def randomised_parafac(tensor,
rank,
n_samples,
n_iter_max=100,
init='svd',
tol=10e-9,
max_stagnation=20,
random_state=None,
verbose=1):
"""Randomised CP decomposition via sampled ALS
Parameters
----------
tensor : ndarray
rank : int
number of components
n_samples : int
number of samples per ALS step
n_iter_max : int
maximum number of iteration
init : {'svd', 'random'}, optional
tol : float, optional
tolerance: the algorithm stops when the variation in
the reconstruction error is less than the tolerance
max_stagnation: int, optional, default is 0
if not zero, the maximum allowed number
of iterations with no decrease in fit
random_state : {None, int, np.random.RandomState}, default is None
verbose : int, optional
level of verbosity
Returns
-------
factors : ndarray list
list of positive factors of the CP decomposition
element `i` is of shape ``(tensor.shape[i], rank)``
References
----------
.. [3] Casey Battaglino, Grey Ballard and Tamara G. Kolda,
"A Practical Randomized CP Tensor Decomposition",
"""
rng = check_random_state(random_state)
factors = initialize_factors(tensor,
rank,
init=init,
random_state=random_state)
rec_errors = []
n_dims = tl.ndim(tensor)
norm_tensor = tl.norm(tensor, 2)
min_error = 0
for iteration in range(n_iter_max):
for mode in range(n_dims):
kr_prod, indices_list = sample_khatri_rao(factors,
n_samples,
skip_matrix=mode,
random_state=rng)
indices_list = [i.tolist() for i in indices_list]
# Keep all the elements of the currently considered mode
indices_list.insert(mode, slice(None, None, None))
# MXNet will not be happy if this is a list insteaf of a tuple
indices_list = tuple(indices_list)
if mode:
sampled_unfolding = tensor[indices_list]
else:
sampled_unfolding = tl.transpose(tensor[indices_list])
pseudo_inverse = tl.dot(tl.transpose(kr_prod), kr_prod)
factor = tl.dot(tl.transpose(kr_prod), sampled_unfolding)
factor = tl.transpose(tl.solve(pseudo_inverse, factor))
factors[mode] = factor
if max_stagnation or tol:
rec_error = tl.norm(tensor - kruskal_to_tensor(factors),
2) / norm_tensor
if not min_error or rec_error < min_error:
min_error = rec_error
stagnation = -1
stagnation += 1
rec_errors.append(rec_error)
if iteration > 1:
if verbose:
print('reconstruction error={}, variation={}.'.format(
rec_errors[-1], rec_errors[-2] - rec_errors[-1]))
if (tol and abs(rec_errors[-2] - rec_errors[-1]) < tol) or \
(stagnation and (stagnation > max_stagnation)):
if verbose:
print('converged in {} iterations.'.format(iteration))
break
return factors
def parafac(tensor, rank, n_iter_max=100, init='svd', svd='numpy_svd',\
normalize_factors=False, orthogonalise=False,\
tol=1e-8, random_state=None,\
verbose=0, return_errors=False,\
sparsity = None,\
l2_reg = 0, mask=None,\
cvg_criterion = 'abs_rec_error',\
fixed_modes = [],
svd_mask_repeats=5,
linesearch = False):
"""CANDECOMP/PARAFAC decomposition via alternating least squares (ALS)
Computes a rank-`rank` decomposition of `tensor` [1]_ such that,
``tensor = [|weights; factors[0], ..., factors[-1] |]``.
Parameters
----------
tensor : ndarray
rank : int
Number of components.
n_iter_max : int
Maximum number of iteration
init : {'svd', 'random'}, optional
Type of factor matrix initialization. See `initialize_factors`.
svd : str, default is 'numpy_svd'
function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
normalize_factors : if True, aggregate the weights of each factor in a 1D-tensor
of shape (rank, ), which will contain the norms of the factors
tol : float, optional
(Default: 1e-6) Relative reconstruction error tolerance. The
algorithm is considered to have found the global minimum when the
reconstruction error is less than `tol`.
random_state : {None, int, np.random.RandomState}
verbose : int, optional
Level of verbosity
return_errors : bool, optional
Activate return of iteration errors
mask : ndarray
array of booleans with the same shape as ``tensor`` should be 0 where
the values are missing and 1 everywhere else. Note: if tensor is
sparse, then mask should also be sparse with a fill value of 1 (or
True). Allows for missing values [2]_
cvg_criterion : {'abs_rec_error', 'rec_error'}, optional
Stopping criterion for ALS, works if `tol` is not None.
If 'rec_error', ALS stops at current iteration if (previous rec_error - current rec_error) < tol.
If 'abs_rec_error', ALS terminates when |previous rec_error - current rec_error| < tol.
sparsity : float or int
If `sparsity` is not None, we approximate tensor as a sum of low_rank_component and sparse_component, where low_rank_component = cp_to_tensor((weights, factors)). `sparsity` denotes desired fraction or number of non-zero elements in the sparse_component of the `tensor`.
fixed_modes : list, default is []
A list of modes for which the initial value is not modified.
The last mode cannot be fixed due to error computation.
svd_mask_repeats: int
If using a tensor with masked values, this initializes using SVD multiple times to
remove the effect of these missing values on the initialization.
linesearch : bool, default is False
Whether to perform line search as proposed by Bro [3].
Returns
-------
CPTensor : (weight, factors)
* weights : 1D array of shape (rank, )
all ones if normalize_factors is False (default),
weights of the (normalized) factors otherwise
* factors : List of factors of the CP decomposition element `i` is of shape
(tensor.shape[i], rank)
* sparse_component : nD array of shape tensor.shape. Returns only if `sparsity` is not None.
errors : list
A list of reconstruction errors at each iteration of the algorithms.
