# Copyright 2023 The JAX Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""Tools to create numpy-style ufuncs."""
from __future__ import annotations
from collections.abc import Callable
from functools import partial
import math
import operator
from typing import Any
import jax
from jax._src.typing import Array, ArrayLike, DTypeLike
from jax._src.lax import lax as lax_internal
import jax._src.numpy.lax_numpy as jnp
from jax._src.numpy.reductions import _moveaxis
from jax._src.numpy.util import check_arraylike, _broadcast_to, _where
from jax._src.numpy.vectorize import vectorize
from jax._src.util import canonicalize_axis, set_module
import numpy as np
_AT_INPLACE_WARNING = """\
Because JAX arrays are immutable, jnp.ufunc.at() cannot operate inplace like
np.ufunc.at(). Instead, you can pass inplace=False and capture the result; e.g.
>>> arr = jnp.add.at(arr, ind, val, inplace=False)
"""
@set_module('jax.numpy')
class ufunc:
"""Universal functions which operation element-by-element on arrays.
JAX implementation of :class:`numpy.ufunc`.
This is a class for JAX-backed implementations of NumPy's ufunc APIs.
Most users will never need to instantiate :class:`ufunc`, but rather
will use the pre-defined ufuncs in :mod:`jax.numpy`.
For constructing your own ufuncs, see :func:`jax.numpy.frompyfunc`.
Examples:
Universal functions are functions that apply element-wise to broadcasted
arrays, but they also come with a number of extra attributes and methods.
As an example, consider the function :obj:`jax.numpy.add`. The object
acts as a function that applies addition to broadcasted arrays in an
element-wise manner:
>>> x = jnp.array([1, 2, 3, 4, 5])
>>> jnp.add(x, 1)
Array([2, 3, 4, 5, 6], dtype=int32)
Each :class:`ufunc` object includes a number of attributes that describe
its behavior:
>>> jnp.add.nin # number of inputs
2
>>> jnp.add.nout # number of outputs
1
>>> jnp.add.identity # identity value, or None if no identity exists
0
Binary ufuncs like :obj:`jax.numpy.add` include number of methods to
apply the function to arrays in different manners.
The :meth:`~ufunc.outer` method applies the function to the
pair-wise outer-product of the input array values:
>>> jnp.add.outer(x, x)
Array([[ 2, 3, 4, 5, 6],
[ 3, 4, 5, 6, 7],
[ 4, 5, 6, 7, 8],
[ 5, 6, 7, 8, 9],
[ 6, 7, 8, 9, 10]], dtype=int32)
The :meth:`ufunc.reduce` method perfoms a reduction over the array.
For example, :meth:`jnp.add.reduce` is equivalent to ``jnp.sum``:
>>> jnp.add.reduce(x)
Array(15, dtype=int32)
The :meth:`ufunc.accumulate` method performs a cumulative reduction
over the array. For example, :meth:`jnp.add.accumulate` is equivalent
to :func:`jax.numpy.cumulative_sum`:
>>> jnp.add.accumulate(x)
Array([ 1, 3, 6, 10, 15], dtype=int32)
The :meth:`ufunc.at` method applies the function at particular indices in the
array; for ``jnp.add`` the computation is similar to :func:`jax.lax.scatter_add`:
>>> jnp.add.at(x, 0, 100, inplace=False)
Array([101, 2, 3, 4, 5], dtype=int32)
And the :meth:`ufunc.reduceat` method performs a number of ``reduce``
operations bewteen specified indices of an array; for ``jnp.add`` the
operation is similar to :func:`jax.ops.segment_sum`:
>>> jnp.add.reduceat(x, jnp.array([0, 2]))
Array([ 3, 12], dtype=int32)
In this case, the first element is ``x[0:2].sum()``, and the second element
is ``x[2:].sum()``.
