def cholesky(kron_a, name='t3f_kronecker_cholesky'):
"""Computes the Cholesky decomposition of a given Kronecker-factorized matrix.
Args:
kron_a: `TensorTrain` or `TensorTrainBatch` object containing a matrix or a
batch of matrices of size N x N, factorized into a Kronecker product of
square matrices (all tt-ranks are 1 and all tt-cores are square). All the
cores must be symmetric positive-definite.
name: string, name of the Op.
Returns:
`TensorTrain` object containing a TT-matrix of size N x N if the argument is
`TensorTrain`
`TensorTrainBatch` object, containing TT-matrices of size N x N if the
argument is `TensorTrainBatch`
Raises:
ValueError if the tt-cores of the provided matrix are not square,
or the tt-ranks are not 1.
"""
if not _is_kron(kron_a):
raise ValueError('The argument should be a Kronecker product '
'(tt-ranks should be 1)')
shapes_defined = kron_a.get_shape().is_fully_defined()
if shapes_defined:
i_shapes = kron_a.get_raw_shape()[0]
j_shapes = kron_a.get_raw_shape()[1]
else:
i_shapes = ops.raw_shape(kron_a)[0]
j_shapes = ops.raw_shape(kron_a)[1]
if shapes_defined:
if i_shapes != j_shapes:
raise ValueError(
'The argument should be a Kronecker product of square '
'matrices (tt-cores must be square)')
is_batch = isinstance(kron_a, TensorTrainBatch)
with tf.name_scope(name):
cho_cores = []
for core_idx in range(kron_a.ndims()):
core = kron_a.tt_cores[core_idx]
if is_batch:
core_cho = tf.linalg.cholesky(core[:, 0, :, :, 0])
core_cho = tf.expand_dims(tf.expand_dims(core_cho, 1), -1)
else:
core_cho = tf.linalg.cholesky(core[0, :, :, 0])
core_cho = tf.expand_dims(tf.expand_dims(core_cho, 0), -1)
cho_cores.append(core_cho)
res_ranks = kron_a.get_tt_ranks()
res_shape = kron_a.get_raw_shape()
if is_batch:
return TensorTrainBatch(cho_cores, res_shape, res_ranks)
else:
return TensorTrain(cho_cores, res_shape, res_ranks)
def slog_determinant(kron_a, name='t3f_kronecker_slog_determinant'):
"""Computes the sign and log-det of a given Kronecker-factorized matrix.
Args:
kron_a: `TensorTrain` or `TensorTrainBatch` object containing a matrix or a
batch of matrices of size N x N, factorized into a Kronecker product of
square matrices (all tt-ranks are 1 and all tt-cores are square).
name: string, name of the Op.
Returns:
Two number or two Tensor with numbers for each element in the batch.
Sign of the determinant and the log-determinant of the given
matrix. If the determinant is zero, then sign will be 0 and logdet will be
-Inf. In all cases, the determinant is equal to sign * np.exp(logdet).
Raises:
ValueError if the tt-cores of the provided matrix are not square,
or the tt-ranks are not 1.
"""
if not _is_kron(kron_a):
raise ValueError('The argument should be a Kronecker product '
'(tt-ranks should be 1)')
shapes_defined = kron_a.get_shape().is_fully_defined()
if shapes_defined:
i_shapes = kron_a.get_raw_shape()[0].as_list()
j_shapes = kron_a.get_raw_shape()[1].as_list()
else:
i_shapes = ops.raw_shape(kron_a)[0].as_list()
j_shapes = ops.raw_shape(kron_a)[1].as_list()
if shapes_defined:
if i_shapes != j_shapes:
raise ValueError(
'The argument should be a Kronecker product of square '
'matrices (tt-cores must be square)')
is_batch = isinstance(kron_a, TensorTrainBatch)
with tf.name_scope(name):
pows = tf.cast(tf.reduce_prod(i_shapes), kron_a.dtype)
logdet = 0.
det_sign = 1.
for core_idx in range(kron_a.ndims()):
core = kron_a.tt_cores[core_idx]
if is_batch:
core_det = tf.linalg.det(core[:, 0, :, :, 0])
else:
core_det = tf.linalg.det(core[0, :, :, 0])
core_abs_det = tf.abs(core_det)
core_det_sign = tf.sign(core_det)
core_pow = pows / i_shapes[core_idx]
logdet += tf.math.log(core_abs_det) * core_pow
det_sign *= core_det_sign**(core_pow)
return det_sign, logdet
def inv(kron_a):
"""Computes the inverse of a given Kronecker-factorized matrix.
