from __future__ import annotations, division
__all__ = [
"LinearOperator",
"aslinearoperator",
]
import logging
from abc import ABC, abstractmethod
import numpy as np
import scipy as sp
from numpy.linalg import solve as np_solve
from scipy.linalg import eigvals, lstsq
from scipy.linalg import solve as sp_solve
from scipy.sparse import csr_matrix
from scipy.sparse.linalg import LinearOperator as spLinearOperator
from scipy.sparse.linalg import eigs as sp_eigs
from scipy.sparse.linalg import eigsh as sp_eigsh
from scipy.sparse.linalg import lobpcg as sp_lobpcg
from scipy.sparse.linalg import lsqr, spsolve
# need to check scipy version since the interface submodule changed into
# _interface from scipy>=1.8.0
sp_version = sp.__version__.split(".")
if int(sp_version[0]) <= 1 and int(sp_version[1]) < 8:
from scipy.sparse.sputils import isintlike, isshape
else:
from scipy.sparse._sputils import isintlike, isshape
from typing import Callable, List, Optional, Sequence, Union
from pylops import get_ndarray_multiplication
from pylops.optimization.basic import cgls
from pylops.utils.backend import get_array_module, get_module, get_sparse_eye
from pylops.utils.decorators import count
from pylops.utils.estimators import trace_hutchinson, trace_hutchpp, trace_nahutchpp
from pylops.utils.typing import DTypeLike, InputDimsLike, NDArray, ShapeLike
logging.basicConfig(format="%(levelname)s: %(message)s", level=logging.WARNING)
class _LinearOperator(ABC):
"""Meta-class for Linear operator"""
@abstractmethod
def _matvec(self, x: NDArray) -> NDArray:
"""Matrix-vector multiplication handler."""
pass
@abstractmethod
def _rmatvec(self, x: NDArray) -> NDArray:
"""Matrix-vector adjoint multiplication handler."""
pass
[docs]class LinearOperator(_LinearOperator):
"""Common interface for performing matrix-vector products.
This class acts as an abstract interface between matrix-like
objects and iterative solvers, providing methods to perform
matrix-vector and adjoint matrix-vector products as as
well as convenience methods such as ``eigs``, ``cond``, and
``conj``.
.. note:: End users of PyLops should not use this class directly but simply
use operators that are already implemented. This class is meant for
developers and it has to be used as the parent class of any new operator
developed within PyLops. Find more details regarding implementation of
new operators at :ref:`addingoperator`.
Parameters
----------
Op : :obj:`scipy.sparse.linalg.LinearOperator` or :obj:`pylops.linearoperator.LinearOperator`
Operator. If other arguments are provided, they will overwrite those obtained from ``Op``.
dtype : :obj:`str`, optional
Type of elements in input array.
shape : :obj:`tuple(int, int)`, optional
Shape of operator. If not provided, obtained from ``dims`` and ``dimsd``.
dims : :obj:`tuple(int, ..., int)`, optional
.. versionadded:: 2.0.0
Dimensions of model. If not provided, ``(self.shape[1],)`` is used.
dimsd : :obj:`tuple(int, ..., int)`, optional
.. versionadded:: 2.0.0
Dimensions of data. If not provided, ``(self.shape[0],)`` is used.
clinear : :obj:`bool`, optional
.. versionadded:: 1.17.0
Operator is complex-linear. Defaults to ``True``.
explicit : :obj:`bool`, optional
Operator contains a matrix that can be solved explicitly
(``True``) or not (``False``). Defaults to ``False``.
forceflat : :obj:`bool`, optional
.. versionadded:: 2.2.0
Force an array to be flattened after matvec/rmatvec if the input is ambiguous
(i.e., is a 1D array both when operating with ND arrays and with 1D arrays).
Defaults to ``None`` for operators that have no ambiguity or to ``True``
for those with ambiguity.
name : :obj:`str`, optional
.. versionadded:: 2.0.0
Name of operator (to be used by :func:`pylops.utils.describe.describe`)
"""
def __init__(
self,
Op: Optional[Union[spLinearOperator, LinearOperator]] = None,
dtype: Optional[DTypeLike] = None,
shape: Optional[ShapeLike] = None,
dims: Optional[ShapeLike] = None,
dimsd: Optional[ShapeLike] = None,
clinear: Optional[bool] = None,
explicit: Optional[bool] = None,
forceflat: Optional[bool] = None,
name: Optional[str] = None,
) -> None:
if Op is not None:
self.Op = Op
# All Operators must have shape and dtype
dtype = self.Op.dtype if dtype is None else dtype
shape = self.Op.shape if shape is None else shape
# Optionally, some operators have other attributes
dims = getattr(Op, "dims", (Op.shape[1],)) if dims is None else dims
dimsd = getattr(Op, "dimsd", (Op.shape[0],)) if dimsd is None else dimsd
clinear = getattr(Op, "clinear", True) if clinear is None else clinear
explicit = (
getattr(self.Op, "explicit", False) if explicit is None else explicit
)
if explicit and hasattr(Op, "A"):
self.A = Op.A
forceflat = (
getattr(self.Op, "forceflat", None) if forceflat is None else forceflat
)
name = getattr(Op, "name", None) if name is None else name
if dtype is not None:
self.dtype = dtype
if shape is not None:
self.shape = shape
if dims is not None:
self.dims = dims
if dimsd is not None:
self.dimsd = dimsd
if clinear is not None:
self.clinear = clinear
if explicit is not None:
self.explicit = explicit
if forceflat is not None:
self.forceflat = forceflat
self.name = name
# counters
self.matvec_count = 0
self.rmatvec_count = 0
self.matmat_count = 0
self.rmatmat_count = 0
@property
def shape(self):
_shape = getattr(self, "_shape", None)
if _shape is None: # Cannot find shape, falling back on dims and dimsd
dims = getattr(self, "_dims", None)
dimsd = getattr(self, "_dimsd", None)
if dims is None or dimsd is None: # Cannot find both dims and dimsd, error
raise AttributeError(
(
f"'{self.__class__.__name__}' object has no attribute 'shape' "
"nor both fallback attributes ('dims', 'dimsd')"
)
)
_shape = (int(np.prod(dimsd)), int(np.prod(dims)))
