transform.py

# Copyright 2021 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
#
#     https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

"""
Sparsify transform
==================

This is an experimental JAX transform that will allow arbitrary JAX functions to accept
sparse matrices as inputs, so long as sparse rules are implemented for the primitives
called by the function.

For example:

>>> import jax.numpy as jnp
>>> from jax import random
>>> from jax.experimental.sparse import BCOO, sparsify

>>> mat = random.uniform(random.PRNGKey(1701), (5, 5))
>>> mat = mat.at[mat < 0.5].set(0)
>>> vec = random.uniform(random.PRNGKey(42), (5,))

>>> def f(mat, vec):
...   return -(jnp.sin(mat) @ vec)
...
>>> f(mat, vec)
DeviceArray([-1.2655463 , -0.52060574, -0.14522289, -0.10817424,
             -0.15574613], dtype=float32)

>>> mat_sparse = BCOO.fromdense(mat)
>>> mat_sparse
BCOO(float32[5, 5], nse=8)

>>> sparsify(f)(mat_sparse, vec)
DeviceArray([-1.2655463 , -0.52060574, -0.14522289, -0.10817424,
             -0.15574613], dtype=float32)
"""

import functools
from typing import (
  Any, Callable, Dict, NamedTuple, List, Optional, Sequence, Tuple, Union)

import numpy as np

from jax import core
from jax import lax
from jax import linear_util as lu
from jax.experimental.sparse.bcoo import bcoo_multiply_dense, bcoo_multiply_sparse
import jax.numpy as jnp
from jax._src.api_util import flatten_fun_nokwargs
from jax.interpreters import partial_eval as pe
from jax.interpreters import xla
from jax.tree_util import tree_flatten, tree_map, tree_unflatten
from jax.util import safe_map, safe_zip, split_list
from jax._src.config import config
from jax._src.lax.control_flow import _check_tree_and_avals
from jax._src.numpy import lax_numpy
from jax._src.util import canonicalize_axis
from jax.experimental import sparse
from jax.experimental.sparse import BCOO

sparse_rules : Dict[core.Primitive, Callable] = {}

_zero_preserving_unary_primitives = [
  lax.abs_p,
  lax.asin_p,
  lax.asinh_p,
  lax.atan_p,
  lax.atanh_p,
  lax.bessel_i1e_p,
  lax.copy_p,
  lax.expm1_p,
  lax.log1p_p,
  lax.neg_p,
  lax.real_p,
  lax.imag_p,
  lax.sign_p,
  lax.sin_p,
  lax.sinh_p,
  lax.sqrt_p,
  lax.tan_p,
  lax.tanh_p,
  lax.convert_element_type_p
]

_densifying_primitives : List[core.Primitive] = [
  lax.acos_p,
  lax.acosh_p,
  lax.bessel_i0e_p,
  lax.cos_p,
  lax.cosh_p,
  lax.eq_p,
  lax.exp_p,
  lax.ge_p,
  lax.gt_p,
  lax.le_p,
  lax.lt_p,
  lax.log_p,
  lax.ne_p,
  lax.xor_p
]

def _raise_unimplemented_primitive(primitive):
  if primitive in _densifying_primitives:
    raise NotImplementedError(f"sparse rule for {primitive} is not implemented because it "
                              "would result in dense output. If this is your intent, use "
                              "sparse.todense() to convert your arguments to dense matrices.")
  raise NotImplementedError(f"sparse rule for {primitive} is not implemented.")


Array = Any
ArrayOrSparse = Any


class SparsifyEnv:
  """Environment for sparse jaxpr evaluation.

  The environment is essentially a collection of buffers and/or tracers
  that may be shared between one or more SparsifyValue objects, which
  represent sparse or dense arrays via indices into the list of buffers.
  """
  _buffers : List[Array]

  def __init__(self, bufs=()):
    self._buffers = list(bufs)

  def _push(self, arr: Array) -> int:
    self._buffers.append(jnp.asarray(arr))  # type: ignore
    return len(self._buffers) - 1

  def data(self, spvalue: 'SparsifyValue') -> Array:
    """Get the data buffer associated with a SparsifyValue."""
    if spvalue.data_ref is None:
      raise RuntimeError("Internal: requested data from spvalue with data_ref=None")
    return self._buffers[spvalue.data_ref]

  def indices(self, spvalue: 'SparsifyValue') -> Array:
    """Get the indices buffer associated with a SparsifyValue."""
    if spvalue.indices_ref is None:
      raise RuntimeError("Internal: requested indices from spvalue with indices_ref=None")
    return self._buffers[spvalue.indices_ref]

