Source code for tensorly.kruskal_tensor

"""
Core operations on Kruskal tensors.
"""

from . import backend as T
from .base import fold, tensor_to_vec
from .tenalg import khatri_rao, multi_mode_dot, inner

import warnings
from collections.abc import Mapping

# Author: Jean Kossaifi

# License: BSD 3 clause

class KruskalTensor(Mapping):
    def __init__(self, kruskal_tensor):
        super().__init__()

        shape, rank = _validate_kruskal_tensor(kruskal_tensor)
        weights, factors = kruskal_tensor

        # Should we allow None weights?
        if weights is None:
            weights = T.ones(rank, **T.context(factors[0]))

        self.shape = shape
        self.rank = rank
        self.factors = factors
        self.weights = weights

    
    def __getitem__(self, index):
        if index == 0:
            return self.weights
        elif index == 1:
            return self.factors
        else: 
            raise IndexError('You tried to access index {} of a Kruskal tensor.\n'
                             'You can only access index 0 and 1 of a Kruskal tensor'
                             '(corresponding respectively to the weights and factors)'.format(index))
    
    def __iter__(self):
        yield self.weights
        yield self.factors
        
    def __len__(self):
        return 2
    
    def __repr__(self):
        message = '(weights, factors) : rank-{} KruskalTensor of shape {} '.format(self.rank, self.shape)
        return message


def _validate_kruskal_tensor(kruskal_tensor):
    """Validates a kruskal_tensor in the form (weights, factors)
    
        Returns the rank and shape of the validated tensor
    
    Parameters
    ----------
    kruskal_tensor : KruskalTensor or (weights, factors)
    
    Returns
    -------
    (shape, rank) : (int tuple, int)
        size of the full tensor and rank of the Kruskal tensor
    """
    if isinstance(kruskal_tensor, KruskalTensor):
        # it's already been validated at creation
        return kruskal_tensor.shape, kruskal_tensor.rank

    weights, factors = kruskal_tensor
            
    if len(factors) < 2:
        raise ValueError('A Kruskal tensor should be composed of at least two factors.'
                         'However, {} factor was given.'.format(len(factors)))

    if T.ndim(factors[0]) == 2:
        rank = int(T.shape(factors[0])[1])
    else:
        rank = 1
    shape = []
    for i, factor in enumerate(factors):
        s = T.shape(factor)
        if len(s) == 2:
            current_mode_size, current_rank = s
        else:
            current_mode_size, current_rank = s, 1

        if current_rank != rank:
            raise ValueError('All the factors of a Kruskal tensor should have the same number of column.'
                             'However, factors[0].shape[1]={} but factors[{}].shape[1]={}.'.format(
                                 rank, i, T.shape(factor)[1]))
        shape.append(current_mode_size)

    if weights is not None and len(weights) != rank:
        raise ValueError('Given factors for a rank-{} Kruskal tensor but len(weights)={}.'.format(
            rank, len(weights)))
        
    return tuple(shape), rank


def kruskal_normalise(kruskal_tensor):
    """Returns kruskal_tensor with factors normalised to unit length

    Turns ``factors = [|U_1, ... U_n|]`` into ``[weights; |V_1, ... V_n|]``,
    where the columns of each `V_k` are normalized to unit Euclidean length
    from the columns of `U_k` with the normalizing constants absorbed into
    `weights`. In the special case of a symmetric tensor, `weights` holds the
    eigenvalues of the tensor.

    Parameters
    ----------
    kruskal_tensor : KruskalTensor = (weight, factors)
        factors is list of matrices, all with the same number of columns
        i.e.::
            for u in U:
                u[i].shape == (s_i, R)

        where `R` is fixed while `s_i` can vary with `i`

    Returns
    -------
    KruskalTensor = (normalisation_weights, normalised_factors)
    """
    _, rank = _validate_kruskal_tensor(kruskal_tensor)
    weights, factors = kruskal_tensor
    
    if weights is None:
        weights = T.ones(rank, **T.context(factors[0]))
    
    normalized_factors = []
    for i, factor in enumerate(factors):
        if i == 0:
            factor = factor*weights
            weights = T.ones(rank, **T.context(factor))
            
        scales = T.norm(factor, axis=0)
        scales_non_zero = T.where(scales==0, T.ones(T.shape(scales), **T.context(factor)), scales)
        weights = weights*scales
        normalized_factors.append(factor / T.reshape(scales_non_zero, (1, -1)))

    return KruskalTensor((weights, normalized_factors))
    

