KernelMatrix¶

class itergp.linops.KernelMatrix(kernel, x0, x1=None, size_keops=1000)¶

Bases: LinearOperator

Kernel matrix.

Linear operator defining (matrix-)multiplication with a kernel matrix \(K=k(X_0,X_1) \in \mathbb{R}^{n_0 \times n_1}\), constructed from two sets of inputs \(X_i \in \mathbb{R}^{n_i \times d}\).

Parameters

kernel (randprocs.kernels.Kernel) – Kernel.
x0 (backend.Array) – Inputs for the first argument of kernel.
x1 (Optional[backend.Array]) – Inputs for the second argument of kernel.
size_keops – At what size of the training data to use KeOps LazyTensor instead of full matrices in memory.

Attributes Summary

`T`
`dtype`	Data type of the linear operator.
`is_lower_triangular`	Whether the `LinearOperator` represents a lower triangular matrix.
`is_positive_definite`	Whether the `LinearOperator` \(L \in \mathbb{R}^{n \times n}\) is (strictly) positive-definite, i.e. \(x^T L x > 0\) for \(x \in \mathbb{R}^n\).
`is_square`	Whether input dimension matches output dimension.
`is_symmetric`	Whether the `LinearOperator` \(L\) is symmetric, i.e. \(L = L^T\).
`is_upper_triangular`	Whether the `LinearOperator` represents an upper triangular matrix.
`kernel`	Covariance function of the kernel matrix.
`ndim`	Number of linear operator dimensions.
`shape`	Shape of the linear operator.
`size`
`x0`	First input(s).
`x1`	Second input(s).

Methods Summary

`__call__`(x[, axis])	Call self as a function.
`astype`(dtype[, order, casting, subok, copy])	Cast a linear operator to a different `dtype`.
`broadcast_matmat`(matmat)	Broadcasting for a (implicitly defined) matrix-matrix product.
`broadcast_matvec`(matvec)	Broadcasting for a (implicitly defined) matrix-vector product.
`broadcast_rmatmat`(rmatmat)
`broadcast_rmatvec`(rmatvec)
`cholesky`([lower])	Computes a Cholesky decomposition of the `LinearOperator`.
`cond`([p])	Compute the condition number of the linear operator.
`det`()	Determinant of the linear operator.
`eigvals`()	Eigenvalue spectrum of the linear operator.
`inv`()	Inverse of the linear operator.
`logabsdet`()	Log absolute determinant of the linear operator.
`rank`()	Rank of the linear operator.
`symmetrize`()	Compute or approximate the closest symmetric `LinearOperator` in the Frobenius norm.
`todense`([cache])	Dense matrix representation of the linear operator.
`trace`()	Trace of the linear operator.
`transpose`(*axes)	Transpose this linear operator.

Attributes Documentation

T¶

dtype¶: Data type of the linear operator.

is_lower_triangular¶

Whether the LinearOperator represents a lower triangular matrix.

If this is None, it is unknown whether the matrix is lower triangular or not.

is_positive_definite¶

Whether the LinearOperator \(L \in \mathbb{R}^{n \times n}\) is (strictly) positive-definite, i.e. \(x^T L x > 0\) for \(x \in \mathbb{R}^n\).

If this is None, it is unknown whether the matrix is positive-definite or not. Only symmetric operators can be positive-definite.

is_square¶: Whether input dimension matches output dimension.

is_symmetric¶

Whether the LinearOperator \(L\) is symmetric, i.e. \(L = L^T\).

If this is None, it is unknown whether the operator is symmetric or not. Only square operators can be symmetric.

is_upper_triangular¶

Whether the LinearOperator represents an upper triangular matrix.

If this is None, it is unknown whether the matrix is upper triangular or not.

kernel¶: Covariance function of the kernel matrix.

ndim¶

Number of linear operator dimensions.

Defined analogously to numpy.ndarray.ndim.

shape¶

Shape of the linear operator.

Defined as a tuple of the output and input dimension of operator.

size¶

x0¶: First input(s).

x1¶: Second input(s).

Methods Documentation

__call__(x, axis=None)¶

Call self as a function.

Parameters

x (ndarray) –
axis (Optional[int]) –

Return type

ndarray

astype(dtype, order='K', casting='unsafe', subok=True, copy=True)¶

Cast a linear operator to a different dtype.

Parameters

dtype (Union[probnum.backend.Dtype, dtype[Any], None, Type[Any], _SupportsDType[dtype[Any]], str, Tuple[Any, int], Tuple[Any, Union[SupportsIndex, Sequence[SupportsIndex]]], List[Any], _DTypeDict, Tuple[Any, Any]]) – Data type to which the linear operator is cast.
order (str) – Memory layout order of the result.
casting (str) – Controls what kind of data casting may occur.
subok (bool) – If True, then sub-classes will be passed-through (default). False is currently not supported for linear operators.
copy (bool) – Whether to return a new linear operator, even if dtype is the same.

Return type

LinearOperator

classmethod broadcast_matmat(matmat)¶

Broadcasting for a (implicitly defined) matrix-matrix product.

