# Copyright 2018 Google LLC
#
# 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.
from functools import partial
import numpy as np
import textwrap
import operator
from typing import Tuple, Union, cast
from jax import jit, custom_jvp
from jax import lax
from jax import ops
from jax._src.lax import linalg as lax_linalg
from jax import dtypes
from .util import _wraps
from . import lax_numpy as jnp
from jax.util import canonicalize_axis
_T = lambda x: jnp.swapaxes(x, -1, -2)
_H = lambda x: jnp.conjugate(jnp.swapaxes(x, -1, -2))
def _promote_arg_dtypes(*args):
"""Promotes `args` to a common inexact type."""
def _to_inexact_type(type):
return type if jnp.issubdtype(type, jnp.inexact) else jnp.float_
inexact_types = [_to_inexact_type(jnp._dtype(arg)) for arg in args]
dtype = dtypes.canonicalize_dtype(jnp.result_type(*inexact_types))
args = [lax.convert_element_type(arg, dtype) for arg in args]
if len(args) == 1:
return args[0]
else:
return args
@_wraps(np.linalg.cholesky)
def cholesky(a):
a = _promote_arg_dtypes(jnp.asarray(a))
return lax_linalg.cholesky(a)
[docs]@_wraps(np.linalg.svd)
def svd(a, full_matrices=True, compute_uv=True):
a = _promote_arg_dtypes(jnp.asarray(a))
return lax_linalg.svd(a, full_matrices, compute_uv)
[docs]@_wraps(np.linalg.matrix_power)
def matrix_power(a, n):
a = _promote_arg_dtypes(jnp.asarray(a))
if a.ndim < 2:
raise TypeError("{}-dimensional array given. Array must be at least "
"two-dimensional".format(a.ndim))
if a.shape[-2] != a.shape[-1]:
raise TypeError("Last 2 dimensions of the array must be square")
try:
n = operator.index(n)
except TypeError as err:
raise TypeError("exponent must be an integer, got {}".format(n)) from err
if n == 0:
return jnp.broadcast_to(jnp.eye(a.shape[-2], dtype=a.dtype), a.shape)
elif n < 0:
a = inv(a)
n = np.abs(n)
if n == 1:
return a
elif n == 2:
return a @ a
elif n == 3:
return (a @ a) @ a
z = result = None
while n > 0:
z = a if z is None else (z @ z)
n, bit = divmod(n, 2)
if bit:
result = z if result is None else (result @ z)
return result
[docs]@_wraps(np.linalg.matrix_rank)
def matrix_rank(M, tol=None):
M = _promote_arg_dtypes(jnp.asarray(M))
if M.ndim > 2:
raise TypeError("array should have 2 or fewer dimensions")
if M.ndim < 2:
return jnp.any(M != 0).astype(jnp.int32)
S = svd(M, full_matrices=False, compute_uv=False)
if tol is None:
tol = S.max() * np.max(M.shape) * jnp.finfo(S.dtype).eps
return jnp.sum(S > tol)
[docs]@custom_jvp
@_wraps(np.linalg.slogdet)
@jit
def slogdet(a):
a = _promote_arg_dtypes(jnp.asarray(a))
dtype = lax.dtype(a)
a_shape = jnp.shape(a)
if len(a_shape) < 2 or a_shape[-1] != a_shape[-2]:
msg = "Argument to slogdet() must have shape [..., n, n], got {}"
raise ValueError(msg.format(a_shape))
lu, pivot, _ = lax_linalg.lu(a)
diag = jnp.diagonal(lu, axis1=-2, axis2=-1)
is_zero = jnp.any(diag == jnp.array(0, dtype=dtype), axis=-1)
parity = jnp.count_nonzero(pivot != jnp.arange(a_shape[-1]), axis=-1)
if jnp.iscomplexobj(a):
sign = jnp.prod(diag / jnp.abs(diag), axis=-1)
else:
sign = jnp.array(1, dtype=dtype)
parity = parity + jnp.count_nonzero(diag < 0, axis=-1)
sign = jnp.where(is_zero,
jnp.array(0, dtype=dtype),
sign * jnp.array(-2 * (parity % 2) + 1, dtype=dtype))
logdet = jnp.where(
is_zero, jnp.array(-jnp.inf, dtype=dtype),
jnp.sum(jnp.log(jnp.abs(diag)), axis=-1))
return sign, jnp.real(logdet)
@slogdet.defjvp
def _slogdet_jvp(primals, tangents):
x, = primals
g, = tangents
sign, ans = slogdet(x)
ans_dot = jnp.trace(solve(x, g), axis1=-1, axis2=-2)
if jnp.issubdtype(jnp._dtype(x), jnp.complexfloating):
sign_dot = (ans_dot - np.real(ans_dot)) * sign
ans_dot = np.real(ans_dot)
else:
sign_dot = jnp.zeros_like(sign)
return (sign, ans), (sign_dot, ans_dot)
def _cofactor_solve(a, b):
"""Equivalent to det(a)*solve(a, b) for nonsingular mat.