self._shape = _shape # Update to not redo everything above on next call
return _shape
@shape.setter
def shape(self, new_shape: ShapeLike) -> None:
new_shape = tuple(new_shape)
if not isshape(new_shape):
raise ValueError(f"invalid shape %{new_shape:r} (must be 2-d)")
dims = getattr(self, "_dims", None)
dimsd = getattr(self, "_dimsd", None)
if dims is not None and dimsd is not None: # Found dims and dimsd
if np.prod(dimsd) != new_shape[0] and np.prod(dims) != new_shape[1]:
raise ValueError("New shape incompatible with dims and dimsd")
elif np.prod(dimsd) != new_shape[0]:
raise ValueError("New shape incompatible with dimsd")
elif np.prod(dims) != new_shape[1]:
raise ValueError("New shape incompatible with dims")
self._shape = new_shape
@shape.deleter
def shape(self):
del self._shape
@property
def dims(self):
_dims = getattr(self, "_dims", None)
if _dims is None:
shape = getattr(self, "_shape", None)
if shape is None:
raise AttributeError(
f"'{self.__class__.__name__}' object has no attributes 'dims' or 'shape'"
)
_dims = (shape[1],)
return _dims
@dims.setter
def dims(self, new_dims: ShapeLike) -> None:
new_dims = tuple(new_dims)
shape = getattr(self, "_shape", None)
if shape is None: # shape not set yet
self._dims = new_dims
else:
if np.prod(new_dims) == self.shape[1]:
self._dims = new_dims
else:
raise ValueError("dims incompatible with shape[1]")
@dims.deleter
def dims(self):
del self._dims
@property
def dimsd(self):
_dimsd = getattr(self, "_dimsd", None)
if _dimsd is None:
shape = getattr(self, "_shape", None)
if shape is None:
raise AttributeError(
f"'{self.__class__.__name__}' object has no attributes 'dimsd' or 'shape'"
)
_dimsd = (shape[0],)
return _dimsd
@dimsd.setter
def dimsd(self, new_dimsd: ShapeLike) -> None:
new_dimsd = tuple(new_dimsd)
shape = getattr(self, "_shape", None)
if shape is None: # shape not set yet
self._dimsd = new_dimsd
else:
if np.prod(new_dimsd) == self.shape[0]:
self._dimsd = new_dimsd
else:
raise ValueError("dimsd incompatible with shape[0]")
@dimsd.deleter
def dimsd(self):
del self._dimsd
@property
def clinear(self):
return getattr(self, "_clinear", True)
@clinear.setter
def clinear(self, new_clinear: bool) -> None:
self._clinear = bool(new_clinear)
@clinear.deleter
def clinear(self):
del self._clinear
@property
def explicit(self):
return getattr(self, "_explicit", False)
@explicit.setter
def explicit(self, new_explicit: bool) -> None:
self._explicit = bool(new_explicit)
@explicit.deleter
def explicit(self):
del self._explicit
@property
def forceflat(self):
return getattr(self, "_forceflat", None)
@forceflat.setter
def forceflat(self, new_forceflat: bool) -> None:
# note that this can also be None so we check before forcing bool
self._forceflat = (
new_forceflat if new_forceflat is None else bool(new_forceflat)
)
@forceflat.deleter
def forceflat(self):
del self._forceflat
@property
def name(self):
return getattr(self, "_name", None)
@name.setter
def name(self, new_name: str) -> None:
self._name = new_name
@name.deleter
def name(self):
del self._name
def __mul__(self, x: Union[float, LinearOperator]) -> LinearOperator:
return self.dot(x)
def __matmul__(self, other):
if np.isscalar(other):
raise ValueError("Scalar not allowed, use * instead")
return self.__mul__(other)
def __rmul__(self, x: float) -> LinearOperator:
if np.isscalar(x):
Op = _ScaledLinearOperator(self, x)
self._copy_attributes(
Op,
exclude=[
"explicit",
"name",
],
)
Op.explicit = False
return Op
else:
return NotImplemented
def __rmatmul__(self, other):
if np.isscalar(other):
raise ValueError("Scalar not allowed, use * instead")
return self.__rmul__(other)
def __pow__(self, p: int) -> LinearOperator:
if np.isscalar(p):
Op = _PowerLinearOperator(self, p)
self._copy_attributes(
Op,
exclude=[
"explicit",
"name",
],
)
Op.explicit = False
return Op
else:
return NotImplemented
def __add__(self, x: LinearOperator) -> LinearOperator:
if isinstance(x, (LinearOperator, spLinearOperator)):
# cast x to pylops linear operator if not already (this is done
# to allow mixing pylops and scipy operators)
Opx = aslinearoperator(x)
Op = _SumLinearOperator(self, Opx)
self._copy_attributes(
Op,
exclude=[
"explicit",
"forceflat",
"name",
],
)
Op.clinear = Op.clinear and Opx.clinear
Op.explicit = False
if self.forceflat is None and Opx.forceflat is None:
Op.forceflat = None
elif self.forceflat is not None and Opx.forceflat is not None:
# Define forceflat only if differing, otherwise raise error
if self.forceflat != Opx.forceflat:
raise ValueError(
f"Operators have conflicting forceflat {Op.forceflat} != {Opx.forceflat}"
)
Op.forceflat = self.forceflat
else: # Only one of them is None
Op.forceflat = (
self.forceflat if self.forceflat is not None else Opx.forceflat
)
# Replace if shape-like
if len(self.dims) == 1:
Op.dims = Opx.dims
if len(self.dimsd) == 1:
Op.dimsd = Opx.dimsd
return Op
else:
return NotImplemented
def __neg__(self) -> LinearOperator:
Op = _ScaledLinearOperator(self, -1)
self._copy_attributes(
Op,
exclude=[
"explicit",
"name",
],
)
Op.explicit = False
return Op
def __sub__(self, x):
return self.__add__(-x)
def __repr__(self):
M, N = self.shape
if self.dtype is None:
dt = "unspecified dtype"
else:
dt = "dtype=" + str(self.dtype)
return "<%dx%d %s with %s>" % (M, N, self.__class__.__name__, dt)
def _copy_attributes(
self,
dest: LinearOperator,
exclude: Optional[List[str]] = None,
) -> None:
"""Copy attributes from one LinearOperator to another"""
if exclude is None:
exclude = ["name"]
attrs = ["dims", "dimsd", "clinear", "explicit", "forceflat", "name"]
if exclude is not None:
for item in exclude:
attrs.remove(item)
for attr in attrs:
if hasattr(self, attr):
setattr(dest, attr, getattr(self, attr))
def _matvec(self, x: NDArray) -> NDArray:
"""Matrix-vector multiplication handler."""
