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"""Sparse matrix norms.
"""
if np.issubdtype(x.dtype, np.complexfloating): sqnorm = abs(x).power(2).sum() else: sqnorm = x.power(2).sum() return sqrt(sqnorm)
""" Norm of a sparse matrix
This function is able to return one of seven different matrix norms, depending on the value of the ``ord`` parameter.
Parameters ---------- x : a sparse matrix Input sparse matrix. ord : {non-zero int, inf, -inf, 'fro'}, optional Order of the norm (see table under ``Notes``). inf means numpy's `inf` object. axis : {int, 2-tuple of ints, None}, optional If `axis` is an integer, it specifies the axis of `x` along which to compute the vector norms. If `axis` is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed. If `axis` is None then either a vector norm (when `x` is 1-D) or a matrix norm (when `x` is 2-D) is returned.
Returns ------- n : float or ndarray
Notes ----- Some of the ord are not implemented because some associated functions like, _multi_svd_norm, are not yet available for sparse matrix.
This docstring is modified based on numpy.linalg.norm. https://github.com/numpy/numpy/blob/master/numpy/linalg/linalg.py
The following norms can be calculated:
===== ============================ ord norm for sparse matrices ===== ============================ None Frobenius norm 'fro' Frobenius norm inf max(sum(abs(x), axis=1)) -inf min(sum(abs(x), axis=1)) 0 abs(x).sum(axis=axis) 1 max(sum(abs(x), axis=0)) -1 min(sum(abs(x), axis=0)) 2 Not implemented -2 Not implemented other Not implemented ===== ============================
The Frobenius norm is given by [1]_:
:math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
References ---------- .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples -------- >>> from scipy.sparse import * >>> import numpy as np >>> from scipy.sparse.linalg import norm >>> a = np.arange(9) - 4 >>> a array([-4, -3, -2, -1, 0, 1, 2, 3, 4]) >>> b = a.reshape((3, 3)) >>> b array([[-4, -3, -2], [-1, 0, 1], [ 2, 3, 4]])
>>> b = csr_matrix(b) >>> norm(b) 7.745966692414834 >>> norm(b, 'fro') 7.745966692414834 >>> norm(b, np.inf) 9 >>> norm(b, -np.inf) 2 >>> norm(b, 1) 7 >>> norm(b, -1) 6
""" if not issparse(x): raise TypeError("input is not sparse. use numpy.linalg.norm")
# Check the default case first and handle it immediately. if axis is None and ord in (None, 'fro', 'f'): return _sparse_frobenius_norm(x)
# Some norms require functions that are not implemented for all types. x = x.tocsr()
if axis is None: axis = (0, 1) elif not isinstance(axis, tuple): msg = "'axis' must be None, an integer or a tuple of integers" try: int_axis = int(axis) except TypeError: raise TypeError(msg) if axis != int_axis: raise TypeError(msg) axis = (int_axis,)
nd = 2 if len(axis) == 2: row_axis, col_axis = axis if not (-nd <= row_axis < nd and -nd <= col_axis < nd): raise ValueError('Invalid axis %r for an array with shape %r' % (axis, x.shape)) if row_axis % nd == col_axis % nd: raise ValueError('Duplicate axes given.') if ord == 2: raise NotImplementedError #return _multi_svd_norm(x, row_axis, col_axis, amax) elif ord == -2: raise NotImplementedError #return _multi_svd_norm(x, row_axis, col_axis, amin) elif ord == 1: return abs(x).sum(axis=row_axis).max(axis=col_axis)[0,0] elif ord == Inf: return abs(x).sum(axis=col_axis).max(axis=row_axis)[0,0] elif ord == -1: return abs(x).sum(axis=row_axis).min(axis=col_axis)[0,0] elif ord == -Inf: return abs(x).sum(axis=col_axis).min(axis=row_axis)[0,0] elif ord in (None, 'f', 'fro'): # The axis order does not matter for this norm. return _sparse_frobenius_norm(x) else: raise ValueError("Invalid norm order for matrices.") elif len(axis) == 1: a, = axis if not (-nd <= a < nd): raise ValueError('Invalid axis %r for an array with shape %r' % (axis, x.shape)) if ord == Inf: M = abs(x).max(axis=a) elif ord == -Inf: M = abs(x).min(axis=a) elif ord == 0: # Zero norm M = (x != 0).sum(axis=a) elif ord == 1: # special case for speedup M = abs(x).sum(axis=a) elif ord in (2, None): M = sqrt(abs(x).power(2).sum(axis=a)) else: try: ord + 1 except TypeError: raise ValueError('Invalid norm order for vectors.') M = np.power(abs(x).power(ord).sum(axis=a), 1 / ord) return M.A.ravel() else: raise ValueError("Improper number of dimensions to norm.") |