Intermediate function used for jvp and vjp of det.
This function borrows heavily from jax.numpy.linalg.solve and
jax.numpy.linalg.slogdet to compute the gradient of the determinant
in a way that is well defined even for low rank matrices.
This function handles two different cases:
* rank(a) == n or n-1
* rank(a) < n-1
For rank n-1 matrices, the gradient of the determinant is a rank 1 matrix.
Rather than computing det(a)*solve(a, b), which would return NaN, we work
directly with the LU decomposition. If a = p @ l @ u, then
det(a)*solve(a, b) =
prod(diag(u)) * u^-1 @ l^-1 @ p^-1 b =
prod(diag(u)) * triangular_solve(u, solve(p @ l, b))
If a is rank n-1, then the lower right corner of u will be zero and the
triangular_solve will fail.
Let x = solve(p @ l, b) and y = det(a)*solve(a, b).
Then y_{n}
x_{n} / u_{nn} * prod_{i=1...n}(u_{ii}) =
x_{n} * prod_{i=1...n-1}(u_{ii})
So by replacing the lower-right corner of u with prod_{i=1...n-1}(u_{ii})^-1
we can avoid the triangular_solve failing.
To correctly compute the rest of y_{i} for i != n, we simply multiply
x_{i} by det(a) for all i != n, which will be zero if rank(a) = n-1.
For the second case, a check is done on the matrix to see if `solve`
returns NaN or Inf, and gives a matrix of zeros as a result, as the
gradient of the determinant of a matrix with rank less than n-1 is 0.
This will still return the correct value for rank n-1 matrices, as the check
is applied *after* the lower right corner of u has been updated.
Args:
a: A square matrix or batch of matrices, possibly singular.
b: A matrix, or batch of matrices of the same dimension as a.
Returns:
det(a) and cofactor(a)^T*b, aka adjugate(a)*b
"""
a = _promote_arg_dtypes(jnp.asarray(a))
b = _promote_arg_dtypes(jnp.asarray(b))
a_shape = jnp.shape(a)
b_shape = jnp.shape(b)
a_ndims = len(a_shape)
if not (a_ndims >= 2 and a_shape[-1] == a_shape[-2]
and b_shape[-2:] == a_shape[-2:]):
msg = ("The arguments to _cofactor_solve must have shapes "
"a=[..., m, m] and b=[..., m, m]; got a={} and b={}")
raise ValueError(msg.format(a_shape, b_shape))
if a_shape[-1] == 1:
return a[0, 0], b
# lu contains u in the upper triangular matrix and l in the strict lower
# triangular matrix.
# The diagonal of l is set to ones without loss of generality.
lu, pivots, permutation = lax_linalg.lu(a)
dtype = lax.dtype(a)
batch_dims = lax.broadcast_shapes(lu.shape[:-2], b.shape[:-2])
x = jnp.broadcast_to(b, batch_dims + b.shape[-2:])
lu = jnp.broadcast_to(lu, batch_dims + lu.shape[-2:])
# Compute (partial) determinant, ignoring last diagonal of LU
diag = jnp.diagonal(lu, axis1=-2, axis2=-1)
parity = jnp.count_nonzero(pivots != jnp.arange(a_shape[-1]), axis=-1)
sign = jnp.asarray(-2 * (parity % 2) + 1, dtype=dtype)
