Linop diagonal (#820)

* Add `LinearOperator.diagonal()`

... which computes the diagonal of the linear operator. Default
implementation multiplies with unit vectors, and subclasses may 
use more efficient implementations, e.g. for Kronecker linear operators.

* Add tests for `LinearOperator.diagonal()`

src/probnum/linops/_arithmetic_fallbacks.py

@@ -40,6 +40,7 @@ def __init__(self, linop: LinearOperator, scalar: ScalarLike):
             transpose=lambda: self._scalar * self._linop.T,
             inverse=self._inv,
             trace=lambda: self._scalar * self._linop.trace(),
+            diagonal=lambda: self._scalar * self._linop.diagonal(),
         )
 
         # Matrix properties
@@ -89,7 +90,6 @@ class SumLinearOperator(LambdaLinearOperator):
     """Sum of linear operators."""
 
     def __init__(self, *summands: LinearOperator):
-
         if not all(summand.shape == summands[0].shape for summand in summands):
             raise ValueError("All summands must have the same shape.")
 
@@ -113,6 +113,9 @@ def __init__(self, *summands: LinearOperator):
             trace=lambda: functools.reduce(
                 operator.add, (summand.trace() for summand in self._summands)
             ),
+            diagonal=lambda: functools.reduce(
+                operator.add, (summand.diagonal() for summand in self._summands)
+            ),
         )
 
         # Matrix properties
@@ -176,7 +179,6 @@ class ProductLinearOperator(LambdaLinearOperator):
     """(Operator) Product of linear operators."""
 
     def __init__(self, *factors: LinearOperator):
-
         if not all(
             lfactor.shape[1] == rfactor.shape[0]
             for lfactor, rfactor in zip(factors[:-1], factors[1:])

src/probnum/linops/_block.py

@@ -132,6 +132,11 @@ def _rank(self) -> np.intp:
             return np.sum([block.rank() for block in self.blocks])
         return super()._rank()
 
+    def _diagonal(self) -> np.ndarray:
+        if self._all_blocks_square:
+            return np.concatenate([block.diagonal() for block in self.blocks])
+        return super()._diagonal()
+
     def _cholesky(self, lower: bool) -> BlockDiagonalMatrix:
         if self._all_blocks_square:
             return BlockDiagonalMatrix(

src/probnum/linops/_kronecker.py

@@ -172,6 +172,11 @@ def _trace(self) -> np.number:
 
         return super()._trace()
 
+    def _diagonal(self) -> np.ndarray:
+        if self.B.is_square:
+            return np.kron(self.A.diagonal(), self.B.diagonal())
+        return super()._diagonal()
+
     def _astype(
         self, dtype: DTypeLike, order: str, casting: str, copy: bool
     ) -> "Kronecker":
@@ -200,7 +205,6 @@ def _matmul_kronecker(self, other: "Kronecker") -> "Kronecker":
     def _add_kronecker(
         self, other: "Kronecker"
     ) -> Union[NotImplementedType, "Kronecker"]:
-
         if self.A is other.A or self.A == other.A:
             return Kronecker(A=self.A, B=self.B + other.B)
 
@@ -212,7 +216,6 @@ def _add_kronecker(
     def _sub_kronecker(
         self, other: "Kronecker"
     ) -> Union[NotImplementedType, "Kronecker"]:
-
         if self.A is other.A or self.A == other.A:
             return Kronecker(A=self.A, B=self.B - other.B)
 
@@ -537,7 +540,6 @@ def _matmul_idkronecker(self, other: "IdentityKronecker") -> "IdentityKronecker"
     def _add_idkronecker(
         self, other: "IdentityKronecker"
     ) -> Union[NotImplementedType, "IdentityKronecker"]:
-
         if self.A.shape == other.A.shape:
             return IdentityKronecker(num_blocks=self._num_blocks, B=self.B + other.B)
 
@@ -546,7 +548,6 @@ def _add_idkronecker(
     def _sub_idkronecker(
         self, other: "IdentityKronecker"
     ) -> Union[NotImplementedType, "IdentityKronecker"]:
-
         if self.A.shape == other.A.shape:
             return IdentityKronecker(num_blocks=self._num_blocks, B=self.B - other.B)
 

src/probnum/linops/_linear_operator.py

@@ -109,6 +109,7 @@ def __init__(
         self._det_cache = None
         self._logabsdet_cache = None
         self._trace_cache = None
+        self._diagonal_cache = None
 
         self._lu_cache = None
         self._cholesky_cache = None
@@ -737,19 +738,7 @@ def _trace(self) -> np.number:
         trace : float
             Trace of the linear operator.
         """
-
-        vec = np.zeros(self.shape[1], dtype=self.dtype)
-
-        vec[0] = 1
-        trace = (self @ vec)[0]
-        vec[0] = 0
-
-        for i in range(1, self.shape[0]):
-            vec[i] = 1
-            trace += (self @ vec)[i]
-            vec[i] = 0
-
-        return trace
+        return np.sum(self.diagonal())
 
     def trace(self) -> np.number:
         r"""Trace of the linear operator.
@@ -777,6 +766,31 @@ def trace(self) -> np.number:
 
