sinkhorn.py

# Copyright 2022 Google LLC.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
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# Lint as: python3
"""A Jax implementation of the Sinkhorn algorithm."""
from typing import Any, Callable, NamedTuple, Optional, Sequence, Tuple

import jax
import jax.numpy as jnp
import numpy as np

from ott.core import anderson as anderson_lib
from ott.core import fixed_point_loop
from ott.core import implicit_differentiation as implicit_lib
from ott.core import initializers as init_lib
from ott.core import linear_problems
from ott.core import momentum as momentum_lib
from ott.core import potentials, unbalanced_functions
from ott.geometry import geometry


class SinkhornState(NamedTuple):
  """Holds the state variables used to solve OT with Sinkhorn."""

  errors: Optional[jnp.ndarray] = None
  fu: Optional[jnp.ndarray] = None
  gv: Optional[jnp.ndarray] = None
  old_fus: Optional[jnp.ndarray] = None
  old_mapped_fus: Optional[jnp.ndarray] = None

  def set(self, **kwargs: Any) -> 'SinkhornState':
    """Return a copy of self, with potential overwrites."""
    return self._replace(**kwargs)

  def solution_error(
      self, ot_prob: linear_problems.LinearProblem, norm_error: Sequence[int],
      lse_mode: bool
  ) -> jnp.ndarray:
    return solution_error(self.fu, self.gv, ot_prob, norm_error, lse_mode)

  def ent_reg_cost(
      self, ot_prob: linear_problems.LinearProblem, lse_mode: bool
  ) -> float:
    return ent_reg_cost(self.fu, self.gv, ot_prob, lse_mode)


def solution_error(
    f_u: jnp.ndarray, g_v: jnp.ndarray, ot_prob: linear_problems.LinearProblem,
    norm_error: Sequence[int], lse_mode: bool
) -> jnp.ndarray:
  """Given two potential/scaling solutions, computes deviation to optimality.

  When the ``ot_prob`` problem is balanced, this is simply deviation to the
  target marginals defined in ``ot_prob.a`` and ``ot_prob.b``. When the problem
  is unbalanced, additional quantities must be taken into account.

  Args:
    f_u: jnp.ndarray, potential or scaling
    g_v: jnp.ndarray, potential or scaling
    ot_prob: linear OT problem
    norm_error: int, p-norm used to compute error.
    lse_mode: True if log-sum-exp operations, False if kernel vector products.

  Returns:
    a positive number quantifying how far from optimality current solution is.
  """
  if ot_prob.is_balanced:
    return marginal_error(
        f_u, g_v, ot_prob.b, ot_prob.geom, 0, norm_error, lse_mode
    )

  # 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 = unbalanced_functions.grad_of_marginal_fit(
        ot_prob.a, f_u, ot_prob.tau_a, ot_prob.epsilon
    )
    grad_b = unbalanced_functions.grad_of_marginal_fit(
        ot_prob.b, g_v, ot_prob.tau_b, ot_prob.epsilon
    )
  else:
    u = ot_prob.geom.potential_from_scaling(f_u)
    v = ot_prob.geom.potential_from_scaling(g_v)
    grad_a = unbalanced_functions.grad_of_marginal_fit(
        ot_prob.a, u, ot_prob.tau_a, ot_prob.epsilon
    )
    grad_b = unbalanced_functions.grad_of_marginal_fit(
        ot_prob.b, v, ot_prob.tau_b, ot_prob.epsilon
    )
  err = marginal_error(f_u, g_v, grad_a, ot_prob.geom, 1, norm_error, lse_mode)
  err += marginal_error(f_u, g_v, grad_b, ot_prob.geom, 0, norm_error, lse_mode)
  return err


def marginal_error(
    f_u: jnp.ndarray,
    g_v: jnp.ndarray,
    target: jnp.ndarray,
    geom: geometry.Geometry,
    axis: int = 0,
    norm_error: Sequence[int] = (1,),
    lse_mode: bool = True
) -> jnp.asarray:
  """Output how far Sinkhorn solution is w.r.t target.

  Args:
    f_u: a vector of potentials or scalings for the first marginal.
    g_v: a vector of potentials or scalings for the second marginal.
    target: target marginal.
    geom: Geometry object.
    axis: axis (0 or 1) along which to compute marginal.
    norm_error: (tuple of int) p's to compute p-norm between marginal/target
    lse_mode: whether operating on scalings or potentials

  Returns:
    Array of floats, quantifying difference between target / marginal.
  """
  if lse_mode:
    marginal = geom.marginal_from_potentials(f_u, g_v, axis=axis)
  else:
    marginal = geom.marginal_from_scalings(f_u, g_v, axis=axis)
  norm_error = jnp.asarray(norm_error)
  error = jnp.sum(
      jnp.abs(marginal - target) ** norm_error[:, jnp.newaxis], axis=1
  ) ** (1.0 / norm_error)
  return error


def ent_reg_cost(
    f: jnp.ndarray, g: jnp.ndarray, ot_prob: linear_problems.LinearProblem,
    lse_mode: bool
) -> float:
  r"""Compute objective of Sinkhorn for OT problem given dual solutions.

  The objective is evaluated for dual solution ``f`` and ``g``, using
  information contained in  ``ot_prob``. The objective is the regularized
  optimal transport cost (i.e. the cost itself plus entropic and unbalanced
  terms). Situations where marginals ``a`` or ``b`` in ot_prob 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:
    f: jnp.ndarray, potential
    g: jnp.ndarray, potential
    ot_prob: linear optimal transport problem.
    lse_mode: bool, whether to compute total mass in lse or kernel mode.