  def dense(self, data):
    """Add a new dense array to the sparsify environment."""
    return SparsifyValue(np.shape(data), self._push(data), None)

  def sparse(self, shape, data=None, indices=None,
             *, data_ref=None, indices_ref=None, indices_sorted=False,
             unique_indices=False):
    """Add a new sparse array to the sparsify environment."""
    if data is not None:
      assert data_ref is None
      data_ref = self._push(data)
    else:
      assert data_ref is not None and data_ref < len(self._buffers)

    if indices is not None:
      assert indices_ref is None
      indices_ref = self._push(indices)
    else:
      assert indices_ref is not None and indices_ref < len(self._buffers)

    return SparsifyValue(shape, data_ref, indices_ref, indices_sorted,
                         unique_indices)


class SparsifyValue(NamedTuple):
  shape: Tuple[int, ...]
  data_ref: Optional[int]
  indices_ref: Optional[int]
  indices_sorted: Optional[bool] = False
  unique_indices: Optional[bool] = False

  @property
  def ndim(self):
    return len(self.shape)

  def is_sparse(self):
    return self.indices_ref is not None

  def is_dense(self):
    return self.indices_ref is None


_is_bcoo = lambda arg: isinstance(arg, BCOO)
_is_spvalue = lambda arg: isinstance(arg, SparsifyValue)


def arrays_to_spvalues(
    spenv: SparsifyEnv,
    args: Any
    ) -> Any:
  """Convert a pytree of (sparse) arrays to an equivalent pytree of spvalues."""
  def array_to_spvalue(arg):
    if isinstance(arg, BCOO):
      return spenv.sparse(arg.shape, arg.data, arg.indices,
                          indices_sorted=arg.indices_sorted,
                          unique_indices=arg.unique_indices)
    else:
      return spenv.dense(arg)
  return tree_map(array_to_spvalue, args, is_leaf=_is_bcoo)


def spvalues_to_arrays(
    spenv: SparsifyEnv,
    spvalues: Any,
    ) -> Any:
  """Convert a pytree of spvalues to an equivalent pytree of (sparse) arrays."""
  def spvalue_to_array(spvalue):
    if spvalue.is_sparse():
      assert spvalue.indices_ref is not None
      return BCOO((spenv.data(spvalue), spenv.indices(spvalue)),
                  shape=spvalue.shape, indices_sorted=spvalue.indices_sorted,
                  unique_indices=spvalue.unique_indices)
    else:
      return spenv.data(spvalue)
  return tree_map(spvalue_to_array, spvalues, is_leaf=_is_spvalue)


def spvalues_to_avals(
    spenv: SparsifyEnv,
    spvalues: Any,
    ) -> Any:
  """Convert a pytree of spvalues to an equivalent pytree of abstract values."""
  def spvalue_to_aval(spvalue):
    data = spenv.data(spvalue)
    return core.ShapedArray(spvalue.shape, data.dtype, data.aval.weak_type)
  return tree_map(spvalue_to_aval, spvalues, is_leaf=_is_spvalue)


#------------------------------------------------------------------------------
# Implementation of sparsify() using tracers.

def popattr(obj, name):
  assert hasattr(obj, name)
  val = getattr(obj, name)
  delattr(obj, name)
  return val

def setnewattr(obj, name, val):
  assert not hasattr(obj, name)
  setattr(obj, name, val)

class SparseTracer(core.Tracer):
  def __init__(self, trace: core.Trace, *, spvalue):
    self._spvalue = spvalue
    self._trace = trace

  @property
  def spenv(self):
    if not hasattr(self._trace.main, 'spenv'):
      raise RuntimeError("Internal: main does not have spenv defined.")
    return self._trace.main.spenv

  @property
  def aval(self):
    return spvalues_to_avals(self.spenv, [self._spvalue])[0]

  def full_lower(self):
    return self

class SparseTrace(core.Trace):
  def pure(self, val: Any):
    if not hasattr(self.main, 'spenv'):
      raise RuntimeError("Internal: main does not have spenv defined.")
    spvalue, = arrays_to_spvalues(self.main.spenv, [val])
    return SparseTracer(self, spvalue=spvalue)

  def lift(self, val: core.Tracer):
    if not hasattr(self.main, 'spenv'):
      raise RuntimeError("Internal: main does not have spenv defined.")
    spvalue, = arrays_to_spvalues(self.main.spenv, [val])
    return SparseTracer(self, spvalue=spvalue)

  def sublift(self, val: SparseTracer):
    return SparseTracer(val._trace, spvalue=val._spvalue)