# partial_det[:, -1] contains the full determinant and
# partial_det[:, -2] contains det(u) / u_{nn}.
partial_det = jnp.cumprod(diag, axis=-1) * sign[..., None]
lu = ops.index_update(lu, ops.index[..., -1, -1], 1.0 / partial_det[..., -2])
permutation = jnp.broadcast_to(permutation, batch_dims + (a_shape[-1],))
iotas = jnp.ix_(*(lax.iota(jnp.int32, b) for b in batch_dims + (1,)))
# filter out any matrices that are not full rank
d = jnp.ones(x.shape[:-1], x.dtype)
d = lax_linalg.triangular_solve(lu, d, left_side=True, lower=False)
d = jnp.any(jnp.logical_or(jnp.isnan(d), jnp.isinf(d)), axis=-1)
d = jnp.tile(d[..., None, None], d.ndim*(1,) + x.shape[-2:])
x = jnp.where(d, jnp.zeros_like(x), x) # first filter
x = x[iotas[:-1] + (permutation, slice(None))]
x = lax_linalg.triangular_solve(lu, x, left_side=True, lower=True,
unit_diagonal=True)
x = jnp.concatenate((x[..., :-1, :] * partial_det[..., -1, None, None],
x[..., -1:, :]), axis=-2)
x = lax_linalg.triangular_solve(lu, x, left_side=True, lower=False)
x = jnp.where(d, jnp.zeros_like(x), x) # second filter
return partial_det[..., -1], x
@custom_jvp
@_wraps(np.linalg.det)
def det(a):
sign, logdet = slogdet(a)
return sign * jnp.exp(logdet)
@det.defjvp
def _det_jvp(primals, tangents):
x, = primals
g, = tangents
y, z = _cofactor_solve(x, g)
return y, jnp.trace(z, axis1=-1, axis2=-2)
[docs]@_wraps(np.linalg.eig)
def eig(a):
a = _promote_arg_dtypes(jnp.asarray(a))
return lax_linalg.eig(a, compute_left_eigenvectors=False)
[docs]@_wraps(np.linalg.eigvals)
def eigvals(a):
return lax_linalg.eig(a, compute_left_eigenvectors=False,
compute_right_eigenvectors=False)[0]
@_wraps(np.linalg.eigh)
def eigh(a, UPLO=None, symmetrize_input=True):
if UPLO is None or UPLO == "L":
lower = True
elif UPLO == "U":
lower = False
else:
msg = "UPLO must be one of None, 'L', or 'U', got {}".format(UPLO)
raise ValueError(msg)
a = _promote_arg_dtypes(jnp.asarray(a))
v, w = lax_linalg.eigh(a, lower=lower, symmetrize_input=symmetrize_input)
return w, v
[docs]@_wraps(np.linalg.eigvalsh)
def eigvalsh(a, UPLO='L'):
w, _ = eigh(a, UPLO)
return w
[docs]@partial(custom_jvp, nondiff_argnums=(1,))
@_wraps(np.linalg.pinv, lax_description=textwrap.dedent("""\
It differs only in default value of `rcond`. In `numpy.linalg.pinv`, the
default `rcond` is `1e-15`. Here the default is
`10. * max(num_rows, num_cols) * jnp.finfo(dtype).eps`.
"""))
def pinv(a, rcond=None):
# Uses same algorithm as
# https://github.com/numpy/numpy/blob/v1.17.0/numpy/linalg/linalg.py#L1890-L1979
a = jnp.conj(a)
if rcond is None:
max_rows_cols = max(a.shape[-2:])
rcond = 10. * max_rows_cols * jnp.finfo(a.dtype).eps
rcond = jnp.asarray(rcond)
u, s, v = svd(a, full_matrices=False)
# Singular values less than or equal to ``rcond * largest_singular_value``
# are set to zero.
cutoff = rcond[..., jnp.newaxis] * jnp.amax(s, axis=-1, keepdims=True)
s = jnp.where(s > cutoff, s, jnp.inf)
res = jnp.matmul(_T(v), jnp.divide(_T(u), s[..., jnp.newaxis]))
return lax.convert_element_type(res, a.dtype)
@pinv.defjvp
def _pinv_jvp(rcond, primals, tangents):
