An Abstract Linear Operator Library for pyTorch

This library implements a generic structure for abstract linear operators and enables a number of standard operations on them: * Arithmetic: A + B, A - B, -A, A @ B all work exactly as expected to combine linear operators. * Indexing: A[k:ell,m:n] works as expected. * Solves: Ax = b can be solved with CG for PSD matrices, minres for symmetric matrices, LSQR (to be implemented), or LSMR (to be implemented). * Trace estimation: The trace of square matrices, can be estimated via Hutch++ and Hutchinson's estimator. * Diamond-Boyd stochastic equilibration * Randomized Nyström Preconditioning * Automatic adjoint operator generation.

Using LinearOperators

The public API of the LinearOperator library is that every LinearOperator has the following properties and methods: ```python

class LinearOperator:

# Properties
shape: tuple[int, int]
T: LinearOperator
supports_operator_matrix: bool
device: torch.Device

# Matrix multiply
def __matmul__(self, b: torch.Tensor) -> torch.Tensor: ...
def __rmatmul__(self, b: torch.Tensor) -> torch.Tensor: ...

def __matmul__(self, b: LinearOperator) -> LinearOperator: ...
def __rmatmul__(self, b: LinearOperator) -> LinearOperator: ...

# Linear Solve Methods
def solve_I_p_lambda_AT_A_x_eq_b(self,
    lambda_: float,
    b: torch.Tensor,
    x0: torch.Tensor | None=None,
    *, precondition: None | Literal['nsytrom'], hot=False) -> torch.Tensor: ...

def solve_A_x_eq_b(self,
    b: torch.Tensor,
    x0: torch.Tensor | None=None) -> torch.Tensor: ...

# Transformations on LinearOperator
def __mul__(self, c: float) -> LinearOperator: ...
def __rmul__(self, c: float) -> LinearOperator: ...

def __truediv__(self, c: float) -> LinearOperator: ...

def __pow__(self, k: int) -> LinearOperator: ...

def __add__(self, c: LinearOperator) -> LinearOperator: ...

def __sub__(self, c: LinearOperator) -> LinearOperator: ...

def __neg__(self) -> LinearOperator: ...

def __pos__(self) -> LinearOperator: ...

def __getitem__(self, key) -> LinearOperator: ...

```

The following functions are available in the root of the library: ```python def operatormatrixproduct(A: LinearOperator, M: torch.Tensor) -> torch.Tensor: ... def aslinearoperator(A: torch.Tensor | LinearOperator) -> LinearOperator: ... def vstack(ops: list[LinearOperator] | tuple[LinearOperator, ...]) -> LinearOperator: ... def hstack(ops: list[LinearOperator] | tuple[LinearOperator, ...]) -> LinearOperator: ...

To be implemented:

def bmat(ops: list[list[LinearOperator]]) -> LinearOperator: ... # Optimizes out ZeroOperator ```

The following functions are available in linops.trace for trace estimation: python def hutchpp(A: lo.LinearOperator, m: int) -> float: ... def hutchinson(A: lo.LinearOperator, m: int) -> float: ...

linops.equilibration contains equilibrate and symmetric_equilibrate. Their public API is not finalized, if you wish to use them it is recommend you read the source code.

Creating Linear Operators

Linear operators can be constructed in the following way: * Creating a sub-class of LinearOperator * Calling one of the following constructors: * IdentityOperator(n: int) * DiagonalOperator(diag: torch.Tensor): where diag is a 1D torch tensor. * MatrixOperator(M: torch.Tensor): where M is a 2D torch tensor. * SelectionOperator(shape: tuple[int, int], idxs: slice | list[int | slice]) * KKTOperator(H: LinearOperator, A: LinearOperator): where H is a square LinearOperator and A is a LinearOperator * VectorJacobianOperator(f: torch.Tensor, x: torch.Tensor): where f is the output of the function being differentiated which has a torch autograd value and x is the vector on which ensures_grad was called. * ZeroOperator(shape: tuple[int, int]) * Combining operators via: * A + B, A - B, A @ B for A, B linear operators * hstack, vstack * A, c A, A / c, v * A, A / v for scalar c and vector v.

Implementing LinearOperators

To implement a LinearOperator the following are mandatory:

• Set _shape: tuple[int, int] to the shape of the operator.
• Set device appropriately, if the operator requires vectors to be on a particular device.
• Implement a method def _matmul_impl(self, v: torch.Tensor) -> torch.Tensor: ... that implements your matrix vector product.

The following are recommended to improve performance:

• If your _matmul_impl method handles matrix inputs correctly, set supports_operator_matrix: bool to True.
• If it is possible to describe the adjoint operator, set _adjoint: LinearOperator to point to the adjoint of your operator. If you do not compute this, then one will be autogenerated by differentiating through your _matmul_impl.

It is suggested that, if possible, you replace any other methods with specialized implementations.