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Proof of Theorem 2
import torch
import torch.nn as nn
m = 121
n = 128
k = 512
Y = torch.randn(m, n)
U = torch.randn(m, m)
X = torch.randn(m, k)
nn.init.orthogonal_(U)
print("||U^TU||_F=", torch.norm(torch.matmul(U, torch.transpose(U, 0, 1)), p='fro').item())
print("||Y^TX||_F=", torch.norm(torch.matmul(torch.transpose(Y, 0, 1), X)).item())
print("||Y^TUX||_F=", torch.norm(torch.matmul(torch.matmul(torch.transpose(Y, 0, 1), U), X)).item())
print("||X||_F=", torch.norm(X).item())
print("||UX||_F=", torch.norm(torch.matmul(U, X)).item())
||U^TU||_F= 11.0
||Y^TX||_F= 2809.56884765625
||Y^TUX||_F= 2826.705078125
||X||_F= 248.32562255859375
||UX||_F= 248.325439453125
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