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 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)