def tensor_to_vector(tt, vertical=True):
"""Converts TT-tensor to TT-matrix with column shape equals to I = (1, ..., 1). If by_columns is False,
then row shape equals to I.
:type tt: `TT`
:param tt: TT-tensor
:type vertical: `bool`
:param vertical: defines, whether tt will be located by columns or by rows
:return: `TT-Matrix`
:rtype: `TT-Matrix`
:raises [ValueError]: if the argument is not a TT-tensor
"""
if not isinstance(tt, TT):
raise ValueError('The argument should be a TT-tensor, not TT-matrix.')
cores = []
for core in tt.tt_cores:
if vertical:
cores.append(
jnp.reshape(core,
(core.shape[0], core.shape[1], 1, core.shape[2]),
order="F"))
else:
cores.append(
jnp.reshape(core,
(core.shape[0], 1, core.shape[1], core.shape[2]),
order="F"))
return TTMatrix(cores)
def _mul_by_scalar(tt, c):
tt = unwrap_tt(tt)
cores = list(tt.tt_cores)
cores[0] = c * cores[0]
if tt.is_tt_matrix:
return TTMatrix(cores)
else:
return TT(cores)
def deltas_to_tangent(deltas: List[jnp.ndarray],
tt: TTTensOrMat) -> TTTensOrMat:
"""Converts deltas representation of tangent space vector to `TT-object`.
Takes as input a list of [dP1, ..., dPd] and returns
dP1 V2 ... Vd + U1 dP2 V3 ... Vd + ... + U1 ... Ud-1 dPd.
This function is hard to use correctly because deltas should obey the
so called gauge conditions. If they don't, the function will silently return
incorrect result. That is why this function is not imported in __init__.
:param deltas: a list of deltas (essentially `TT-cores`) obeying the gauge
conditions.
:param tt: object on which the tangent space tensor represented
by delta is projected.
:type tt: `TT-Tensor` or `TT-Matrix`
:return: object constructed from deltas, that is from the tangent
space at point `tt`.
:rtype: `TT-Tensor` or `TT-Matrix`
"""
cores = []
dtype = tt.dtype
left = orthogonalize(tt)
right = orthogonalize(left, left_to_right=False)
left_rank_dim = 0
right_rank_dim = 3 if tt.is_tt_matrix else 2
for i in range(tt.ndim):
left_tt_core = left.tt_cores[i]
right_tt_core = right.tt_cores[i]
if i == 0:
tangent_core = jnp.concatenate((deltas[i], left_tt_core),
axis=right_rank_dim)
elif i == tt.ndim - 1:
tangent_core = jnp.concatenate((right_tt_core, deltas[i]),
axis=left_rank_dim)
else:
rank_1 = right.tt_ranks[i]
rank_2 = left.tt_ranks[i + 1]
if tt.is_tt_matrix:
mode_size_n = tt.raw_tensor_shape[0][i]
mode_size_m = tt.raw_tensor_shape[1][i]
shape = [rank_1, mode_size_n, mode_size_m, rank_2]
else:
mode_size_n = tt.shape[i]
shape = [rank_1, mode_size_n, rank_2]
zeros = jnp.zeros(shape, dtype=dtype)
upper = jnp.concatenate((right_tt_core, zeros),
axis=right_rank_dim)
lower = jnp.concatenate((deltas[i], left_tt_core),
axis=right_rank_dim)
tangent_core = jnp.concatenate((upper, lower), axis=left_rank_dim)
cores.append(tangent_core)
if tt.is_tt_matrix:
return TTMatrix(cores)
else:
return TT(cores)
def matrix(rng, shape, tt_rank=2, batch_shape=None, dtype=jnp.float32):
"""Generate a random `TT-Matrix` of the given shape and `TT-rank`.
:param rng: JAX PRNG key
:type rng: random state is described by two unsigned 32-bit integers
:param shape: desired tensor shape. Also supports omitting one of the dimensions
matrix(..., shape=((2, 2, 2), None))
and
matrix(..., shape=(None, (2, 2, 2)))
will create an 8-element column/row vector.
