def hooi(X, rank, **kwargs):
"""
Compute Tucker decomposition of a tensor using Higher-Order Orthogonal
Iterations.
Parameters
----------
X : tensor_mixin
The tensor to be decomposed
rank : array_like
The rank of the decomposition for each mode of the tensor.
The length of ``rank`` must match the number of modes of ``X``.
init : {'random', 'nvecs'}, optional
The initialization method to use.
- random : Factor matrices are initialized randomly.
- nvecs : Factor matrices are initialzed via HOSVD.
default : 'nvecs'
Examples
--------
Create dense tensor
>>> T = np.zeros((3, 4, 2))
>>> T[:, :, 0] = [[ 1, 4, 7, 10], [ 2, 5, 8, 11], [3, 6, 9, 12]]
>>> T[:, :, 1] = [[13, 16, 19, 22], [14, 17, 20, 23], [15, 18, 21, 24]]
>>> T = dtensor(T)
Compute Tucker decomposition of ``T`` with n-rank [2, 3, 1] via higher-order
orthogonal iterations
>>> Y = hooi(T, [2, 3, 1], init='nvecs')
Shape of the core tensor matches n-rank of the decomposition.
>>> Y['core'].shape
(2, 3, 1)
>>> Y['U'][1].shape
(3, 2)
References
----------
.. [1] L. De Lathauwer, B. De Moor, J. Vandewalle: On the best rank-1 and
rank-(R_1, R_2, \ldots, R_N) approximation of higher order tensors;
IEEE Trans. Signal Process. 49 (2001), pp. 2262-2271
"""
# init options
ainit = kwargs.pop('init', __DEF_INIT)
maxIter = kwargs.pop('maxIter', __DEF_MAXITER)
conv = kwargs.pop('conv', __DEF_CONV)
dtype = kwargs.pop('dtype', X.dtype)
if not len(kwargs) == 0:
raise ValueError('Unknown keywords (%s)' % (kwargs.keys()))
ndims = X.ndim
if isNumberType(rank):
rank = rank * ones(ndims)
normX = norm(X)
U = __init(ainit, X, ndims, rank, dtype)
fit = 0
exectimes = []
for itr in xrange(maxIter):
tic = time.clock()
fitold = fit
for n in range(ndims):
Utilde = ttm(X, U, n, transp=True, without=True)
U[n] = nvecs(Utilde, n, rank[n])
# compute core tensor to get fit
core = ttm(Utilde, U, n, transp=True)
# since factors are orthonormal, compute fit on core tensor
normresidual = sqrt(normX ** 2 - norm(core) ** 2)
# fraction explained by model
fit = 1 - (normresidual / normX)
fitchange = abs(fitold - fit)
exectimes.append(time.clock() - tic)
_log.debug(
'[%3d] fit: %.5f | delta: %7.1e | secs: %.5f'
% (itr, fit, fitchange, exectimes[-1])
)
if itr > 1 and fitchange < conv:
break
return core, U
def als(X, rank, dtype=np.float, **kwargs):
"""
Alternating least-sqaures algorithm to compute the CP decomposition.
Parameters
----------
X : tensor_mixin
The tensor to be decomposed.
rank : int
Tensor rank of the decomposition.
init : {'random', 'nvecs'}, optional
The initialization method to use.
- random : Factor matrices are initialized randomly.
- nvecs : Factor matrices are initialzed via HOSVD.
(default 'nvecs')
max_iter : int, optional
Maximium number of iterations of the ALS algorithm.
(default 500)
fit_method : {'full', None}
The method to compute the fit of the factorization
- 'full' : Compute least-squares fit of the dense approximation of.
X and X.
- None : Do not compute the fit of the factorization, but iterate
until ``max_iter`` (Useful for large-scale tensors).
(default 'full')
conv : float
Convergence tolerance on difference of fit between iterations
(default 1e-5)
Returns
-------
P : ktensor
Rank ``rank`` factorization of X. ``P.U[i]`` corresponds to the factor
matrix for the i-th mode. ``P.lambda[i]`` corresponds to the weight
of the i-th mode.
fit : float
Fit of the factorization compared to ``X``
itr : int
Number of iterations that were needed until convergence
exectimes : ndarray of floats
Time needed for each single iteration
Examples
--------
Create random dense tensor
>>> from sktensor import dtensor
>>> U = [np.random.rand(i,3) for i in (20, 10, 14)]
>>> T = dtensor(ktensor(U).toarray())
Compute rank-3 CP decomposition of ``T`` with ALS
>>> P, fit, itr, _ = als(T, 3)
Result is a decomposed tensor stored as a Kruskal operator
>>> type(P)
<class 'sktensor.ktensor.ktensor'>
Factorization should be close to original data
>>> np.allclose(T, P.totensor())
True
References
----------
.. [1] Kolda, T. G. & Bader, B. W.
Tensor Decompositions and Applications.
SIAM Rev. 51, 455–500 (2009).
.. [2] Harshman, R. A.
Foundations of the PARAFAC procedure: models and conditions for an 'explanatory' multimodal factor analysis.
UCLA Working Papers in Phonetics 16, (1970).
.. [3] Carroll, J. D., Chang, J. J.
Analysis of individual differences in multidimensional scaling via an N-way generalization of 'Eckart-Young' decomposition.
Psychometrika 35, 283–319 (1970).