Args:
kron_a: `TensorTrain` or `TensorTrainBatch` object containing a matrix or a
batch of matrices of size N x N, factorized into a Kronecker product of
square matrices (all tt-ranks are 1 and all tt-cores are square).
Returns:
`TensorTrain` object containing a TT-matrix of size N x N if the argument is
`TensorTrain`
`TensorTrainBatch` object, containing TT-matrices of size N x N if the
argument is `TensorTrainBatch`
Raises:
ValueError if the tt-cores of the provided matrix are not square,
or the tt-ranks are not 1.
"""
if not _is_kron(kron_a):
raise ValueError('The argument should be a Kronecker product '
'(tt-ranks should be 1)')
shapes_defined = kron_a.get_shape().is_fully_defined()
if shapes_defined:
i_shapes = kron_a.get_raw_shape()[0]
j_shapes = kron_a.get_raw_shape()[1]
else:
i_shapes = ops.raw_shape(kron_a)[0]
j_shapes = ops.raw_shape(kron_a)[1]
if shapes_defined:
if i_shapes != j_shapes:
raise ValueError(
'The argument should be a Kronecker product of square '
'matrices (tt-cores must be square)')
is_batch = isinstance(kron_a, TensorTrainBatch)
inv_cores = []
for core_idx in range(kron_a.ndims()):
core = kron_a.tt_cores[core_idx]
if is_batch:
core_inv = tf.matrix_inverse(core[:, 0, :, :, 0])
core_inv = tf.expand_dims(tf.expand_dims(core_inv, 1), -1)
else:
core_inv = tf.matrix_inverse(core[0, :, :, 0])
core_inv = tf.expand_dims(tf.expand_dims(core_inv, 0), -1)
inv_cores.append(core_inv)
res_ranks = kron_a.get_tt_ranks()
res_shape = kron_a.get_raw_shape()
if is_batch:
return TensorTrainBatch(inv_cores, res_shape, res_ranks)
else:
return TensorTrain(inv_cores, res_shape, res_ranks)
def determinant(kron_a, name='t3f_kronecker_determinant'):
"""Computes the determinant of a given Kronecker-factorized matrix.
Note, that this method can suffer from overflow.
Args:
kron_a: `TensorTrain` or `TensorTrainBatch` object containing a matrix or a
batch of matrices of size N x N, factorized into a Kronecker product of
square matrices (all tt-ranks are 1 and all tt-cores are square).
name: string, name of the Op.
Returns:
A number or a Tensor with numbers for each element in the batch.
The determinant of the given matrix.
Raises:
ValueError if the tt-cores of the provided matrix are not square,
or the tt-ranks are not 1.
"""
if not _is_kron(kron_a):
raise ValueError(
'The argument should be a Kronecker product (tt-ranks '
'should be 1)')
shapes_defined = kron_a.get_shape().is_fully_defined()
if shapes_defined:
i_shapes = kron_a.get_raw_shape()[0].as_list()
j_shapes = kron_a.get_raw_shape()[1].as_list()
else:
i_shapes = ops.raw_shape(kron_a)[0].as_list()
j_shapes = ops.raw_shape(kron_a)[1].as_list()
if shapes_defined:
if i_shapes != j_shapes:
raise ValueError(
'The argument should be a Kronecker product of square '
'matrices (tt-cores must be square)')
is_batch = isinstance(kron_a, TensorTrainBatch)
with tf.name_scope(name):
pows = tf.cast(tf.reduce_prod(i_shapes), kron_a.dtype)
cores = kron_a.tt_cores
det = 1
for core_idx in range(kron_a.ndims()):
core = cores[core_idx]
if is_batch:
core_det = tf.linalg.det(core[:, 0, :, :, 0])
else:
core_det = tf.linalg.det(core[0, :, :, 0])
core_pow = pows / i_shapes[core_idx]
det *= tf.pow(core_det, core_pow)
return det
def slog_determinant(kron_a):
"""Computes the sign and log-det of a given Kronecker-factorized matrix.
Args:
kron_a: `TensorTrain` object containing a matrix of size N x N,
factorized into a Kronecker product of square matrices (all
tt-ranks are 1 and all tt-cores are square).
Returns:
Two numbers, sign of the determinant and the log-determinant of the given
matrix. If the determinant is zero, then sign will be 0 and logdet will be
-Inf. In all cases, the determinant is equal to sign * np.exp(logdet).
