def __init__(self, input_size, ranks, output_size, verbose=1, **kwargs):
super(TRL, self).__init__(**kwargs)
self.ranks = list(ranks)
self.verbose = verbose
if isinstance(output_size, int):
self.input_size = [input_size]
else:
self.input_size = list(input_size)
if isinstance(output_size, int):
self.output_size = [output_size]
else:
self.output_size = list(output_size)
self.n_outputs = int(np.prod(output_size[1:]))
# Core of the regression tensor weights
self.core = nn.Parameter(tl.zeros(self.ranks), requires_grad=True)
self.bias = nn.Parameter(tl.zeros(1), requires_grad=True)
weight_size = list(self.input_size[1:]) + list(self.output_size[1:])
# Add and register the factors
self.factors = []
for index, (in_size, rank) in enumerate(zip(weight_size, ranks)):
self.factors.append(
nn.Parameter(tl.zeros((in_size, rank)), requires_grad=True))
self.register_parameter('factor_{}'.format(index),
self.factors[index])
self.core.data.uniform_(-0.1, 0.1)
for f in self.factors:
f.data.uniform_(-0.1, 0.1)
def simplex_prox(tensor, parameter):
"""
Projects the input tensor on the simplex of radius parameter.
Parameters
----------
tensor : ndarray
parameter : float
Returns
-------
ndarray
References
----------
.. [1]: Held, Michael, Philip Wolfe, and Harlan P. Crowder.
"Validation of subgradient optimization."
Mathematical programming 6.1 (1974): 62-88.
"""
_, col = tl.shape(tensor)
tensor = tl.clip(tensor, 0, tl.max(tensor))
tensor_sort = tl.sort(tensor, axis=0, descending=True)
to_change = tl.sum(tl.where(
tensor_sort > (tl.cumsum(tensor_sort, axis=0) - parameter), 1.0, 0.0),
axis=0)
difference = tl.zeros(col)
for i in range(col):
if to_change[i] > 0:
difference = tl.index_update(
difference, tl.index[i],
tl.cumsum(tensor_sort, axis=0)[int(to_change[i] - 1), i])
difference = (difference - parameter) / to_change
return tl.clip(tensor - difference, a_min=0)
def monotonicity_prox(tensor, decreasing=False):
"""
This function projects each column of the input array on the set of arrays so that
x[1] <= x[2] <= ... <= x[n] (decreasing=False)
or
x[1] >= x[2] >= ... >= x[n] (decreasing=True)
is satisfied columnwise.
Parameters
----------
tensor : ndarray
decreasing : If it is True, function returns columnwise
monotone decreasing tensor. Otherwise, returned array
will be monotone increasing.
Default: True
Returns
-------
ndarray
A tensor of which columns' are monotonic.
References
----------
.. [1]: G. Chierchia, E. Chouzenoux, P. L. Combettes, and J.-C. Pesquet
"The Proximity Operator Repository. User's guide"
"""
if tl.ndim(tensor) == 1:
tensor = tl.reshape(tensor, [tl.shape(tensor)[0], 1])
elif tl.ndim(tensor) > 2:
raise ValueError(
"Monotonicity prox doesn't support an input which has more than 2 dimensions."
)
tensor_mon = tl.copy(tensor)
if decreasing:
tensor_mon = tl.flip(tensor_mon, axis=0)
row, column = tl.shape(tensor_mon)
cum_sum = tl.cumsum(tensor_mon, axis=0)
for j in range(column):
assisted_tensor = tl.zeros([row, row])
for i in range(row):
if i == 0:
assisted_tensor = tl.index_update(
assisted_tensor, tl.index[i, i:], cum_sum[i:, j] /
tl.tensor(tl.arange(row - i) + 1, **tl.context(tensor)))
else:
assisted_tensor = tl.index_update(
assisted_tensor, tl.index[i, i:],
(cum_sum[i:, j] - cum_sum[i - 1, j]) /
tl.tensor(tl.arange(row - i) + 1, **tl.context(tensor)))
tensor_mon = tl.index_update(tensor_mon, tl.index[:, j],
tl.max(assisted_tensor, axis=0))
for i in reversed(range(row - 1)):
if tensor_mon[i, j] > tensor_mon[i + 1, j]:
tensor_mon = tl.index_update(tensor_mon, tl.index[i, j],
tensor_mon[i + 1, j])
if decreasing:
tensor_mon = tl.flip(tensor_mon, axis=0)
return tensor_mon
def __init__(self,
input_size,
rank: int,
output_size,
verbose=1,
**kwargs):
super(CPRL, self).__init__(**kwargs)
self.rank = [rank] * (len(input_size))
self.verbose = verbose
if isinstance(input_size, int):
self.input_size = [input_size]
else:
self.input_size = list(input_size)
if isinstance(output_size, int):
self.output_size = [output_size]
else:
self.output_size = list(output_size)
self.n_outputs = int(np.prod(output_size[1:]))
self.bias = nn.Parameter(tl.zeros(1), requires_grad=True)
# add and register factors
self.factors = []
ranks = self.rank
factor_size = list(input_size)[1:] + list(output_size)[1:]
for index, (in_size, rank) in enumerate(zip(factor_size, ranks)):
self.factors.append(
nn.Parameter(tl.zeros((in_size, rank)), requires_grad=True))
self.register_parameter(f'factor_{index}', self.factors[index])
self.weights = nn.Parameter(tl.zeros((rank, )), requires_grad=True)
# init params
for f in self.factors:
f.data.uniform_(-.1, .1)
self.weights.data.uniform_(-.1, 1)
def cov(A, B):
"""Computes the mode 1 (mode 0 in python) contraction of 2 matrices."""
assert A.shape[0] == B.shape[0], "A and B need to have the same shape on axis 0"
dimension_A = A.shape[1:]
dimension_B = B.shape[1:]
dimensions = list(dimension_A) + list(dimension_B)
rmode_A = len(dimension_A)
dim = A.shape[0]
C = tl.zeros(dimensions)
indices = []
for mode in dimensions:
indices.append(range(mode))
for idx in product(*indices):
idx_A, idx_B = list(idx[:rmode_A]), list(idx[rmode_A:])
C[idx] = np.sum(
[A[tuple([i] + idx_A)] * B[tuple([i] + idx_B)] for i in range(dim)]
)
return C
def sparsify_tensor(tensor, card):
"""Zeros out all elements in the `tensor` except `card` elements with maximum absolute values.
Parameters
----------
tensor : ndarray
card : int
Desired number of non-zero elements in the `tensor`
Returns
-------
ndarray of shape tensor.shape
"""
if card >= np.prod(tensor.shape):
return tensor
bound = tl.sort(tl.abs(tensor), axis = None)[-card]
return tl.where(tl.abs(tensor) < bound, tl.zeros(tensor.shape, **tl.context(tensor)), tensor)
def test_tr_to_tensor():
# Create ground truth TR factors
factors = [tl.randn((2, 4, 3)), tl.randn((3, 5, 2)), tl.randn((2, 6, 2))]
# Create tensor
tensor = tl.zeros((4, 5, 6))
for i in range(4):
for j in range(5):
for k in range(6):
product = tl.dot(
tl.dot(factors[0][:, i, :], factors[1][:, j, :]),
factors[2][:, k, :])
# TODO: add trace to backend instead of this
tensor = tl.index_update(
tensor, tl.index[i, j, k],
tl.sum(product * tl.eye(product.shape[0])))
# Check that TR factors re-assemble to the original tensor
assert_array_almost_equal(tensor, tr_to_tensor(factors))
def _fit_2d(self, X, Y):
"""
Compute the HOPLS for X and Y wrt the parameters R, Ln and Km for the special case mode_Y = 2.