if self.Op is not None:
return self.Op._matvec(x)
def _rmatvec(self, x: NDArray) -> NDArray:
"""Matrix-vector adjoint multiplication handler."""
if self.Op is not None:
return self.Op._rmatvec(x)
def _matmat(self, X: NDArray) -> NDArray:
"""Matrix-matrix multiplication handler.
Modified version of scipy _matmat to avoid having trailing dimension
in col when provided to matvec
"""
ncp = get_array_module(X)
if sp.sparse.issparse(X):
y = ncp.vstack([self.matvec(col.toarray().reshape(-1)) for col in X.T]).T
else:
y = ncp.vstack([self.matvec(col.reshape(-1)) for col in X.T]).T
return y
def _rmatmat(self, X: NDArray) -> NDArray:
"""Matrix-matrix adjoint multiplication handler.
Modified version of scipy _rmatmat to avoid having trailing dimension
in col when provided to rmatvec
"""
ncp = get_array_module(X)
if sp.sparse.issparse(X):
y = ncp.vstack([self.rmatvec(col.toarray().reshape(-1)) for col in X.T]).T
else:
y = ncp.vstack([self.rmatvec(col.reshape(-1)) for col in X.T]).T
return y
def _adjoint(self) -> LinearOperator:
Op = _AdjointLinearOperator(self)
self._copy_attributes(Op, exclude=["dims", "dimsd", "explicit", "name"])
Op.explicit = False
Op.dims = self.dimsd
Op.dimsd = self.dims
return Op
def _transpose(self) -> LinearOperator:
Op = _TransposedLinearOperator(self)
self._copy_attributes(Op, exclude=["dims", "dimsd", "explicit", "name"])
Op.explicit = False
Op.dims = self.dimsd
Op.dimsd = self.dims
return Op
def adjoint(self):
return self._adjoint()
H: Callable[[LinearOperator], LinearOperator] = property(adjoint)
def transpose(self):
return self._transpose()
T: Callable[[LinearOperator], LinearOperator] = property(transpose)
@count(forward=True)
def matvec(self, x: NDArray) -> NDArray:
"""Matrix-vector multiplication.
Modified version of scipy matvec which does not consider the case
where the input vector is ``np.matrix`` (the use ``np.matrix`` is now
discouraged in numpy's documentation).
Parameters
----------
x : :obj:`numpy.ndarray`
Input array of shape (N,) or (N,1)
Returns
-------
y : :obj:`numpy.ndarray`
Output array of shape (M,) or (M,1)
"""
M, N = self.shape
if x.shape != (N,) and x.shape != (N, 1):
raise ValueError(
f"Dimension mismatch. Got {x.shape}, but expected ({N},) or ({N}, 1)."
)
y = self._matvec(x)
if x.ndim == 1:
y = y.reshape(M)
elif x.ndim == 2:
y = y.reshape(M, 1)
else:
raise ValueError("Invalid shape returned by user-defined matvec()")
return y
@count(forward=False)
def rmatvec(self, x: NDArray) -> NDArray:
"""Adjoint matrix-vector multiplication.
Modified version of scipy rmatvec which does not consider the case
where the input vector is ``np.matrix`` (the use ``np.matrix`` is now
discouraged in numpy's documentation).
Parameters
----------
y : :obj:`numpy.ndarray`
Input array of shape (M,) or (M,1)
Returns
-------
x : :obj:`numpy.ndarray`
Output array of shape (N,) or (N,1)
"""
M, N = self.shape
if x.shape != (M,) and x.shape != (M, 1):
raise ValueError(
f"Dimension mismatch. Got {x.shape}, but expected ({M},) or ({M}, 1)."
)
y = self._rmatvec(x)
if x.ndim == 1:
y = y.reshape(N)
elif x.ndim == 2:
y = y.reshape(N, 1)
else:
raise ValueError("Invalid shape returned by user-defined rmatvec()")
return y
@count(forward=True, matmat=True)
def matmat(self, X: NDArray) -> NDArray:
"""Matrix-matrix multiplication.
Modified version of scipy matmat which does not consider the case
where the input vector is ``np.matrix`` (the use ``np.matrix`` is now
discouraged in numpy's documentation).
Parameters
----------
x : :obj:`numpy.ndarray`
Input array of shape (N,K)
Returns
-------
y : :obj:`numpy.ndarray`
Output array of shape (M,K)
"""
if X.ndim != 2:
raise ValueError(f"Expected 2-d ndarray or matrix, not {X.ndim}-d ndarray")
if X.shape[0] != self.shape[1]:
raise ValueError(f"Dimension mismatch: {self.shape}, {X.shape}")
Y = self._matmat(X)
return Y
@count(forward=False, matmat=True)
def rmatmat(self, X: NDArray) -> NDArray:
"""Matrix-matrix multiplication.
Modified version of scipy rmatmat which does not consider the case
where the input vector is ``np.matrix`` (the use ``np.matrix`` is now
discouraged in numpy's documentation).
