Source code for ott.solvers.linear.sinkhorn_lr

# Copyright OTT-JAX
# 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
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# See the License for the specific language governing permissions and
# limitations under the License.
from typing import (

import jax
import jax.numpy as jnp
import jax.scipy as jsp

from ott.geometry import geometry, low_rank, pointcloud
from ott.initializers.linear import initializers_lr as init_lib
from ott.math import fixed_point_loop
from ott.math import utils as mu
from ott.problems.linear import linear_problem
from ott.solvers.linear import sinkhorn

__all__ = ["LRSinkhorn", "LRSinkhornOutput"]

class LRSinkhornState(NamedTuple):
  """State of the Low Rank Sinkhorn algorithm."""

  q: jnp.ndarray
  r: jnp.ndarray
  g: jnp.ndarray
  gamma: float
  costs: jnp.ndarray
  errors: jnp.ndarray
  crossed_threshold: bool

  def compute_error(  # noqa: D102
      self, previous_state: "LRSinkhornState"
  ) -> float:
    err_1 = mu.js(self.q, previous_state.q, c=1.)
    err_2 = mu.js(self.r, previous_state.r, c=1.)
    err_3 = mu.js(self.g, previous_state.g, c=1.)

    return ((1. / self.gamma) ** 2) * (err_1 + err_2 + err_3)

  def reg_ot_cost(  # noqa: D102
      ot_prob: linear_problem.LinearProblem,
      use_danskin: bool = False
  ) -> float:
    return compute_reg_ot_cost(self.q, self.r, self.g, ot_prob, use_danskin)

  def solution_error(  # noqa: D102
      self, ot_prob: linear_problem.LinearProblem, norm_error: Tuple[int, ...],
      lse_mode: bool
  ) -> jnp.ndarray:
    return solution_error(self.q, self.r, ot_prob, norm_error, lse_mode)

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

def compute_reg_ot_cost(
    q: jnp.ndarray,
    r: jnp.ndarray,
    g: jnp.ndarray,
    ot_prob: linear_problem.LinearProblem,
    use_danskin: bool = False
) -> float:
  """Compute the regularized OT cost.

    q: first factor of solution
    r: second factor of solution
    g: weights of solution
    ot_prob: linear problem
    use_danskin: if True, use Danskin's theorem :cite:`danskin:67,bertsekas:71`
      to avoid computing the gradient of the cost function.

    regularized OT cost
  q = jax.lax.stop_gradient(q) if use_danskin else q
  r = jax.lax.stop_gradient(r) if use_danskin else r
  g = jax.lax.stop_gradient(g) if use_danskin else g
  return jnp.sum(ot_prob.geom.apply_cost(r, axis=1) * q * (1. / g)[None, :])

def solution_error(
    q: jnp.ndarray, r: jnp.ndarray, ot_prob: linear_problem.LinearProblem,
    norm_error: Tuple[int, ...], lse_mode: bool
) -> jnp.ndarray:
  """Compute solution error.

  Since only balanced case is available for LR, this is marginal deviation.

    q: first factor of solution.
    r: second factor of solution.
    ot_prob: linear problem.
    norm_error: int, p-norm used to compute error.
    lse_mode: True if log-sum-exp operations, False if kernel vector products.

    one or possibly many numbers quantifying deviation to true marginals.
  del lse_mode
  norm_error = jnp.array(norm_error)
  # Update the error
  err = jnp.sum(
      jnp.abs(jnp.sum(q, axis=1) - ot_prob.a) ** norm_error[:, None], axis=1
  ) ** (1.0 / norm_error)
  err += jnp.sum(
      jnp.abs(jnp.sum(r, axis=1) - ot_prob.b) ** norm_error[:, None], axis=1
  ) ** (1.0 / norm_error)
  err += jnp.sum(
      jnp.abs(jnp.sum(q, axis=0) - jnp.sum(r, axis=0)) ** norm_error[:, None],
  ) ** (1.0 / norm_error)