"""
N = len(X.shape)
normX = norm(X)
# init options
ainit = kwargs.pop('init', __DEF_INIT)
maxiter = kwargs.pop('maxIter', __DEF_MAXITER)
fit_method = kwargs.pop('fit_method', __DEF_FIT_METHOD)
conv = kwargs.pop('conv', __DEF_CONV)
if not len(kwargs) == 0:
raise ValueError('Unknown keywords (%s)' % (kwargs.keys()))
U = __init(ainit, X, N, rank, dtype)
fit = 0
exectimes = []
for itr in xrange(maxiter):
tic = time.clock()
fitold = fit
for n in range(N):
Unew = X.uttkrp(U, n)
Y = ones((rank, rank), dtype=dtype)
for i in (range(n) + range(n + 1, N)):
Y = Y * dot(U[i].T, U[i])
Unew = Unew.dot(pinv(Y))
# Normalize
if itr == 0:
lmbda = sqrt((Unew ** 2).sum(axis=0))
else:
lmbda = Unew.max(axis=0)
lmbda[lmbda < 1] = 1
U[n] = Unew / lmbda
P = ktensor(U, lmbda)
if fit_method == 'full':
normresidual = normX ** 2 + P.norm() ** 2 - 2 * P.innerprod(X)
fit = 1 - (normresidual / normX ** 2)
else:
fit = itr
fitchange = abs(fitold - fit)
exectimes.append(time.clock() - tic)
_log.debug(
'[%3d] fit: %.5f | delta: %7.1e | secs: %.5f' %
(itr, fit, fitchange, exectimes[-1])
)
if itr > 0 and fitchange < conv:
break
return P, fit, itr, array(exectimes)
def als(X, rank, **kwargs):
"""
Alternating least-sqaures algorithm to compute the CP decomposition.
Parameters
----------
X : tensor_mixin
The tensor to be decomposed.
rank : int
Tensor rank of the decomposition.
init : {'random', 'nvecs'}, optional
The initialization method to use.
- random : Factor matrices are initialized randomly.
- nvecs : Factor matrices are initialzed via HOSVD.
(default 'nvecs')
max_iter : int, optional
Maximium number of iterations of the ALS algorithm.
(default 500)
fit_method : {'full', None}
The method to compute the fit of the factorization
- 'full' : Compute least-squares fit of the dense approximation of.
X and X.
- None : Do not compute the fit of the factorization, but iterate
until ``max_iter`` (Useful for large-scale tensors).
(default 'full')
conv : float
Convergence tolerance on difference of fit between iterations
(default 1e-5)
Returns
-------
P : ktensor
Rank ``rank`` factorization of X. ``P.U[i]`` corresponds to the factor
matrix for the i-th mode. ``P.lambda[i]`` corresponds to the weight
of the i-th mode.
fit : float
Fit of the factorization compared to ``X``
itr : int
Number of iterations that were needed until convergence
exectimes : ndarray of floats
Time needed for each single iteration
Examples
--------
Create random dense tensor
>>> from sktensor import dtensor, ktensor
>>> U = [np.random.rand(i,3) for i in (20, 10, 14)]
>>> T = dtensor(ktensor(U).toarray())
Compute rank-3 CP decomposition of ``T`` with ALS
>>> P, fit, itr, _ = als(T, 3)
Result is a decomposed tensor stored as a Kruskal operator
>>> type(P)
<class 'sktensor.ktensor.ktensor'>
Factorization should be close to original data
>>> np.allclose(T, P.totensor())
True
References
----------
.. [1] Kolda, T. G. & Bader, B. W.
Tensor Decompositions and Applications.
SIAM Rev. 51, 455–500 (2009).
.. [2] Harshman, R. A.
Foundations of the PARAFAC procedure: models and conditions for an 'explanatory' multimodal factor analysis.
UCLA Working Papers in Phonetics 16, (1970).
.. [3] Carroll, J. D., Chang, J. J.
Analysis of individual differences in multidimensional scaling via an N-way generalization of 'Eckart-Young' decomposition.
Psychometrika 35, 283–319 (1970).
"""
# init options
ainit = kwargs.pop('init', _DEF_INIT)
maxiter = kwargs.pop('max_iter', _DEF_MAXITER)
fit_method = kwargs.pop('fit_method', _DEF_FIT_METHOD)
conv = kwargs.pop('conv', _DEF_CONV)
dtype = kwargs.pop('dtype', _DEF_TYPE)
if not len(kwargs) == 0:
raise ValueError('Unknown keywords (%s)' % (kwargs.keys()))
N = X.ndim
normX = norm(X)
U = _init(ainit, X, N, rank, dtype)
fit = 0
exectimes = []
for itr in range(maxiter):
tic = time.clock()
fitold = fit
for n in range(N):
Unew = X.uttkrp(U, n)
Y = ones((rank, rank), dtype=dtype)
for i in (list(range(n)) + list(range(n + 1, N))):
Y = Y * dot(U[i].T, U[i])
Unew = Unew.dot(pinv(Y))
# Normalize
if itr == 0:
lmbda = sqrt((Unew**2).sum(axis=0))
else:
lmbda = Unew.max(axis=0)
lmbda[lmbda < 1] = 1
U[n] = Unew / lmbda
P = ktensor(U, lmbda)
if fit_method == 'full':
normresidual = normX**2 + P.norm()**2 - 2 * P.innerprod(X)
fit = 1 - (normresidual / normX**2)
else:
fit = itr
fitchange = abs(fitold - fit)
exectimes.append(time.clock() - tic)
_log.debug('[%3d] fit: %.5f | delta: %7.1e | secs: %.5f' %
(itr, fit, fitchange, exectimes[-1]))
if itr > 0 and fitchange < conv:
break
return P, fit, itr, array(exectimes)