Parameters
----------
x : :obj:`numpy.ndarray`
Input array of shape (M,K)
Returns
-------
y : :obj:`numpy.ndarray`
Output array of shape (N,K)
"""
if X.ndim != 2:
raise ValueError(f"Expected 2-d ndarray or matrix, not {X.ndim}-d ndarray")
if X.shape[0] != self.shape[0]:
raise ValueError(f"Dimension mismatch: {self.shape}, {X.shape}")
Y = self._rmatmat(X)
return Y
def dot(self, x: NDArray) -> NDArray:
"""Matrix-matrix or matrix-vector multiplication.
Parameters
----------
x : :obj:`numpy.ndarray`
Input array (or matrix)
Returns
-------
y : :obj:`numpy.ndarray`
Output array (or matrix) that represents
the result of applying the linear operator on x.
"""
if isinstance(x, (LinearOperator, spLinearOperator)):
# cast x to pylops linear operator if not already (this is done
# to allow mixing pylops and scipy operators)
Opx = aslinearoperator(x)
Op = _ProductLinearOperator(self, Opx)
self._copy_attributes(Op, exclude=["dims", "explicit", "forceflat", "name"])
Op.clinear = Op.clinear and Opx.clinear
Op.explicit = False
if self.forceflat is None and Opx.forceflat is None:
Op.forceflat = None
elif self.forceflat is not None and Opx.forceflat is not None:
# Define forceflat only if differing, otherwise raise error
if self.forceflat != Opx.forceflat:
raise ValueError(
f"Operators have conflicting forceflat {Op.forceflat} != {Opx.forceflat}"
)
Op.forceflat = self.forceflat
else: # Only one of them is None
Op.forceflat = (
self.forceflat if self.forceflat is not None else Opx.forceflat
)
Op.dims = Opx.dims
return Op
elif np.isscalar(x):
Op = _ScaledLinearOperator(self, x)
self._copy_attributes(
Op,
exclude=["explicit", "name"],
)
Op.explicit = False
return Op
else:
if not get_ndarray_multiplication() and (
x.ndim > 2 or (x.ndim == 2 and x.shape[0] != self.shape[1])
):
raise ValueError(
"Operator can only be applied 1D vectors or 2D matrices. "
"Enable ndarray multiplication with pylops.set_ndarray_multiplication(True)."
)
is_dims_shaped = x.shape == self.dims
is_dims_shaped_matrix = len(x.shape) > 1 and x.shape[:-1] == self.dims
if is_dims_shaped:
# (dims1, ..., dimsK) => (dims1 * ... * dimsK,) == self.shape
x = x.ravel()
if is_dims_shaped_matrix and not self.forceflat:
# (dims1, ..., dimsK, P) => (dims1 * ... * dimsK, P)
x = x.reshape((-1, x.shape[-1]))
if x.ndim == 1:
y = self.matvec(x)
if (
is_dims_shaped
and not self.forceflat
and get_ndarray_multiplication()
):
y = y.reshape(self.dimsd)
return y
elif x.ndim == 2:
y = self.matmat(x)
if (
is_dims_shaped_matrix
and not self.forceflat
and get_ndarray_multiplication()
):
y = y.reshape((*self.dimsd, -1))
return y
else:
raise ValueError(
(
"Wrong shape.\nFor vector multiplication, expects either a 1d "
"array or, an ndarray of size `dims` when `dims` and `dimsd` "
"both are available.\n"
"For matrix multiplication, expects a 2d array with its first "
f"dimension is equal to {self.shape[1]}.\n"
f"Instead, received an array of shape {x.shape}."
)
)
def div(
self,
y: NDArray,
niter: int = 100,
densesolver: str = "scipy",
) -> NDArray:
r"""Solve the linear problem :math:`\mathbf{y}=\mathbf{A}\mathbf{x}`.
Overloading of operator ``/`` to improve expressivity of `Pylops`
when solving inverse problems.
Parameters
----------
y : :obj:`np.ndarray`
Data
niter : :obj:`int`, optional
Number of iterations (to be used only when ``explicit=False``)
densesolver : :obj:`str`, optional
Use scipy (``scipy``) or numpy (``numpy``) dense solver
Returns
-------
xest : :obj:`np.ndarray`
Estimated model
"""
xest = self.__truediv__(y, niter=niter, densesolver=densesolver)
return xest
def __truediv__(
self,
y: NDArray,
niter: int = 100,
densesolver: str = "scipy",
) -> NDArray:
if self.explicit is True:
if sp.sparse.issparse(self.A):
# use scipy solver for sparse matrices
xest = spsolve(self.A, y)
elif isinstance(self.A, np.ndarray):
# use scipy solvers for dense matrices (used for backward
# compatibility, could be switched to numpy equivalents)
if self.A.shape[0] == self.A.shape[1]:
if densesolver == "scipy":
xest = sp_solve(self.A, y)
else:
xest = np_solve(self.A, y)
else:
xest = lstsq(self.A, y)[0]
else:
# use numpy/cupy solvers for dense matrices
ncp = get_array_module(y)
if self.A.shape[0] == self.A.shape[1]:
xest = ncp.linalg.solve(self.A, y)
else:
xest = ncp.linalg.lstsq(self.A, y)[0]
else:
if isinstance(y, np.ndarray):
# numpy backend
xest = lsqr(self, y, iter_lim=niter, atol=1e-8, btol=1e-8)[0]
else:
# cupy backend
ncp = get_array_module(y)
xest = cgls(
self,
y,
x0=ncp.zeros(int(self.shape[1]), dtype=self.dtype),
niter=niter,
)[0]
return xest
def todense(
self,
backend: str = "numpy",
) -> NDArray:
r"""Return dense matrix.
The operator is converted into its dense matrix equivalent. In order
to do so, square or tall operators are applied to an identity matrix
whose number of rows and columns is equivalent to the number of
columns of the operator. Conversely, for skinny operators, the
transpose operator is applied to an identity matrix
whose number of rows and columns is equivalent to the number of
rows of the operator and the resulting matrix is transposed
(and complex conjugated).
Note that this operation may be costly for operators with large number
of rows and columns and it should be used mostly as a way to inspect
the structure of the matricial equivalent of the operator.