References
----------
.. [1] T.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications",
SIAM REVIEW, vol. 51, n. 3, pp. 455-500, 2009.
.. [2] Tomasi, Giorgio, and Rasmus Bro. "PARAFAC and missing values."
Chemometrics and Intelligent Laboratory Systems 75.2 (2005): 163-180.
.. [3] R. Bro, "Multi-Way Analysis in the Food Industry: Models, Algorithms, and
Applications", PhD., University of Amsterdam, 1998
"""
rank = validate_cp_rank(tl.shape(tensor), rank=rank)
if orthogonalise and not isinstance(orthogonalise, int):
orthogonalise = n_iter_max
if linesearch:
acc_pow = 2.0 # Extrapolate to the iteration^(1/acc_pow) ahead
acc_fail = 0 # How many times acceleration have failed
max_fail = 4 # Increase acc_pow with one after max_fail failure
weights, factors = initialize_cp(tensor, rank, init=init, svd=svd,
random_state=random_state,
normalize_factors=normalize_factors)
if mask is not None and init == "svd":
for _ in range(svd_mask_repeats):
tensor = tensor*mask + tl.cp_to_tensor((weights, factors), mask=1-mask)
weights, factors = initialize_cp(tensor, rank, init=init, svd=svd, random_state=random_state, normalize_factors=normalize_factors)
rec_errors = []
norm_tensor = tl.norm(tensor, 2)
Id = tl.eye(rank, **tl.context(tensor))*l2_reg
if tl.ndim(tensor)-1 in fixed_modes:
warnings.warn('You asked for fixing the last mode, which is not supported.\n The last mode will not be fixed. Consider using tl.moveaxis()')
fixed_modes.remove(tl.ndim(tensor)-1)
modes_list = [mode for mode in range(tl.ndim(tensor)) if mode not in fixed_modes]
if sparsity:
sparse_component = tl.zeros_like(tensor)
if isinstance(sparsity, float):
sparsity = int(sparsity * np.prod(tensor.shape))
else:
sparsity = int(sparsity)
for iteration in range(n_iter_max):
if orthogonalise and iteration <= orthogonalise:
factors = [tl.qr(f)[0] if min(tl.shape(f)) >= rank else f for i, f in enumerate(factors)]
if linesearch and iteration % 2 == 0:
factors_last = [tl.copy(f) for f in factors]
weights_last = tl.copy(weights)
if verbose > 1:
print("Starting iteration", iteration + 1)
for mode in modes_list:
if verbose > 1:
print("Mode", mode, "of", tl.ndim(tensor))
pseudo_inverse = tl.tensor(np.ones((rank, rank)), **tl.context(tensor))
for i, factor in enumerate(factors):
if i != mode:
pseudo_inverse = pseudo_inverse*tl.dot(tl.conj(tl.transpose(factor)), factor)
pseudo_inverse += Id
if not iteration and weights is not None:
# Take into account init weights
mttkrp = unfolding_dot_khatri_rao(tensor, (weights, factors), mode)
else:
mttkrp = unfolding_dot_khatri_rao(tensor, (None, factors), mode)
factor = tl.transpose(tl.solve(tl.conj(tl.transpose(pseudo_inverse)),
tl.transpose(mttkrp)))
if normalize_factors:
scales = tl.norm(factor, 2, axis=0)
weights = tl.where(scales==0, tl.ones(tl.shape(scales), **tl.context(factor)), scales)
factor = factor / tl.reshape(weights, (1, -1))
factors[mode] = factor
# Will we be performing a line search iteration
if linesearch and iteration % 2 == 0 and iteration > 5:
line_iter = True
else:
line_iter = False
# Calculate the current unnormalized error if we need it
if (tol or return_errors) and line_iter is False:
unnorml_rec_error, tensor, norm_tensor = error_calc(tensor, norm_tensor, weights, factors, sparsity, mask, mttkrp)
else:
if mask is not None:
tensor = tensor*mask + tl.cp_to_tensor((weights, factors), mask=1-mask)
# Start line search if requested.
if line_iter is True:
jump = iteration ** (1.0 / acc_pow)
new_weights = weights_last + (weights - weights_last) * jump
new_factors = [factors_last[ii] + (factors[ii] - factors_last[ii])*jump for ii in range(tl.ndim(tensor))]
new_rec_error, new_tensor, new_norm_tensor = error_calc(tensor, norm_tensor, new_weights, new_factors, sparsity, mask)
if (new_rec_error / new_norm_tensor) < rec_errors[-1]:
factors, weights = new_factors, new_weights
tensor, norm_tensor = new_tensor, new_norm_tensor
unnorml_rec_error = new_rec_error
acc_fail = 0
if verbose:
print("Accepted line search jump of {}.".format(jump))
else:
unnorml_rec_error, tensor, norm_tensor = error_calc(tensor, norm_tensor, weights, factors, sparsity, mask, mttkrp)
acc_fail += 1
if verbose:
print("Line search failed for jump of {}.".format(jump))
if acc_fail == max_fail:
acc_pow += 1.0
acc_fail = 0
if verbose:
print("Reducing acceleration.")
