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)
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)
def cg_step(i, state): z = operator.apply(state.p) alpha = state.gamma / util.dot(state.p, z) x = state.x + alpha * state.p r = state.r - alpha * z gamma = util.l2norm_squared(r) beta = gamma / state.gamma p = r + beta * state.p return i + 1, cg_state(i + 1, x, r, p, gamma)
def cg_step(i, state):
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
gamma = util.l2norm_squared(r)
beta = gamma / state.gamma
p = r + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
def testL2Norm(self):
with self.test_session():
x_np = np.array([[2], [-3.], [5.]])
x_norm_np = np.linalg.norm(x_np)
x_normalized_np = x_np / x_norm_np
x = constant_op.constant(x_np)
l2norm = util.l2norm(x)
l2norm_squared = util.l2norm_squared(x)
x_normalized, x_norm = util.l2normalize(x)
self.assertAllClose(l2norm.eval(), x_norm_np)
self.assertAllClose(l2norm_squared.eval(), np.square(x_norm_np))
self.assertAllClose(x_norm.eval(), x_norm_np)
self.assertAllClose(x_normalized.eval(), x_normalized_np)
def testL2Norm(self):
with self.test_session():
x_np = np.array([[2], [-3.], [5.]])
x_norm_np = np.linalg.norm(x_np)
x_normalized_np = x_np / x_norm_np
x = constant_op.constant(x_np)
l2norm = util.l2norm(x)
l2norm_squared = util.l2norm_squared(x)
x_normalized, x_norm = util.l2normalize(x)
self.assertAllClose(l2norm.eval(), x_norm_np)
self.assertAllClose(l2norm_squared.eval(), np.square(x_norm_np))
self.assertAllClose(x_norm.eval(), x_norm_np)
self.assertAllClose(x_normalized.eval(), x_normalized_np)
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
righ-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)
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
righ-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 tf.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 tf.name_scope(name):
n = operator.shape[1:]
rhs = tf.expand_dims(rhs, -1)
s0 = operator.apply_adjoint(rhs)
gamma0 = util.l2norm_squared(s0)
tol = tol * tol * gamma0
x = tf.expand_dims(tf.zeros(n, dtype=rhs.dtype.base_dtype), -1)
i = tf.constant(0, dtype=tf.int32)
state = cgls_state(i=i, x=x, r=rhs, p=s0, gamma=gamma0)
_, state = tf.while_loop(stopping_criterion, cgls_step, [i, state])
return cgls_state(
state.i,
x=tf.squeeze(state.x),
r=tf.squeeze(state.r),
p=tf.squeeze(state.p),
gamma=state.gamma)
def conjugate_gradient(operator,
rhs,
tol=1e-4,
max_iter=20,
name="conjugate_gradient"):
r"""Conjugate gradient solver.
Solves a linear system of equations `A*x = rhs` for selfadjoint, positive
definite matrix `A` and righ-hand side vector `rhs`, using an iterative,
matrix-free algorithm where the action of the matrix A is represented by
`operator`. The iteration terminates when either the number of iterations
exceeds `max_iter` or when the residual norm has been reduced to `tol`
times its initial value, i.e. \\(||rhs - A x_k|| <= tol ||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 N x N matrix A, `shape` must contain
`[N, N]`.
- dtype: The datatype of input to and output from `apply`.
- 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`.
rhs: A rank-1 `Tensor` of shape `[N]` 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]`. `A`-conjugate basis vector.
- gamma: \\(||r||_2^2\\)
"""
# ephemeral class holding CG state.
cg_state = collections.namedtuple("CGState", ["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 cg_step(i, state):
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
gamma = util.l2norm_squared(r)
beta = gamma / state.gamma
p = r + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
with ops.name_scope(name):
n = operator.shape[1:]
rhs = array_ops.expand_dims(rhs, -1)
gamma0 = util.l2norm_squared(rhs)
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 = cg_state(i=i, x=x, r=rhs, p=rhs, gamma=gamma0)
_, state = control_flow_ops.while_loop(stopping_criterion, cg_step,
[i, state])
return cg_state(
state.i,
x=array_ops.squeeze(state.x),
r=array_ops.squeeze(state.r),
p=array_ops.squeeze(state.p),
gamma=state.gamma)
def conjugate_gradient(operator,
rhs,
tol=1e-4,
max_iter=20,
name="conjugate_gradient"):
r"""Conjugate gradient solver.
Solves a linear system of equations `A*x = rhs` for selfadjoint, positive
definite matrix `A` and righ-hand side vector `rhs`, using an iterative,
matrix-free algorithm where the action of the matrix A is represented by
`operator`. The iteration terminates when either the number of iterations
exceeds `max_iter` or when the residual norm has been reduced to `tol`
times its initial value, i.e. \\(||rhs - A x_k|| <= tol ||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 N x N matrix A, `shape` must contain
`[N, N]`.
- dtype: The datatype of input to and output from `apply`.
- 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`.
rhs: A rank-1 `Tensor` of shape `[N]` 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]`. `A`-conjugate basis vector.
- gamma: \\(||r||_2^2\\)
"""
# ephemeral class holding CG state.
cg_state = collections.namedtuple("CGState", ["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 id:1554 gh:1555
def cg_step(i, state):
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
gamma = util.l2norm_squared(r)
beta = gamma / state.gamma
p = r + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
with ops.name_scope(name):
n = operator.shape[1:]
rhs = array_ops.expand_dims(rhs, -1)
gamma0 = util.l2norm_squared(rhs)
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 = cg_state(i=i, x=x, r=rhs, p=rhs, gamma=gamma0)
_, state = control_flow_ops.while_loop(stopping_criterion, cg_step,
[i, state])
return cg_state(state.i,
x=array_ops.squeeze(state.x),
r=array_ops.squeeze(state.r),
p=array_ops.squeeze(state.p),
gamma=state.gamma)