Parameters
----------
backend : :obj:`str`, optional
Backend used to densify matrix (``numpy`` or ``cupy`` or ``jax``). Note that
this must be consistent with how the operator has been created.
Returns
-------
matrix : :obj:`numpy.ndarray` or :obj:`cupy.ndarray`
Dense matrix.
"""
ncp = get_module(backend)
Op = self
# Create identity matrix
shapemin = min(Op.shape)
if shapemin <= 1e3:
# use numpy for small matrices (faster but heavier on memory)
identity = ncp.eye(shapemin, dtype=self.dtype)
else:
# use scipy for large matrices (slower but lighter on memory)
identity = get_sparse_eye(ncp.ones(1))(shapemin, dtype=self.dtype).tocsc()
# Apply operator
if Op.shape[1] == shapemin:
matrix = Op.matmat(identity)
else:
matrix = ncp.conj(Op.rmatmat(identity)).T
return matrix
def tosparse(self) -> NDArray:
r"""Return sparse matrix.
The operator in converted into its sparse (CSR) matrix equivalent. In order
to do so, the operator is applied to series of unit vectors with length equal
to the number of coloumns in the original operator.
Returns
-------
matrix : :obj:`scipy.sparse.csr_matrix`
Sparse matrix.
"""
Op = self
(_, n) = self.shape
# stores non-zero data for the sparse matrix creation
entries = []
indices = []
# loop through columns of self
for i in range(n):
# make a unit vector for the ith column
unit_i = np.zeros(n)
unit_i[i] = 1
# multiply unit vector to self and find the non-zeros
res_i: NDArray = Op * unit_i
rows_nz = np.where(res_i != 0)[0]
# append the non-zero values and indices to the lists
for j in rows_nz:
indices.append([i, j])
entries_i = res_i[rows_nz]
for e in entries_i:
entries.append(e)
# post process the entries / indices for scipy.sparse.csr_matrix
entries = np.array(entries)
indices = np.array(indices)
i, j = indices[:, 0], indices[:, 1]
# construct a sparse, CSR matrix from the entries / indices data.
matrix = csr_matrix((entries, (j, i)), shape=self.shape, dtype=self.dtype)
return matrix
def eigs(
self,
neigs: Optional[int] = None,
symmetric: bool = False,
niter: Optional[int] = None,
uselobpcg: bool = False,
**kwargs_eig: Union[int, float, str],
) -> NDArray:
r"""Most significant eigenvalues of linear operator.
Return an estimate of the most significant eigenvalues
of the linear operator. If the operator has rectangular
shape (``shape[0]!=shape[1]``), eigenvalues are first
computed for the square operator :math:`\mathbf{A^H}\mathbf{A}`
and the square-root values are returned.
Parameters
----------
neigs : :obj:`int`
Number of eigenvalues to compute (if ``None``, return all). Note
that for ``explicit=False``, only :math:`N-1` eigenvalues can be
computed where :math:`N` is the size of the operator in the
model space
symmetric : :obj:`bool`, optional
Operator is symmetric (``True``) or not (``False``). User should
set this parameter to ``True`` only when it is guaranteed that the
operator is real-symmetric or complex-hermitian matrices
niter : :obj:`int`, optional
Number of iterations for eigenvalue estimation
uselobpcg : :obj:`bool`, optional
Use :func:`scipy.sparse.linalg.lobpcg`
**kwargs_eig
Arbitrary keyword arguments for :func:`scipy.sparse.linalg.eigs`,
:func:`scipy.sparse.linalg.eigsh`, or
:func:`scipy.sparse.linalg.lobpcg`
Returns
-------
eigenvalues : :obj:`numpy.ndarray`
Operator eigenvalues.
Raises
------
ValueError
If ``uselobpcg=True`` for a non-symmetric square matrix with
complex type
Notes
-----
Depending on the size of the operator, whether it is explicit or not
and the number of eigenvalues requested, different algorithms are
used by this routine.
More precisely, when only a limited number of eigenvalues is requested
the :func:`scipy.sparse.linalg.eigsh` method is used in case of
``symmetric=True`` and the :func:`scipy.sparse.linalg.eigs` method
is used ``symmetric=False``. However, when the matrix is represented
explicitly within the linear operator (``explicit=True``) and all the
eigenvalues are requested the :func:`scipy.linalg.eigvals` is used
instead.
Finally, when only a limited number of eigenvalues is required,
it is also possible to explicitly choose to use the
``scipy.sparse.linalg.lobpcg`` method via the ``uselobpcg`` input
parameter flag.
Most of these algorithms are a port of ARPACK [1]_, a Fortran package
which provides routines for quickly finding eigenvalues/eigenvectors
of a matrix. As ARPACK requires only left-multiplication
by the matrix in question, eigenvalues/eigenvectors can also be
estimated for linear operators when the dense matrix is not available.
.. [1] http://www.caam.rice.edu/software/ARPACK/
"""
if self.explicit and isinstance(self.A, np.ndarray):
if self.shape[0] == self.shape[1]:
if neigs is None or neigs == self.shape[1]:
eigenvalues = eigvals(self.A)
else:
if not symmetric and np.iscomplexobj(self) and uselobpcg:
raise ValueError(
"cannot use scipy.sparse.linalg.lobpcg "
"for non-symmetric square matrices of "
"complex type..."