rec_error = unnorml_rec_error / norm_tensor
rec_errors.append(rec_error)
if tol:
if iteration >= 1:
rec_error_decrease = rec_errors[-2] - rec_errors[-1]
if verbose:
print("iteration {}, reconstruction error: {}, decrease = {}, unnormalized = {}".format(iteration, rec_error, rec_error_decrease, unnorml_rec_error))
if cvg_criterion == 'abs_rec_error':
stop_flag = abs(rec_error_decrease) < tol
elif cvg_criterion == 'rec_error':
stop_flag = rec_error_decrease < tol
else:
raise TypeError("Unknown convergence criterion")
if stop_flag:
if verbose:
print("PARAFAC converged after {} iterations".format(iteration))
break
else:
if verbose:
print('reconstruction error={}'.format(rec_errors[-1]))
cp_tensor = CPTensor((weights, factors))
if sparsity:
sparse_component = sparsify_tensor(tensor -\
cp_to_tensor((weights, factors)),\
sparsity)
cp_tensor = (cp_tensor, sparse_component)
if return_errors:
return cp_tensor, rec_errors
else:
return cp_tensor
def hals_nnls(UtM,
UtU,
V=None,
n_iter_max=500,
tol=10e-8,
sparsity_coefficient=None,
normalize=False,
nonzero_rows=False,
exact=False):
"""
Non Negative Least Squares (NNLS)
Computes an approximate solution of a nonnegative least
squares problem (NNLS) with an exact block-coordinate descent scheme.
M is m by n, U is m by r, V is r by n.
All matrices are nonnegative componentwise.
This algorithm is defined in [1], as an accelerated version of the HALS algorithm.
It features two accelerations: an early stop stopping criterion, and a
complexity averaging between precomputations and loops, so as to use large
precomputations several times.
This function is made for being used repetively inside an
outer-loop alternating algorithm, for instance for computing nonnegative
matrix Factorization or tensor factorization.
Parameters
----------
UtM: r-by-n array
Pre-computed product of the transposed of U and M, used in the update rule
UtU: r-by-r array
Pre-computed product of the transposed of U and U, used in the update rule
V: r-by-n initialization matrix (mutable)
Initialized V array
By default, is initialized with one non-zero entry per column
corresponding to the closest column of U of the corresponding column of M.
n_iter_max: Postivie integer
Upper bound on the number of iterations
Default: 500
tol : float in [0,1]
early stop criterion, while err_k > delta*err_0. Set small for
almost exact nnls solution, or larger (e.g. 1e-2) for inner loops
of a PARAFAC computation.
Default: 10e-8
sparsity_coefficient: float or None
The coefficient controling the sparisty level in the objective function.
If set to None, the problem is solved unconstrained.
Default: None
nonzero_rows: boolean
True if the lines of the V matrix can't be zero,
False if they can be zero
Default: False
exact: If it is True, the algorithm gives a results with high precision but it needs high computational cost.
If it is False, the algorithm gives an approximate solution
Default: False
Returns
-------
V: array
a r-by-n nonnegative matrix \approx argmin_{V >= 0} ||M-UV||_F^2
rec_error: float
number of loops authorized by the error stop criterion
iteration: integer
final number of update iteration performed
complexity_ratio: float
number of loops authorized by the stop criterion
Notes
-----
We solve the following problem :math:`\\min_{V >= 0} ||M-UV||_F^2`
The matrix V is updated linewise. The update rule for this resolution is::
.. math::
\\begin{equation}
V[k,:]_(j+1) = V[k,:]_(j) + (UtM[k,:] - UtU[k,:]\\times V_(j))/UtU[k,k]
\\end{equation}
with j the update iteration.
This problem can also be defined by adding a sparsity coefficient,
enhancing sparsity in the solution [2]. In this sparse version, the update rule becomes::
.. math::
\\begin{equation}
V[k,:]_(j+1) = V[k,:]_(j) + (UtM[k,:] - UtU[k,:]\\times V_(j) - sparsity_coefficient)/UtU[k,k]
\\end{equation}
References
----------
.. [1]: N. Gillis and F. Glineur, Accelerated Multiplicative Updates and
Hierarchical ALS Algorithms for Nonnegative Matrix Factorization,
Neural Computation 24 (4): 1085-1105, 2012.
.. [2] J. Eggert, and E. Korner. "Sparse coding and NMF."
2004 IEEE International Joint Conference on Neural Networks
(IEEE Cat. No. 04CH37541). Vol. 4. IEEE, 2004.
"""
rank, n_col_M = tl.shape(UtM)
if V is None: # checks if V is empty
V = tl.solve(UtU, UtM)
V = tl.clip(V, a_min=0, a_max=None)
# Scaling
scale = tl.sum(UtM * V) / tl.sum(UtU * tl.dot(V, tl.transpose(V)))
V = V * scale
if exact:
n_iter_max = 50000
tol = 10e-16
for iteration in range(n_iter_max):
rec_error = 0
rec_error0 = 0
for k in range(rank):
if UtU[k, k]:
if sparsity_coefficient is not None: # Modifying the function for sparsification
deltaV = tl.where(
(UtM[k, :] - tl.dot(UtU[k, :], V) -
sparsity_coefficient) / UtU[k, k] > -V[k, :],
(UtM[k, :] - tl.dot(UtU[k, :], V) -
sparsity_coefficient) / UtU[k, k], -V[k, :])
V = tl.index_update(V, tl.index[k, :], V[k, :] + deltaV)
else: # without sparsity
deltaV = tl.where(
(UtM[k, :] - tl.dot(UtU[k, :], V)) / UtU[k, k] >
-V[k, :],
(UtM[k, :] - tl.dot(UtU[k, :], V)) / UtU[k, k],
-V[k, :])
V = tl.index_update(V, tl.index[k, :], V[k, :] + deltaV)
rec_error = rec_error + tl.dot(deltaV, tl.transpose(deltaV))
# Safety procedure, if columns aren't allow to be zero
if nonzero_rows and tl.all(V[k, :] == 0):
V[k, :] = tl.eps(V.dtype) * tl.max(V)
elif nonzero_rows:
raise ValueError("Column " + str(k) +
" of U is zero with nonzero condition")
if normalize:
norm = tl.norm(V[k, :])
if norm != 0:
V[k, :] /= norm
else:
sqrt_n = 1 / n_col_M**(1 / 2)
V[k, :] = [sqrt_n for i in range(n_col_M)]
if iteration == 1:
rec_error0 = rec_error
numerator = tl.shape(V)[0] * tl.shape(V)[1] + tl.shape(V)[1] * rank
denominator = tl.shape(V)[0] * rank + tl.shape(V)[0]
complexity_ratio = 1 + (numerator / denominator)
if exact:
if rec_error < tol * rec_error0:
break
else:
if rec_error < tol * rec_error0 or iteration > 1 + 0.5 * complexity_ratio:
break
return V, rec_error, iteration, complexity_ratio
def parafac(tensor,
rank,
n_iter_max=100,
init='svd',
tol=1e-8,
orthogonalise=False,
random_state=None,
verbose=False,
return_errors=False):
"""CANDECOMP/PARAFAC decomposition via alternating least squares (ALS)
Computes a rank-`rank` decomposition of `tensor` [1]_ such that,
``tensor = [| factors[0], ..., factors[-1] |]``.