Convenience function / decorator to broadcast the definition of a matrix-matrix product to vectors. This can be used to easily construct a new linear operator only from a matrix-matrix product.

Parameters: matmat (Callable[[ndarray], ndarray]) –
Return type: Callable[[ndarray], ndarray]

classmethod broadcast_matvec(matvec)¶

Broadcasting for a (implicitly defined) matrix-vector product.

Convenience function / decorator to broadcast the definition of a matrix-vector product. This can be used to easily construct a new linear operator only from a matrix-vector product.

Parameters: matvec (Callable[[ndarray], ndarray]) –
Return type: Callable[[ndarray], ndarray]

classmethod broadcast_rmatmat(rmatmat)¶

Parameters: rmatmat (Callable[[ndarray], ndarray]) –
Return type: Callable[[ndarray], ndarray]

classmethod broadcast_rmatvec(rmatvec)¶

Parameters: rmatvec (Callable[[ndarray], ndarray]) –
Return type: Callable[[ndarray], ndarray]

cholesky(lower=True)¶

Computes a Cholesky decomposition of the LinearOperator.

The Cholesky decomposition of a symmetric positive-definite matrix \(A \in \mathbb{R}^{n \times n}\) is given by \(A = L L^T\), where the unique Cholesky factor \(L \in \mathbb{R}^{n \times n}\) of \(A\) is a lower triangular matrix with a positive diagonal.

As a side-effect, this method will set the value of is_positive_definite to True, if the computation of the Cholesky factorization succeeds. Otherwise, is_positive_definite will be set to False.

The result of this computation will be cached. If cholesky() is first called with lower=True first and afterwards with lower=False or vice-versa, the method simply returns the transpose of the cached value.

Parameters

lower (bool) – If this is set to False, this method computes and returns the upper triangular Cholesky factor \(U = L^T\), for which \(A = U^T U\). By default (True), the method computes the lower triangular Cholesky factor \(L\).

Returns

The lower or upper Cholesky factor of the LinearOperator, depending on the value of the parameter lower. The result will have its properties is_upper_triangular/is_lower_triangular set accordingly.

Return type

cholesky_factor

Raises

numpy.linalg.LinAlgError – If the LinearOperator is not symmetric, i.e. if is_symmetric is not set to True.
numpy.linalg.LinAlgError – If the LinearOperator is not positive definite.

cond(p=None)¶

Compute the condition number of the linear operator.

The condition number of the linear operator with respect to the p norm. It measures how much the solution \(x\) of the linear system \(Ax=b\) changes with respect to small changes in \(b\).

Parameters

p ({None, 1, , 2, , inf, 'fro'}, optional) –

Order of the norm:

p	norm for matrices
None	2-norm, computed directly via singular value decomposition
’fro’	Frobenius norm
np.inf	max(sum(abs(x), axis=1))
1	max(sum(abs(x), axis=0))
2	2-norm (largest sing. value)

Returns

The condition number of the linear operator. May be infinite.

Return type

cond

det()¶

Determinant of the linear operator.

Return type: inexact

eigvals()¶

Eigenvalue spectrum of the linear operator.

Return type: ndarray

inv()¶

Inverse of the linear operator.

Return type: LinearOperator

logabsdet()¶

Log absolute determinant of the linear operator.

Return type: inexact

rank()¶

Rank of the linear operator.

Return type: int64

symmetrize()¶

Compute or approximate the closest symmetric LinearOperator in the Frobenius norm.

The closest symmetric matrix to a given square matrix \(A\) in the Frobenius norm is given by

\[\operatorname{sym}(A) := \frac{1}{2} (A + A^T).\]

However, for efficiency reasons, it is preferrable to approximate this operator in some cases. For example, a Kronecker product \(K = A \otimes B\) is more efficiently symmetrized by means of

\begin{equation} \operatorname{sym}(A) \otimes \operatorname{sym}(B) = \operatorname{sym}(K) + \frac{1}{2} \left( \frac{1}{2} \left( A \otimes B^T + A^T \otimes B \right) - \operatorname{sym}(K) \right). \end{equation}

Returns: The closest symmetric LinearOperator in the Frobenius norm, or an approximation, which makes a reasonable trade-off between accuracy and efficiency (see above). The resulting LinearOperator will have its is_symmetric property set to True.
Return type: symmetrized_linop
Raises: numpy.linalg.LinAlgError – If this method is called on a non-square LinearOperator.

todense(cache=True)¶

Dense matrix representation of the linear operator.

This method can be computationally very costly depending on the shape of the linear operator. Use with caution.

Returns: matrix – Matrix representation of the linear operator.
Return type: np.ndarray
Parameters: cache (bool) –

trace()¶

Trace of the linear operator.

Computes the trace of a square linear operator \(\text{tr}(A) = \sum_{i-1}^n A_{ii}\).

Returns: trace – Trace of the linear operator.
Return type: float
Raises: LinAlgError : – If trace() is called on a non-square matrix.

transpose(*axes)¶

Transpose this linear operator.

Can be abbreviated self.T instead of self.transpose().

Parameters: axes (Union[int, Tuple[int]]) –
Return type: LinearOperator