# The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems
# Whose Variables Separate. Author(s): G. H. Golub and V. Pereyra. SIAM
# Journal on Numerical Analysis, Vol. 10, No. 2 (Apr., 1973), pp. 413-432.
# (via https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse#Derivative)
a, = primals
a_dot, = tangents
p = pinv(a, rcond=rcond)
m, n = a.shape[-2:]
# TODO(phawkins): on TPU, we would need to opt into high precision here.
# TODO(phawkins): consider if this can be simplified in the Hermitian case.
p_dot = -p @ a_dot @ p
p_dot = p_dot + p @ _H(p) @ _H(a_dot) @ (jnp.eye(m, dtype=a.dtype) - a @ p)
p_dot = p_dot + (jnp.eye(n, dtype=a.dtype) - p @ a) @ _H(a_dot) @ _H(p) @ p
return p, p_dot
@_wraps(np.linalg.inv)
def inv(a):
if jnp.ndim(a) < 2 or a.shape[-1] != a.shape[-2]:
raise ValueError("Argument to inv must have shape [..., n, n], got {}."
.format(jnp.shape(a)))
return solve(
a, lax.broadcast(jnp.eye(a.shape[-1], dtype=lax.dtype(a)), a.shape[:-2]))
@partial(jit, static_argnums=(1, 2, 3))
def _norm(x, ord, axis: Union[None, Tuple[int, ...], int], keepdims):
x = _promote_arg_dtypes(jnp.asarray(x))
x_shape = jnp.shape(x)
ndim = len(x_shape)
if axis is None:
# NumPy has an undocumented behavior that admits arbitrary rank inputs if
# `ord` is None: https://github.com/numpy/numpy/issues/14215
if ord is None:
return jnp.sqrt(jnp.sum(jnp.real(x * jnp.conj(x)), keepdims=keepdims))
axis = tuple(range(ndim))
elif isinstance(axis, tuple):
axis = tuple(canonicalize_axis(x, ndim) for x in axis)
else:
axis = (canonicalize_axis(axis, ndim),)
num_axes = len(axis)
if num_axes == 1:
if ord is None or ord == 2:
return jnp.sqrt(jnp.sum(jnp.real(x * jnp.conj(x)), axis=axis,
keepdims=keepdims))
elif ord == jnp.inf:
return jnp.amax(jnp.abs(x), axis=axis, keepdims=keepdims)
elif ord == -jnp.inf:
return jnp.amin(jnp.abs(x), axis=axis, keepdims=keepdims)
elif ord == 0:
return jnp.sum(x != 0, dtype=jnp.finfo(lax.dtype(x)).dtype,
axis=axis, keepdims=keepdims)
elif ord == 1:
# Numpy has a special case for ord == 1 as an optimization. We don't
# really need the optimization (XLA could do it for us), but the Numpy
# code has slightly different type promotion semantics, so we need a
# special case too.
return jnp.sum(jnp.abs(x), axis=axis, keepdims=keepdims)
else:
abs_x = jnp.abs(x)
ord = lax._const(abs_x, ord)
out = jnp.sum(abs_x ** ord, axis=axis, keepdims=keepdims)
return jnp.power(out, 1. / ord)
elif num_axes == 2:
row_axis, col_axis = cast(Tuple[int, ...], axis)
if ord is None or ord in ('f', 'fro'):
return jnp.sqrt(jnp.sum(jnp.real(x * jnp.conj(x)), axis=axis,
keepdims=keepdims))
elif ord == 1:
if not keepdims and col_axis > row_axis:
col_axis -= 1
return jnp.amax(jnp.sum(jnp.abs(x), axis=row_axis, keepdims=keepdims),
axis=col_axis, keepdims=keepdims)
elif ord == -1:
if not keepdims and col_axis > row_axis:
col_axis -= 1
return jnp.amin(jnp.sum(jnp.abs(x), axis=row_axis, keepdims=keepdims),
axis=col_axis, keepdims=keepdims)
elif ord == jnp.inf:
if not keepdims and row_axis > col_axis:
row_axis -= 1
return jnp.amax(jnp.sum(jnp.abs(x), axis=col_axis, keepdims=keepdims),
axis=row_axis, keepdims=keepdims)
elif ord == -jnp.inf:
if not keepdims and row_axis > col_axis:
row_axis -= 1