Parameters:
X: tensorly Tensor, The target tensor of shape [i1, ... iN], N = 2.
Y: tensorly Tensor, The target tensor of shape [j1, ... jM], M >= 3.
Returns:
G: Tensor, The core Tensor of the HOPLS for X, of shape (R, L2, ..., LN).
P: List, The N-1 loadings of X.
D: Tensor, The core Tensor of the HOPLS for Y, of shape (R, K2, ..., KN).
Q: List, The N-1 loadings of Y.
ts: Tensor, The latent vectors of the HOPLS, of shape (i1, R).
"""
# Initialization
Er, Fr = X, Y
P, T, W, Q = [], [], [], []
D = tl.zeros((self.R, self.R))
G = []
# Beginning of the algorithm
# Gr, _ = tucker(Er, ranks=[1] + self.Ln)
for r in range(self.R):
if torch.norm(Er) > self.epsilon and torch.norm(Fr) > self.epsilon:
# computing the covariance
Cr = mode_dot(Er, Fr.t(), 0)
# HOOI tucker decomposition of C
Gr_C, latents = tucker(Cr, rank=[1] + self.Ln)
# Getting P and Q loadings
qr = latents[0]
qr /= torch.norm(qr)
# Pr = latents[1:]
Pr = [a / torch.norm(a) for a in latents[1:]]
P.append(Pr)
tr = multi_mode_dot(Er, Pr, list(range(1, len(Pr) + 1)), transpose=True)
# Gr_pi = torch.pinverse(matricize(Gr))
# tr = torch.mm(matricize(tr), Gr_pi)
GrC_pi = torch.pinverse(matricize(Gr_C))
tr = torch.mm(matricize(tr), GrC_pi)
tr /= torch.norm(tr)
# recomposition of the core tensor of Y
ur = torch.mm(Fr, qr)
dr = torch.mm(ur.t(), tr)
D[r, r] = dr
Pkron = kronecker([Pr[self.N - n - 1] for n in range(self.N)])
# P.append(torch.mm(matricize(Gr), Pkron.t()).t())
# W.append(torch.mm(Pkron, Gr_pi))
Q.append(qr)
T.append(tr)
Gd = tl.tucker_to_tensor([Er, [tr] + Pr], transpose_factors=True)
Gd_pi = torch.pinverse(matricize(Gd))
W.append(torch.mm(Pkron, Gd_pi))
# Deflation
# X_hat = torch.mm(torch.cat(T, dim=1), torch.cat(P, dim=1).t())
# Er = X - np.reshape(X_hat, (Er.shape), order="F")
Er = Er - tl.tucker_to_tensor([Gd, [tr] + Pr])
Fr = Fr - dr * torch.mm(tr, qr.t())
else:
break
Q = torch.cat(Q, dim=1)
T = torch.cat(T, dim=1)
# P = torch.cat(P, dim=1)
W = torch.cat(W, dim=1)
self.model = (P, Q, D, T, W)
return self
def tensor_train_cross(input_tensor, rank, tol=1e-4, n_iter_max=100):
"""TT (tensor-train) decomposition via cross-approximation (TTcross) [1]
Decomposes `input_tensor` into a sequence of order-3 tensors of given rank. (factors/cores)
Rather than directly decompose the whole tensor, we sample fibers based on skeleton decomposition.
We initialize a random tensor-train and sweep from left to right and right to left.
On each core, we shape the core as a matrix and choose the fibers indices by finding maximum-volume submatrix and update the core.
Advantage: faster
The main advantage of TTcross is that it doesn't need to evaluate all the entries of the tensor.
For a tensor_shape^tensor_order tensor, SVD needs O(tensor_shape^tensor_order) runtime, but TTcross' runtime is linear in tensor_shape and tensor_order, which makes it feasible in high dimension.
Disadvantage: less accurate
TTcross may underestimate the error, since it only evaluates partial entries of the tensor.
Besides, in contrast to its practical fast performance, there is no theoretical guarantee of it convergence.
Parameters
----------
input_tensor : tensorly.tensor
The tensor to decompose.
rank : {int, int list}
maximum allowable TT rank of the factors
if int, then this is the same for all the factors
if int list, then rank[k] is the rank of the kth factor
tol : float
accuracy threshold for outer while-loop
n_iter_max : int
maximum iterations of outer while-loop (the 'crosses' or 'sweeps' sampled)
Returns
-------
factors : TT factors
order-3 tensors of the TT decomposition
Examples
--------
Generate a 5^3 tensor, and decompose it into tensor-train of 3 factors, with rank = [1,3,3,1]
>>> tensor = tl.tensor(np.arange(5**3).reshape(5,5,5))
>>> rank = [1, 3, 3, 1]
>>> factors = tensor_train_cross(tensor, rank)
print the first core:
>>> print(factors[0])
.[[[ 24. 0. 4.]
[ 49. 25. 29.]
[ 74. 50. 54.]
[ 99. 75. 79.]
[124. 100. 104.]]]
Notes
-----
Pseudo-code [2]:
1. Initialization tensor_order cores and column indices
2. while (error > tol)
3. update the tensor-train from left to right:
for Core 1 to Core tensor_order
approximate the skeleton-decomposition by QR and maxvol
4. update the tensor-train from right to left:
for Core tensor_order to Core 1
approximate the skeleton-decomposition by QR and maxvol
5. end while
Acknowledgement: the main body of the code is modified based on TensorToolbox by Daniele Bigoni
References
----------
.. [1] Ivan Oseledets and Eugene Tyrtyshnikov. Tt-cross approximation for multidimensional arrays.
LinearAlgebra and its Applications, 432(1):70–88, 2010.
.. [2] Sergey Dolgov and Robert Scheichl. A hybrid alternating least squares–tt cross algorithm for parametricpdes.
arXiv preprint arXiv:1707.04562, 2017.
"""
# Check user input for errors
tensor_shape = tl.shape(input_tensor)
tensor_order = tl.ndim(input_tensor)
if isinstance(rank, int):
rank = [rank] * (tensor_order + 1)
elif tensor_order + 1 != len(rank):
message = 'Provided incorrect number of ranks. Should verify len(rank) == tl.ndim(tensor)+1, but len(rank) = {} while tl.ndim(tensor) + 1 = {}'.format(
len(rank), tensor_order)
raise (ValueError(message))
# Make sure iter's not a tuple but a list
rank = list(rank)
# Initialize rank
if rank[0] != 1:
print(
'Provided rank[0] == {} but boundary conditions dictate rank[0] == rank[-1] == 1: setting rank[0] to 1.'.format(
rank[0]))
rank[0] = 1
if rank[-1] != 1:
print(
'Provided rank[-1] == {} but boundary conditions dictate rank[0] == rank[-1] == 1: setting rank[-1] to 1.'.format(
rank[0]))