  def process_primitive(self, primitive, tracers, params):
    spenv = popattr(self.main, 'spenv')
    spvalues = [t._spvalue for t in tracers]
    if any(spvalue.is_sparse() for spvalue in spvalues):
      if primitive not in sparse_rules:
        _raise_unimplemented_primitive(primitive)
      out_spvalues = sparse_rules[primitive](spenv, *(t._spvalue for t in tracers), **params)
    else:
      out_bufs = primitive.bind(*(spenv.data(spvalue) for spvalue in spvalues), **params)
      out_spvalues = arrays_to_spvalues(spenv, out_bufs if primitive.multiple_results else [out_bufs])
    setnewattr(self.main, 'spenv', spenv)
    out_tracers = tuple(SparseTracer(self, spvalue=spvalue) for spvalue in out_spvalues)
    return out_tracers if primitive.multiple_results else out_tracers[0]

  def process_call(self, call_primitive, f: lu.WrappedFun, tracers, params):
    spenv = popattr(self.main, 'spenv')
    spvalues = tuple(t._spvalue for t in tracers)
    in_bufs = spenv._buffers
    fun, out_spvalues = sparsify_subtrace(f, self.main, spvalues)
    if any(params['donated_invars']):
      raise NotImplementedError("sparsify does not support donated_invars")
    params = dict(params, donated_invars=tuple(False for buf in in_bufs))
    bufs_out = call_primitive.bind(fun, *in_bufs, **params)
    setnewattr(self.main, 'spenv', SparsifyEnv(bufs_out))
    return [SparseTracer(self, spvalue=spvalue) for spvalue in out_spvalues()]

@lu.transformation_with_aux
def sparsify_subtrace(main, spvalues, *bufs):
  setnewattr(main, 'spenv', SparsifyEnv(bufs))
  trace = main.with_cur_sublevel()
  in_tracers = [SparseTracer(trace, spvalue=spvalue) for spvalue in spvalues]
  outs = yield in_tracers, {}
  out_traces = [trace.full_raise(out) for out in outs]
  buffers = popattr(main, 'spenv')._buffers
  yield buffers, [out._spvalue for out in out_traces]

def sparsify_fun(wrapped_fun, args: List[ArrayOrSparse]):
  with core.new_main(SparseTrace) as main:
    spenv = SparsifyEnv()
    spvalues = arrays_to_spvalues(spenv, args)
    in_bufs = spenv._buffers
    fun, out_spvalues = sparsify_subtrace(wrapped_fun, main, spvalues)
    out_bufs = fun.call_wrapped(*in_bufs)
    spenv = SparsifyEnv(out_bufs)
    del main
  return spvalues_to_arrays(spenv, out_spvalues())

def _sparsify_with_tracer(fun):
  """Implementation of sparsify() using tracers."""
  @functools.wraps(fun)
  def _wrapped(*args):
    args_flat, in_tree = tree_flatten(args, is_leaf=_is_bcoo)
    wrapped_fun, out_tree = flatten_fun_nokwargs(lu.wrap_init(fun), in_tree)
    out = sparsify_fun(wrapped_fun, args_flat)
    return tree_unflatten(out_tree(), out)
  return _wrapped

#------------------------------------------------------------------------------
# Implementation of sparsify() using a jaxpr interpreter.

def eval_sparse(
    jaxpr: core.Jaxpr,
    consts: Sequence[Array],  # all consts are dense
    spvalues: Sequence[SparsifyValue],  # mix of sparse and dense pointers into spenv
    spenv: SparsifyEnv,
) -> Sequence[SparsifyValue]:
  env : Dict[core.Var, SparsifyValue] = {}

  def read(var: core.Var) -> Union[Array, SparsifyValue]:
    # all literals are dense
    if isinstance(var, core.Literal):
      return spenv.dense(var.val)
    else:
      return env[var]

  def write_buffer(var: core.Var, a: Array) -> None:
    if isinstance(var, core.DropVar):
      return
    env[var] = spenv.dense(a)

  def write(var: core.Var, a: SparsifyValue) -> None:
    if isinstance(var, core.DropVar):
      return
    assert a is not None
    env[var] = a

  safe_map(write_buffer, jaxpr.constvars, consts)
  safe_map(write, jaxpr.invars, spvalues)

  for eqn in jaxpr.eqns:
    prim = eqn.primitive
    invals = safe_map(read, eqn.invars)