Parameters
----------
tensor : ndarray
rank : int
Number of components.
n_iter_max : int
Maximum number of iteration
init : {'svd', 'random'}, optional
Type of factor matrix initialization. See `initialize_factors`.
tol : float, optional
(Default: 1e-6) Relative reconstruction error tolerance. The
algorithm is considered to have found the global minimum when the
reconstruction error is less than `tol`.
random_state : {None, int, np.random.RandomState}
verbose : int, optional
Level of verbosity
return_errors : bool, optional
Activate return of iteration errors
Returns
-------
factors : ndarray list
List of factors of the CP decomposition element `i` is of shape
(tensor.shape[i], rank)
errors : list
A list of reconstruction errors at each iteration of the algorithms.
References
----------
.. [1] tl.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications",
SIAM REVIEW, vol. 51, n. 3, pp. 455-500, 2009.
"""
if orthogonalise and not isinstance(orthogonalise, int):
orthogonalise = n_iter_max
factors = initialize_factors(tensor,
rank,
init=init,
random_state=random_state)
rec_errors = []
norm_tensor = tl.norm(tensor, 2)
for iteration in range(n_iter_max):
if orthogonalise and iteration <= orthogonalise:
factor = [tl.qr(factor)[0] for factor in factors]
for mode in range(tl.ndim(tensor)):
pseudo_inverse = tl.tensor(np.ones((rank, rank)),
**tl.context(tensor))
for i, factor in enumerate(factors):
if i != mode:
pseudo_inverse = pseudo_inverse * tl.dot(
tl.transpose(factor), factor)
factor = tl.dot(unfold(tensor, mode),
khatri_rao(factors, skip_matrix=mode))
factor = tl.transpose(
tl.solve(tl.transpose(pseudo_inverse), tl.transpose(factor)))
factors[mode] = factor
if tol:
rec_error = tl.norm(tensor - kruskal_to_tensor(factors),
2) / norm_tensor
rec_errors.append(rec_error)
if iteration > 1:
if verbose:
print('reconstruction error={}, variation={}.'.format(
rec_errors[-1], rec_errors[-2] - rec_errors[-1]))
if tol and abs(rec_errors[-2] - rec_errors[-1]) < tol:
if verbose:
print('converged in {} iterations.'.format(iteration))
break
if return_errors:
return factors, rec_errors
else:
return factors
def active_set_nnls(Utm, UtU, x=None, n_iter_max=100, tol=10e-8):
"""
Active set algorithm for non-negative least square solution.
Computes an approximate non-negative solution for Ux=m linear system.
Parameters
----------
Utm : vectorized ndarray
Pre-computed product of the transposed of U and m
UtU : ndarray
Pre-computed Kronecker product of the transposed of U and U
x : init
Default: None
n_iter_max : int
Maximum number of iteration
Default: 100
tol : float
Early stopping criterion
Returns
-------
x : ndarray
Notes
-----
This function solves following problem:
.. math::
\\begin{equation}
\\min_{x} ||Ux - m||^2
\\end{equation}
According to [1], non-negativity-constrained least square estimation problem becomes:
.. math::
\\begin{equation}
x' = (Utm) - (UTU)\\times x
\\end{equation}
Reference
----------
[1] : Bro, R., & De Jong, S. (1997). A fast non‐negativity‐constrained
least squares algorithm. Journal of Chemometrics: A Journal of
the Chemometrics Society, 11(5), 393-401.
"""
if tl.get_backend() == 'tensorflow':
raise ValueError(
"Active set is not supported with the tensorflow backend. Consider using fista method with tensorflow."