)
if symmetric and uselobpcg:
X = np.random.rand(self.shape[0], neigs).astype(self.dtype)
eigenvalues = sp_lobpcg(
self.A, X=X, maxiter=niter, **kwargs_eig
)[0]
elif symmetric:
eigenvalues = sp_eigsh(
self.A, k=neigs, maxiter=niter, **kwargs_eig
)[0]
else:
eigenvalues = sp_eigs(
self.A, k=neigs, maxiter=niter, **kwargs_eig
)[0]
else:
if neigs is None or neigs == self.shape[1]:
eigenvalues = np.sqrt(eigvals(np.dot(np.conj(self.A.T), self.A)))
else:
if uselobpcg:
X = np.random.rand(self.shape[1], neigs).astype(self.dtype)
eigenvalues = np.sqrt(
sp_lobpcg(
np.dot(np.conj(self.A.T), self.A),
X=X,
maxiter=niter,
**kwargs_eig,
)[0]
)
else:
eigenvalues = np.sqrt(
sp_eigsh(
np.dot(np.conj(self.A.T), self.A),
k=neigs,
maxiter=niter,
**kwargs_eig,
)[0]
)
else:
if neigs is None or neigs >= self.shape[1]:
neigs = self.shape[1] - 2
if self.shape[0] == self.shape[1]:
if not symmetric and np.iscomplexobj(self) and uselobpcg:
raise ValueError(
"cannot use scipy.sparse.linalg.lobpcg for "
"non symmetric square matrices of "
"complex type..."
)
if symmetric and uselobpcg:
X = np.random.rand(self.shape[0], neigs).astype(self.dtype)
eigenvalues = sp_lobpcg(self, X=X, maxiter=niter, **kwargs_eig)[0]
elif symmetric:
eigenvalues = sp_eigsh(self, k=neigs, maxiter=niter, **kwargs_eig)[
0
]
else:
eigenvalues = sp_eigs(self, k=neigs, maxiter=niter, **kwargs_eig)[0]
else:
if uselobpcg:
X = np.random.rand(self.shape[1], neigs).astype(self.dtype)
eigenvalues = np.sqrt(
sp_lobpcg(self.H * self, X=X, maxiter=niter, **kwargs_eig)[0]
)
else:
eigenvalues = np.sqrt(
sp_eigs(self.H * self, k=neigs, maxiter=niter, **kwargs_eig)[0]
)
return -np.sort(-eigenvalues)
def cond(
self,
uselobpcg: bool = False,
**kwargs_eig: Union[int, float, str],
) -> NDArray:
r"""Condition number of linear operator.
Return an estimate of the condition number of the linear operator as
the ratio of the largest and lowest estimated eigenvalues.
Parameters
----------
uselobpcg : :obj:`bool`, optional
Use :func:`scipy.sparse.linalg.lobpcg` to compute eigenvalues
**kwargs_eig
Arbitrary keyword arguments for :func:`scipy.sparse.linalg.eigs`,
:func:`scipy.sparse.linalg.eigsh`, or
:func:`scipy.sparse.linalg.lobpcg`
Returns
-------
eigenvalues : :obj:`numpy.ndarray`
Operator eigenvalues.
Notes
-----
The condition number of a matrix (or linear operator) can be estimated
as the ratio of the largest and lowest estimated eigenvalues:
.. math::
k= \frac{\lambda_{max}}{\lambda_{min}}
The condition number provides an indication of the rate at which the
solution of the inversion of the linear operator :math:`A` will
change with respect to a change in the data :math:`y`.
Thus, if the condition number is large, even a small error in :math:`y`
may cause a large error in :math:`x`. On the other hand, if the
condition number is small then the error in :math:`x` is not much
bigger than the error in :math:`y`. A problem with a low condition
number is said to be *well-conditioned*, while a problem with a high
condition number is said to be *ill-conditioned*.
"""
if not uselobpcg:
cond = (
self.eigs(neigs=1, which="LM", **kwargs_eig).item()
/ self.eigs(neigs=1, which="SM", **kwargs_eig).item()
)
else:
cond = (
self.eigs(neigs=1, uselobpcg=True, largest=True, **kwargs_eig).item()
/ self.eigs(neigs=1, uselobpcg=True, largest=False, **kwargs_eig).item()
)
return cond
def conj(self) -> LinearOperator:
"""Complex conjugate operator
Returns
-------
conjop : :obj:`pylops.LinearOperator`
Complex conjugate operator
"""
conjop = _ConjLinearOperator(self)
return conjop
def apply_columns(self, cols: InputDimsLike) -> LinearOperator:
"""Apply subset of columns of operator
This method can be used to wrap a LinearOperator and mimic the action
of a subset of columns of the operator on a reduced model in forward
mode, and retrieve only the result of a subset of rows in adjoint mode.
Note that unless the operator has ``explicit=True``, this is not
optimal as the entire forward and adjoint passes of the original
operator will have to be perfomed. It can however be useful for the
implementation of solvers such as Orthogonal Matching Pursuit (OMP)
that iteratively build a solution by evaluate only a subset of the
columns of the operator.
Parameters
----------
cols : :obj:`list`
Columns to be selected
Returns
-------
colop : :obj:`pylops.LinearOperator`
Apply column operator
"""
colop = _ColumnLinearOperator(self, cols)
return colop
def toreal(self, forw: bool = True, adj: bool = True) -> LinearOperator:
"""Real operator
Parameters
----------
forw : :obj:`bool`, optional
Apply real to output of forward pass
adj : :obj:`bool`, optional
Apply real to output of adjoint pass
Returns
-------
realop : :obj:`pylops.LinearOperator`
Real operator
"""
realop = _RealImagLinearOperator(self, forw, adj, True)
return realop
def toimag(self, forw: bool = True, adj: bool = True) -> LinearOperator:
"""Imag operator
Parameters
----------
forw : :obj:`bool`, optional
Apply imag to output of forward pass
adj : :obj:`bool`, optional
Apply imag to output of adjoint pass
Returns
-------
imagop : :obj:`pylops.LinearOperator`
Imag operator
"""
imagop = _RealImagLinearOperator(self, forw, adj, False)
return imagop
def trace(
self,
neval: Optional[int] = None,
method: Optional[str] = None,
backend: str = "numpy",
**kwargs_trace,
) -> float:
r"""Trace of linear operator.
Returns the trace (or its estimate) of the linear operator.
Parameters
----------
neval : :obj:`int`, optional
Maximum number of matrix-vector products compute. Default depends
``method``.
method : :obj:`str`, optional
Should be one of the following:
- **explicit**: If the operator is not explicit, will convert to
dense first.
