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In this paper we explore avenues for improving the reliability of dimensionality reduction methods such as Non-Negative Matrix Factorization (NMF) as interpretive exploratory data analysis tools. We first explore the difficulties of the optimization problem underlying NMF, showing for the first time that non-trivial NMF solutions always exist and that the optimization problem is actually convex, by using the theory of Completely Positive Factorization. We subsequently explore four novel approaches to finding globally-optimal NMF solutions using various ideas from convex optimization. We then develop a new method, isometric NMF (isoNMF), which preserves non-negativity while also providing an isometric embedding, simultaneously achieving two properties which are helpful for interpretation. Though it results in a more difficult optimization problem, we show experimentally that the resulting method is scalable and even achieves more compact spectra than standard NMF.
In the non-negative matrix factorization (NMF) problem, the input is an $mtimes n$ matrix $M$ with non-negative entries and the goal is to factorize it as $Mapprox AW$. The $mtimes k$ matrix $A$ and the $ktimes n$ matrix $W$ are both constrained to h
Extracting genetic information from a full range of sequencing data is important for understanding diseases. We propose a novel method to effectively explore the landscape of genetic mutations and aggregate them to predict cancer type. We used multin
The Baum-Welsh algorithm together with its derivatives and variations has been the main technique for learning Hidden Markov Models (HMM) from observational data. We present an HMM learning algorithm based on the non-negative matrix factorization (NM
Community structures detection is one of the fundamental problems in complex network analysis towards understanding the topology structures of the network and the functions of it. Nonnegative matrix factorization (NMF) is a widely used method for com
Hypertension is a heterogeneous syndrome in need of improved subtyping using phenotypic and genetic measurements so that patients in different subtypes share similar pathophysiologic mechanisms and respond more uniformly to targeted treatments. Exist