Source code for tensorly.tt_tensor

"""
Core operations on tensors in Tensor-Train (TT) format, also known as Matrix-Product-State (MPS)
"""

import tensorly as tl
from ._factorized_tensor import FactorizedTensor
import numpy as np
import warnings


def _validate_tt_tensor(tt_tensor):
    factors = tt_tensor
    n_factors = len(factors)

    if isinstance(tt_tensor, TTTensor):
        # it's already been validated at creation
        return tt_tensor.shape, tt_tensor.rank
    elif isinstance(tt_tensor, (float, int)):  # 0-order tensor
        return 0, 0

    rank = []
    shape = []
    for index, factor in enumerate(factors):
        current_rank, current_shape, next_rank = tl.shape(factor)

        # Check that factors are third order tensors
        if not tl.ndim(factor) == 3:
            raise ValueError(
                "TT expresses a tensor as third order factors (tt-cores).\n"
                f"However, tl.ndim(factors[{index}]) = {tl.ndim(factor)}"
            )
        # Consecutive factors should have matching ranks
        if index and tl.shape(factors[index - 1])[2] != current_rank:
            raise ValueError(
                "Consecutive factors should have matching ranks\n"
                " -- e.g. tl.shape(factors[0])[2]) == tl.shape(factors[1])[0])\n"
                f"However, tl.shape(factor[{index-1}])[2] == {tl.shape(factors[index - 1])[2]} but"
                f" tl.shape(factor[{index}])[0] == {current_rank} "
            )
        # Check for boundary conditions
        if (index == 0) and current_rank != 1:
            raise ValueError(
                "Boundary conditions dictate factor[0].shape[0] == 1."
                f"However, got factor[0].shape[0] = {current_rank}."
            )
        if (index == n_factors - 1) and next_rank != 1:
            raise ValueError(
                "Boundary conditions dictate factor[-1].shape[2] == 1."
                f"However, got factor[{n_factors}].shape[2] = {next_rank}."
            )

        shape.append(current_shape)
        rank.append(current_rank)

    # Add last rank (boundary condition)
    rank.append(next_rank)

    return tuple(shape), tuple(rank)


