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Register a new userDropping Symmetry for Fast Symmetric Nonnegative Matrix Factorization
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Symmetric nonnegative matrix factorization (NMF), a special but important class of the general NMF, is demonstrated to be useful for data analysis and in particular for various clustering tasks. Unfortunately, designing fast algorithms for Symmetric NMF is not as easy as for the nonsymmetric counterpart, the latter admitting the splitting property that allows efficient alternating-type algorithms. To overcome this issue, we transfer the symmetric NMF to a nonsymmetric one, then we can adopt the idea from the state-of-the-art algorithms for nonsymmetric NMF to design fast algorithms solving symmetric NMF. We rigorously establish that solving nonsymmetric reformulation returns a solution for symmetric NMF and then apply fast alternating based algorithms for the corresponding reformulated problem. Furthermore, we show these fast algorithms admit strong convergence guarantee in the sense that the generated sequence is convergent at least at a sublinear rate and it converges globally to a critical point of the symmetric NMF. We conduct experiments on both synthetic data and image clustering to support our result.
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ICU mortality risk prediction is a tough yet important task. On one hand, due to the complex temporal data collected, it is difficult to identify the effective features and interpret them easily; on the other hand, good prediction can help clinicians take timely actions to prevent the mortality. These correspond to the interpretability and accuracy problems. Most existing methods lack of the interpretability, but recently Subgraph Augmented Nonnegative Matrix Factorization (SANMF) has been successfully applied to time series data to provide a path to interpret the features well. Therefore, we adopted this approach as the backbone to analyze the patient data. One limitation of the raw SANMF method is its poor prediction ability due to its unsupervised nature. To deal with this problem, we proposed a supervised SANMF algorithm by integrating the logistic regression loss function into the NMF framework and solved it with an alternating optimization procedure. We used the simulation data to verify the effectiveness of this method, and then we applied it to ICU mortality risk prediction and demonstrated its superiority over other conventional supervised NMF methods.
Unsupervised learning of the Dawid-Skene (D&S) model from noisy, incomplete and crowdsourced annotations has been a long-standing challenge, and is a critical step towards reliably labeling massive data. A recent work takes a coupled nonnegative matrix factorization (CNMF) perspective, and shows appealing features: It ensures the identifiability of the D&S model and enjoys low sample complexity, as only the estimates of the co-occurrences of annotator labels are involved. However, the identifiability holds only when certain somewhat restrictive conditions are met in the context of crowdsourcing. Optimizing the CNMF criterion is also costly -- and convergence assurances are elusive. This work recasts the pairwise co-occurrence based D&S model learning problem as a symmetric NMF (SymNMF) problem -- which offers enhanced identifiability relative to CNMF. In practice, the SymNMF model is often (largely) incomplete, due to the lack of co-labeled items by some annotators. Two lightweight algorithms are proposed for co-occurrence imputation. Then, a low-complexity shifted rectified linear unit (ReLU)-empowered SymNMF algorithm is proposed to identify the D&S model. Various performance characterizations (e.g., missing co-occurrence recoverability, stability, and convergence) and evaluations are also presented.
Fully unsupervised topic models have found fantastic success in document clustering and classification. However, these models often suffer from the tendency to learn less-than-meaningful or even redundant topics when the data is biased towards a set of features. For this reason, we propose an approach based upon the nonnegative matrix factorization (NMF) model, deemed textit{Guided NMF}, that incorporates user-designed seed word supervision. Our experimental results demonstrate the promise of this model and illustrate that it is competitive with other methods of this ilk with only very little supervision information.
We present a general-purpose data compression algorithm, Regularized L21 Semi-NonNegative Matrix Factorization (L21 SNF). L21 SNF provides robust, parts-based compression applicable to mixed-sign data for which high fidelity, individualdata point reconstruction is paramount. We derive a rigorous proof of convergenceof our algorithm. Through experiments, we show the use-case advantages presentedby L21 SNF, including application to the compression of highly overdeterminedsystems encountered broadly across many general machine learning processes.
Inthischapterwediscusshowtolearnanoptimalmanifoldpresentationto regularize nonegative matrix factorization (NMF) for data representation problems. NMF,whichtriestorepresentanonnegativedatamatrixasaproductoftwolowrank nonnegative matrices, has been a popular method for data representation due to its ability to explore the latent part-based structure of data. Recent study shows that lots of data distributions have manifold structures, and we should respect the manifold structure when the data are represented. Recently, manifold regularized NMF used a nearest neighbor graph to regulate the learning of factorization parameter matrices and has shown its advantage over traditional NMF methods for data representation problems. However, how to construct an optimal graph to present the manifold prop- erly remains a difficultproblem due to the graph modelselection, noisy features, and nonlinear distributed data. In this chapter, we introduce three effective methods to solve these problems of graph construction for manifold regularized NMF. Multiple graph learning is proposed to solve the problem of graph model selection, adaptive graph learning via feature selection is proposed to solve the problem of constructing a graph from noisy features, while multi-kernel learning-based graph construction is used to solve the problem of learning a graph from nonlinearly distributed data.
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