"""
[文档]
def __init__(self, func: Callable[..., Any], /,
nin: int, nout: int, *,
name: str | None = None,
nargs: int | None = None,
identity: Any = None,
call: Callable[..., Any] | None = None,
reduce: Callable[..., Any] | None = None,
accumulate: Callable[..., Any] | None = None,
at: Callable[..., Any] | None = None,
reduceat: Callable[..., Any] | None = None,
):
self.__doc__ = func.__doc__
self.__name__ = name or func.__name__
# We want ufunc instances to work properly when marked as static,
# and for this reason it's important that their properties not be
# mutated. We prevent this by storing them in a dunder attribute,
# and accessing them via read-only properties.
self.__static_props = {
'func': func,
'nin': operator.index(nin),
'nout': operator.index(nout),
'nargs': operator.index(nargs or nin),
'identity': identity,
'call': call,
'reduce': reduce,
'accumulate': accumulate,
'at': at,
'reduceat': reduceat,
}
_func = property(lambda self: self.__static_props['func'])
nin = property(lambda self: self.__static_props['nin'])
nout = property(lambda self: self.__static_props['nout'])
nargs = property(lambda self: self.__static_props['nargs'])
identity = property(lambda self: self.__static_props['identity'])
def __hash__(self) -> int:
# In both __hash__ and __eq__, we do not consider call, reduce, etc.
# because they are considered implementation details rather than
# necessary parts of object identity.
return hash((self._func, self.__name__, self.identity,
self.nin, self.nout, self.nargs))
def __eq__(self, other: Any) -> bool:
return isinstance(other, ufunc) and (
(self._func, self.__name__, self.identity, self.nin, self.nout, self.nargs) ==
(other._func, other.__name__, other.identity, other.nin, other.nout, other.nargs))
def __repr__(self) -> str:
return f"<jnp.ufunc '{self.__name__}'>"
def __call__(self, *args: ArrayLike, out: None = None, where: None = None) -> Any:
check_arraylike(self.__name__, *args)
if out is not None:
raise NotImplementedError(f"out argument of {self}")
if where is not None:
raise NotImplementedError(f"where argument of {self}")
call = self.__static_props['call'] or self._call_vectorized
return call(*args)
@partial(jax.jit, static_argnames=['self'])
def _call_vectorized(self, *args):
return vectorize(self._func)(*args)
@partial(jax.jit, static_argnames=['self', 'axis', 'dtype', 'out', 'keepdims'])
def reduce(self, a: ArrayLike, axis: int = 0,
dtype: DTypeLike | None = None,
out: None = None, keepdims: bool = False, initial: ArrayLike | None = None,
where: ArrayLike | None = None) -> Array:
"""Reduction operation derived from a binary function.
JAX implementation of :meth:`numpy.ufunc.reduce`.
Args:
a: Input array.
axis: integer specifying the axis over which to reduce. default=0
dtype: optionally specify the type of the output array.
out: Unused by JAX
keepdims: If True, reduced axes are left in the result with size 1.
If False (default) then reduced axes are squeezed out.
initial: int or array, Default=None. Initial value for the reduction.
where: boolean mask, default=None. The elements to be used in the sum. Array
should be broadcast compatible to the input.
Returns:
array containing the result of the reduction operation.
