def svd_jvp_rule(primals, tangents, full_matrices, compute_uv):
A, = primals
dA, = tangents
s, U, Vt = svd_p.bind(A, full_matrices=False, compute_uv=True)
if compute_uv and full_matrices:
# TODO: implement full matrices case, documented here: https://people.maths.ox.ac.uk/gilesm/files/NA-08-01.pdf
raise NotImplementedError(
"Singular value decomposition JVP not implemented for full matrices"
)
k = s.shape[-1]
Ut, V = _H(U), _H(Vt)
s_dim = s[..., None, :]
dS = np.matmul(np.matmul(Ut, dA), V)
ds = np.real(np.diagonal(dS, 0, -2, -1))
F = 1 / (np.square(s_dim) - np.square(_T(s_dim)) +
np.eye(k, dtype=A.dtype))
F = F - np.eye(k, dtype=A.dtype)
dSS = s_dim * dS
SdS = _T(s_dim) * dS
dU = np.matmul(U, F * (dSS + _T(dSS)))
dV = np.matmul(V, F * (SdS + _T(SdS)))
m, n = A.shape[-2:]
if m > n:
dU = dU + np.matmul(
np.eye(m, dtype=A.dtype) - np.matmul(U, Ut), np.matmul(dA,
V)) / s_dim
if n > m:
dV = dV + np.matmul(
np.eye(n, dtype=A.dtype) - np.matmul(V, Vt), np.matmul(_H(dA),
U)) / s_dim
return (s, U, Vt), (ds, dU, _T(dV))
def svd_jvp_rule(primals, tangents, full_matrices, compute_uv):
if full_matrices:
#TODO: implement full matrices case, documented here: https://people.maths.ox.ac.uk/gilesm/files/NA-08-01.pdf
raise NotImplementedError(
"Singular value decomposition JVP not implemented for full matrices"
)
A, = primals
dA, = tangents
s, U, Vt = svd_p.bind(A, full_matrices=False, compute_uv=True)
k = s.shape[-1]
Ut, V = np.conj(U).T, np.conj(Vt).T
s_dim = s[..., None, :]
dS = np.dot(np.dot(Ut, dA), V)
ds = np.real(np.diag(dS))
F = 1 / (np.square(s_dim) - np.square(s_dim.T) + np.eye(k)) - np.eye(k)
dSS = s_dim * dS
SdS = s_dim.T * dS
dU = np.dot(U, F * (dSS + dSS.T))
dV = np.dot(V, F * (SdS + SdS.T))
m, n = A.shape[-2], A.shape[-1]
if m > n:
dU = dU + np.dot(np.eye(m) - np.dot(U, Ut), np.dot(dA, V)) / s_dim
if n > m:
dV = dV + np.dot(np.eye(n) - np.dot(V, Vt), np.dot(np.conj(dA).T,
U)) / s_dim
return core.pack((s, U, Vt)), core.pack((ds, dU, dV.T))
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 eigh_jvp_rule(primals, tangents, lower):
# Derivative for eigh in the simplest case of distinct eigenvalues.
# This is classic nondegenerate perurbation theory, but also see
# https://people.maths.ox.ac.uk/gilesm/files/NA-08-01.pdf
# The general solution treating the case of degenerate eigenvalues is
# considerably more complicated. Ambitious readers may refer to the general
# methods below or refer to degenerate perturbation theory in physics.
# https://www.win.tue.nl/analysis/reports/rana06-33.pdf and
# https://people.orie.cornell.edu/aslewis/publications/99-clarke.pdf
a, = primals
a_dot, = tangents
v, w = eigh_p.bind(symmetrize(a), lower=lower)
# for complex numbers we need eigenvalues to be full dtype of v, a:
w = w.astype(a.dtype)
eye_n = np.eye(a.shape[-1], dtype=a.dtype)
# carefully build reciprocal delta-eigenvalue matrix, avoiding NaNs.
Fmat = np.reciprocal(eye_n + w[..., np.newaxis, :] -
w[..., np.newaxis]) - eye_n
# eigh impl doesn't support batch dims, but future-proof the grad.
dot = partial(lax.dot if a.ndim == 2 else lax.batch_matmul,
precision=lax.Precision.HIGHEST)
vdag_adot_v = dot(dot(_H(v), a_dot), v)
dv = dot(v, np.multiply(Fmat, vdag_adot_v))
dw = np.diagonal(vdag_adot_v, axis1=-2, axis2=-1)
return (v, w), (dv, dw)
def cholesky_jvp_rule(primals, tangents):
x, = primals
sigma_dot, = tangents
L = cholesky_p.bind(x)
# Forward-mode rule from https://arxiv.org/pdf/1602.07527.pdf
phi = lambda X: np.tril(X) / (1 + np.eye(X.shape[-1], dtype=X.dtype))
tmp = triangular_solve(L, sigma_dot,
left_side=False, transpose_a=True, lower=True)
L_dot = lax.batch_matmul(L, phi(triangular_solve(
L, tmp, left_side=True, transpose_a=False, lower=True)))
return L, L_dot
def qr_jvp_rule(primals, tangents, full_matrices):
# See j-towns.github.io/papers/qr-derivative.pdf for a terse derivation.
x, = primals
dx, = tangents
q, r = qr_p.bind(x, full_matrices=False)
*_, m, n = x.shape
if full_matrices or m < n:
raise NotImplementedError(
"Unimplemented case of QR decomposition derivative")
dx_rinv = triangular_solve(r, dx) # Right side solve by default
qt_dx_rinv = jnp.matmul(_H(q), dx_rinv)
qt_dx_rinv_lower = jnp.tril(qt_dx_rinv, -1)
do = qt_dx_rinv_lower - _H(qt_dx_rinv_lower) # This is skew-symmetric
# The following correction is necessary for complex inputs
do = do + jnp.eye(n, dtype=do.dtype) * (qt_dx_rinv - jnp.real(qt_dx_rinv))
dq = jnp.matmul(q, do - qt_dx_rinv) + dx_rinv
dr = jnp.matmul(qt_dx_rinv - do, r)
return (q, r), (dq, dr)
def phi(X):
l = np.tril(X)
return l / (np._constant_like(X, 1) +
np.eye(X.shape[-1], dtype=X.dtype))