Raises:
ValueError if the tt-cores of the provided matrix are not square,
or the tt-ranks are not 1.
"""
if not _is_kron(kron_a):
raise ValueError('The argument should be a Kronecker product '
'(tt-ranks should be 1)')
shapes_defined = kron_a.get_shape().is_fully_defined()
if shapes_defined:
i_shapes = kron_a.get_raw_shape()[0]
j_shapes = kron_a.get_raw_shape()[1]
else:
i_shapes = ops.raw_shape(kron_a)[0]
j_shapes = ops.raw_shape(kron_a)[1]
if shapes_defined:
if i_shapes != j_shapes:
raise ValueError('The argument should be a Kronecker product of square '
'matrices (tt-cores must be square)')
pows = tf.cast(tf.reduce_prod(i_shapes), kron_a.dtype)
logdet = 0.
det_sign = 1.
for core_idx in range(kron_a.ndims()):
core = kron_a.tt_cores[core_idx]
core_det = tf.matrix_determinant(core[0, :, :, 0])
core_abs_det = tf.abs(core_det)
core_det_sign = tf.sign(core_det)
core_pow = pows / i_shapes[core_idx].value
logdet += tf.log(core_abs_det) * core_pow
det_sign *= core_det_sign**(core_pow)
return det_sign, logdet
def determinant(kron_a):
"""Computes the determinant of a given Kronecker-factorized matrix.
Note, that this method can suffer from overflow.
Args:
kron_a: `TensorTrain` object containing a matrix of size N x N,
factorized into a Kronecker product of square matrices (all
tt-ranks are 1 and all tt-cores are square).
Returns:
Number, the determinant of the given matrix.
Raises:
ValueError if the tt-cores of the provided matrix are not square,
or the tt-ranks are not 1.
"""
if not _is_kron(kron_a):
raise ValueError('The argument should be a Kronecker product (tt-ranks '
'should be 1)')
shapes_defined = kron_a.get_shape().is_fully_defined()
if shapes_defined:
i_shapes = kron_a.get_raw_shape()[0]
j_shapes = kron_a.get_raw_shape()[1]
else:
i_shapes = ops.raw_shape(kron_a)[0]
j_shapes = ops.raw_shape(kron_a)[1]
if shapes_defined:
if i_shapes != j_shapes:
raise ValueError('The argument should be a Kronecker product of square '
'matrices (tt-cores must be square)')
pows = tf.cast(tf.reduce_prod(i_shapes), kron_a.dtype)
cores = kron_a.tt_cores
det = 1
for core_idx in range(kron_a.ndims()):
core = cores[core_idx]
core_det = tf.matrix_determinant(core[0, :, :, 0])
core_pow = pows / i_shapes[core_idx].value
det *= tf.pow(core_det, core_pow)
return det
def cholesky(kron_a):
"""Computes the Cholesky decomposition of a given Kronecker-factorized matrix.
Args:
kron_a: `TensorTrain` object containing a matrix of size N x N,
factorized into a Kronecker product of square matrices (all
tt-ranks are 1 and all tt-cores are square). All the cores
must be symmetric positive-definite.
Returns:
`TensorTrain` object, containing a TT-matrix of size N x N.
Raises:
ValueError if the tt-cores of the provided matrix are not square,
or the tt-ranks are not 1.
"""
if not _is_kron(kron_a):
raise ValueError('The argument should be a Kronecker product '
'(tt-ranks should be 1)')
shapes_defined = kron_a.get_shape().is_fully_defined()
if shapes_defined:
i_shapes = kron_a.get_raw_shape()[0]
j_shapes = kron_a.get_raw_shape()[1]
else:
i_shapes = ops.raw_shape(kron_a)[0]
j_shapes = ops.raw_shape(kron_a)[1]
if shapes_defined:
if i_shapes != j_shapes:
raise ValueError('The argument should be a Kronecker product of square '
'matrices (tt-cores must be square)')
cho_cores = []
for core_idx in range(kron_a.ndims()):
core = kron_a.tt_cores[core_idx]
core_cho = tf.cholesky(core[0, :, :, 0])
cho_cores.append(tf.expand_dims(tf.expand_dims(core_cho, 0), -1))
res_ranks = kron_a.get_tt_ranks()
res_shape = kron_a.get_raw_shape()
return TensorTrain(cho_cores, res_shape, res_ranks)