 TENSORBOX (2013), a Matlab toolbox for tensor decompositions is now available to download.
 Tensor Deflation for CANDECOMP/PARAFAC.
Part 1: Alternating Subspace Update Algorithm
CANDECOMP/PARAFAC (CP) approximates multiway
data by sum of rank1 tensors. Unlike matrix decomposition,
the procedure which estimates the best rankR tensor
approximation through R sequential best rank1 approximations
does not work for tensors, because the deflation does not always
reduce the tensor rank. In this paper, we propose a novel deflation
method for the problem. When one factor matrix of a rank
R CP decomposition is of full column rank, the decomposition
can be performed through (R1) rank1 reductions. For decomposition of order3 tensors of size
R x R x R and rankR, estimation of one rank1 tensor has a
computational cost of O(R^3) per iteration which is lower than the
cost O(R^4) of the ALS algorithm for the overall CP decomposition.
Ref:
AnhHuy Phan, Petr Tichavský and Andrzej Cichocki, Tensor Deflation for CANDECOMP/PARAFAC. Part 1: Alternating Subspace Update Algorithm
, IEEE, Transaction on Signal Processing, 2015.
 Tensor Deflation for CANDECOMP/PARAFAC.
Part 2: Initialization and Error Analysis
Part 2 of the study presents several initialization algorithms suitable for the
algorithm proposed in Part 1, and error analysis of the tensor deflation algorithm, which
shows that there is a marginal loss of accuracy of the deflation
algorithm compared to the ordinary CP decomposition.
Ref:
AnhHuy Phan, Petr Tichavský and Andrzej Cichocki, Tensor Deflation for CANDECOMP/PARAFAC. Part 2: Initialization and Error Analysis
, IEEE, Transaction on Signal Processing, 2015.
 Tensor Deflation for CANDECOMP/PARAFAC.
Part 3: Rank Splitting
In this paper, we extend
the rank1 tensor deflation to block deflation problem. When at least two factor
matrices have full column rank, one can extract two rank1
tensors simultaneously, and rank of the data tensor is reduced
by 2. For decomposition of order3 tensors of size R x R x R
and rankR, the block deflation has a complexity of O(R^3)
per iteration which is lower than the cost O(R^4) of the ALS
algorithm for the overall CPD.
Ref:
AnhHuy Phan, Petr Tichavský and Andrzej Cichocki, Tensor Deflation for CANDECOMP/PARAFAC. Part 3: Rank Splitting
, 2015.
 Matlab code and demo for the fast damped GaussNewton algorithms for CANDECOMP/PARAFAC (CP)
Approximate Hessian is often huge and impractical to directly invert. Such task normally costs O(R^{3}T^{3}) where
T = I_{1} + I_{2} + ... + I_{N}.
The fLM algorithms invert the aproximate Hessian with a cost of O(N^{3}R^{6}) or O(NR^{6}).
Ref: Anh Huy Phan, Petr Tichavský, and Andrzej Cichocki, Low Complexity Damped GaussNewton Algorithms for CANDECOMP/PARAFAC, SIAM. J. Matrix Anal. & Appl. 34, 126 (2013).
Matlab code of the fLM algorithm is provided in the TENSORBOX.
 Matlab code and demo for the fast CP gradients for CANDECOMP/PARAFAC (CP)
The fast CP gradient can accelerate CP algorithm such ALS, OPT, dGN 8 times fast for 4D tensors, and even up to 30 times for higher dimensional tensors.
Ref:
Anh Huy Phan, Petr Tichavský, and Andrzej Cichocki, Fast Alternating LS Algorithms for High Order
CANDECOMP/PARAFAC Tensor Factorizations, IEEE, Transaction on Signal Processing, accepted for publication, 2013.
 CramerRao Induced bound for CANDECOMP/PARAFAC Decomposition
This paper presents a CramerRao lower bound (CRLB) on the variance of unbiased estimates of factor matrices in CANDECOMP/PARAFAC (CP) tensor decomposition. A novel expression needs less operations for computing
the bound, O(NR^{6}), than the best existing stateofthe art algorithm, O(N^{3}R^{6}) operations.
Insightful expressions are derived for tensors of
rank 1 and rank 2 of arbitrary dimension and for tensors of arbitrary dimension and rank, where two
factor matrices have orthogonal columns.
The results can be used as a gauge of performance of different approximate CP decomposition
algorithms, prediction of their accuracy, and for checking stability of a given decomposition of a
tensor. A novel expression is derived for a Hessian
matrix needed in popular damped GaussNewton method for solving the CP decomposition of tensors
with missing elements.
Ref:
Petr Tichavský, Anh Huy Phan, Zbynek Koldovský, CramerRaoInduced Bounds for CANDECOMP/PARAFAC tensor decomposition
, IEEE, Transaction on Signal Processing, pp. 19861997, vol. 61, no. 8, 2013.
 Matlab code and demo for the fast algorithm for high order CANDECOMP/PARAFAC (CP) through tensor reshaping
The method exploits the uniqueness of CPD and the relation of
a tensor in Kruskal form and its unfolded tensor. The FCP algorithm decomposes an unfolded tensor with
lower order, e.g., order3 tensor. On basis of the order3 estimated tensor, a structured Kruskal tensor
of the same dimension as the data tensor is then generated, and decomposed to find the final solution
using fast algorithms for the structured CPD. Using CRIB, we design tensor unfolding strategy which ensures
as little deterioration of accuracy as possible.
Ref:
Anh Huy Phan, Petr Tichavský, and Andrzej Cichocki, CANDECOMP/PARAFAC Decomposition of Highorder Tensors Through Tensor Reshaping, IEEE, Transaction on Signal Processing, accepted for publication, 2013.
 NFEA: Tensor Toolbox for Feature Extraction and Applications
(ver. for BCI)
TENSORBOX: A Matlab toolbox for tensor decompositions is now available.