  Returns:
    The regularized transport cost.
  """
  supp_a = ot_prob.a > 0
  supp_b = ot_prob.b > 0
  fa = ot_prob.geom.potential_from_scaling(ot_prob.a)
  if ot_prob.tau_a == 1.0:
    div_a = jnp.sum(jnp.where(supp_a, ot_prob.a * (f - fa), 0.0))
  else:
    rho_a = ot_prob.epsilon * (ot_prob.tau_a / (1 - ot_prob.tau_a))
    div_a = -jnp.sum(
        jnp.where(
            supp_a, ot_prob.a * unbalanced_functions.phi_star(-(f - fa), rho_a),
            0.0
        )
    )

  gb = ot_prob.geom.potential_from_scaling(ot_prob.b)
  if ot_prob.tau_b == 1.0:
    div_b = jnp.sum(jnp.where(supp_b, ot_prob.b * (g - gb), 0.0))
  else:
    rho_b = ot_prob.epsilon * (ot_prob.tau_b / (1 - ot_prob.tau_b))
    div_b = -jnp.sum(
        jnp.where(
            supp_b, ot_prob.b * unbalanced_functions.phi_star(-(g - gb), rho_b),
            0.0
        )
    )

  # Using https://arxiv.org/pdf/1910.12958.pdf (24)
  if lse_mode:
    total_sum = jnp.sum(ot_prob.geom.marginal_from_potentials(f, g))
  else:
    u = ot_prob.geom.scaling_from_potential(f)
    v = ot_prob.geom.scaling_from_potential(g)
    total_sum = jnp.sum(ot_prob.geom.marginal_from_scalings(u, v))
  return div_a + div_b + ot_prob.epsilon * (
      jnp.sum(ot_prob.a) * jnp.sum(ot_prob.b) - total_sum
  )


class SinkhornOutput(NamedTuple):
  """Implements the problems.Transport interface, for a Sinkhorn solution."""

  f: Optional[jnp.ndarray] = None
  g: Optional[jnp.ndarray] = None
  errors: Optional[jnp.ndarray] = None
  reg_ot_cost: Optional[float] = None
  ot_prob: Optional[linear_problems.LinearProblem] = None

  def set(self, **kwargs: Any) -> 'SinkhornOutput':
    """Return a copy of self, with potential overwrites."""
    return self._replace(**kwargs)

  def set_cost(
      self, ot_prob: linear_problems.LinearProblem, lse_mode: bool,
      use_danskin: bool
  ) -> 'SinkhornOutput':
    f = jax.lax.stop_gradient(self.f) if use_danskin else self.f
    g = jax.lax.stop_gradient(self.g) if use_danskin else self.g
    return self.set(reg_ot_cost=ent_reg_cost(f, g, ot_prob, lse_mode))

  @property
  def linear(self) -> bool:
    return isinstance(self.ot_prob, linear_problems.LinearProblem)

  @property
  def geom(self) -> geometry.Geometry:
    return self.ot_prob.geom

  @property
  def a(self) -> jnp.ndarray:
    return self.ot_prob.a

  @property
  def b(self) -> jnp.ndarray:
    return self.ot_prob.b

  @property
  def linear_output(self) -> bool:
    return True

  @property
  def converged(self) -> bool:
    if self.errors is None:
      return False
    return jnp.logical_and(
        jnp.any(self.errors == -1), jnp.all(jnp.isfinite(self.errors))
    )

  @property
  def scalings(self) -> Tuple[jnp.ndarray, jnp.ndarray]:
    u = self.ot_prob.geom.scaling_from_potential(self.f)
    v = self.ot_prob.geom.scaling_from_potential(self.g)
    return u, v

  @property
  def matrix(self) -> jnp.ndarray:
    """Transport matrix if it can be instantiated."""
    try:
      return self.ot_prob.geom.transport_from_potentials(self.f, self.g)
    except ValueError:
      return self.ot_prob.geom.transport_from_scalings(*self.scalings)

  def apply(self, inputs: jnp.ndarray, axis: int = 0) -> jnp.ndarray:
    """Apply the transport to a ndarray; axis=1 for its transpose."""
    try:
      return self.ot_prob.geom.apply_transport_from_potentials(
          self.f, self.g, inputs, axis=axis
      )
    except ValueError:
      u, v = self.scalings
      return self.ot_prob.geom.apply_transport_from_scalings(
          u, v, inputs, axis=axis
      )

  def marginal(self, axis: int) -> jnp.ndarray:
    return self.ot_prob.geom.marginal_from_potentials(self.f, self.g, axis=axis)

  def cost_at_geom(self, other_geom: geometry.Geometry) -> float:
    """Return reg-OT cost for matrix, evaluated at other cost matrix."""
    return (
        jnp.sum(self.matrix * other_geom.cost_matrix) -
        self.geom.epsilon * jnp.sum(jax.scipy.special.entr(self.matrix))
    )

  def transport_mass(self) -> float:
    """Sum of transport matrix."""
    return self.marginal(0).sum()

  def to_dual_potentials(self) -> potentials.EntropicPotentials:
    """Return the entropic map estimator."""
    return potentials.EntropicPotentials(
        self.f, self.g, self.geom, self.a, self.b
    )


@jax.tree_util.register_pytree_node_class
class Sinkhorn:
  """A Sinkhorn solver for linear reg-OT problem.

  A Sinkhorn solver takes a linear OT problem object as an input and returns a
  SinkhornOutput object that contains all the information required to compute
  transports. See :func:`~ott.core.sinkhorn.sinkhorn` for a functional wrapper.