Examples:
Consider the following array:
>>> x = jnp.array([[1, 2, 3],
... [4, 5, 6]])
:meth:`jax.numpy.add.reduce` is equivalent to :func:`jax.numpy.sum`
along ``axis=0``:
>>> jnp.add.reduce(x)
Array([5, 7, 9], dtype=int32)
>>> x.sum(0)
Array([5, 7, 9], dtype=int32)
Similarly, :meth:`jax.numpy.logical_and.reduce` is equivalent to
:func:`jax.numpy.all`:
>>> jnp.logical_and.reduce(x > 2)
Array([False, False, True], dtype=bool)
>>> jnp.all(x > 2, axis=0)
Array([False, False, True], dtype=bool)
Some reductions do not correspond to any built-in aggregation function;
for example here is the reduction of :func:`jax.numpy.bitwise_or` along
the first axis of ``x``:
>>> jnp.bitwise_or.reduce(x, axis=1)
Array([3, 7], dtype=int32)
"""
check_arraylike(f"{self.__name__}.reduce", a)
if self.nin != 2:
raise ValueError("reduce only supported for binary ufuncs")
if self.nout != 1:
raise ValueError("reduce only supported for functions returning a single value")
if out is not None:
raise NotImplementedError(f"out argument of {self.__name__}.reduce()")
if initial is not None:
check_arraylike(f"{self.__name__}.reduce", initial)
if where is not None:
check_arraylike(f"{self.__name__}.reduce", where)
if self.identity is None and initial is None:
raise ValueError(f"reduction operation {self.__name__!r} does not have an identity, "
"so to use a where mask one has to specify 'initial'.")
if lax_internal._dtype(where) != bool:
raise ValueError(f"where argument must have dtype=bool; got dtype={lax_internal._dtype(where)}")
reduce = self.__static_props['reduce'] or self._reduce_via_scan
return reduce(a, axis=axis, dtype=dtype, keepdims=keepdims, initial=initial, where=where)
def _reduce_via_scan(self, arr: ArrayLike, axis: int | None = 0, dtype: DTypeLike | None = None,
keepdims: bool = False, initial: ArrayLike | None = None,
where: ArrayLike | None = None) -> Array:
assert self.nin == 2 and self.nout == 1
arr = lax_internal.asarray(arr)
if initial is None:
initial = self.identity
if dtype is None:
dtype = jax.eval_shape(self._func, lax_internal._one(arr), lax_internal._one(arr)).dtype
if where is not None:
where = _broadcast_to(where, arr.shape)
if isinstance(axis, tuple):
axis = tuple(canonicalize_axis(a, arr.ndim) for a in axis)
raise NotImplementedError("tuple of axes")
elif axis is None:
if keepdims:
final_shape = (1,) * arr.ndim
else:
final_shape = ()
arr = arr.ravel()
if where is not None:
where = where.ravel()
axis = 0
else:
axis = canonicalize_axis(axis, arr.ndim)
if keepdims:
final_shape = (*arr.shape[:axis], 1, *arr.shape[axis + 1:])
else:
final_shape = (*arr.shape[:axis], *arr.shape[axis + 1:])
# TODO: handle without transpose?
if axis != 0:
arr = _moveaxis(arr, axis, 0)
if where is not None:
where = _moveaxis(where, axis, 0)
if initial is None and arr.shape[0] == 0:
raise ValueError("zero-size array to reduction operation {self.__name__} which has no ideneity")
def body_fun(i, val):
if where is None:
return self(val, arr[i].astype(dtype))
else:
return _where(where[i], self(val, arr[i].astype(dtype)), val)
start_value: ArrayLike
if initial is None:
start_index = 1
start_value = arr[0]
else:
start_index = 0
start_value = initial
start_value = _broadcast_to(lax_internal.asarray(start_value).astype(dtype), arr.shape[1:])
result = jax.lax.fori_loop(start_index, arr.shape[0], body_fun, start_value)
if keepdims:
result = result.reshape(final_shape)
return result
@partial(jax.jit, static_argnames=['self', 'axis', 'dtype'])
def accumulate(self, a: ArrayLike, axis: int = 0, dtype: DTypeLike | None = None,
out: None = None) -> Array:
"""Accumulate operation derived from binary ufunc.
JAX implementation of :func:`numpy.ufunc.accumulate`.
Args:
a: N-dimensional array over which to accumulate.
axis: integer axis over which accumulation will be performed (default = 0)
dtype: optionally specify the type of the output array.
out: Unused by JAX
Returns:
An array containing the accumulated result.