# list col_idx: column indices (right indices) for skeleton-decomposition: indicate which columns used in each core.
# list row_idx: row indices (left indices) for skeleton-decomposition: indicate which rows used in each core.
# Initialize indice: random selection of column indices
random_seed = None
rng = check_random_state(random_seed)
col_idx = [None] * tensor_order
for k_col_idx in range(tensor_order - 1):
col_idx[k_col_idx] = []
for i in range(rank[k_col_idx + 1]):
newidx = tuple([rng.randint(tensor_shape[j]) for j in range(k_col_idx + 1, tensor_order)])
while newidx in col_idx[k_col_idx]:
newidx = tuple([rng.randint(tensor_shape[j]) for j in range(k_col_idx + 1, tensor_order)])
col_idx[k_col_idx].append(newidx)
# Initialize the cores of tensor-train
factor_old = [tl.zeros((rank[k], tensor_shape[k], rank[k + 1]),
**tl.context(input_tensor)) for k in range(tensor_order)]
factor_new = [tl.tensor(rng.random_sample((rank[k], tensor_shape[k], rank[k + 1])),
**tl.context(input_tensor)) for k in range(tensor_order)]
iter = 0
error = tl.norm(tt_to_tensor(factor_old) - tt_to_tensor(factor_new), 2)
threshold = tol * tl.norm(tt_to_tensor(factor_new), 2)
for iter in range(n_iter_max):
if error < threshold:
break
factor_old = factor_new
factor_new = [None for i in range(tensor_order)]
######################################
# left-to-right step
left_to_right_fiberlist = []
# list row_idx: list of (tensor_order-1) of lists of left indices
row_idx = [[()]]
for k in range(tensor_order - 1):
(next_row_idx, fibers_list) = left_right_ttcross_step(input_tensor, k, rank, row_idx, col_idx)
# update row indices
left_to_right_fiberlist.extend(fibers_list)
row_idx.append(next_row_idx)
# end left-to-right step
###############################################
###############################################
# right-to-left step
right_to_left_fiberlist = []
# list col_idx: list (tensor_order-1) of lists of right indices
col_idx = [None] * tensor_order
col_idx[-1] = [()]
for k in range(tensor_order, 1, -1):
(next_col_idx, fibers_list, Q_skeleton) = right_left_ttcross_step(input_tensor, k, rank, row_idx, col_idx)
# update col indices
right_to_left_fiberlist.extend(fibers_list)
col_idx[k - 2] = next_col_idx
# Compute cores
try:
factor_new[k - 1] = tl.transpose(Q_skeleton)
factor_new[k - 1] = tl.reshape(factor_new[k - 1], (rank[k - 1], tensor_shape[k - 1], rank[k]))
except:
# The rank should not be larger than the input tensor's size
raise (ValueError("The rank is too large compared to the size of the tensor. Try with small rank."))
# Add the last core
idx = (slice(None, None, None),) + tuple(zip(*col_idx[0]))
core = input_tensor[idx]
core = tl.reshape(core, (tensor_shape[0], 1, rank[1]))
core = tl.transpose(core, (1, 0, 2))
factor_new[0] = core
# end right-to-left step
################################################
# check the error for while-loop
error = tl.norm(tt_to_tensor(factor_old) - tt_to_tensor(factor_new), 2)
threshold = tol * tl.norm(tt_to_tensor(factor_new), 2)
print("It: " + str(iter) + "; error: " + str(error) + "; threshold: " + str(threshold))
# check convergence
if iter >= n_iter_max:
print('Maximum number of iterations reached.')
if tl.norm(tt_to_tensor(factor_old) - tt_to_tensor(factor_new), 2) > tol * tl.norm(tt_to_tensor(factor_new), 2):
print('Low Rank Approximation algorithm did not converge.')
return factor_new
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 constrained_parafac(tensor, rank, n_iter_max=100, n_iter_max_inner=10,
init='svd', svd='numpy_svd',
tol_outer=1e-8, tol_inner=1e-6, random_state=None,
verbose=0, return_errors=False,
cvg_criterion='abs_rec_error',
fixed_modes=None, non_negative=None, l1_reg=None,
l2_reg=None, l2_square_reg=None, unimodality=None, normalize=None,
simplex=None, normalized_sparsity=None, soft_sparsity=None,
smoothness=None, monotonicity=None, hard_sparsity=None):
"""CANDECOMP/PARAFAC decomposition via alternating optimization of
alternating direction method of multipliers (AO-ADMM):
Computes a rank-`rank` decomposition of `tensor` [1]_ such that::
tensor = [|weights; factors[0], ..., factors[-1] |],
where factors are either penalized or constrained according to the user-defined constraint.
In order to compute the factors efficiently, the ADMM algorithm
introduces an auxilliary factor which is called factor_aux in the function.
Parameters
----------
tensor : ndarray
rank : int
Number of components.
n_iter_max : int
Maximum number of iteration for outer loop
n_iter_max_inner : int
Number of iteration for inner loop
init : {'svd', 'random', cptensor}, 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
tol_outer : float, optional
(Default: 1e-8) Relative reconstruction error tolerance for outer loop. The
algorithm is considered to have found a local minimum when the
reconstruction error is less than `tol_outer`.
tol_inner : float, optional
(Default: 1e-6) Absolute reconstruction error tolerance for factor update during inner loop, i.e. ADMM optimization.
random_state : {None, int, np.random.RandomState}
verbose : int, optional
Level of verbosity
return_errors : bool, optional
Activate return of iteration errors
non_negative : bool or dictionary
This constraint is clipping negative values to '0'. If it is True non-negative constraint is applied to all modes.
l1_reg : float or list or dictionary, optional
l2_reg : float or list or dictionary, optional
l2_square_reg : float or list or dictionary, optional
unimodality : bool or dictionary, optional
If it is True unimodality constraint is applied to all modes.
normalize : bool or dictionary, optional
This constraint divides all the values by maximum value of the input array. If it is True normalize constraint
is applied to all modes.
simplex : float or list or dictionary, optional
normalized_sparsity : float or list or dictionary, optional
soft_sparsity : float or list or dictionary, optional
smoothness : float or list or dictionary, optional
monotonicity : bool or dictionary, optional
hard_sparsity : float or list or dictionary, optional
cvg_criterion : {'abs_rec_error', 'rec_error'}, optional
Stopping criterion if `tol` is not None.
If 'rec_error', algorithm stops at current iteration if ``(previous rec_error - current rec_error) < tol``.
If 'abs_rec_error', algorithm terminates when `|previous rec_error - current rec_error| < tol`.
fixed_modes : list, default is None
A list of modes for which the initial value is not modified.
The last mode cannot be fixed due to error computation.
Returns
-------
CPTensor : (weight, factors)
* weights : 1D array of shape (rank, )
* factors : 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] T.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications", SIAM
REVIEW, vol. 51, n. 3, pp. 455-500, 2009.
.. [2] Huang, Kejun, Nicholas D. Sidiropoulos, and Athanasios P. Liavas.
"A flexible and efficient algorithmic framework for constrained matrix and tensor factorization." IEEE
Transactions on Signal Processing 64.19 (2016): 5052-5065.