return jnp.amin(jnp.sum(jnp.abs(x), axis=col_axis, keepdims=keepdims),
axis=row_axis, keepdims=keepdims)
elif ord in ('nuc', 2, -2):
x = jnp.moveaxis(x, axis, (-2, -1))
if ord == 2:
reducer = jnp.amax
elif ord == -2:
reducer = jnp.amin
else:
reducer = jnp.sum
y = reducer(svd(x, compute_uv=False), axis=-1)
if keepdims:
result_shape = list(x_shape)
result_shape[axis[0]] = 1
result_shape[axis[1]] = 1
y = jnp.reshape(y, result_shape)
return y
else:
raise ValueError("Invalid order '{}' for matrix norm.".format(ord))
else:
raise ValueError(
"Invalid axis values ({}) for jnp.linalg.norm.".format(axis))
@_wraps(np.linalg.norm)
def norm(x, ord=None, axis=None, keepdims=False):
return _norm(x, ord, axis, keepdims)
[docs]@_wraps(np.linalg.qr)
def qr(a, mode="reduced"):
if mode in ("reduced", "r", "full"):
full_matrices = False
elif mode == "complete":
full_matrices = True
else:
raise ValueError("Unsupported QR decomposition mode '{}'".format(mode))
a = _promote_arg_dtypes(jnp.asarray(a))
q, r = lax_linalg.qr(a, full_matrices)
if mode == "r":
return r
return q, r
[docs]@_wraps(np.linalg.solve)
@jit
def solve(a, b):
a, b = _promote_arg_dtypes(jnp.asarray(a), jnp.asarray(b))
return lax_linalg._solve(a, b)
[docs]@_wraps(np.linalg.lstsq, lax_description=textwrap.dedent("""\
It has two important differences:
1. In `numpy.linalg.lstsq`, the default `rcond` is `-1`, and warns that in the future
the default will be `None`. Here, the default rcond is `None`.
2. In `np.linalg.lstsq` the returned residuals are empty for low-rank or over-determined
solutions. Here, the residuals are returned in all cases, to make the function
compatible with jit. The non-jit compatible numpy behavior can be recovered by
passing numpy_resid=True.
The lstsq function does not currently have a custom JVP rule, so the gradient is
poorly behaved for some inputs, particularly for low-rank `a`.
"""))
def lstsq(a, b, rcond=None, *, numpy_resid=False):
# TODO: add lstsq to lax_linalg and implement this function via those wrappers.
# TODO: add custom jvp rule for more robust lstsq differentiation
a, b = _promote_arg_dtypes(a, b)
if a.shape[0] != b.shape[0]:
raise ValueError("Leading dimensions of input arrays must match")
b_orig_ndim = b.ndim
if b_orig_ndim == 1:
b = b[:, None]
if a.ndim != 2:
raise TypeError(
f"{a.ndim}-dimensional array given. Array must be two-dimensional")
if b.ndim != 2:
raise TypeError(
f"{b.ndim}-dimensional array given. Array must be one or two-dimensional")
m, n = a.shape
dtype = a.dtype
if rcond is None:
rcond = jnp.finfo(dtype).eps * max(n, m)
elif rcond < 0:
rcond = jnp.finfo(dtype).eps
u, s, vt = svd(a, full_matrices=False)
mask = s >= rcond * s[0]
rank = mask.sum()
safe_s = jnp.where(mask, s, 1)
s_inv = jnp.where(mask, 1 / safe_s, 0)[:, jnp.newaxis]
uTb = jnp.matmul(u.conj().T, b, precision=lax.Precision.HIGHEST)
x = jnp.matmul(vt.conj().T, s_inv * uTb, precision=lax.Precision.HIGHEST)
# Numpy returns empty residuals in some cases. To allow compilation, we
# default to returning full residuals in all cases.
if numpy_resid and (rank < n or m <= n):
resid = jnp.asarray([])
else:
b_estimate = jnp.matmul(a, x, precision=lax.Precision.HIGHEST)
resid = norm(b - b_estimate, axis=0) ** 2
if b_orig_ndim == 1:
x = x.ravel()
return x, resid, rank, s