[docs]def kruskal_to_tensor(kruskal_tensor, mask=None): """Turns the Khatri-product of matrices into a full tensor ``factor_matrices = [|U_1, ... U_n|]`` becomes a tensor shape ``(U[1].shape[0], U[2].shape[0], ... U[-1].shape[0])`` Parameters ---------- kruskal_tensor : KruskalTensor = (weight, factors) factors is a list of factor matrices, all with the same number of columns i.e. for all matrix U in factor_matrices: U has shape ``(s_i, R)``, where R is fixed and s_i varies with i mask : ndarray a mask to be applied to the final tensor. It should be broadcastable to the shape of the final tensor, that is ``(U[1].shape[0], ... U[-1].shape[0])``. Returns ------- ndarray full tensor of shape ``(U[1].shape[0], ... U[-1].shape[0])`` Notes ----- This version works by first computing the mode-0 unfolding of the tensor and then refolding it. There are other possible and equivalent alternate implementation, e.g. summing over r and updating an outer product of vectors. """ shape, rank = _validate_kruskal_tensor(kruskal_tensor) weights, factors = kruskal_tensor if weights is None: weights = 1 if mask is None: full_tensor = T.dot(factors[0]*weights, T.transpose(khatri_rao(factors, skip_matrix=0))) else: full_tensor = T.sum(khatri_rao([factor[0]*weights]+factors[1:], mask=mask), axis=1) return fold(full_tensor, 0, shape)
[docs]def kruskal_to_unfolded(kruskal_tensor, mode): """Turns the khatri-product of matrices into an unfolded tensor turns ``factors = [|U_1, ... U_n|]`` into a mode-`mode` unfolding of the tensor Parameters ---------- kruskal_tensor : KruskalTensor = (weight, factors) factors is a list of matrices, all with the same number of columns ie for all u in factor_matrices: u[i] has shape (s_u_i, R), where R is fixed mode: int mode of the desired unfolding Returns ------- ndarray unfolded tensor of shape (tensor_shape[mode], -1) Notes ----- Writing factors = [U_1, ..., U_n], we exploit the fact that ``U_k = U[k].dot(khatri_rao(U_1, ..., U_k-1, U_k+1, ..., U_n))`` """ _validate_kruskal_tensor(kruskal_tensor) weights, factors = kruskal_tensor if weights is not None: return T.dot(factors[mode]*weights, T.transpose(khatri_rao(factors, skip_matrix=mode))) else: return T.dot(factors[mode], T.transpose(khatri_rao(factors, skip_matrix=mode)))
[docs]def kruskal_to_vec(kruskal_tensor): """Turns the khatri-product of matrices into a vector (the tensor ``factors = [|U_1, ... U_n|]`` is converted into a raveled mode-0 unfolding) Parameters ---------- kruskal_tensor : KruskalTensor = (weight, factors) factors is a list of matrices, all with the same number of columns i.e.:: for u in U: u[i].shape == (s_i, R) where `R` is fixed while `s_i` can vary with `i` Returns ------- ndarray vectorised tensor """ return tensor_to_vec(kruskal_to_tensor(kruskal_tensor))
[docs]def kruskal_mode_dot(kruskal_tensor, matrix_or_vector, mode, keep_dim=False, copy=False): """n-mode product of a Kruskal tensor and a matrix or vector at the specified mode Parameters ---------- kruskal_tensor : tl.KruskalTensor or (core, factors) matrix_or_vector : ndarray 1D or 2D array of shape ``(J, i_k)`` or ``(i_k, )`` matrix or vectors to which to n-mode multiply the tensor mode : int Returns ------- KruskalTensor = (core, factors) `mode`-mode product of `tensor` by `matrix_or_vector` * of shape :math:`(i_1, ..., i_{k-1}, J, i_{k+1}, ..., i_N)` if matrix_or_vector is a matrix * of shape :math:`(i_1, ..., i_{k-1}, i_{k+1}, ..., i_N)` if matrix_or_vector is a vector See also -------- kruskal_multi_mode_dot : chaining several mode_dot in one call """ shape, _ = _validate_kruskal_tensor(kruskal_tensor) weights, factors = kruskal_tensor contract = False if T.ndim(matrix_or_vector) == 2: # Tensor times matrix # Test for the validity of the operation if