownload可能です。

.ipynb形式：「morii_testcode_test_GithubGist_to_multi.ipynb」
・OneDriveから
https://1drv.ms/u/s!AncYIM5QjoV1gq4V6GtV2oVvmdm0Jg?e=2BSbVM
・GoogleDriveから
https://drive.google.com/file/d/1AMqcEJ9q4c6ZZ4UZDnJG0VKDN4McB_O8/view?usp=sharing

.py形式：「morii_testcode_test_GithubGist_to_multi.py」
・OneDriveから
https://1drv.ms/u/s!AncYIM5QjoV1gq4Wlew3mDDALentZw?e=K8tNBK
・GoogleDriveから
https://drive.google.com/file/d/1_T45kts2wSXVFsCUeyqd1RGWWiFaBt9G/view?usp=sharing

Q1. A_1 とB_1 間の距離行列C_1を求めてください.(出力は8×8のテンソルになるはずです)

import torch
A_1=torch.randn(8,3)
B_1=torch.randn(8,3)

print(f'A_1は\n{A_1}')
print(f'B_1は\n{B_1}')

A_1は
tensor([[-1.2416, -2.0468, 0.3050],
[ 0.0556, -0.3844, 0.6254],
[ 0.8345, -1.0627, -1.2592],
[-0.6727, 0.6312, 0.4476],
[-0.3870, -0.9692, -0.9783],
[-0.0361, 1.5460, -0.6113],
[ 0.4282, 0.8991, 0.4301],
[-1.4248, 0.5365, -0.6753]])

B_1は
tensor([[-1.7169, -0.2868, -0.1328],
[-2.0779, -0.2973, 1.1613],
[-0.5776, 0.4738, 0.6828],
[-1.6807, 0.4168, -0.4185],
[-0.8357, 0.9086, 1.1973],
[-0.7952, 0.7565, 1.5547],
[-0.1374, 0.0659, 0.0473],
[ 0.1833, 0.2078, 0.3087]])

C_1=torch.cdist(A_1, B_1, p=2)

print(f'C_1は\n{C_1}')

C_1は
tensor([[1.8749, 2.1198, 2.6339, 2.6049, 3.1137, 3.1015, 2.3978, 2.6672],
[1.9304, 2.2015, 1.0681, 2.1786, 1.6713, 1.6996, 0.7578, 0.6836],
[2.8949, 3.8635, 2.8507, 3.0367, 3.5651, 3.7260, 1.9812, 2.1205],
[1.5067, 1.8292, 0.2985, 1.3462, 0.8158, 1.1208, 0.8754, 0.9651],
[1.7173, 2.8087, 2.2086, 1.9769, 2.9087, 3.0920, 1.4784, 1.8349],
[2.5324, 3.2724, 1.7656, 2.0043, 2.0777, 2.4271, 1.6231, 1.6386],
[2.5149, 2.8717, 1.1209, 2.3239, 1.4786, 1.6678, 1.0773, 0.7433],
[1.0284, 2.1202, 1.6019, 0.3819, 1.9980, 2.3276, 1.5495, 1.9137]])

Q2. 行列A_1,B_1,C_1を使って、Sinkhoon アルゴリズムを実装してください。(出力は1次元のスカラーになります。)

from sinkhorn import SinkhornSolver

epsilon = 0.001
solver = SinkhornSolver(epsilon=epsilon, iterations=10000)
cost, pi = solver.forward(A_1, B_1)

Finished computing transport plan in 1 iterations

C_dist = solver._compute_cost_pytorch_cdist(A_1, B_1)

print(f'C_dist\n{C_dist}')

C_dist
tensor([[[1.8749, 2.1198, 2.6339, 2.6049, 3.1137, 3.1015, 2.3978, 2.6672]],

        [[1.9304, 2.2015, 1.0681, 2.1786, 1.6713, 1.6996, 0.7578, 0.6836]],

        [[2.8949, 3.8635, 2.8507, 3.0367, 3.5651, 3.7260, 1.9812, 2.1205]],

        [[1.5067, 1.8292, 0.2985, 1.3462, 0.8158, 1.1208, 0.8754, 0.9651]],

        [[1.7173, 2.8087, 2.2086, 1.9769, 2.9087, 3.0920, 1.4784, 1.8349]],

        [[2.5324, 3.2724, 1.7656, 2.0043, 2.0777, 2.4271, 1.6231, 1.6386]],

        [[2.5149, 2.8717, 1.1209, 2.3239, 1.4786, 1.6678, 1.0773, 0.7433]],

        [[1.0284, 2.1202, 1.6019, 0.3819, 1.9980, 2.3276, 1.5495, 1.9137]]])

print(f'cost\n{cost}')

cost
1.6988214254379272

Q3. AとBは10個のテンソルが集約されたテンソルです。距離行列Cを求めてください。(出力は10×8×8になります。)

