# General Linear Group

$$\operatorname{GL^+}(n)$$ is the manifold of invertible matrices of positive determinant

$\operatorname{GL^+}(n) = \{X \in \mathbb{R}^{n\times n}\:\mid\:\det(X) > 0\}$

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{GL^+}(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 $$f\colon \mathbb{R} \to (0, \infty)$$ is applied element-wise to the diagonal. By default, the softmax function is used

\begin{split}\begin{align*} \operatorname{softmax} \colon \mathbb{R} &\to (0, \infty) \\ x &\mapsto \log(1+\exp(x)) + \varepsilon \end{align*}\end{split}

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)

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