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def subspace(A: torch.Tensor) -> torch.Tensor:
"""
Compute orthonormal bases of the subspace
Args:
A: bases of the linear subspace (n_bases, dim)
Return:
Orthonormal bases
Example:
>>> A = torch.rand(10, 300)
>>> subspace(A)
"""
return torch.linalg.qr(A.t()).Q.t()
def intersection(SA: torch.Tensor, SB: torch.Tensor, threshold: float = 1e-2) -> torch.Tensor:
"""
Compute bases of the intersection
Args:
SA, SB: bases of the linear subspace (n_bases, dim)
Return:
Bases of intersection
Example:
>>> A = torch.rand(10, 300)
>>> B = torch.rand(20, 300)
>>> intersection(A, B)
"""
assert threshold > 1e-6
if SA.shape[0] > SB.shape[0]:
return intersection(SB, SA, threshold)
# orthonormalize
SA = subspace(SA)
SB = subspace(SB)
# compute canonical angles
u, s, v = torch.linalg.svd(SA @ SB.t())
# extract the basis that the canonical angle is zero
u = u[:, (s - 1.0).abs() < threshold]
return (SA.t() @ u).t()
def sum_space(SA: torch.Tensor, SB: torch.Tensor) -> torch.Tensor:
"""
Compute bases of the sum space
Args:
SA, SB: bases of the linear subspace (n_bases, dim)
Return:
Bases of sum space
Example:
>>> A = torch.rand(10, 300)
>>> B = torch.rand(20, 300)
>>> sum_space(A, B)
"""
M = torch.cat([SA, SB], dim=0)
return subspace(M)
def orthogonal_complement(SA: torch.Tensor, threshold: float = 1e-2) -> torch.Tensor:
"""
Compute bases of the orthogonal complement
Args:
SA: bases of the linear subspace (n_bases, dim)
Return:
Bases of the orthogonal complement
Example:
>>> A = torch.rand(10, 300)
>>> orthogonal_complement(A)
"""
assert threshold > 1e-6
u, s, v = torch.linalg.svd(SA.t())
# compute rank
rank = (s > threshold).sum()
return u[:, rank:].T
def soft_membership(A: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
"""
Compute membership degree of the vector v for the subspace A
Args:
A: bases of the linear subspace (n_bases, dim)
v: vector (dim,)
Return:
soft membership degree
Example:
>>> A = torch.tensor([[1,0,0], [0,1,0]])
>>> v = torch.tensor([1,0,0])
>>> soft_membership(A, v)
1.0
>>> A = torch.tensor([[1,0,0], [0,1,0]])
>>> v = torch.tensor([0,0,1])
>>> soft_membership(A, v)
0.0
"""
v = v.reshape(1, len(v))
v = subspace(v)
A = subspace(A)
# The cosine of the angles between a subspace and a vector are singular values
u, s, v = torch.linalg.svd(A @ v.t())
s[s > 1] = 1
# Return the maximum cosine of the canonical angles, i.e., the soft membership.
return torch.max(s)
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