Examples:
Consider the following array:
>>> x = jnp.array([[1, 2, 3],
... [4, 5, 6]])
:meth:`jax.numpy.add.accumulate` is equivalent to
:func:`jax.numpy.cumsum` along the specified axis:
>>> jnp.add.accumulate(x, axis=1)
Array([[ 1, 3, 6],
[ 4, 9, 15]], dtype=int32)
>>> jnp.cumsum(x, axis=1)
Array([[ 1, 3, 6],
[ 4, 9, 15]], dtype=int32)
Similarly, :meth:`jax.numpy.multiply.accumulate` is equivalent to
:func:`jax.numpy.cumprod` along the specified axis:
>>> jnp.multiply.accumulate(x, axis=1)
Array([[ 1, 2, 6],
[ 4, 20, 120]], dtype=int32)
>>> jnp.cumprod(x, axis=1)
Array([[ 1, 2, 6],
[ 4, 20, 120]], dtype=int32)
For other binary ufuncs, the accumulation is an operation not available
via standard APIs. For example, :meth:`jax.numpy.bitwise_or.accumulate`
is essentially a bitwise cumulative ``any``:
>>> jnp.bitwise_or.accumulate(x, axis=1)
Array([[1, 3, 3],
[4, 5, 7]], dtype=int32)
"""
if self.nin != 2:
raise ValueError("accumulate only supported for binary ufuncs")
if self.nout != 1:
raise ValueError("accumulate only supported for functions returning a single value")
if out is not None:
raise NotImplementedError(f"out argument of {self.__name__}.accumulate()")
accumulate = self.__static_props['accumulate'] or self._accumulate_via_scan
return accumulate(a, axis=axis, dtype=dtype)
def _accumulate_via_scan(self, arr: ArrayLike, axis: int = 0,
dtype: DTypeLike | None = None) -> Array:
assert self.nin == 2 and self.nout == 1
check_arraylike(f"{self.__name__}.accumulate", arr)
arr = lax_internal.asarray(arr)
if dtype is None:
dtype = jax.eval_shape(self._func, lax_internal._one(arr), lax_internal._one(arr)).dtype
if axis is None or isinstance(axis, tuple):
raise ValueError("accumulate does not allow multiple axes")
axis = canonicalize_axis(axis, np.ndim(arr))
arr = _moveaxis(arr, axis, 0)
def scan_fun(carry, _):
i, x = carry
y = _where(i == 0, arr[0].astype(dtype), self(x.astype(dtype), arr[i].astype(dtype)))
return (i + 1, y), y
_, result = jax.lax.scan(scan_fun, (0, arr[0].astype(dtype)), None, length=arr.shape[0])
return _moveaxis(result, 0, axis)
@partial(jax.jit, static_argnums=[0], static_argnames=['inplace'])
def at(self, a: ArrayLike, indices: Any, b: ArrayLike | None = None, /, *,
inplace: bool = True) -> Array:
"""Update elements of an array via the specified unary or binary ufunc.
JAX implementation of :func:`numpy.ufunc.at`.
Note:
:meth:`numpy.ufunc.at` mutates arrays in-place. JAX arrays are immutable,
so :meth:`jax.numpy.ufunc.at` cannot replicate these semantics. Instead, JAX
will return the updated value, but requires explicitly passing ``inplace=False``
as a reminder of this difference.
Args:
a: N-dimensional array to update
indices: index, slice, or tuple of indices and slices.
b: array of values for binary ufunc updates.
inplace: must be set to False to indicate that an updated copy will be returned.
Returns:
an updated copy of the input array.