"""
rank = validate_cp_rank(tl.shape(tensor), rank=rank)
_, _ = validate_constraints(non_negative=non_negative, l1_reg=l1_reg, l2_reg=l2_reg, l2_square_reg=l2_square_reg,
unimodality=unimodality, normalize=normalize, simplex=simplex,
normalized_sparsity=normalized_sparsity, soft_sparsity=soft_sparsity,
smoothness=smoothness, monotonicity=monotonicity, hard_sparsity=hard_sparsity,
n_const=tl.ndim(tensor))
weights, factors = initialize_constrained_parafac(tensor, rank, init=init, svd=svd,
random_state=random_state, non_negative=non_negative,
l1_reg=l1_reg, l2_reg=l2_reg, l2_square_reg=l2_square_reg,
unimodality=unimodality, normalize=normalize,
simplex=simplex,
normalized_sparsity=normalized_sparsity,
soft_sparsity=soft_sparsity,
smoothness=smoothness, monotonicity=monotonicity,
hard_sparsity=hard_sparsity)
rec_errors = []
norm_tensor = tl.norm(tensor, 2)
if fixed_modes is None:
fixed_modes = []
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]
# ADMM inits
dual_variables = []
factors_aux = []
for i in range(len(factors)):
dual_variables.append(tl.zeros(tl.shape(factors[i])))
factors_aux.append(tl.transpose(tl.zeros(tl.shape(factors[i]))))
for iteration in range(n_iter_max):
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.transpose(factor), factor)
mttkrp = unfolding_dot_khatri_rao(tensor, (None, factors), mode)
factors[mode], factors_aux[mode], dual_variables[mode] = admm(mttkrp, pseudo_inverse, factors[mode], dual_variables[mode],
n_iter_max=n_iter_max_inner, n_const=tl.ndim(tensor),
order=mode, non_negative=non_negative, l1_reg=l1_reg,
l2_reg=l2_reg, l2_square_reg=l2_square_reg,
unimodality=unimodality, normalize=normalize,
simplex=simplex, normalized_sparsity=normalized_sparsity,
soft_sparsity=soft_sparsity,
smoothness=smoothness, monotonicity=monotonicity,
hard_sparsity=hard_sparsity, tol=tol_inner)
factors_norm = cp_norm((weights, factors))
iprod = tl.sum(tl.sum(mttkrp * factors[-1], axis=0) * weights)
rec_error = tl.sqrt(tl.abs(norm_tensor ** 2 + factors_norm ** 2 - 2 * iprod)) / norm_tensor
rec_errors.append(rec_error)
constraint_error = 0
for mode in modes_list:
constraint_error += tl.norm(factors[mode] - tl.transpose(factors_aux[mode])) / tl.norm(factors[mode])
if tol_outer:
if iteration >= 1:
rec_error_decrease = rec_errors[-2] - rec_errors[-1]
if verbose:
print("iteration {}, reconstruction error: {}, decrease = {}".format(iteration,
rec_error,
rec_error_decrease))
if constraint_error < tol_outer:
break
if cvg_criterion == 'abs_rec_error':
stop_flag = abs(rec_error_decrease) < tol_outer
elif cvg_criterion == 'rec_error':
stop_flag = rec_error_decrease < tol_outer
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 return_errors:
return cp_tensor, rec_errors
else:
return cp_tensor
def fista(UtM,
UtU,
x=None,
n_iter_max=100,
non_negative=True,
sparsity_coef=0,
lr=None,
tol=10e-8):
"""
Fast Iterative Shrinkage Thresholding Algorithm (FISTA)
Computes an approximate (nonnegative) solution for Ux=M linear system.
Parameters
----------
UtM : ndarray
Pre-computed product of the transposed of U and M
UtU : ndarray
Pre-computed product of the transposed of U and U
x : init
Default: None
n_iter_max : int
Maximum number of iteration
Default: 100
non_negative : bool, default is False
if True, result will be non-negative
lr : float
learning rate
Default : None
sparsity_coef : float or None
tol : float
stopping criterion
Returns
-------
x : approximate solution such that Ux = M
Notes
-----
We solve the following problem :math: `1/2 ||m - Ux ||_2^2 + \\lambda |x|_1`
Reference
----------
[1] : Beck, A., & Teboulle, M. (2009). A fast iterative
shrinkage-thresholding algorithm for linear inverse problems.
SIAM journal on imaging sciences, 2(1), 183-202.
"""
if sparsity_coef is None:
sparsity_coef = 0
if x is None:
x = tl.zeros(tl.shape(UtM), **tl.context(UtM))
if lr is None:
lr = 1 / (tl.partial_svd(UtU)[1][0])
# Parameters
momentum_old = tl.tensor(1.0)
norm_0 = 0.0
x_update = tl.copy(x)
for iteration in range(n_iter_max):
if isinstance(UtU, list):
x_gradient = -UtM + tl.tenalg.multi_mode_dot(
x_update, UtU, transpose=False) + sparsity_coef
else:
x_gradient = -UtM + tl.dot(UtU, x_update) + sparsity_coef
if non_negative is True:
x_gradient = tl.where(lr * x_gradient < x_update, x_gradient,
x_update / lr)
x_new = x_update - lr * x_gradient
momentum = (1 + tl.sqrt(1 + 4 * momentum_old**2)) / 2
x_update = x_new + ((momentum_old - 1) / momentum) * (x_new - x)
momentum_old = momentum
x = tl.copy(x_new)
norm = tl.norm(lr * x_gradient)
if iteration == 1:
norm_0 = norm
if norm < tol * norm_0:
break
return x
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):
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 parafac2(
tensor_slices,
rank,
n_iter_max=2000,
init='random',
svd='numpy_svd',
normalize_factors=False,
tol=1e-8,
absolute_tol=1e-13,
nn_modes=None,
random_state=None,
verbose=False,
return_errors=False,
n_iter_parafac=5,
):
r"""PARAFAC2 decomposition [1]_ of a third order tensor via alternating least squares (ALS)
Computes a rank-`rank` PARAFAC2 decomposition of the third-order tensor defined by
`tensor_slices`. The decomposition is on the form :math:`(A [B_i] C)` such that the
i-th frontal slice, :math:`X_i`, of :math:`X` is given by
.. math::
X_i = B_i diag(a_i) C^T,
where :math:`diag(a_i)` is the diagonal matrix whose nonzero entries are equal to
the :math:`i`-th row of the :math:`I \times R` factor matrix :math:`A`, :math:`B_i`
is a :math:`J_i \times R` factor matrix such that the cross product matrix :math:`B_{i_1}^T B_{i_1}`
is constant for all :math:`i`, and :math:`C` is a :math:`K \times R` factor matrix.
To compute this decomposition, we reformulate the expression for :math:`B_i` such that
.. math::
B_i = P_i B,
where :math:`P_i` is a :math:`J_i \times R` orthogonal matrix and :math:`B` is a
:math:`R \times R` matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element
:math:`X_{ijk}` is given by
.. math::
X_{ijk} = \sum_{r=1}^R A_{ir} B_{ijr} C_{kr},
with the same constraints hold for :math:`B_i` as above.
Parameters
----------
tensor_slices : ndarray or list of ndarrays
Either a third order tensor or a list of second order tensors that may have different number of rows.
