def _use_cholesky(u, params):
"""Uses Cholesky decomposition."""
a, b, c = params
_, n = u.shape
x = c * u.T.conj() @ u + jnp.eye(n)
# `y` is lower triangular.
y = lax_linalg.cholesky(x, symmetrize_input=False)
z = lax_linalg.triangular_solve(y,
u.T,
left_side=True,
lower=True,
conjugate_a=True).conj()
z = lax_linalg.triangular_solve(y,
z,
left_side=True,
lower=True,
transpose_a=True,
conjugate_a=True).T.conj()
e = b / c
u = e * u + (a - e) * z
return u
def _use_cholesky(u, m, n, params):
"""QDWH iteration using Cholesky decomposition.
Args:
u: a matrix, with static (padded) shape M x N
m, n: the dynamic shape of the matrix, where m <= M and n <= N.
params: the QDWH parameters.
"""
a, b, c = params
_, N = u.shape
x = c * (u.T.conj() @ u) + jnp.eye(N, dtype=jnp.dtype(u))
# Pads the lower-right corner with the identity matrix to prevent the Cholesky
# decomposition from failing due to the matrix not being PSD if padded with
# zeros.
x = _mask(x, (n, n), jnp.eye(N, dtype=x.dtype))
# `y` is lower triangular.
y = lax_linalg.cholesky(x, symmetrize_input=False)
z = lax_linalg.triangular_solve(y,
u.T,
left_side=True,
lower=True,
conjugate_a=True).conj()
z = lax_linalg.triangular_solve(y,
z,
left_side=True,
lower=True,
transpose_a=True,
conjugate_a=True).T.conj()
e = b / c
u = e * u + (a - e) * z
return u
def _cho_solve(c, b, lower):
c, b = np_linalg._promote_arg_dtypes(jnp.asarray(c), jnp.asarray(b))
lax_linalg._check_solve_shapes(c, b)
b = lax_linalg.triangular_solve(c, b, left_side=True, lower=lower,
transpose_a=not lower, conjugate_a=not lower)
b = lax_linalg.triangular_solve(c, b, left_side=True, lower=lower,
transpose_a=lower, conjugate_a=lower)
return b
def _solve_triangular(a, b, trans, lower, unit_diagonal):
if trans == 0 or trans == "N":
transpose_a, conjugate_a = False, False
elif trans == 1 or trans == "T":
transpose_a, conjugate_a = True, False
elif trans == 2 or trans == "C":
transpose_a, conjugate_a = True, True
else:
raise ValueError(f"Invalid 'trans' value {trans}")
a, b = _promote_dtypes_inexact(jnp.asarray(a), jnp.asarray(b))
# lax_linalg.triangular_solve only supports matrix 'b's at the moment.
b_is_vector = jnp.ndim(a) == jnp.ndim(b) + 1
if b_is_vector:
b = b[..., None]
out = lax_linalg.triangular_solve(a,
b,
left_side=True,
lower=lower,
transpose_a=transpose_a,
conjugate_a=conjugate_a,
unit_diagonal=unit_diagonal)
if b_is_vector:
return out[..., 0]
else:
return out
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 = lu.at[..., -1, -1].set(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