matrix_or_vector.shape[1] != shape[mode]: raise ValueError( 'shapes {0} and {1} not aligned in mode-{2} multiplication: {3} (mode {2}) != {4} (dim 1 of matrix)'.format( shape, matrix_or_vector.shape, mode, shape[mode], matrix_or_vector.shape[1] )) elif T.ndim(matrix_or_vector) == 1: # Tensor times vector if matrix_or_vector.shape[0] != shape[mode]: raise ValueError( 'shapes {0} and {1} not aligned for mode-{2} multiplication: {3} (mode {2}) != {4} (vector size)'.format( shape, matrix_or_vector.shape, mode, shape[mode], matrix_or_vector.shape[0] )) if not keep_dim: contract = True # Contract over that mode else: raise ValueError('Can only take n_mode_product with a vector or a matrix.') if copy: factors = [T.copy(f) for f in factors] weights = T.copy(weights) if contract: factor = factors.pop(mode) factor = T.dot(matrix_or_vector, factor) mode = max(mode - 1, 0) factors[mode] *= factor else: factors[mode] = T.dot(matrix_or_vector, factors[mode]) return KruskalTensor((weights, factors))
[docs]def unfolding_dot_khatri_rao(tensor, kruskal_tensor, mode): """mode-n unfolding times khatri-rao product of factors Parameters ---------- tensor : tl.tensor tensor to unfold factors : tl.tensor list list of matrices of which to the khatri-rao product mode : int mode on which to unfold `tensor` Returns ------- mttkrp dot(unfold(tensor, mode), khatri-rao(factors)) Notes ----- This is a variant of:: unfolded = unfold(tensor, mode) kr_factors = khatri_rao(factors, skip_matrix=mode) mttkrp2 = tl.dot(unfolded, kr_factors) Multiplying with the Khatri-Rao product is equivalent to multiplying, for each rank, with the kronecker product of each factor. In code:: mttkrp_parts = [] for r in range(rank): component = tl.tenalg.multi_mode_dot(tensor, [f[:, r] for f in factors], skip=mode) mttkrp_parts.append(component) mttkrp = tl.stack(mttkrp_parts, axis=1) return mttkrp This can be done by taking n-mode-product with the full factors (faster but more memory consuming):: projected = multi_mode_dot(tensor, factors, skip=mode, transpose=True) ndims = T.ndim(tensor) res = [] for i in range(factors[0].shape[1]): index = tuple([slice(None) if k == mode else i for k in range(ndims)]) res.append(projected[index]) return T.stack(res, axis=-1) The same idea could be expressed using einsum:: ndims = tl.ndim(tensor) tensor_idx = ''.join(chr(ord('a') + i) for i in range(ndims)) rank = chr(ord('a') + ndims + 1) op = tensor_idx for i in range(ndims): if i != mode: op += ',' + ''.join([tensor_idx[i], rank]) else: result = ''.join([tensor_idx[i], rank]) op += '->' + result factors = [f for (i, f) in enumerate(factors) if i != mode] return tl_einsum(op, tensor, *factors) """ mttkrp_parts = [] _, rank = _validate_kruskal_tensor(kruskal_tensor) weights, factors = kruskal_tensor for r in range(rank): component = multi_mode_dot(tensor, [f[:, r] for f in factors], skip=mode) mttkrp_parts.append(component) if weights is None: return T.stack(mttkrp_parts, axis=1) else: return T.stack(mttkrp_parts, axis=1)*T.reshape(weights, (1, -1))
def kruskal_norm(kruskal_tensor): """Returns the l2 norm of a Kruskal tensor Parameters ---------- kruskal_tensor : tl.KruskalTensor or (core, factors) Returns ------- l2-norm : int Notes ----- This is ||kruskal_to_tensor(factors)||^2 You can see this using the fact that khatria-rao(A, B)^T x khatri-rao(A, B) = A^T x A * B^T x B """ _ = _validate_kruskal_tensor(kruskal_tensor) weights, factors = kruskal_tensor norm = 1 for factor in factors: norm *= T.dot(T.transpose(factor), factor) if weights is not None: #norm = T.dot(T.dot(weights, norm), weights) norm = norm * (T.reshape(weights, (-1, 1))*T.reshape(weights, (1, -1))) # We sum even if weigths is not None # as e.g. MXNet would return a 1D tensor, not a 0D tensor return T.sqrt(T.sum(norm))