Examples:
Add numbers to specified indices:
>>> x = jnp.ones(10, dtype=int)
>>> indices = jnp.array([2, 5, 7])
>>> values = jnp.array([10, 20, 30])
>>> jnp.add.at(x, indices, values, inplace=False)
Array([ 1, 1, 11, 1, 1, 21, 1, 31, 1, 1], dtype=int32)
This is roughly equivalent to JAX's :meth:`jax.numpy.ndarray.at` method
called this way:
>>> x.at[indices].add(values)
Array([ 1, 1, 11, 1, 1, 21, 1, 31, 1, 1], dtype=int32)
"""
if inplace:
raise NotImplementedError(_AT_INPLACE_WARNING)
at = self.__static_props['at'] or self._at_via_scan
return at(a, indices) if b is None else at(a, indices, b)
def _at_via_scan(self, a: ArrayLike, indices: Any, *args: Any) -> Array:
assert len(args) in {0, 1}
check_arraylike(f"{self.__name__}.at", a, *args)
dtype = jax.eval_shape(self._func, lax_internal._one(a), *(lax_internal._one(arg) for arg in args)).dtype
a = lax_internal.asarray(a).astype(dtype)
args = tuple(lax_internal.asarray(arg).astype(dtype) for arg in args)
indices = jnp._eliminate_deprecated_list_indexing(indices)
if not indices:
return a
shapes = [np.shape(i) for i in indices if not isinstance(i, slice)]
shape = shapes and jax.lax.broadcast_shapes(*shapes)
if not shape:
return a.at[indices].set(self(a.at[indices].get(), *args))
if args:
arg = _broadcast_to(args[0], (*shape, *args[0].shape[len(shape):]))
args = (arg.reshape(math.prod(shape), *args[0].shape[len(shape):]),)
indices = [idx if isinstance(idx, slice) else _broadcast_to(idx, shape).ravel() for idx in indices]
def scan_fun(carry, x):
i, a = carry
idx = tuple(ind if isinstance(ind, slice) else ind[i] for ind in indices)
a = a.at[idx].set(self(a.at[idx].get(), *(arg[i] for arg in args)))
return (i + 1, a), x
carry, _ = jax.lax.scan(scan_fun, (0, a), None, len(indices[0]))
return carry[1]
@partial(jax.jit, static_argnames=['self', 'axis', 'dtype'])
def reduceat(self, a: ArrayLike, indices: Any, axis: int = 0,
dtype: DTypeLike | None = None, out: None = None) -> Array:
"""Reduce an array between specified indices via a binary ufunc.
JAX implementation of :meth:`numpy.ufunc.reduceat`
Args:
a: N-dimensional array to reduce
indices: a 1-dimensional array of increasing integer values which encodes
segments of the array to be reduced.
axis: integer specifying the axis along which to reduce: default=0.
dtype: optionally specify the dtype of the output array.
out: unused by JAX
Returns:
An array containing the reduced values.
Examples:
The ``reduce`` method lets you efficiently compute reduction operations
over array segments. For example:
>>> x = jnp.array([1, 2, 3, 4, 5, 6, 7, 8])
>>> indices = jnp.array([0, 2, 5])
>>> jnp.add.reduce(x, indices)
Array([ 3, 12, 21], dtype=int32)
This is more-or-less equivalent to the following:
>>> jnp.array([x[0:2].sum(), x[2:5].sum(), x[5:].sum()])
Array([ 3, 12, 21], dtype=int32)
For some binary ufuncs, JAX provides similar APIs within :mod:`jax.ops`.
For example, :meth:`jax.add.reduceat` is similar to :func:`jax.ops.segment_sum`,
although in this case the segments are defined via an array of segment ids:
>>> segments = jnp.array([0, 0, 1, 1, 1, 2, 2, 2])
>>> jax.ops.segment_sum(x, segments)
Array([ 3, 12, 21], dtype=int32)
"""
if self.nin != 2:
raise ValueError("reduceat only supported for binary ufuncs")
if self.nout != 1:
raise ValueError("reduceat only supported for functions returning a single value")
if out is not None:
raise NotImplementedError(f"out argument of {self.__name__}.reduceat()")
reduceat = self.__static_props['reduceat'] or self._reduceat_via_scan
return reduceat(a, indices, axis=axis, dtype=dtype)
def _reduceat_via_scan(self, a: ArrayLike, indices: Any, axis: int = 0,
dtype: DTypeLike | None = None) -> Array:
check_arraylike(f"{self.__name__}.reduceat", a, indices)
a = lax_internal.asarray(a)
idx_tuple = jnp._eliminate_deprecated_list_indexing(indices)
assert len(idx_tuple) == 1
indices = idx_tuple[0]
if a.ndim == 0:
raise ValueError(f"reduceat: a must have 1 or more dimension, got {a.shape=}")
if indices.ndim != 1:
raise ValueError(f"reduceat: indices must be one-dimensional, got {indices.shape=}")
if dtype is None:
dtype = a.dtype
if axis is None or isinstance(axis, (tuple, list)):
raise ValueError("reduceat requires a single integer axis.")