  Args:
    lse_mode: ``True`` for log-sum-exp computations, ``False`` for kernel
      multiplication.
    threshold: tolerance used to stop the Sinkhorn iterations. This is
      typically the deviation between a target marginal and the marginal of the
      current primal solution when either or both tau_a and tau_b are 1.0
      (balanced or semi-balanced problem), or the relative change between two
      successive solutions in the unbalanced case.
    norm_error: power used to define p-norm of error for marginal/target.
    inner_iterations: the Sinkhorn error is not recomputed at each
      iteration but every inner_num_iter instead.
    min_iterations: the minimum number of Sinkhorn iterations carried
      out before the error is computed and monitored.
    max_iterations: the maximum number of Sinkhorn iterations. If
      ``max_iterations`` is equal to ``min_iterations``, sinkhorn iterations are
      run by default using a :func:`jax.lax.scan` loop rather than a custom,
      unroll-able :func:`jax.lax.while_loop` that monitors convergence.
      In that case the error is not monitored and the ``converged``
      flag will return ``False`` as a consequence.
    momentum: a Momentum instance. See ott.core.momentum
    anderson: an AndersonAcceleration instance. See ott.core.anderson.
    implicit_diff: instance used to solve implicit differentiation. Unrolls
      iterations if None.
    parallel_dual_updates: updates potentials or scalings in parallel if True,
      sequentially (in Gauss-Seidel fashion) if False.
    use_danskin: when ``True``, it is assumed the entropy regularized cost is
      is evaluated using optimal potentials that are frozen, i.e. whose
      gradients have been stopped. This is useful when carrying out first order
      differentiation, and is only valid (as with ``implicit_differentiation``)
      when the algorithm has converged with a low tolerance.
    jit: if True, automatically jits the function upon first call.
      Should be set to False when used in a function that is jitted by the user,
      or when computing gradients (in which case the gradient function
      should be jitted by the user)
    initializer: how to compute the initial potentials/scalings.
  """

  def __init__(
      self,
      lse_mode: bool = True,
      threshold: float = 1e-3,
      norm_error: int = 1,
      inner_iterations: int = 10,
      min_iterations: int = 0,
      max_iterations: int = 2000,
      momentum: Optional[momentum_lib.Momentum] = None,
      anderson: Optional[anderson_lib.AndersonAcceleration] = None,
      parallel_dual_updates: bool = False,
      use_danskin: Optional[bool] = None,
      implicit_diff: Optional[implicit_lib.ImplicitDiff
                             ] = implicit_lib.ImplicitDiff(),  # noqa: E124
      initializer: init_lib.SinkhornInitializer = init_lib.DefaultInitializer(),
      jit: bool = True
  ):
    self.lse_mode = lse_mode
    self.threshold = threshold
    self.inner_iterations = inner_iterations
    self.min_iterations = min_iterations
    self.max_iterations = max_iterations
    self._norm_error = norm_error
    if momentum is not None:
      self.momentum = momentum_lib.Momentum(
          momentum.start, momentum.value, self.inner_iterations
      )
    else:
      self.momentum = momentum_lib.Momentum(
          inner_iterations=self.inner_iterations
      )
    self.anderson = anderson
    self.implicit_diff = implicit_diff
    self.parallel_dual_updates = parallel_dual_updates
    self.initializer = initializer
    self.jit = jit

    # Force implicit_differentiation to True when using Anderson acceleration,
    # Reset all momentum parameters.
    if anderson:
      self.implicit_diff = (
          implicit_lib.ImplicitDiff()
          if self.implicit_diff is None else self.implicit_diff
      )
      self.momentum = momentum_lib.Momentum(start=0, value=1.0)

    # By default, use Danskin theorem to differentiate
    # the objective when using implicit_lib.
    self.use_danskin = ((self.implicit_diff is not None)
                        if use_danskin is None else use_danskin)

  def __call__(
      self,
      ot_prob: linear_problems.LinearProblem,
      init: Tuple[Optional[jnp.ndarray], Optional[jnp.ndarray]] = (None, None),
  ) -> SinkhornOutput:
    """Run Sinkhorn algorithm.

    Args:
      ot_prob: Linear OT problem.
      init: Initial dual potentials/scalings f_u and g_v, respectively.
        Any `None` values will be initialized using the initializer.

    Returns:
      The Sinkhorn output.
    """
    init_dual_a, init_dual_b = self.initializer(
        ot_prob, *init, lse_mode=self.lse_mode
    )
    run_fn = jax.jit(run) if self.jit else run
    return run_fn(ot_prob, self, (init_dual_a, init_dual_b))

  def lse_step(
      self, ot_prob: linear_problems.LinearProblem, state: SinkhornState,
      iteration: int
  ) -> SinkhornState:
    """Sinkhorn LSE update."""
    w = self.momentum.weight(state, iteration)
    old_gv = state.gv

    new_gv = ot_prob.tau_b * ot_prob.geom.update_potential(
        state.fu, state.gv, jnp.log(ot_prob.b), iteration, axis=0
    )
    gv = self.momentum(w, state.gv, new_gv, self.lse_mode)
    new_fu = ot_prob.tau_a * ot_prob.geom.update_potential(
        state.fu,
        old_gv if self.parallel_dual_updates else gv,
        jnp.log(ot_prob.a),
        iteration,
        axis=1
    )
    fu = self.momentum(w, state.fu, new_fu, self.lse_mode)
    return state.set(fu=fu, gv=gv)

  def kernel_step(
      self, ot_prob: linear_problems.LinearProblem, state: SinkhornState,
      iteration: int
  ) -> SinkhornState:
    """Sinkhorn multiplicative update."""
    w = self.momentum.weight(state, iteration)
    old_gv = state.gv
    new_gv = ot_prob.geom.update_scaling(
        state.fu, ot_prob.b, iteration, axis=0
    ) ** ot_prob.tau_b
    gv = self.momentum(w, state.gv, new_gv, self.lse_mode)
    new_fu = ot_prob.geom.update_scaling(
        old_gv if self.parallel_dual_updates else gv,
        ot_prob.a,
        iteration,
        axis=1
    ) ** ot_prob.tau_a
    fu = self.momentum(w, state.fu, new_fu, self.lse_mode)
    return state.set(fu=fu, gv=gv)

  def one_iteration(
      self, ot_prob: linear_problems.LinearProblem, state: SinkhornState,
      iteration: int, compute_error: bool
  ) -> SinkhornState:
    """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:
      ot_prob: the transport problem definition
      state: SinkhornState named tuple.
      iteration: the current iteration of the Sinkhorn loop.
      compute_error: flag to indicate this iteration computes/stores an error