)
if x is None:
x_vec = tl.zeros(tl.shape(UtU)[1], **tl.context(UtU))
else:
x_vec = tl.base.tensor_to_vec(x)
x_gradient = Utm - tl.dot(UtU, x_vec)
passive_set = x_vec > 0
active_set = x_vec <= 0
support_vec = tl.zeros(tl.shape(x_vec), **tl.context(x_vec))
for iteration in range(n_iter_max):
if iteration > 0 or tl.all(x_vec == 0):
indice = tl.argmax(x_gradient)
passive_set = tl.index_update(passive_set, tl.index[indice], True)
active_set = tl.index_update(active_set, tl.index[indice], False)
# To avoid singularity error when initial x exists
try:
passive_solution = tl.solve(UtU[passive_set, :][:, passive_set],
Utm[passive_set])
indice_list = []
for i in range(tl.shape(support_vec)[0]):
if passive_set[i]:
indice_list.append(i)
support_vec = tl.index_update(
support_vec, tl.index[int(i)],
passive_solution[len(indice_list) - 1])
else:
support_vec = tl.index_update(support_vec,
tl.index[int(i)], 0)
# Start from zeros if solve is not achieved
except:
x_vec = tl.zeros(tl.shape(UtU)[1])
support_vec = tl.zeros(tl.shape(x_vec), **tl.context(x_vec))
passive_set = x_vec > 0
active_set = x_vec <= 0
if tl.any(active_set) == True:
indice = tl.argmax(x_gradient)
passive_set = tl.index_update(passive_set, tl.index[indice],
True)
active_set = tl.index_update(active_set, tl.index[indice],
False)
passive_solution = tl.solve(UtU[passive_set, :][:, passive_set],
Utm[passive_set])
indice_list = []
for i in range(tl.shape(support_vec)[0]):
if passive_set[i]:
indice_list.append(i)
support_vec = tl.index_update(
support_vec, tl.index[int(i)],
passive_solution[len(indice_list) - 1])
else:
support_vec = tl.index_update(support_vec,
tl.index[int(i)], 0)
# update support vector if it is necessary
if tl.min(support_vec[passive_set]) <= 0:
for i in range(len(passive_set)):
alpha = tl.min(
x_vec[passive_set][support_vec[passive_set] <= 0] /
(x_vec[passive_set][support_vec[passive_set] <= 0] -
support_vec[passive_set][support_vec[passive_set] <= 0]))
update = alpha * (support_vec - x_vec)
x_vec = x_vec + update
passive_set = x_vec > 0
active_set = x_vec <= 0
passive_solution = tl.solve(
UtU[passive_set, :][:, passive_set], Utm[passive_set])
indice_list = []
for i in range(tl.shape(support_vec)[0]):
if passive_set[i]:
indice_list.append(i)
support_vec = tl.index_update(
support_vec, tl.index[int(i)],
passive_solution[len(indice_list) - 1])
else:
support_vec = tl.index_update(support_vec,
tl.index[int(i)], 0)
if tl.any(passive_set) != True or tl.min(
support_vec[passive_set]) > 0:
break
# set x to s
x_vec = tl.clip(support_vec, 0, tl.max(support_vec))
# gradient update
x_gradient = Utm - tl.dot(UtU, x_vec)
if tl.any(active_set) != True or tl.max(x_gradient[active_set]) <= tol:
break
return x_vec
def maxvol(A):
""" Find the rxr submatrix of maximal volume in A(nxr), n>=r
We want to decompose matrix A as
A = A[:,J] * (A[I,J])^-1 * A[I,:]
This algorithm helps us find this submatrix A[I,J] from A, which has the largest determinant.
We greedily find vector of max norm, and subtract its projection from the rest of rows.
Parameters
----------
A: matrix
The matrix to find maximal volume
Returns
-------
row_idx: list of int
is the list or rows of A forming the matrix with maximal volume,
A_inv: matrix
is the inverse of the matrix with maximal volume.
References
----------
S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, N. L. Zamarashkin.
How to find a good submatrix.Goreinov, S. A., et al.
Matrix Methods: Theory, Algorithms and Applications: Dedicated to the Memory of Gene Golub. 2010. 247-256.
Ali Çivril, Malik Magdon-Ismail
On selecting a maximum volume sub-matrix of a matrix and related problems
Theoretical Computer Science. Volume 410, Issues 47–49, 6 November 2009, Pages 4801-4811
"""
(n, r) = tl.shape(A)
# The index of row of the submatrix
row_idx = tl.zeros(r)
# Rest of rows / unselected rows
rest_of_rows = tl.tensor(list(range(n)), dtype=tl.int64)
# Find r rows iteratively
i = 0
A_new = A
while i < r:
mask = list(range(tl.shape(A_new)[0]))
# Compute the square of norm of each row
rows_norms = tl.sum(A_new**2, axis=1)
# If there is only one row of A left, let's just return it. MxNet is not robust about this case.
if tl.shape(rows_norms) == ():
row_idx[i] = rest_of_rows
break
# If a row is 0, we delete it.
if any(rows_norms == 0):
zero_idx = tl.argmin(rows_norms, axis=0)
mask.pop(zero_idx)
rest_of_rows = rest_of_rows[mask]
A_new = A_new[mask, :]
continue
# Find the row of max norm
max_row_idx = tl.argmax(rows_norms, axis=0)
max_row = A[rest_of_rows[max_row_idx], :]
# Compute the projection of max_row to other rows
# projection a to b is computed as: <a,b> / sqrt(|a|*|b|)
projection = tl.dot(A_new, tl.transpose(max_row))
normalization = tl.sqrt(rows_norms[max_row_idx] * rows_norms)
# make sure normalization vector is of the same shape of projection (causing bugs for MxNet)
normalization = tl.reshape(normalization, tl.shape(projection))
projection = projection / normalization
# Subtract the projection from A_new: b <- b - a * projection
A_new = A_new - A_new * tl.reshape(projection, (tl.shape(A_new)[0], 1))
# Delete the selected row
mask.pop(max_row_idx)
A_new = A_new[mask, :]
# update the row_idx and rest_of_rows
row_idx[i] = rest_of_rows[max_row_idx]
rest_of_rows = rest_of_rows[mask]
i = i + 1
row_idx = tl.tensor(row_idx, dtype=tl.int64)
inverse = tl.solve(A[row_idx, :],
tl.eye(tl.shape(A[row_idx, :])[0], **tl.context(A)))
row_idx = tl.to_numpy(row_idx)
return row_idx, inverse
def parafac(tensor, rank, n_iter_max=100, init='svd', svd='numpy_svd',\
normalize_factors=False, orthogonalise=False,\
tol=1e-8, random_state=None,\
verbose=0, return_errors=False,\
non_negative=False,\
sparsity = None,\
l2_reg = 0, mask=None,\
cvg_criterion = 'abs_rec_error',\
fixed_modes = []):
"""CANDECOMP/PARAFAC decomposition via alternating least squares (ALS)
Computes a rank-`rank` decomposition of `tensor` [1]_ such that,
``tensor = [|weights; factors[0], ..., factors[-1] |]``.