  return err

[docs]class LRSinkhornOutput(NamedTuple): """Implement the problems.Transport interface, for a LR Sinkhorn solution.""" q: jnp.ndarray r: jnp.ndarray g: jnp.ndarray costs: jnp.ndarray # TODO(michalk8): must be called `errors`, because of `store_inner_errors` # in future, enforce via class hierarchy errors: jnp.ndarray ot_prob: linear_problem.LinearProblem # TODO(michalk8): Optional is an artifact of the current impl., refactor reg_ot_cost: Optional[float] = None
[docs] def set(self, **kwargs: Any) -> "LRSinkhornOutput": """Return a copy of self, with potential overwrites.""" return self._replace(**kwargs)
[docs] def set_cost( # noqa: D102 self, ot_prob: linear_problem.LinearProblem, lse_mode: bool, use_danskin: bool = False ) -> "LRSinkhornOutput": del lse_mode return self.set(reg_ot_cost=self.compute_reg_ot_cost(ot_prob, use_danskin))
[docs] def compute_reg_ot_cost( # noqa: D102 self, ot_prob: linear_problem.LinearProblem, use_danskin: bool = False, ) -> float: return compute_reg_ot_cost(self.q, self.r, self.g, ot_prob, use_danskin)
@property def linear(self) -> bool: # noqa: D102 return isinstance(self.ot_prob, linear_problem.LinearProblem) @property def geom(self) -> geometry.Geometry: # noqa: D102 return self.ot_prob.geom @property def a(self) -> jnp.ndarray: # noqa: D102 return self.ot_prob.a @property def b(self) -> jnp.ndarray: # noqa: D102 return self.ot_prob.b @property def linear_output(self) -> bool: # noqa: D102 return True @property def converged(self) -> bool: # noqa: D102 return jnp.logical_and( jnp.any(self.costs == -1), jnp.all(jnp.isfinite(self.costs)) ) @property def matrix(self) -> jnp.ndarray: """Transport matrix if it can be instantiated.""" return (self.q * self._inv_g) @ self.r.T
[docs] def apply(self, inputs: jnp.ndarray, axis: int = 0) -> jnp.ndarray: """Apply the transport to a array; axis=1 for its transpose.""" q, r = (self.q, self.r) if axis == 1 else (self.r, self.q) # for `axis=0`: (batch, m), (m, r), (r,), (r, n) return ((inputs @ r) * self._inv_g) @ q.T
[docs] def marginal(self, axis: int) -> jnp.ndarray: # noqa: D102 length = self.q.shape[0] if axis == 0 else self.r.shape[0] return self.apply(jnp.ones(length,), axis=axis)
[docs] def cost_at_geom(self, other_geom: geometry.Geometry) -> float: """Return OT cost for current solution, evaluated at any cost matrix.""" return jnp.sum(self.q * other_geom.apply_cost(self.r, axis=1) * self._inv_g)
[docs] def transport_cost_at_geom(self, other_geom: geometry.Geometry) -> float: """Return (by recomputing it) bare transport cost of current solution.""" return self.cost_at_geom(other_geom)
@property def primal_cost(self) -> float: """Return (by recomputing it) transport cost of current solution.""" return self.transport_cost_at_geom(other_geom=self.geom) @property def transport_mass(self) -> float: """Sum of transport matrix.""" return self.marginal(0).sum() @property def _inv_g(self) -> jnp.ndarray: return 1. / self.g
[docs]@jax.tree_util.register_pytree_node_class class LRSinkhorn(sinkhorn.Sinkhorn): r"""A Low-Rank Sinkhorn solver for linear reg-OT problems. The algorithm is described in :cite:`scetbon:21` and the implementation contained here is adapted from `LOT <>`_. The algorithm minimizes a non-convex problem. It therefore requires special care to initialization and convergence. Convergence is evaluated on successive evaluations of the objective. The algorithm is only provided for the balanced case. Args: rank: Rank constraint on the coupling to minimize the linear OT problem gamma: The (inverse of) gradient step size used by mirror descent. gamma_rescale: Whether to rescale :math:`\gamma` every iteration as described in :cite:`scetbon:22b`. epsilon: Entropic regularization added on top of low-rank problem. initializer: How to initialize the :math:`Q`, :math:`R` and :math:`g` factors. Valid options are `'random'`, `'rank2'`, `'k-means'`, and `'generalized-k-means`. If `None`, :class:`~ott.initializers.linear.initializers_lr.KMeansInitializer` is used when the linear problem's geometry is :class:`~ott.geometry.pointcloud.PointCloud` or :class:`~ott.geometry.low_rank.LRCGeometry`. Otherwise, use :class:`~ott.initializers.linear.initializers_lr.RandomInitializer`. lse_mode: Whether to run computations in lse or kernel mode. At the moment, only ``lse_mode = True`` is implemented. inner_iterations: Number of inner iterations used by the algorithm before re-evaluating progress. use_danskin: Use Danskin theorem to evaluate gradient of objective w.r.t. input parameters. Only `True` handled at this moment. implicit_diff: Whether to use implicit differentiation. Currently, only ``implicit_diff = False`` is implemented. kwargs_dys: Keyword arguments passed to :meth:`dykstra_update`. kwargs_init: Keyword arguments for :class:`~ott.initializers.linear.initializers_lr.LRInitializer`. kwargs: Keyword arguments for :class:`~ott.solvers.linear.sinkhorn.Sinkhorn`. """ def __init__( self, rank: int, gamma: float = 10., gamma_rescale: bool = True, epsilon: float = 0., initializer: Optional[Union[Literal["random", "rank2", "k-means", "generalized-k-means"], init_lib.LRInitializer]] = "random", lse_mode: bool = True, inner_iterations: int = 10, use_danskin: bool = True, implicit_diff: bool = False, kwargs_dys: Optional[Mapping[str, Any]] = None, kwargs_init: Optional[Mapping[str, Any]] = None, **kwargs: Any, ): assert lse_mode, "Kernel mode not yet implemented." assert not implicit_diff, "Implicit diff. not yet implemented." super().__init__( lse_mode=lse_mode, inner_iterations=inner_iterations, use_danskin=use_danskin, implicit_diff=implicit_diff, **kwargs ) self.rank = rank self.gamma = gamma self.gamma_rescale = gamma_rescale self.epsilon = epsilon self.initializer = initializer # can be `None` self.kwargs_dys = {} if kwargs_dys is None else kwargs_dys self.kwargs_init = {} if kwargs_init is None else kwargs_init def __call__( self, ot_prob: linear_problem.LinearProblem, init: Tuple[Optional[jnp.ndarray], Optional[jnp.ndarray], Optional[jnp.ndarray]] = (None, None, None), rng: jax.random.PRNGKeyArray = jax.random.PRNGKey(0), **kwargs: Any, ) -> LRSinkhornOutput: """Run low-rank Sinkhorn. Args: ot_prob: Linear OT problem. init: Initial values for the low-rank factors: - :attr:`~ott.solvers.linear.sinkhorn_lr.LRSinkhornOutput.q`. - :attr:`~ott.solvers.linear.sinkhorn_lr.LRSinkhornOutput.r`. - :attr:`~ott.solvers.linear.sinkhorn_lr.LRSinkhornOutput.g`. Any `None` values will be initialized using the initializer. rng: Random key for seeding. kwargs: Additional arguments when calling the initializer. Returns: The low-rank Sinkhorn output. """ assert ot_prob.is_balanced, "Unbalanced case is not implemented." initializer = self.create_initializer(ot_prob) init = initializer(ot_prob, *init, rng=rng, **kwargs) return run(ot_prob, self, init) def _lr_costs( self, ot_prob: linear_problem.LinearProblem, state: LRSinkhornState, ) -> Tuple[jnp.ndarray, jnp.ndarray, jnp.ndarray, float]: log_q, log_r, log_g = ( mu.safe_log(state.q), mu.safe_log(state.r), mu.safe_log(state.g) ) grad_q = ot_prob.geom.apply_cost(state.r, axis=1) / state.g[None, :] grad_r = ot_prob.geom.apply_cost(state.q) / state.g[None, :] diag_qcr = jnp.sum( state.q * ot_prob.geom.apply_cost(state.r, axis=1), axis=0 ) grad_g = -diag_qcr / (state.g ** 2) if self.is_entropic: grad_q += self.epsilon * log_q grad_r += self.epsilon * log_r grad_g += self.epsilon * log_g if self.gamma_rescale: norm_q = jnp.max(jnp.abs(grad_q)) ** 2 norm_r = jnp.max(jnp.abs(grad_r)) ** 2 norm_g = jnp.max(jnp.abs(grad_g)) ** 2 gamma = self.gamma / jnp.max(jnp.array([norm_q, norm_r, norm_g])) else: gamma = self.gamma c_q = grad_q - (1. / gamma) * log_q c_r = grad_r - (1. / gamma) * log_r h = -grad_g + (1. / gamma) * log_g return c_q, c_r, h, gamma
