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Explicable recommendation system is proved to be conducive to improving the persuasiveness of the recommendation system, enabling users to trust the system more and make more intelligent decisions. Nonnegative Matrix Factorization (NMF) produces interpretable solutions for many applications including collaborative filtering as it’s nonnegativity. However, the latent features make it difficult to interpret recommendation results to users because we don’t know the specific meaning of features that users are interested in and the extent to which the items or users belong to these features. To overcome this difficulty, we develop a novel method called Partially Explainable Nonnegative Matrix Factorization (PE-NMF) by employing explicit data to replace part latent variables of item-feature matrix, by which users can learn more about the features of the items and then to make ideal decisions and recommendations. The objective function of PE-NMF is composed of two parts: one part corresponding to explicit features and the other part is about implicit features. We develop an iterative method to minimize the objective function and derive the iterative update rules, with which the objective function can be proved to be decreasing. Finally, the experiments are executed on Yelp, Amazon and Dianping datasets, and the experimental results demonstrate PE-NMF keeps a high prediction performance on both rating prediction and top-N recommendation that compare to fully explainable nonnegative matrix factorization (FE-NMF), which is obtained by using explicit opinions instead of item-feature matrix. Also PE-NMF holds almost the same recommendation ability as NMF.

Matrix factorization is widely used in recommendation systems, text mining, face recognition and computer vision. As one of the most popular methods, nonnegative matrix factorization and its incremental variants have attracted much attention. The existing incremental algorithms are established based on the assumption of samples are independent and only update the new latent variable of weighting coefficient matrix when the new sample comes, which may lead to inferior solutions. To address this issue, we investigate a novel incremental nonnegative matrix factorization algorithm based on correlation and graph regularizer (ICGNMF). The correlation is mainly used for finding out those correlated rows to be updated, that is, we assume that samples are dependent on each other. We derive the updating rules for ICGNMF by considering the correlation. We also present tests on widely used image datasets, and show ICGNMF reduces the error by comparing other methods.

Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a matrix with a product of a nonnegative, low-rank matrix and it transpose. To design faster and more scalable algorithms for SymNMF we develop two randomized algorithms for its computation. The first method uses randomized matrix sketching to compute an initial low-rank approximation to the input matrix and proceeds to uses this as a low-rank input to rapidly compute a SymNMF. The second methods uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. Here, we prove theoretically that leverage score sampling can approximately solve constrained least squares problems to e-accuracy. Finally we demonstrate both methods work in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speed ups for both large dense and large sparse problems.

Modern electron microscopy enables the acquisition of extremely large datasets, necessitating optimized machine learning techniques, such as dimensionality reduction and clustering, to extract material insights. We propose a novel nonnegative matrix factorization (NMF) technique that integrates domain-specific constraints inherent to electron microscopy, including spatial resolution and continuous intensity features without downward-convex peaks. This constrained NMF was applied to four-dimensional (4D) scanning transmission electron microscopy (STEM). Using the constrained NMF, both simulated and actual experimental data were successfully decomposed into interpretable diffractions and maps that cannot be achieved using principal component analysis (PCA) and primitive NMF methods. Additionally, hierarchical clustering was optimized based on diffraction similarity, which is a combination of a polar coordinate transformation and uniaxial cross-correlation. Then, nanometer-sized crystalline precipitates embedded in an amorphous metallic glass, ZrCuAl, were successfully detected and classified according to their diffraction patterns. The present scheme is broadly applicable across various characterization techniques, including hyperspectral imaging, and effectively mitigates the known artifacts found in conventional machine learning techniques that rely solely on mathematical constraints without domain-specific knowledge.

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an m -by- n nonnegative matrix X and a factorization rank r , find, if possible, an m -by- r nonnegative matrix W and an r -by- n nonnegative matrix H such that X = W H . In this paper, we propose two heuristics for exact NMF, one inspired from simulated annealing and the other from the greedy randomized adaptive search procedure. We show empirically that these two heuristics are able to compute exact nonnegative factorizations for several classes of nonnegative matrices (namely, linear Euclidean distance matrices, slack matrices, unique-disjointness matrices, and randomly generated matrices) and as such demonstrate their superiority over standard multi-start strategies. We also consider a hybridization between these two heuristics that allows us to combine the advantages of both methods. Finally, we discuss the use of these heuristics to gain insight on the behavior of the nonnegative rank, i.e., the minimum factorization rank such that an exact NMF exists. In particular, we disprove a conjecture on the nonnegative rank of a Kronecker product, propose a new upper bound on the extension complexity of generic n -gons and conjecture the exact value of (i) the extension complexity of regular n -gons and (ii) the nonnegative rank of a submatrix of the slack matrix of the correlation polytope.

