def ent_reg_cost(geom: geometry.Geometry, a: jnp.ndarray, b: jnp.ndarray,
tau_a: float, tau_b: float, f: jnp.ndarray,
g: jnp.ndarray) -> jnp.ndarray:
"""Computes objective of regularized OT given dual solutions f,g."""
# In all sums below, jnp.where handle situations in which some coordinates of
# a and b are zero. For those coordinates, their potential is -inf.
# This leads to -inf - -inf or -inf x 0 operations which result in NaN.
# These contributions are discarded when computing the objective.
if tau_a == 1.0:
div_a = jnp.sum(
jnp.where(a > 0, (f - geom.potential_from_scaling(a)) * a, 0))
else:
rho_a = geom.epsilon * (tau_a / (1 - tau_a))
div_a = jnp.sum(
jnp.where(
a > 0,
a * (rho_a - (rho_a + geom.epsilon / 2) *
jnp.exp(-(f - geom.potential_from_scaling(a)) / rho_a)),
0))
if tau_b == 1.0:
div_b = jnp.sum(
jnp.where(b > 0, (g - geom.potential_from_scaling(b)) * b, 0))
else:
rho_b = geom.epsilon * (tau_b / (1 - tau_b))
div_b = jnp.sum(
jnp.where(
b > 0,
b * (rho_b - (rho_b + geom.epsilon / 2) *
jnp.exp(-(g - geom.potential_from_scaling(b)) / rho_b)),
0))
# Using https://arxiv.org/pdf/1910.12958.pdf Eq. 30
return div_a + div_b + geom.epsilon * jnp.sum(a) * jnp.sum(b)
def marginal_error(geom: geometry.Geometry, a: jnp.ndarray, b: jnp.ndarray,
tau_a: float, tau_b: float, f_u: jnp.ndarray,
g_v: jnp.ndarray, norm_error: int, lse_mode) -> jnp.ndarray:
"""Conputes marginal error, the stopping criterion used to terminate Sinkhorn.
Args:
geom: a Geometry object.
a: jnp.ndarray<float>[num_a,] or jnp.ndarray<float>[batch,num_a] weights.
b: jnp.ndarray<float>[num_b,] or jnp.ndarray<float>[batch,num_b] weights.
tau_a: float, ratio lam/(lam+eps) between KL divergence regularizer to first
marginal and itself + epsilon regularizer used in the unbalanced
formulation.
tau_b: float, ratio lam/(lam+eps) between KL divergence regularizer to first
marginal and itself + epsilon regularizer used in the unbalanced
formulation.
f_u: jnp.ndarray, potential or scaling
g_v: jnp.ndarray, potential or scaling
norm_error: int, p-norm used to compute error.
lse_mode: True if log-sum-exp operations, False if kernel vector producs.
Returns:
a positive number quantifying how far from convergence the algorithm stands.
"""
if tau_a == 1.0 and tau_b == 1.0:
err = geom.error(f_u, g_v, b, 0, norm_error, lse_mode)
else:
# In the unbalanced case, we compute the norm of the gradient.
# the gradient is equal to the marginal of the current plan minus
# the gradient of < z, rho_z(exp^(-h/rho_z) -1> where z is either a or b
# and h is either f or g. Note this is equal to z if rho_z → inf, which
# is the case when tau_z → 1.0
if lse_mode:
grad_a = grad_of_marginal_fit(a, f_u, tau_a, geom.epsilon)
grad_b = grad_of_marginal_fit(b, g_v, tau_b, geom.epsilon)
else:
grad_a = grad_of_marginal_fit(a, geom.potential_from_scaling(f_u),
tau_a, geom.epsilon)
grad_b = grad_of_marginal_fit(b, geom.potential_from_scaling(g_v),
tau_b, geom.epsilon)
err = geom.error(f_u, g_v, grad_a, 1, norm_error, lse_mode)
err += geom.error(f_u, g_v, grad_b, 0, norm_error, lse_mode)
return err
def ent_reg_cost(geom: geometry.Geometry,
a: jnp.ndarray,
b: jnp.ndarray,
tau_a: float,
tau_b: float,
f: jnp.ndarray,
g: jnp.ndarray,
lse_mode: bool) -> jnp.ndarray:
r"""Computes objective of regularized OT given dual solutions ``f``, ``g``.