- **hutchinson**: see :obj:`pylops.utils.trace_hutchinson`
- **hutch++**: see :obj:`pylops.utils.trace_hutchpp`
- **na-hutch++**: see :obj:`pylops.utils.trace_nahutchpp`
Defaults to 'explicit' for explicit operators, and 'Hutch++' for
the rest.
backend : :obj:`str`, optional
Backend used to densify matrix (``numpy`` or ``cupy``). Note that
this must be consistent with how the operator has been created.
**kwargs_trace
Arbitrary keyword arguments passed to
:obj:`pylops.utils.trace_hutchinson`,
:obj:`pylops.utils.trace_hutchpp`, or
:obj:`pylops.utils.trace_nahutchpp`
Returns
-------
trace : :obj:`float`
Operator trace.
Raises
------
ValueError
If the operator has rectangular shape (``shape[0] != shape[1]``)
NotImplementedError
If the ``method`` is not one of the available methods.
"""
if self.shape[0] != self.shape[1]:
raise ValueError("operator is not square.")
ncp = get_module(backend)
if method is None:
method = "explicit" if self.explicit else "hutch++"
method_l = method.lower()
if method_l == "explicit":
A = self.A if self.explicit else self.todense(backend=backend)
return ncp.trace(A)
elif method_l == "hutchinson":
return trace_hutchinson(self, neval=neval, backend=backend, **kwargs_trace)
elif method_l == "hutch++":
return trace_hutchpp(self, neval=neval, backend=backend, **kwargs_trace)
elif method_l == "na-hutch++":
return trace_nahutchpp(self, neval=neval, backend=backend, **kwargs_trace)
else:
raise NotImplementedError(f"method {method} not available.")
def reset_count(self) -> None:
"""Reset counters
When invoked all counters are set back to 0.
"""
self.matvec_count = 0
self.rmatvec_count = 0
self.matmat_count = 0
self.rmatmat_count = 0
def _get_dtype(
operators: Sequence[LinearOperator],
dtypes: Optional[Sequence[DTypeLike]] = None,
) -> DTypeLike:
if dtypes is None:
dtypes = []
for obj in operators:
if obj is not None and hasattr(obj, "dtype"):
dtypes.append(obj.dtype)
return np.result_type(*dtypes)
class _ScaledLinearOperator(LinearOperator):
"""Scaled Linear Operator"""
def __init__(
self,
A: LinearOperator,
alpha: float,
) -> None:
if not isinstance(A, LinearOperator):
raise ValueError("LinearOperator expected as A")
if not np.isscalar(alpha):
raise ValueError("scalar expected as alpha")
if isinstance(alpha, complex) and not np.iscomplexobj(
np.ones(1, dtype=A.dtype)
):
# if the scalar is of complex type but not the operator, find out type
dtype = _get_dtype([A], [type(alpha)])
else:
# if both the scalar and operator are of real or complex type, use type
# of the operator
dtype = A.dtype
super(_ScaledLinearOperator, self).__init__(dtype=dtype, shape=A.shape)
self.args = (A, alpha)
def _matvec(self, x: NDArray) -> NDArray:
return self.args[1] * self.args[0].matvec(x)
def _rmatvec(self, x: NDArray) -> NDArray:
return np.conj(self.args[1]) * self.args[0].rmatvec(x)
def _rmatmat(self, x: NDArray) -> NDArray:
return np.conj(self.args[1]) * self.args[0].rmatmat(x)
def _matmat(self, x: NDArray) -> NDArray:
return self.args[1] * self.args[0].matmat(x)
def _adjoint(self) -> LinearOperator:
A, alpha = self.args
return A.H * np.conj(alpha)
class _ConjLinearOperator(LinearOperator):
"""Complex conjugate linear operator"""
def __init__(self, Op: LinearOperator) -> None:
if not isinstance(Op, LinearOperator):
raise TypeError("Op must be a LinearOperator")
super(_ConjLinearOperator, self).__init__(Op, shape=Op.shape)
self.Op = Op
def _matvec(self, x: NDArray) -> NDArray:
return (self.Op._matvec(x.conj())).conj()
def _rmatvec(self, x: NDArray) -> NDArray:
return (self.Op._rmatvec(x.conj())).conj()
def _adjoint(self) -> LinearOperator:
return _ConjLinearOperator(self.Op.H)
class _ColumnLinearOperator(LinearOperator):
"""Column selector linear operator
Produces the forward and adjoint passes with a subset of columns of the
original operator
"""
def __init__(
self,
Op: LinearOperator,
cols: InputDimsLike,
) -> None:
if not isinstance(Op, LinearOperator):
raise TypeError("Op must be a LinearOperator")
super(_ColumnLinearOperator, self).__init__(Op, explicit=Op.explicit)
self.Op = Op
self.cols = cols
self._shape = (Op.shape[0], len(cols))
self._dims = len(cols)
if self.explicit:
self.Opcol = Op.A[:, cols]
def _matvec(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
if self.explicit:
y = self.Opcol @ x
else:
y = ncp.zeros(int(self.Op.shape[1]), dtype=self.dtype)
y[self.cols] = x
y = self.Op._matvec(y)
return y
def _rmatvec(self, x: NDArray) -> NDArray:
if self.explicit:
y = self.Opcol.T.conj() @ x
else:
y = self.Op._rmatvec(x)
y = y[self.cols]
return y
class _AdjointLinearOperator(LinearOperator):
"""Adjoint of Linear Operator"""
def __init__(self, A: LinearOperator):
shape = (A.shape[1], A.shape[0])
super(_AdjointLinearOperator, self).__init__(shape=shape, dtype=A.dtype)
self.A = A
self.args = (A,)
def _matvec(self, x: NDArray) -> NDArray:
return self.A._rmatvec(x)
def _rmatvec(self, x: NDArray) -> NDArray:
return self.A._matvec(x)
def _matmat(self, X: NDArray) -> NDArray:
return self.A._rmatmat(X)
def _rmatmat(self, X: NDArray) -> NDArray:
return self.A._matmat(X)