         return self._trace_cache
 
+    def _diagonal(self) -> np.ndarray:
+        """Diagonal of the linear operator.
+
+        You may implement this method in a subclass.
+        """
+        D = np.min(self.shape)
+        diag = np.zeros(D, dtype=self.dtype)
+        vec = np.zeros(self.shape[1], dtype=self.dtype)
+
+        for i in range(D):
+            vec[i] = 1
+            diag[i] = (self @ vec)[i]
+            vec[i] = 0
+
+        return diag
+
+    def diagonal(self) -> np.ndarray:
+        """Diagonal of the linear operator."""
+        if self._diagonal_cache is None:
+            self._diagonal_cache = self._diagonal()
+
+            self._diagonal_cache.setflags(write=False)
+
+        return self._diagonal_cache
+
     ####################################################################################
     # Matrix Decompositions
     ####################################################################################
@@ -1337,6 +1351,7 @@ def __init__(
         det: Optional[Callable[[], np.inexact]] = None,
         logabsdet: Optional[Callable[[], np.floating]] = None,
         trace: Optional[Callable[[], np.number]] = None,
+        diagonal: Optional[Callable[[], np.ndarray]] = None,
     ):
         super().__init__(shape, dtype)
 
@@ -1357,6 +1372,7 @@ def __init__(
         self._det_fn = det
         self._logabsdet_fn = logabsdet
         self._trace_fn = trace
+        self._diagonal_fn = diagonal
 
     def _matmul(self, x: np.ndarray) -> np.ndarray:
         return self._matmul_fn(x)
@@ -1429,6 +1445,12 @@ def _trace(self) -> np.number:
 
         return self._trace_fn()
 
+    def _diagonal(self) -> np.ndarray:
+        if self._diagonal_fn is None:
+            return super()._diagonal()
+
+        return self._diagonal_fn()
+
 
 class TransposedLinearOperator(LambdaLinearOperator):
     """Transposition of a linear operator."""
@@ -1457,6 +1479,7 @@ def __init__(
             det=self._linop.det,
             logabsdet=self._linop.logabsdet,
             trace=self._linop.trace,
+            diagonal=self._linop.diagonal,
         )
 
     def _astype(
@@ -1561,6 +1584,7 @@ def __init__(
             det=lambda: self._linop.det().astype(self._inexact_dtype),
             logabsdet=lambda: self._linop.logabsdet().astype(self._inexact_dtype),
             trace=lambda: self._linop.trace().astype(dtype),
+            diagonal=lambda: self._linop.diagonal().astype(dtype),
         )
 
     def _astype(
@@ -1591,20 +1615,23 @@ def __init__(self, A: Union[ArrayLike, scipy.sparse.spmatrix]):
             matmul = LinearOperator.broadcast_matmat(lambda x: self.A @ x)
             todense = self.A.toarray
             trace = lambda: self.A.diagonal().sum()
+            diagonal = self.A.diagonal
         else:
             self.A = np.asarray(A)
             self.A.setflags(write=False)
 
             matmul = lambda x: self.A @ x
             todense = lambda: self.A
             trace = lambda: np.trace(self.A)
+            diagonal = lambda: np.diagonal(self.A)
 
         super().__init__(
             self.A.shape,
             self.A.dtype,
             matmul=matmul,
             todense=todense,
             trace=trace,
+            diagonal=diagonal,
         )
 
     def _transpose(self) -> "Matrix":
@@ -1691,6 +1718,7 @@ def __init__(
             trace=lambda: probnum.utils.as_numpy_scalar(
                 self.shape[0], dtype=self.dtype
             ),
+            diagonal=lambda: np.ones(shape[0], dtype=self.dtype),
         )
 
         # Matrix properties

src/probnum/linops/_scaling.py

@@ -312,6 +312,9 @@ def _cond_isotropic(self, p: Union[None, int, float, str]) -> np.floating:
 
         return np.linalg.cond(self.todense(cache=False), p=p)
 
+    def _diagonal(self) -> np.ndarray:
+        return self.factors
+
     def _cholesky(self, lower: bool = True) -> Scaling:
         if self._scalar is not None:
             if self._scalar <= 0:
@@ -347,6 +350,7 @@ def __init__(self, shape, dtype=np.float64):
         det = lambda: np.zeros(shape=(), dtype=dtype)
 
         trace = lambda: np.zeros(shape=(), dtype=dtype)
+        diagonal = lambda: np.zeros(shape=(np.min(shape),), dtype=dtype)
 
         def matmul(x: np.ndarray) -> np.ndarray:
             target_shape = list(x.shape)
@@ -363,6 +367,7 @@ def matmul(x: np.ndarray) -> np.ndarray:
             eigvals=eigvals,
             det=det,
             trace=trace,
+            diagonal=diagonal,
         )
 
         # Matrix properties

tests/test_linops/test_linops.py

@@ -381,6 +381,18 @@ def test_trace(linop: pn.linops.LinearOperator, matrix: np.ndarray):
             linop.trace()
 
 
+@pytest_cases.parametrize_with_cases("linop,matrix", cases=case_modules)
+def test_diagonal(linop: pn.linops.LinearOperator, matrix: np.ndarray):
+    linop_diagonal = linop.diagonal()
+    matrix_diagonal = np.diagonal(matrix)
+
+    assert isinstance(linop_diagonal, np.ndarray)
+    assert linop_diagonal.shape == matrix_diagonal.shape
+    assert linop_diagonal.dtype == matrix_diagonal.dtype
+
+    np.testing.assert_allclose(linop_diagonal, matrix_diagonal)
+
+
 @pytest_cases.parametrize_with_cases("linop,matrix", cases=case_modules)
 def test_transpose(linop: pn.linops.LinearOperator, matrix: np.ndarray):
     matrix_transpose = matrix.transpose()