Note that the second mode factor matrices are allowed to change over the first mode, not the
third mode as some other implementations use (see note below).
rank : int
Number of components.
n_iter_max : int, optional
(Default: 2000) Maximum number of iteration
.. versionchanged:: 0.6.1
Previously, the default maximum number of iterations was 100.
init : {'svd', 'random', CPTensor, Parafac2Tensor}
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 : bool (optional)
If True, aggregate the weights of each factor in a 1D-tensor
of shape (rank, ), which will contain the norms of the factors. Note that
there may be some inaccuracies in the component weights.
tol : float, optional
(Default: 1e-8) Relative reconstruction error decrease tolerance. The
algorithm is considered to have converged when
:math:`\left|\| X - \hat{X}_{n-1} \|^2 - \| X - \hat{X}_{n} \|^2\right| < \epsilon \| X - \hat{X}_{n-1} \|^2`.
That is, when the relative change in sum of squared error is less
than the tolerance.
.. versionchanged:: 0.6.1
Previously, the stopping condition was
:math:`\left|\| X - \hat{X}_{n-1} \| - \| X - \hat{X}_{n} \|\right| < \epsilon`.
absolute_tol : float, optional
(Default: 1e-13) Absolute reconstruction error tolearnce. The algorithm
is considered to have converged when
:math:`\left|\| X - \hat{X}_{n-1} \|^2 - \| X - \hat{X}_{n} \|^2\right| < \epsilon_\text{abs}`.
That is, when the relative sum of squared error is less than the specified tolerance.
The absolute tolerance is necessary for stopping the algorithm when used on noise-free
data that follows the PARAFAC2 constraint.
If None, then the machine precision + 1000 will be used.
nn_modes: None, 'all' or array of integers
(Default: None) Used to specify which modes to impose non-negativity constraints on.
We cannot impose non-negativity constraints on the the B-mode (mode 1) with the ALS
algorithm, so if this mode is among the constrained modes, then a warning will be shown
(see notes for more info).
random_state : {None, int, np.random.RandomState}
verbose : int, optional
Level of verbosity
return_errors : bool, optional
Activate return of iteration errors
n_iter_parafac : int, optional
Number of PARAFAC iterations to perform for each PARAFAC2 iteration
Returns
-------
Parafac2Tensor : (weight, factors, projection_matrices)
* 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)
* projection_matrices : List of projection matrices used to create evolving
factors.
errors : list
A list of reconstruction errors at each iteration of the algorithms.
References
----------
.. [1] Kiers, H.A.L., ten Berge, J.M.F. and Bro, R. (1999),
PARAFAC2—Part I. A direct fitting algorithm for the PARAFAC2 model.
J. Chemometrics, 13: 275-294.
Notes
-----
This formulation of the PARAFAC2 decomposition is slightly different from the one in [1]_.
The difference lies in that here, the second mode changes over the first mode, whereas in
[1]_, the second mode changes over the third mode. We made this change since that means
that the function accept both lists of matrices and a single nd-array as input without
any reordering of the modes.
Because of the reformulation above, :math:`B_i = P_i B`, the :math:`B_i` matrices
cannot be constrained to be non-negative with ALS. If this mode is constrained to be
non-negative, then :math:`B` will be non-negative, but not the orthogonal `P_i` matrices.
Consequently, the `B_i` matrices are unlikely to be non-negative.
"""
weights, factors, projections = initialize_decomposition(
tensor_slices, rank, init=init, svd=svd, random_state=random_state)
rec_errors = []
norm_tensor = tl.sqrt(
sum(tl.norm(tensor_slice, 2)**2 for tensor_slice in tensor_slices))
svd_fun = _get_svd(svd)
if absolute_tol is None:
absolute_tol = tl.eps(factors[0].dtype) * 1000
# If nn_modes is set, we use HALS, otherwise, we use the standard parafac implementation.
if nn_modes is None:
def parafac_updates(X, w, f):
return parafac(X,
rank,
n_iter_max=n_iter_parafac,
init=(w, f),
svd=svd,
orthogonalise=False,
verbose=verbose,
return_errors=False,
normalize_factors=False,
mask=None,
random_state=random_state,
tol=1e-100)[1]
else:
if nn_modes == 'all' or 1 in nn_modes:
warn(
"Mode `1` of PARAFAC2 fitted with ALS cannot be constrained to be truly non-negative. See the documentation for more info."
)
def parafac_updates(X, w, f):
return non_negative_parafac_hals(X,
rank,
n_iter_max=n_iter_parafac,
init=(w, f),
svd=svd,
nn_modes=nn_modes,
verbose=verbose,
return_errors=False,
tol=1e-100)[1]
projected_tensor = tl.zeros([factor.shape[0] for factor in factors],
**T.context(factors[0]))
for iteration in range(n_iter_max):
if verbose:
print("Starting iteration", iteration)
factors[1] = factors[1] * T.reshape(weights, (1, -1))
weights = T.ones(weights.shape, **tl.context(tensor_slices[0]))
projections = _compute_projections(tensor_slices,
factors,
svd_fun,
out=projections)
projected_tensor = _project_tensor_slices(tensor_slices,
projections,
out=projected_tensor)
factors = parafac_updates(projected_tensor, weights, factors)
if normalize_factors:
new_factors = []
for factor in factors:
norms = T.norm(factor, axis=0)
norms = tl.where(
tl.abs(norms) <= tl.eps(factor.dtype),
tl.ones(tl.shape(norms), **tl.context(factors[0])), norms)
weights = weights * norms
new_factors.append(factor / (tl.reshape(norms, (1, -1))))
factors = new_factors
if tol:
rec_error = _parafac2_reconstruction_error(
tensor_slices, (weights, factors, projections))
rec_error /= norm_tensor
rec_errors.append(rec_error)
if iteration >= 1:
if verbose:
print('PARAFAC2 reconstruction error={}, variation={}.'.
format(rec_errors[-1],
rec_errors[-2] - rec_errors[-1]))
if abs(rec_errors[-2]**2 - rec_errors[-1]**2) < (
tol *
rec_errors[-2]**2) or rec_errors[-1]**2 < absolute_tol:
if verbose:
print('converged in {} iterations.'.format(iteration))
break
else:
if verbose:
print('PARAFAC2 reconstruction error={}'.format(
rec_errors[-1]))
parafac2_tensor = Parafac2Tensor((weights, factors, projections))
if return_errors:
return parafac2_tensor, rec_errors
else:
return parafac2_tensor
def test_clips_all_negative_tensor_correctly():
# Regression test for bug found with the pytorch backend
negative_valued_tensor = tl.zeros((10, 10)) - 0.1
clipped_tensor = tl.clip(negative_valued_tensor, 0)
assert tl.all(clipped_tensor == 0)
def parafac2(tensor_slices,
rank,
n_iter_max=100,
init='random',
svd='numpy_svd',
normalize_factors=False,
tol=1e-8,
random_state=None,
verbose=False,
return_errors=False,
n_iter_parafac=5):
r"""PARAFAC2 decomposition [1]_ of a third order tensor via alternating least squares (ALS)
Computes a rank-`rank` PARAFAC2 decomposition of the third-order tensor defined by
`tensor_slices`. The decomposition is on the form :math:`(A [B_i] C)` such that the
i-th frontal slice, :math:`X_i`, of :math:`X` is given by
.. math::
X_i = B_i diag(a_i) C^T,
where :math:`diag(a_i)` is the diagonal matrix whose nonzero entries are equal to
the :math:`i`-th row of the :math:`I \times R` factor matrix :math:`A`, :math:`B_i`
is a :math:`J_i \times R` factor matrix such that the cross product matrix :math:`B_{i_1}^T B_{i_1}`
is constant for all :math:`i`, and :math:`C` is a :math:`K \times R` factor matrix.
