The candecomp parafac cp method decomposes a threeway array into a prespecified number r of rank1 arrays, by minimizing the sum of squares of the residual array. How can we efficiently decompose a tensor into sparse factors, when the data do not. To be able to use the cp decomposition for the approximation of functionrelated tensors, robust and ef. Candecompparafac cp decomposes a tensor as a sum of rankone tensors, and the tucker decomposition is a higherorder form of principal components analysis. In this paper, stability of the candecompparafac cp tensor decomposition is addressed. Overview of constrained parafac models eurasip journal. Michael roger honeycomb decomposition book, yellow cover with. We consider the decomposition of a tensor in a sum of rank1 terms. Tensor decomposition and coupled cluster theory in the. The candecompparafac canonical decomposition parallel factor analysis, ab. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. The candecompparafac cp method decomposes a threeway array into a prespecified number r of rank1 arrays, by minimizing the sum of squares of the residual array.
There are many other tensor decompositions, including indscal, parafac2, candelinc, dedicom, and paratuck2 as well as nonnegative variants of all of the above. In general, algorithms for order3 candecomp parafac cp, also coined canonical polyadic decomposition cpd, are easily to implement and can be extended to. Fast and guaranteed tensor decomposition via sketching. Stability of candecompparafac tensor decomposition ieee. There are many other tensor decompositions, including indscal, parafac2, candelinc, dedicom, and paratuck2 as well as nonnegative vari ants of all of the above. Using a candecompparafac tensor decomposition of the data, we illustrate the usefulness of exploiting the natural threedimensional structure of temporal link data. Rank, decomposition, and uniqueness for 3way and nway. We demonstrate here how to perform a canonical polyadic decomposition. Degeneracy in candecompparafac and indscal explained for.
The transformational matrix is constructed by applying the analysis of matrix product. Tensor decompositions and applications sandia national. Michael roger honeycomb decomposition book, yellow cover. Understanding the candecompparafac tensor decomposition, aka. Analysis of individual differences in multidimensional scaling via an nway generalization of eckartyoung decomposition, psychometrika, springer. Canonical decomposition candecomp also known as parallel factor analysis parafac is a wellknown multiway model in highdimensional data modeling. The basic building block of chemometrics intechopen. The existence of the bound reveals necessary conditions for. For the tensorbased methods, we consider the candecomp parafac cp tensor decomposition carroll and chang 1970. On kruskals uniqueness condition for the candecompparafac. Randomized online cp decomposition ieee conference. Siam journal on matrix analysis and applications 35. It is done by deriving the cramerrao lower bound crlb on variance of an unbiased estimate of the tensor. Computing the unique candecompparafac decomposition of.
Furthermore, nonnegative tensor factorization ntf is applied to pattern recognition by introducing cp candecompparafac decomposition and the block coordinate descent bcd algorithm for the selected data set. Using a candecomp parafac tensor decomposition of the data, we illustrate the usefulness of exploiting the natural threedimensional structure of temporal link data. Cp decomposition of a higherorder tensor is a powerful multilinear algebra, thus denoising observed data and identification of multiple sources can both be accomplished by the cp decomposition. In proceedings of the 2014 ieee international conference on acoustics, speech and signal processing icassp. Pdf stability of candecompparafac tensor decomposition. Harshman 1970, which does not collapse the data but instead retains its natural threedimensional structure. I will take a tensor decomposition perspective on the relation between vcc and vci. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Given a tensor x of rank r, its parafac decomposition. In this paper, we propose fast and randomized tensor cp decomposition algorithms based on sketching. The decomposition book is made with 100% postconsumer waste. Candecompparafac cp decomposes a threeway array x of order i. Computing a compressed approximate matrix decomposition. The purpose of this tutorial is to dive deep into the canonical polyadic tensor decomposition also known as candecomp, parafac, or just cp, giving attendees the mathematical and algorithmic tools to understand existing methods and have a strong foundation for developing their own tools.
For this reason, the tensor rank decomposition is sometimes historically referred to as parafac or candecomp. An algorithm for leastsquares estimation of nonlinear. Like most of the candecompparafacdecomposition based methods, for each candidate rank, the factor matrices of the corresponding candecompparafac model i. Like most of the candecomp parafac decomposition based methods, for each candidate rank, the factor matrices of the corresponding candecomp parafac model i. In this case, trying to compute the approximation results in diverging cp components.
Candecomp parafac cp decomposes a tensor as a sum of rankone tensors, and the tucker decomposition is a higherorder form of principal components analysis. A pure tensor is the tensor product of a collection of vectors. The current standard approaches include nonnegative matrix factorization lee and seung 1999, 2001 for matrices and candecomp parafac cp decomposition harshman 1970 or tucker decomposition tucker 1966 for tensors. Low complexity damped gaussnewton algorithms for candecomp. Candecompparafac decomposition of highorder tensors. But before spelling it out in mathematical terms, lets start with a simple toy example using the r. Accelerated canonical polyadic decomposition by using mode.