Parameters
----------
tensor : ndarray
rank : int
Number of components.
n_iter_max : int
Maximum number of iteration
init : {'svd', 'random'}, optional
Type of factor matrix initialization. See `initialize_factors`.
svd : str, default is 'numpy_svd'
function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
normalize_factors : if True, aggregate the weights of each factor in a 1D-tensor
of shape (rank, ), which will contain the norms of the factors
tol : float, optional
(Default: 1e-6) Relative reconstruction error tolerance. The
algorithm is considered to have found the global minimum when the
reconstruction error is less than `tol`.
random_state : {None, int, np.random.RandomState}
verbose : int, optional
Level of verbosity
return_errors : bool, optional
Activate return of iteration errors
mask : ndarray
array of booleans with the same shape as ``tensor`` should be 0 where
the values are missing and 1 everywhere else. Note: if tensor is
sparse, then mask should also be sparse with a fill value of 1 (or
True). Allows for missing values [2]_
cvg_criterion : {'abs_rec_error', 'rec_error'}, optional
Stopping criterion for ALS, works if `tol` is not None.
If 'rec_error', ALS stops at current iteration if (previous rec_error - current rec_error) < tol.
If 'abs_rec_error', ALS terminates when |previous rec_error - current rec_error| < tol.
sparsity : float or int
If `sparsity` is not None, we approximate tensor as a sum of low_rank_component and sparse_component, where low_rank_component = kruskal_to_tensor((weights, factors)). `sparsity` denotes desired fraction or number of non-zero elements in the sparse_component of the `tensor`.
fixed_modes : list, default is []
A list of modes for which the initial value is not modified.
The last mode cannot be fixed due to error computation.
Returns
-------
KruskalTensor : (weight, factors)
* weights : 1D array of shape (rank, )
all ones if normalize_factors is False (default),
weights of the (normalized) factors otherwise
* factors : List of factors of the CP decomposition element `i` is of shape
(tensor.shape[i], rank)
* sparse_component : nD array of shape tensor.shape. Returns only if `sparsity` is not None.
errors : list
A list of reconstruction errors at each iteration of the algorithms.
References
----------
.. [1] T.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications",
SIAM REVIEW, vol. 51, n. 3, pp. 455-500, 2009.
.. [2] Tomasi, Giorgio, and Rasmus Bro. "PARAFAC and missing values."
Chemometrics and Intelligent Laboratory Systems 75.2 (2005): 163-180.
"""
epsilon = 10e-12
if mask is not None and init == "svd":
message = "Masking occurs after initialization. Therefore, random initialization is recommended."
warnings.warn(message, Warning)
if orthogonalise and not isinstance(orthogonalise, int):
orthogonalise = n_iter_max
factors = initialize_factors(tensor,
rank,
init=init,
svd=svd,
random_state=random_state,
normalize_factors=normalize_factors)
rec_errors = []
norm_tensor = tl.norm(tensor, 2)
weights = tl.ones(rank, **tl.context(tensor))
Id = tl.eye(rank, **tl.context(tensor)) * l2_reg
if tl.ndim(tensor) - 1 in fixed_modes:
warnings.warn(
'You asked for fixing the last mode, which is not supported.\n The last mode will not be fixed. Consider using tl.moveaxis()'
)
fixed_modes.remove(tl.ndim(tensor) - 1)
modes_list = [
mode for mode in range(tl.ndim(tensor)) if mode not in fixed_modes
]
if sparsity:
sparse_component = tl.zeros_like(tensor)
if isinstance(sparsity, float):
sparsity = int(sparsity * np.prod(tensor.shape))
else:
sparsity = int(sparsity)
for iteration in range(n_iter_max):
if orthogonalise and iteration <= orthogonalise:
factors = [
tl.qr(f)[0] if min(tl.shape(f)) >= rank else f
for i, f in enumerate(factors)
]
if verbose > 1:
print("Starting iteration", iteration + 1)
for mode in modes_list:
if verbose > 1:
print("Mode", mode, "of", tl.ndim(tensor))
pseudo_inverse = tl.tensor(np.ones((rank, rank)),
**tl.context(tensor))
for i, factor in enumerate(factors):
if i != mode:
pseudo_inverse = pseudo_inverse * tl.dot(
tl.conj(tl.transpose(factor)), factor)
pseudo_inverse += Id
if mask is not None:
tensor = tensor * mask + tl.kruskal_to_tensor(
(None, factors), mask=1 - mask)
mttkrp = unfolding_dot_khatri_rao(tensor, (None, factors), mode)
factor = tl.transpose(
tl.solve(tl.conj(tl.transpose(pseudo_inverse)),
tl.transpose(mttkrp)))
if normalize_factors:
weights = tl.norm(factor, order=2, axis=0)
weights = tl.where(
tl.abs(weights) <= tl.eps(tensor.dtype),
tl.ones(tl.shape(weights), **tl.context(factors[0])),
weights)
factor = factor / (tl.reshape(weights, (1, -1)))
factors[mode] = factor
if tol:
if sparsity:
low_rank_component = kruskal_to_tensor((weights, factors))
sparse_component = sparsify_tensor(tensor - low_rank_component,
sparsity)
unnorml_rec_error = tl.norm(
tensor - low_rank_component - sparse_component, 2)
else:
# ||tensor - rec||^2 = ||tensor||^2 + ||rec||^2 - 2*<tensor, rec>
factors_norm = kruskal_norm((weights, factors))
# mttkrp and factor for the last mode. This is equivalent to the
# inner product <tensor, factorization>
iprod = tl.sum(tl.sum(mttkrp * factor, axis=0) * weights)
unnorml_rec_error = tl.sqrt(