To compute this decomposition, we reformulate the expression for :math:`B_i` such that
.. math::
B_i = P_i B,
where :math:`P_i` is a :math:`J_i \times R` orthogonal matrix and :math:`B` is a
:math:`R \times R` matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element
:math:`X_{ijk}` is given by
.. math::
X_{ijk} = \sum_{r=1}^R A_{ir} B_{ijr} C_{kr},
with the same constraints hold for :math:`B_i` as above.
Parameters
----------
tensor_slices : ndarray or list of ndarrays
Either a third order tensor or a list of second order tensors that may have different number of rows.
Note that the second mode factor matrices are allowed to change over the first mode, not the
third mode as some other implementations use (see note below).
rank : int
Number of components.
n_iter_max : int
Maximum number of iteration
init : {'svd', 'random', CPTensor, Parafac2Tensor}
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 : bool (optional)
If True, aggregate the weights of each factor in a 1D-tensor
of shape (rank, ), which will contain the norms of the factors. Note that
there may be some inaccuracies in the component weights.
tol : float, optional
(Default: 1e-8) 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
n_iter_parafac: int, optional
Number of PARAFAC iterations to perform for each PARAFAC2 iteration
Returns
-------
Parafac2Tensor : (weight, factors, projection_matrices)
* 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)
* projection_matrices : List of projection matrices used to create evolving
factors.
errors : list
A list of reconstruction errors at each iteration of the algorithms.
References
----------
.. [1] Kiers, H.A.L., ten Berge, J.M.F. and Bro, R. (1999),
PARAFAC2—Part I. A direct fitting algorithm for the PARAFAC2 model.
J. Chemometrics, 13: 275-294.
Notes
-----
This formulation of the PARAFAC2 decomposition is slightly different from the one in [1]_.
The difference lies in that here, the second mode changes over the first mode, whereas in
[1]_, the second mode changes over the third mode. We made this change since that means
that the function accept both lists of matrices and a single nd-array as input without
any reordering of the modes.
"""
weights, factors, projections = initialize_decomposition(
tensor_slices, rank, init=init, svd=svd, random_state=random_state)
rec_errors = []
norm_tensor = tl.sqrt(
sum(tl.norm(tensor_slice, 2)**2 for tensor_slice in tensor_slices))
svd_fun = _get_svd(svd)
projected_tensor = tl.zeros([factor.shape[0] for factor in factors],
**T.context(factors[0]))
for iteration in range(n_iter_max):
if verbose:
print("Starting iteration", iteration)
factors[1] = factors[1] * T.reshape(weights, (1, -1))
weights = T.ones(weights.shape, **tl.context(tensor_slices[0]))
projections = _compute_projections(tensor_slices,
factors,
svd_fun,
out=projections)
projected_tensor = _project_tensor_slices(tensor_slices,
projections,
out=projected_tensor)
_, factors = parafac(projected_tensor,
rank,
n_iter_max=n_iter_parafac,
init=(weights, factors),
svd=svd,
orthogonalise=False,
verbose=verbose,
return_errors=False,
normalize_factors=False,
mask=None,
random_state=random_state,
tol=1e-100)
if normalize_factors:
new_factors = []
for factor in factors:
norms = T.norm(factor, axis=0)
norms = tl.where(
tl.abs(norms) <= tl.eps(factor.dtype),
tl.ones(tl.shape(norms), **tl.context(factors[0])), norms)
weights = weights * norms
new_factors.append(factor / (tl.reshape(norms, (1, -1))))
factors = new_factors
if tol:
rec_error = _parafac2_reconstruction_error(
tensor_slices, (weights, factors, projections))
rec_error /= norm_tensor
rec_errors.append(rec_error)
if iteration >= 1:
if verbose:
print('PARAFAC2 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
else:
if verbose:
print('PARAFAC2 reconstruction error={}'.format(
rec_errors[-1]))
parafac2_tensor = Parafac2Tensor((weights, factors, projections))
if return_errors:
return parafac2_tensor, rec_errors
else:
return parafac2_tensor
def gcp(X, R, type='normal', func=None, grad=None, lower=None,\
opt='lbfgsb', mask=None, maxiters=1000, \
init='random', printitn=10, state=None, factr=1e7, pgtol=1e-4, \
fsamp=None, gsamp=None, oversample=1.1, sampler='uniform', \
fsampler=None, rate=1e-3, decay=0.1, maxfails=1, epciters=1000, \
festtol=-math.inf, beta1=0.9, beta2=0.999, epsilon=1e-8):
"""Generalized CANDECOMP/PARAFAC (GCP) decomposition via all-at-once optimization (OPT) [1]
Computes a rank-'R' decomposition of 'tensor' such that::
tensor = [|weights; factors[0], ..., factors[-1] |].
GCP-OPT allows the use of a variety of statistically motivated loss functions
suited to the data held in a tensor (i.e. continuous, discrete, binary, etc)
Parameters
----------
X : ndarray
Tensor to factorize
**COMING SOON**
Sparse tensor support
R : int
Rank of decomposition (Number of components).
type : str,
Type of objective function used
Options include:
'normal' or 'gaussian' - Gaussian for real-valued data (DEFAULT)
'binary' or 'bernoulli-odds' - Bernoulli w/ odds link for binary data
'bernoulli-logit' - Bernoulli w/ logit link for binary data
'count' or 'poisson' - Poisson for count data
'poisson-log' - Poisson w/ log link for count data
'rayleigh' - Rayleigh distribution for real-valued data
'gamma' - Gamma distribution for non-negative real-valued data
**COMING SOON**:
'huber (DELTA) - Similar to Gaussian, for real-valued data
'negative-binomial (r)' - Negative binomial for count data
'beta (BETA)' - Beta divergence for non-negative real-valued data
'user-specified' - Customized objective function provided by user
func: lambda function
User specified custom objective function, eg. lambda x, m: (m-x)**2
grad: lambda function
User specified custom gradient function, eg. lambda x, m: 2*(m-x)
lower: 0 or -inf
Lower bound for custom objective/gradient
opt : str
Optimization method
Options include:
'lbfgsb' - Bound-constrained limited-memory BFGS
'sgd' - Stochastic gradient descent (SGD)
**COMING SOON**
'adam' - Momentum-based SGD method
'adagrad' - Adaptive gradient algorithm, well suited for sparse data
If 'tensor' is dense, all 4 options can be used, 'lbfgsb' by default.
**COMING SOON** - Sparse format support
If 'tensor' is sparse, only 'sgd', 'adam' and 'adagrad' can be used, 'adam' by
default.
Each method has specific parameters, see documentation
mask : ndarray
Specifies a mask, 0's for missing/incomplete entries, 1's elsewhere, with
the same shape as 'tensor'.
**COMING SOON** - Missing/incomplete data simulation.
maxiters : int
Maximum number of outer iterations, 1000 by default.
init : {'random', 'svd', cptensor}
Initialization for factor matrices, 'random' by default.
Options include:
'random' - random initialization from a uniform distribution on [0,1)
'svd' - initialize the `m`th factor matrix using the `rank` left
singular vectors of the `m`th unfolding of the input tensor.
cptensor - initialization provided by user. NOTE: weights are pulled
in the last factor and then the weights are set to "1" for
the output tensor.