A=torch.randn(10,8,3)
B=torch.randn(10,8,3)
C=torch.cdist(A, B, p=2)

print(C)

tensor([[[3.3452, 3.5884, 2.6145, 3.5548, 2.6811, 1.7614, 4.8377, 3.2827],
         [2.7055, 2.8352, 3.2904, 3.4297, 3.2852, 3.5755, 3.8415, 3.5525],
         [2.8125, 3.0205, 1.4525, 2.8069, 2.1878, 0.3542, 4.1632, 2.1684],
         [2.7713, 3.0012, 2.2180, 3.0812, 2.3985, 1.7458, 4.2431, 2.8260],
         [2.8290, 2.6322, 2.2476, 3.3099, 4.0334, 2.9723, 2.6648, 1.8386],
         [1.8975, 1.9757, 2.3589, 1.1350, 2.3386, 3.7782, 2.4262, 2.0510],
         [0.9523, 0.7452, 2.4652, 1.8895, 2.9092, 3.7592, 1.3614, 2.1590],
         [2.3381, 2.3968, 2.0447, 1.6194, 2.5042, 3.3260, 2.8266, 1.7473]],

        [[1.3115, 1.6532, 3.0833, 1.0828, 2.1806, 2.0015, 1.7661, 0.9011],
         [1.5730, 2.5654, 3.9633, 0.3884, 2.4030, 1.8086, 2.4826, 0.4883],
         [1.9457, 2.7556, 3.9811, 0.8660, 1.8056, 1.8166, 2.4628, 0.9498],
         [2.3601, 4.1735, 4.9313, 2.3770, 2.9711, 4.1579, 4.1577, 3.0857],
         [2.7012, 2.3534, 3.7331, 1.7409, 2.1794, 0.5135, 1.8746, 0.9426],
         [2.1261, 1.5404, 2.7603, 1.6741, 1.2132, 1.6354, 1.0849, 1.2387],
         [1.7516, 2.0816, 2.8682, 2.7619, 3.5830, 3.9498, 2.8420, 2.9162],
         [1.8000, 1.4290, 1.8932, 2.7812, 2.7963, 3.7957, 2.1804, 2.8766]],

        [[3.0585, 2.9258, 2.9229, 1.4963, 2.1865, 1.5238, 2.9494, 1.1623],
         [2.4140, 2.8110, 2.6675, 2.7249, 3.4203, 2.4400, 1.6703, 2.7754],
         [2.4229, 2.9906, 2.1859, 1.9546, 2.4580, 1.9566, 2.0477, 1.5294],
         [1.5616, 2.3468, 1.9568, 1.8826, 2.4227, 2.0639, 1.0216, 2.6526],
         [1.3946, 3.6217, 0.4893, 2.3777, 1.6518, 3.2633, 1.6216, 3.1761],
         [1.9462, 4.6370, 1.2176, 3.8646, 3.2498, 4.5615, 1.9073, 4.4238],
         [1.3844, 2.4626, 1.9043, 1.2222, 0.7378, 2.4471, 1.8722, 3.2822],
...
         [2.0949, 2.8157, 4.1192, 2.0881, 2.6393, 1.0169, 1.6152, 1.5368],
         [1.8283, 2.0573, 2.3762, 1.9086, 3.7325, 2.7268, 2.0280, 3.4565],
         [0.9737, 2.8433, 1.9362, 2.7126, 3.1949, 2.4475, 2.1220, 3.6938],
         [3.0246, 2.2267, 3.0546, 2.2309, 4.6683, 3.7015, 2.8659, 4.0444]]])

Q4. A,B,Cを使って、sinkhoonアルゴリズムを実装してください。(出力は各テンソルに対応するスカラーが集約された10次元のテンソルになります。)

epsilon = 0.00001
solver = SinkhornSolver(epsilon=epsilon, iterations=100000)
cost, pi = solver.forward(A, B)

print(f'cost\n',cost)

cost
tensor([nan, nan, nan, nan, nan, nan, nan, nan, nan, nan])

C_dist = solver._compute_cost_pytorch_cdist(A, B)

print(C_dist)

tensor([[[[3.3452, 3.5884, 2.6145, 3.5548, 2.6811, 1.7614, 4.8377, 3.2827]],

         [[2.7055, 2.8352, 3.2904, 3.4297, 3.2852, 3.5755, 3.8415, 3.5525]],

         [[2.8125, 3.0205, 1.4525, 2.8069, 2.1878, 0.3542, 4.1632, 2.1684]],

         [[2.7713, 3.0012, 2.2180, 3.0812, 2.3985, 1.7458, 4.2431, 2.8260]],

         [[2.8290, 2.6322, 2.2476, 3.3099, 4.0334, 2.9723, 2.6648, 1.8386]],

         [[1.8975, 1.9757, 2.3589, 1.1350, 2.3386, 3.7782, 2.4262, 2.0510]],

         [[0.9523, 0.7452, 2.4652, 1.8895, 2.9092, 3.7592, 1.3614, 2.1590]],

         [[2.3381, 2.3968, 2.0447, 1.6194, 2.5042, 3.3260, 2.8266, 1.7473]]],


        [[[1.3115, 1.6532, 3.0833, 1.0828, 2.1806, 2.0015, 1.7661, 0.9011]],

         [[1.5730, 2.5654, 3.9633, 0.3884, 2.4030, 1.8086, 2.4826, 0.4883]],

         [[1.9457, 2.7556, 3.9811, 0.8660, 1.8056, 1.8166, 2.4628, 0.9498]],

         [[2.3601, 4.1735, 4.9313, 2.3770, 2.9711, 4.1579, 4.1577, 3.0857]],
...

         [[0.9737, 2.8433, 1.9362, 2.7126, 3.1949, 2.4475, 2.1220, 3.6938]],

         [[3.0246, 2.2267, 3.0546, 2.2309, 4.6683, 3.7015, 2.8659, 4.0444]]]])