This paper provides a theoretical explanation on the clustering aspect of nonnegative matrix factorization (NMF). We prove that even without imposing orthogonality nor sparsity constraint on the basis and or coefficient matrix, NMF still can give clustering results, thus providing a theoretical support for many works, e.g., [1] and [2], that show the superiority of the standard NMF as a clustering method.
@cite_10 provides the theoretical analysis on the equivalences between orthogonal NMF to @math -means clustering for both rectangular data matrices and symmetric matrices. However as their proofs utilize the zero gradient conditions, the hidden assumptions (setting the Lagrange multipliers to zeros) are not revealed there. Actually it can be easily shown that their approach is the KKT conditions applied to the unconstrained version of eq. . Thus there is no guarantee that minimizing eq. by using the zero gradient conditions leads to the stationary point located on the nonnegative orthant as required by the objective.
@cite_10
2022712430 2132538571 2076566842 2135029798
Nonnegative Matrix Factorization (NMF) is an effective dimension reduction method for non-negative dyadic data, and has proven to be useful in many areas, such as text mining, bioinformatics and image processing. NMF is usually formulated as a constrained non-convex optimization problem, and many algorithms have been developed for solving it. Recently, a coordinate descent method, called FastHals, has been proposed to solve least squares NMF and is regarded as one of the state-of-the-art techniques for the problem. In this paper, we first show that FastHals has an inefficiency in that it uses a cyclic coordinate descent scheme and thus, performs unneeded descent steps on unimportant variables. We then present a variable selection scheme that uses the gradient of the objective function to arrive at a new coordinate descent method. Our new method is considerably faster in practice and we show that it has theoretical convergence guarantees. Moreover when the solution is sparse, as is often the case in real applications, our new method benefits by selecting important variables to update more often, thus resulting in higher speed. As an example, on a text dataset RCV1, our method is 7 times faster than FastHals, and more than 15 times faster when the sparsity is increased by adding an L1 penalty. We also develop new coordinate descent methods when error in NMF is measured by KL-divergence by applying the Newton method to solve the one-variable sub-problems. Experiments indicate that our algorithm for minimizing the KL-divergence is faster than the Lee & Seung multiplicative rule by a factor of 10 on the CBCL image dataset. Nonnegative matrix factorization (NMF) is useful to find basis information of nonnegative data. Currently, multiplicative updates are a simple and popular way to find the factorization. However, for the common NMF approach of minimizing the Euclidean distance between approximate and true values, no proof has shown that multiplicative updates converge to a stationary point of the NMF optimization problem. Stationarity is important as it is a necessary condition of a local minimum. This paper discusses the difficulty of proving the convergence. We propose slight modifications of existing updates and prove their convergence. Techniques invented in this paper may be applied to prove the convergence for other bound-constrained optimization problems. Nonnegative matrix factorization (NMF) determines a lower rank approximation of a matrix @math where an integer @math is given and nonnegativity is imposed on all components of the factors @math and @math . NMF has attracted much attention for over a decade and has been successfully applied to numerous data analysis problems. In applications where the components of the data are necessarily nonnegative, such as chemical concentrations in experimental results or pixels in digital images, NMF provides a more relevant interpretation of the results since it gives nonsubtractive combinations of nonnegative basis vectors. In this paper, we introduce an algorithm for NMF based on alternating nonnegativity constrained least squares (NMF ANLS) and the active set-based fast algorithm for nonnegativity constrained least squares with multiple right-hand side vectors, and we discuss its convergence properties and a rigorous convergence criterion based on the Karush-Kuhn-Tucker (KKT) conditions. In addition, we also describe algorithms for sparse NMFs and regularized NMF. We show how we impose a sparsity constraint on one of the factors by @math -norm minimization and discuss its convergence properties. Our algorithms are compared to other commonly used NMF algorithms in the literature on several test data sets in terms of their convergence behavior. Non-negative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence. The monotonic convergence of both algorithms can be proven using an auxiliary function analogous to that used for proving convergence of the Expectation-Maximization algorithm. The algorithms can also be interpreted as diagonally rescaled gradient descent, where the rescaling factor is optimally chosen to ensure convergence.