    if any(val.is_sparse() for val in invals):
      if prim not in sparse_rules:
        _raise_unimplemented_primitive(prim)
      out = sparse_rules[prim](spenv, *invals, **eqn.params)
    else:
      if prim is xla.xla_call_p:
        # TODO(vanderplas,frostig): workaround for binding call primitives
        # within a jaxpr interpreter
        params = eqn.params.copy()
        fun = lu.wrap_init(core.jaxpr_as_fun(pe.ClosedJaxpr(params.pop('call_jaxpr'), ())))
        out_bufs = prim.bind(fun, *(spenv.data(val) for val in invals), **params)
      else:
        out_bufs = prim.bind(*(spenv.data(val) for val in invals), **eqn.params)
      out_bufs = out_bufs if prim.multiple_results else [out_bufs]
      out = []
      for buf, outvar in safe_zip(out_bufs, eqn.outvars):
        if isinstance(outvar, core.DropVar):
          out.append(None)
        else:
          out.append(spenv.dense(buf))
    safe_map(write, eqn.outvars, out)

  return safe_map(read, jaxpr.outvars)

def sparsify_raw(f):
  def wrapped(spenv: SparsifyEnv, *spvalues: SparsifyValue, **params: Any) -> Tuple[Sequence[SparsifyValue], bool]:
    spvalues_flat, in_tree = tree_flatten(spvalues, is_leaf=_is_spvalue)
    in_avals_flat = spvalues_to_avals(spenv, spvalues_flat)
    wrapped_fun, out_tree = flatten_fun_nokwargs(lu.wrap_init(f, params), in_tree)
    jaxpr, out_avals_flat, consts = pe.trace_to_jaxpr_dynamic(wrapped_fun, in_avals_flat)
    result = eval_sparse(jaxpr, consts, spvalues_flat, spenv)
    if len(out_avals_flat) != len(result):
      raise Exception("Internal: eval_sparse does not return expected number of arguments. "
                      "Got {result} for avals {out_avals_flat}")
    return result, out_tree()
  return wrapped

def _sparsify_with_interpreter(f):
  """Implementation of sparsify() using jaxpr interpreter."""
  f_raw = sparsify_raw(f)
  @functools.wraps(f)
  def wrapped(*args, **params):
    spenv = SparsifyEnv()
    spvalues = arrays_to_spvalues(spenv, args)
    spvalues_out, out_tree = f_raw(spenv, *spvalues, **params)
    out = spvalues_to_arrays(spenv, spvalues_out)
    return tree_unflatten(out_tree, out)
  return wrapped

def sparsify(f, use_tracer=False):
  """Experimental sparsification transform.

  Examples:

    Decorate JAX functions to make them compatible with :class:`jax.experimental.sparse.BCOO`
    matrices:

    >>> from jax.experimental import sparse

    >>> @sparse.sparsify
    ... def f(M, v):
    ...   return 2 * M.T @ v

    >>> M = sparse.BCOO.fromdense(jnp.arange(12).reshape(3, 4))

    >>> v = jnp.array([3, 4, 2])

    >>> f(M, v)
    DeviceArray([ 64,  82, 100, 118], dtype=int32)
  """
  if use_tracer:
    return _sparsify_with_tracer(f)
  else:
    return _sparsify_with_interpreter(f)


#------------------------------------------------------------------------------
# Sparse rules for various primitives

def _ensure_unique_indices(spenv, spvalue):
  """Return an spvalue representation with deduplicated indices."""
  if spvalue.is_dense() or spvalue.unique_indices:
    return spvalue
  arr = spvalues_to_arrays(spenv, spvalue)
  arr = arr.sum_duplicates(nse=arr.nse)
  return arrays_to_spvalues(spenv, arr)

def _zero_preserving_unary_op(prim):
  def func(spenv, *spvalues, **kwargs):
    assert len(spvalues) == 1
    # Since unary operations don't commute with addition, we need to ensure
    # that indices are unique before applying the operator elementwise.
    spvalue = _ensure_unique_indices(spenv, spvalues[0])
    buf = spenv.data(spvalue)
    buf_out = prim.bind(buf, **kwargs)
    if spvalues[0].is_sparse():
      out_spvalue = spenv.sparse(spvalue.shape, buf_out,
                                 indices_ref=spvalue.indices_ref,
                                 indices_sorted=spvalue.indices_sorted,
                                 unique_indices=spvalue.unique_indices)
    else:
      out_spvalue = spenv.dense(buf)
    return (out_spvalue,)
  return func

for _prim in _zero_preserving_unary_primitives:
  sparse_rules[_prim] = _zero_preserving_unary_op(_prim)

def _dot_general_sparse(spenv, *spvalues, dimension_numbers, precision, preferred_element_type):
  # TODO(jakevdp): pass along these unused configurations?
  del precision, preferred_element_type  # unused
  result = sparse.bcoo_dot_general(*spvalues_to_arrays(spenv, spvalues),
                                   dimension_numbers=dimension_numbers)
  return arrays_to_spvalues(spenv, [result])