Nonnegative matrix factorization is comprehensively used in recommendation systems. In an effort to reduce the recommended cost of newly added samples, incremental nonnegative matrix factorization and its variants have been extensively studied in recommendation systems. However, the recommendation performance is incapable of particular applications in terms of data sparsity and sample diversity. In this paper, we propose a new incremental recommend algorithm by improving incremental nonnegative matrix factorization based on three-way decision, called Three-way Decision Recommendations Based on Incremental Non-negative Matrix Factorization (3WD-INMF), in which the concept of positive, negative, and boundary regions are employed to update the new coming samples’ features. Finally, experiments on six public data sets demonstrate the error induced by 3WD-INMF is decreasing as the addition of new samples and deliver state-of-the-art performance compared with existing recommendation algorithms. The results indicate our method is more reasonable and efficient by leveraging the idea of three-way decision to perform the recommendation decision process.

Due to the limited spatial resolution of remote hyperspectral sensors, pixels are usually highly mixed in the hyperspectral images. Endmember extraction refers to the process identifying the pure endmember signatures from the mixture, which is an important step towards the utilization of hyperspectral data. Nonnegative matrix factorization (NMF) is a widely used method of endmember extraction due to its effectiveness and convenience. While most NMF-based methods have single-layer structures, which may have difficulties in effectively learning the structures of highly mixed and complex data. On the other hand, multilayer algorithms have shown great advantages in learning data features and been widely studied in many fields. In this paper, we presented a L1 sparsityconstrained multilayer NMF method for endmember extraction of highly mixed data. Firstly, the multilayer NMF structure was obtained by unfolding NMF into a certain number of layers. In each layer, the abundance matrix was decomposed into the endmember matrix and abundance matrix of the next layer. Besides, to improve the performance of NMF, we incorporated sparsity constraints to the multilayer NMF model by adding a L1 regularizer of the abundance matrix to each layer. At last, a layer-wise optimization method based on NeNMF was proposed to train the multilayer NMF structure. Experiments were conducted on both synthetic data and real data. The results demonstrate that our proposed algorithm can achieve better results than several state-of-art approaches.

Exemplar-based sparse representation is a nonparametric framework for voice conversion. In this framework, a target spectrum is generated as a weighted linear combination of a set of basis spectra, namely exemplars, extracted from the training data. This framework adopts coupled source-target dictionaries consisting of acoustically aligned source-target exemplars, and assumes they can share the same activation matrix. At runtime, a source spectrogram is factorized as a product of the source dictionary and the common activation matrix, which is applied to the target dictionary to generate the target spectrogram. In practice, either low-resolution mel-scale filter bank energies or high-resolution spectra are adopted in the source dictionary. Low-resolution features are flexible in capturing the temporal information without increasing the computational cost and the memory occupation significantly, while high-resolution spectra contain significant spectral details. In this paper, we propose a joint nonnegative matrix factorization technique to find the common activation matrix using low- and high-resolution features at the same time. In this way, the common activation matrix is able to benefit from low- and high-resolution features directly. We conducted experiments on the VOICES database to evaluate the performance of the proposed method. Both objective and subjective evaluations confirmed the effectiveness of the proposed methods.

In recent years, nonnegative matrix factorization (NMF) has attracted significant amount of attentions in image processing, text mining, speech processing and related fields. Although NMF has been applied in several application successfully, its simple application on image processing has a few caveats. For example, NMF costs considerable computational resources when performing on large databases. In this paper, we propose two enhanced NMF algorithms for image processing to save the computational costs. One is modified rank-one residue iteration (MRRI) algorithm , the other is element-wisely residue iteration (ERI) algorithm. Here we combine CAPG (a NMF algorithm proposed by Lin), MRRI and ERI with two-dimensional nonnegative matrix factorization (2DNMF) for image processing. The main difference between NMF and 2DNMF is that the former first aligns images into one-dimensional (1D) vectors and then represents them with a set of 1D bases, while the latter regards images as 2D matrices and represents them with a set of 2D bases. The three combined algorithms are named CAPG-2DNMF, MRRI-2DNMF and ERI-2DNMF. The computational complexity and convergence analyses of proposed algorithms are also presented in this paper. Three public databases are used to test the three NMF algorithms and the three combinations, the results of which show the enhancement performance of our proposed algorithms (MRRI and ERI algorithms) over the CAPG algorithm. MRRI and ERI have similar performance. The three combined algorithms have better image reconstruction quality and less running time than their corresponding 1DNMF algorithms under the same compression ratio. We also do some experiments on a real-captured image database and get similar conclusions.