Furthermore, nonnegative tensor factorization ntf is applied to pattern recognition by introducing cp candecomp parafac decomposition and the block coordinate descent bcd algorithm for the selected data set. An efficient algorithm for parafac of threeway data with. Understanding the candecompparafac tensor decomposition. Candecompparafac decomposition of highorder tensors through. The data recorded on such an array present the socalled multiple invariances, which can be linked to the candecompparafac cp model. Canonical polyadic or candecompparafac, cp decompositions cpd are widely applied to analyze high order tensors. In general, algorithms for order3 candecomp parafac cp, also coined canonical polyadic decomposition cpd, are easy to implement and can be extended to higher order cpd. The standard alternating least squares algorithm for the cp decomposition cpals involves a series of highly overdetermined linear least squares problems. Candecomp parafac cp decomposes a tensor as a sum of rankone tensors, and the tucker decomposition is a higherorder form of principal component analysis. Then, they get the reduced qb system which can match the first several expansion coefficients of the original output. Candecomp parafac tensor decomposition petr tichavsky. Unfortunately, in some cases, the model suffers from the socalled degeneracy, that is a solution with diverging and uninterpretable components. Through several numerical experiments, we demonstrate that both matrix and tensorbased techniques are effective for.
It is done by deriving the cramerrao lower bound crlb on variance of an unbiased estimate of the tensor parameters, i. The candecompparafac decomposition cpd of the tensor whose maximal dimension is greater than its rank is considered. The two fundamental tensor decompositions are the candecompparafac cp and tucker decomposition, which are shown as follows. On the nonexistence of optimal solutions and the occurrence. Sidiropoulos b a heijmans institute of psychological research, university of groningen, grote kruisstraat 21, 9712 ts groningen, the netherlands. We derive the upper bound of rank under which computing the cpd is equivalent to solving a structured polynomial system that is determined by the full rank factorization of the matricization of the tensor. Cp decomposition quick summary of the math behind it the cp decomposition factorizes a tensor into a sum of outer products of vectors. On kruskals uniqueness condition for the candecompparafac decomposition alwin stegeman a. Nonnegative candecomp parafac decomposition given a nonnegative tensor, the problem seeks nonnegative matrices such that or, where denotes outer product.
Link prediction on evolving data using matrix and tensor factorizations evrim acar. The two fundamental tensor decompositions are the candecomp parafac cp and tucker decomposition, which are shown as follows. Both the candecompparafac cp 35, 73 and tucker 187 tensor decompositions can be considered higherorder generalization of the matrix singular value decomposition svd and principal component analysis pca. Nonnegative candecompparafac decomposition given a nonnegative tensor, the problem seeks nonnegative matrices such that or, where denotes outer product. Cp decomposition and its application in noise reduction and. Well, there are several types of tensor decomposition, but in this blog post i will introduce only the candecompparafac decomposition. In general, algorithms for order3 candecompparafac cp, also coined canonical polyadic decomposition cpd, are easily to implement. Given a nonnegative tensor, the problem seeks nonnegative matrices such that or, where denotes outer product. The practical use of cp is sometimes complicated by the occurrence of socalled degenerate sequences of solutions, in which several rank1 arrays become highly correlated in all three modes and some elements of the rank1. In general, algorithms for order3 candecompparafac cp, also coined canonical polyadic decomposition cpd, are easily to implement and can be extended to. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo sition. Fitting a candecompparafac cp decomposition also known as canonical polyadic decomposition to a multiway array or higherorder tensor, is equivalent to finding a best lowrank approximation to the multiway array or higherorder tensor, where the rank is defined as the outerproduct rank. Candecomp parafac cp decomposition has been widely used to deal with multiway data. The candecomp parafac canonical decomposition parallel factor analysis, ab.
In this paper, stability of the candecomp parafac cp tensor decomposition is addressed. Tensor candecompparafac cp decomposition has wide applications in statistical learning of latent variable models and in data mining. To avoid degeneracy, orthogonality constraints are usually applied to one. Jan 15, 2007 introduction the decomposition of threeway or threemode arrays into rankone threeway outer products was proposed independently by carroll and chang 1 who called it candecomp and harsh man 2 who called it parafac. A pilot implementation has been made for the candecompparafac tensor decomposition within our existing vccvci codes. The decomposition is modeled as the optimization problem. Full references including those not matched with items on ideas. Using a candecompparafac tensor decomposition of the. Approaches that use candecomp parafac for parametric modeling of a noisy observation require an estimate of the number of signal components rank of the data as well.