[docs] def tt_to_tensor(factors): """Returns the full tensor whose TT decomposition is given by 'factors' Re-assembles 'factors', which represent a tensor in TT/Matrix-Product-State format into the corresponding full tensor Parameters ---------- factors : list of 3D-arrays TT factors (TT-cores) Returns ------- output_tensor : ndarray tensor whose TT/MPS decomposition was given by 'factors' """ if isinstance(factors, (float, int)): # 0-order tensor return factors full_shape = [f.shape[1] for f in factors] full_tensor = tl.reshape(factors[0], (full_shape[0], -1)) for factor in factors[1:]: rank_prev, _, rank_next = factor.shape factor = tl.reshape(factor, (rank_prev, -1)) full_tensor = tl.dot(full_tensor, factor) full_tensor = tl.reshape(full_tensor, (-1, rank_next)) return tl.reshape(full_tensor, full_shape)
[docs] def tt_to_unfolded(factors, mode): """Returns the unfolding matrix of a tensor given in TT (or Tensor-Train) format Reassembles a full tensor from 'factors' and returns its unfolding matrix with mode given by 'mode' Parameters ---------- factors: list of 3D-arrays TT factors mode: int unfolding matrix to be computed along this mode Returns ------- 2-D array unfolding matrix at mode given by 'mode' """ return tl.unfold(tt_to_tensor(factors), mode)
[docs] def tt_to_vec(factors): """Returns the tensor defined by its TT format ('factors') into its vectorized format Parameters ---------- factors: list of 3D-arrays TT factors Returns ------- 1-D array vectorized format of tensor defined by 'factors' """ return tl.tensor_to_vec(tt_to_tensor(factors))
def _tt_n_param(tensor_shape, rank): """Number of parameters of a MPS decomposition for a given `rank` and full `tensor_shape`. Parameters ---------- tensor_shape : int tuple shape of the full tensor to decompose (or approximate) rank : tuple rank of the MPS decomposition Returns ------- n_params : int Number of parameters of a MPS decomposition of rank `rank` of a full tensor of shape `tensor_shape` """ factor_params = [] for i, s in enumerate(tensor_shape): factor_params.append(rank[i] * s * rank[i + 1]) return np.sum(factor_params) def validate_tt_rank( tensor_shape, rank="same", constant_rank=False, rounding="round", allow_overparametrization=True, ): """Returns the rank of a TT Decomposition Parameters ---------- tensor_shape : tupe shape of the tensor to decompose rank : {'same', float, tuple, int}, default is same way to determine the rank, by default 'same' if 'same': rank is computed to keep the number of parameters (at most) the same if float, computes a rank so as to keep rank percent of the original number of parameters if int or tuple, just returns rank constant_rank : bool, default is False * if True, the *same* rank will be chosen for each modes * if False (default), the rank of each mode will be proportional to the corresponding tensor_shape *used only if rank == 'same' or 0 < rank <= 1* rounding = {'round', 'floor', 'ceil'} allow_overparametrization : bool, default is True if False, the rank must be realizable through iterative application of SVD (used in tensorly.decomposition.tensor_train) Returns ------- rank : int tuple rank of the decomposition """ if rounding == "ceil": rounding_fun = np.ceil elif rounding == "floor": rounding_fun = np.floor elif rounding == "round": rounding_fun = np.round else: raise ValueError(f"Rounding should be round, floor or ceil, but got {rounding}") if rank == "same": rank = float(1) if isinstance(rank, float) and constant_rank: # Choose the *same* rank for each mode n_param_tensor = np.prod(tensor_shape) * rank order = len(tensor_shape) if order == 2: rank = (1, n_param_tensor / (tensor_shape[0] + tensor_shape[1]), 1) warnings.warn( f"Determining the tt-rank for the trivial case of a matrix (order 2 tensor) of shape {tensor_shape}, not a higher-order tensor." ) # R_k I_k R_{k+1} = R^2 I_k a = np.sum(tensor_shape[1:-1]) # Border rank of 1, R_0 = R_N = 1 # First and last factor of size I_0 R and I_N R b = np.sum(tensor_shape[0] + tensor_shape[-1]) # We want the number of params of decomp (=sum of params of factors) # To be equal to c = \prod_k I_k c = -n_param_tensor delta = np.sqrt(b**2 - 4 * a * c) # We get the non-negative solution solution = int(rounding_fun((-b + delta) / (2 * a))) rank = rank = (1,) + (solution,) * (order - 1) + (1,) elif isinstance(rank, float): # Choose a rank proportional to the size of each mode # The method is similar to the above one for constant_rank == True order = len(tensor_shape) avg_dim = [ (tensor_shape[i] + tensor_shape[i + 1]) / 2 for i in range(order - 1) ] if len(avg_dim) > 1: a = sum( avg_dim[i - 1] * tensor_shape[i] * avg_dim[i] for i in range(1, order - 1) ) else: warnings.warn( f"Determining the tt-rank for the trivial case of a matrix (order 2 tensor) of shape {tensor_shape}, not a higher-order tensor." ) a = avg_dim[0] ** 2 * tensor_shape[0] b = tensor_shape[0] * avg_dim[0] + tensor_shape[-1] * avg_dim[-1] c = -np.prod(tensor_shape) * rank delta = np.sqrt(b**2 - 4 * a * c) # We get the non-negative solution fraction_param = (-b + delta) / (2 * a) rank = tuple([max(int(rounding_fun(d * fraction_param)), 1) for d in avg_dim]) rank = (1,) + rank + (1,) else: # Check user input for potential errors n_dim = len(tensor_shape) if isinstance(rank, int): rank = [1] + [rank] * (n_dim - 1) + [1] elif n_dim + 1 != len(rank): message = f"Provided incorrect number of ranks. Should verify len(rank) == tl.ndim(tensor)+1, but len(rank) = {len(rank)} while tl.ndim(tensor) + 1 = {n_dim+1}" raise (ValueError(message)) # Initialization if rank[0] != 1: message = "Provided rank[0] == {} but boundary conditions dictate rank[0] == rank[-1] == 1.".format( rank[0] ) raise ValueError(message) if rank[-1] != 1: message = "Provided rank[-1] == {} but boundary conditions dictate rank[0] == rank[-1] == 1.".format( rank[-1] ) raise ValueError(message) if allow_overparametrization: return list(rank) else: validated_rank = [1] for i, s in enumerate(tensor_shape[:-1]): n_row = int(rank[i] * s) n_column = np.prod(tensor_shape[(i + 1) :]) # n_column of unfolding validated_rank.append(min(n_row, n_column, rank[i + 1])) validated_rank.append(1) return validated_rank class TTTensor(FactorizedTensor): def __init__(self, factors, inplace=False): super().__init__() # Will raise an error if invalid shape, rank = _validate_tt_tensor(factors) self.shape = tuple(shape) self.rank = tuple(rank) self.factors = factors def __getitem__(self, index): return self.factors[index] def __setitem__(self, index, value): self.factors[index] = value def __iter__(self): for index in range(len(self)): yield self[index] def __len__(self): return len(self.factors) def __repr__(self): message = f"factors list : rank-{self.rank} matrix-product-state tensor of shape {self.shape} " return message def to_tensor(self): return tt_to_tensor(self) def to_unfolding(self, mode): return tt_to_unfolded(self, mode) def to_vec(self): return tt_to_vec(self)
[docs] def pad_tt_rank(factor_list, n_padding=1, pad_boundaries=False): """Pads the factors of a Tensor-Train so as to increase its rank without changing its reconstruction The tensor-train (ring) will be padded with 0s to increase its rank only but not the underlying tensor it represents. Parameters ---------- factor_list : tensor list n_padding : int, default is 1 how much to increase the rank (bond dimension) by pad_boundaries : bool, default is False if True, also pad the boundaries (useful for a tensor-ring) should be False for a tensor-train to keep the boundary rank to be 1 Returns ------- padded_factor_list """ new_factors = [] n_factors = len(factor_list) for i, factor in enumerate(factor_list): n_padding_left = n_padding_right = n_padding if (i == 0) and not pad_boundaries: n_padding_left = 0 elif (i == n_factors - 1) and not pad_boundaries: n_padding_right = 0 r1, *s, r2 = tl.shape(factor) new_factor = tl.zeros( (r1 + n_padding_left, *s, r2 + n_padding_right), **tl.context(factor) ) new_factors.append(tl.index_update(new_factor, tl.index[:r1, ..., :r2], factor)) return new_factors