def _lu_jvp_rule(primals, tangents):
a, = primals
a_dot, = tangents
lu, pivots = lu_p.bind(a)
if a_dot is ad_util.zero:
return (core.pack(
(lu, pivots)), ad.TangentTuple((ad_util.zero, ad_util.zero)))
a_shape = np.shape(a)
m, n = a_shape[-2:]
dtype = lax.dtype(a)
k = min(m, n)
permutation = lu_pivots_to_permutation(pivots, m)
batch_dims = a_shape[:-2]
iotas = np.ix_(*(lax.iota(np.int32, b) for b in batch_dims + (1, )))
x = a_dot[iotas[:-1] + (permutation, slice(None))]
# Differentiation of Matrix Functionals Using Triangular Factorization
# F. R. De Hoog, R. S. Anderssen, and M. A. Lukas
#
# LU = A
# ==> L'U + LU' = A'
# ==> inv(L) . L' + U' . inv(U) = inv(L) A' inv(U)
# ==> L' = L . tril(inv(L) . A' . inv(U), -1)
# U' = triu(inv(L) . A' . inv(U)) . U
ndims = len(a_shape)
l_padding = [(0, 0, 0)] * ndims
l_padding[-1] = (0, m - k, 0)
zero = np._constant_like(lu, 0)
l = lax.pad(np.tril(lu[..., :, :k], -1), zero, l_padding)
l = l + np.eye(m, m, dtype=dtype)
u_eye = lax.pad(np.eye(n - k, n - k, dtype=dtype), zero,
((k, 0, 0), (k, 0, 0)))
u_padding = [(0, 0, 0)] * ndims
u_padding[-2] = (0, n - k, 0)
u = lax.pad(np.triu(lu[..., :k, :]), zero, u_padding) + u_eye
la = triangular_solve(l,
x,
left_side=True,
transpose_a=False,
lower=True,
unit_diagonal=True)
lau = triangular_solve(u,
la,
left_side=False,
transpose_a=False,
lower=False)
l_dot = np.matmul(l, np.tril(lau, -1))
u_dot = np.matmul(np.triu(lau), u)
lu_dot = l_dot + u_dot
return (lu, pivots), (lu_dot, ad_util.zero)
def lu_jvp_rule(primals, tangents):
a, = primals
a_dot, = tangents
lu, pivots = lu_p.bind(a)
a_shape = np.shape(a)
m, n = a_shape[-2:]
dtype = lax._dtype(a)
k = min(m, n)
# TODO(phawkins): use a gather rather than a matrix multiplication here.
permutation = lu_pivots_to_permutation(pivots, m)
p = np.array(permutation[:, None] == np.arange(m), dtype=dtype)
x = np.matmul(p, a_dot)
# Differentiation of Matrix Functionals Using Triangular Factorization
# F. R. De Hoog, R. S. Anderssen, and M. A. Lukas
#
# LU = A
# ==> L'U + LU' = A'
# ==> inv(L) . L' + U' . inv(U) = inv(L) A' inv(U)
# ==> L' = L . tril(inv(L) . A' . inv(U), -1)
# U' = triu(inv(L) . A' . inv(U)) . U
ndims = len(a_shape)
l_padding = [(0, 0, 0)] * ndims
l_padding[-1] = (0, m - k, 0)
zero = np._constant_like(lu, 0)
l = lax.pad(np.tril(lu[..., :, :k], -1), zero, l_padding)
l = l + np.eye(m, m, dtype=dtype)
u_eye = lax.pad(np.eye(n - k, n - k, dtype=dtype), zero,
((k, 0, 0), (k, 0, 0)))
u_padding = [(0, 0, 0)] * ndims
u_padding[-2] = (0, n - k, 0)
u = lax.pad(np.triu(lu[..., :k, :]), zero, u_padding) + u_eye
la = triangular_solve(l, x, left_side=True, transpose_a=False, lower=True)
lau = triangular_solve(u,
la,
left_side=False,
transpose_a=False,
lower=False)
l_dot = np.matmul(l, np.tril(lau, -1))
u_dot = np.matmul(np.triu(lau), u)
lu_dot = l_dot + u_dot
return core.pack((lu, pivots)), ad.TangentTuple((lu_dot, ad_util.zero))
def triangular_solve_jvp_rule_a(g_a, ans, a, b, left_side, lower, transpose_a,
conjugate_a, unit_diagonal):
m, n = b.shape[-2:]
k = 1 if unit_diagonal else 0
g_a = np.tril(g_a, k=-k) if lower else np.triu(g_a, k=k)
g_a = lax.neg(g_a)
g_a = np.swapaxes(g_a, -1, -2) if transpose_a else g_a
g_a = np.conj(g_a) if conjugate_a else g_a
dot = partial(lax.dot if g_a.ndim == 2 else lax.batch_matmul,
precision=lax.Precision.HIGHEST)
def a_inverse(rhs):
return triangular_solve(a, rhs, left_side, lower, transpose_a,
conjugate_a, unit_diagonal)
# triangular_solve is about the same cost as matrix multplication (~n^2 FLOPs
# for matrix/vector inputs). Order these operations in whichever order is
# cheaper.
if left_side:
assert g_a.shape[-2:] == a.shape[-2:] == (m, m) and ans.shape[-2:] == (
m, n)
if m > n:
return a_inverse(dot(g_a, ans)) # A^{-1} (∂A X)
else:
return dot(a_inverse(g_a), ans) # (A^{-1} ∂A) X
else:
assert g_a.shape[-2:] == a.shape[-2:] == (n, n) and ans.shape[-2:] == (
m, n)
if m < n:
return a_inverse(dot(ans, g_a)) # (X ∂A) A^{-1}
else:
return dot(ans, a_inverse(g_a)) # X (∂A A^{-1})
def triangular_solve_jvp_rule_a(g_a, ans, a, b, left_side, lower, transpose_a,
conjugate_a, unit_diagonal):
k = 1 if unit_diagonal else 0
g_a = np.tril(g_a, k=-k) if lower else np.triu(g_a, k=k)
g_a = lax.neg(g_a)
g_a = np.swapaxes(g_a, -1, -2) if transpose_a else g_a
g_a = np.conj(g_a) if conjugate_a else g_a
tmp = triangular_solve(a, g_a, left_side, lower, transpose_a, conjugate_a,
unit_diagonal)
dot = lax.dot if g_a.ndim == 2 else lax.batch_matmul
if left_side:
return dot(tmp, ans)
else:
return dot(ans, tmp)
def qr(x, full_matrices=True):
q, r = qr_p.bind(x, full_matrices=full_matrices)
return q, np.triu(r)