The objective is evaluated for dual solution ``f`` and ``g``, using inputs
``geom``, ``a`` and ``b``, in addition to parameters ``tau_a``, ``tau_b``.
Situations where ``a`` or ``b`` have zero coordinates are reflected in
minus infinity entries in their corresponding dual potentials. To avoid NaN
that may result when multiplying 0's by infinity values, ``jnp.where`` is
used to cancel these contributions.
Args:
geom: a Geometry object.
a: jnp.ndarray<float>[num_a,] or jnp.ndarray<float>[batch,num_a] weights.
b: jnp.ndarray<float>[num_b,] or jnp.ndarray<float>[batch,num_b] weights.
tau_a: float, ratio lam/(lam+eps) between KL divergence regularizer to first
marginal and itself + epsilon regularizer used in the unbalanced
formulation.
tau_b: float, ratio lam/(lam+eps) between KL divergence regularizer to first
marginal and itself + epsilon regularizer used in the unbalanced
formulation.
f: jnp.ndarray, potential
g: jnp.ndarray, potential
lse_mode: bool, whether to compute total mass in lse or kernel mode.
Returns:
a float, the regularized transport cost.
"""
supp_a = a > 0
supp_b = b > 0
if tau_a == 1.0:
div_a = jnp.sum(
jnp.where(supp_a, a * (f - geom.potential_from_scaling(a)), 0.0))
else:
rho_a = geom.epsilon * (tau_a / (1 - tau_a))
div_a = - jnp.sum(jnp.where(
supp_a,
a * phi_star(-(f - geom.potential_from_scaling(a)), rho_a),
0.0))
if tau_b == 1.0:
div_b = jnp.sum(
jnp.where(supp_b, b * (g - geom.potential_from_scaling(b)), 0.0))
else:
rho_b = geom.epsilon * (tau_b / (1 - tau_b))
div_b = - jnp.sum(jnp.where(
supp_b,
b * phi_star(-(g - geom.potential_from_scaling(b)), rho_b),
0.0))
# Using https://arxiv.org/pdf/1910.12958.pdf (24)
if lse_mode:
total_sum = jnp.sum(geom.marginal_from_potentials(f, g))
else:
total_sum = jnp.sum(geom.marginal_from_scalings(
geom.scaling_from_potential(f), geom.scaling_from_potential(g)))
return div_a + div_b + geom.epsilon * (jnp.sum(a) * jnp.sum(b) - total_sum)
def _sinkhorn_iterations(
tau_a: float,
tau_b: float,
inner_iterations: int,
min_iterations: int,
max_iterations: int,
momentum_default: float,
chg_momentum_from: int,
lse_mode: bool,
implicit_differentiation: bool,
linear_solve_kwargs: Mapping[str, Union[Callable, float]],
parallel_dual_updates: bool,
init_dual_a: jnp.ndarray,
init_dual_b: jnp.ndarray,
threshold: float,
norm_error: Sequence[int],
geom: geometry.Geometry,
a: jnp.ndarray,
b: jnp.ndarray) -> Tuple[jnp.ndarray, jnp.ndarray, jnp.ndarray]:
"""The jittable Sinkhorn loop, that uses a custom backward or not.
Args:
tau_a: float, ratio lam/(lam+eps) between KL divergence regularizer to first
marginal and itself + epsilon regularizer used in the unbalanced
formulation.
tau_b: float, ratio lam/(lam+eps) between KL divergence regularizer to first
marginal and itself + epsilon regularizer used in the unbalanced
formulation.
inner_iterations: (int32) the Sinkhorn error is not recomputed at each
iteration but every inner_num_iter instead.
min_iterations: (int32) the minimum number of Sinkhorn iterations.
max_iterations: (int32) the maximum number of Sinkhorn iterations.