[docs] def dykstra_update( self, c_q: jnp.ndarray, c_r: jnp.ndarray, h: jnp.ndarray, gamma: float, ot_prob: linear_problem.LinearProblem, min_entry_value: float = 1e-6, tolerance: float = 1e-3, min_iter: int = 0, inner_iter: int = 10, max_iter: int = 10000 ) -> Tuple[jnp.ndarray, jnp.ndarray, jnp.ndarray]: """Run Dykstra's algorithm.""" # shortcuts for problem's definition. r = self.rank n, m = ot_prob.geom.shape loga, logb = jnp.log(ot_prob.a), jnp.log(ot_prob.b) h_old = h g1_old, g2_old = jnp.zeros(r), jnp.zeros(r) f1, f2 = jnp.zeros(n), jnp.zeros(m) w_gi, w_gp = jnp.zeros(r), jnp.zeros(r) w_q, w_r = jnp.zeros(r), jnp.zeros(r) err = jnp.inf state_inner = f1, f2, g1_old, g2_old, h_old, w_gi, w_gp, w_q, w_r, err constants = c_q, c_r, loga, logb def cond_fn( iteration: int, constants: Tuple[jnp.ndarray, ...], state_inner: Tuple[jnp.ndarray, ...] ) -> bool: del iteration, constants *_, err = state_inner return err > tolerance def _softm( f: jnp.ndarray, g: jnp.ndarray, c: jnp.ndarray, axis: int ) -> jnp.ndarray: return jsp.special.logsumexp( gamma * (f[:, None] + g[None, :] - c), axis=axis ) def body_fn( iteration: int, constants: Tuple[jnp.ndarray, ...], state_inner: Tuple[jnp.ndarray, ...], compute_error: bool ) -> Tuple[jnp.ndarray, ...]: # TODO(michalk8): in the future, use `NamedTuple` f1, f2, g1_old, g2_old, h_old, w_gi, w_gp, w_q, w_r, err = state_inner c_q, c_r, loga, logb = constants # First Projection f1 = jnp.where( jnp.isfinite(loga), (loga - _softm(f1, g1_old, c_q, axis=1)) / gamma + f1, loga ) f2 = jnp.where( jnp.isfinite(logb), (logb - _softm(f2, g2_old, c_r, axis=1)) / gamma + f2, logb ) h = h_old + w_gi h = jnp.maximum(jnp.log(min_entry_value) / gamma, h) w_gi += h_old - h h_old = h # Update couplings g_q = _softm(f1, g1_old, c_q, axis=0) g_r = _softm(f2, g2_old, c_r, axis=0) # Second Projection h = (1. / 3.) * (h_old + w_gp + w_q + w_r) h += g_q / (3. * gamma) h += g_r / (3. * gamma) g1 = h + g1_old - g_q / gamma g2 = h + g2_old - g_r / gamma w_q = w_q + g1_old - g1 w_r = w_r + g2_old - g2 w_gp = h_old + w_gp - h q, r, _ = recompute_couplings(f1, g1, c_q, f2, g2, c_r, h, gamma) g1_old = g1 g2_old = g2 h_old = h err = jax.lax.cond( jnp.logical_and(compute_error, iteration >= min_iter), lambda: solution_error(q, r, ot_prob, self.norm_error, self.lse_mode)[ 0], lambda: err ) return f1, f2, g1_old, g2_old, h_old, w_gi, w_gp, w_q, w_r, err def recompute_couplings( f1: jnp.ndarray, g1: jnp.ndarray, c_q: jnp.ndarray, f2: jnp.ndarray, g2: jnp.ndarray, c_r: jnp.ndarray, h: jnp.ndarray, gamma: float, ) -> Tuple[jnp.ndarray, jnp.ndarray, jnp.ndarray]: q = jnp.exp(gamma * (f1[:, None] + g1[None, :] - c_q)) r = jnp.exp(gamma * (f2[:, None] + g2[None, :] - c_r)) g = jnp.exp(gamma * h) return q, r, g state_inner = fixed_point_loop.fixpoint_iter_backprop( cond_fn, body_fn, min_iter, max_iter, inner_iter, constants, state_inner ) f1, f2, g1_old, g2_old, h_old, _, _, _, _, _ = state_inner return recompute_couplings(f1, g1_old, c_q, f2, g2_old, c_r, h_old, gamma)
[docs] def lse_step( self, ot_prob: linear_problem.LinearProblem, state: LRSinkhornState, iteration: int ) -> LRSinkhornState: """LR Sinkhorn LSE update.""" c_q, c_r, h, gamma = self._lr_costs(ot_prob, state) q, r, g = self.dykstra_update( c_q, c_r, h, gamma, ot_prob, **self.kwargs_dys ) return state.set(q=q, g=g, r=r, gamma=gamma)
[docs] def kernel_step( self, ot_prob: linear_problem.LinearProblem, state: LRSinkhornState, iteration: int ) -> NoReturn: """Not implemented.""" # TODO(cuturi): kernel step not implemented. raise NotImplementedError("Not implemented.")
[docs] def one_iteration( self, ot_prob: linear_problem.LinearProblem, state: LRSinkhornState, iteration: int, compute_error: bool ) -> LRSinkhornState: """Carries out one low-rank 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: LRSinkhornState named tuple. iteration: the current iteration of the Sinkhorn outer loop. compute_error: flag to indicate this iteration computes/stores an error Returns: The updated state. """ previous_state = state it = iteration // self.inner_iterations 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) # re-computes error if compute_error is True, else set it to inf. cost = jax.lax.cond( jnp.logical_and(compute_error, iteration >= self.min_iterations), lambda: state.reg_ot_cost(ot_prob), lambda: jnp.inf ) error = state.compute_error(previous_state) crossed_threshold = jnp.logical_or( state.crossed_threshold, jnp.logical_and( state.errors[it - 1] >= self.threshold, error < self.threshold ) ) return state.set([it].set(cost),[it].set(error), crossed_threshold=crossed_threshold, )
