General Linear Group

\(\operatorname{SL}(n)\) is the manifold of matrices of determinant equal to 1.

\[\operatorname{SL}(n) = \{X \in \mathbb{R}^{n\times n}\:\mid\:\det(X) = 1\}\]

It is realized via an SVD-like factorization:

\[\begin{split}\begin{align*} \pi \colon \operatorname{SO}(n) \times \mathbb{R}^n \times \operatorname{SO}(n) &\to \operatorname{SL}(n) \\ (U, \Sigma, V) &\mapsto Uf(\Sigma)V^\intercal \end{align*}\end{split}\]

where we have identified the vector \(\Sigma\) with a diagonal matrix in \(\mathbb{R}^{n \times n}\). The function \(f\colon \mathbb{R} \to (\varepsilon, \infty)\) is applied element-wise to the diagonal for a small \(\varepsilon > 0\). By default, a combination of the softplus function

\[\begin{split}\begin{align*} \operatorname{softplus} \colon \mathbb{R} &\to (\varepsilon, \infty) \\ x &\mapsto \log(1+\exp(x)) + \varepsilon \end{align*}\end{split}\]

composed with the normalization function

\[\begin{split}\begin{align*} \operatorname{g} \colon \mathbb{R}^n &\to (\varepsilon, \infty)^n \\ (x_1, \dots, x_n) &\mapsto \left(\frac{x_i}{\sqrt[\leftroot{-2}\uproot{2}n]{\prod_i x_i}}\right)_i \end{align*}\end{split}\]

to ensure that the product of all the singular values is equal to 1.

class geotorch.SL(size, f='softplus', triv='expm')[source]

Manifold of special linear matrices

  • 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)[source]

Returns a randomly sampled matrix on the manifold by sampling a matrix according to init_ and projecting it onto the manifold.

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 = SL(layer.weight.size(), rank=6)
>>> geotorch.register_parametrization(layer, "weight", M)
>>> layer.weight = M.sample()
  • 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=0.005)[source]

Checks that a given matrix is in the manifold.

  • 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: 5e-3