While the rank of a matrix can be found in polynomial time using the svd algorithm, the rank of a tensor is an nphard problem. Extended from nmf based methods, a matrixvector nonnegative tensor factorization ntf model is proposed in this paper for spectral unmixing. Ntt communication science laboratories, kyoto, japan takeuchi. These python scripts are to study nonnegative tensor factorization ntf. Nonnegative matrix factorization nmf is a dimensionreduction technique based on a lowrank approximation of the feature space. A sparse tensor is a tensor where only a small fraction of the elements are nonzero. The mult algorithm is sensitive to initial values, which makes it a good choice when using replicates to find w and h from multiple random starting values now perform the factorization using alternating leastsquares algorithm, which converges faster and more consistently. To download the software, proceed first to the tensor toolbox license and registration page. Nonnegative matrix and tensor factorizations guide books. Given a nonnegative tensor, the problem seeks a nonnegative core tensor and nonnegative matrices such that or, where denotes modetensormatrix product given a tensor and a matrix, the product is defined by. Nonnegative tensor factorization with missing data for.
The nonnegative tensor matrix factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. Nonnegative matrix and tensor factorization, which extract datadependent nonnegative basis functions, have been commonly applied for the analysis of such data for data compression, visualization, and detection of hidden information factors. This procedure may result in information loss since the local object structure is lost due to vectorization. Even worse, with matrices there is a fundamental relationship between rank1 and rankk approximations. The matlab tensor toolbox provides the following classes for manipulating dense, sparse, and structured tensors using matlabs objectoriented features. Ntf can be interpreted as generalized nonnegative matrix factorizationnmf. A tensor is the generalization of a vector or matrix to multidimensional objects, i. The nonnegative matrix factorization toolbox in matlab. This matlab function factors the nonnegative nbym matrix a into nonnegative factors w nbyk and h kbym. The factors w and h are chosen to minimize the rootmeansquared residual d between a and wh. Such factorization is only available if the coefficients of the input are convertible to floatingpoint numbers, that is, if. Nonnegative tensor factorization canonical decomposition parafac based on the matlab version written by jingu kim jingu. Nonnegative matrix and tensor factorization nmf, ntf.
The rescal decomposition decomposes a tensor xof shape n n m, into a factor matrix aof shape n rand a core tensor rof shape r r m, where each of the mfrontal slices of xcan be seen as a binary adjacencymatrix between. The decomposition is modeled as the optimization problem. Nonnegative matrix and tensor factorization lecture notes article pdf available in ieee signal processing magazine 251. W,h nmfv,winit,hinit,tol,timelimit,maxiter outputinput parameters are. We motivate the use of nntf in three areas of data analysis. Another mat lab nmf toolbox has been written by hansen. Description w,h nnmfa,k factors the nonnegative nbym matrix a into nonnegative factors w nbyk and h kbym. The threedimensional 3d tensor of an image cube is decomposed to the spectral signatures and abundance matrix using nonnegative tensor factorization ntf methods. Ntf can be interpreted as generalized nonnegative matrix factorization nmf. We derive algorithms for finding a nonnegative ndimensional tensor factorization nntf which includes the nonnegative matrix factorization nmf as a particular case when n 2. Nonnegative matrix and tensor factorization lecture notes. Tensor factorization arises in many machine learning applications, such knowledge base modeling and parameter estimation in latent variable models.
This package includes matlab implementations of fast optimization algorithms for computing. The increasing interest in nonnegative matrix and tensor factorizations, as well as decompositions and sparse representation of data, will ensure that this book is essential reading for engineers, scientists, researchers, industry practitioners and graduate students across signal and image processing. Sparse nonnegative tensor factorization using columnwise. Tensor factorization via matrix factorization our guarantees are independent of the algorithm used for diagonalizing the projection matrices. Because the rootmeansquared residual d may have local minima, repeated factorizations may yield different w and h. Run 100 times more iterations, beginning from the initial w0 and h0 identified above. Throughout the paper, a nonnegative matrix or a nonnegative tensor refers to a matrix or a tensor with only nonnegative elements. However, the optimization process for the core tensor dominates the compu. Nonnegative matrix factorization matlab nnmf mathworks. However, numerical methods for tensor factorization have not reached the level of maturity of matrix factorization methods. This book provides a broad survey of models and efficient algorithms for nonnegative matrix factorization nmf. Nonnegative matrix factorization with alternating nonnegativityconstrained least squares and block principal pivoting active set methods. Nonnegative matrix and tensor factorization algorithms toolbox.
Newtonbased optimization for kullbackleibler nonnegative tensor. Nonnegative tensor factorization with applications to. Orthogonal nonnegative matrix trifactorizations for. The optimization aspects of our method, on the other hand, depend on the choice of joint diagonalization subroutine. Nmfntf and their extensions are increasingly used as tools in signal and image processing, and data analysis, having.
