Source code for probnum.utils.linalg._inner_product

"""Functions defining useful inner products."""
from __future__ import annotations

from typing import TYPE_CHECKING, Optional, Union

import numpy as np

    from probnum import linops

[docs]def inner_product( v: np.ndarray, w: np.ndarray, A: Optional[Union[np.ndarray, linops.LinearOperator]] = None, ) -> np.ndarray: r"""Inner product :math:`\langle v, w \rangle_A := v^T A w`. For n-d arrays the function computes the inner product over the last axis of the two arrays ``v`` and ``w``. Parameters ---------- v First array. w Second array. A Symmetric positive (semi-)definite matrix defining the geometry. Returns ------- inprod : Inner product(s) of ``v`` and ``w``. Notes ----- Note that the broadcasting behavior of :func:`inner_product` differs from :func:`numpy.inner`. Rather it follows the broadcasting rules of :func:`numpy.matmul` in that n-d arrays are treated as stacks of vectors. """ v_T = v[..., None, :] w = w[..., :, None] if A is None: vw_inprod = v_T @ w else: vw_inprod = v_T @ (A @ w) return np.squeeze(vw_inprod, axis=(-2, -1))
[docs]def induced_norm( v: np.ndarray, A: Optional[Union[np.ndarray, linops.LinearOperator]] = None, axis: int = -1, ) -> np.ndarray: r"""Induced norm :math:`\lVert v \rVert_A := \sqrt{v^T A v}`. Computes the induced norm over the given axis of the array. Parameters ---------- v Array. A Symmetric positive (semi-)definite linear operator defining the geometry. axis Specifies the axis along which to compute the vector norms. Returns ------- norm : Vector norm of ``v`` along the given ``axis``. """ if A is None: return np.linalg.norm(v, ord=2, axis=axis, keepdims=False) v = np.moveaxis(v, axis, -1) w = np.squeeze(A @ v[..., :, None], axis=-1) return np.sqrt(np.sum(v * w, axis=-1))