:type shape: tuple
:param tt_rank: desired `TT-ranks` of `TT-Matrix`
:type tt_rank: single number for equal `TT-ranks` or array specifying all `TT-ranks`
:param batch_shape: desired batch shape of `TT-Matrix`
:type batch_shape: tuple
:param dtype: type of elements in `TT-Matrix`
:type dtype: `dtype`
:return: generated `TT-Matrix`
:rtype: TTMatrix
:raises [ValueError]: if shape is (None, None)
"""
if shape == (None, None):
raise ValueError("At least one of shape elements must not be None")
if None in shape:
shape = [
np.array(shape[0])
if shape[0] is not None else np.ones(len(shape[1]), dtype=int),
np.array(shape[1])
if shape[1] is not None else np.ones(len(shape[0]), dtype=int),
]
else:
shape = [np.array(shape[0]), np.array(shape[1])]
tt_rank = np.array(tt_rank)
batch_shape = list(batch_shape) if batch_shape else []
num_dims = shape[0].size
if tt_rank.size == 1:
tt_rank = tt_rank * np.ones(num_dims - 1)
tt_rank = np.insert(tt_rank, 0, 1)
tt_rank = np.append(tt_rank, 1)
tt_rank = tt_rank.astype(int)
tt_cores = []
rng_arr = jax.random.split(rng, num_dims)
for i in range(num_dims):
curr_core_shape = [
tt_rank[i], shape[0][i], shape[1][i], tt_rank[i + 1]
]
curr_core_shape = batch_shape + curr_core_shape
tt_cores.append(
jax.random.normal(rng_arr[i], curr_core_shape, dtype=dtype))
return TTMatrix(tt_cores)
def vectorized_func(*args, **kwargs):
tt_arg = args[0] # TODO: what if only kwargs are present?
if tt_arg.num_batch_dims == 0:
return func(*args, **kwargs)
else:
if num_batch_args is not None:
num_non_batch_args = len(args) + len(
kwargs) - num_batch_args
in_axis = [0] * num_batch_args + [None
] * num_non_batch_args
num_args = num_batch_args
else:
num_args = len(args) + len(kwargs)
in_axis = [0] * num_args
if num_args > 1 and (isinstance(args[1], TTMatrix)
or isinstance(args[1], TT)):
if args[0].is_tt_matrix != args[1].is_tt_matrix:
raise ValueError(
'Types of the arguments are different.')
if not are_batches_broadcastable(args[0], args[1]):
raise ValueError(
'The batch sizes are different and not 1, '
'broadcasting is not available.')
broadcast_shape = np.maximum(list(args[0].batch_shape),
list(args[1].batch_shape))
new_args = list(args)
if args[0].is_tt_matrix:
for i, tt in enumerate(args[:2]):
new_cores = []
for core in tt.tt_cores:
core = jnp.broadcast_to(
core,
list(broadcast_shape) +
list(core.shape[-4:]))
new_cores.append(core)
new_args[i] = TTMatrix(new_cores)
else:
for i, tt in enumerate(args[:2]):
new_cores = []
for core in tt.tt_cores:
core = jnp.broadcast_to(
core,
list(broadcast_shape) +
list(core.shape[-3:]))
new_cores.append(core)
new_args[i] = TT(new_cores)
else:
new_args = args
vmapped = func
for _ in range(tt_arg.num_batch_dims):
vmapped = jax.vmap(vmapped, in_axis)
return vmapped(*new_args, **kwargs)
def transpose(tt):
"""Transpose a TT-matrix or a batch of TT-matrices.
:type tt: `TT-Matrix` object containing TT-matrix or batch of TT-matrices
:param tt: TT-matrix or batch of TT-matrices
:return: `TT-Matrix` object containing TT-matrix or batch of TT-matrices
:rtype: `TT-Matrix` object containing TT-matrix or batch of TT-matrices
:raises [ValueError]: if the argument is not a TT-matrix
"""
if not isinstance(tt, TTMatrix) or not tt.is_tt_matrix:
raise ValueError('The argument should be a TT-matrix.')