    Returns:
      The updated state.
    """
    # When running updates in parallel (Gauss-Seidel mode), old_g_v will be
    # used to update f_u, rather than the latest g_v computed in this loop.
    # Unused otherwise.
    if self.anderson:
      state = self.anderson.update(state, iteration, ot_prob, self.lse_mode)

    if self.lse_mode:  # In lse_mode, run additive updates.
      state = self.lse_step(ot_prob, state, iteration)
    else:
      state = self.kernel_step(ot_prob, state, iteration)

    if self.anderson:
      state = self.anderson.update_history(state, ot_prob, self.lse_mode)

    # re-computes error if compute_error is True, else set it to inf.
    err = jnp.where(
        jnp.logical_and(compute_error, iteration >= self.min_iterations),
        state.solution_error(ot_prob, self.norm_error, self.lse_mode), jnp.inf
    )
    errors = (state.errors.at[iteration // self.inner_iterations, :].set(err))
    return state.set(errors=errors)

  def _converged(self, state: SinkhornState, iteration: int) -> bool:
    err = state.errors[iteration // self.inner_iterations - 1, 0]
    return jnp.logical_and(iteration > 0, err < self.threshold)

  def _diverged(self, state: SinkhornState, iteration: int) -> bool:
    err = state.errors[iteration // self.inner_iterations - 1, 0]
    return jnp.logical_not(jnp.isfinite(err))

  def _continue(self, state: SinkhornState, iteration: int) -> bool:
    """Continue while not(converged) and not(diverged)."""
    return jnp.logical_and(
        jnp.logical_not(self._diverged(state, iteration)),
        jnp.logical_not(self._converged(state, iteration))
    )

  @property
  def outer_iterations(self) -> int:
    """Upper bound on number of times inner_iterations are carried out.

    This integer can be used to set constant array sizes to track the algorithm
    progress, notably errors.
    """
    return np.ceil(self.max_iterations / self.inner_iterations).astype(int)

  def init_state(
      self, ot_prob: linear_problems.LinearProblem, init: Tuple[jnp.ndarray,
                                                                jnp.ndarray]
  ) -> SinkhornState:
    """Return the initial state of the loop."""
    fu, gv = init
    errors = -jnp.ones((self.outer_iterations, len(self.norm_error)),
                       dtype=fu.dtype)
    state = SinkhornState(errors=errors, fu=fu, gv=gv)
    return self.anderson.init_maps(ot_prob, state) if self.anderson else state

  def output_from_state(
      self, ot_prob: linear_problems.LinearProblem, state: SinkhornState
  ) -> SinkhornOutput:
    """Create an output from a loop state.

    Note:
      When differentiating the regularized OT cost, and assuming Sinkhorn has
      run to convergence, Danskin's (or the envelope)
      `theorem <https://en.wikipedia.org/wiki/Danskin%27s_theorem>`_
      states that the resulting OT cost as a function of any of the inputs
      (``geometry``, ``a``, ``b``) behaves locally as if the dual optimal
      potentials were frozen and did not vary with those inputs.

      Notice this is only valid, as when using ``implicit_differentiation``
      mode, if the Sinkhorn algorithm outputs potentials that are near optimal.
      namely when the threshold value is set to a small tolerance.

      The flag ``use_danskin`` controls whether that assumption is made. By
      default, that flag is set to the value of ``implicit_differentiation`` if
      not specified. If you wish to compute derivatives of order 2 and above,
      set ``use_danskin`` to ``False``.

    Args:
      ot_prob: the transport problem.
      state: a SinkhornState.

    Returns:
      A SinkhornOutput.
    """
    geom = ot_prob.geom
    f = state.fu if self.lse_mode else geom.potential_from_scaling(state.fu)
    g = state.gv if self.lse_mode else geom.potential_from_scaling(state.gv)
    errors = state.errors[:, 0]
    return SinkhornOutput(f=f, g=g, errors=errors)

  @property
  def norm_error(self) -> Tuple[int, ...]:
    """Powers used to compute the p-norm between marginal/target."""
    # To change momentum adaptively, one needs errors in ||.||_1 norm.
    # In that case, we add this exponent to the list of errors to compute,
    # notably if that was not the error requested by the user.
    if self.momentum and self.momentum.start > 0 and self._norm_error != 1:
      return self._norm_error, 1
    return self._norm_error,

  def tree_flatten(self):
    aux = vars(self).copy()
    aux['norm_error'] = aux.pop('_norm_error')
    aux.pop('threshold')
    return [self.threshold], aux