@property def norm_error(self) -> Tuple[int]: # noqa: D102 return self._norm_error, @property def is_entropic(self) -> bool: """Whether entropy regularization is used.""" return self.epsilon > 0.
[docs] def create_initializer( self, prob: linear_problem.LinearProblem ) -> init_lib.LRInitializer: """Create a low-rank Sinkhorn initializer. Args: prob: Linear OT problem used to determine the initializer. Returns: Low-rank initializer. """ if isinstance(self.initializer, init_lib.LRInitializer): initializer = self.initializer elif self.initializer is None: kind = "k-means" if isinstance( prob.geom, (pointcloud.PointCloud, low_rank.LRCGeometry) ) else "random" initializer = init_lib.LRInitializer.from_solver( self, kind=kind, **self.kwargs_init ) else: initializer = init_lib.LRInitializer.from_solver( self, kind=self.initializer, **self.kwargs_init ) assert initializer.rank == self.rank, \ f"Expected initializer of rank `{self.rank}`, " \ f"found `{initializer.rank}`." return initializer
[docs] def init_state( self, ot_prob: linear_problem.LinearProblem, init: Tuple[jnp.ndarray, jnp.ndarray, jnp.ndarray] ) -> LRSinkhornState: """Return the initial state of the loop.""" q, r, g = init return LRSinkhornState( q=q, r=r, g=g, gamma=self.gamma, costs=-jnp.ones(self.outer_iterations), errors=-jnp.ones(self.outer_iterations), crossed_threshold=False, )
[docs] def output_from_state( self, ot_prob: linear_problem.LinearProblem, state: LRSinkhornState ) -> LRSinkhornOutput: """Create an output from a loop state. Args: ot_prob: the transport problem. state: a LRSinkhornState. Returns: A LRSinkhornOutput. """ return LRSinkhornOutput( q=state.q, r=state.r, g=state.g, ot_prob=ot_prob, costs=state.costs, errors=state.errors, )
def _converged(self, state: LRSinkhornState, iteration: int) -> bool: def conv_crossed(prev_err: float, curr_err: float) -> bool: return jnp.logical_and( prev_err < self.threshold, curr_err < self.threshold ) def conv_not_crossed(prev_err: float, curr_err: float) -> bool: return jnp.logical_and(curr_err < prev_err, curr_err < self.threshold) # for convergence error, we consider 2 possibilities: # 1. we either crossed the convergence threshold; in this case we require # that the previous error was also below the threshold # 2. we haven't crossed the threshold; in this case, we can be below or # above the threshold: # if we're above, we wait until we reach the convergence threshold and # then, the above condition applies # if we're below and we improved w.r.t. the previous iteration, # we have converged; otherwise we continue, since we may be stuck # in a local minimum (e.g., during the initial iterations) it = iteration // self.inner_iterations return jax.lax.cond( state.crossed_threshold, conv_crossed, conv_not_crossed, state.errors[it - 2], state.errors[it - 1] ) def _diverged(self, state: LRSinkhornState, iteration: int) -> bool: it = iteration // self.inner_iterations return jnp.logical_and( jnp.logical_not(jnp.isfinite(state.errors[it - 1])), jnp.logical_not(jnp.isfinite(state.costs[it - 1])) )
def run( ot_prob: linear_problem.LinearProblem, solver: LRSinkhorn, init: Tuple[Optional[jnp.ndarray], Optional[jnp.ndarray], Optional[jnp.ndarray]], ) -> LRSinkhornOutput: """Run loop of the solver, outputting a state upgraded to an output.""" out = sinkhorn.iterations(ot_prob, solver, init) out = out.set_cost( ot_prob, lse_mode=solver.lse_mode, use_danskin=solver.use_danskin ) return out.set(ot_prob=ot_prob)