sparse_rules[lax.dot_general_p] = _dot_general_sparse

def _transpose_sparse(spenv, *spvalues, permutation):
  permutation = tuple(permutation)
  args = spvalues_to_arrays(spenv, spvalues)
  shape = args[0].shape
  mat_transposed = sparse.bcoo_transpose(args[0], permutation=permutation)
  out_shape = tuple(shape[i] for i in permutation)

  n_batch = args[0].indices.ndim - 2
  n_sparse = args[0].indices.shape[-1]
  batch_dims_unchanged = (permutation[:n_batch] == tuple(range(n_batch)))
  dense_dims_unchanged = (permutation[n_batch + n_sparse:] == tuple(range(n_batch + n_sparse, len(shape))))
  sparse_dims_unchanged = (permutation[n_batch:n_batch + n_sparse] == tuple(range(n_batch, n_batch + n_sparse)))

  # Data is unchanged if batch & dense dims are not permuted
  kwds = {}
  if batch_dims_unchanged and dense_dims_unchanged:
    kwds['data_ref'] = spvalues[0].data_ref
  else:
    kwds['data'] = mat_transposed.data

  # Indices unchanged if batch & sparse dims are not permuted
  if batch_dims_unchanged and sparse_dims_unchanged:
    kwds['indices_ref'] = spvalues[0].indices_ref
  else:
    kwds['indices'] = mat_transposed.indices

  kwds['indices_sorted'] = mat_transposed.indices_sorted
  kwds['unique_indices'] = mat_transposed.unique_indices
  spvalue = spenv.sparse(out_shape, **kwds)
  return (spvalue,)

sparse_rules[lax.transpose_p] = _transpose_sparse

def _add_sparse(spenv, *spvalues):
  X, Y = spvalues
  if X.is_sparse() and Y.is_sparse():
    if X.shape != Y.shape:
      raise NotImplementedError("Addition between sparse matrices of different shapes.")
    if X.indices_ref == Y.indices_ref:
      out_data = lax.add(spenv.data(X), spenv.data(Y))
      if config.jax_enable_checks:
        assert X.indices_sorted == Y.indices_sorted
        assert X.unique_indices == Y.unique_indices
      out_spvalue = spenv.sparse(X.shape, out_data, indices_ref=X.indices_ref,
                                 indices_sorted=X.indices_sorted,
                                 unique_indices=X.unique_indices)
    elif spenv.indices(X).ndim != spenv.indices(Y).ndim or spenv.data(X).ndim != spenv.data(Y).ndim:
      raise NotImplementedError("Addition between sparse matrices with different batch/dense dimensions.")
    else:
      out_indices = lax.concatenate([spenv.indices(X), spenv.indices(Y)], dimension=spenv.indices(X).ndim - 2)
      out_data = lax.concatenate([spenv.data(X), spenv.data(Y)], dimension=spenv.indices(X).ndim - 2)
      out_spvalue = spenv.sparse(X.shape, out_data, out_indices)
  else:
    raise NotImplementedError("Addition between sparse and dense array.")

  return (out_spvalue,)

sparse_rules[lax.add_p] = _add_sparse

def _sub_sparse(spenv, *spvalues):
  X, Y = spvalues
  if X.is_sparse() and Y.is_sparse():
    return _add_sparse(spenv, X, *sparse_rules[lax.neg_p](spenv, Y))
  else:
    raise NotImplementedError("Subtraction between sparse and dense array.")

sparse_rules[lax.sub_p] = _sub_sparse

def _mul_sparse(spenv, *spvalues):
  X, Y = spvalues
  if X.is_sparse() and Y.is_sparse():
    if X.indices_ref == Y.indices_ref and X.unique_indices:
      if config.jax_enable_checks:
        assert X.indices_sorted == Y.indices_sorted
        assert X.unique_indices == Y.unique_indices
      out_data = lax.mul(spenv.data(X), spenv.data(Y))
      out_spvalue = spenv.sparse(X.shape, out_data, indices_ref=X.indices_ref,
                                 indices_sorted=X.indices_sorted,
                                 unique_indices=True)
    else:
      X_promoted, Y_promoted = spvalues_to_arrays(spenv, spvalues)
      mat = bcoo_multiply_sparse(X_promoted, Y_promoted)
      out_spvalue = spenv.sparse(mat.shape, mat.data, mat.indices)
  else:
    if Y.is_sparse():
      X, Y = Y, X
    X_promoted = spvalues_to_arrays(spenv, X)
    out_data = bcoo_multiply_dense(X_promoted, spenv.data(Y))
    out_spvalue = spenv.sparse(X.shape, out_data, indices_ref=X.indices_ref,
                               indices_sorted=X.indices_sorted,
                               unique_indices=X.unique_indices)