  @classmethod
  def tree_unflatten(cls, aux_data, children):
    return cls(**aux_data, threshold=children[0])


def run(
    ot_prob: linear_problems.LinearProblem, solver: Sinkhorn,
    init: Tuple[jnp.ndarray, ...]
) -> SinkhornOutput:
  """Run loop of the solver, outputting a state upgraded to an output."""
  iter_fun = _iterations_implicit if solver.implicit_diff else iterations
  out = iter_fun(ot_prob, solver, init)
  # Be careful here, the geom and the cost are injected at the end, where it
  # does not interfere with the implicit differentiation.
  out = out.set_cost(ot_prob, solver.lse_mode, solver.use_danskin)
  return out.set(ot_prob=ot_prob)


def iterations(
    ot_prob: linear_problems.LinearProblem, solver: Sinkhorn,
    init: Tuple[jnp.ndarray, ...]
) -> SinkhornOutput:
  """Jittable Sinkhorn loop. args contain initialization variables."""

  def cond_fn(
      iteration: int, const: Tuple[linear_problems.LinearProblem, Sinkhorn],
      state: SinkhornState
  ) -> bool:
    _, solver = const
    return solver._continue(state, iteration)

  def body_fn(
      iteration: int, const: Tuple[linear_problems.LinearProblem, Sinkhorn],
      state: SinkhornState, compute_error: bool
  ) -> SinkhornState:
    ot_prob, solver = const
    return solver.one_iteration(ot_prob, state, iteration, compute_error)

  # 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 solver.implicit_diff:
    fix_point = fixed_point_loop.fixpoint_iter
  else:
    fix_point = fixed_point_loop.fixpoint_iter_backprop

  const = ot_prob, solver
  state = solver.init_state(ot_prob, init)
  state = fix_point(
      cond_fn, body_fn, solver.min_iterations, solver.max_iterations,
      solver.inner_iterations, const, state
  )
  return solver.output_from_state(ot_prob, state)


def _iterations_taped(
    ot_prob: linear_problems.LinearProblem, solver: Sinkhorn,
    init: Tuple[jnp.ndarray, ...]
) -> Tuple[SinkhornOutput, Tuple[jnp.ndarray, jnp.ndarray,
                                 linear_problems.LinearProblem, Sinkhorn]]:
  """Run forward pass of the Sinkhorn algorithm storing side information."""
  state = iterations(ot_prob, solver, init)
  return state, (state.f, state.g, ot_prob, solver)


def _iterations_implicit_bwd(res, gr):
  """Run Sinkhorn in backward mode, using implicit differentiation.

  Args:
    res: residual data sent from fwd pass, used for computations below. In this
      case consists in the output itself, as well as inputs against which we
      wish to differentiate.
    gr: gradients w.r.t outputs of fwd pass, here w.r.t size f, g, errors. Note
      that differentiability w.r.t. errors is not handled, and only f, g is
      considered.

  Returns:
    a tuple of gradients: PyTree for geom, one jnp.ndarray for each of a and b.
  """
  f, g, ot_prob, solver = res
  gr = gr[:2]
  return (
      *solver.implicit_diff.gradient(ot_prob, f, g, solver.lse_mode, gr), None,
      None
  )


# Sets threshold, norm_errors, geom, a and b to be differentiable, as those are
# non static. Only differentiability w.r.t. geom, a and b will be used.
_iterations_implicit = jax.custom_vjp(iterations)
_iterations_implicit.defvjp(_iterations_taped, _iterations_implicit_bwd)


def make(
    tau_a: float = 1.0,
    tau_b: float = 1.0,
    threshold: float = 1e-3,
    norm_error: int = 1,
    inner_iterations: int = 10,
    min_iterations: int = 0,
    max_iterations: int = 2000,
    momentum: float = 1.0,
    chg_momentum_from: int = 0,
    anderson_acceleration: int = 0,
    refresh_anderson_frequency: int = 1,
    lse_mode: bool = True,
    implicit_differentiation: bool = True,
    implicit_solver_fun=jax.scipy.sparse.linalg.cg,
    implicit_solver_ridge_kernel: float = 0.0,
    implicit_solver_ridge_identity: float = 0.0,
    implicit_solver_symmetric: bool = False,
    precondition_fun: Optional[Callable[[float], float]] = None,
    parallel_dual_updates: bool = False,
    use_danskin: bool = None,
    initializer: init_lib.SinkhornInitializer = init_lib.DefaultInitializer(),
    jit: bool = False
) -> Sinkhorn:
  """For backward compatibility."""
  del tau_a, tau_b
  if not implicit_differentiation:
    implicit_diff = None
  else:
    implicit_diff = implicit_lib.ImplicitDiff(
        solver_fun=implicit_solver_fun,
        ridge_kernel=implicit_solver_ridge_kernel,
        ridge_identity=implicit_solver_ridge_identity,
        symmetric=implicit_solver_symmetric,
        precondition_fun=precondition_fun
    )

  if anderson_acceleration > 0:
    anderson = anderson_lib.AndersonAcceleration(
        memory=anderson_acceleration, refresh_every=refresh_anderson_frequency
    )
  else:
    anderson = None

  return Sinkhorn(
      lse_mode=lse_mode,
      threshold=threshold,
      norm_error=norm_error,
      inner_iterations=inner_iterations,
      min_iterations=min_iterations,
      max_iterations=max_iterations,
      momentum=momentum_lib.Momentum(start=chg_momentum_from, value=momentum),
      anderson=anderson,
      implicit_diff=implicit_diff,
      parallel_dual_updates=parallel_dual_updates,
      use_danskin=use_danskin,
      initializer=initializer,
      jit=jit
  )


def sinkhorn(
    geom: geometry.Geometry,
    a: Optional[jnp.ndarray] = None,
    b: Optional[jnp.ndarray] = None,
    tau_a: float = 1.0,
    tau_b: float = 1.0,
    init_dual_a: Optional[jnp.ndarray] = None,
    init_dual_b: Optional[jnp.ndarray] = None,
    **kwargs: Any,
):
  r"""Solve regularized OT problem using Sinkhorn iterations.