Initializations all result in a cptensor where the weights are one.
printitn : int
Print every n iterations; 0 for no printing, 10 by default.
state : {None, int, np.random.RandomState}
Seed for reproducable random number generation
factr : float
(L-BFGS-B parameter)
Tolerance on the change of objective values. Defaults to 1e7.
pgtol : float
(L-BFGS-B parameter)
Projected gradient tolerance. Defaults to 1e-5
sampler : {uniform, stratified, semi-stratified}
Type of sampling to use for stochastic gradient (SGD/ADAM/ADAGRAD).
Defaults to 'uniform' for dense tensors.
Defaults to 'stratified' for sparse tensors.
Options include:
'uniform' - Uniform random sampling
**COMING SOON**
'stratified' - Stratified sampling, targets sparse data. Zero and
nonzero values sampled separately.
'semi-stratified' - Similar to stratified sampling, but is more
computationally efficient (See papers referenced).
gsamp : int
Number of samples for stochastic gradient (SGD/ADAM/ADAGRAD parameter).
Generally set to be O(sum(shape)*R).
**COMING SOON**
For stratified or semi-stratified, this may be two numbers:
- the number of nnz samples
- the number of zero samples.
If only one number is specified, then this value is used for both nnzs and
zeros (total number of samples is 2x specified value in this case).
fsampler : {'uniform', 'stratified', custom}
Type of sampling for estimating objective function (SGD/ADAM/ADAGRAD parameter).
Options include:
'uniform' - Uniform random sampling
**COMING SOON**
'stratified' - Stratified sampling, targets sparse data. Zero and
nonzero values sampled separately.
custom - User-defined sampler (lambda function). Custom option
is primarily useful in reusing sampled elements across
multiple tests.
fsamp : int
(SGD/ADAM/ADAGRAD parameter)
Number of samples to estimate objective function.
This should generally be somewhat large since we want this sample to generate a
reliable estimate of the true function value.
oversample : float
(Stratified sampling parameter)
Factor to oversample when implicitly sampling zeros in the sparse case.
Defaults to 1.1. Only adjust for very small tensors.
rate : float
(SGD/ADAM parameter)
Initial learning rate. Defaults to 1e-3.
decay : float
(SGD/ADAM parameter)
Amount to decrease learning rate when progress stagnates, i.e. no change in
objective function between epochs. Defaults to 0.1.
maxfails : int
(SGD/ADAM parameter)
Number of times to decrease the learning rate.
Defaults to 1, may be set to zero.
epciters : int
(SGD/ADAM parameter)
Iterations per epoch. Defaults to 1000.
festtol : float
(SGD/ADAM parameter)
Quit estimation of function if it goes below this level.
Defaults to -inf.
beta1 : float
(ADAM parameter) - generally doesn't need to be changed
Defaults to 0.9
beta2 : float
(ADAM parameter) - generally doesn't need to be changed
Defaults to 0.999
epsilon : float
(ADAM parameter) - generally doesn't need to be changed
Defaults to 1e-8
Returns
-------
Mfin : CPTensor
Canonical polyadic decomposition of input tensor X
Reference
---------
[1] D. Hong, T. G. Kolda, J. A. Duersch, Generalized Canonical
Polyadic Tensor Decomposition, SIAM Review, 62:133-163, 2020,
https://doi.org/10.1137/18M1203626
[2] T. G. Kolda, D. Hong, Stochastic Gradients for Large-Scale Tensor
Decomposition. SIAM J. Mathematics of Data Science, 2:1066-1095,
2020, https://doi.org/10.1137/19m1266265
"""
# Timer - Setup (outside optimization)
start_setup0 = time.perf_counter()
# Initial setup
nd = tl.ndim(X)
sz = tl.shape(X)
tsz = X.size
X_context = tl.context(X)
vecsz = 0
for i in range(nd):
# tsz *= sz[i]
vecsz += sz[i]
vecsz *= R
W = mask
# Random set-up
if state is not None:
state = tl.check_random_state(state)
# capture stats(nnzs, zeros, missing)
nnonnzeros = 0
X = tl.tensor_to_vec(X)
for i in X:
if i > 0:
nnonnzeros += 1
X = tl.reshape(X, sz)
nzeros = tsz - nnonnzeros
nmissing = 0
if W is not None:
W = tl.tensor_to_vec(W)
for i in range(tl.shape(W)[0]):
if W[i] > 0: nmissing += 1 # TODO: is this right??
W = tl.reshape(W, sz)
# Dictionary for storing important information regarding the decomposition problem
info = {}
info['tsz'] = tsz
info[
'nmissing'] = 0 # TODO: revisit once missing value functionality incorporated
info['nnonnzeros'] = nnonnzeros
info[
'nzeros'] = nzeros # TODO: revisit once missing value functionality incorporated
# Set up function, gradient, and bounds
fh, gh, lb = validate_type(type, X)
info['type'] = type
info['fh'] = fh
info['gh'] = gh
info['lb'] = lb
# initialize CP-tensor and make a copy to work with so as to have the starting guess
M0 = initialize_cp(X, R, init=init, random_state=state)
wghts0 = tl.copy(M0[0])
fcts0 = []
for i in range(nd):
f = tl.copy(M0[1][i])
fcts0.append(f)
M = CPTensor((wghts0, fcts0))
# Lambda weights are assumed to be all ones throughout, check initial guess satisfies assumption
if not tl.all(M[0]):
print("Initialization of CP tensor has failed (lambda weight(s) != 1.")
sys.exit(1)
# check optimization method
if validate_opt(opt):
print("Choose optimization method from: {lbfgsb, sgd}")
sys.exit(1)
use_stoc = False
if opt != 'lbfgsb':
use_stoc = True
info['opt'] = opt
# set up for stochastic optimization (e.g. sgd, adam, adagrad)
if use_stoc:
# set up fsampler, gsampler ---> uniform sampling only for now
# TODO : add stratified, semi-stratified and user-specified sampling options
if not sampler == "uniform":
print(
"Only uniform sampling currently supported for stochastic optimization."