tl.abs(norm_tensor**2 + factors_norm**2 - 2 * iprod))
rec_error = unnorml_rec_error / norm_tensor
rec_errors.append(rec_error)
if iteration >= 1:
rec_error_decrease = rec_errors[-2] - rec_errors[-1]
if verbose:
print(
"iteration {}, reconstraction error: {}, decrease = {}, unnormalized = {}"
.format(iteration, rec_error, rec_error_decrease,
unnorml_rec_error))
if cvg_criterion == 'abs_rec_error':
stop_flag = abs(rec_error_decrease) < tol
elif cvg_criterion == 'rec_error':
stop_flag = rec_error_decrease < tol
else:
raise TypeError("Unknown convergence criterion")
if stop_flag:
if verbose:
print("PARAFAC converged after {} iterations".format(
iteration))
break
else:
if verbose:
print('reconstruction error={}'.format(rec_errors[-1]))
kruskal_tensor = KruskalTensor((weights, factors))
if sparsity:
sparse_component = sparsify_tensor(tensor -\
kruskal_to_tensor((weights, factors)),\
sparsity)
kruskal_tensor = (kruskal_tensor, sparse_component)
if return_errors:
return kruskal_tensor, rec_errors
else:
return kruskal_tensor
def her_CPRAND5(tensor,
rank,
n_samples,
factors=None,
exact_err=True,
it_max=100,
err_it_max=20,
tol=1e-7,
beta=0.1,
eta=3,
gamma=1.01,
gamma_bar=1.005,
list_factors=False,
time_rec=False):
"""
different err sample taking mean value
"""
beta_bar = 1
N = tl.ndim(tensor) # order of tensor
norm_tensor = tl.norm(tensor) # norm of tensor
if list_factors == True: list_fac = []
if (time_rec == True): list_time = []
if (factors == None): factors = svd_init_fac(tensor, rank)
# Initialization of factor hat matrice by factor matrices
factors_hat = factors
if list_factors == True: list_fac.append(copy.deepcopy(factors))
list_F_hat_bf = []
weights = None
it = 0
err_it = 0
cpt = 0
n_samples_err = 400 # assuming that miu max = 1
########################################
######### error initialization #########
########################################
F_hat_bf, ind_bf = err_rand(tensor, None, factors, n_samples_err)
list_F_hat_bf.append(F_hat_bf)
F_hat_bf_ex = err(tensor, None, factors) # exact cost
rng = tl.check_random_state(None)
error = [F_hat_bf / norm_tensor]
error_ex = [F_hat_bf_ex / norm_tensor]
min_err = error[len(error) - 1]
while (min_err > tol and it < it_max and err_it < err_it_max):
if time_rec == True: tic = time.time()
for n in range(N):
Zs, indices = sample_khatri_rao(factors_hat,
n_samples,
skip_matrix=n,
random_state=rng)
indices_list = [i.tolist() for i in indices]
indices_list.insert(n, slice(None, None, None))
indices_list = tuple(indices_list)
V = tl.dot(tl.transpose(Zs), Zs)
# J'ai du mal avec la syntaxe tensor[indices_list],
# Ca renvoie une matrices et non un tenseur?
if (n == 0): sampled_unfolding = tensor[indices_list]
else: sampled_unfolding = tl.transpose(tensor[indices_list])
W = tl.dot(sampled_unfolding, Zs)
factor_bf = factors[n]
# update
factors[n] = tl.transpose(
tl.solve(V, tl.transpose(W))
) # solve needs a squared full rank matrix, if rank>nb_sampls ok
# if (n==N-1) : F_hat_new=tl.norm(tl.dot(Zs,tl.transpose(factors[n]))-sampled_unfolding,2) # cost update
# extrapolate
factors_hat[n] = factors[n] + beta * (factors[n] - factor_bf)
########################################
######### error update #########
########################################
F_hat_new, _ = err_rand(tensor,
weights,
factors,
n_samples_err,
indices_list=ind_bf)
#
# added
# a new sample
ind_bf = [
np.random.choice(tl.shape(m)[0], n_samples_err) for m in factors
]
ind_bf = [i.tolist() for i in ind_bf]
ind_bf = tuple(ind_bf)
list_F_hat_bf.append(F_hat_new)
if (F_hat_new > F_hat_bf):
factors_hat = factors
beta_bar = beta
beta = beta / eta
cpt = cpt + 1
else:
factors = factors_hat
beta_bar = min(1, beta_bar * gamma_bar)
beta = min(beta_bar, gamma * beta)
########################################
######### update for next it #########
########################################
it = it + 1
if it < 10:
F_hat_bf = np.mean(list_F_hat_bf)
else:
F_hat_bf = np.mean(list_F_hat_bf[(len(list_F_hat_bf) -
10):(len(list_F_hat_bf) - 1)])
if list_factors == True: list_fac.append(copy.deepcopy(factors))
error.append(F_hat_new / norm_tensor)
if (error[len(error) - 1] < min_err):
min_err = error[len(error) - 1] # err update
else:
err_it = err_it + 1
if time_rec == True:
toc = time.time()
list_time.append(toc - tic)
error_ex.append(err(tensor, None, factors) /
norm_tensor) # exact cost update
# weights,factors=tl.cp_normalize((None,factors))
if list_factors == True and time_rec == True:
return (weights, factors, it, error_ex, error, cpt / it, list_fac,
list_time)
if list_factors == True:
return (weights, factors, it, error_ex, error, cpt / it, list_fac)
if time_rec == True:
return (weights, factors, it, error_ex, error, cpt / it, list_time)
return (weights, factors, it, error_ex, error, cpt / it)
def her_CPRAND(tensor,
rank,
n_samples,
n_samples_err=400,
factors=None,
exact_err=False,
it_max=100,
err_it_max=20,
tol=1e-7,
beta=0.1,
eta=3,
gamma=1.01,
gamma_bar=1.005,
list_factors=False,
time_rec=False,
filter=10):
"""
herCPRAND for CP-decomposition
same err sample taking mean value of last filter values
Parameters
----------
tensor : tensor
rank : int
n_samples : int
sample size
n_samples_err : int, optional
sample size used for error estimation. The default is 400.
