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 core.pack((lu, pivots)), ad.TangentTuple((lu_dot, ad_util.zero))
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 dA is ad_util.zero:
return (core.pack((s, U, Vt)),
ad.TangentTuple(ad_util.zero, ad_util.zero, ad_util.zero))
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")
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 (s, U, Vt), (ds, dU, dV.T)
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)
if a_dot is ad_util.zero:
return core.pack((v, w)), ad.TangentTuple(ad_util.zero, ad_util.zero)
# 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 - w[..., np.newaxis]) - eye_n
# eigh impl doesn't support batch dims, but future-proof the grad.
dot = lax.dot if a.ndim == 2 else lax.batch_matmul
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)
return (v, w), (dv, dw)
def _scan_jvp(primals, tangents, forward, length, jaxpr):
consts, init, xs = primals
consts_dot, init_dot, xs_dot = tangents
consts_aval, carry_aval, x_aval = jaxpr.in_avals
_, y_aval = jaxpr.out_aval
consts_nonzeros = ad.get_nonzeros(consts_dot)
init_nonzeros = ad.get_nonzeros(init_dot)
xs_nonzeros = ad.get_nonzeros(xs_dot) # same as x_nonzeros b/c arrays
carry_nonzeros = init_nonzeros
for _ in range(1000):
nonzeros = (consts_nonzeros, carry_nonzeros, xs_nonzeros)
jaxpr_jvp, nonzeros_out = ad.jvp_jaxpr(jaxpr,
nonzeros,
instantiate=(carry_nonzeros,
False))
carry_nonzeros_out, ys_nonzeros = nonzeros_out
if carry_nonzeros_out == carry_nonzeros:
break
else:
carry_nonzeros = _binary_lattice_join(carry_nonzeros_out,
carry_nonzeros)
else:
raise FixedPointError
# convert_zeros is like strip_zeros but uses explicit lattice information to
# instantiate zeros in some cases, namely in init_dot based on the fixed point
nonzero_init_dot = _convert_zeros(carry_nonzeros, init, init_dot)
nonzero_consts_dot = _convert_zeros(consts_nonzeros, consts, consts_dot)
nonzero_xs_dot = _convert_zeros(xs_nonzeros, xs, xs_dot)
consts_dual = core.pack((consts, nonzero_consts_dot))
init_dual = core.pack((init, nonzero_init_dot))
xs_dual = core.pack((xs, nonzero_xs_dot))
carry_out_dual, ys_dual = scan_p.bind(consts_dual,
init_dual,
xs_dual,
forward=forward,
length=length,
jaxpr=jaxpr_jvp)
ys, ys_dot = ys_dual
ys_dot = ad.put_zeros(ad.TangentTuple, ys_nonzeros, ys_dot)
carry_out, carry_out_dot = carry_out_dual
carry_out_dot = ad.put_zeros(ad.TangentTuple, carry_nonzeros_out,
carry_out_dot)
return core.pack((carry_out, ys)), ad.TangentTuple((carry_out_dot, ys_dot))
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 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)
if dx is ad_util.zero:
return core.pack((q, r)), ad.TangentTuple(ad_util.zero, ad_util.zero)
if full_matrices or np.shape(x)[-2] < np.shape(x)[-1]:
raise NotImplementedError
dx_rinv = triangular_solve(r, dx) # Right side solve by default
qt_dx_rinv = np.matmul(_T(q), dx_rinv)
qt_dx_rinv_lower = np.tril(qt_dx_rinv, -1)
domega = qt_dx_rinv_lower - _T(qt_dx_rinv_lower) # This is skew-symmetric
dq = np.matmul(q, domega - qt_dx_rinv) + dx_rinv
dr = np.matmul(qt_dx_rinv - domega, r)
return (q, r), (dq, dr)