  The Sinkhorn algorithm is a fixed point iteration that solves a regularized
  optimal transport (reg-OT) problem between two measures.
  The optimization variables are a pair of vectors (called potentials, or
  scalings when parameterized as exponentials of the former). Calling this
  function returns therefore a pair of optimal vectors. In addition to these,
  it also returns the objective value achieved by these optimal vectors;
  a vector of size ``max_iterations/inner_iterations`` that records the vector
  of values recorded to monitor convergence, throughout the execution of the
  algorithm (padded with `-1` if convergence happens before), as well as a
  boolean to signify whether the algorithm has converged within the number of
  iterations specified by the user.

  The reg-OT problem is specified by two measures, of respective sizes ``n`` and
  ``m``. From the viewpoint of the ``sinkhorn`` function, these two measures are
  only seen through a triplet (``geom``, ``a``, ``b``), where ``geom`` is a
  ``Geometry`` object, and ``a`` and ``b`` are weight vectors of respective
  sizes ``n`` and ``m``. Starting from two initial values for those potentials
  or scalings (both can be defined by the user by passing value in
  ``init_dual_a`` or ``init_dual_b``), the Sinkhorn algorithm will use
  elementary operations that are carried out by the ``geom`` object.

  Some maths:
    Given a geometry ``geom``, which provides a cost matrix :math:`C` with its
    regularization parameter :math:`\varepsilon`, (or a kernel matrix :math:`K`)
    the reg-OT problem consists in finding two vectors `f`, `g` of size ``n``,
    ``m`` that maximize the following criterion.

    .. math::

      \arg\max_{f, g}{- \langle a, \phi_a^{*}(-f) \rangle -  \langle b,
      \phi_b^{*}(-g) \rangle - \varepsilon \langle e^{f/\varepsilon},
      e^{-C/\varepsilon} e^{-g/\varepsilon}} \rangle

    where :math:`\phi_a(z) = \rho_a z(\log z - 1)` is a scaled entropy, and
    :math:`\phi_a^{*}(z) = \rho_a e^{z/\varepsilon}`, its Legendre transform.

    That problem can also be written, instead, using positive scaling vectors
    `u`, `v` of size ``n``, ``m``, handled with the kernel
    :math:`K := e^{-C/\varepsilon}`,

    .. math::

      \arg\max_{u, v >0} - \langle a,\phi_a^{*}(-\varepsilon\log u) \rangle +
      \langle b, \phi_b^{*}(-\varepsilon\log v) \rangle - \langle u, K v \rangle

    Both of these problems corresponds, in their *primal* formulation, to
    solving the unbalanced optimal transport problem with a variable matrix
    :math:`P` of size ``n`` x ``m``:

    .. math::

      \arg\min_{P>0} \langle P,C \rangle -\varepsilon \text{KL}(P | ab^T)
      + \rho_a \text{KL}(P\mathbf{1}_m | a) + \rho_b \text{KL}(P^T \mathbf{1}_n
      | b)

    where :math:`KL` is the generalized Kullback-Leibler divergence.

    The very same primal problem can also be written using a kernel :math:`K`
    instead of a cost :math:`C` as well:

    .. math::

      \arg\min_{P} \varepsilon KL(P|K) + \rho_a \text{KL}(P\mathbf{1}_m | a) +
      \rho_b \text{KL}(P^T \mathbf{1}_n | b)

    The *original* OT problem taught in linear programming courses is recovered
    by using the formulation above relying on the cost :math:`C`, and letting
    :math:`\varepsilon \rightarrow 0`, and :math:`\rho_a, \rho_b \rightarrow
    \infty`.
    In that case the entropy disappears, whereas the :math:`KL` regularization
    above become constraints on the marginals of :math:`P`: This results in a
    standard min cost flow problem. This problem is not handled for now in this
    toolbox, which focuses exclusively on the case :math:`\varepsilon > 0`.

    The *balanced* regularized OT problem is recovered for finite
    :math:`\varepsilon > 0` but letting :math:`\rho_a, \rho_b \rightarrow
    \infty`. This problem can be shown to be equivalent to a matrix scaling
    problem, which can be solved using the Sinkhorn fixed-point algorithm.
    To handle the case :math:`\rho_a, \rho_b \rightarrow \infty`, the
    ``sinkhorn`` function uses parameters :math:`tau\_a := \rho_a /
    (\varepsilon + \rho_a)` and :math:`tau\_b := \rho_b / (\varepsilon +
    \rho_b)` instead. Setting either of these parameters to 1 corresponds to
    setting the corresponding :math:`\rho_a, \rho_b` to :math:`\infty`.

    The Sinkhorn algorithm solves the reg-OT problem by seeking optimal `f`, `g`
    potentials (or alternatively their parametrization as positive scalings
    `u`, `v`), rather than solving the primal problem in :math:`P`.
    This is mostly for efficiency (potentials and scalings have a ``n + m``
    memory footprint, rather than ``n m`` required to store `P`). This is also
    because both problems are, in fact, equivalent, since the optimal transport
    math:`P^*` can be recovered from optimal potentials :math:`f^*`, :math:`g^*`
    or scalings :math:`u^*`, :math:`v^*`, using the geometry's cost or kernel
    matrix respectively:

    .. math::

      P^* = \exp\left(\frac{f^*\mathbf{1}_m^T + \mathbf{1}_n g^{*T} -
      C}{\varepsilon}\right) \text{ or } P^* = \text{diag}(u^*) K
      \text{diag}(v^*)

    By default, the Sinkhorn algorithm solves this dual problem in `f, g` or
    `u, v` using block coordinate ascent, i.e. devising an update for each `f`
    and `g` (resp. `u` and `v`) that cancels their respective gradients, one at
    a time. These two iterations are repeated ``inner_iterations`` times, after
    which the norm of these gradients will be evaluated and compared with the
    ``threshold`` value. The iterations are then repeated as long as that error
    exceeds ``threshold``.

