A novel trilinear decomposition algorithm for second-order linear calibration

Chen, Zengping; Wu, Hai‐Long; Jiang, Jian‐Hui; Li, Yang; Yu, Ru‐Qin

doi:10.1016/s0169-7439(00)00081-2

Cited by 188 publications

(56 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…alternating trilinear decomposition (ATLD) [18], alternating coupled vector resolution (ACOVER) [19], alternating slice-wise diagonalization (ASD) [20], alternating coupled matrices resolution (ACOMAR) [21] and self-weighted alternating trilinear decomposition (SWATLD) [22]. It will be reported in the near future that GRAM compares well with these methods (N. M. Faber and P. K. Hopke, in preparation).…”

Section: Discussionmentioning

confidence: 99%

“…This is certainly a requirement for a valid comparison. Second, a critical step in the derivation of (10) is the observation that Q is reproduced by the reconstructed profiles X and Ŷ in the least squares sense; that is, analogous to (6), Q = X Ŷ T Ê Q (see manipulations leading to (22) in Reference [9]). Thus, for (10) to be valid, it is not required that X and Ŷ reproduce the true profiles X and Y well separately.…”

Section: Generalized Rank Annihilation Methodsmentioning

confidence: 99%

See 1 more Smart Citation

The price paid for the second‐order advantage when using the generalized rank annihilation method (GRAM)

Faber¹

2001

Journal of Chemometrics

View full text Add to dashboard Cite

SUMMARYIn a ground-breaking paper, Linder and Sundberg developed a statistical framework for the calibration of bilinear data (Chemometrics Intell. Lab. Syst. 1998; 42: 159-178). Within this framework they formulated three different predictor construction methods (J. Chemometrics accepted), namely a so-called naive method, a least squares (LS) method and a refined version of the latter that takes account of the calibration uncertainty. They showed that the naive method is statistically less efficient than the others under the assumption of white noise. In the current work a close relationship is established between the generalized rank annihilation method (GRAM) and the naive method by comparing expressions for prediction variance. The main conclusion is that the relatively poor efficiency of GRAM is the price one pays for obtaining the second-order advantage with a single calibration sample.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Generalized Rank Annihilation Methodsmentioning

confidence: 99%

The price paid for the second‐order advantage when using the generalized rank annihilation method (GRAM)

Faber¹

2001

Journal of Chemometrics

View full text Add to dashboard Cite

show abstract

“…The algorithm returns the factor matrices and the elements of the resulting diagonal tensor λ. Other alternation-based algorithms for PARAFAC models include the alternating slice-wise diagonalization (ASD) ] and the self-weighted alternating trilinear diagonalization (SWA-TLD) [Chen et al 2000] algorithms: they improve fitting using objective functions that are not based on least squares.…”

Section: Obtaining Tensor Decompositionsmentioning

confidence: 99%

Multiresolution Tensor Decompositions with Mode Hierarchies

Schifanella

Candan

Sapino

2014

ACM Trans. Knowl. Discov. Data

View full text Add to dashboard Cite

Tensors (multidimensional arrays) are widely used for representing high-order dimensional data, in applications ranging from social networks, sensor data, and Internet traffic. Multiway data analysis techniques, in particular tensor decompositions, allow extraction of hidden correlations among multiway data and thus are key components of many data analysis frameworks. Intuitively, these algorithms can be thought of as multiway clustering schemes, which consider multiple facets of the data in identifying clusters, their weights, and contributions of each data element. Unfortunately, algorithms for fitting multiway models are, in general, iterative and very time consuming. In this article, we observe that, in many applications, there is a priori background knowledge (or metadata) about one or more domain dimensions. This metadata is often in the form of a hierarchy that clusters the elements of a given data facet (or mode). We investigate whether such single-mode data hierarchies can be used to boost the efficiency of tensor decomposition process, without significant impact on the final decomposition quality. We consider each domain hierarchy as a guide to help provide higher-or lower-resolution views of the data in the tensor on demand and we rely on these metadata-induced multiresolution tensor representations to develop a multiresolution approach to tensor decomposition. In this article, we focus on an alternating least squares (ALS)-based implementation of the two most important decomposition models such as the PARAllel FACtors (PARAFAC, which decomposes a tensor into a diagonal tensor and a set of factor matrices) and the Tucker (which produces as result a core tensor and a set of dimension-subspaces matrices). Experiment results show that, when the available metadata is used as a rough guide, the proposed multiresolution method helps fit both PARAFAC and Tucker models with consistent (under different parameters settings) savings in execution time and memory consumption, while preserving the quality of the decomposition.

show abstract

“…Faber, Bro, and Hopke [15] compared ALS with a number of competing algorithms: direct trilinear decomposition (DTLD) [16,17,14,18,29,31,42], alternating trilinear decomposition (ATLD) [50], self-weighted alternating trilinear decomposition (SWATLD) [10,11], pseudo alternating least squares (PALS) [9], alternating coupled vectors resolution (ACOVER) [23], alternating slice-wise diagonalization (ASD) [22], and alternating coupled matrices resolution (ACOMAR) [30]. It is shown that while none of the algorithms is better than ALS in terms of the quality of solution, ASD may be an alternative to ALS when the computation time is a priority.…”

Section: Introductionmentioning

confidence: 99%

An optimization approach for fitting canonical tensor decompositions.

Report

Acar²,

Kolda³

et al. 2009

View full text Add to dashboard Cite

Tensor decompositions are higher-order analogues of matrix decompositions and have proven to be powerful tools for data analysis. In particular, we are interested in the canonical tensor decomposition, otherwise known as the CANDECOMP/PARAFAC decomposition (CPD), which expresses a tensor as the sum of component rank-one tensors and is used in a multitude of applications such as chemometrics, signal processing, neuroscience, and web analysis. The task of computing the CPD, however, can be difficult. The typical approach is based on alternating least squares (ALS) optimization, which can be remarkably fast but is not very accurate. Previously, nonlinear least squares (NLS) methods have also been recommended; existing NLS methods are accurate but slow. In this paper, we propose the use of gradient-based optimization methods. We discuss the mathematical calculation of the derivatives and further show that they can be computed efficiently, at the same cost as one iteration of ALS. Computational experiments demonstrate that the gradient-based optimization methods are much more accurate than ALS and orders of magnitude faster than NLS.3

show abstract

A novel trilinear decomposition algorithm for second-order linear calibration

Cited by 188 publications

References 21 publications

The price paid for the second‐order advantage when using the generalized rank annihilation method (GRAM)

The price paid for the second‐order advantage when using the generalized rank annihilation method (GRAM)

Multiresolution Tensor Decompositions with Mode Hierarchies

An optimization approach for fitting canonical tensor decompositions.

Contact Info

Product

Resources

About