sphere_np

CLASS sphere_np(var_shape)

This manifold class defines the shpere with unit Frobenius Norm, i.e.

\[ \{X \in \mathbb{R}^{m\times s \times ...}: \sum X_{i,j,...}^2 = 1 \}. \]

Parameters:

• var_shape (tuple(int, int)) – The shape of the variables of the manifold. The length of var_shape should equal to \(2\).

Attributes:

A(x) (callable)

The constraint dissolving mapping \(\mathcal{A}(x)\). A(X) is set as (2*X)/( 1 + np.sum( X**2 ) ).

C(X) (callable)

Describe the constraints \(c\). C(X) = 0 returns np.sum( X **2 ) - 1.

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_autograd.

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_autograd.

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.

To-dos

• Add supports for Nd-array variables.