arXiv subjects

Andrzej Cichocki

Publications and source records attributed to Andrzej Cichocki.

79 records · Page 5Linked to original sources

arXiv2013

Accelerated Canonical Polyadic Decomposition by Using Mode Reduction

Canonical Polyadic (or CANDECOMP/PARAFAC, CP) decompositions (CPD) are widely applied to analyze high order tensors. Existing CPD methods use alternating least square (ALS) iterations and hence need to unfold tensors to each of the $N$ modes frequently, which is one major bottleneck of efficiency for large-scale data and especially when $N$ is large. To overcome this problem, in this paper we proposed a new CPD method which converts the original $N$th ($N>3$) order tensor to a 3rd-order tensor first. Then the full CPD is realized by decomposing this mode reduced tensor followed by a Khatri-Rao product projection procedure. This way is quite efficient as unfolding to each of the $N$ modes are avoided, and dimensionality reduction can also be easily incorporated to further improve the efficiency. We show that, under mild conditions, any $N$th-order CPD can be converted into a 3rd-order case but without destroying the essential uniqueness, and theoretically gives the same results as direct $N$-way CPD methods. Simulations show that, compared with state-of-the-art CPD methods, the proposed method is more efficient and escape from local solutions more easily.

math.NA↗

arXiv2013

Tensor Decompositions: A New Concept in Brain Data Analysis?

Matrix factorizations and their extensions to tensor factorizations and decompositions have become prominent techniques for linear and multilinear blind source separation (BSS), especially multiway Independent Component Analysis (ICA), NonnegativeMatrix and Tensor Factorization (NMF/NTF), Smooth Component Analysis (SmoCA) and Sparse Component Analysis (SCA). Moreover, tensor decompositions have many other potential applications beyond multilinear BSS, especially feature extraction, classification, dimensionality reduction and multiway clustering. In this paper, we briefly overview new and emerging models and approaches for tensor decompositions in applications to group and linked multiway BSS/ICA, feature extraction, classification andMultiway Partial Least Squares (MPLS) regression problems. Keywords: Multilinear BSS, linked multiway BSS/ICA, tensor factorizations and decompositions, constrained Tucker and CP models, Penalized Tensor Decompositions (PTD), feature extraction, classification, multiway PLS and CCA.

math.NA↗

arXiv2013

Convergence analysis of the FOCUSS algorithm

FOCal Underdetermined System Solver (FOCUSS) is a powerful tool for sparse representation and underdetermined inverse problems, which is extremely easy to implement. In this paper, we give a comprehensive convergence analysis on the FOCUSS algorithm towards establishing a systematic convergence theory by providing three primary contributions as follows. First, we give a rigorous derivation for this algorithm exploiting the auxiliary function. Then, we prove its convergence. Third, we systematically study its convergence rate with respect to the sparsity parameter p and demonstrate its convergence rate by numerical experiments.

cs.IT↗

arXiv2012

CANDECOMP/PARAFAC Decomposition of High-order Tensors Through Tensor Reshaping

In general, algorithms for order-3 CANDECOMP/-PARAFAC (CP), also coined canonical polyadic decomposition (CPD), are easily to implement and can be extended to higher order CPD. Unfortunately, the algorithms become computationally demanding, and they are often not applicable to higher order and relatively large scale tensors. In this paper, by exploiting the uniqueness of CPD and the relation of a tensor in Kruskal form and its unfolded tensor, we propose a fast approach to deal with this problem. Instead of directly factorizing the high order data tensor, the method decomposes an unfolded tensor with lower order, e.g., order-3 tensor. On basis of the order-3 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. In addition, strategies to unfold tensors are suggested and practically verified in the paper.

math.NA↗

arXiv2012

Low Complexity Damped Gauss-Newton Algorithms for CANDECOMP/PARAFAC

The damped Gauss-Newton (dGN) algorithm for CANDECOMP/PARAFAC (CP) decomposition can handle the challenges of collinearity of factors and different magnitudes of factors; nevertheless, for factorization of an $N$-D tensor of size $I_1\times I_N$ with rank $R$, the algorithm is computationally demanding due to construction of large approximate Hessian of size $(RT \times RT)$ and its inversion where $T = \sum_n I_n$. In this paper, we propose a fast implementation of the dGN algorithm which is based on novel expressions of the inverse approximate Hessian in block form. The new implementation has lower computational complexity, besides computation of the gradient (this part is common to both methods), requiring the inversion of a matrix of size $NR^2\times NR^2$, which is much smaller than the whole approximate Hessian, if $T \gg NR$. In addition, the implementation has lower memory requirements, because neither the Hessian nor its inverse never need to be stored in their entirety. A variant of the algorithm working with complex valued data is proposed as well. Complexity and performance of the proposed algorithm is compared with those of dGN and ALS with line search on examples of difficult benchmark tensors.

math.NA↗

arXiv2012

Higher-Order Partial Least Squares (HOPLS): A Generalized Multi-Linear Regression Method

A new generalized multilinear regression model, termed the Higher-Order Partial Least Squares (HOPLS), is introduced with the aim to predict a tensor (multiway array) $\tensor{Y}$ from a tensor $\tensor{X}$ through projecting the data onto the latent space and performing regression on the corresponding latent variables. HOPLS differs substantially from other regression models in that it explains the data by a sum of orthogonal Tucker tensors, while the number of orthogonal loadings serves as a parameter to control model complexity and prevent overfitting. The low dimensional latent space is optimized sequentially via a deflation operation, yielding the best joint subspace approximation for both $\tensor{X}$ and $\tensor{Y}$. Instead of decomposing $\tensor{X}$ and $\tensor{Y}$ individually, higher order singular value decomposition on a newly defined generalized cross-covariance tensor is employed to optimize the orthogonal loadings. A systematic comparison on both synthetic data and real-world decoding of 3D movement trajectories from electrocorticogram (ECoG) signals demonstrate the advantages of HOPLS over the existing methods in terms of better predictive ability, suitability to handle small sample sizes, and robustness to noise.

cs.AI↗

arXiv2012

On Fast Computation of Gradients for CANDECOMP/PARAFAC Algorithms

Product between mode-$n$ unfolding $\bY_{(n)}$ of an $N$-D tensor $\tY$ and Khatri-Rao products of $(N-1)$ factor matrices $\bA^{(m)}$, $m = 1,..., n-1, n+1, ..., N$ exists in algorithms for CANDECOMP/PARAFAC (CP). If $\tY$ is an error tensor of a tensor approximation, this product is the gradient of a cost function with respect to factors, and has the largest workload in most CP algorithms. In this paper, a fast method to compute this product is proposed. Experimental verification shows that the fast CP gradient can accelerate the CP_ALS algorithm 2 times and 8 times faster for factorizations of 3-D and 4-D tensors, and the speed-up ratios can be 20-30 times for higher dimensional tensors.

math.NA↗