308 lines
8.8 KiB
Python
308 lines
8.8 KiB
Python
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import numpy as np
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from scipy._lib._util import _asarray_validated
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__all__ = ["logsumexp", "softmax", "log_softmax"]
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def logsumexp(a, axis=None, b=None, keepdims=False, return_sign=False):
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"""Compute the log of the sum of exponentials of input elements.
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Parameters
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----------
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a : array_like
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Input array.
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axis : None or int or tuple of ints, optional
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Axis or axes over which the sum is taken. By default `axis` is None,
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and all elements are summed.
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.. versionadded:: 0.11.0
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b : array-like, optional
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Scaling factor for exp(`a`) must be of the same shape as `a` or
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broadcastable to `a`. These values may be negative in order to
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implement subtraction.
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.. versionadded:: 0.12.0
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keepdims : bool, optional
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If this is set to True, the axes which are reduced are left in the
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result as dimensions with size one. With this option, the result
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will broadcast correctly against the original array.
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.. versionadded:: 0.15.0
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return_sign : bool, optional
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If this is set to True, the result will be a pair containing sign
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information; if False, results that are negative will be returned
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as NaN. Default is False (no sign information).
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.. versionadded:: 0.16.0
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Returns
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-------
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res : ndarray
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The result, ``np.log(np.sum(np.exp(a)))`` calculated in a numerically
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more stable way. If `b` is given then ``np.log(np.sum(b*np.exp(a)))``
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is returned. If ``return_sign`` is True, ``res`` contains the log of
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the absolute value of the argument.
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sgn : ndarray
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If ``return_sign`` is True, this will be an array of floating-point
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numbers matching res containing +1, 0, -1 (for real-valued inputs)
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or a complex phase (for complex inputs). This gives the sign of the
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argument of the logarithm in ``res``.
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If ``return_sign`` is False, only one result is returned.
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See Also
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--------
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numpy.logaddexp, numpy.logaddexp2
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Notes
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-----
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NumPy has a logaddexp function which is very similar to `logsumexp`, but
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only handles two arguments. `logaddexp.reduce` is similar to this
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function, but may be less stable.
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.special import logsumexp
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>>> a = np.arange(10)
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>>> logsumexp(a)
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9.4586297444267107
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>>> np.log(np.sum(np.exp(a)))
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9.4586297444267107
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With weights
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>>> a = np.arange(10)
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>>> b = np.arange(10, 0, -1)
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>>> logsumexp(a, b=b)
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9.9170178533034665
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>>> np.log(np.sum(b*np.exp(a)))
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9.9170178533034647
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Returning a sign flag
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>>> logsumexp([1,2],b=[1,-1],return_sign=True)
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(1.5413248546129181, -1.0)
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Notice that `logsumexp` does not directly support masked arrays. To use it
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on a masked array, convert the mask into zero weights:
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>>> a = np.ma.array([np.log(2), 2, np.log(3)],
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... mask=[False, True, False])
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>>> b = (~a.mask).astype(int)
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>>> logsumexp(a.data, b=b), np.log(5)
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1.6094379124341005, 1.6094379124341005
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"""
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a = _asarray_validated(a, check_finite=False)
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if b is not None:
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a, b = np.broadcast_arrays(a, b)
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if np.any(b == 0):
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a = a + 0. # promote to at least float
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a[b == 0] = -np.inf
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# Scale by real part for complex inputs, because this affects
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# the magnitude of the exponential.
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a_max = np.amax(a.real, axis=axis, keepdims=True)
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if a_max.ndim > 0:
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a_max[~np.isfinite(a_max)] = 0
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elif not np.isfinite(a_max):
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a_max = 0
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if b is not None:
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b = np.asarray(b)
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tmp = b * np.exp(a - a_max)
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else:
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tmp = np.exp(a - a_max)
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# suppress warnings about log of zero
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with np.errstate(divide='ignore'):
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s = np.sum(tmp, axis=axis, keepdims=keepdims)
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if return_sign:
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# For complex, use the numpy>=2.0 convention for sign.
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if np.issubdtype(s.dtype, np.complexfloating):
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sgn = s / np.where(s == 0, 1, abs(s))
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else:
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sgn = np.sign(s)
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s = abs(s)
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out = np.log(s)
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if not keepdims:
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a_max = np.squeeze(a_max, axis=axis)
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out += a_max
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if return_sign:
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return out, sgn
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else:
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return out
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def softmax(x, axis=None):
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r"""Compute the softmax function.
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The softmax function transforms each element of a collection by
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computing the exponential of each element divided by the sum of the
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exponentials of all the elements. That is, if `x` is a one-dimensional
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numpy array::
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softmax(x) = np.exp(x)/sum(np.exp(x))
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Parameters
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----------
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x : array_like
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Input array.