axis = canonicalize_axis(axis, a.ndim)
out = jnp.take(a, indices, axis=axis)
ind = jax.lax.expand_dims(jnp.append(indices, a.shape[axis]),
list(np.delete(np.arange(out.ndim), axis)))
ind_start = jax.lax.slice_in_dim(ind, 0, ind.shape[axis] - 1, axis=axis)
ind_end = jax.lax.slice_in_dim(ind, 1, ind.shape[axis], axis=axis)
def loop_body(i, out):
return _where((i > ind_start) & (i < ind_end),
self(out, jnp.take(a, jax.lax.expand_dims(i, (0,)), axis=axis)),
out)
return jax.lax.fori_loop(0, a.shape[axis], loop_body, out)
@partial(jax.jit, static_argnums=[0])
def outer(self, A: ArrayLike, B: ArrayLike, /) -> Array:
"""Apply the function to all pairs of values in ``A`` and ``B``.
JAX implementation of :meth:`numpy.ufunc.outer`.
Args:
A: N-dimensional array
B: N-dimensional array
Returns:
An array of shape `tuple(*A.shape, *B.shape)`
Examples:
A times-table for integers 1...10 created via
:meth:`jax.numpy.multiply.outer`:
>>> x = jnp.arange(1, 11)
>>> print(jnp.multiply.outer(x, x))
[[ 1 2 3 4 5 6 7 8 9 10]
[ 2 4 6 8 10 12 14 16 18 20]
[ 3 6 9 12 15 18 21 24 27 30]
[ 4 8 12 16 20 24 28 32 36 40]
[ 5 10 15 20 25 30 35 40 45 50]
[ 6 12 18 24 30 36 42 48 54 60]
[ 7 14 21 28 35 42 49 56 63 70]
[ 8 16 24 32 40 48 56 64 72 80]
[ 9 18 27 36 45 54 63 72 81 90]
[ 10 20 30 40 50 60 70 80 90 100]]
For input arrays with ``N`` and ``M`` dimensions respectively, the output
will have dimesion ``N + M``:
>>> x = jnp.ones((1, 3, 5))
>>> y = jnp.ones((2, 4))
>>> jnp.add.outer(x, y).shape
(1, 3, 5, 2, 4)
"""
if self.nin != 2:
raise ValueError("outer only supported for binary ufuncs")
if self.nout != 1:
raise ValueError("outer only supported for functions returning a single value")
check_arraylike(f"{self.__name__}.outer", A, B)
_ravel = lambda A: jax.lax.reshape(A, (np.size(A),))
result = jax.vmap(jax.vmap(self, (None, 0)), (0, None))(_ravel(A), _ravel(B))
return result.reshape(*np.shape(A), *np.shape(B))
[文档]
def frompyfunc(func: Callable[..., Any], /, nin: int, nout: int,
*, identity: Any = None) -> ufunc:
"""Create a JAX ufunc from an arbitrary JAX-compatible scalar function.
Args:
func : a callable that takes `nin` scalar arguments and returns `nout` outputs.
nin: integer specifying the number of scalar inputs
nout: integer specifying the number of scalar outputs
identity: (optional) a scalar specifying the identity of the operation, if any.
Returns:
wrapped : jax.numpy.ufunc wrapper of func.
"""
return ufunc(func, nin, nout, identity=identity)
