# Copyright 2016 The TensorFlow Authors. All Rights Reserved.
#
# 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
# 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
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# ==============================================================================
"""Solvers for linear least-squares."""

from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

import collections

from tensorflow.contrib.solvers.python.ops import util
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import control_flow_ops
from tensorflow.python.ops import math_ops


def cgls(operator, rhs, tol=1e-6, max_iter=20, name="cgls"):
  r"""Conjugate gradient least squares solver.

  Solves a linear least squares problem \\(||A x - rhs||_2\\) for a single
  right-hand side, using an iterative, matrix-free algorithm where the action of
  the matrix A is represented by `operator`. The CGLS algorithm implicitly
  applies the symmetric conjugate gradient algorithm to the normal equations
  \\(A^* A x = A^* rhs\\). The iteration terminates when either
  the number of iterations exceeds `max_iter` or when the norm of the conjugate
  residual (residual of the normal equations) have been reduced to `tol` times
  its initial initial value, i.e.
  \\(||A^* (rhs - A x_k)|| <= tol ||A^* rhs||\\).

  Args:
    operator: An object representing a linear operator with attributes:
      - shape: Either a list of integers or a 1-D `Tensor` of type `int32` of
        length 2. `shape[0]` is the dimension on the domain of the operator,
        `shape[1]` is the dimension of the co-domain of the operator. On other
        words, if operator represents an M x N matrix A, `shape` must contain
        `[M, N]`.
      - dtype: The datatype of input to and output from `apply` and
        `apply_adjoint`.
      - apply: Callable object taking a vector `x` as input and returning a
        vector with the result of applying the operator to `x`, i.e. if
       `operator` represents matrix `A`, `apply` should return `A * x`.
      - apply_adjoint: Callable object taking a vector `x` as input and
        returning a vector with the result of applying the adjoint operator
        to `x`, i.e. if `operator` represents matrix `A`, `apply_adjoint` should
        return `conj(transpose(A)) * x`.

    rhs: A rank-1 `Tensor` of shape `[M]` containing the right-hand size vector.
    tol: A float scalar convergence tolerance.
    max_iter: An integer giving the maximum number of iterations.
    name: A name scope for the operation.


  Returns:
    output: A namedtuple representing the final state with fields:
      - i: A scalar `int32` `Tensor`. Number of iterations executed.
      - x: A rank-1 `Tensor` of shape `[N]` containing the computed solution.
      - r: A rank-1 `Tensor` of shape `[M]` containing the residual vector.
      - p: A rank-1 `Tensor` of shape `[N]`. The next descent direction.
      - gamma: \\(||A^* r||_2^2\\)
  """
  # ephemeral class holding CGLS state.
  cgls_state = collections.namedtuple("CGLSState",
                                      ["i", "x", "r", "p", "gamma"])

  def stopping_criterion(i, state):
    return math_ops.logical_and(i < max_iter, state.gamma > tol)

  # TODO(rmlarsen): add preconditioning
  def cgls_step(i, state):
    q = operator.apply(state.p)
    alpha = state.gamma / util.l2norm_squared(q)
    x = state.x + alpha * state.p
    r = state.r - alpha * q
    s = operator.apply_adjoint(r)
    gamma = util.l2norm_squared(s)
    beta = gamma / state.gamma
    p = s + beta * state.p
    return i + 1, cgls_state(i + 1, x, r, p, gamma)

  with ops.name_scope(name):
    n = operator.shape[1:]
    rhs = array_ops.expand_dims(rhs, -1)
    s0 = operator.apply_adjoint(rhs)
    gamma0 = util.l2norm_squared(s0)
    tol = tol * tol * gamma0
    x = array_ops.expand_dims(
        array_ops.zeros(
            n, dtype=rhs.dtype.base_dtype), -1)
    i = constant_op.constant(0, dtype=dtypes.int32)
    state = cgls_state(i=i, x=x, r=rhs, p=s0, gamma=gamma0)
    _, state = control_flow_ops.while_loop(stopping_criterion, cgls_step,
                                           [i, state])
    return cgls_state(
        state.i,
        x=array_ops.squeeze(state.x),
        r=array_ops.squeeze(state.r),
        p=array_ops.squeeze(state.p),
        gamma=state.gamma)
