stiefel_torch

CLASS stiefel_torch(var_shape, device = torch.device('cpu'), dtype = torch.float64)

This manifold class defines the Stiefel manifold, i.e.

\[ \{X \in \mathbb{R}^{n\times p}: X^T X = I_p \}. \]

Parameters:

• var_shape (tuple of ints) – The shape of the variables of the manifold. var_shape = (*self.dim, self._n, self_p)

• device (PyTorch device) – The object representing the device on which a torch.Tensor is or will be allocated.

• dtype (PyTorch dtype) – The object that represents the data type of a torch.Tensor.

The variables in stiefel_torch is not restricted in 2D-tensors, they can be Nd-tensors whose .size() is var_shape. In that cases, for each index (i,j,k,...), we have torch.matmul( X(i,j,k,...,:,:).T ,X(i,j,k,...,:,:)) is a \(p\times p\)-dimensional identity matrix.

Attributes:

A(x) (callable)

The constraint dissolving mapping \(\mathcal{A}(x)\). A(X) is set as 1.5 * X - torch.matmul(X , (torch.matmul(X.transpose(-2,-1), X)/2)).

C(X) (callable)

Describe the constraints \(c\). C(X) returns torch.matmul(X.transpose(-2,-1), X) - self.Ip.

_parameter() (OrdDict)

The ordered dictionary that contains all the variables that changes when device and dtype changes. It contains

self._parameters['Ip'] = torch.diag_embed(torch.ones((*self.dim, self._p), device=self.device, dtype=self.dtype))

m2v(x) (callable)

Flatten the variable of the manifold.

v2m(x) (callable)

Recover flattened variables to its original shape as variable_shape.

Init_point(Xinit = None) (callable)

Generate the initial point.

tensor2array(x) (callable)

Transfer the variable of the manifold to the numpy Nd-array while keep its shape. Default settings are provided in the core.backbone_torch.

array2tensor(x) (callable)

Transfer the numpy Nd-array to the variable of the manifold while keep its shape. Default settings are provided in the core.backbone_torch.

JC(x, lambda) (callable)

The Jacobian of C(x).

JC_transpose(x, lambda) (callable)

The transpose of \(J_c(x)\), expressed by matrix-vector production.

JA(x, d) (callable)

The transposed Jacobian of \(\mathcal{A}(x)\).

JA_transpose(x, d) (callable)

The transpose (or adjoint) of JA(x), i.e. \(\lim_{t \to 0} \frac{1}{t}(J_A(x+td) -J_A(x)) \).

C_quad_penalty(x) (callable)

Returns the quadratical penalty term \(||c(x)||^2\).

hessA(X, U, D) (callable)

Returns the Hessian of \(\mathcal{A}(x)\) in a tensor-vector product form.

hess_feas(X, D) (callable)

Returns the hessian-vector product of \(\frac{1}{2} ||c(x)||^2\).

Feas_eval(X) (callable)

Returns the feasibility of \(x\), measured by value of \(||c(x)||\).

Post_process(X) (callable)

Return the post-processing for X to achieve a point with better feasibility.

generate_cdf_fun(obj_fun, beta) (callable)

Return the function value of the constraint dissolving function. obj_fun is a callable function that returns the value of \(f\) at \(x\). beta is a float object that refers to the penalty parameter in the constraint dissolving function.

generate_cdf_grad(obj_grad, beta) (callable)

Return the gradient of the constraint dissolving function. obj_grad is a callable function that returns the gradient of \(f\) at \(x\). beta is a float object that refers to the penalty parameter in the constraint dissolving function.

generate_cdf_hess(obj_grad, obj_hvp, beta) (callable)

Return the hessian of the constraint dissolving function. obj_grad is a callable function that returns the gradient of \(f\) at \(x\). obj_hvp is the hessian-vector product of \(f\) at \(x\), i.e., \(\nabla^2 h(x)[d]\). beta is a float object that refers to the penalty parameter in the constraint dissolving function.