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**1 - 3**of**3**### Journal of Global Optimization manuscript No. (will be inserted by the editor) SymNMF: Nonnegative Low-Rank Approximation of a Similarity Matrix for Graph Clustering

"... Abstract Nonnegative matrix factorization (NMF) provides a lower rank approx-imation of a matrix by a product of two nonnegative factors. NMF has been shown to produce clustering results that are often superior to those by other methods such as K-means. In this paper, we provide further interpretati ..."

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Abstract Nonnegative matrix factorization (NMF) provides a lower rank approx-imation of a matrix by a product of two nonnegative factors. NMF has been shown to produce clustering results that are often superior to those by other methods such as K-means. In this paper, we provide further interpretation of NMF as a clustering method and study an extended formulation for graph clustering called Symmetric NMF (SymNMF). In contrast to NMF that takes a data matrix as an input, SymNMF takes a nonnegative similarity matrix as an input, and a symmet-ric nonnegative lower rank approximation is computed. We show that SymNMF is related to spectral clustering, justify SymNMF as a general graph clustering method, and discuss the strengths and shortcomings of SymNMF and spectral clustering. We propose two optimization algorithms for SymNMF and discuss their convergence properties and computational efficiencies. Our experiments on docu-ment clustering, image clustering, and image segmentation support SymNMF as a graph clustering method that captures latent linear and nonlinear relationships in the data.

### Nonlocal Total Variation with Primal Dual Algorithm and Stable Simplex Clustering in Unsupervised Hyperspectral Imagery Analysis

"... We focus on implementing a nonlocal total variational method for unsupervised classification of hyperspectral imagery. We minimize the energy directly using a primal dual algorithm, which we modified for the non-local gradient and weighted centroid recalculation. By squaring the labeling function in ..."

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We focus on implementing a nonlocal total variational method for unsupervised classification of hyperspectral imagery. We minimize the energy directly using a primal dual algorithm, which we modified for the non-local gradient and weighted centroid recalculation. By squaring the labeling function in the fidelity term before re-calculating the cluster centroids, we can implement an unsupervised clustering method with random initialization. We stabilize this method with stable simplex clustering. To better differentiate clusters, we use a linear combination of the cosine and Euclidean distance between spectral signatures instead of the traditional cosine distance. Finally, we speed up the calculation using a k-d tree and approximate nearest neighbor search algorithm for calculation of the weight matrix for distances between pixel signatures. We implement our method on six different datasets and compare results to traditional clustering methods like k-means, non-negative matrix factorization, and the graph-based MBO scheme.

### 1First Order Methods for Robust Non-negative Matrix Factorization for Large Scale Noisy Data

"... Nonnegative matrix factorization (NMF) has been shown to be identifiable under the separability assumption, under which all the columns(or rows) of the input data matrix belong to the convex cone generated by only a few of these columns(or rows) [1]. In real applications, however, such separability ..."

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Nonnegative matrix factorization (NMF) has been shown to be identifiable under the separability assumption, under which all the columns(or rows) of the input data matrix belong to the convex cone generated by only a few of these columns(or rows) [1]. In real applications, however, such separability assumption is hard to satisfy. Following [4] and [5], in this paper, we look at the Linear Programming (LP) based reformulation to locate the extreme rays of the convex cone but in a noisy setting. Furthermore, in order to deal with the large scale data, we employ First-Order Methods (FOM) to mitigate the computational complexity of LP, which primarily results from a large number of constraints. We show the performance of the algorithm on real and synthetic data sets.