  return (out_spvalue,)

sparse_rules[lax.mul_p] = _mul_sparse

def _reduce_sum_sparse(spenv, *spvalues, axes):
  X, = spvalues
  X_promoted = spvalues_to_arrays(spenv, X)
  mat = sparse.bcoo_reduce_sum(X_promoted, axes=axes)
  out_shape = mat.shape
  if out_shape == ():
    out_spvalue = spenv.dense(mat.data.sum())
  else:
    out_spvalue = spenv.sparse(out_shape, mat.data, mat.indices)
  return (out_spvalue,)

sparse_rules[lax.reduce_sum_p] = _reduce_sum_sparse

def _broadcast_in_dim_sparse(spenv, *spvalues, shape, broadcast_dimensions):
  operand, = spvalues
  operand_promoted = spvalues_to_arrays(spenv, operand)
  mat = sparse.bcoo_broadcast_in_dim(operand_promoted, shape=shape,
                                     broadcast_dimensions=broadcast_dimensions)
  return (spenv.sparse(shape, mat.data, mat.indices),)

sparse_rules[lax.broadcast_in_dim_p] = _broadcast_in_dim_sparse

def _concatenate_sparse(spenv, *spvalues, dimension):
  operands = spvalues_to_arrays(spenv, spvalues)
  result = sparse.bcoo_concatenate(operands, dimension=dimension)
  return arrays_to_spvalues(spenv, (result,))

sparse_rules[lax.concatenate_p] = _concatenate_sparse

def _squeeze_sparse(spenv, *spvalues, dimensions):
  arr, = spvalues
  dimensions = tuple(canonicalize_axis(dim, arr.ndim) for dim in dimensions)
  if any(arr.shape[dim] != 1 for dim in dimensions):
    raise ValueError("cannot select an axis to squeeze out which has size not equal to one, "
                     f"got shape={arr.shape} and dimensions={dimensions}")
  data = spenv.data(arr)
  indices = spenv.indices(arr)
  n_sparse = indices.shape[-1]
  n_batch = indices.ndim - 2
  batch_dims = tuple(d for d in dimensions if d < n_batch)
  sparse_dims = np.array([i for i in range(n_sparse) if i + n_batch not in dimensions], dtype=int)
  dense_dims = tuple(d - n_sparse + 1 for d in dimensions if d >= n_batch + n_sparse)
  data_out = lax.squeeze(data, batch_dims + dense_dims)
  indices_out = lax.squeeze(indices[..., sparse_dims], batch_dims)
  out_shape = tuple(s for i, s in enumerate(arr.shape) if i not in dimensions)
  return (spenv.sparse(out_shape, data_out, indices_out),)

sparse_rules[lax.squeeze_p] = _squeeze_sparse

def _reshape_sparse(spenv, *spvalues, new_sizes, dimensions):
  operand, = spvalues_to_arrays(spenv, spvalues)
  result = sparse.bcoo_reshape(operand, new_sizes=new_sizes, dimensions=dimensions)
  return arrays_to_spvalues(spenv, (result,))

sparse_rules[lax.reshape_p] = _reshape_sparse

def _sparsify_jaxpr(spenv, jaxpr, *spvalues):
  # TODO(jakevdp): currently this approach discards all information about
  #   shared data & indices when generating the sparsified jaxpr. The
  #   current approach produces valid sparsified while loops, but they
  #   don't work in corner cases (see associated TODO in sparsify_test.py)
  out_tree = None

  @lu.wrap_init
  def wrapped(*args_flat):
    # TODO(frostig,jakevdp): This closes over `spenv`, which can bring
    # in buffers from the "outer scope" as constants. Is this a
    # problem for primitives like cond and while_loop, which always
    # convert constvars to invars when staging out their subjaxprs?
    nonlocal out_tree
    args = tree_unflatten(in_tree, args_flat)
    spvalues = arrays_to_spvalues(spenv, args)
    result = eval_sparse(jaxpr.jaxpr, jaxpr.consts, spvalues, spenv)
    out = spvalues_to_arrays(spenv, result)
    out_flat, out_tree = tree_flatten(out)
    return out_flat

  args = spvalues_to_arrays(spenv, spvalues)
  args_flat, in_tree = tree_flatten(args)
  avals_flat = [core.raise_to_shaped(core.get_aval(arg)) for arg in args_flat]
  sp_jaxpr, _, consts = pe.trace_to_jaxpr_dynamic(wrapped, avals_flat)
  sp_jaxpr = pe.ClosedJaxpr(sp_jaxpr, consts)
  return sp_jaxpr, out_tree