class _TransposedLinearOperator(LinearOperator):
"""Transposition of Linear Operator"""
def __init__(self, A: LinearOperator):
shape = (A.shape[1], A.shape[0])
super(_TransposedLinearOperator, self).__init__(shape=shape, dtype=A.dtype)
self.A = A
self.args = (A,)
def _matvec(self, x: NDArray) -> NDArray:
return np.conj(self.A._rmatvec(np.conj(x)))
def _rmatvec(self, x: NDArray) -> NDArray:
return np.conj(self.A._matvec(np.conj(x)))
def _matmat(self, X: NDArray) -> NDArray:
return np.conj(self.A._rmatmat(np.conj(X)))
def _rmatmat(self, X: NDArray) -> NDArray:
return np.conj(self.A._matmat(np.conj(X)))
class _ProductLinearOperator(LinearOperator):
"""Product of Linear Operators"""
def __init__(self, A: LinearOperator, B: LinearOperator):
if not isinstance(A, LinearOperator) or not isinstance(B, LinearOperator):
raise ValueError(
f"both operands have to be a LinearOperator{type(A)} {type(B)}"
)
if A.shape[1] != B.shape[0]:
raise ValueError("cannot add %r and %r: shape mismatch" % (A, B))
super().__init__(dtype=_get_dtype([A, B]), shape=(A.shape[0], B.shape[1]))
self.args = (A, B)
def _matvec(self, x: NDArray) -> NDArray:
return self.args[0].matvec(self.args[1].matvec(x))
def _rmatvec(self, x: NDArray) -> NDArray:
return self.args[1].rmatvec(self.args[0].rmatvec(x))
def _rmatmat(self, X: NDArray) -> NDArray:
return self.args[1].rmatmat(self.args[0].rmatmat(X))
def _matmat(self, X: NDArray) -> NDArray:
return self.args[0].matmat(self.args[1].matmat(X))
def _adjoint(self):
A, B = self.args
return B.H * A.H
class _SumLinearOperator(LinearOperator):
def __init__(
self,
A: LinearOperator,
B: LinearOperator,
) -> None:
if not isinstance(A, LinearOperator) or not isinstance(B, LinearOperator):
raise ValueError("both operands have to be a LinearOperator")
if A.shape != B.shape:
raise ValueError("cannot add %r and %r: shape mismatch" % (A, B))
self.args = (A, B)
super(_SumLinearOperator, self).__init__(
dtype=_get_dtype([A, B]), shape=A.shape
)
def _matvec(self, x: NDArray) -> NDArray:
return self.args[0].matvec(x) + self.args[1].matvec(x)
def _rmatvec(self, x: NDArray) -> NDArray:
return self.args[0].rmatvec(x) + self.args[1].rmatvec(x)
def _rmatmat(self, x: NDArray) -> NDArray:
return self.args[0].rmatmat(x) + self.args[1].rmatmat(x)
def _matmat(self, x: NDArray) -> NDArray:
return self.args[0].matmat(x) + self.args[1].matmat(x)
def _adjoint(self) -> LinearOperator:
A, B = self.args
return A.H + B.H
class _PowerLinearOperator(LinearOperator):
def __init__(self, A: LinearOperator, p: int) -> None:
if not isinstance(A, LinearOperator):
raise ValueError("LinearOperator expected as A")
if A.shape[0] != A.shape[1]:
raise ValueError("square LinearOperator expected, got %r" % A)
if not isintlike(p) or p < 0:
raise ValueError("non-negative integer expected as p")
super(_PowerLinearOperator, self).__init__(dtype=A.dtype, shape=A.shape)
self.args = (A, p)
def _power(self, fun: Callable, x: NDArray) -> NDArray:
res = np.array(x, copy=True)
for _ in range(self.args[1]):
res = fun(res)
return res
def _matvec(self, x: NDArray) -> NDArray:
return self._power(self.args[0].matvec, x)
def _rmatvec(self, x: NDArray) -> NDArray:
return self._power(self.args[0].rmatvec, x)
def _rmatmat(self, x: NDArray) -> NDArray:
return self._power(self.args[0].rmatmat, x)
def _matmat(self, x: NDArray) -> NDArray:
return self._power(self.args[0].matmat, x)
def _adjoint(self) -> LinearOperator:
A, p = self.args
return A.H**p
class _RealImagLinearOperator(LinearOperator):
"""Real-Imag linear operator
Computes forward and adjoint passes of an operator Op and returns only
its real (or imaginary) component. Note that for the adjoint step the
output must be complex conjugated (i.e. opposite of the imaginary part is
returned)
"""
def __init__(
self,
Op: LinearOperator,
forw: bool = True,
adj: bool = True,
real: bool = True,
) -> None:
if not isinstance(Op, LinearOperator):
raise TypeError("Op must be a LinearOperator")
super(_RealImagLinearOperator, self).__init__(Op, shape=Op.shape)
self.Op = Op
self.real = real
self.forw = forw
self.adj = adj
self.dtype = np.array(0, dtype=self.Op.dtype).real.dtype
def _matvec(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = self.Op._matvec(x)
if self.forw:
if self.real:
y = ncp.real(y)
else:
y = ncp.imag(y)
return y
def _rmatvec(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = self.Op._rmatvec(x)
if self.adj:
if self.real:
y = ncp.real(y)
else:
y = -ncp.imag(y)
return y
def aslinearoperator(Op: Union[spLinearOperator, LinearOperator]) -> LinearOperator:
"""Return Op as a LinearOperator.
Converts any operator compatible with pylops definition of LinearOperator into a pylops
LinearOperator. This can be used for example when `Op` is a scipy operator to ensure
that the returned operator has all properties and methods of the pylops class.
Parameters
----------
Op : :obj:`pylops.LinearOperator` or any other Operator
Operator of any type
Returns
-------
Op : :obj:`pylops.LinearOperator`
Operator of type :obj:`pylops.LinearOperator`
"""
if isinstance(Op, LinearOperator):
return Op
else:
return LinearOperator(Op)