)
sys.exit(1)
fsampler_type = sampler
gsampler_type = sampler
# setup fsampler
f_samp = fsamp
if f_samp == None:
upper = np.maximum(math.ceil(tsz / 10), 10 ^ 6)
f_samp = np.minimum(upper, tsz)
# set up lambda function/function handle for uniform sampling
fsampler = lambda: tl_sample_uniform(X, f_samp)
fsampler_str = "{} with {} samples".format(fsampler_type, f_samp)
# setup gsampler
g_samp = gsamp
if g_samp == None:
upper = np.maximum(1000, math.ceil(10 * tsz / maxiters))
g_samp = np.minimum(upper, tsz)
# setup lambda function/function handle for uniform sampling
gsampler = lambda: tl_sample_uniform(X, g_samp)
gsampler_str = "{} with {} samples".format(gsampler_type, g_samp)
# capture the info
info['fsampler'] = fsampler_str
info['gsampler'] = gsampler_str
info['fsamp'] = f_samp
info['gsamp'] = g_samp
time_setup0 = time.perf_counter() - start_setup0
# Welcome message
if printitn > 0:
print("GCP-OPT-{} (Generalized CP Tensor Decomposition)".format(opt))
print("------------------------------------------------")
print("Tensor size:\t\t\t\t{} ({} total entries)".format(sz, tsz))
if nmissing > 0:
print("Missing entries: {} ({})".format(nmissing,
100 * nmissing / tsz))
print("Generalized function type:\t{}".format(type))
print("Objective function:\t\t\t{}".format(
inspect.getsource(fh).strip()))
print("Gradient function:\t\t\t{}".format(
inspect.getsource(gh).strip()))
print("Lower bound of factors:\t\t{}".format(lb))
print("Optimization method:\t\t{}".format(opt))
if use_stoc:
print("Max iterations (epochs): {}".format(maxiters))
print("Iterations per epoch: {}".format(epciters))
print("Learning rate / decay / maxfails: {} {} {}".format(
rate, decay, maxfails))
print("Function Sampler: {}".format(fsampler_str))
print("Gradient Sampler: {}".format(gsampler_str))
else:
print("Max iterations:\t\t\t\t{}".format(maxiters))
print("Projected gradient tol:\t\t{}\n".format(pgtol))
# Make like a zombie and start decomposing
Mfin = None
# L-BFGS-B optimization
if opt == 'lbfgsb':
# Timer - Setup (inside optimization)
start_setup1 = time.perf_counter()
# set up bounds for l-bfgs-b if lb = 0
bounds = None
if lb == 0:
lb = tl.zeros(tsz)
ub = math.inf * tl.ones(tsz)
fcn = lambda x: tl_gcp_fg(vec2factors(x, sz, R, X_context), X, fh, gh)
m = factors2vec(M[1])
# capture params for l-bfgs-b
lbfgsb_params = {}
lbfgsb_params['x0'] = factors2vec(M0.factors)
lbfgsb_params['printEvery'] = printitn
lbfgsb_params['maxIts'] = maxiters
lbfgsb_params['maxTotalIts'] = maxiters * 10
lbfgsb_params['factr'] = factr
lbfgsb_params['pgtol'] = pgtol
time_setup1 = time.perf_counter() - start_setup1
if printitn > 0:
print("Begin main loop")
# Timer - Main operation
start_main = time.perf_counter()
x, f, info_dict = fmin_l_bfgs_b(fcn, m, approx_grad=False, bounds=None, \
pgtol=pgtol, factr=factr, maxiter=maxiters)
time_main = time.perf_counter() - start_main
# capture info
info['fcn'] = fcn
info['lbfgsbopts'] = lbfgsb_params
info['lbfgsbout'] = info_dict
info['finalf'] = f
if printitn > 0:
print("\nFinal objective: {}".format(f))
print("Setup time: {}".format(time_setup0 + time_setup1))
print("Main loop time: {}".format(time_main))
print("Outer iterations:"
) # TODO: access this value (see manpage for fmin_l_bfgs_b)
print("Total iterations: {}".format(info_dict['nit']))
print("L-BFGS-B exit message: {} ({})".format(
info_dict['task'], info_dict['warnflag']))
Mfin = vec2factors(x, sz, R, X_context)
# Stochastic optimization
else:
# Timer - Setup (inside optimization)
start_setup1 = time.perf_counter()
if opt == "adam" or opt == "adagrad":
print("{} not currently supported".format(opt))
sys.exit(1)
# prepare for sgd
# initialize moments
m = []
v = []
# Extract samples for estimating function value (i.e. call fsampler), these never change
fsubs, fvals, fwgts = fsampler()
# Compute initial estimated function value
fest = tl_gcp_fg_est(M, fh, gh, fsubs, fvals, fwgts, True, False,
False, False)
# Set up loop variables
nfails = 0
titers = 0
M_weights = tl.copy(M[0])
M_factors = []
for k in range(nd):
M_factors.append(tl.copy(M[1][k]))
Msave = CPTensor(
(M_weights, M_factors)) # save a copy of the initial model
msave = m
vsave = v
fest_prev = fest[0]
# Tracing the progress in function value by epoch
fest_trace = tl.zeros(maxiters + 1)
step_trace = tl.zeros(maxiters + 1)
time_trace = tl.zeros(maxiters + 1)
fest_trace[0] = fest[0]
# Print status
if printitn > 0:
print("Begin main loop")
print("Initial f-est: {}".format(fest[0]))
time_setup1 = time.perf_counter() - start_setup1
start_main = time.perf_counter()
time_trace[0] = time.perf_counter() - start_setup0
# Main loop - outer iteration
for nepoch in range(maxiters):
step = (decay**nfails) * rate
# Main loop - inner iteration
for iter in range(epciters):
# Tracking iterations
titers = titers + 1
# Select subset for stochastic gradient (i.e. call gsampler)
gsubs, gvals, gwts = gsampler()
# Compute gradients for each mode
Gest = tl_gcp_fg_est(M, fh, gh, gsubs, gvals, gwts, False,
True, False, False)
# Check for inf gradient
for g in Gest[0]:
g_max = tl.max(g)
g_min = tl.min(g)
if math.isinf(g_max) or math.isinf(g_min):
print(
"Infinite gradient encountered! (epoch = {}, iter = {})"
.format(nepoch, iter))
# TODO : add functionality for ADAM and ADAGRAD optimization
# Take gradient step
for k in range(nd):
M.factors[k] = M.factors[k] - step * Gest[0][k]
# Estimate objective (i.e. call tl_gcp_fg_est)
fest = tl_gcp_fg_est(M, fh, gh, fsubs, fvals, fwgts, True, False,
False, False)
# Save trace (fest & step)
fest_trace[nepoch + 1] = fest[0]
step_trace[nepoch + 1] = step
# Check convergence condition
failed_epoch = False
if fest[0] > fest_prev:
failed_epoch = True
if failed_epoch:
nfails += 1
festtol_test = False
if fest[0] < festtol:
festtol_test = True
# Reporting
if printitn > 0 and (nepoch % printitn == 0 or failed_epoch
or festtol_test):
print("Epoch {}: f-est = {}, step = {}".format(
nepoch, fest[0], step),
end='')
if failed_epoch:
print(
", nfails = {} (resetting to solution from last epoch)"
.format(nfails))
print("")
# Rectify failed epoch or save current solution
if failed_epoch:
M = Msave
m = msave
v = vsave
fest[0] = fest_prev
titers = titers - epciters
else:
Msave = CPTensor((tl.copy(M.weights), tl.copy(M.factors)))
msave = m
vsave = v
fest_prev = fest[0]
time_trace[nepoch] = time.perf_counter() - start_setup0
if (nfails > maxfails) or festtol_test:
break
Mfin = M
time_main = time.perf_counter() - start_main
# capture info
info['fest_trace'] = fest_trace
info['step_trace'] = step_trace
info['time_trace'] = time_trace
info['nepoch'] = nepoch
# Report end of main loop
if printitn > 0:
print("End Main Loop")
print("")
print("Final f-east: {}".format(fest[0]))
print("Setup time: {0:0.6f}".format(time_setup0 + time_setup1))
print("Main loop time: {0:0.6f}".format(time_main))
print("Total iterations: {}".format(nepoch * epciters))
# Wrap up / capture remaining info
info['mainTime'] = time_main
info['setupTime0'] = time_setup0
info['setupTime1'] = time_setup1
info['setupTime'] = time_setup0 + time_setup1
return Mfin