A library to define abstract linear operators, and associated algebra and matrix-free algorithms, that works with pyTorch Tensors.
Project description
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 withCG
for PSD matrices,minres
for symmetric matrices,LSQR
(to be implemented), orLSMR
(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 LinearOperator
s
The public API of the LinearOperator
library is that every LinearOperator
has the
following properties and methods:
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:
def operator_matrix_product(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:
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)
: wherediag
is a 1D torch tensor.MatrixOperator(M: torch.Tensor)
: whereM
is a 2D torch tensor.SelectionOperator(shape: tuple[int, int], idxs: slice | list[int | slice])
KKTOperator(H: LinearOperator, A: LinearOperator)
: whereH
is a squareLinearOperator
andA
is aLinearOperator
VectorJacobianOperator(f: torch.Tensor, x: torch.Tensor)
: wheref
is the output of the function being differentiated which has a torch autograd value andx
is the vector on whichensures_grad
was called.ZeroOperator(shape: tuple[int, int])
- Combining operators via:
A + B
,A - B
,A @ B
forA
,B
linear operatorshstack
,vstack
A
,c A
,A / c
,v * A
,A / v
for scalarc
and vectorv
.
Implementing LinearOperator
s
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, setsupports_operator_matrix: bool
toTrue
. - 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.
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