transposed_tt_cores = [
jnp.transpose(tt.tt_cores[core_idx], (0, 2, 1, 3))
for core_idx in range(tt.ndim)
]
return TTMatrix(transposed_tt_cores)
def new_func(*args):
are_tt_matrix_inputs = args[0].is_tt_matrix
tt_einsum_ = tt_einsum.resolve_i_or_ij(are_tt_matrix_inputs)
is_fusing = any([isinstance(tt, WrappedTT) for tt in args])
if is_fusing:
# Have to use a different name to make upper level tt_einsum visible.
tt_einsum_, args = _fuse_tt_einsums(tt_einsum_, args)
einsum = tt_einsum_.to_vanilla_einsum()
num_batch_dims = args[0].num_batch_dims
# TODO: support broadcasting.
res_batch_shape = list(args[0].batch_shape)
# TODO: do in parallel w.r.t. cores.
# TODO: use optimal einsum.
res_cores = []
for i in range(len(args[0].tt_cores)):
curr_input_cores = [tt.tt_cores[i] for tt in args]
core = oe.contract(einsum, *curr_input_cores, backend='jax')
shape = core.shape[num_batch_dims:]
num_left_rank_dims = len(tt_einsum_.output[0])
num_tensor_dims = len(tt_einsum_.output[1])
split_points = (num_left_rank_dims, num_left_rank_dims + num_tensor_dims)
new_shape = np.split(shape, split_points)
left_rank = np.prod(new_shape[0])
right_rank = np.prod(new_shape[2])
new_shape = [left_rank] + new_shape[1].tolist() + [right_rank]
new_shape = res_batch_shape + new_shape
res_cores.append(core.reshape(new_shape))
if are_tt_matrix_inputs:
res = TTMatrix(res_cores)
else:
res = TT(res_cores)
if is_fusing:
res = WrappedTT(res, args, tt_einsum_)
return res
def add(tt_a, tt_b):
"""Returns a `TT-object` corresponding to elementwise sum `tt_a + tt_b`.
The shapes of `tt_a` and `tt_b` should coincide.
Supports broadcasting, e.g. you can add a tensor train with
batch size 7 and a tensor train with batch size 1:
``tt_batch.add(tt_single.batch_loc[np.newaxis])``
where ``tt_single.batch_loc[np.newaxis]``
creates a singleton batch dimension.
:type tt_a: `TT-Tensor` or `TT-Matrix`
:param tt_a: first argument
:type tt_b: `TT-Tensor` or `TT-Matrix`
:param tt_b: second argument
:rtype: `TT-Tensor` or `TT-Matrix`
:return: `tt_a + tt_b`
:raises [ValueError]: if the arguments shapes do not coincide
"""
tt_a = unwrap_tt(tt_a)
tt_b = unwrap_tt(tt_b)
if not are_shapes_equal(tt_a, tt_b):
raise ValueError('Types of the arguments or their tensor '
'shapes are different, addition is not '
'available.')
if not are_batches_broadcastable(tt_a, tt_b):
raise ValueError('The batch sizes are different and not 1, '
'broadcasting is not available.')
if tt_a.is_tt_matrix:
tt_cores = _add_matrix_cores(tt_a, tt_b)
return TTMatrix(tt_cores)
else:
tt_cores = _add_tensor_cores(tt_a, tt_b)
return TT(tt_cores)
def _orthogonalize_tt_cores_right_to_left(tt):
"""Orthogonalize TT-cores of a TT-object.
Args:
tt: TT-tensor or TT-matrix.
Returns:
TT-tensor or TT-matrix.
"""
# Right to left orthogonalization.
num_dims = tt.ndim
if tt.is_tt_matrix:
raw_shape = tt.raw_tensor_shape
else:
raw_shape = tt.shape
tt_ranks = tt.tt_ranks
prev_rank = tt_ranks[num_dims]
# Copy cores references so we can change the cores.
tt_cores = list(tt.tt_cores)
for core_idx in range(num_dims - 1, 0, -1):
curr_core = tt_cores[core_idx]
# TT-ranks could have changed on the previous iteration, so `tt_ranks` can
# be outdated for the current TT-rank, but should be valid for the next
# TT-rank.
curr_rank = prev_rank
prev_rank = tt_ranks[core_idx]
if tt.is_tt_matrix:
curr_mode_left = raw_shape[0][core_idx]
curr_mode_right = raw_shape[1][core_idx]
curr_mode = curr_mode_left * curr_mode_right
else:
curr_mode = raw_shape[core_idx]
qr_shape = (prev_rank, curr_mode * curr_rank)
curr_core = jnp.reshape(curr_core, qr_shape)
curr_core, triang = jnp.linalg.qr(curr_core.T)