  Note on Sinkhorn updates:
    The boolean flag ``lse_mode`` sets whether the algorithm is run in either:

    - log-sum-exp mode (``lse_mode=True``), in which case it is directly
      defined in terms of updates to `f` and `g`, using log-sum-exp
      computations. This requires access to the cost matrix :math:`C`, as it is
      stored, or possibly computed on the fly by ``geom``.

    - kernel mode (``lse_mode=False``), in which case it will require access
      to a matrix vector multiplication operator :math:`z \rightarrow K z`,
      where :math:`K` is either instantiated from :math:`C` as
      :math:`\exp(-C/\varepsilon)`, or provided directly. In that case, rather
      than optimizing on :math:`f` and :math:`g`, it is more convenient to
      optimize on their so called scaling formulations,
      :math:`u := \exp(f / \varepsilon)` and :math:`v := \exp(g / \varepsilon)`.
      While faster (applying matrices is faster than applying ``lse`` repeatedly
      over lines), this mode is also less stable numerically, notably for
      smaller :math:`\varepsilon`.

    In the source code, the variables ``f_u`` or ``g_v`` can be either regarded
    as potentials (real) or scalings (positive) vectors, depending on the choice
    of ``lse_mode`` by the user. Once optimization is carried out, we only
    return dual variables in potential form, i.e. ``f`` and ``g``.

    In addition to standard Sinkhorn updates, the user can also use heavy-ball
    type updates using a ``momentum`` parameter in ]0,2[. We also implement a
    strategy that tries to set that parameter adaptively at
    ``chg_momentum_from`` iterations, as a function of progress in the error,
    as proposed in the literature.

    Another upgrade to the standard Sinkhorn updates provided to the users lies
    in using Anderson acceleration. This can be parameterized by setting the
    otherwise null ``anderson`` to a positive integer. When selected,the
    algorithm will recompute, every ``refresh_anderson_frequency`` (set by
    default to 1) an extrapolation of the most recently computed ``anderson``
    iterates. When using that option, notice that differentiation (if required)
    can only be carried out using implicit differentiation, and that all
    momentum related parameters are ignored.

    The ``parallel_dual_updates`` flag is set to ``False`` by default. In that
    setting, ``g_v`` is first updated using the latest values for ``f_u`` and
    ``g_v``, before proceeding to update ``f_u`` using that new value for
    ``g_v``. When the flag is set to ``True``, both ``f_u`` and ``g_v`` are
    updated simultaneously. Note that setting that choice to ``True`` requires
    using some form of averaging (e.g. ``momentum=0.5``). Without this, and on
    its own ``parallel_dual_updates`` won't work.

  Differentiation:
    The optimal solutions ``f`` and ``g`` and the optimal objective
    (``reg_ot_cost``) outputted by the Sinkhorn algorithm can be differentiated
    w.r.t. relevant inputs ``geom``, ``a`` and ``b`` using, by default, implicit
    differentiation of the optimality conditions (``implicit_differentiation``
    set to ``True``). This choice has two consequences.

    - The termination criterion used to stop Sinkhorn (cancellation of
      gradient of objective w.r.t. ``f_u`` and ``g_v``) is used to differentiate
      ``f`` and ``g``, given a change in the inputs. These changes are computed
      by solving a linear system. The arguments starting with
      ``implicit_solver_*`` allow to define the linear solver that is used, and
      to control for two types or regularization (we have observed that,
      depending on the architecture, linear solves may require higher ridge
      parameters to remain stable). The optimality conditions in Sinkhorn can be
      analyzed as satisfying a ``z=z'`` condition, which are then
      differentiated. It might be beneficial (e.g., as in :cite:`cuturi:20a`)
      to use a preconditioning function ``precondition_fun`` to differentiate
      instead ``h(z) = h(z')``.

    - The objective ``reg_ot_cost`` returned by Sinkhorn uses the so-called
      envelope (or Danskin's) theorem. In that case, because it is assumed that
      the gradients of the dual variables ``f_u`` and ``g_v`` w.r.t. dual
      objective are zero (reflecting the fact that they are optimal), small
      variations in ``f_u`` and ``g_v`` due to changes in inputs (such as
      ``geom``, ``a`` and ``b``) are considered negligible. As a result,
      ``stop_gradient`` is applied on dual variables ``f_u`` and ``g_v`` when
      evaluating the ``reg_ot_cost`` objective. Note that this approach is
      `invalid` when computing higher order derivatives. In that case the
      ``use_danskin`` flag must be set to ``False``.

    An alternative yet more costly way to differentiate the outputs of the
    Sinkhorn iterations is to use unrolling, i.e. reverse mode differentiation
    of the Sinkhorn loop. This is possible because Sinkhorn iterations are
    wrapped in a custom fixed point iteration loop, defined in
    ``fixed_point_loop``, rather than a standard while loop. This is to ensure
    the end result of this fixed point loop can also be differentiated, if
    needed, using standard JAX operations. To ensure backprop differentiability,
    the ``fixed_point_loop.fixpoint_iter_backprop`` loop does checkpointing of
    state variables (here ``f_u`` and ``g_v``) every ``inner_iterations``, and
    backpropagates automatically, block by block, through blocks of
    ``inner_iterations`` at a time.