def _while_sparse(spenv, *spvalues, cond_jaxpr, cond_nconsts, body_jaxpr, body_nconsts):
  cond_const_spvalues, body_const_spvalues, init_val_spvalues = split_list(
    spvalues, [cond_nconsts, body_nconsts])

  cond_sp_jaxpr, _ = _sparsify_jaxpr(spenv, cond_jaxpr, *cond_const_spvalues, *init_val_spvalues)
  body_sp_jaxpr, out_tree = _sparsify_jaxpr(spenv, body_jaxpr, *body_const_spvalues, *init_val_spvalues)

  cond_consts, _ = tree_flatten(spvalues_to_arrays(spenv, cond_const_spvalues))
  body_consts, _ = tree_flatten(spvalues_to_arrays(spenv, body_const_spvalues))
  init_vals, _ = tree_flatten(spvalues_to_arrays(spenv, init_val_spvalues))

  out_flat = lax.while_p.bind(*cond_consts, *body_consts, *init_vals,
                              cond_nconsts=len(cond_consts), cond_jaxpr=cond_sp_jaxpr,
                              body_nconsts=len(body_consts), body_jaxpr=body_sp_jaxpr)
  return arrays_to_spvalues(spenv, tree_unflatten(out_tree, out_flat))

sparse_rules[lax.while_p] = _while_sparse

def _xla_call_sparse(spenv, *spvalues, call_jaxpr, donated_invars, **params):
  if any(donated_invars):
    raise NotImplementedError("sparse xla_call with donated_invars")
  sp_call_jaxpr, out_tree = _sparsify_jaxpr(spenv, pe.ClosedJaxpr(call_jaxpr, ()), *spvalues)
  fun = lu.wrap_init(core.jaxpr_as_fun(sp_call_jaxpr))
  args_flat, _ = tree_flatten(spvalues_to_arrays(spenv, spvalues))
  donated_invars = tuple(False for arg in args_flat)
  out_flat = xla.xla_call_p.bind(fun, *args_flat, donated_invars=donated_invars, **params)
  return arrays_to_spvalues(spenv, tree_unflatten(out_tree, out_flat))

sparse_rules[xla.xla_call_p] = _xla_call_sparse

def _duplicate_for_sparse_spvalues(spvalues, params):
  for spvalue, param in safe_zip(spvalues, params):
    yield from [param, param] if spvalue.is_sparse() else [param]

def _scan_sparse(spenv, *spvalues, jaxpr, num_consts, num_carry, **params):
  const_spvalues, carry_spvalues, xs_spvalues = split_list(
    spvalues, [num_consts, num_carry])
  if xs_spvalues:
    # TODO(jakevdp): we don't want to pass xs_spvalues, we want to pass one row
    # of xs spvalues. How to do this?
    raise NotImplementedError("sparse rule for scan with x values.")
  sp_jaxpr, _ = _sparsify_jaxpr(spenv, jaxpr, *const_spvalues, *carry_spvalues, *xs_spvalues)

  consts, _ = tree_flatten(spvalues_to_arrays(spenv, const_spvalues))
  carry, carry_tree = tree_flatten(spvalues_to_arrays(spenv, carry_spvalues))
  xs, xs_tree = tree_flatten(spvalues_to_arrays(spenv, xs_spvalues))

  # params['linear'] has one entry per arg; expand it to match the sparsified args.
  const_linear, carry_linear, xs_linear = split_list(
    params.pop('linear'), [num_consts, num_carry])
  sp_linear = tuple([
    *_duplicate_for_sparse_spvalues(const_spvalues, const_linear),
    *_duplicate_for_sparse_spvalues(carry_spvalues, carry_linear),
    *_duplicate_for_sparse_spvalues(xs_spvalues, xs_linear)])

  out = lax.scan_p.bind(*consts, *carry, *xs, jaxpr=sp_jaxpr, linear=sp_linear,
                        num_consts=len(consts), num_carry=len(carry), **params)
  carry_out = tree_unflatten(carry_tree, out[:len(carry)])
  xs_out = tree_unflatten(xs_tree, out[len(carry):])
  return arrays_to_spvalues(spenv, carry_out + xs_out)