In this paper, we propose a new method for cp tensor factorization that uses random projections to reduce the problem. Nonnegative tensor factorization for contaminant source. The factorization uses an iterative method starting with random initial values for w and h. Most subroutines enjoy local quadratic convergence rates,14,15 and so does our method. Nonnegative matrix and tensor factorizations by andrzej. A novel unsupervised machine learning ml method based on nonnegative tensor factorization ntf combined with a custom kmeans clustering called ntfk has been developed ntfk has been applied to identify contaminant sources based on highdimensional tensor datasets representing spatial and temporal variations of observed geochemical species the ntfk approach is an extension of our. Recently, in order to remedy this disadvantage of nmf methods, nonnegative tensor factorization ntf algorithms that can be applied directly to the tensor representation. Zhang w, sun h, liu x and guo x temporal qosaware web service recommendation via nonnegative tensor factorization proceedings of the 23rd international conference on world wide web, 585596 vaca c, mantrach a, jaimes a and saerens m a timebased collective factorization for topic discovery and monitoring in news proceedings of the 23rd. This page provides matlab software for efficient nonnegative matrix factorization nmf algorithms based on alternating nonnegativity constrained least squares. A 1way tensor is a vector and a 2way tensor is a matrix. These python scripts are to study nonnegative tensor factorizationntf. Graphical illustration of the rescal tensor factorization into the factor matrix a and the core tensor r 11.
Nonnegative tensor factorizations using an alternating. Ntfk performs a novel unsupervised machine learning ml method based on tensor decomposition coupled with sparsity and nonnegativity constraints ntfk has been applied to extract the temporal and spatial footprints of the features in multidimensional datasets in the form of multiway arrays or tensors ntfk executes the decomposition factorization of a given tensor. Nonnegative tensor factorization, nonnegative candecompparafac, poisson tensor factorization, leeseung multiplicative updates, majorizationminimization algorithms 1. Learn more about nonnegative matrix factorization, multiplicative update, parfor. This includes nmfs various extensions and modifications, especially nonnegative tensor factorizations ntf and nonnegative tucker decompositions ntd. Nmf and ntf are lowrank approximation methods for matrices and tensors in which the lowrank factors are constrained to have only nonnegative elements. Nmf is very common decomposition method, which is useful to see essentials from dataset, but the method can be just applied to matrix data expressed by 2d. Plain, sparse, and regularized nmfs are all included and can be easily selected.
The matlab code of nonnegative matrixtensor decomposition and completion, along with a few demos, are accessible from the authors homepages. Nonnegative tensor factorization with applications to statistics and computer vision matrix and n 2 tensor. A complex numeric factorization is a factorization into linear factors whose coefficients are floatingpoint numbers. Nonnegative tensor factorization, based on the matlab. This page provides matlab software for efficient nonnegative matrix factorization. Besides providing a reduction in the number of features, nmf guarantees that the features are nonnegative, producing additive models that respect, for example, the nonnegativity of physical quantities. We built a thirdorder tensor \\mathscro\ with cooccurrences of patients, diagnoses, and prescriptions from intensive care unit icu ehrs.
Nonnegative multiple tensor factorization koh takeuchi. A multilevel approach for nonnegative matrix factorization. The same authors provide matlab code in their book 7 about nonnegative matrixtensor factorization. In recent years, nonnegative matrix factorization nmf has received considerable interest from the data mining and. Matrixvector nonnegative tensor factorization for blind.
Citations for tensor toolbox for matlab sandia national laboratories. Compared to the existing stateoftheart algorithms, the proposed algorithms demonstrate superior performance in both speed and solution quality. Nonnegative matrix and tensor factorizations andrzej. Tensors have found application in a variety of elds, ranging from chemometrics to signal processing and beyond. Many applications in computer vision, biomedical informatics, and graphics deal with data in the matrix or tensor form. Discriminative and distinct phenotyping by constrained. Perform nonnegative matrix factorization using the multiplicative and alternating leastsquares algorithms.
Sometimes the algorithm converges to a solution of lower rank than k, which may indicate that the result is not optimal w and h are normalized so that the rows of h have unit length. Algorithms for nonnegative tensor factorization springerlink. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. Tensor or multiway array decompositions have become popular tools for analyzing and summarizing multiway data in part due to the recent increased storage and computation capabilities2. Under some mild assumptions, every limit point is a stationary point of nmf, see 21. Introduction the nonnegative matrix factorization nmf has been shown recently to be useful for many applications in environment, pattern. The nonnegative matrix factorization toolbox in matlab developed by yifeng li. Nonnegative matrix and tensor factorization nmf, ntf with any beta divergence in matlab search form the following matlab project contains the source code and matlab examples used for nonnegative matrix and tensor factorization nmf, ntf with any beta divergence. In this case, it is more efficient to store just the nonzeros and their indices.