Positive Semidefinite Fixed Rank Matrices

\(\operatorname{PSSDFixedRank}(n,r)\) is the manifold of positive semidefinite matrices with rank equal to \(r\), for a given \(r \leq n\):

\[\operatorname{PSSDFixedRank}(n,r) = \{X \in \mathbb{R}^{n\times n}\:\mid\:X \succeq 0,\,\operatorname{rank}(X) = r\}\]

It is realized via an eigenvalue-like factorization:

\[\begin{split}\begin{align*} \pi \colon \operatorname{St}(n,r) \times \mathbb{R}^r &\to \operatorname{PSSDFixedRank}(n,r) \\ (Q, \Lambda) &\mapsto Qf(\Lambda)Q^\intercal \end{align*}\end{split}\]

where we have identified the vector \(\Lambda\) with a diagonal matrix in \(\mathbb{R}^{r \times r}\). 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.

Note

For practical applications, it will be almost always more convenient to use the class geotorch.PSSDLowRank, as it is less restrictive, and most of the times it will converge to a max-rank solution anyway.

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

Manifold of symmetric positive semidefinite matrices of rank \(r\).

Parameters

  • size (torch.size) – Size of the tensor to be parametrized

  • rank (int) – Rank of the matrices. It has to be less or equal to \(\min(\texttt{size}[-1], \texttt{size}[-2])\)

  • 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 \(Q\) in the eigenvalue decomposition. It can be one of ["expm", "cayley"] or a custom callable. Default: "expm"

sample(init_=<function xavier_normal_>, eps=5e-06)[source]

Returns a randomly sampled matrix on the manifold as

\[WW^\intercal \qquad W_{i,j} \sim \texttt{init_}\]

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 as:

>>> layer = nn.Linear(20, 20)
>>> M = PSSD(layer.weight.size())
>>> 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 eigenvalue of the sampled matrix. Default: 5e-6

in_manifold(X, eps=1e-06)

Checks that a matrix is in the manifold.

Parameters

  • X (torch.Tensor) – The matrix or batch of matrices of shape (*, n, n) to check.

  • eps (float) – Optional. Threshold at which the singular values are considered to be zero. Default: 1e-6