General Linear Group
\(\operatorname{GL^+}(n)\) is the manifold of invertible matrices of positive determinant
It is realized via an SVD-like factorization:
where we have identified the vector \(\Sigma\) with a diagonal matrix in \(\mathbb{R}^{n \times n}\). The function \(f\colon \mathbb{R} \to (0, \infty)\) is applied element-wise to the diagonal. By default, the softplus function is used
where we use a small \(\varepsilon > 0\) for numerical stability.
- class geotorch.GLp(size, f='softplus', triv='expm')[source]
Manifold of invertible matrices
- Parameters
size (torch.size) – Size of the tensor to be parametrized
f (str or callable or pair of callables) –
Optional. Either:
"softplus"
A callable that maps real numbers to the interval \((0, \infty)\)
A pair of callables such that the first maps the real numbers to \((0, \infty)\) and the second is a (right) inverse of the first
Default:
"softplus"
triv (str or callable) – Optional. A map that maps skew-symmetric matrices onto the orthogonal matrices surjectively. This is used to optimize the \(U\) and \(V\) in the SVD. It can be one of
["expm", "cayley"]
or a custom callable. Default:"expm"
- sample(init_=<function xavier_normal_>, eps=5e-06, factorized=False)
Returns a randomly sampled matrix on the manifold by sampling a matrix according to
init_
and projecting it onto the manifold.If the sampled matrix has more than self.rank small singular values, the smallest ones are clamped to be at least
eps
in absolute value.The output of this method can be used to initialize a parametrized tensor that has been parametrized with this or any other manifold as:
>>> layer = nn.Linear(20, 20) >>> M = FixedRank(layer.weight.size(), rank=6) >>> geotorch.register_parametrization(layer, "weight", M) >>> layer.weight = M.sample()
- Parameters
init_ (callable) – Optional. A function that takes a tensor and fills it in place according to some distribution. See torch.init. Default:
torch.nn.init.xavier_normal_
eps (float) – Optional. Minimum singular value of the sampled matrix. Default:
5e-6
- in_manifold(X, eps=1e-05)
Checks that a given matrix is in the manifold.
- Parameters
X (torch.Tensor or tuple) – The input matrix or matrices of shape
(*, n, k)
.eps (float) – Optional. Threshold at which the singular values are considered to be zero Default:
1e-5