curr_core = curr_core.T
triang = triang.T
triang_shape = triang.shape
# The TT-rank could have changed: if qr_shape is e.g. 4 x 10, than q would
# be of size 4 x 4 and r would be 4 x 10, which means that the next rank
# should be changed to 4.
prev_rank = triang_shape[1]
if tt.is_tt_matrix:
new_core_shape = (prev_rank, curr_mode_left, curr_mode_right, curr_rank)
else:
new_core_shape = (prev_rank, curr_mode, curr_rank)
tt_cores[core_idx] = jnp.reshape(curr_core, new_core_shape)
prev_core = jnp.reshape(tt_cores[core_idx - 1], (-1, triang_shape[0]))
tt_cores[core_idx - 1] = jnp.matmul(prev_core, triang)
if tt.is_tt_matrix:
first_core_shape = (1, raw_shape[0][0], raw_shape[1][0], prev_rank)
else:
first_core_shape = (1, raw_shape[0], prev_rank)
tt_cores[0] = jnp.reshape(tt_cores[0], first_core_shape)
if tt.is_tt_matrix:
return TTMatrix(tt_cores)
else:
return TT(tt_cores)
def _orthogonalize_tt_cores_left_to_right(tt):
"""Orthogonalize TT-cores of a TT-object.
Args:
tt: TT-tensor or TT-matrix.
TT-tensor or TT-matrix.
"""
# Left to right orthogonalization.
num_dims = tt.ndim
if tt.is_tt_matrix:
raw_shape = tt.raw_tensor_shape
else:
raw_shape = tt.shape
tt_ranks = tt.tt_ranks
next_rank = tt_ranks[0]
# Copy cores references so we can change the cores.
tt_cores = list(tt.tt_cores)
for core_idx in range(num_dims - 1):
curr_core = tt_cores[core_idx]
# TT-ranks could have changed on the previous iteration, so `tt_ranks` can
# be outdated for the current TT-rank, but should be valid for the next
# TT-rank.
curr_rank = next_rank
next_rank = tt_ranks[core_idx + 1]
if tt.is_tt_matrix:
curr_mode_left = raw_shape[0][core_idx]
curr_mode_right = raw_shape[1][core_idx]
curr_mode = curr_mode_left * curr_mode_right
else:
curr_mode = raw_shape[core_idx]
qr_shape = (curr_rank * curr_mode, next_rank)
curr_core = jnp.reshape(curr_core, qr_shape)
curr_core, triang = jnp.linalg.qr(curr_core)
triang_shape = triang.shape
# The TT-rank could have changed: if qr_shape is e.g. 4 x 10, than q would
# be of size 4 x 4 and r would be 4 x 10, which means that the next rank
# should be changed to 4.
next_rank = triang_shape[0]
if tt.is_tt_matrix:
new_core_shape = (curr_rank, curr_mode_left, curr_mode_right, next_rank)
else:
new_core_shape = (curr_rank, curr_mode, next_rank)
tt_cores[core_idx] = jnp.reshape(curr_core, new_core_shape)
next_core = jnp.reshape(tt_cores[core_idx + 1], (triang_shape[1], -1))
tt_cores[core_idx + 1] = jnp.matmul(triang, next_core)
if tt.is_tt_matrix:
last_core_shape = (next_rank, raw_shape[0][-1], raw_shape[1][-1], 1)
else:
last_core_shape = (next_rank, raw_shape[-1], 1)
tt_cores[-1] = jnp.reshape(tt_cores[-1], last_core_shape)
if tt.is_tt_matrix:
return TTMatrix(tt_cores)
else:
return TT(tt_cores)
def round(tt, max_tt_rank=None, epsilon=None):
"""Tensor Train rounding procedure, returns a `TT-object` with smaller `TT-ranks`.
:param tt: argument which ranks would be reduced
:type tt: `TT-Tensor` or `TT-Matrix`
:type max_tt_rank: int or list of ints
:param max_tt_rank:
- If a number, than defines the maximal `TT-rank` of the result.
- If a list of numbers, than `max_tt_rank` length should be d+1
(where d is the number of dimensions) and `max_tt_rank[i]`
defines the maximal (i+1)-th `TT-rank` of the result.