sparse_rules[lax.scan_p] = _scan_sparse

def _cond_sparse(spenv, pred, *operands, branches, linear, **params):
  sp_branches, treedefs = zip(*(_sparsify_jaxpr(spenv, jaxpr, *operands)
                                for jaxpr in branches))
  _check_tree_and_avals("sparsified true_fun and false_fun output",
                        treedefs[0], sp_branches[0].out_avals,
                        treedefs[1], sp_branches[1].out_avals)
  sp_linear = tuple(_duplicate_for_sparse_spvalues(operands, linear))
  args, _ = tree_flatten(spvalues_to_arrays(spenv, (pred, *operands)))
  out_flat = lax.cond_p.bind(*args, branches=sp_branches, linear=sp_linear, **params)
  out = tree_unflatten(treedefs[0], out_flat)
  return arrays_to_spvalues(spenv, out)

sparse_rules[lax.cond_p] = _cond_sparse

def _todense_sparse_rule(spenv, spvalue, *, tree):
  del tree  # TODO(jakvdp): we should assert that tree is PytreeDef(*)
  out = spvalues_to_arrays(spenv, spvalue).todense()
  return (spenv.dense(out),)

sparse_rules[sparse.todense_p] = _todense_sparse_rule

def _slice_sparse_rule(spenv, *operands, **params):
  args = spvalues_to_arrays(spenv, operands)
  out = sparse.bcoo_slice(*args, **params)
  return arrays_to_spvalues(spenv, [out])

sparse_rules[lax.slice_p] = _slice_sparse_rule

def _dynamic_slice_sparse_rule(spenv, *operands, **params):
  args = spvalues_to_arrays(spenv, operands)
  out = sparse.bcoo_dynamic_slice(args[0], args[1:], **params)
  return arrays_to_spvalues(spenv, [out])

sparse_rules[lax.dynamic_slice_p] = _dynamic_slice_sparse_rule


#------------------------------------------------------------------------------
# BCOO methods derived from sparsify
# defined here to avoid circular imports

def _sum(self, *args, **kwargs):
  """Sum array along axis."""
  return sparsify(lambda x: x.sum(*args, **kwargs))(self)

def _reshape(self, *args, **kwargs):
  """Sum array along axis."""
  return sparsify(lambda x: x.reshape(*args, **kwargs))(self)

def _sparse_rewriting_take(arr, idx, indices_are_sorted=False, unique_indices=False,
                           mode=None, fill_value=None):
  # mirrors lax_numpy._rewriting_take.

  # Handle some special cases, falling back if error messages might differ.
  if (arr.ndim > 0 and isinstance(idx, (int, np.integer)) and
      not isinstance(idx, (bool, np.bool_)) and isinstance(arr.shape[0], int)):
    if 0 <= idx < arr.shape[0]:
      return sparsify(lambda arr: lax.index_in_dim(arr, idx, keepdims=False))(arr)
  if (arr.ndim > 0 and isinstance(arr.shape[0], int) and
      isinstance(idx, slice) and
      (type(idx.start) is int or idx.start is None) and
      (type(idx.stop)  is int or idx.stop is  None) and
      (type(idx.step)  is int or idx.step is  None)):
    n = arr.shape[0]
    start = idx.start if idx.start is not None else 0
    stop  = idx.stop  if idx.stop  is not None else n
    step  = idx.step  if idx.step  is not None else 1
    if (0 <= start < n and 0 <= stop <= n and 0 < step and
        (start, stop, step) != (0, n, 1)):
      return sparsify(lambda arr: lax.slice_in_dim(arr, start, stop, step))(arr)

  treedef, static_idx, dynamic_idx = lax_numpy._split_index_for_jit(idx, arr.shape)
  result = sparsify(
      lambda arr, idx: lax_numpy._gather(arr, treedef, static_idx, idx, indices_are_sorted,
                                         unique_indices, mode, fill_value))(arr, dynamic_idx)
  # Account for a corner case in the rewriting_take implementation.
  if not isinstance(result, BCOO) and np.size(result) == 0:
    result = BCOO.fromdense(result)
  return result

_swap_args = lambda f: lambda a, b: f(b, a)

_bcoo_methods = {
  'reshape': _reshape,
  'sum': _sum,
  "__neg__": sparsify(jnp.negative),
  "__pos__": sparsify(jnp.positive),
  "__matmul__": sparsify(jnp.matmul),
  "__rmatmul__": sparsify(_swap_args(jnp.matmul)),
  "__mul__": sparsify(jnp.multiply),
  "__rmul__": sparsify(_swap_args(jnp.multiply)),
  "__add__": sparsify(jnp.add),
  "__radd__": sparsify(_swap_args(jnp.add)),
  "__sub__": sparsify(jnp.subtract),
  "__rsub__": sparsify(_swap_args(jnp.subtract)),
  "__getitem__": _sparse_rewriting_take,
}

for method, impl in _bcoo_methods.items():
  setattr(BCOO, method, impl)