factors : list of matrices, optional
an initial factor matrices. The default is None.
exact_err : boolean, optional
whether use err or err_rand_fast for terminaison criterion. The default is False.
it_max : int, optional
maximal number of iteration. The default is 100.
err_it_max : int, optional
maximal of iteration if terminaison critirion is not improved. The default is 20.
tol : float, optional
error tolerance. The default is 1e-7.
beta : float, optional
extrapolation parameter. The default is 0.5.
eta : float, optional
decrease coefficient of beta. The default is 1.5.
gamma : float, optional
increase coefficient of beta. The default is 1.05.
gamma_bar : float, optional
increase coeefficient of beta_bar. The default is 1.01.
list_factors : boolean, optional
If true, then return factor matrices of each iteration. The default is False.
time_rec : boolean, optional
If true, return computation time of each iteration. The default is False.
filter : int, optional
The filter size used for the mean value
Returns
-------
the CP decomposition, number of iteration, error and restart pourcentage.
list_fac and list_time are optional.
"""
beta_bar = 1
N = tl.ndim(tensor) # order of tensor
norm_tensor = tl.norm(tensor) # norm of tensor
if list_factors == True: list_fac = []
if (time_rec == True): list_time = []
if (factors == None): factors = svd_init_fac(tensor, rank)
# Initialization of factor hat matrice by factor matrices
factors_hat = factors
if list_factors == True: list_fac.append(copy.deepcopy(factors))
list_F_hat_bf = []
weights = None
it = 0
err_it = 0
cpt = 0
########################################
######### error initialization #########
########################################
if (exact_err == True):
F_hat_bf = err(tensor, weights, factors)
else:
F_hat_bf, ind_bf = err_rand(tensor, None, factors, n_samples_err)
list_F_hat_bf.append(F_hat_bf)
rng = tl.check_random_state(None)
error = [F_hat_bf / norm_tensor]
min_err = error[len(error) - 1]
while (min_err > tol and it < it_max and err_it < err_it_max):
if time_rec == True: tic = time.time()
for n in range(N):
Zs, indices = sample_khatri_rao(factors_hat,
n_samples,
skip_matrix=n,
random_state=rng)
indices_list = [i.tolist() for i in indices]
indices_list.insert(n, slice(None, None, None))
indices_list = tuple(indices_list)
V = tl.dot(tl.transpose(Zs), Zs)
if (n == 0): sampled_unfolding = tensor[indices_list]
else: sampled_unfolding = tl.transpose(tensor[indices_list])
W = tl.dot(sampled_unfolding, Zs)
factor_bf = factors[n]
# update
factors[n] = tl.transpose(
tl.solve(V, tl.transpose(W))
) # solve needs a squared full rank matrix, if rank>nb_sampls ok
# extrapolate
factors_hat[n] = factors[n] + beta * (factors[n] - factor_bf)
########################################
######### error update #########
########################################
if (exact_err == False):
F_hat_new, _ = err_rand(tensor,
weights,
factors,
n_samples_err,
indices_list=ind_bf)
else:
F_hat_new = err(tensor, weights, factors)
list_F_hat_bf.append(F_hat_new)
if (F_hat_new > F_hat_bf):
factors_hat = factors
beta_bar = beta
beta = beta / eta
cpt = cpt + 1
else:
factors = factors_hat
beta_bar = min(1, beta_bar * gamma_bar)
beta = min(beta_bar, gamma * beta)
########################################
######### update for next it #########
########################################
it = it + 1
if (exact_err == False):
if it < filter:
F_hat_bf = np.mean(list_F_hat_bf)
else:
F_hat_bf = np.mean(list_F_hat_bf[(len(list_F_hat_bf) -
filter):(len(list_F_hat_bf) -
1)])
else:
F_hat_bf = F_hat_new
if list_factors == True: list_fac.append(copy.deepcopy(factors))
error.append(F_hat_new / norm_tensor)
if (error[len(error) - 1] < min_err):
min_err = error[len(error) - 1] # err update
else:
err_it = err_it + 1
if time_rec == True:
toc = time.time()
list_time.append(toc - tic)
if list_factors == True and time_rec == True:
return (weights, factors, it, error, cpt / it, list_fac, list_time)
if list_factors == True:
return (weights, factors, it, error, cpt / it, list_fac)
if time_rec == True:
return (weights, factors, it, error, cpt / it, list_time)
return (weights, factors, it, error, cpt / it)