  Note:
    * The Sinkhorn algorithm may not converge within the maximum number of
      iterations for possibly several reasons:

      1. the regularizer (defined as ``epsilon`` in the geometry ``geom``
         object) is too small. Consider either switching to ``lse_mode=True``
         (at the price of a slower execution), increasing ``epsilon``, or,
         alternatively, if you are unable or unwilling to increase  ``epsilon``,
         either increase ``max_iterations`` or ``threshold``.
      2. the probability weights ``a`` and ``b`` do not have the same total
         mass, while using a balanced (``tau_a=tau_b=1.0``) setup.
         Consider either normalizing ``a`` and ``b``, or set either ``tau_a``
         and/or ``tau_b<1.0``.
      3. OOMs issues may arise when storing either cost or kernel matrices that
         are too large in ``geom``. In the case where, the ``geom`` geometry is
         a ``PointCloud``, some of these issues might be solved by setting the
         ``online`` flag to ``True``. This will trigger a re-computation on the
         fly of the cost/kernel matrix.

    * The weight vectors ``a`` and ``b`` can be passed on with coordinates that
      have zero weight. This is then handled by relying on simple arithmetic for
      ``inf`` values that will likely arise (due to :math:`\log 0` when
      ``lse_mode`` is ``True``, or divisions by zero when ``lse_mode`` is
      ``False``). Whenever that arithmetic is likely to produce ``NaN`` values
      (due to ``-inf * 0``, or ``-inf - -inf``) in the forward pass, we use
      ``jnp.where`` conditional statements to carry ``inf`` rather than ``NaN``
      values. In the reverse mode differentiation, the inputs corresponding to
      these 0 weights (a location `x`, or a row in the corresponding cost/kernel
      matrix), and the weight itself will have ``NaN`` gradient values. This is
      reflects that these gradients are undefined, since these points were not
      considered in the optimization and have therefore no impact on the output.

  Args:
    geom: a Geometry object.
    a: The first marginal. If `None`, it will be uniform.
    b: The second marginal. If `None`, it will be uniform.
    tau_a: ratio rho/(rho+eps) between KL divergence regularizer to first
     marginal and itself + epsilon regularizer used in the unbalanced
     formulation.
    tau_b: ratio rho/(rho+eps) between KL divergence regularizer to first
     marginal and itself + epsilon regularizer used in the unbalanced
     formulation.
    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: tolerance used to stop the Sinkhorn iterations. This is
      typically the deviation between a target marginal and the marginal of the
      current primal solution when either or both tau_a and tau_b are 1.0
      (balanced or semi-balanced problem), or the relative change between two
      successive solutions in the unbalanced case.
    norm_error:
      power used to define p-norm of error for marginal/target.
    inner_iterations:the Sinkhorn error is not recomputed at each
      iteration but every inner_num_iter instead.
    min_iterations: the minimum number of Sinkhorn iterations carried
      out before the error is computed and monitored.
    max_iterations: the maximum number of Sinkhorn iterations. If
      ``max_iterations`` is equal to ``min_iterations``, sinkhorn iterations are
      run by default using a :func:`jax.lax.scan` loop rather than a custom,
      unroll-able :func:`jax.lax.while_loop` that monitors convergence.
      In that case, the error is not monitored and the ``converged`` flag
      will return ``False`` as a consequence.
    momentum:
      a float in [0,2].
    chg_momentum_from: if positive, momentum is recomputed using the
      adaptive rule provided in :cite:`lehmann:21`
      after that number of iterations.
    anderson_acceleration: int, if 0 (default), no acceleration. If positive,
      use Anderson acceleration on the dual sinkhorn (in log/potential form), as
      described `here <https://en.wikipedia.org/wiki/Anderson_acceleration>`_
      and advocated in :cite:`chizat:20`, with a memory of size equal
      to ``anderson_acceleration``. In that case, differentiation is
      necessarily handled implicitly (``implicit_differentiation`` is set to
      ``True``) and all ``momentum`` related parameters are ignored.
    refresh_anderson_frequency: int, when using ``anderson_acceleration``,
      recompute direction periodically every int sinkhorn iterations.
    lse_mode: ``True`` for log-sum-exp computations, ``False`` for kernel
      multiplication.
    implicit_differentiation: ``True`` if using implicit differentiation,
      ``False`` if unrolling Sinkhorn iterations.
    linear_solve_kwargs: parametrization of linear solver when using implicit
      differentiation. Arguments currently accepted appear in the optional
      arguments of ``apply_inv_hessian``, namely ``linear_solver_fun``,  a
      Callable that specifies the linear solver, as well as ``ridge_kernel`` and
      ``ridge_identity``, to be added to enforce stability of linear solve.
    parallel_dual_updates: updates potentials or scalings in parallel if True,
      sequentially (in Gauss-Seidel fashion) if False.
    use_danskin: when ``True``, it is assumed the entropy regularized cost is
      is evaluated using optimal potentials that are frozen, i.e. whose
      gradients have been stopped. This is useful when carrying out first order
      differentiation, and is only valid (as with ``implicit_differentiation``)
      when the algorithm has converged with a low tolerance.
    jit: if True, automatically jits the function upon first call.
      Should be set to False when used in a function that is jitted by the user,
      or when computing gradients (in which case the gradient function
      should be jitted by the user).
    kwargs: Additional keyword arguments (see above).

  Returns:
    a ``SinkhornOutput`` named tuple. The tuple contains two optimal potential
    vectors ``f`` and ``g``, the objective ``reg_ot_cost`` evaluated at those
    solutions, an array of ``errors`` to monitor convergence every
    ``inner_iterations`` and a flag ``converged`` that is ``True`` if the
    algorithm has converged within the number of iterations that was predefined
    by the user.
  """
  sink = make(**kwargs)
  ot_prob = linear_problems.LinearProblem(geom, a, b, tau_a, tau_b)
  return sink(ot_prob, (init_dual_a, init_dual_b))