The following two versions are equivalent
- ``max_tt_rank = r``
- ``max_tt_rank = [1] + [r] * (d-1) + [1]``
:type epsilon: float or None
:param epsilon:
- If the `TT-ranks` are not restricted (`max_tt_rank=None`), then
the result would be guarantied to be `epsilon`-close to `tt`
in terms of relative Frobenius error:
`||res - tt||_F / ||tt||_F <= epsilon`
- If the `TT-ranks` are restricted, providing a loose `epsilon` may
reduce the `TT-ranks` of the result. E.g.
``round(tt, max_tt_rank=100, epsilon=0.9)``
will probably return you a `TT-Tensor` with `TT-ranks` close to 1, not 100.
Note that providing a nontrivial (= not equal to `None`) epsilon will make
the `TT-ranks` of the result change depending on the data,
which will prevent you from using ``jax.jit``
for speeding up the computations.
:return: `TT-object` with reduced `TT-ranks`
:rtype: `TT-Tensor` or `TT-Matrix`
:raises:
ValueError if `max_tt_rank` is less than 0, if `max_tt_rank` is not a number and
not a vector of length d + 1 where d is the number of dimensions of
the input tensor, if `epsilon` is less than 0.
"""
if max_tt_rank is None:
max_tt_rank = np.iinfo(np.int32).max
num_dims = tt.ndim
max_tt_rank = np.array(max_tt_rank).astype(np.int32)
if np.any(max_tt_rank < 1):
raise ValueError('Maximum TT-rank should be greater or equal to 1.')
if epsilon is not None:
raise NotImplementedError('Epsilon is not supported yet.')
if max_tt_rank.size == 1:
max_tt_rank = (max_tt_rank * np.ones(num_dims + 1)).astype(jnp.int32)
elif max_tt_rank.size != num_dims + 1:
raise ValueError('max_tt_rank should be a number or a vector of size (d+1) '
'where d is the number of dimensions of the tensor.')
if tt.is_tt_matrix:
raw_shape = tt.raw_tensor_shape
else:
raw_shape = tt.shape
tt_cores = orthogonalize(tt).tt_cores
# Copy cores references so we can change the cores.
tt_cores = list(tt_cores)
ranks = [1] * (num_dims + 1)
# Right to left SVD compression.
for core_idx in range(num_dims - 1, 0, -1):
curr_core = tt_cores[core_idx]
if tt.is_tt_matrix:
curr_mode_left = raw_shape[0][core_idx]
curr_mode_right = raw_shape[1][core_idx]
curr_mode = curr_mode_left * curr_mode_right
else:
curr_mode = raw_shape[core_idx]
columns = curr_mode * ranks[core_idx + 1]
curr_core = jnp.reshape(curr_core, [-1, columns])
rows = curr_core.shape[0]
if max_tt_rank[core_idx] == 1:
ranks[core_idx] = 1
else:
ranks[core_idx] = min(max_tt_rank[core_idx], rows, columns)
u, s, v = jnp.linalg.svd(curr_core, full_matrices=False)
u = u[:, 0:ranks[core_idx]]
s = s[0:ranks[core_idx]]
v = v[0:ranks[core_idx], :]
if tt.is_tt_matrix:
core_shape = (ranks[core_idx], curr_mode_left, curr_mode_right,
ranks[core_idx + 1])
else:
core_shape = (ranks[core_idx], curr_mode, ranks[core_idx + 1])
tt_cores[core_idx] = jnp.reshape(v, core_shape)
prev_core_shape = (-1, rows)
tt_cores[core_idx - 1] = jnp.reshape(tt_cores[core_idx - 1], prev_core_shape)
tt_cores[core_idx - 1] = jnp.matmul(tt_cores[core_idx - 1], u)
tt_cores[core_idx - 1] = jnp.matmul(tt_cores[core_idx - 1], jnp.diag(s))
if tt.is_tt_matrix:
core_shape = (ranks[0], raw_shape[0][0], raw_shape[1][0], ranks[1])
else:
core_shape = (ranks[0], raw_shape[0], ranks[1])
tt_cores[0] = jnp.reshape(tt_cores[0], core_shape)
if tt.is_tt_matrix:
return TTMatrix(tt_cores)
else:
return TT(tt_cores)