In multilinear algebra, the tensor rank decomposition or canonical polyadic decomposition. Sparse parallelizable candecompparafac tensor decomposition. Numerical experiments show that the designed method has good performance in terms of computation speed and solution quality. Cp decomposition compresses an input tensor into a sum of rankone components, and tucker decomposition approximates an input. The candecompparafac cp model is a wellknown tool for summarizing a threeway array by extracting a limited number of components.
We address the uniqueness problem in estimating the directionsofarrival doas of multiple narrowband and fully polarized signals impinging on a passive sensor array composed of identical vector sensors. The decomposition book line by michael roger is a new spin on an old concept. Using a candecompparafac tensor decomposition of the data, we illustrate the usefulness of. In 3, we discuss the cp decomposition, its connection to tensor rank and.
Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decomposition. It is well known that an optimal solution for cp need not exist. It is done by deriving the cramerrao lower bound crlb. Such a best rankr approximation may not exist due to the fact that the set of multiway arrays with rank at most r is not closed. Compared with matrix decomposition, tensor decomposition can better explore multidimensional data features by treating data as a whole to reserve its intrinsic structure. Nowadays, with the fast development of multisensor technology, more and more data appe higherorder nonnegative candecomp parafac tensor decomposition using proximal algorithm ieee conference publication. Link prediction on evolving data using matrix and tensor. Pdf in this paper, stability of the candecompparafac cp tensor decomposition is addressed. On kruskals uniqueness condition for the candecomp. Temporal link prediction using matrix and tensor factorizations daniel m. The tensor rank decomposition expresses a tensor as a minimumlength linear combination of rank1 tensors. The candecomp parafac cp model decomposes a threeway array into a prespecified number of r factors and a residual array by minimizing the sum of squares of the latter. The trilinear decomposition comes from the model structure and sometimes data itself implies that because of its their natural decomposition in 3 modes. Cp decomposition and its application in noise reduction.
Temporal link prediction using matrix and tensor factorizations. This decomposition is generally referred to as canonical polyadic decomposition, and sometimes as parafac. Unfortunately, the algorithms become computationally demanding, and they are often not applicable to higher order and relatively large scale tensors. The candecomp parafac cp model is a wellknown tool for summarizing a threeway array by extracting a limited number of components. Besides, they propose another mor method based on the candecomp parafac decomposition. Candecompparafac cp decomposes a tensor as a sum of rankone tensors, and the tucker decomposition is a higherorder form of principal component analysis. A, b, c, are first estimated via als algorithm and then according to an estimate of the data with that candidate rank is calculated.
A practical randomized cp tensor decomposition siam. The parafac decomposition was independently proposed by carroll and chang under the name canonical decomposition candecomp and then called candecompparafac cp in. The decomposition book is made with 100% postconsumer waste recycled paper, printed with soy ink, and made in the usa. On fast computation of gradients for candecompparafac algorithms. The parafac model is a generalization of the pca itself bilinear, to arrays of higher order i. For example, for a 3way tensor, the cp decomposition can be written as where and, are vectors of appropriate dimensions, and where the notation denotes the outer product for tensors. Nowadays, with the fast development of multisensor technology, more and more data appe higherorder nonnegative candecompparafac tensor decomposition using proximal algorithm ieee conference publication. It is done by deriving the cramerrao lower bound crlb on.
The candecompparafac cp model assumes that a threeway array containing, for example, scores of cases on variables measured at several occasions is the sum of a systematic part. The current standard approaches include nonnegative matrix factorization lee and seung 1999, 2001 for matrices and candecompparafac cp decomposition harshman 1970 or tucker decomposition tucker 1966 for tensors. Tensor decomposition is a powerful tool for analyzing multiway data. Partial uniqueness in candecompparafac request pdf. For realtime or largescale tensors, based on the ideas of randomizedsampling cp decomposition algorithm and online cp decomposition algorithm, a novel cp decomposition algorithm called randomized online cp decomposition rocp is proposed in this paper. The candecompparafac cp decomposition is a leading method for the analysis of multiway data. In general, algorithms for order3 candecomp parafac cp, also coined canonical polyadic decomposition cpd, are easy to implement. In this study, based on tensor decomposition and matrix product, the authors investigate two modelorder reduction mor methods for the quadraticbilinear qb systems which are equivalently transformed from the nonlinear inputoutput systems. Numerical studies on models systems, formaldehyde, and thiadiazole will be used to illustrate the theoretical concepts. A candecompparafac perspective on uniqueness of doa. In order to fix this discrepancy, many authors have adopted the cp acronym, which stands both for candecompparafac and for canonical polyadic. Since the quadratic term coefficient of the qb system can be considered as the matricisation of a tensor, they propose two computationally efficient. The practical use of cp is sometimes complicated by the occurrence of socalled. Sustainability free fulltext application of nonnegative.
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