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axis : int or tuple of ints, optional
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Axis to compute values along. Default is None and softmax will be
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computed over the entire array `x`.
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Returns
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-------
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s : ndarray
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An array the same shape as `x`. The result will sum to 1 along the
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specified axis.
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Notes
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-----
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The formula for the softmax function :math:`\sigma(x)` for a vector
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:math:`x = \{x_0, x_1, ..., x_{n-1}\}` is
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.. math:: \sigma(x)_j = \frac{e^{x_j}}{\sum_k e^{x_k}}
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The `softmax` function is the gradient of `logsumexp`.
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The implementation uses shifting to avoid overflow. See [1]_ for more
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details.
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.. versionadded:: 1.2.0
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References
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----------
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.. [1] P. Blanchard, D.J. Higham, N.J. Higham, "Accurately computing the
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log-sum-exp and softmax functions", IMA Journal of Numerical Analysis,
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Vol.41(4), :doi:`10.1093/imanum/draa038`.
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.special import softmax
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>>> np.set_printoptions(precision=5)
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>>> x = np.array([[1, 0.5, 0.2, 3],
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... [1, -1, 7, 3],
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... [2, 12, 13, 3]])
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...
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Compute the softmax transformation over the entire array.
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>>> m = softmax(x)
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>>> m
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array([[ 4.48309e-06, 2.71913e-06, 2.01438e-06, 3.31258e-05],
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[ 4.48309e-06, 6.06720e-07, 1.80861e-03, 3.31258e-05],
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[ 1.21863e-05, 2.68421e-01, 7.29644e-01, 3.31258e-05]])
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>>> m.sum()
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1.0
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Compute the softmax transformation along the first axis (i.e., the
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columns).
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>>> m = softmax(x, axis=0)
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>>> m
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array([[ 2.11942e-01, 1.01300e-05, 2.75394e-06, 3.33333e-01],
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[ 2.11942e-01, 2.26030e-06, 2.47262e-03, 3.33333e-01],
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[ 5.76117e-01, 9.99988e-01, 9.97525e-01, 3.33333e-01]])
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>>> m.sum(axis=0)
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array([ 1., 1., 1., 1.])
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Compute the softmax transformation along the second axis (i.e., the rows).
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>>> m = softmax(x, axis=1)
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>>> m
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array([[ 1.05877e-01, 6.42177e-02, 4.75736e-02, 7.82332e-01],
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[ 2.42746e-03, 3.28521e-04, 9.79307e-01, 1.79366e-02],
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[ 1.22094e-05, 2.68929e-01, 7.31025e-01, 3.31885e-05]])
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>>> m.sum(axis=1)
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array([ 1., 1., 1.])
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"""
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x = _asarray_validated(x, check_finite=False)
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x_max = np.amax(x, axis=axis, keepdims=True)
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exp_x_shifted = np.exp(x - x_max)
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return exp_x_shifted / np.sum(exp_x_shifted, axis=axis, keepdims=True)
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def log_softmax(x, axis=None):
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r"""Compute the logarithm of the softmax function.
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In principle::
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log_softmax(x) = log(softmax(x))
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but using a more accurate implementation.
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Parameters
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----------
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x : array_like
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Input array.
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axis : int or tuple of ints, optional
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Axis to compute values along. Default is None and softmax will be
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computed over the entire array `x`.
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Returns
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-------
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s : ndarray or scalar
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An array with the same shape as `x`. Exponential of the result will
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sum to 1 along the specified axis. If `x` is a scalar, a scalar is
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returned.
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Notes
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-----
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`log_softmax` is more accurate than ``np.log(softmax(x))`` with inputs that
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make `softmax` saturate (see examples below).
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.. versionadded:: 1.5.0
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.special import log_softmax
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>>> from scipy.special import softmax
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>>> np.set_printoptions(precision=5)
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>>> x = np.array([1000.0, 1.0])
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>>> y = log_softmax(x)
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>>> y
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array([ 0., -999.])
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>>> with np.errstate(divide='ignore'):
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... y = np.log(softmax(x))
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...
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>>> y
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array([ 0., -inf])
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"""
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x = _asarray_validated(x, check_finite=False)
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x_max = np.amax(x, axis=axis, keepdims=True)
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if x_max.ndim > 0:
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x_max[~np.isfinite(x_max)] = 0
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elif not np.isfinite(x_max):
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x_max = 0
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tmp = x - x_max
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exp_tmp = np.exp(tmp)
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# suppress warnings about log of zero
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with np.errstate(divide='ignore'):
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s = np.sum(exp_tmp, axis=axis, keepdims=True)
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out = np.log(s)
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out = tmp - out
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return out
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