momentum_default: float, a float between ]0,2[
chg_momentum_from: int, # of iterations after which momentum is computed
lse_mode: True for log-sum-exp computations, False for kernel
multiplication.
implicit_differentiation: if True, do not backprop through the Sinkhorn
loop, but use the implicit function theorem on the fixed point optimality
conditions.
linear_solve_kwargs: parameterization of linear solver when using implicit
differentiation.
parallel_dual_updates: updates potentials or scalings in parallel if True,
sequentially (in Gauss-Seidel fashion) if False.
init_dual_a: optional initialization for potentials/scalings w.r.t.
first marginal (``a``) of reg-OT problem.
init_dual_b: optional initialization for potentials/scalings w.r.t.
second marginal (``b``) of reg-OT problem.
threshold: (float) the relative threshold on the Sinkhorn error to stop the
Sinkhorn iterations.
norm_error: t-uple of int, p-norms of marginal / target errors to track
geom: a Geometry object.
a: jnp.ndarray<float>[num_a,] or jnp.ndarray<float>[batch,num_a] weights.
b: jnp.ndarray<float>[num_b,] or jnp.ndarray<float>[batch,num_b] weights.
Returns:
f: potential
g: potential
errors: ndarray of errors
"""
# Initializing solutions
f_u, g_v = init_dual_a, init_dual_b
# Delete arguments not used in forward pass.
del linear_solve_kwargs
# Defining the Sinkhorn loop, by setting initializations, body/cond.
errors = -jnp.ones((np.ceil(max_iterations / inner_iterations).astype(int),
len(norm_error)))
const = (geom, a, b, threshold)
def cond_fn(iteration, const, state):
threshold = const[-1]
errors = state[0]
err = errors[iteration // inner_iterations-1, 0]
return jnp.logical_or(iteration == 0,
jnp.logical_and(jnp.isfinite(err), err > threshold))
def get_momentum(errors, idx):
"""momentum formula, https://arxiv.org/pdf/2012.12562v1.pdf, p.7 and (5)."""
error_ratio = jnp.minimum(errors[idx - 1, -1] / errors[idx - 2, -1], .99)
power = 1.0 / inner_iterations
return 2.0 / (1.0 + jnp.sqrt(1.0 - error_ratio ** power))
def body_fn(iteration, const, state, compute_error):
"""Carries out sinkhorn iteration.
Depending on lse_mode, these iterations can be either in:
- log-space for numerical stability.
- scaling space, using standard kernel-vector multiply operations.
Args:
iteration: iteration number
const: tuple of constant parameters that do not change throughout the
loop, here the geometry and the marginals a, b.
state: potential/scaling variables updated in the loop & error log.
compute_error: flag to indicate this iteration computes/stores an error
Returns:
state variables, i.e. errors and updated f_u, g_v potentials.
"""
geom, a, b, _ = const
errors, f_u, g_v = state
# compute momentum term if needed, using previously seen errors.
w = jax.lax.stop_gradient(jnp.where(iteration >= (
inner_iterations * chg_momentum_from + min_iterations),
get_momentum(errors, chg_momentum_from),
momentum_default))
# Sinkhorn updates using momentum, in either scaling or potential form.
if parallel_dual_updates:
old_g_v = g_v
if lse_mode:
new_g_v = tau_b * geom.update_potential(f_u, g_v, jnp.log(b),
iteration, axis=0)
g_v = (1.0 - w) * jnp.where(jnp.isfinite(g_v), g_v, 0.0) + w * new_g_v
new_f_u = tau_a * geom.update_potential(
f_u, old_g_v if parallel_dual_updates else g_v,
jnp.log(a), iteration, axis=1)
f_u = (1.0 - w) * jnp.where(jnp.isfinite(f_u), f_u, 0.0) + w * new_f_u
else:
new_g_v = geom.update_scaling(f_u, b, iteration, axis=0) ** tau_b
g_v = jnp.where(g_v > 0, g_v, 1) ** (1.0 - w) * new_g_v ** w
new_f_u = geom.update_scaling(
old_g_v if parallel_dual_updates else g_v,
a, iteration, axis=1) ** tau_a
f_u = jnp.where(f_u > 0, f_u, 1) ** (1.0 - w) * new_f_u ** w
# re-computes error if compute_error is True, else set it to inf.
err = jnp.where(
jnp.logical_and(compute_error, iteration >= min_iterations),
marginal_error(geom, a, b, tau_a, tau_b, f_u, g_v, norm_error,
lse_mode),
jnp.inf)
errors = jax.ops.index_update(
errors, jax.ops.index[iteration // inner_iterations, :], err)
return errors, f_u, g_v
# Run the Sinkhorn loop. choose either a standard fixpoint_iter loop if
# differentiation is implicit, otherwise switch to the backprop friendly
# version of that loop if unrolling to differentiate.
if implicit_differentiation:
fix_point = fixed_point_loop.fixpoint_iter
else:
fix_point = fixed_point_loop.fixpoint_iter_backprop
errors, f_u, g_v = fix_point(
cond_fn, body_fn, min_iterations, max_iterations, inner_iterations, const,
(errors, f_u, g_v))
f = f_u if lse_mode else geom.potential_from_scaling(f_u)
g = g_v if lse_mode else geom.potential_from_scaling(g_v)
return f, g, errors[:, 0]
def ent_reg_cost(geom: geometry.Geometry, a: jnp.ndarray, b: jnp.ndarray,
tau_a: float, tau_b: float, f: jnp.ndarray,
g: jnp.ndarray) -> jnp.ndarray:
"""Computes objective of regularized OT given dual solutions f,g.
In all sums below, jnp.where handle situations in which some coordinates of
a and b are zero. For those coordinates, their potential is -inf.
This leads to -inf - -inf or -inf x 0 operations which result in NaN.
These contributions are discarded when computing the objective.
Args:
geom: a Geometry object.
a: jnp.ndarray<float>[num_a,] or jnp.ndarray<float>[batch,num_a] weights.
b: jnp.ndarray<float>[num_b,] or jnp.ndarray<float>[batch,num_b] weights.
tau_a: float, ratio lam/(lam+eps) between KL divergence regularizer to first
marginal and itself + epsilon regularizer used in the unbalanced
formulation.
tau_b: float, ratio lam/(lam+eps) between KL divergence regularizer to first
marginal and itself + epsilon regularizer used in the unbalanced
formulation.
f: jnp.ndarray, potential
g: jnp.ndarray, potential
Returns:
a float, the regularized transport cost.
"""
if tau_a == 1.0:
div_a = jnp.sum(
jnp.where(a > 0, (f - geom.potential_from_scaling(a)) * a, 0.0))
else:
rho_a = geom.epsilon * (tau_a / (1 - tau_a))
div_a = jnp.sum(
jnp.where(
a > 0,
a * (rho_a - (rho_a + geom.epsilon / 2) *
jnp.exp(-(f - geom.potential_from_scaling(a)) / rho_a)),
0.0))
if tau_b == 1.0:
div_b = jnp.sum(
jnp.where(b > 0, (g - geom.potential_from_scaling(b)) * b, 0.0))
else:
rho_b = geom.epsilon * (tau_b / (1 - tau_b))
div_b = jnp.sum(
jnp.where(
b > 0,
b * (rho_b - (rho_b + geom.epsilon / 2) *
jnp.exp(-(g - geom.potential_from_scaling(b)) / rho_b)),
0.0))
# Using https://arxiv.org/pdf/1910.12958.pdf (30), corrected with (15)
# The total mass of the coupling is computed in scaling space. This avoids
# differentiation issues linked with the automatic differention of
# jnp.exp(jnp.logsumexp(...)) when some of those logs appear as -inf.
# Because we are computing total mass it is irrelevant to have underflow since
# this would simply result in near 0 contributions, which, unlike Sinkhorn
# iterations, do not appear next in a numerator.
total_sum = jnp.sum(
geom.marginal_from_scalings(geom.scaling_from_potential(f),
geom.scaling_from_potential(g)))
return div_a + div_b + geom.epsilon * (jnp.sum(a) * jnp.sum(b) - total_sum)